{"node_types":["book","person","chapter","exercise_set","problem","problem_form","problem_shape","concept","method","theorem","law","quantity","unit","instrument","experiment","excerpt","equation","capability"],"edge_types":["written_by","part_of","taught_in","practices","quoted_from","explains","appears_in","states","relates","instance_of","needs","prerequisite_of","special_case_of","generalizes","uses","inverse_of","contrasts_with","measures","unit_of","named_after","discovered_by","related_to"],"concept_layer":["book","person","chapter","concept","method","theorem","law","quantity","unit","instrument","experiment"],"counts":{"nodes":19904,"links":58517},"nodes":[["planck-treatise-on-thermodynamics-1903/eq-5bbc3f5dc7",16,"Planck 1903, p. 93: \\Phi_{1}' + \\Phi_{2}' + \\dots + \\Phi_{n}' < \\Phi_{1} + \\Phi_{2} + \\dots + \\Phi_{n}"],["form/401ec1147e",5,"solve: Eq(6*x + 2 + 16/x, 10*x + 2)"],["form/4aef1a5eaa",5,"solve: Eq(x + log(x)/log(10), x*log(x)/log(10))"],["wentworth-first-steps-in-algebra-1894/ex-6",3,"Wentworth 1894, Exercise 6"],["wentworth-first-steps-in-algebra-1894/ex-7",3,"Wentworth 1894, Exercise 7"],["wentworth-first-steps-in-algebra-1894/ex-8",3,"Wentworth 1894, Exercise 8"],["boyden-first-book-in-algebra-1895/ch-simultaneous-equations",2,"Boyden 1895, SIMULTANEOUS EQUATIONS","../books/boyden-first-book-in-algebra-1895/ch/ch-simultaneous-equations/index.html"],["blackburn-elements-plane-trigonometry-1863/ch-vi",2,"Blackburn 1863, ch. VI: OF LOGARITHMIC TABLES","../books/blackburn-elements-plane-trigonometry-1863/ch/ch-vi/index.html"],["blackburn-elements-plane-trigonometry-1863/ch-x",2,"Blackburn 1863, ch. X: THE SINE AND COSINE OF THE SUM AND DIFFERENCE OF TWO ANGLES","../books/blackburn-elements-plane-trigonometry-1863/ch/ch-x/index.html"],["planck-treatise-on-thermodynamics-1903/ch-quantity-of-heat",2,"Planck 1903, Quantity of Heat","../books/planck-treatise-on-thermodynamics-1903/ch/ch-quantity-of-heat/index.html"],["hardy-course-of-pure-mathematics-1921/ex-iv",3,"Hardy 1921, Exercise IV"],["shape/a3cf4c3068",6,"solve: Eq(N*x + N + N/x, N*x + N)"],["thompson-calculus-made-easy-1914/ch-ix",2,"Thompson 1914, ch. IX: Introducing a Useful Dodge","../books/thompson-calculus-made-easy-1914/ch/ch-ix/index.html"],["hardy-course-of-pure-mathematics-1921/ex-v",3,"Hardy 1921, Exercise V"],["wentworth-plane-geometry-1899/x-544009887d",15,"Wentworth 1899, scan 145: In every proportion the product of the extremes is ..."],["thompson-calculus-made-easy-1914/eq-7cf8c6a05e",16,"Thompson 1914, p. 207: \\text{area of $1$~strip} = dS = y · dx."],["wentworth-first-steps-in-algebra-1894",0,"Wentworth, The First Steps in Algebra (1894)","../books/wentworth-first-steps-in-algebra-1894/index.html"],["dickson-theory-of-equations-1922/ex-page108",3,"Dickson 1922, Exercise Page108"],["wentworth-first-steps-in-algebra-1894/ex-2",3,"Wentworth 1894, Exercise 2"],["form/61428ebf9a",5,"identity: a - x - (a**2 + x**2)/(a - x)"],["shape/0c99fbb649",6,"identity: 0"],["shape/75aa99f1e6",6,"identity: a - x - (a**N + x**N)/(a - x)"],["wentworth-first-steps-in-algebra-1894/ex-5",3,"Wentworth 1894, Exercise 5"],["wentworth-first-steps-in-algebra-1894/ch-i",2,"Wentworth 1894, ch. I: Introduction","../books/wentworth-first-steps-in-algebra-1894/ch/ch-i/index.html"],["wentworth-first-steps-in-algebra-1894/ex-4",3,"Wentworth 1894, Exercise 4"],["shape/6cf0b4f833",6,"solve: Eq(x + log(x)/log(N), x*log(x)/log(N))"],["blackburn-elements-plane-trigonometry-1863/eq-221cce7152",16,"Blackburn 1863, p. 49: DE &= R \\sin (\\theta + \\phi)"],["thompson-calculus-made-easy-1914/eq-73d243f09c",16,"Thompson 1914, p. 207: \\text{total area~$S$} = \\int dS = \\int y\\, dx."],["blackburn-elements-plane-trigonometry-1863/eq-18342c1601",16,"Blackburn 1863, p. 49: \\sin (\\theta + \\phi) &= \\sin\\theta \\cos\\phi + \\cos\\theta \\sin\\phi"],["thompson-calculus-made-easy-1914/eq-dd9cc42873",16,"Thompson 1914, p. 85: \\frac{dy}{dx} = 2ax"],["thompson-calculus-made-easy-1914/eq-04869f075a",16,"Thompson 1914, p. 210: \\int^{x=x_2}_{x=x_1} y\\, dx = y_2 - y_1,"],["shape/bb1eae1f2b",6,"identity: 2/(N + x)"],["hardy-course-of-pure-mathematics-1921/ex-xx",3,"Hardy 1921, Exercise XX"],["boyden-first-book-in-algebra-1895/ex-58",3,"Boyden 1895, Exercise 58"],["thompson-calculus-made-easy-1914/eq-7cccfd51b2",16,"Thompson 1914, p. 79: \\dfrac{dy}{dx}=0"],["theorem/line-through-two-points-of-a-plane-lies-in-the-plane",9,"line through two points of a plane lies in the plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-line-through-two-points-of-a-plane-lies-in-the-plane"],["wentworth-first-steps-in-algebra-1894/ex-47/2",4,"Wentworth 1894, Exercise 47 (2)"],["hardy-course-of-pure-mathematics-1921/ex-vi",3,"Hardy 1921, Exercise VI"],["hardy-course-of-pure-mathematics-1921/ex-vii",3,"Hardy 1921, Exercise VII"],["blackburn-elements-plane-trigonometry-1863/eq-e35d999d91",16,"Blackburn 1863, p. 49: \\cos (\\theta + \\phi) &= \\cos\\theta \\cos\\phi - \\sin\\theta \\sin\\phi"],["wentworth-first-steps-in-algebra-1894/eq-312216bfb2",16,"Wentworth 1894, p. 98: \\frac{a}{b} × \\frac{c}{d} × \\frac{e}{f} = \\frac{ac}{bd} × \\frac{e}{f} = \\frac{ace}{bdf}"],["hardy-course-of-pure-mathematics-1921/x-4a8e547287",15,"Hardy 1921, p. 149: An infinite aggregate of numbers does not necessarily possess ..."],["de-morgan-elementary-illustrations-calculus-1899/ch-connexion-of-the-integral-with-the-differential-calculus",2,"De Morgan 1899, Connexion of the Integral with the Differential Calculus","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-connexion-of-the-integral-with-the-differential-calculus/index.html"],["concept/mantissa",7,"mantissa","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-mantissa"],["maxwell-elementary-treatise-electricity-1888/eq-eb1c306935",16,"Maxwell 1888, scan 49: {E_r}' = -n{E_s}'"],["concept/method-multiplying-fractions",7,"method: multiplying fractions"],["wentworth-first-steps-in-algebra-1894/eq-250437650b",16,"Wentworth 1894, p. 122: y = 10 - x"],["blackburn-elements-plane-trigonometry-1863/eq-a25882ff41",16,"Blackburn 1863, p. 50: \\sin (\\theta - \\phi) &= \\sin\\theta \\cos\\phi - \\cos\\theta \\sin\\phi"],["theorem/logarithm-of-a-product",9,"logarithm of a product","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-logarithm-of-a-product"],["hardy-course-of-pure-mathematics-1921/ex-viii",3,"Hardy 1921, Exercise VIII"],["dickson-theory-of-equations-1922/ex-page112",3,"Dickson 1922, Exercise Page112"],["de-morgan-elementary-illustrations-calculus-1899/x-f92a3e12c8",15,"De Morgan 1899, p. 102: It is not necessary that we should be able ..."],["wentworth-first-steps-in-algebra-1894/x-55f57fabd2",15,"Wentworth 1894, p. 1: In counting separate objects or in measuring magnitudes, the ..."],["theorem/sine-subtraction-formula",9,"sine subtraction formula"],["quantity/total-area",11,"total area","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-quantity-total-area"],["boyden-first-book-in-algebra-1895/ex-57",3,"Boyden 1895, Exercise 57"],["blackburn-elements-plane-trigonometry-1863/eq-138592b15b",16,"Blackburn 1863, p. 50: \\cos (\\theta - \\phi) &= \\cos\\phi \\cos\\theta + \\sin\\phi \\sin\\theta"],["thompson-calculus-made-easy-1914/ch-xiv",2,"Thompson 1914, ch. XIV: On true Compound Interest and the Law of Organic Growth","../books/thompson-calculus-made-easy-1914/ch/ch-xiv/index.html"],["theorem/cosine-subtraction-formula",9,"cosine subtraction formula"],["blackburn-elements-plane-trigonometry-1863/eq-07d3a230a5",16,"Blackburn 1863, p. 51: R - r = 2\\delta"],["shape/4b45bf774a",6,"identity: 1/(x + 1) + 1/(x - 1)"],["form/34c1c50cc2",5,"identity: -(5*a**5 - 10*a**3)/(5*a**3)"],["thompson-calculus-made-easy-1914/eq-0cc50021bc",16,"Thompson 1914, p. 175: y &= u×v"],["cap/other:resultant_elimination",17,"other:resultant_elimination"],["theorem/logarithm-of-a-quotient",9,"logarithm of a quotient","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-logarithm-of-a-quotient"],["dickson-theory-of-equations-1922/ex-page113",3,"Dickson 1922, Exercise Page113"],["blackburn-elements-plane-trigonometry-1863/eq-2cffb15855",16,"Blackburn 1863, p. 51: r_1 = \\dfrac{R + r}{2}"],["blackburn-elements-plane-trigonometry-1863/eq-6d0771fbad",16,"Blackburn 1863, p. 51: R_1^2 &= r_1R"],["blackburn-elements-plane-trigonometry-1863/eq-fb0cb3d355",16,"Blackburn 1863, p. 51: \\delta_1 = \\frac{\\delta}{4} - \\frac{\\delta_1^2}{r_1}"],["blackburn-elements-plane-trigonometry-1863/eq-e1dd45729f",16,"Blackburn 1863, p. 52: \\delta + \\delta_1 + \\delta_2 + \\dots < \\frac{4}{3}\\, \\delta"],["theorem/equality-axioms",9,"equality axioms","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-theorem-equality-axioms"],["de-morgan-elementary-illustrations-calculus-1899/x-fdc09fa29e",15,"De Morgan 1899, p. 103: That is, the effect of the operation or set ..."],["shape/016c631008",6,"evaluate: pi*N*x + N"],["wentworth-first-steps-in-algebra-1894/eq-a8d55786ea",16,"Wentworth 1894, p. 122: x - y = 2"],["form/1a53b06f2f",5,"evaluate: -400"],["shape/bf75670fe0",6,"extremum: x**N + (a - x)**N"],["shape/fd68122a58",6,"differentiate: a*tan(x**c)**b"],["form/163bb5c643",5,"differentiate: (x**3)**(1/a)"],["form/94ce3205b6",5,"evaluate: pi*x**2*(-2*pi*x + 6) at r=2/pi"],["shape/a6738dfeb9",6,"evaluate: pi*x**N*(pi*N*x + N)"],["maxwell-elementary-treatise-electricity-1888/eq-ba8287018e",16,"Maxwell 1888, scan 108: Q &= K(P-p) + HP"],["planck-treatise-on-thermodynamics-1903/eq-6fb45885ba",16,"Planck 1903, p. 93: (\\Phi_{1}' + \\Phi_{2}' + \\dots + \\Phi_{n}') - \\Phi_{1} - \\Phi_{2} - \\dots - \\Phi_{n-1}\\Add{.}"],["wentworth-first-steps-in-algebra-1894/ex-14",3,"Wentworth 1894, Exercise 14"],["shape/ad71b0aa13",6,"identity: -(-a*b + a*b**N + a**N*b)/(a*b)"],["dickson-theory-of-equations-1922/ch-ii",2,"Dickson 1922, ch. II: Elementary Theorems on the Roots of an Equation","../books/dickson-theory-of-equations-1922/ch/ch-ii/index.html"],["wentworth-first-steps-in-algebra-1894/ex-13",3,"Wentworth 1894, Exercise 13"],["concept/identity",7,"identity","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-identity"],["wentworth-first-steps-in-algebra-1894/ex-10",3,"Wentworth 1894, Exercise 10"],["wentworth-first-steps-in-algebra-1894/ex-11",3,"Wentworth 1894, Exercise 11"],["wentworth-first-steps-in-algebra-1894/ex-12",3,"Wentworth 1894, Exercise 12"],["form/00efe830a1",5,"solve: Eq((x - 5)*(x + 5), 24)"],["wentworth-first-steps-in-algebra-1894/ch-ii",2,"Wentworth 1894, ch. II: Simple Equations","../books/wentworth-first-steps-in-algebra-1894/ch/ch-ii/index.html"],["thompson-calculus-made-easy-1914/eq-de7b3de8c1",16,"Thompson 1914, p. 67: \\frac{dy}{du} = \\frac{3}{2} u^{\\efrac{1}{2}}"],["thompson-calculus-made-easy-1914/eq-84ea8aa82d",16,"Thompson 1914, p. 67: \\frac{du}{dx} = 2x"],["thompson-calculus-made-easy-1914/eq-20028de417",16,"Thompson 1914, p. 68: \\frac{dy}{dx} = \\frac{dy}{du}×\\frac{du}{dx}"],["form/b7d1be001c",5,"identity: 2*a*b/3"],["form/640908419a",5,"evaluate: 1225"],["form/d439270550",5,"solve: Eq(x**4 - 2*x**3 - 21*x**2 + 22*x + 40, 0)"],["form/08773a8eb6",5,"solve: Eq(x**2, -a)"],["maxwell-elementary-treatise-electricity-1888/eq-b7538c7c44",16,"Maxwell 1888, scan 108: q &= K(p-P) + hp"],["de-morgan-elementary-illustrations-calculus-1899/x-7f7251efdf",15,"De Morgan 1899, p. 103: Hence \\dfrac{dx}{dy} as deduced from the second, and \\dfrac{dy}{dx} ..."],["maxwell-elementary-treatise-electricity-1888/eq-1d080cba6f",16,"Maxwell 1888, scan 50: Q = W + Q'"],["hardy-course-of-pure-mathematics-1921/x-b7a05b230e",15,"Hardy 1921, p. 287: The distinction between the definite and the indefinite integral ..."],["blackburn-elements-plane-trigonometry-1863/x-0a39a94bd9",15,"Blackburn 1863, p. 33: Logarithms of ordinary numbers may be defined to be ..."],["boyden-first-book-in-algebra-1895/x-08a02029e7",15,"Boyden 1895: This is the reverse of the case under Multiplication ..."],["boyden-first-book-in-algebra-1895/ex-35/1",4,"Boyden 1895, Exercise 35 (1)"],["thompson-calculus-made-easy-1914/eq-63e8456a71",16,"Thompson 1914, p. 210: \\text{area~$S$} = b(x_2 - x_1) + \\frac{a}{3}(x_2^3 - x_1^3)."],["blackburn-elements-plane-trigonometry-1863/eq-787c1b8b8d",16,"Blackburn 1863, p. 23: \\cos(-\\theta) &= +\\cos\\theta."],["slaught-lennes-solid-geometry-1919/ch-book-iii",2,"Slaught & Lennes 1919, ch. BOOK III: Prisms and Cylinders","../books/slaught-lennes-solid-geometry-1919/ch/ch-book-iii/index.html"],["person/jaenisch",1,"Jaenisch","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-jaenisch"],["planck-treatise-on-thermodynamics-1903/eq-5a433ac025",16,"Planck 1903, p. 46: U_{2} - U_{1} = Q + W"],["de-morgan-elementary-illustrations-calculus-1899/ch-the-notation-of-the-differential-calculus",2,"De Morgan 1899, The Notation of the Differential Calculus","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-the-notation-of-the-differential-calculus/index.html"],["maxwell-elementary-treatise-electricity-1888/eq-dcedd21e9f",16,"Maxwell 1888, scan 108: P_1=P-\\frac{K}{K+H}p"],["maxwell-elementary-treatise-electricity-1888/eq-31be88cd1e",16,"Maxwell 1888, scan 108: Q_1=(K+H)P_1"],["macfarlane-vector-analysis-quaternions-1906/ch-addition-of-coplanar-vectors",2,"Macfarlane 1906, Addition of Coplanar Vectors","../books/macfarlane-vector-analysis-quaternions-1906/ch/ch-addition-of-coplanar-vectors/index.html"],["wentworth-first-steps-in-algebra-1894/ch-iii",2,"Wentworth 1894, ch. III: Positive and Negative Numbers","../books/wentworth-first-steps-in-algebra-1894/ch/ch-iii/index.html"],["wentworth-first-steps-in-algebra-1894/ex-15",3,"Wentworth 1894, Exercise 15"],["wentworth-first-steps-in-algebra-1894/ex-16",3,"Wentworth 1894, Exercise 16"],["form/50b6c89261",5,"integrate: x**2*cos(x)"],["shape/0287583a34",6,"integrate: x**N*cos(x)"],["dickson-theory-of-equations-1922/ex-page26",3,"Dickson 1922, Exercise Page26"],["dickson-theory-of-equations-1922/ex-page25",3,"Dickson 1922, Exercise Page25"],["person/n-j-lennes",1,"N. J. Lennes"],["dickson-theory-of-equations-1922/ex-page27",3,"Dickson 1922, Exercise Page27"],["form/722d5fc787",5,"differentiate: a + b*x**2 + c*x**4"],["maxwell-elementary-treatise-electricity-1888/eq-b34956cafd",16,"Maxwell 1888, scan 108: q_1=-KP_1"],["de-morgan-elementary-illustrations-calculus-1899/x-5b87b971e4",15,"De Morgan 1899, p. 104: There is no very obvious analogy between \\dfrac{d^{2} y}{dx^{2}} ..."],["macfarlane-vector-analysis-quaternions-1906/eq-c0cf769126",16,"Macfarlane 1906: B = b\\beta"],["blackburn-elements-plane-trigonometry-1863/eq-2ee5101c5c",16,"Blackburn 1863, p. 21: \\sin\\theta = \\dfrac{BD}{R}"],["hardy-course-of-pure-mathematics-1921/ex-li",3,"Hardy 1921, Exercise LI"],["hardy-course-of-pure-mathematics-1921/ex-lii",3,"Hardy 1921, Exercise LII"],["dickson-theory-of-equations-1922/ex-page115/1",4,"Dickson 1922, Exercise Page115 (1)"],["hardy-course-of-pure-mathematics-1921/ex-liii",3,"Hardy 1921, Exercise LIII"],["blackburn-elements-plane-trigonometry-1863/eq-b78f2b353e",16,"Blackburn 1863, p. 21: \\cos\\theta = \\sin\\left(\\frac{\\pi}{2} - \\theta\\right)."],["thompson-calculus-made-easy-1914/eq-8702c5480b",16,"Thompson 1914, p. 218: \\text{mean~$y$} = \\frac{1}{x_1} \\int^{x=x_1}_{x=0} y · dx."],["hardy-course-of-pure-mathematics-1921/ex-liv",3,"Hardy 1921, Exercise LIV"],["form/8fbe9e2ed1",5,"solve: Eq(x**3 + 4*x**2 - 7, 0)"],["blackburn-elements-plane-trigonometry-1863/eq-2b466fdea5",16,"Blackburn 1863, p. 21: \\sin\\theta = \\cos\\left(\\frac{\\pi}{2} - \\theta\\right)."],["dickson-theory-of-equations-1922/ex-page28",3,"Dickson 1922, Exercise Page28"],["form/76c17d815a",5,"solve: (Eq(a + b + c, 11), Eq(2*a - 6*b - c, 0), Eq(3*a + 4*b + 2*c, 0))"],["thompson-calculus-made-easy-1914/eq-ca965ed37d",16,"Thompson 1914, p. 217: dA = 2 \\pi r\\, dr."],["thompson-calculus-made-easy-1914/eq-2461951643",16,"Thompson 1914, p. 217: A &= \\pi R^2."],["planck-treatise-on-thermodynamics-1903/eq-9588d311c6",16,"Planck 1903, p. 47: U_{2} = U_{1}"],["wentworth-first-steps-in-algebra-1894/ex-19",3,"Wentworth 1894, Exercise 19"],["wentworth-first-steps-in-algebra-1894/eq-e2f6bfbee5",16,"Wentworth 1894, p. 71: 3a^{2} - 6ab &= 3a(a - 2b)"],["shape/61aef0fbe6",6,"solve: (Eq(a + b + c, N), Eq(N*a + N*b - c, 0), Eq(N*a + N*b + N*c, 0))"],["maxwell-elementary-treatise-electricity-1888/eq-bc144d0269",16,"Maxwell 1888, scan 108: p_2=-\\frac{K}{K+h}P_1"],["wentworth-first-steps-in-algebra-1894/ch-iv",2,"Wentworth 1894, ch. IV: Addition and Subtraction","../books/wentworth-first-steps-in-algebra-1894/ch/ch-iv/index.html"],["de-morgan-elementary-illustrations-calculus-1899/x-03b4a33dd5",15,"De Morgan 1899, p. 104: Therefore dy, dy_{1}, etc., are not equal; whence arises ..."],["wentworth-first-steps-in-algebra-1894/ex-18",3,"Wentworth 1894, Exercise 18"],["wentworth-first-steps-in-algebra-1894/ex-20",3,"Wentworth 1894, Exercise 20"],["shape/1594659120",6,"identity: (N*a*b + N*a*b**N + a**N*b**N)/(a*b)"],["form/c0683e0a50",5,"identity: (3*a**2 + 6*a)/(a**2 + 4*a + 4)"],["dickson-theory-of-equations-1922/ex-page126",3,"Dickson 1922, Exercise Page126"],["shape/1f12a9cbd9",6,"identity: (N*a + N*a**N)/(N*a + N + a**N)"],["todhunter-spherical-trigonometry-1886/ch-area-of-a-spherical-triangle-spherical-excess",2,"Todhunter 1886, Area of a Spherical Triangle. Spherical Excess","../books/todhunter-spherical-trigonometry-1886/ch/ch-area-of-a-spherical-triangle-spherical-excess/index.html"],["dickson-theory-of-equations-1922/ex-page136/3c",4,"Dickson 1922, Exercise Page136 (3c)"],["maxwell-elementary-treatise-electricity-1888/eq-aaf0e2683b",16,"Maxwell 1888, scan 108: Q_2=-Kp_2"],["thompson-calculus-made-easy-1914/eq-a3b75dc585",16,"Thompson 1914, p. 214: \\pi(r_2^2 - r_1^2)"],["blackburn-elements-plane-trigonometry-1863/x-88af92efa9",15,"Blackburn 1863, p. 34: Equation (5) shews that the logarithm increases with the ..."],["thompson-calculus-made-easy-1914/eq-68f78bd090",16,"Thompson 1914, p. 207: dS = y · dx"],["todhunter-spherical-trigonometry-1886/eq-eb8673a8f4",16,"Todhunter 1886, scan 77: \\dfrac{\\text{area of lune}}{\\text{surface of sphere}} = \\dfrac{A}{2\\pi}\\,"],["hardy-course-of-pure-mathematics-1921/ex-xc",3,"Hardy 1921, Exercise XC"],["wentworth-first-steps-in-algebra-1894/eq-c17fe3aa07",16,"Wentworth 1894, p. 72: 4x^{3} + 12x^{2} - 8x &= 4x(x^{2} + 3x - 2)"],["planck-treatise-on-thermodynamics-1903/eq-3f572b1d00",16,"Planck 1903, p. 108: -\\frac{Q}{\\theta} \\geq 0"],["wentworth-first-steps-in-algebra-1894/eq-3df2422d67",16,"Wentworth 1894, p. 74: (x + y)(x - y) = x^{2} - y^{2}"],["maxwell-elementary-treatise-electricity-1888/eq-96c158bdfd",16,"Maxwell 1888, scan 51: W = Q' - Q"],["thompson-calculus-made-easy-1914/eq-075c515c62",16,"Thompson 1914, p. 175: y &= f(u, v)"],["concept/method-constructing-a-regular-tetrahedron",7,"method: constructing a regular tetrahedron"],["hardy-course-of-pure-mathematics-1921/ex-xciv",3,"Hardy 1921, Exercise XCIV"],["todhunter-spherical-trigonometry-1886/ch-spherical-geometry",2,"Todhunter 1886, Spherical Geometry","../books/todhunter-spherical-trigonometry-1886/ch/ch-spherical-geometry/index.html"],["planck-treatise-on-thermodynamics-1903/x-8686f4354a",15,"Planck 1903, p. 32: It was, in general, customary to take as the ..."],["hardy-course-of-pure-mathematics-1921/ex-xl",3,"Hardy 1921, Exercise XL"],["law/kepler-s-laws",10,"Kepler's laws","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-law-kepler-s-laws"],["wentworth-first-steps-in-algebra-1894/ch-v",2,"Wentworth 1894, ch. V: Multiplication and Division","../books/wentworth-first-steps-in-algebra-1894/ch/ch-v/index.html"],["slaught-lennes-solid-geometry-1919/ch-book-ii",2,"Slaught & Lennes 1919, ch. BOOK II: Regular Polyhedrons","../books/slaught-lennes-solid-geometry-1919/ch/ch-book-ii/index.html"],["wentworth-first-steps-in-algebra-1894/ex-71/6",4,"Wentworth 1894, Exercise 71 (6)"],["slaught-lennes-solid-geometry-1919/eq-8d0dfcd620",16,"Slaught & Lennes 1919, p. 53: AD = AC"],["hardy-course-of-pure-mathematics-1921/ex-xli",3,"Hardy 1921, Exercise XLI"],["theorem/chain-rule",9,"Chain rule","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-chain-rule"],["maxwell-elementary-treatise-electricity-1888/eq-b3a4853c56",16,"Maxwell 1888, scan 108: q_2=(K+h)p_2"],["maxwell-elementary-treatise-electricity-1888/eq-a111219051",16,"Maxwell 1888, scan 109: P_3=-\\frac{K}{K+H}p_2"],["maxwell-elementary-treatise-electricity-1888/eq-15f5747cb8",16,"Maxwell 1888, scan 109: Q_3=(K+H)P_3"],["form/345c98ed8a",5,"solve: (Eq(a + b + c + d, 1), Eq(4*a + b + 2*c + 3*d, 11), Eq(10*a + b + 3*c + 6*d, 26), Eq(20*a + b + 4*c + 10*d, 47))"],["planck-treatise-on-thermodynamics-1903/eq-df10b04ea3",16,"Planck 1903, p. 108: Q \\leq 0"],["thompson-calculus-made-easy-1914/eq-3108f28f11",16,"Thompson 1914, p. 175: dy_v &= v\\, du"],["planck-treatise-on-thermodynamics-1903/eq-03350b4e50",16,"Planck 1903, p. 108: W \\geq 0"],["theorem/there-are-exactly-five-regular-polyhedrons",9,"there are exactly five regular polyhedrons","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-there-are-exactly-five-regular-polyhedrons"],["form/7ee24a05cd",5,"differentiate: (a + x**c)/(b + x**(-c))"],["hardy-course-of-pure-mathematics-1921/x-0987a6a14b",15,"Hardy 1921, p. 248: The integrals of the inverse sine and tangent and ..."],["thompson-calculus-made-easy-1914/ex-ii/1",4,"Thompson 1914, Exercise II (1)"],["maxwell-elementary-treatise-electricity-1888/eq-3b38c6e471",16,"Maxwell 1888, scan 109: q_3=-KP_3"],["wentworth-first-steps-in-algebra-1894/ex-30",3,"Wentworth 1894, Exercise 30"],["wentworth-first-steps-in-algebra-1894/ex-24",3,"Wentworth 1894, Exercise 24"],["whitehead-introduction-to-mathematics-1911/x-9ac1b0cb5e",15,"Whitehead 1911, p. 236: Furthermore, particular things such as the Houses of Parliament, ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-3ff2a7fc5c",16,"De Morgan 1899, p. 87: y = \\text{common log}~x"],["form/e52531eeeb",5,"differentiate: a*x**3 + 6"],["wentworth-first-steps-in-algebra-1894/ex-28",3,"Wentworth 1894, Exercise 28"],["wentworth-first-steps-in-algebra-1894/ex-29",3,"Wentworth 1894, Exercise 29"],["hardy-course-of-pure-mathematics-1921/eq-ee4e71bd85",16,"Hardy 1921, p. 300: f(b) = f(a) + \\tfrac{1}{6}(b - a) [f'(a) + f'(b) + 4f'\\{\\tfrac{1}{2}(a + b)\\}] - \\tfrac{1}{2880}(b - a)^{5} f^{(5)}(\\DPt"],["thompson-calculus-made-easy-1914/eq-37ba606fc7",16,"Thompson 1914, p. 175: dy_u &= u\\, dv"],["thompson-calculus-made-easy-1914/eq-39bc7f55e9",16,"Thompson 1914, p. 176: \\frac{\\partial y}{\\partial u} &= v"],["thompson-calculus-made-easy-1914/eq-e010fe9d8c",16,"Thompson 1914, p. 215: pv^n = c"],["whitehead-introduction-to-mathematics-1911/x-6e7dee6b77",15,"Whitehead 1911, p. 237: The answer is that the triangles are in all ..."],["thompson-calculus-made-easy-1914/eq-746f2d64f2",16,"Thompson 1914, p. 176: \\frac{\\partial y}{\\partial v} &= u"],["thompson-calculus-made-easy-1914/eq-6fef4e3d05",16,"Thompson 1914, p. 176: dy_v &= \\frac{\\partial y}{\\partial u}\\, du"],["maxwell-elementary-treatise-electricity-1888/eq-4848cfcebd",16,"Maxwell 1888, scan 101: \\overline{CA} = ma"],["maxwell-elementary-treatise-electricity-1888/eq-861bb630c8",16,"Maxwell 1888, scan 101: \\overline{AP} : \\overline{PB} : : \\overline{AC} : \\overline{PC}"],["thompson-calculus-made-easy-1914/eq-ce98597298",16,"Thompson 1914, p. 215: y &= b\\epsilon^{-x}."],["wentworth-first-steps-in-algebra-1894/eq-f064f31fcf",16,"Wentworth 1894, p. 157: 3a^{2}b + 3ab^{2} + b^{3} = (3a^{2} + 3ab + b^{2})b"],["boyden-first-book-in-algebra-1895/ex-35/2",4,"Boyden 1895, Exercise 35 (2)"],["planck-treatise-on-thermodynamics-1903/eq-9754c8c091",16,"Planck 1903, p. 47: \\theta_{2} = \\theta_{1}"],["planck-treatise-on-thermodynamics-1903/eq-eb89a07b3e",16,"Planck 1903, p. 49: \\left(\\frac{\\dd U}{\\dd V}\\right)_{\\theta} = 0\\Add{.}"],["concept/rational-expression",7,"rational expression","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-rational-expression"],["wentworth-first-steps-in-algebra-1894/ch-vii",2,"Wentworth 1894, ch. VII: Factors","../books/wentworth-first-steps-in-algebra-1894/ch/ch-vii/index.html"],["wentworth-first-steps-in-algebra-1894/eq-9ea7d1a1f8",16,"Wentworth 1894, p. 76: \\frac{a^{3} - b^{3}}{a - b} = a^{2} + ab + b^{2}"],["maxwell-elementary-treatise-electricity-1888/eq-ee099e4ac7",16,"Maxwell 1888, scan 109: Q_1=(K_1+H_1)P"],["thompson-calculus-made-easy-1914/eq-1339629dbb",16,"Thompson 1914, p. 176: dy_u &= \\frac{\\partial y}{\\partial v}\\, dv"],["blackburn-elements-plane-trigonometry-1863/eq-00af57c3af",16,"Blackburn 1863, p. 3: \\dfrac{\\pi}{2}"],["thompson-calculus-made-easy-1914/eq-dd75bd9097",16,"Thompson 1914, p. 176: dy = \\frac{\\partial y}{\\partial u}\\, du + \\dfrac{\\partial y}{\\partial v}\\, dv"],["shape/a1e8ad7fc3",6,"solve: (Eq(a + b + c + d, 1), Eq(N*a + N*c + N*d + b, N), Eq(N*a + N*c + N*d + b, N), Eq(N*a + N*c + N*d + b, N))"],["blackburn-elements-plane-trigonometry-1863/ch-i",2,"Blackburn 1863, ch. I: OF THE MENSURATION OF THE CIRCLE","../books/blackburn-elements-plane-trigonometry-1863/ch/ch-i/index.html"],["blackburn-elements-plane-trigonometry-1863/eq-f24cdb358c",16,"Blackburn 1863, p. 3: \\text{the circumference $÷$ the diameter $= \\pi$,}"],["blackburn-elements-plane-trigonometry-1863/eq-ee246409b8",16,"Blackburn 1863, p. 3: \\text{and the circumference $= 2\\pi R$.}"],["blackburn-elements-plane-trigonometry-1863/eq-30d8edb714",16,"Blackburn 1863, p. 4: AOC : AOB :: AC : AB"],["blackburn-elements-plane-trigonometry-1863/ch-vii",2,"Blackburn 1863, ch. VII: SOLUTION OF TRIANGLES","../books/blackburn-elements-plane-trigonometry-1863/ch/ch-vii/index.html"],["wentworth-first-steps-in-algebra-1894/x-7b54d1c573",15,"Wentworth 1894, p. 71: The factors of a monomial may be found by ..."],["maxwell-elementary-treatise-electricity-1888/eq-9fe54f8926",16,"Maxwell 1888, scan 109: Q_2=-K_2 P"],["thompson-calculus-made-easy-1914/eq-18c7d9b8d9",16,"Thompson 1914, p. 176: dy = \\left(\\dfrac{dy}{du}\\right)\\, du + \\left(\\dfrac{dy}{dv}\\right)\\, dv"],["hardy-course-of-pure-mathematics-1921/eq-d0dbb97785",16,"Hardy 1921, p. 300: f(b) = f(a) + \\tfrac{1}{2}(b - a) \\{f'(a) + f'(b)\\} - \\tfrac{1}{12}(b - a)^{2} \\{f''(b) - f''(a)\\} + \\tfrac{1}{720}(b - "],["thompson-calculus-made-easy-1914/eq-7e6337cc3a",16,"Thompson 1914, p. 176: w = 2ax^2 + 3bxy + 4cy^3"],["thompson-calculus-made-easy-1914/eq-03b870d573",16,"Thompson 1914, p. 176: \\frac{\\partial w}{\\partial x} &= 4ax + 3by"],["shape/0e741257c5",6,"solve: Eq(N*x**N + N + x**N, 0)"],["blackburn-elements-plane-trigonometry-1863/eq-a259018857",16,"Blackburn 1863, p. 4: AOC = \\frac{2}{\\pi} × \\text{a right angle},"],["thompson-calculus-made-easy-1914/eq-c54307323e",16,"Thompson 1914, p. 176: \\frac{\\partial w}{\\partial y} &= 3bx + 12cy^2"],["shape/d4c61e1220",6,"integrate: N*x"],["concept/level-curve",7,"level curve","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-level-curve"],["thompson-calculus-made-easy-1914/eq-88fdf5281e",16,"Thompson 1914, p. 215: &= b(1-\\epsilon^{-a})."],["planck-treatise-on-thermodynamics-1903/eq-2fff2c26d4",16,"Planck 1903, p. 49: Q = 0"],["form/2d19e8ac3e",5,"identity: (x**4 + 2*x**3 - 7*x**2 - 8*x + 12)/(x**2 - 3*x + 2)"],["theorem/polyhedral-angle-has-at-least-three-face-angles",9,"polyhedral angle has at least three face angles","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-polyhedral-angle-has-at-least-three-face-angles"],["wentworth-first-steps-in-algebra-1894/ex-49",3,"Wentworth 1894, Exercise 49"],["wentworth-first-steps-in-algebra-1894/ex-41",3,"Wentworth 1894, Exercise 41"],["wentworth-first-steps-in-algebra-1894/ch-viii",2,"Wentworth 1894, ch. VIII: Common Factors and Multiples","../books/wentworth-first-steps-in-algebra-1894/ch/ch-viii/index.html"],["shape/704fc98efd",6,"differentiate: N*x + N*x**N + N"],["blackburn-elements-plane-trigonometry-1863/ch-ii",2,"Blackburn 1863, ch. II: OF THE AREA OF A TRIANGLE AND OF THE INSCRIBED CIRCLE","../books/blackburn-elements-plane-trigonometry-1863/ch/ch-ii/index.html"],["wentworth-first-steps-in-algebra-1894/ex-50",3,"Wentworth 1894, Exercise 50"],["form/ebcca8141a",5,"evaluate: 31*x**2/5 - 9*x/2 + 12 at t=4"],["blackburn-elements-plane-trigonometry-1863/eq-a4e77e2c7a",16,"Blackburn 1863, p. 12: \\text{Then, numerically, the Area} = rs."],["shape/354f6abeed",6,"evaluate: N*x + N*x**N + N"],["blackburn-elements-plane-trigonometry-1863/eq-505da7b5c9",16,"Blackburn 1863, p. 12: Ab = Ac = s - a"],["form/36722d0964",5,"evaluate: 31*x**2/5 - 9*x/2 + 12"],["hardy-course-of-pure-mathematics-1921/eq-e8695e73d5",16,"Hardy 1921, p. 301: \\begin{vmatrix} f(a) & f(b)\\\\ g(a) & g(b) \\end{vmatrix} = (b - a) \\begin{vmatrix} f(a) & f'(\\beta)\\\\ g(a) & g'(\\beta) \\e"],["hardy-course-of-pure-mathematics-1921/ch-appendix-i",2,"Hardy 1921, ch. Appendix I: The Proof that every Equation has a Root","../books/hardy-course-of-pure-mathematics-1921/ch/ch-appendix-i/index.html"],["form/0b72d86480",5,"factor: -a**3*b**2 + a**3*c**2 + a**2*b**3 - a**2*c**3 - b**3*c**2 + b**2*c**3"],["wentworth-first-steps-in-algebra-1894/ex-43",3,"Wentworth 1894, Exercise 43"],["thompson-calculus-made-easy-1914/eq-a7416eabd3",16,"Thompson 1914, p. 222: \\sqrt[2] {\\frac{1}{l} \\int^l_0 y^2\\, dx}."],["hardy-course-of-pure-mathematics-1921/ex-xcviii",3,"Hardy 1921, Exercise XCVIII"],["thompson-calculus-made-easy-1914/eq-b62da49d8a",16,"Thompson 1914, p. 177: dw = (4ax+3by)\\, dx + (3bx+12cy^2)\\, dy"],["hardy-course-of-pure-mathematics-1921/ex-misc-x",3,"Hardy 1921, Exercise Misc-X"],["concept/common-root",7,"common root","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-common-root"],["concept/extraneous-factor",7,"extraneous factor","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-extraneous-factor"],["wentworth-first-steps-in-algebra-1894/ex-45",3,"Wentworth 1894, Exercise 45"],["wentworth-first-steps-in-algebra-1894/ex-47",3,"Wentworth 1894, Exercise 47"],["wentworth-first-steps-in-algebra-1894/ex-63",3,"Wentworth 1894, Exercise 63"],["wentworth-first-steps-in-algebra-1894/ex-64",3,"Wentworth 1894, Exercise 64"],["wentworth-first-steps-in-algebra-1894/ex-48",3,"Wentworth 1894, Exercise 48"],["dickson-theory-of-equations-1922/ex-page136/3d",4,"Dickson 1922, Exercise Page136 (3d)"],["wentworth-first-steps-in-algebra-1894/ex-44",3,"Wentworth 1894, Exercise 44"],["wentworth-first-steps-in-algebra-1894/ex-46",3,"Wentworth 1894, Exercise 46"],["form/d3a2b9c99a",5,"differentiate: 34*x**3/5 - 54*x/5"],["thompson-calculus-made-easy-1914/eq-68f039c7ec",16,"Thompson 1914, p. 223: \\text{quadratic mean} = \\frac{1}{\\sqrt 3}\\, al."],["maxwell-elementary-treatise-electricity-1888/eq-9539378b27",16,"Maxwell 1888, scan 109: Q_1 + Q_2 = {Q_1}' + {Q_2}'"],["wentworth-first-steps-in-algebra-1894/x-e44972fef4",15,"Wentworth 1894, p. 76: In like manner we can resolve into factors any ..."],["concept/mean-value",7,"mean value"],["de-morgan-elementary-illustrations-calculus-1899/ch-orders-of-infinity",2,"De Morgan 1899, Orders of Infinity","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-orders-of-infinity/index.html"],["boyden-first-book-in-algebra-1895/ex-1/4",4,"Boyden 1895, Exercise 1 (4)"],["de-morgan-elementary-illustrations-calculus-1899/eq-543f0584e1",16,"De Morgan 1899, p. 61: nt' = t"],["wentworth-plane-geometry-1899/x-cb4b6795d4",15,"Wentworth 1899, scan 144: If three quantities are in continued proportion, the second ..."],["form/2e6e98b3a0",5,"extremum: x**3 + x**2 - 10*x + 8"],["maxwell-elementary-treatise-electricity-1888/eq-9be60f31cf",16,"Maxwell 1888, scan 109: {P_1}' = {P_2}' = P'"],["maxwell-elementary-treatise-electricity-1888/eq-23dca126d4",16,"Maxwell 1888, scan 109: (K_1+H_1-K_2)P=(K_1+H_1+K_2+H_2)P'"],["method/grouping-digits",8,"grouping digits","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-method-grouping-digits"],["hardy-course-of-pure-mathematics-1921/x-c9099c9e7a",15,"Hardy 1921, p. 177: An ‘infinity’ is the kind of discontinuity of most ..."],["wentworth-plane-geometry-1899/x-1628dabbfa",15,"Wentworth 1899, scan 144: The fourth proportional to three given quantities is the ..."],["de-morgan-elementary-illustrations-calculus-1899/ch-the-differential-coefficient-considered-with-respect-to-its-magnitude",2,"De Morgan 1899, The Differential Coefficient Considered with Respect to its Magnitude","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-the-differential-coefficient-considered-with-respect-to-its-magnitude/index.html"],["blackburn-elements-plane-trigonometry-1863/eq-78d14cf27b",16,"Blackburn 1863, p. 12: Bc = Ba = s-b"],["form/d42e28105c",5,"differentiate: 4*x/5 - 1 + 4/(x**2 + 4)"],["shape/df18246067",6,"differentiate: N*x + N/(N + x**N) - 1"],["form/4cd2a7a068",5,"evaluate: 4*x/5 - 1 + 4/(x**2 + 4) at t=10"],["shape/1018eee0a9",6,"evaluate: N*x + N/(N + x**N) - 1"],["thompson-calculus-made-easy-1914/eq-e26bc583ff",16,"Thompson 1914, p. 223: \\dfrac{2}{\\sqrt 3}=1.155"],["de-morgan-elementary-illustrations-calculus-1899/eq-6981964a41",16,"De Morgan 1899, p. 62: n · \\frac{(n + 1)}{2}\\, v't' = \\frac{n^{2} v't' + nv't'}{2}"],["thompson-calculus-made-easy-1914/eq-a401ac6fd4",16,"Thompson 1914, p. 223: \\text{quadratic mean} = \\sqrt[2]{\\dfrac{l^{2a}}{2a+1}}."],["thompson-calculus-made-easy-1914/ch-epilogue-and-apologue",2,"Thompson 1914, Epilogue and Apologue","../books/thompson-calculus-made-easy-1914/ch/ch-epilogue-and-apologue/index.html"],["theorem/ratio-of-series-of-the-same-order",9,"ratio of series of the same order","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-ratio-of-series-of-the-same-order"],["wentworth-first-steps-in-algebra-1894/ex-55",3,"Wentworth 1894, Exercise 55"],["wentworth-first-steps-in-algebra-1894/ex-56",3,"Wentworth 1894, Exercise 56"],["wentworth-first-steps-in-algebra-1894/ex-57",3,"Wentworth 1894, Exercise 57"],["wentworth-first-steps-in-algebra-1894/ex-58",3,"Wentworth 1894, Exercise 58"],["wentworth-first-steps-in-algebra-1894/ex-59",3,"Wentworth 1894, Exercise 59"],["wentworth-first-steps-in-algebra-1894/ex-60",3,"Wentworth 1894, Exercise 60"],["wentworth-first-steps-in-algebra-1894/ex-61",3,"Wentworth 1894, Exercise 61"],["wentworth-first-steps-in-algebra-1894/ex-62",3,"Wentworth 1894, Exercise 62"],["form/af0f21575e",5,"solve: Eq(x**4 - 4*x**3 + 4*x - 1, 0)"],["thompson-calculus-made-easy-1914/eq-e50a14f883",16,"Thompson 1914, p. 220: \\tfrac{1}{2} \\int^{\\theta=\\theta_2}_{\\theta=\\theta_1} r^2\\, d\\theta."],["wentworth-first-steps-in-algebra-1894/ex-54",3,"Wentworth 1894, Exercise 54"],["wentworth-first-steps-in-algebra-1894/ex-53",3,"Wentworth 1894, Exercise 53"],["de-morgan-elementary-illustrations-calculus-1899/eq-c4e4f44716",16,"De Morgan 1899, p. 61: nv' = v"],["form/28ca5d02e8",5,"solve: (Eq(a, 7*x), Eq(a + x, 64))"],["wentworth-first-steps-in-algebra-1894/ex-52",3,"Wentworth 1894, Exercise 52"],["shape/959e79fcbd",6,"solve: Eq(N*x + N*x**N + x**N - 1, 0)"],["de-morgan-elementary-illustrations-calculus-1899/x-7ed41c3747",15,"De Morgan 1899, p. 104: The resulting values of y, or y, y_{1}, etc., ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-6e5039d9c6",16,"De Morgan 1899, p. 62: \\frac{1}{2}v(t + t')"],["concept/numerical-quadratic-equation",7,"numerical quadratic equation","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-numerical-quadratic-equation"],["thompson-calculus-made-easy-1914/eq-9f1d1909d1",16,"Thompson 1914, p. 177: z = x^y"],["shape/08c3c128ba",6,"extremum: N*x + N + 2*x**N"],["concept/formula",7,"formula","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-formula"],["thompson-calculus-made-easy-1914/eq-cb13b3f4e7",16,"Thompson 1914, p. 220: r=a(1+\\cos \\theta)"],["de-morgan-elementary-illustrations-calculus-1899/eq-7676464158",16,"De Morgan 1899, p. 62: \\frac{1}{2}vt"],["de-morgan-elementary-illustrations-calculus-1899/eq-db9ced9a57",16,"De Morgan 1899, p. 62: v = gt"],["thompson-calculus-made-easy-1914/eq-fd1a2eb2a8",16,"Thompson 1914, p. 19: \\frac{dy}{dx} = 2x"],["thompson-calculus-made-easy-1914/eq-51925fb234",16,"Thompson 1914, p. 21: \\frac{dy}{dx} = 3x^2"],["form/0ba5cf8595",5,"identity: (a**2 + 5*a)/(a**2 + 4*a - 5)"],["planck-treatise-on-thermodynamics-1903/eq-304f06735f",16,"Planck 1903, p. 120: \\left(\\frac{\\dd p}{\\dd v}\\right)_{\\theta} = -\\frac{1014000}{0.00000295 · v}"],["planck-treatise-on-thermodynamics-1903/eq-b16c6bcf63",16,"Planck 1903, p. 125: \\Delta \\theta = \\frac{\\theta \\left(\\dfrac{\\dd v}{\\dd \\theta}\\right)_{p} - v}{c_{p}}\\, \\Delta p"],["thompson-calculus-made-easy-1914/eq-9549b92db4",16,"Thompson 1914, p. 220: &= \\frac{a^2(3\\pi+8)}{8}."],["dickson-theory-of-equations-1922/ex-page120/1",4,"Dickson 1922, Exercise Page120 (1)"],["wentworth-first-steps-in-algebra-1894/ex-66",3,"Wentworth 1894, Exercise 66"],["shape/ba622928fe",6,"identity: (N*a + a**N)/(N*a + N + a**N)"],["thompson-calculus-made-easy-1914/eq-e1976ae675",16,"Thompson 1914, p. 177: \\dfrac{\\partial z}{\\partial x} &= yx^{y-1}"],["blackburn-elements-plane-trigonometry-1863/eq-500b5b38e9",16,"Blackburn 1863, p. 6: \\dfrac{1}{2}\\, R × \\text{circumference} = \\pi R^2"],["de-morgan-elementary-illustrations-calculus-1899/ch-recapitulation-of-results-reached-in-the-theory-of-functions",2,"De Morgan 1899, Recapitulation of Results Reached in the Theory of Functions","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-recapitulation-of-results-reached-in-the-theory-of-functions/index.html"],["wentworth-first-steps-in-algebra-1894/ex-68",3,"Wentworth 1894, Exercise 68"],["wentworth-first-steps-in-algebra-1894/ex-69",3,"Wentworth 1894, Exercise 69"],["wentworth-first-steps-in-algebra-1894/ex-70",3,"Wentworth 1894, Exercise 70"],["de-morgan-elementary-illustrations-calculus-1899/eq-4ae42ee0fc",16,"De Morgan 1899, p. 74: y = \\phi(x)"],["wentworth-first-steps-in-algebra-1894/ex-67",3,"Wentworth 1894, Exercise 67"],["wentworth-first-steps-in-algebra-1894/x-82f185e255",15,"Wentworth 1894, p. 147: A heavy body falling from a height falls 16.1 ..."],["thompson-calculus-made-easy-1914/eq-b5e5f54d1a",16,"Thompson 1914, p. 221: y^2 = r^2 - x^2."],["concept/perfect-number",7,"perfect number","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-perfect-number"],["blackburn-elements-plane-trigonometry-1863/eq-805c551708",16,"Blackburn 1863, p. 6: = \\dfrac{\\pi}{4}"],["thompson-calculus-made-easy-1914/eq-e8f74d6dc7",16,"Thompson 1914, p. 177: \\dfrac{\\partial z}{\\partial y} &= x^y × \\log_\\epsilon x"],["macfarlane-vector-analysis-quaternions-1906/eq-00278da789",16,"Macfarlane 1906: \\gamma^c = \\cos c + \\sin c \\cdot \\gamma^\\frac{\\pi}{2}"],["form/c57d7ddc2e",5,"differentiate: sqrt(a**2 + x**2)"],["shape/6f2536a01b",6,"differentiate: (a**N + x**N)**N"],["blackburn-elements-plane-trigonometry-1863/eq-61e3fce184",16,"Blackburn 1863, p. 8: r' = \\frac{R + r}{2}"],["blackburn-elements-plane-trigonometry-1863/eq-b35d2bcc52",16,"Blackburn 1863, p. 8: R' = \\sqrt{r' · R}"],["thompson-calculus-made-easy-1914/eq-4890a57bfb",16,"Thompson 1914, p. 21: \\frac{dy}{dx} = 4x^3"],["quantity/rate-of-interest",11,"rate of interest","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-quantity-rate-of-interest"],["planck-treatise-on-thermodynamics-1903/x-fe8c8c8404",15,"Planck 1903, p. 37: It is reckoned positive when heat is set free ..."],["dickson-theory-of-equations-1922/eq-8583a9fea2",16,"Dickson 1922, p. 47: (x-1)(x-\\omega)(x-\\omega^2) \\equiv x^3 - 1"],["planck-treatise-on-thermodynamics-1903/eq-1feca011d2",16,"Planck 1903, p. 108: dU = Q + W"],["blackburn-elements-plane-trigonometry-1863/eq-51c08a8c4d",16,"Blackburn 1863, p. 8: EC: EF:: EF: EG"],["blackburn-elements-plane-trigonometry-1863/eq-7bd8cbfc6a",16,"Blackburn 1863, p. 10: \\dfrac{1}{3} (r + 2R)"],["blackburn-elements-plane-trigonometry-1863/eq-566f2e71f9",16,"Blackburn 1863, p. 34: \\log 10 = 1"],["theorem/mutually-bounded-increasing-sequences-have-the-same-limit",9,"mutually bounded increasing sequences have the same limit","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-mutually-bounded-increasing-sequences-have-the-same-limit"],["theorem/mutually-bounded-decreasing-sequences-have-the-same-limit",9,"mutually bounded decreasing sequences have the same limit","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-mutually-bounded-decreasing-sequences-have-the-same-limit"],["de-morgan-elementary-illustrations-calculus-1899/ch-approximations-by-the-differential-calculus",2,"De Morgan 1899, Approximations by the Differential Calculus","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-approximations-by-the-differential-calculus/index.html"],["dickson-theory-of-equations-1922/ex-page120/3",4,"Dickson 1922, Exercise Page120 (3)"],["de-morgan-elementary-illustrations-calculus-1899/eq-040a23ef8e",16,"De Morgan 1899, p. 75: \\phi x + \\phi' x\\, dx"],["wentworth-first-steps-in-algebra-1894/ex-73",3,"Wentworth 1894, Exercise 73"],["form/39b2bd52bd",5,"identity: (a**2 - 14*a - 15)/(a**2 - 12*a - 45)"],["blackburn-elements-plane-trigonometry-1863/eq-ce2039ce2d",16,"Blackburn 1863, p. 11: \\pi = \\frac{20000000000}{6366197723}"],["form/596a7cefda",5,"identity: 4*a**4 - 2*a**3*x + 2*a*x**3 - 3*x**4"],["wentworth-first-steps-in-algebra-1894/ex-74",3,"Wentworth 1894, Exercise 74"],["dickson-theory-of-equations-1922/ex-page120/2",4,"Dickson 1922, Exercise Page120 (2)"],["wentworth-first-steps-in-algebra-1894/ch-xii",2,"Wentworth 1894, ch. XII: Quadratic Equations","../books/wentworth-first-steps-in-algebra-1894/ch/ch-xii/index.html"],["theorem/radius-of-the-inscribed-circle-from-the-sides",9,"radius of the inscribed circle from the sides","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-radius-of-the-inscribed-circle-from-the-sides"],["shape/a556f0b9b4",6,"identity: a*x**N - c*x + c*x**N - x"],["theorem/sum-of-a-geometrical-progression",9,"sum of a geometrical progression","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-theorem-sum-of-a-geometrical-progression"],["wentworth-first-steps-in-algebra-1894/x-5c26feff53",15,"Wentworth 1894, p. 148: A series of numbers is said to be in ..."],["wentworth-first-steps-in-algebra-1894/ex-25/17",4,"Wentworth 1894, Exercise 25 (17)"],["form/3a7852d379",5,"identity: a*x**4 - a*x**3 + b*x**4 + b*x**3 - c*x - 2*x"],["thompson-calculus-made-easy-1914/eq-8dc090ccf1",16,"Thompson 1914, p. 79: \\dfrac{dy}{dx}= 0"],["slaught-lennes-solid-geometry-1919/ch-book-v",2,"Slaught & Lennes 1919, ch. BOOK V: The Sphere","../books/slaught-lennes-solid-geometry-1919/ch/ch-book-v/index.html"],["concept/distance-from-a-point-to-a-plane",7,"distance from a point to a plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-distance-from-a-point-to-a-plane"],["maxwell-elementary-treatise-electricity-1888/eq-0107660d5c",16,"Maxwell 1888, scan 109: K_1 + H_1 = K_2"],["maxwell-elementary-treatise-electricity-1888/eq-4e95d4778c",16,"Maxwell 1888, scan 112: a &= (P+R+\\alpha+\\eta)A-PB-RD-\\eta C"],["wentworth-first-steps-in-algebra-1894/x-27e69d9a62",15,"Wentworth 1894, p. 148: Hence the nth term will be ar^{n - 1}."],["de-morgan-elementary-illustrations-calculus-1899/x-db7b930150",15,"De Morgan 1899, p. 104: The limiting ratio of d^{2} y to (dx)^{2}, expressed ..."],["hardy-course-of-pure-mathematics-1921/eq-07ce19e33a",16,"Hardy 1921, p. 368: \\lim_{\\xi\\to\\infty} \\left(1 + \\frac{x}{\\xi}\\right)^{\\xi} = \\lim_{\\xi\\to -\\infty} \\left(1 + \\frac{x}{\\xi}\\right)^{\\xi} = "],["hardy-course-of-pure-mathematics-1921/x-89a956fdd3",15,"Hardy 1921, p. 422: Here both (1 + tz)^{m} and (1 + tz)^{m-1} ..."],["theorem/convex-line-is-less-than-any-line-enveloping-it",9,"convex line is less than any line enveloping it","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-convex-line-is-less-than-any-line-enveloping-it"],["thompson-calculus-made-easy-1914/eq-bb308f801a",16,"Thompson 1914, p. 177: dz = yx^{y-1}\\, dx + x^y \\log_\\epsilon x \\, dy"],["form/041ae4efdc",5,"differentiate: 1/sqrt(a + x)"],["form/6bce9e0d89",5,"differentiate: sin(2*x)"],["concept/fermat-number",7,"Fermat number","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-fermat-number"],["concept/odd-magic-square",7,"odd magic square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-odd-magic-square"],["theorem/nth-term-of-an-arithmetical-progression",9,"nth term of an arithmetical progression","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-theorem-nth-term-of-an-arithmetical-progression"],["hardy-course-of-pure-mathematics-1921/eq-aa3e300157",16,"Hardy 1921, p. 368: n(1 - x^{-1/n}) < \\log x < n(x^{1/n} - 1)"],["thompson-calculus-made-easy-1914/eq-79ec7d2af9",16,"Thompson 1914, p. 177: V=\\frac{1}{3} \\pi r^2 h"],["theorem/sum-of-face-angles-of-a-polyhedral-angle-is-less-than-360-degrees",9,"sum of face angles of a polyhedral angle is less than 360 degrees","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-sum-of-face-angles-of-a-polyhedral-angle-is-less-than-360-degrees"],["person/benjamin-franklin",1,"Benjamin Franklin","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-benjamin-franklin"],["person/william-gilbert",1,"William Gilbert","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-william-gilbert"],["wentworth-first-steps-in-algebra-1894/ex-76",3,"Wentworth 1894, Exercise 76"],["wentworth-first-steps-in-algebra-1894/ex-75",3,"Wentworth 1894, Exercise 75"],["cap/other:derivative with respect to a function of the variable (ratio dy5/dy2, chain rule)",17,"other:derivative with respect to a function of the variable (ratio dy5/dy2, chain rule)"],["form/6128736d93",5,"differentiate: sqrt(-x**2 + 1)/(-x + 1)"],["concept/doubly-even-magic-square",7,"doubly-even magic square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-doubly-even-magic-square"],["de-morgan-elementary-illustrations-calculus-1899/eq-b4aa171034",16,"De Morgan 1899, p. 77: x^{2} + x - 4 = 0"],["concept/singly-even-magic-square",7,"singly-even magic square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-singly-even-magic-square"],["hardy-course-of-pure-mathematics-1921/eq-9dea1624e7",16,"Hardy 1921, p. 368: \\left(1 + \\frac{y}{n}\\right)^{n} < x < \\left(1 - \\frac{y}{n}\\right)^{-n}"],["shape/409db907a1",6,"identity: N*x + 2*b*x**N - c*x"],["de-morgan-elementary-illustrations-calculus-1899/eq-bbfbf427cb",16,"De Morgan 1899, p. 78: \\tan x = ax"],["shape/e1363e3e0f",6,"differentiate: (-x**N + 1)**N/(-x + 1)"],["form/0cd0b05d96",5,"evaluate: x**3 + x**2 - 10*x + 8 at x=(-1 - sqrt(31))/3"],["thompson-calculus-made-easy-1914/eq-6ea96ed433",16,"Thompson 1914, p. 177: \\frac{\\partial V}{\\partial r} &= \\dfrac{2\\pi}{3} rh"],["de-morgan-elementary-illustrations-calculus-1899/eq-924ea37426",16,"De Morgan 1899, p. 88: \\frac{d^{2} y}{dx^{2}}"],["shape/5beb801cc3",6,"differentiate: sin(N*x)"],["boyden-first-book-in-algebra-1895/ch-addition",2,"Boyden 1895, ADDITION","../books/boyden-first-book-in-algebra-1895/ch/ch-addition/index.html"],["planck-treatise-on-thermodynamics-1903/eq-a853c4143d",16,"Planck 1903, p. 168: v &= \\frac{\\lambda v_{1} + \\mu v_{2} + \\nu v_{3}}{\\lambda + \\mu + \\nu}"],["dickson-theory-of-equations-1922/eq-47be3a5cd5",16,"Dickson 1922, p. 157: x^2 + y^2 = R^2"],["wentworth-first-steps-in-algebra-1894/ex-78",3,"Wentworth 1894, Exercise 78"],["wentworth-first-steps-in-algebra-1894/ex-81",3,"Wentworth 1894, Exercise 81"],["wentworth-first-steps-in-algebra-1894/ex-79",3,"Wentworth 1894, Exercise 79"],["boyden-first-book-in-algebra-1895/ex-15",3,"Boyden 1895, Exercise 15"],["wentworth-first-steps-in-algebra-1894/ex-80",3,"Wentworth 1894, Exercise 80"],["thompson-calculus-made-easy-1914/eq-7768d9fff6",16,"Thompson 1914, p. 78: \\dfrac{dy}{dx} = 1"],["person/andr-marie-amp-re",1,"André-Marie Ampère","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-andr-marie-amp-re"],["planck-treatise-on-thermodynamics-1903/eq-dd8269b127",16,"Planck 1903, p. 95: -\\frac{Q}{\\theta}"],["maxwell-elementary-treatise-electricity-1888/eq-1af5b2593f",16,"Maxwell 1888, scan 191: \\overline{V} = \\frac{KV + K'V'}{K + K'}\\text{.}"],["concept/theorem-common-potential-after-contact",7,"theorem: common potential after contact"],["planck-treatise-on-thermodynamics-1903/eq-863ead6ae6",16,"Planck 1903, p. 96: -\\tsum \\frac{Q}{\\theta} \\geq 0"],["wentworth-first-steps-in-algebra-1894/ch-xv",2,"Wentworth 1894, ch. XV: Square and Cube Roots","../books/wentworth-first-steps-in-algebra-1894/ch/ch-xv/index.html"],["thompson-calculus-made-easy-1914/eq-1a76990960",16,"Thompson 1914, p. 83: y=x+b"],["planck-treatise-on-thermodynamics-1903/eq-45f847b5bb",16,"Planck 1903, p. 96: \\tsum \\frac{Q}{\\theta} \\leq 0"],["dickson-theory-of-equations-1922/eq-341c4382d1",16,"Dickson 1922, p. 86: p^3 + 6p^2 + 10p - 1 = 0"],["thompson-calculus-made-easy-1914/eq-792f35c5da",16,"Thompson 1914, p. 199: \\int x^n\\, dx = \\dfrac{1}{n+1} x^{n+1}."],["wentworth-first-steps-in-algebra-1894/eq-67581977ce",16,"Wentworth 1894, p. 126: y + 10 &= 3(x - 10)"],["boyden-first-book-in-algebra-1895/ex-1/5",4,"Boyden 1895, Exercise 1 (5)"],["maxwell-elementary-treatise-electricity-1888/eq-324b4c6e29",16,"Maxwell 1888, scan 191: V = \\overline{V} + \\frac{K'}{K} (\\overline{V} - V')\\text{.}"],["concept/perfect-cube",7,"perfect cube","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-perfect-cube"],["wentworth-first-steps-in-algebra-1894/eq-9df268b59e",16,"Wentworth 1894, p. 122: x + y &= 10"],["thompson-calculus-made-easy-1914/eq-b6f255250c",16,"Thompson 1914, p. 177: \\frac{\\partial V}{\\partial h} &= \\dfrac{\\pi}{3} r^2"],["thompson-calculus-made-easy-1914/eq-13d074fc0b",16,"Thompson 1914, p. 199: y = a \\log_\\epsilon x + C."],["wentworth-first-steps-in-algebra-1894/eq-510c997133",16,"Wentworth 1894, p. 129: 10x + y + 18 &= 10y + x"],["wentworth-first-steps-in-algebra-1894/eq-485175ce55",16,"Wentworth 1894, p. 129: x - y &= -2"],["blackburn-elements-plane-trigonometry-1863/eq-31d17b22a1",16,"Blackburn 1863, p. 10: \\frac{2000000}{636621} < \\pi < \\frac{2000000}{636617}"],["concept/series-of-positive-terms",7,"series of positive terms","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-series-of-positive-terms"],["form/d450737137",5,"solve: Eq(3*x, 24)"],["planck-treatise-on-thermodynamics-1903/eq-63290a0a99",16,"Planck 1903, p. 96: W = -p\\, dV"],["person/heinrich-hertz",1,"Heinrich Hertz","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-heinrich-hertz"],["planck-treatise-on-thermodynamics-1903/eq-eb734c0854",16,"Planck 1903, p. 96: \\tsum \\frac{Q}{\\theta} = 0"],["planck-treatise-on-thermodynamics-1903/eq-5ac1e4955a",16,"Planck 1903, p. 96: \\tsum \\frac{dU + p\\, dV}{\\theta} = 0"],["person/hugh-blackburn",1,"Hugh Blackburn"],["maxwell-elementary-treatise-electricity-1888/eq-8fbe8426c8",16,"Maxwell 1888, scan 192: Q = V'a\\text{,}"],["planck-treatise-on-thermodynamics-1903/eq-3448721ed2",16,"Planck 1903, p. 97: \\int_{1}^{2} \\frac{dU + p\\, dV}{\\theta}"],["wentworth-first-steps-in-algebra-1894/eq-55d59c2fa2",16,"Wentworth 1894, p. 77: a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})"],["dickson-theory-of-equations-1922/ex-page13/1",4,"Dickson 1922, Exercise Page13 (1)"],["form/72d185e3fd",5,"evaluate: x**4 - 3*x**2 - x - 6 at x=-3"],["hardy-course-of-pure-mathematics-1921/eq-906a72d767",16,"Hardy 1921, p. 369: \\lim n(1 - x^{-1/n}) = \\lim n(x^{1/n} - 1) = \\log x"],["maxwell-elementary-treatise-electricity-1888/eq-738d3c1502",16,"Maxwell 1888, scan 112: b &= (P+Q+\\beta+\\xi)B-PA-QC-\\xi D"],["shape/b67cc2dfe3",6,"evaluate: N*x**N + N - x + x**N"],["theorem/lengths-of-similar-arcs-are-proportional-to-their-chords",9,"lengths of similar arcs are proportional to their chords","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-lengths-of-similar-arcs-are-proportional-to-their-chords"],["maxwell-elementary-treatise-electricity-1888/eq-3c0f77dde6",16,"Maxwell 1888, scan 112: c &= (Q+S+\\gamma+\\eta)C-QB-SD-\\eta A"],["maxwell-elementary-treatise-electricity-1888/eq-f01c164397",16,"Maxwell 1888, scan 192: V + V' = 0\\text{,}"],["concept/approximating-sequence-of-polygons",7,"approximating sequence of polygons","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-approximating-sequence-of-polygons"],["maxwell-elementary-treatise-electricity-1888/eq-ae8310db6c",16,"Maxwell 1888, scan 112: d &= (R+S+\\delta+\\xi)D-RA-SC-\\xi B"],["maxwell-elementary-treatise-electricity-1888/eq-6248eca347",16,"Maxwell 1888, scan 112: a &= (P+R+\\alpha+\\eta)A-\\eta C"],["maxwell-elementary-treatise-electricity-1888/eq-a653d55727",16,"Maxwell 1888, scan 112: b &= \\hphantom{(P+R+\\alpha}-PA-QC"],["maxwell-elementary-treatise-electricity-1888/eq-204ec80eaa",16,"Maxwell 1888, scan 112: c &= (Q+S+\\gamma+\\eta)C-\\eta A"],["maxwell-elementary-treatise-electricity-1888/eq-378c477504",16,"Maxwell 1888, scan 192: -Va = Q"],["theorem/acute-angle-with-projection-is-the-least-angle-with-lines-in-the-plane",9,"acute angle with projection is the least angle with lines in the plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-acute-angle-with-projection-is-the-least-angle-with-lines-in-the-plane"],["shape/b0bd73aac5",6,"solve: (Eq(N*x, N), Eq(N*x - a, N*a + N*x))"],["shape/db015ca2d2",6,"differentiate: (2*N*x**N)**N"],["shape/83d40b1696",6,"integrate: cos(x)"],["maxwell-elementary-treatise-electricity-1888/eq-3a6ae9ec79",16,"Maxwell 1888, scan 112: a' + c' = a + c"],["thompson-calculus-made-easy-1914/x-ec720cd927",15,"Thompson 1914, p. 67: Sometimes one is stumped by finding that the expression ..."],["maxwell-elementary-treatise-electricity-1888/eq-57b10d2396",16,"Maxwell 1888, scan 112: b' = b"],["wentworth-first-steps-in-algebra-1894/ex-10/1",4,"Wentworth 1894, Exercise 10 (1)"],["maxwell-elementary-treatise-electricity-1888/eq-3d973a64e0",16,"Maxwell 1888, scan 112: A' = C' = y"],["maxwell-elementary-treatise-electricity-1888/eq-0f829495e9",16,"Maxwell 1888, scan 112: a' &= (P+R+\\alpha)y-PB'"],["hardy-course-of-pure-mathematics-1921/ch-appendix-ii",2,"Hardy 1921, ch. Appendix II: A Note on Double Limit Problems","../books/hardy-course-of-pure-mathematics-1921/ch/ch-appendix-ii/index.html"],["dickson-theory-of-equations-1922/ex-page89/8",4,"Dickson 1922, Exercise Page89 (8)"],["form/5f7244a8a4",5,"solve: Eq(x**3 - 7*x - 7, 0)"],["form/9b2b60cc77",5,"solve: Eq(9*x, 270)"],["blackburn-elements-plane-trigonometry-1863",0,"Blackburn, Elements of Plane Trigonometry (1863)","../books/blackburn-elements-plane-trigonometry-1863/index.html"],["maxwell-elementary-treatise-electricity-1888/eq-7c42fa0c4c",16,"Maxwell 1888, scan 112: b' &= (P+Q+\\beta+\\xi)B' - (P+Q)y"],["maxwell-elementary-treatise-electricity-1888/eq-3550752c46",16,"Maxwell 1888, scan 112: c' &= (Q+S+\\gamma)y-QB'"],["maxwell-elementary-treatise-electricity-1888/eq-5d880364aa",16,"Maxwell 1888, scan 113: (P+R+Q+S+\\alpha+\\gamma)y-(P+Q)B'=(P+R+\\alpha)A+(Q+S+\\gamma)C"],["whitehead-introduction-to-mathematics-1911/ch-xvi",2,"Whitehead 1911, ch. XVI: Geometry","../books/whitehead-introduction-to-mathematics-1911/ch/ch-xvi/index.html"],["hardy-course-of-pure-mathematics-1921/ch-ix",2,"Hardy 1921, ch. IX: THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE","../books/hardy-course-of-pure-mathematics-1921/ch/ch-ix/index.html"],["dickson-theory-of-equations-1922/ex-page2/3",4,"Dickson 1922, Exercise Page2 (3)"],["form/c19fb85092",5,"extremum: -c*x**2 + b*x/a"],["wentworth-first-steps-in-algebra-1894/ex-26",3,"Wentworth 1894, Exercise 26"],["de-morgan-elementary-illustrations-calculus-1899/ch-the-integral-calculus",2,"De Morgan 1899, The Integral Calculus","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-the-integral-calculus/index.html"],["shape/236c3d6549",6,"extremum: -c*x**N + b*x/a"],["form/db8b9ad78a",5,"evaluate: -x*sqrt(x*(-x + 10))/6"],["maxwell-elementary-treatise-electricity-1888/eq-212db2de95",16,"Maxwell 1888, scan 113: (P+Q+\\beta+\\xi)B'-(P+Q)y = -PA-QC"],["hardy-course-of-pure-mathematics-1921/eq-a1914f46a6",16,"Hardy 1921, p. 375: D_{x}(\\log x)^{1-s} = \\frac{1 - s}{x(\\log x)^{s}}"],["concept/sexagesimal-system",7,"sexagesimal system","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-sexagesimal-system"],["blackburn-elements-plane-trigonometry-1863/x-f3aab8c4be",15,"Blackburn 1863, p. 42: These relations give the simplest logarithmic solution of a ..."],["wentworth-first-steps-in-algebra-1894/ex-72",3,"Wentworth 1894, Exercise 72"],["theorem/circumferences-and-areas-of-circles-vary-as-radii-and-squares-of-radii",9,"circumferences and areas of circles vary as radii and squares of radii","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-circumferences-and-areas-of-circles-vary-as-radii-and-squares-of-radii"],["thompson-calculus-made-easy-1914/x-0da2a79318",15,"Thompson 1914, p. 67: Thus, the equationdodge y = (x^2+a^2)^32 is awkward to ..."],["maxwell-elementary-treatise-electricity-1888/ch-v",2,"Maxwell 1888, ch. V: FARADAY'S LAW OF LINES OF INDUCTION","../books/maxwell-elementary-treatise-electricity-1888/ch/ch-v/index.html"],["method/constructing-a-plane-parallel-to-a-line",8,"constructing a plane parallel to a line","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-constructing-a-plane-parallel-to-a-line"],["wentworth-first-steps-in-algebra-1894/ex-2/3",4,"Wentworth 1894, Exercise 2 (3)"],["blackburn-elements-plane-trigonometry-1863/x-4e4f4c5f69",15,"Blackburn 1863, p. 44: The formulæ used here were discovered by William Purser ..."],["form/527c455724",5,"identity: 60"],["slaught-lennes-solid-geometry-1919/x-ea4e45500d",15,"Slaught & Lennes 1919, p. 188: Now the greater the number of sides the more ..."],["hardy-course-of-pure-mathematics-1921/eq-73521a181a",16,"Hardy 1921, p. 375: D_{x}\\log\\log x = \\frac{1}{x\\log x}"],["wentworth-first-steps-in-algebra-1894/ex-2/4",4,"Wentworth 1894, Exercise 2 (4)"],["wentworth-first-steps-in-algebra-1894/ex-2/5",4,"Wentworth 1894, Exercise 2 (5)"],["theorem/dirichlet-s-test",9,"Dirichlet's test","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-dirichlet-s-test"],["cap/other:root_bound_method",17,"other:root_bound_method"],["maxwell-elementary-treatise-electricity-1888/eq-e329b44aa5",16,"Maxwell 1888, scan 113: B'\\{(P+Q)(R+S)+(P+Q)(\\alpha+\\beta+\\gamma+\\xi)+(R+S+\\alpha+\\gamma)(\\beta+\\xi)\\}=\\{Q(R+\\alpha)-P(S+\\gamma)\\}(A-C)"],["shape/0e467024fe",6,"solve: (Eq(a, N + b), Eq(b, N*a))"],["unit/second-of-arc",12,"second of arc","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-unit-second-of-arc"],["form/a0466c2863",5,"identity: (x**5 - x**4 + 1)/(x**2 - x - 1)"],["blackburn-elements-plane-trigonometry-1863/ch-iv",2,"Blackburn 1863, ch. IV: OF THE UNIT OF ANGULAR MAGNITUDE","../books/blackburn-elements-plane-trigonometry-1863/ch/ch-iv/index.html"],["blackburn-elements-plane-trigonometry-1863/x-947e75438f",15,"Blackburn 1863, p. 44: There is also a simple verification of the process ..."],["wentworth-plane-geometry-1899/x-489e486a75",15,"Wentworth 1899, scan 145: If the product of two quantities is equal to ..."],["shape/d3b795f959",6,"identity: (N*a + N*b)**2"],["shape/8ad2cc6d17",6,"identity: 1/(-x + x**N - 1)"],["dickson-theory-of-equations-1922/ex-page2/1",4,"Dickson 1922, Exercise Page2 (1)"],["dickson-theory-of-equations-1922/ex-page23/1",4,"Dickson 1922, Exercise Page23 (1)"],["hardy-course-of-pure-mathematics-1921/eq-0a53fa3f89",16,"Hardy 1921, p. 301: \\begin{vmatrix} f(a) & f(b) & f(c)\\\\ g(a) & g(b) & g(c)\\\\ h(a) & h(b) & h(c) \\end{vmatrix} = \\tfrac{1}{2} (b - c)(c - a)"],["wentworth-first-steps-in-algebra-1894/ex-27",3,"Wentworth 1894, Exercise 27"],["form/b6c78ae19d",5,"differentiate: x**13"],["slaught-lennes-solid-geometry-1919/x-0c5cdcc596",15,"Slaught & Lennes 1919, p. 191: For practical purposes the lengths of such segments are ..."],["shape/e640d11928",6,"differentiate: x**N"],["maxwell-elementary-treatise-electricity-1888/eq-d6fdc1e1db",16,"Maxwell 1888, scan 113: B' = 0"],["slaught-lennes-solid-geometry-1919/x-c7eb4f04a9",15,"Slaught & Lennes 1919, p. 193: Not every infinite sequence serves to single out a ..."],["blackburn-elements-plane-trigonometry-1863/ch-viii",2,"Blackburn 1863, ch. VIII: OF TRIGONOMETRICAL SURVEYING","../books/blackburn-elements-plane-trigonometry-1863/ch/ch-viii/index.html"],["todhunter-spherical-trigonometry-1886/ex-ix",3,"Todhunter 1886, Exercise IX"],["slaught-lennes-solid-geometry-1919/x-bc0ff198d8",15,"Slaught & Lennes 1919, p. 193: That is, 1 is the smallest number beyond which ..."],["form/a45bd46bc5",5,"identity: 3*I"],["shape/32abb0c463",6,"identity: I*N"],["dickson-theory-of-equations-1922/ex-page2/2",4,"Dickson 1922, Exercise Page2 (2)"],["form/ba55f36788",5,"differentiate: x**(2*a)"],["todhunter-spherical-trigonometry-1886/ex-ix/4",4,"Todhunter 1886, Exercise IX (4)"],["thompson-calculus-made-easy-1914/eq-a46e74cc91",16,"Thompson 1914, p. 252: \\frac{1}{2} x^2 + C"],["dickson-theory-of-equations-1922/ex-page141/3",4,"Dickson 1922, Exercise Page141 (3)"],["thompson-calculus-made-easy-1914/eq-cd7d495dd0",16,"Thompson 1914, p. 252: ax + C"],["dickson-theory-of-equations-1922/ex-page141/4",4,"Dickson 1922, Exercise Page141 (4)"],["thompson-calculus-made-easy-1914/ex-iii/11",4,"Thompson 1914, Exercise III (11)"],["form/a6b62b7451",5,"differentiate: a + b*x + c*x**2"],["todhunter-spherical-trigonometry-1886/ex-ix/6",4,"Todhunter 1886, Exercise IX (6)"],["shape/2e8ae5a320",6,"differentiate: a + b*x + c*x**N"],["form/02ad43c05d",5,"evaluate: x**3 + 3*x at x=1"],["form/b7ba480807",5,"evaluate: x**3 + 3*x at x=2"],["form/4a4169809f",5,"solve: Eq(x**2 - 14*x - 51, 0)"],["form/3fd789b6fb",5,"identity: (x + 4)*(x + 7)"],["wentworth-first-steps-in-algebra-1894/ex-9",3,"Wentworth 1894, Exercise 9"],["form/28e7143729",5,"differentiate: a*x**b"],["cap/other:implicit differentiation",17,"other:implicit differentiation"],["shape/28e7143729",6,"differentiate: a*x**b"],["shape/15ed8c991c",6,"differentiate: exp(x)"],["thompson-calculus-made-easy-1914/ex-iii/4",4,"Thompson 1914, Exercise III (4)"],["todhunter-spherical-trigonometry-1886/ex-ix/10",4,"Todhunter 1886, Exercise IX (10)"],["form/d27f9b9d96",5,"differentiate: a/(b*x + c*x**2 + 1)"],["shape/5cba5f35a9",6,"solve: Eq(N*x/(N + x) + N*(N + x)/x, N)"],["form/2800caaa1a",5,"identity: 4*I*(5 + 5*I)"],["form/fa29812fc4",5,"differentiate: sqrt(c*d/x)/(sqrt(pi)*a*b)"],["thompson-calculus-made-easy-1914/ex-iii/1b",4,"Thompson 1914, Exercise III (1b)"],["form/90f01aca80",5,"differentiate: a*x**2 + b*x + c"],["shape/4d8584bd0b",6,"differentiate: a*x**N + b*x + c"],["cap/core.ratio.partials",17,"core.ratio.partials"],["todhunter-spherical-trigonometry-1886/ex-v/1",4,"Todhunter 1886, Exercise V (1)"],["form/ace3545107",5,"differentiate: (-34*x**2 + 197*x)*(-83*x**3 + 22*x + 7)"],["form/ea0a0c1d4d",5,"partial_ratio: 2*a*x**3/(y**3*(-b**2 + 1))"],["shape/aefdd1f76f",6,"partial_ratio: N*a*x**N*y**N/(-b**N + 1)"],["shape/6b3f4891a8",6,"differentiate: (N*x + N*x**N)*(N*x + N*x**N + N)"],["shape/0cda917cd8",6,"differentiate: a/(b*x + c*x**N + 1)"],["form/5ef2305585",5,"differentiate: 2*pi*x"],["todhunter-spherical-trigonometry-1886/ex-v/5b",4,"Todhunter 1886, Exercise V (5b)"],["thompson-calculus-made-easy-1914/eq-716bfee8b4",16,"Thompson 1914, p. 252: \\frac{1}{2} ax^2 + C"],["shape/8b5769d18f",6,"identity: I*N*(N + I*N)"],["dickson-theory-of-equations-1922/ex-page2/4",4,"Dickson 1922, Exercise Page2 (4)"],["form/9361f41381",5,"differentiate: pi*x**2"],["thompson-calculus-made-easy-1914/ex-iii/1a",4,"Thompson 1914, Exercise III (1a)"],["todhunter-spherical-trigonometry-1886/ex-v/8",4,"Todhunter 1886, Exercise V (8)"],["form/15ed8c991c",5,"differentiate: exp(x)"],["cap/other:infinite_product",17,"other:infinite_product"],["thompson-calculus-made-easy-1914/eq-53ccb9dfbe",16,"Thompson 1914, p. 177: dV = \\dfrac{2\\pi}{3} rh\\, dV + \\dfrac{\\pi}{3} r^2\\, dh"],["form/6c6c0b28ed",5,"differentiate: a*x - b*x**2/2"],["shape/0c2879764e",6,"differentiate: N*b*x**N + a*x"],["todhunter-spherical-trigonometry-1886/ex-v/11",4,"Todhunter 1886, Exercise V (11)"],["form/8ca8d88bc5",5,"differentiate: (a + x)**2"],["todhunter-spherical-trigonometry-1886/ex-v/12",4,"Todhunter 1886, Exercise V (12)"],["shape/d50962fcea",6,"differentiate: (a + x)**N"],["thompson-calculus-made-easy-1914/eq-b0aeeaeb3b",16,"Thompson 1914, p. 178: y &= F(x+at) + f(x-at)"],["form/192468d11f",5,"differentiate: (a + x)**3"],["todhunter-spherical-trigonometry-1886/ex-v/13",4,"Todhunter 1886, Exercise V (13)"],["thompson-calculus-made-easy-1914/eq-3d360f534f",16,"Thompson 1914, p. 252: \\frac{1}{3} x^3 + C"],["maxwell-elementary-treatise-electricity-1888/eq-015e300532",16,"Maxwell 1888, scan 113: P : Q :: R + \\alpha : S + \\gamma"],["form/4d38d9989e",5,"differentiate: (x + 3)*(x + 5)"],["person/h-e-slaught",1,"H. E. Slaught"],["thompson-calculus-made-easy-1914/eq-b745cc87f3",16,"Thompson 1914, p. 252: \\dfrac{1}{n+1} x^{n+1} + C"],["wentworth-first-steps-in-algebra-1894/eq-9127fb2210",16,"Wentworth 1894, p. 127: x + 10 &= 2(y - 10)"],["form/4b45bf774a",5,"identity: 1/(x + 1) + 1/(x - 1)"],["form/60c55fc958",5,"identity: -2/3"],["form/a8399fb7a3",5,"differentiate: a*(b*x + c*x**2 + 1)"],["thompson-calculus-made-easy-1914/ex-ix/12",4,"Thompson 1914, Exercise IX (12)"],["form/14f5f20a07",5,"differentiate2: (a + x**2)/(a + x)"],["shape/5fbfe81a50",6,"differentiate2: (a + x**N)/(a + x)"],["todhunter-spherical-trigonometry-1886/ex-vi/2a",4,"Todhunter 1886, Exercise VI (2a)"],["thompson-calculus-made-easy-1914/ex-iv/3a",4,"Thompson 1914, Exercise IV (3a)"],["form/3dba0670e2",5,"differentiate: x**4/24 + x**3/6 + x**2/2 + x + 1"],["shape/7dac6f8c0a",6,"differentiate: 3*N*x**N + x + 1"],["shape/4e3152904f",6,"identity: (a**N*x**N - b**N)/(a*x - b)"],["shape/ece92fb04c",6,"differentiate: N*x + N*(N + x)**N + N"],["thompson-calculus-made-easy-1914/ex-iv/1a",4,"Thompson 1914, Exercise IV (1a)"],["form/80f94b8018",5,"identity: (-a**6*b**6 + x**12)/(-a**2*b**2 + x**4)"],["shape/1c07793d3b",6,"differentiate: N*x + N*x**N"],["thompson-calculus-made-easy-1914/eq-180be9d000",16,"Thompson 1914, p. 252: \\log_\\epsilon x + C"],["todhunter-spherical-trigonometry-1886/ex-vi/3",4,"Todhunter 1886, Exercise VI (3)"],["dickson-theory-of-equations-1922/ex-page142/1",4,"Dickson 1922, Exercise Page142 (1)"],["form/8d089a6fb1",5,"differentiate: a + b*e + (c + d*e)/x"],["form/0926e8e7c4",5,"differentiate: 12*x**2 + 17*x"],["thompson-calculus-made-easy-1914/ex-ix/1a",4,"Thompson 1914, Exercise IX (1a)"],["thompson-calculus-made-easy-1914/eq-e1186b8e51",16,"Thompson 1914, p. 252: \\int u\\, dx ± \\int v\\, dx ± \\int w\\, dx"],["thompson-calculus-made-easy-1914/eq-a87f82215e",16,"Thompson 1914, p. 252: u\\, \\dfrac{dv}{dx} + v\\, \\dfrac{du}{dx}"],["form/5c0556266c",5,"extremum: x**2/(x + 1)"],["shape/27092d424f",6,"extremum: x**N/(x + 1)"],["form/00b66b0c47",5,"evaluate: x**2/(x + 1) at x=-2"],["todhunter-spherical-trigonometry-1886/ex-vi/7",4,"Todhunter 1886, Exercise VI (7)"],["thompson-calculus-made-easy-1914/eq-917cb1e041",16,"Thompson 1914, p. 178: \\frac{\\partial^2 y}{\\partial x^2} &= F''(w) + f''(v)"],["wentworth-first-steps-in-algebra-1894/eq-2e70b5e188",16,"Wentworth 1894, p. 98: \\dfrac{b}{a} × \\dfrac{a}{b} = \\dfrac{ba}{ab} = 1"],["form/6dbe3a32a4",5,"evaluate: x**2/(x + 1) at x=0"],["todhunter-spherical-trigonometry-1886/ex-vi/10",4,"Todhunter 1886, Exercise VI (10)"],["thompson-calculus-made-easy-1914/eq-13d61cb689",16,"Thompson 1914, p. 252: \\dfrac{v\\, \\dfrac{du}{dx} - u\\, \\dfrac{dv}{dx}}{v^2}"],["concept/line-of-projection",7,"line of projection","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-line-of-projection"],["concept/method-dividing-by-a-fraction",7,"method: dividing by a fraction"],["form/80e6118ee4",5,"differentiate: 10/(-x + 8) + 10/x"],["shape/1ef81075fa",6,"differentiate: N/(N - x) + N/x"],["blackburn-elements-plane-trigonometry-1863/ch-ix",2,"Blackburn 1863, ch. IX: OF PROJECTIONS","../books/blackburn-elements-plane-trigonometry-1863/ch/ch-ix/index.html"],["todhunter-spherical-trigonometry-1886/ex-vii/2",4,"Todhunter 1886, Exercise VII (2)"],["de-morgan-elementary-illustrations-calculus-1899/eq-5ff27c6a1e",16,"De Morgan 1899, p. 50: Bb : B'b :: OA : OB :: a : b"],["thompson-calculus-made-easy-1914/eq-9ed2ae84b9",16,"Thompson 1914, p. 178: \\frac{\\partial^2 y}{\\partial t^2} &= F''(w)a^2 + f''(v)a^2"],["form/b5df6d3e14",5,"identity: 1/(x + 3) + 1/(x - 2)"],["form/8ff4ec4a07",5,"extremum: 2*pi*x*(a - x)"],["shape/4a8c1c0aec",6,"extremum: pi*N*x*(a - x)"],["todhunter-spherical-trigonometry-1886/ex-vii/5",4,"Todhunter 1886, Exercise VII (5)"],["form/54dab9a798",5,"identity: -1/(x + 3) + 7/(-x**2 + 9) - 1/(-x + 3)"],["dickson-theory-of-equations-1922/ex-page152",3,"Dickson 1922, Exercise Page152"],["thompson-calculus-made-easy-1914/eq-783a6856db",16,"Thompson 1914, p. 178: \\frac{\\partial^2 y}{\\partial t^2} &= a^2\\, \\frac{\\partial^2 y}{\\partial x^2}"],["thompson-calculus-made-easy-1914/eq-d86c1470a1",16,"Thompson 1914, p. 179: A = \\sqrt{s(s-x)(s-y)(s-30+x+y)}"],["todhunter-spherical-trigonometry-1886/ex-vii/10",4,"Todhunter 1886, Exercise VII (10)"],["thompson-calculus-made-easy-1914/eq-e35c85441c",16,"Thompson 1914, p. 252: ux - \\int x\\, du + C"],["wentworth-first-steps-in-algebra-1894/eq-ff019a38b5",16,"Wentworth 1894, p. 91: \\dfrac{x^{3} - 1}{x - 1} = x^{2} + x + 1"],["thompson-calculus-made-easy-1914/eq-d481f6a022",16,"Thompson 1914, p. 179: A = \\sqrt{15P}"],["todhunter-spherical-trigonometry-1886/ex-viii/1",4,"Todhunter 1886, Exercise VIII (1)"],["thompson-calculus-made-easy-1914/eq-9d7bc865d0",16,"Thompson 1914, p. 252: \\epsilon^x + C"],["concept/method-dividing-a-polynomial-by-a-polynomial",7,"method: dividing a polynomial by a polynomial"],["wentworth-first-steps-in-algebra-1894/ch-xiii",2,"Wentworth 1894, ch. XIII: Arithmetical Progression","../books/wentworth-first-steps-in-algebra-1894/ch/ch-xiii/index.html"],["form/d280a5a181",5,"differentiate: 24*x**2/5 - 16*x/5 + 21/10"],["blackburn-elements-plane-trigonometry-1863/eq-4a098e8ee6",16,"Blackburn 1863, p. 23: \\cos(\\pi - \\theta) &= -\\cos\\theta;"],["todhunter-spherical-trigonometry-1886/ex-viii/4",4,"Todhunter 1886, Exercise VIII (4)"],["thompson-calculus-made-easy-1914/eq-57d855071e",16,"Thompson 1914, p. 252: x(\\log_\\epsilon x - 1) + C"],["boyden-first-book-in-algebra-1895/ex-26/27",4,"Boyden 1895, Exercise 26 (27)"],["form/c2559725b0",5,"differentiate: (x - 125)**(1/3)/10 + 1/2"],["shape/9d16a64ed9",6,"differentiate: N*(N + x)**N + N"],["shape/311b98b343",6,"evaluate: N/(x + x**N + 1)"],["thompson-calculus-made-easy-1914/ex-vi",3,"Thompson 1914, Exercise VI"],["thompson-calculus-made-easy-1914/ex-vi/1",4,"Thompson 1914, Exercise VI (1)"],["todhunter-spherical-trigonometry-1886/ex-viii/9",4,"Todhunter 1886, Exercise VIII (9)"],["form/b1072c17fd",5,"evaluate: 3/(x**2 + x + 1) at x=-Rational(1,2)"],["form/2032bb5b6d",5,"identity: (-a**6 + a**4 + a**2*x**2 + x**6 + x**4)/(-a**2 + x**2 + 1)"],["form/6b4e6e6f25",5,"differentiate: sqrt(x**2 + 1)"],["shape/604b79544c",6,"differentiate: (x**N + 1)**N"],["thompson-calculus-made-easy-1914/ex-vi/2",4,"Thompson 1914, Exercise VI (2)"],["form/a59c7fdd76",5,"evaluate: 3/(x**2 + x + 1)"],["todhunter-spherical-trigonometry-1886/ex-viii/12",4,"Todhunter 1886, Exercise VIII (12)"],["shape/a2037fe079",6,"identity: (a**N*x**N + 2*x**N)/(-a**N + x**N + 1)"],["theorem/sine-of-the-sum-of-two-angles",9,"sine of the sum of two angles","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-sine-of-the-sum-of-two-angles"],["thompson-calculus-made-easy-1914/eq-fa4e0ffcef",16,"Thompson 1914, p. 179: P &= (15-x)(15-y)(x+y-15)"],["form/b8c9ace604",5,"solve: Eq(x**3 - 3*x**2 + 8/3, 0)"],["todhunter-spherical-trigonometry-1886/ex-viii/15",4,"Todhunter 1886, Exercise VIII (15)"],["thompson-calculus-made-easy-1914/eq-8815df15f1",16,"Thompson 1914, p. 252: 0.4343x (\\log_\\epsilon x - 1) + C"],["cap/core.trig",17,"core.trig"],["hardy-course-of-pure-mathematics-1921/eq-a8d1fa8567",16,"Hardy 1921, p. 301: A(x^{n}/n!) \\leq F(x) \\leq B(x^{n}/n!)"],["shape/994f920d41",6,"identity: (a + a**N + 1)/(a**N + 1 + 1/a)"],["todhunter-spherical-trigonometry-1886/ex-xi/1",4,"Todhunter 1886, Exercise XI (1)"],["theorem/cosine-of-the-difference-of-two-angles",9,"cosine of the difference of two angles","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-cosine-of-the-difference-of-two-angles"],["thompson-calculus-made-easy-1914/eq-d1e3dd27fc",16,"Thompson 1914, p. 179: dP = \\dfrac{\\partial P}{\\partial x}\\, dx + \\dfrac{\\partial P}{\\partial y}\\, dy"],["boyden-first-book-in-algebra-1895/ex-35/3",4,"Boyden 1895, Exercise 35 (3)"],["thompson-calculus-made-easy-1914/ex-x/11",4,"Thompson 1914, Exercise X (11)"],["todhunter-spherical-trigonometry-1886/ex-xi/4",4,"Todhunter 1886, Exercise XI (4)"],["method/deriving-sum-and-difference-formulae-by-projection",8,"deriving sum and difference formulae by projection","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-method-deriving-sum-and-difference-formulae-by-projection"],["thompson-calculus-made-easy-1914/eq-7e987cde51",16,"Thompson 1914, p. 179: \\dfrac{\\partial P}{\\partial x} = 0 \\quad\\text{and}\\quad \\dfrac{\\partial P}{\\partial y} = 0"],["shape/e5b56fa2dd",6,"factor: -a**N*b**N + x**N"],["todhunter-spherical-trigonometry-1886/ex-xii/5",4,"Todhunter 1886, Exercise XII (5)"],["thompson-calculus-made-easy-1914/eq-4f74e70f1f",16,"Thompson 1914, p. 179: 2xy - 30x + y^2 - 45y + 450 &= 0"],["thompson-calculus-made-easy-1914/eq-085a4861bd",16,"Thompson 1914, p. 179: 2xy - 30y + x^2 - 45x + 450 &= 0"],["form/3ce769e8ae",5,"extremum: 2*x + 1 + 5/x**2"],["todhunter-spherical-trigonometry-1886/ex-xii/8",4,"Todhunter 1886, Exercise XII (8)"],["thompson-calculus-made-easy-1914/eq-7403babbc9",16,"Thompson 1914, p. 252: \\dfrac{a^x}{\\log_\\epsilon a} + C"],["thompson-calculus-made-easy-1914/eq-66c2a74851",16,"Thompson 1914, p. 125: \\frac{3}{x+1} - \\frac{1}{x-1} + \\frac{2}{x+3}"],["form/e524447f90",5,"extremum: 5*x/(x**2 + 2)"],["shape/1aa7427df8",6,"extremum: N*x/(N + x**N)"],["todhunter-spherical-trigonometry-1886/ex-xii/10",4,"Todhunter 1886, Exercise XII (10)"],["concept/order-of-parts",7,"order of parts","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-order-of-parts"],["thompson-calculus-made-easy-1914/eq-48ff44bf6a",16,"Thompson 1914, p. 252: -\\cos x + C"],["thompson-calculus-made-easy-1914/eq-2f1cb9c34a",16,"Thompson 1914, p. 252: \\sin x + C"],["form/bd9ea3ff85",5,"evaluate: x/2 + 3*x/(x**2 - 3) + 5 at x=-sqrt(6 + 3*sqrt(5))"],["form/b2e88bcf01",5,"evaluate: x/2 + 3*x/(x**2 - 3) + 5 at x=sqrt(6 + 3*sqrt(5))"],["thompson-calculus-made-easy-1914/ex-xi/4",4,"Thompson 1914, Exercise XI (4)"],["form/c31c1235bd",5,"differentiate: -3**x - 3**sin(x) + x**3 + 3*sin(x + 3)"],["thompson-calculus-made-easy-1914/ex-xi/5",4,"Thompson 1914, Exercise XI (5)"],["form/ce7ef1e780",5,"evaluate: 1000/x at v=650**(Rational(1,3))"],["shape/4ca8118169",6,"evaluate: N/x"],["shape/46cd7333c4",6,"differentiate: N*sin(N + x) - N**x - N**sin(x) + x**N"],["concept/method-partial-fractions",7,"method: partial fractions"],["todhunter-spherical-trigonometry-1886/ex-xiii/4",4,"Todhunter 1886, Exercise XIII (4)"],["thompson-calculus-made-easy-1914/eq-1896337166",16,"Thompson 1914, p. 252: -\\log_\\epsilon \\cos x + C"],["thompson-calculus-made-easy-1914/eq-40daf6d98c",16,"Thompson 1914, p. 126: \\frac{x-1}{x^2+1} - \\frac{2}{x+1}"],["form/276e9a2f73",5,"solve: Eq(-(x + 2)/(x - 3) + (x + 4)/(x - 4), 1)"],["form/16d940ae12",5,"evaluate: (10*x**3 + 13000)/x at v=650**(Rational(1,3))"],["thompson-calculus-made-easy-1914/ex-xi/1",4,"Thompson 1914, Exercise XI (1)"],["form/88ea4e1135",5,"differentiate: a*x**2 + 2*log(x)"],["cap/cas.partfrac",17,"cas.partfrac"],["thompson-calculus-made-easy-1914/ex-xi/2",4,"Thompson 1914, Exercise XI (2)"],["shape/4a02b2518b",6,"differentiate: N*log(x) + a*x**N"],["theorem/three-non-collinear-points-determine-a-plane",9,"three non-collinear points determine a plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-three-non-collinear-points-determine-a-plane"],["form/36e251414e",5,"extremum: 4*x**3 - x**2 - 2*x + 1"],["thompson-calculus-made-easy-1914/ex-xiv/15a",4,"Thompson 1914, Exercise XIV (15a)"],["cap/cas.pdiv",17,"cas.pdiv"],["todhunter-spherical-trigonometry-1886/ex-xiii/10",4,"Todhunter 1886, Exercise XIII (10)"],["form/754a5590e6",5,"solve: Eq(x**3 - 144*x + 665, 0)"],["thompson-calculus-made-easy-1914/ex-xii",3,"Thompson 1914, Exercise XII"],["todhunter-spherical-trigonometry-1886/ex-xiii/13",4,"Todhunter 1886, Exercise XIII (13)"],["thompson-calculus-made-easy-1914/ex-xii/1",4,"Thompson 1914, Exercise XII (1)"],["concept/convex-polyhedron",7,"convex polyhedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-convex-polyhedron"],["form/877e703f4c",5,"differentiate: b*(exp(a*x) - exp(-a*x))"],["shape/877e703f4c",6,"differentiate: b*(exp(a*x) - exp(-a*x))"],["thompson-calculus-made-easy-1914/ex-xii/2",4,"Thompson 1914, Exercise XII (2)"],["form/6cd5ec1bac",5,"extremum: x*cos(x)"],["todhunter-spherical-trigonometry-1886/ex-xiii/16",4,"Todhunter 1886, Exercise XIII (16)"],["shape/6cd5ec1bac",6,"extremum: x*cos(x)"],["form/c1ea02dbab",5,"evaluate: x*cos(x)"],["shape/c1ea02dbab",6,"evaluate: x*cos(x)"],["thompson-calculus-made-easy-1914/ex-xiv/15b",4,"Thompson 1914, Exercise XIV (15b)"],["form/78e190d601",5,"extremum: exp(a + x)/(a*x)"],["shape/78e190d601",6,"extremum: exp(a + x)/(a*x)"],["form/dd3a0bee8f",5,"differentiate: log(a**x)"],["shape/dd3a0bee8f",6,"differentiate: log(a**x)"],["thompson-calculus-made-easy-1914/ex-xix/1",4,"Thompson 1914, Exercise XIX (1)"],["todhunter-spherical-trigonometry-1886/ex-xv/1",4,"Todhunter 1886, Exercise XV (1)"],["cap/cas.integrate.subst",17,"cas.integrate.subst"],["shape/4521846089",6,"integrate: (a**N - x**N)**N"],["form/cd84fa94ac",5,"integrate: sqrt(a**2 - x**2)"],["thompson-calculus-made-easy-1914/eq-2c669761f2",16,"Thompson 1914, p. 179: P = (15-x)^2 (2x-15) = 2x^3 - 75x^2 + 900x - 3375"],["thompson-calculus-made-easy-1914/ex-xiii/91",4,"Thompson 1914, Exercise XIII (91)"],["form/78788b2cbc",5,"integrate: x*log(x)"],["form/849be908d7",5,"evaluate: 100*exp(-57/125)"],["shape/025e25b3c3",6,"evaluate: N*exp(N)"],["thompson-calculus-made-easy-1914/ex-xix/2",4,"Thompson 1914, Exercise XIX (2)"],["shape/78788b2cbc",6,"integrate: x*log(x)"],["thompson-calculus-made-easy-1914/eq-801bf7fc83",16,"Thompson 1914, p. 179: 6x^2 - 150x + 900 = 0"],["todhunter-spherical-trigonometry-1886/ex-xv/5",4,"Todhunter 1886, Exercise XV (5)"],["planck-treatise-on-thermodynamics-1903/eq-af58ce4572",16,"Planck 1903, p. 108: d\\Phi + d\\Phi_{0} \\geq 0"],["form/e026201ea7",5,"evaluate: x**3 + 3*x at x=1/2"],["thompson-calculus-made-easy-1914/eq-fdfbc1d57f",16,"Thompson 1914, p. 128: \\frac{2}{x+1} - \\frac{2}{(x+1)^2} + \\frac{1}{x-2}"],["form/6b5d05ce9b",5,"evaluate: 24/log(2)"],["todhunter-spherical-trigonometry-1886/ex-xv/9",4,"Todhunter 1886, Exercise XV (9)"],["thompson-calculus-made-easy-1914/ex-xix/3",4,"Thompson 1914, Exercise XIX (3)"],["form/78a5e2728f",5,"integrate: x**a*log(x)"],["cap/cas.integrate.parts",17,"cas.integrate.parts"],["form/3df0e2f47f",5,"evaluate: -291819*x/250000000 + 5019*(x - 15)**2/500000000 + 72575457/50000000 at t=25"],["shape/78a5e2728f",6,"integrate: x**a*log(x)"],["form/87457e4cdd",5,"differentiate: x**x"],["thompson-calculus-made-easy-1914/ex-xix/4",4,"Thompson 1914, Exercise XIX (4)"],["form/e027877202",5,"integrate: 2*x + 3"],["form/2b20fb3f33",5,"integrate: exp(x)*cos(exp(x))"],["shape/2b20fb3f33",6,"integrate: exp(x)*cos(exp(x))"],["todhunter-spherical-trigonometry-1886/ex-xv/10",4,"Todhunter 1886, Exercise XV (10)"],["shape/1329c3b3c2",6,"integrate: N*x + N"],["thompson-calculus-made-easy-1914/ex-xix/5",4,"Thompson 1914, Exercise XIX (5)"],["form/a0fe6f240f",5,"integrate: cos(log(x))/x"],["form/c908e8ae92",5,"differentiate: exp(x**x)"],["shape/a791b1221d",6,"identity: (N + N/x + x)/(N*x**N + N/x + 1)"],["shape/a0fe6f240f",6,"integrate: cos(log(x))/x"],["thompson-calculus-made-easy-1914/ex-xvi/5",4,"Thompson 1914, Exercise XVI (5)"],["shape/db3c70b634",6,"differentiate: -a*cos(x)"],["todhunter-spherical-trigonometry-1886/ex-xv/12",4,"Todhunter 1886, Exercise XV (12)"],["thompson-calculus-made-easy-1914/ex-xiv/1b",4,"Thompson 1914, Exercise XIV (1b)"],["thompson-calculus-made-easy-1914/eq-8fac84a73c",16,"Thompson 1914, p. 179: \\dfrac{d^2 P}{dx^2} = 12x - 150"],["form/0e637f1ae0",5,"differentiate: sin(x)**2"],["cap/other:construction",17,"other:construction"],["shape/7a3a46c2a7",6,"differentiate: sin(x)**N"],["thompson-calculus-made-easy-1914/ex-xiv/1c",4,"Thompson 1914, Exercise XIV (1c)"],["cap/cas.trig",17,"cas.trig"],["concept/method-second-derivative-test",7,"method: second derivative test"],["form/1ff4eccec7",5,"evaluate: log(950/249)/10"],["thompson-calculus-made-easy-1914/ex-xiv",3,"Thompson 1914, Exercise XIV"],["todhunter-spherical-trigonometry-1886/ex-xv/19",4,"Todhunter 1886, Exercise XV (19)"],["thompson-calculus-made-easy-1914/ex-xiv/1a",4,"Thompson 1914, Exercise XIV (1a)"],["form/db3c70b634",5,"differentiate: -a*cos(x)"],["shape/9f6e15f51d",6,"differentiate: 2*N*x**N + x"],["form/ce97d915dd",5,"evaluate: x**3 + x**2 - 10*x + 8"],["hardy-course-of-pure-mathematics-1921/ex-app-i/1",4,"Hardy 1921, Exercise App-I (1)"],["hardy-course-of-pure-mathematics-1921/ex-app-i/2",4,"Hardy 1921, Exercise App-I (2)"],["thompson-calculus-made-easy-1914/ex-xvii/1",4,"Thompson 1914, Exercise XVII (1)"],["form/49e32b4f9e",5,"integrate: 2*sqrt(a)*sqrt(x)"],["shape/88146a2c90",6,"integrate: N*a**N*x**N"],["thompson-calculus-made-easy-1914/eq-dd26224cde",16,"Thompson 1914, p. 252: x · \\arcsin x + \\sqrt{1 - x^2} + C"],["form/9874da5212",5,"differentiate: sin(x)**3"],["hardy-course-of-pure-mathematics-1921/ex-app-i/3",4,"Hardy 1921, Exercise App-I (3)"],["hardy-course-of-pure-mathematics-1921/ex-app-i/4",4,"Hardy 1921, Exercise App-I (4)"],["thompson-calculus-made-easy-1914/eq-5788f5155d",16,"Thompson 1914, p. 252: x · \\arccos x - \\sqrt{1 - x^2} + C"],["thompson-calculus-made-easy-1914/eq-4b49668077",16,"Thompson 1914, p. 252: x · \\arctan x - \\frac{1}{2} \\log_\\epsilon (1 + x^2) + C"],["shape/6318c3cde0",6,"solve: Eq(N*b*e**N*g*x/c, N*d**N*e*f**N*g/(a*c))"],["boyden-first-book-in-algebra-1895/ex-39",3,"Boyden 1895, Exercise 39"],["form/b16b5c1bce",5,"differentiate: sin(a**x)"],["shape/b16b5c1bce",6,"differentiate: sin(a**x)"],["hardy-course-of-pure-mathematics-1921/ex-app-i/5",4,"Hardy 1921, Exercise App-I (5)"],["hardy-course-of-pure-mathematics-1921/ex-app-i/7",4,"Hardy 1921, Exercise App-I (7)"],["thompson-calculus-made-easy-1914/ex-xix/6",4,"Thompson 1914, Exercise XIX (6)"],["shape/c5d5ee2296",6,"evaluate: N*x + N + 2*x**N"],["shape/65a079fb79",6,"hcf: (x**N - 1, N*x + N + x**N, -x + 1, x - 1)"],["thompson-calculus-made-easy-1914/eq-4134466ca3",16,"Thompson 1914, p. 253: \\cosh x + C"],["wentworth-first-steps-in-algebra-1894/ex-77",3,"Wentworth 1894, Exercise 77"],["form/15c23c1051",5,"differentiate: sin(x)*sin(2*x)"],["hardy-course-of-pure-mathematics-1921/ex-app-i/8",4,"Hardy 1921, Exercise App-I (8)"],["hardy-course-of-pure-mathematics-1921/ex-app-i/9",4,"Hardy 1921, Exercise App-I (9)"],["form/f85579f712",5,"integrate: x**2*exp(x)"],["hardy-course-of-pure-mathematics-1921/ex-app-i/10",4,"Hardy 1921, Exercise App-I (10)"],["shape/104f6460de",6,"integrate: x**N*exp(x)"],["thompson-calculus-made-easy-1914/eq-1fb045da24",16,"Thompson 1914, p. 253: \\sinh x + C"],["thompson-calculus-made-easy-1914/ex-xix/7",4,"Thompson 1914, Exercise XIX (7)"],["form/d592f5eb83",5,"integrate: log(x)**a/x"],["hardy-course-of-pure-mathematics-1921/ex-i/3",4,"Hardy 1921, Exercise I (3)"],["hardy-course-of-pure-mathematics-1921/ex-i/4",4,"Hardy 1921, Exercise I (4)"],["shape/d592f5eb83",6,"integrate: log(x)**a/x"],["thompson-calculus-made-easy-1914/ex-xix/8",4,"Thompson 1914, Exercise XIX (8)"],["form/21acf686f8",5,"integrate: 1/(x*log(x))"],["shape/21acf686f8",6,"integrate: 1/(x*log(x))"],["form/0a514f051d",5,"differentiate: 1/cos(x)"],["form/68e48f0fb6",5,"integrate: a**2*(3*x**(1/3) + 3*sqrt(x))"],["shape/e3081b24c6",6,"integrate: 2*N*a**N*x**N"],["form/142cf36c3e",5,"integrate: sin(x)/3 - 1/6"],["shape/3a06c0d2b3",6,"integrate: N*sin(x) + N"],["hardy-course-of-pure-mathematics-1921/ex-ii/4",4,"Hardy 1921, Exercise II (4)"],["hardy-course-of-pure-mathematics-1921/ex-ii/5",4,"Hardy 1921, Exercise II (5)"],["form/59088302d9",5,"solve: Eq(x**4 - 4*x**3 - 8*x**2 + 13*x + 10, 0)"],["wentworth-first-steps-in-algebra-1894/eq-da05d08912",16,"Wentworth 1894, p. 91: \\dfrac{x^{3} - 1}{x + 1} = x^{2} - x + 1 - \\dfrac{2}{x + 1}"],["thompson-calculus-made-easy-1914/eq-e4baf7e24c",16,"Thompson 1914, p. 253: \\log_\\epsilon \\cosh x + C"],["hardy-course-of-pure-mathematics-1921/ex-iii/2",4,"Hardy 1921, Exercise III (2)"],["hardy-course-of-pure-mathematics-1921/ex-iii/3",4,"Hardy 1921, Exercise III (3)"],["shape/b5114df6af",6,"identity: (-a + d)*((-a + d)**N + (-b + e)**N + (-c + f)**N)**N + (-b + e)*((-a + d)**N + (-b + e)**N + (-c + f)**N)**N + (-c + f)*((-a + d)**N + (-b + e)**N + (-c + f)**N)**N"],["thompson-calculus-made-easy-1914/ex-xv/52",4,"Thompson 1914, Exercise XV (52)"],["thompson-calculus-made-easy-1914/eq-4b95faecad",16,"Thompson 1914, p. 253: \\log_\\epsilon (x+a) + C"],["hardy-course-of-pure-mathematics-1921/ex-iii/4",4,"Hardy 1921, Exercise III (4)"],["hardy-course-of-pure-mathematics-1921/ex-iii/5",4,"Hardy 1921, Exercise III (5)"],["thompson-calculus-made-easy-1914/ex-xv",3,"Thompson 1914, Exercise XV"],["thompson-calculus-made-easy-1914/ex-xv/1a",4,"Thompson 1914, Exercise XV (1a)"],["boyden-first-book-in-algebra-1895/ex-35/4",4,"Boyden 1895, Exercise 35 (4)"],["thompson-calculus-made-easy-1914/eq-5cc9a8b86e",16,"Thompson 1914, p. 180: S=xy + \\dfrac{2V}{x} + \\dfrac{2V}{y}"],["form/a773e7377b",5,"differentiate: -2*a**2*x - 2*a*x**3 + a/3 + x**3/3"],["hardy-course-of-pure-mathematics-1921/ex-iv/1",4,"Hardy 1921, Exercise IV (1)"],["hardy-course-of-pure-mathematics-1921/ex-iv/2",4,"Hardy 1921, Exercise IV (2)"],["hardy-course-of-pure-mathematics-1921/ex-iv/3",4,"Hardy 1921, Exercise IV (3)"],["hardy-course-of-pure-mathematics-1921/ex-iv/5",4,"Hardy 1921, Exercise IV (5)"],["boyden-first-book-in-algebra-1895/ex-26/28",4,"Boyden 1895, Exercise 26 (28)"],["unit/second",12,"second","../books/ball-mathematical-recreations-1905/terms/index.html#t-unit-second"],["hardy-course-of-pure-mathematics-1921/ex-iv/6",4,"Hardy 1921, Exercise IV (6)"],["hardy-course-of-pure-mathematics-1921/ex-iv/7",4,"Hardy 1921, Exercise IV (7)"],["hardy-course-of-pure-mathematics-1921/ex-iv/8",4,"Hardy 1921, Exercise IV (8)"],["theorem/section-of-a-cone-parallel-to-its-base-is-circular",9,"section of a cone parallel to its base is circular","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-section-of-a-cone-parallel-to-its-base-is-circular"],["wentworth-first-steps-in-algebra-1894/ch-x",2,"Wentworth 1894, ch. X: Fractional Equations","../books/wentworth-first-steps-in-algebra-1894/ch/ch-x/index.html"],["form/ce4354f22f",5,"identity: (-a**6 - a**4 - a**2*x**2 + x**6 - x**4)/(-a**2 + x**2 - 1)"],["shape/a131cd51b0",6,"identity: (-a**N*x**N - 2*a**N)/(-a**N + x**N - 1)"],["form/e57452696b",5,"factor: -a**3 + b**3*x**3"],["shape/5333dd6c22",6,"factor: -a**N + b**N*x**N"],["shape/6eb1e9fc29",6,"evaluate: N*x + N"],["hardy-course-of-pure-mathematics-1921/ex-iv/9",4,"Hardy 1921, Exercise IV (9)"],["thompson-calculus-made-easy-1914/eq-4ac9f44ca4",16,"Thompson 1914, p. 23: \\frac{dy}{dx} = 5x^4"],["form/1d93a9026a",5,"extremum: 2*a*x + a + x"],["form/140deb31b7",5,"identity: 91"],["shape/1a70a99c02",6,"extremum: N*a*x + a + x"],["hardy-course-of-pure-mathematics-1921/ex-ix/3",4,"Hardy 1921, Exercise IX (3)"],["hardy-course-of-pure-mathematics-1921/ex-ix/4",4,"Hardy 1921, Exercise IX (4)"],["thompson-calculus-made-easy-1914/eq-051f9f812b",16,"Thompson 1914, p. 180: dS = \\frac{\\partial S}{\\partial x}\\, dx + \\frac{\\partial S}{\\partial y}\\, dy"],["hardy-course-of-pure-mathematics-1921/eq-bec150828e",16,"Hardy 1921, p. 301: \\Delta_{h}^{n}\\phi(x) = \\sum_{r=0}^{n}(-1)^{r} \\binom{n}{r} \\phi(x + rh) = (-h)^{n} \\phi^{(n)}(\\xi)"],["hardy-course-of-pure-mathematics-1921/ex-l/1a",4,"Hardy 1921, Exercise L (1a)"],["shape/46a78e44f0",6,"integrate: (N*x + N + x**N)**N/x"],["form/bbf8011b8c",5,"integrate: 1/((x - 1)*sqrt(x**2 + 1))"],["thompson-calculus-made-easy-1914/ex-xvi/4a",4,"Thompson 1914, Exercise XVI (4a)"],["shape/1b40d8f1b8",6,"integrate: (x**N + 1)**N/(x - 1)"],["hardy-course-of-pure-mathematics-1921/ex-l/1c",4,"Hardy 1921, Exercise L (1c)"],["form/da7eb4f9ae",5,"integrate: 1/((x + 1)*sqrt(-x**2 + 2*x + 1))"],["shape/e4b2dd651f",6,"integrate: (N*x - x**N + 1)**N/(x + 1)"],["dickson-theory-of-equations-1922/ex-page94/1",4,"Dickson 1922, Exercise Page94 (1)"],["form/2c98918b61",5,"evaluate: 4*x**3 - x**2 - 2*x + 1 at x=1/2"],["hardy-course-of-pure-mathematics-1921/ex-l/2",4,"Hardy 1921, Exercise L (2)"],["thompson-calculus-made-easy-1914/ex-xvi/1",4,"Thompson 1914, Exercise XVI (1)"],["hardy-course-of-pure-mathematics-1921/ex-l/3",4,"Hardy 1921, Exercise L (3)"],["method/substitution",8,"substitution","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-method-substitution"],["thompson-calculus-made-easy-1914/eq-8dad9dc515",16,"Thompson 1914, p. 180: y - \\frac{2V}{x^2} = 0"],["shape/c4414cac4a",6,"solve: Eq(N*x + N*(N - x)*(N*x - 1) + N + x**N, 0)"],["form/1172078479",5,"extremum: 4*a/x + x**2"],["thompson-calculus-made-easy-1914/ex-iii",3,"Thompson 1914, Exercise III"],["shape/c3c560cc37",6,"extremum: N*a/x + x**N"],["hardy-course-of-pure-mathematics-1921/ex-l/4",4,"Hardy 1921, Exercise L (4)"],["hardy-course-of-pure-mathematics-1921/ex-l/5",4,"Hardy 1921, Exercise L (5)"],["hardy-course-of-pure-mathematics-1921/ex-l/6",4,"Hardy 1921, Exercise L (6)"],["hardy-course-of-pure-mathematics-1921/eq-0858e2a43c",16,"Hardy 1921, p. 301: \\{\\Delta_{h}^{n}\\phi(x)\\}/h^{n} \\to (-1)^{n}\\phi^{(n)}(x)"],["thompson-calculus-made-easy-1914/eq-35de85f576",16,"Thompson 1914, p. 253: \\log_\\epsilon (x + \\sqrt{a^2 + x^2}) + C"],["form/464696f0f5",5,"integrate: 3/x**4"],["shape/47d297a695",6,"integrate: N*x**N"],["hardy-course-of-pure-mathematics-1921/ex-l/7",4,"Hardy 1921, Exercise L (7)"],["hardy-course-of-pure-mathematics-1921/ex-l/8",4,"Hardy 1921, Exercise L (8)"],["form/0dbcabc6c3",5,"integrate: 1/(sqrt(5*x**2 + 2*x - 7)*(5*x**2 + 12*x + 8))"],["shape/405683bd9e",6,"integrate: (N*x + N*x**N + N)**N/(N*x + N*x**N + N)"],["hardy-course-of-pure-mathematics-1921/ex-l/9a",4,"Hardy 1921, Exercise L (9a)"],["form/dc2f870b9f",5,"integrate: (x + 1)/((2*x**2 - 2*x + 1)*sqrt(3*x**2 - 2*x + 1))"],["hardy-course-of-pure-mathematics-1921/eq-e62004f273",16,"Hardy 1921, p. 301: x^{n-m}\\, \\Delta_{h}^{n} x^{m} \\to m(m - 1) \\dots (m - n + 1)h^{n}"],["whitehead-introduction-to-mathematics-1911/ch-v",2,"Whitehead 1911, ch. V: The Symbolism of Mathematics","../books/whitehead-introduction-to-mathematics-1911/ch/ch-v/index.html"],["form/5dc4ac4e99",5,"integrate: a*x/2 + b*x**2/3 + c*x**3/4"],["thompson-calculus-made-easy-1914/eq-e4ac49f6b8",16,"Thompson 1914, p. 129: \\frac{(8x - 5)}{(2x^2 - 1)^2} + \\frac{8(x - 1)}{2x^2 - 1} - \\frac{4}{x + 1}"],["hardy-course-of-pure-mathematics-1921/ex-l/10",4,"Hardy 1921, Exercise L (10)"],["thompson-calculus-made-easy-1914/eq-e07086df09",16,"Thompson 1914, p. 180: x - \\frac{2V}{y^2} = 0"],["thompson-calculus-made-easy-1914/eq-b6db8f54bf",16,"Thompson 1914, p. 180: S = x^2 + \\dfrac{4V}{x}"],["thompson-calculus-made-easy-1914/eq-f9c00b6b7a",16,"Thompson 1914, p. 180: \\dfrac{dS}{dx}= 2x - \\dfrac{4V}{x^2} =0"],["shape/fc379ac4b2",6,"solve: Eq(N*sin(x) + x, pi*N)"],["form/042636e76a",5,"integrate: cos(a*x)**2"],["shape/cd24a9863e",6,"solve: Eq((a + b*x)*(c + d), (a + b)*(c + d*x))"],["hardy-course-of-pure-mathematics-1921/ex-li/1",4,"Hardy 1921, Exercise LI (1)"],["form/3872d5e6d9",5,"integrate: sin(x)**3*cos(2*x)**2"],["thompson-calculus-made-easy-1914/ex-xviii",3,"Thompson 1914, Exercise XVIII"],["thompson-calculus-made-easy-1914/eq-36ea38e0d0",16,"Thompson 1914, p. 253: \\dfrac{x}{\\sqrt{a^2 + x^2}} + C"],["hardy-course-of-pure-mathematics-1921/ex-li/2a",4,"Hardy 1921, Exercise LI (2a)"],["hardy-course-of-pure-mathematics-1921/eq-23c0182ab2",16,"Hardy 1921, p. 301: x\\sqrt{x} \\{\\sqrt{x} - 2\\sqrtp{x + 1} + \\sqrtp{x + 2}\\} \\to -\\tfrac{1}{4}"],["form/914578a937",5,"integrate: 1/(-x + 1)"],["form/3538c18372",5,"hcf: (-b**3 + x**3, a*b - a*x - b*c + c*x)"],["hardy-course-of-pure-mathematics-1921/ex-li/2b",4,"Hardy 1921, Exercise LI (2b)"],["form/f667a874e4",5,"integrate: sin(a*x)*sin(b*x)"],["shape/f667a874e4",6,"integrate: sin(a*x)*sin(b*x)"],["hardy-course-of-pure-mathematics-1921/ex-li/2c",4,"Hardy 1921, Exercise LI (2c)"],["form/2ec4ad3c2d",5,"integrate: sin(b*x)*cos(a*x)"],["shape/2ec4ad3c2d",6,"integrate: sin(b*x)*cos(a*x)"],["thompson-calculus-made-easy-1914/eq-d6eadd8bd5",16,"Thompson 1914, p. 180: x = \\sqrt[3]{2V}"],["hardy-course-of-pure-mathematics-1921/eq-3636549117",16,"Hardy 1921, p. 302: y = \\phi(x) = x + a_{2}x^{2} + a_{3}x^{3} + (a_{4} + \\epsilon_{x})x^{4}"],["slaught-lennes-solid-geometry-1919",0,"Slaught & Lennes, Solid Geometry with Problems and Applications (1919)","../books/slaught-lennes-solid-geometry-1919/index.html"],["form/8a3343f8b2",5,"evaluate: Integral(sin(x), (x, 0, pi))"],["hardy-course-of-pure-mathematics-1921/ex-li/2d",4,"Hardy 1921, Exercise LI (2d)"],["hardy-course-of-pure-mathematics-1921/ex-li/2e",4,"Hardy 1921, Exercise LI (2e)"],["dickson-theory-of-equations-1922/ex-page99/1",4,"Dickson 1922, Exercise Page99 (1)"],["shape/a96beb95ed",6,"hcf: (-b**N + x**N, a*b - a*x - b*c + c*x)"],["form/014ab07b17",5,"identity: (-4*a + x)/(-2*a + x) - (-4*a**2 + x**2)/(2*a*x + x**2)"],["form/dd540762e7",5,"solve: Eq(x - 8, x/4 + 1/4)"],["form/6b2667d334",5,"evaluate: Integral(2*x**(5/2), (x, 0, 1))"],["person/isaac-todhunter",1,"Isaac Todhunter"],["hardy-course-of-pure-mathematics-1921/ex-li/2f",4,"Hardy 1921, Exercise LI (2f)"],["form/a79e7d99df",5,"integrate: cos(x)**4"],["hardy-course-of-pure-mathematics-1921/ex-li/2g",4,"Hardy 1921, Exercise LI (2g)"],["shape/7094966d39",6,"integrate: cos(x)*cos(N*x)**2"],["hardy-course-of-pure-mathematics-1921/ex-li/2i",4,"Hardy 1921, Exercise LI (2i)"],["thompson-calculus-made-easy-1914/eq-7ae3912f20",16,"Thompson 1914, p. 129: \\frac{4}{(x + 1)^2} - \\frac{3}{(x + 1)^3}"],["form/868a797a27",5,"evaluate: Integral(pi*(x**2 + 1), (x, 0, 4))"],["blackburn-elements-plane-trigonometry-1863/ch-iii",2,"Blackburn 1863, ch. III: OF SYMBOLS OF QUANTITY","../books/blackburn-elements-plane-trigonometry-1863/ch/ch-iii/index.html"],["blackburn-elements-plane-trigonometry-1863/eq-e03b6609e5",16,"Blackburn 1863, p. 15: = (a - b)"],["shape/dc3aae225c",6,"evaluate: Integral(pi*(x**N + 1), (x, 0, N))"],["form/1d1a86b535",5,"integrate: sin(x)**7*cos(x)**5"],["shape/97b23761e1",6,"integrate: sin(x)**N*cos(x)**N"],["thompson-calculus-made-easy-1914/eq-19814313f2",16,"Thompson 1914, p. 253: -\\dfrac{1}{a} \\cos ax + C"],["wentworth-first-steps-in-algebra-1894/eq-bbce9cd735",16,"Wentworth 1894, p. 119: i = prt"],["maxwell-elementary-treatise-electricity-1888/eq-ca9420bca2",16,"Maxwell 1888, scan 47: \\frac{dQ_p}{dP} = E"],["form/79e7404484",5,"integrate: x**2*sin(x)**2*sin(2*x)**2"],["hardy-course-of-pure-mathematics-1921/ex-ii/1",4,"Hardy 1921, Exercise II (1)"],["hardy-course-of-pure-mathematics-1921/ex-lii/1a",4,"Hardy 1921, Exercise LII (1a)"],["form/55fd1283d6",5,"integrate: x*sin(x)"],["shape/55fd1283d6",6,"integrate: x*sin(x)"],["hardy-course-of-pure-mathematics-1921/ex-lii/1c",4,"Hardy 1921, Exercise LII (1c)"],["shape/c53864e045",6,"integrate: x**N*sin(x)**N*sin(N*x)**N"],["thompson-calculus-made-easy-1914/eq-c3d3cb6f67",16,"Thompson 1914, p. 253: \\dfrac{1}{a} \\sin ax + C"],["thompson-calculus-made-easy-1914/eq-1422eebb5b",16,"Thompson 1914, p. 253: -\\dfrac{1}{a} \\log_\\epsilon \\cos ax + C"],["hardy-course-of-pure-mathematics-1921/ex-lii/1e",4,"Hardy 1921, Exercise LII (1e)"],["form/a57d5db635",5,"evaluate: Integral(a*sin(x) + a*sin(3*x), (x, 0, 2*pi))/(2*pi)"],["form/1185f41330",5,"integrate: x*sin(x)**2*cos(x)**4"],["shape/31e00d27f0",6,"integrate: x*sin(x)**N*cos(x)**N"],["hardy-course-of-pure-mathematics-1921/ex-lii/1f",4,"Hardy 1921, Exercise LII (1f)"],["form/a8f11a08a2",5,"integrate: x**3*sin(x/3)**3"],["shape/6b8eae871e",6,"integrate: x**N*sin(N*x)**N"],["thompson-calculus-made-easy-1914/eq-42fc8fbd29",16,"Thompson 1914, p. 253: \\dfrac{x}{2} - \\dfrac{\\sin 2x}{4} + C"],["maxwell-elementary-treatise-electricity-1888/eq-f4d3757be2",16,"Maxwell 1888, scan 47: E_t{P_t}' = 0\\quad\\text{and}\\quad {E_t}'P_t = 0"],["theorem/intersecting-planes-meet-in-a-straight-line",9,"intersecting planes meet in a straight line","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-intersecting-planes-meet-in-a-straight-line"],["hardy-course-of-pure-mathematics-1921/ex-lii/2",4,"Hardy 1921, Exercise LII (2)"],["hardy-course-of-pure-mathematics-1921/ex-lii/3",4,"Hardy 1921, Exercise LII (3)"],["theorem/line-perpendicular-to-two-intersecting-lines-is-perpendicular-to-their-plane",9,"line perpendicular to two intersecting lines is perpendicular to their plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-line-perpendicular-to-two-intersecting-lines-is-perpendicular-to-their-plane"],["thompson-calculus-made-easy-1914/x-f2bbf3126b",15,"Thompson 1914, p. 9: We classify all quantities into two classes: constants and ..."],["cap/other:polar-coordinate area",17,"other:polar-coordinate area"],["blackburn-elements-plane-trigonometry-1863/ch-v",2,"Blackburn 1863, ch. V: CIRCULAR FUNCTIONS, OR TRIGONOMETRICAL RATIOS","../books/blackburn-elements-plane-trigonometry-1863/ch/ch-v/index.html"],["form/3f30d532d3",5,"evaluate: Integral(pi*x**3*(-x + 10)/36, (x, 0, 10))"],["slaught-lennes-solid-geometry-1919/eq-508d936ec1",16,"Slaught & Lennes 1919, p. 62: \\textit{Volume} = \\textit{Length}\\/ × \\textit{Width}\\/ × \\textit{Height}."],["theorem/plane-determined-by-a-line-and-point-two-intersecting-lines-or-two-parallel-lines",9,"plane determined by a line and point, two intersecting lines, or two parallel lines","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-plane-determined-by-a-line-and-point-two-intersecting-lines-or-two-parallel-lines"],["shape/f4e8e60837",6,"evaluate: Integral(pi*N*x**N*(N - x), (x, 0, N))"],["thompson-calculus-made-easy-1914/x-07e1886cc5",15,"Thompson 1914, p. 12: So we see that making dx an increase of ..."],["slaught-lennes-solid-geometry-1919/eq-7462ffcf12",16,"Slaught & Lennes 1919, p. 62: \\textbf{Volume} = \\textbf{Length} × \\textbf{Width} × \\textbf{Height}."],["concept/versine",7,"versine","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-versine"],["ball-mathematical-recreations-1905/x-c03998c72c",15,"Ball 1905, scan 73: The mathematical theory for a board of 9 cells ..."],["form/7df925c690",5,"hcf: (x**3 - x, x**4 - x, x**3 + 9*x**2 - 10*x)"],["shape/dd5b3b99a5",6,"hcf: (-x + x**N, -x + x**N, N*x + N*x**N + x**N)"],["concept/formula-for-volume-of-a-rectangular-solid",7,"formula for volume of a rectangular solid"],["maxwell-elementary-treatise-electricity-1888/eq-89e6314fac",16,"Maxwell 1888, scan 64: E_{BA} = -E_{AB}\\text{,}"],["theorem/through-a-point-there-is-one-plane-perpendicular-to-a-line",9,"through a point there is one plane perpendicular to a line","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-through-a-point-there-is-one-plane-perpendicular-to-a-line"],["theorem/through-a-point-there-is-one-line-perpendicular-to-a-plane",9,"through a point there is one line perpendicular to a plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-through-a-point-there-is-one-line-perpendicular-to-a-plane"],["ball-mathematical-recreations-1905/x-9c88e04306",15,"Ball 1905, scan 74: Let P be any point on a cubic. Let ..."],["ball-mathematical-recreations-1905/x-2d0dac69f5",15,"Ball 1905, scan 74: Sylvester stated that 9 counters can be placed in ..."],["ball-mathematical-recreations-1905/x-6c2ff588f1",15,"Ball 1905, scan 75: To those who have never looked into the matter ..."],["ball-mathematical-recreations-1905/x-5209d90349",15,"Ball 1905, scan 76: A cube has six faces, and if six colours ..."],["concept/spherical-zone",7,"spherical zone","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-spherical-zone"],["theorem/bounded-monotone-sequence-has-a-limit",9,"bounded monotone sequence has a limit","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-bounded-monotone-sequence-has-a-limit"],["thompson-calculus-made-easy-1914/eq-6abe9e26a5",16,"Thompson 1914, p. 253: \\dfrac{x}{2} + \\dfrac{\\sin 2x}{4} + C"],["thompson-calculus-made-easy-1914/eq-aefa094327",16,"Thompson 1914, p. 253: -\\frac{\\cos x}{n} \\sin^{n-1} x + \\frac{n-1}{n} \\int \\sin^{n-2} x\\, dx + C"],["thompson-calculus-made-easy-1914/x-2b02b9edc9",15,"Thompson 1914, p. 14: For example x^2 + 3 = 2y - 7 ..."],["shape/9ad079e3fd",6,"identity: -(N*a**N + x**N)/(N*a*x + x**N) + 1"],["ball-mathematical-recreations-1905/x-37f9eaa338",15,"Ball 1905, scan 80: Three beautiful ladies have for husbands three men, who ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-c714e5cdd1",16,"De Morgan 1899, p. 77: \\phi x = 0"],["hardy-course-of-pure-mathematics-1921/ex-liii/1a",4,"Hardy 1921, Exercise LIII (1a)"],["ball-mathematical-recreations-1905/x-b03826d5b1",15,"Ball 1905, scan 83: To obtain a solution we observe that we can ..."],["thompson-calculus-made-easy-1914/eq-48ec2075ee",16,"Thompson 1914, p. 253: \\log_\\epsilon \\tan \\dfrac{x}{2} + C"],["hardy-course-of-pure-mathematics-1921/x-38700e4624",15,"Hardy 1921, p. 72: In other words, addition of displacements obeys the commutative ..."],["thompson-calculus-made-easy-1914/eq-a2035197e1",16,"Thompson 1914, p. 253: -\\cotan x + C"],["thompson-calculus-made-easy-1914/ch-v",2,"Thompson 1914, ch. V: Next Stage. What to do with Constants","../books/thompson-calculus-made-easy-1914/ch/ch-v/index.html"],["thompson-calculus-made-easy-1914/eq-8fbafa85db",16,"Thompson 1914, p. 27: \\frac{dy}{dx} = 3x^2."],["thompson-calculus-made-easy-1914/eq-f60c095eab",16,"Thompson 1914, p. 165: y= \\sin \\theta"],["quantity/right-angle",11,"right angle","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-quantity-right-angle"],["thompson-calculus-made-easy-1914/eq-01e611e566",16,"Thompson 1914, p. 166: dy = \\sin(\\theta + d \\theta)- \\sin \\theta"],["thompson-calculus-made-easy-1914/eq-85259a47d9",16,"Thompson 1914, p. 166: \\sin M - \\sin N = 2 \\cos\\frac{M+N}{2}·\\sin\\frac{M-N}{2}"],["thompson-calculus-made-easy-1914/eq-14ef531dc7",16,"Thompson 1914, p. 253: \\log_\\epsilon \\tan x + C"],["boyden-first-book-in-algebra-1895/ex-35/6",4,"Boyden 1895, Exercise 35 (6)"],["form/03c34c94eb",5,"identity: 3*(a**3*b**9)**(1/3)"],["person/benjamin-thompson-count-rumford",1,"Benjamin Thompson, Count Rumford","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-benjamin-thompson-count-rumford"],["person/humphry-davy",1,"Humphry Davy","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-humphry-davy"],["boyden-first-book-in-algebra-1895/ex-35/5",4,"Boyden 1895, Exercise 35 (5)"],["form/8f7c5418a0",5,"factor: -8*a**3 + 27*x**3"],["form/70a3fd35c5",5,"solve: (Eq(2*a, b + 120), Eq(5*a, 3*b))"],["shape/c17e85683c",6,"solve: (Eq(N*a, N + b), Eq(N*a, N*b))"],["thompson-calculus-made-easy-1914/eq-d1abf927ed",16,"Thompson 1914, p. 166: dy = \\cos \\theta · d \\theta"],["hardy-course-of-pure-mathematics-1921/ex-liii/2a",4,"Hardy 1921, Exercise LIII (2a)"],["thompson-calculus-made-easy-1914/eq-1cac35fd85",16,"Thompson 1914, p. 168: \\cos \\theta=\\sin\\left(\\dfrac{\\pi}{2}-\\theta\\right)"],["thompson-calculus-made-easy-1914/eq-85327723ec",16,"Thompson 1914, p. 253: \\frac{1}{2} \\cos(m - n)x - \\frac{1}{2} \\cos(m + n)x + C"],["hardy-course-of-pure-mathematics-1921/x-790e555930",15,"Hardy 1921, p. 334: Some care has occasionally to be exercised in applying ..."],["concept/product-of-trigonometric-functions",7,"product of trigonometric functions"],["de-morgan-elementary-illustrations-calculus-1899/eq-f1f1cc4e27",16,"De Morgan 1899, p. 131: AB = a"],["form/93a27b3907",5,"factor: -64*a**3 + 8*x**3"],["boyden-first-book-in-algebra-1895/ex-35/21",4,"Boyden 1895, Exercise 35 (21)"],["theorem/trihedral-angles-equal-by-equal-face-angles",9,"trihedral angles equal by equal face angles","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-trihedral-angles-equal-by-equal-face-angles"],["thompson-calculus-made-easy-1914/eq-81b74fcbf1",16,"Thompson 1914, p. 168: \\frac{dy}{d\\theta} = -\\sin \\theta"],["hardy-course-of-pure-mathematics-1921/eq-fa326fef07",16,"Hardy 1921, p. 302: x = \\psi(y) = y - a_{2}y^{2} + (2a_{2}^{2} - a_{3})y^{3} - (5a_{2}^{3} - 5a_{2}a_{3} + a_{4} + \\epsilon_{y})y^{4}"],["thompson-calculus-made-easy-1914/eq-14856c66bc",16,"Thompson 1914, p. 169: \\frac{dy}{d\\theta} = \\sec^2 \\theta"],["de-morgan-elementary-illustrations-calculus-1899/eq-491358a244",16,"De Morgan 1899, p. 131: n\\, dx = a"],["de-morgan-elementary-illustrations-calculus-1899/eq-a51b80cd06",16,"De Morgan 1899, p. 131: wb\\, dx"],["slaught-lennes-solid-geometry-1919/eq-7aa76b6a93",16,"Slaught & Lennes 1919, p. 188: a = bh"],["slaught-lennes-solid-geometry-1919/eq-e3873c2f19",16,"Slaught & Lennes 1919, p. 189: d = \\sqrt{2}"],["shape/69c15da644",6,"solve: Eq(x, N*x + N)"],["slaught-lennes-solid-geometry-1919/x-96f111d37b",15,"Slaught & Lennes 1919, p. 26: If two straight lines are cut by three parallel ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-85e33b8c02",16,"De Morgan 1899, p. 131: x^{2} × bw\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/eq-f489c02f1f",16,"De Morgan 1899, p. 132: bwx^{2}\\, dx + \\alpha"],["thompson-calculus-made-easy-1914/eq-8758ad57f0",16,"Thompson 1914, p. 172: -(1+\\cot^2 \\theta) = -\\cosec^2 \\theta"],["thompson-calculus-made-easy-1914/eq-c2be6d43f6",16,"Thompson 1914, p. 172: \\frac{1}{\\sin\\theta} × \\cos\\theta = \\cot\\theta"],["thompson-calculus-made-easy-1914/eq-d4e35ee42f",16,"Thompson 1914, p. 169: \\theta = 2\\pi\\frac{t}{T}"],["thompson-calculus-made-easy-1914/eq-7c62bbffbf",16,"Thompson 1914, p. 169: \\theta = 360\\frac{t}{T}"],["thompson-calculus-made-easy-1914/eq-81081b9e3f",16,"Thompson 1914, p. 170: n = \\dfrac{1}{T}"],["thompson-calculus-made-easy-1914/eq-57b95c695b",16,"Thompson 1914, p. 253: \\dfrac{x}{2} - \\dfrac{\\sin 2ax}{4a} + C"],["experiment/porous-plug-experiment",14,"porous plug experiment","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-experiment-porous-plug-experiment"],["thompson-calculus-made-easy-1914/eq-ec5dfb218c",16,"Thompson 1914, p. 253: \\dfrac{x}{2} + \\dfrac{\\sin 2ax}{4a} + C"],["thompson-calculus-made-easy-1914/eq-be5eb2b5eb",16,"Thompson 1914, p. 161: I = I_0\\epsilon^{-Kl}"],["slaught-lennes-solid-geometry-1919/eq-19bdc9073b",16,"Slaught & Lennes 1919, p. 198: \\dfrac{AD}{AB} = \\dfrac{AE}{AC}"],["thompson-calculus-made-easy-1914/eq-056fe6f33d",16,"Thompson 1914, p. 170: \\theta=2\\pi nt."],["thompson-calculus-made-easy-1914/eq-352d01f370",16,"Thompson 1914, p. 170: y = \\sin 2\\pi nt."],["thompson-calculus-made-easy-1914/eq-191d4fb27e",16,"Thompson 1914, p. 170: \\frac{dy}{dt} = \\frac{dy}{d\\theta} · \\frac{d\\theta}{dt}"],["concept/intensity",7,"intensity","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-intensity"],["thompson-calculus-made-easy-1914/eq-ff6779bcde",16,"Thompson 1914, p. 170: \\frac{d(\\cos 2\\pi nt)}{dt} = -2\\pi n · \\sin 2\\pi nt"],["thompson-calculus-made-easy-1914/eq-31b38e8267",16,"Thompson 1914, p. 170: \\frac{d^2(\\DPtypo{\\cos \\theta}{\\sin \\theta})}{d\\theta^2} = -\\sin \\theta"],["thompson-calculus-made-easy-1914/x-fb108dc83a",15,"Thompson 1914, p. 79: For a horizontal line, or a horizontal place in ..."],["thompson-calculus-made-easy-1914/eq-80a36b9cff",16,"Thompson 1914, p. 162: Q = Q_0 \\epsilon^{-\\lambda t}"],["form/dbcecce2a6",5,"identity: 21"],["hardy-course-of-pure-mathematics-1921/ex-app-i",3,"Hardy 1921, Exercise App-I"],["wentworth-first-steps-in-algebra-1894/eq-329d346ad7",16,"Wentworth 1894, p. 120: p = \\frac{i}{rt}"],["thompson-calculus-made-easy-1914/eq-c8b8556bed",16,"Thompson 1914, p. 171: \\frac{d^2(\\cos\\theta)}{d\\theta^2} = -\\cos\\theta"],["thompson-calculus-made-easy-1914/eq-61cca4f3ed",16,"Thompson 1914, p. 172: \\frac{dy}{dx}=\\cos(x+a)"],["thompson-calculus-made-easy-1914/eq-0a0eb7c28f",16,"Thompson 1914, p. 172: \\frac{dy}{d\\theta}=3 \\sec^2 3\\theta"],["hardy-course-of-pure-mathematics-1921/eq-62f12eea15",16,"Hardy 1921, p. 302: \\frac{\\phi(x)\\psi(x) - x^{2}}{x^{4}} \\to a_{2}^{2}"],["theorem/projection-of-a-broken-line",9,"projection of a broken line","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-projection-of-a-broken-line"],["theorem/joule-s-law",9,"Joule's law","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-theorem-joule-s-law"],["thompson-calculus-made-easy-1914/ex-ix",3,"Thompson 1914, Exercise IX"],["wentworth-first-steps-in-algebra-1894/eq-a67b1d4d0e",16,"Wentworth 1894, p. 120: p = \\frac{a}{1 + rt}"],["slaught-lennes-solid-geometry-1919/eq-3fff19f93b",16,"Slaught & Lennes 1919, p. 26: \\dfrac{AE}{EB} = \\dfrac{CG}{GD}"],["concept/centre-of-curvature",7,"centre of curvature","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-centre-of-curvature"],["de-morgan-elementary-illustrations-calculus-1899/x-a95b3b9aac",15,"De Morgan 1899, p. 121: Let us suppose that \\psi a is the function ..."],["hardy-course-of-pure-mathematics-1921/eq-0f2c31532a",16,"Hardy 1921, p. 302: -(\\xi - x)/y' = (\\eta - y)/x' = (x'^{2} + y'^{2})/(x'y'' - x''y')"],["concept/vertical-angle",7,"vertical angle","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-vertical-angle"],["boyden-first-book-in-algebra-1895/ex-26/29",4,"Boyden 1895, Exercise 26 (29)"],["concept/angle-of-elevation",7,"angle of elevation","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-angle-of-elevation"],["wentworth-plane-geometry-1899/x-04c4105e8a",15,"Wentworth 1899, scan 146: If four quantities are in proportion, they are in ..."],["hardy-course-of-pure-mathematics-1921/eq-70b6709b57",16,"Hardy 1921, p. 302: (x'^{2} + y'^{2})^{3/2}/(x'y'' - x''y')"],["thompson-calculus-made-easy-1914/eq-2d8b0c74c7",16,"Thompson 1914, p. 127: \\frac{3x^2 - 2x + 1}{(x+1)^2(x-2)} = \\frac{x-1}{(x+1)^2} + \\frac{1}{x+1} + \\frac{1}{x-2}"],["instrument/theodolite",13,"theodolite","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-instrument-theodolite"],["concept/altitude-of-a-cylinder",7,"altitude of a cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-altitude-of-a-cylinder"],["instrument/sextant",13,"sextant","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-instrument-sextant"],["hardy-course-of-pure-mathematics-1921/eq-4dd0f77bfa",16,"Hardy 1921, p. 302: 3a(\\xi + x) + 2x^{2} = 0"],["quantity/height",11,"height","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-quantity-height"],["concept/theorem-parallel-planes-intercept-proportional-segments",7,"theorem: parallel planes intercept proportional segments"],["blackburn-elements-plane-trigonometry-1863/x-ee412313e9",15,"Blackburn 1863, p. 44: In Geodesy, or the application of this part of ..."],["form/c29f3751e6",5,"identity: (-16*a**8 + 81*x**12)/(-8*a**6 + 12*a**4*x**3 - 18*a**2*x**6 + 27*x**9)"],["shape/a7b0ea0fcd",6,"identity: (N*a**N + N*x**N)/(2*N*a**N*x**N + N*a**N + N*x**N)"],["form/fbcbaf0b08",5,"identity: 24"],["slaught-lennes-solid-geometry-1919/eq-3e79b28097",16,"Slaught & Lennes 1919, p. 201: \\dfrac{k^2A}{A} = k^2 = \\dfrac{r'^2}{r^2}"],["slaught-lennes-solid-geometry-1919/ch-appendix-iii",2,"Slaught & Lennes 1919, ch. Appendix III: VARIABLES. LIMITS","../books/slaught-lennes-solid-geometry-1919/ch/ch-appendix-iii/index.html"],["concept/base-of-a-cylinder",7,"base of a cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-base-of-a-cylinder"],["shape/2ef9f7ea91",6,"identity: N*(a**N*b**N)**N"],["concept/right-cylinder",7,"right cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-right-cylinder"],["hardy-course-of-pure-mathematics-1921/ex-liii/2b",4,"Hardy 1921, Exercise LIII (2b)"],["blackburn-elements-plane-trigonometry-1863/x-c2d4ed49ec",15,"Blackburn 1863, p. 45: The horizontal angle between two objects is the angle ..."],["shape/b84960f39f",6,"integrate: cot(x)"],["thompson-calculus-made-easy-1914/x-a0a4648296",15,"Thompson 1914, p. 97: So, writing \\dfrac{dy}{dx} = 0 does not mean that ..."],["todhunter-spherical-trigonometry-1886/x-b1fdcc90aa",15,"Todhunter 1886, scan 112: We shall now shew that a small circle can ..."],["dickson-theory-of-equations-1922/eq-dbac901c70",16,"Dickson 1922, p. 107: \\Delta = -D"],["shape/1d6328c777",6,"identity: N*b**N*(a**N + c)**N*Abs(c)*Abs(a**N + c)"],["concept/tangent-plane-to-a-cylinder",7,"tangent plane to a cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-tangent-plane-to-a-cylinder"],["slaught-lennes-solid-geometry-1919/ch-appendix-i",2,"Slaught & Lennes 1919, ch. Appendix I: SIMILAR SOLIDS","../books/slaught-lennes-solid-geometry-1919/ch/ch-appendix-i/index.html"],["dickson-theory-of-equations-1922/eq-5ad187574c",16,"Dickson 1922, p. 102: D = 0"],["blackburn-elements-plane-trigonometry-1863/x-d47d9b8142",15,"Blackburn 1863, p. 46: If the angle of elevation of C be likewise ..."],["thompson-calculus-made-easy-1914/eq-b71f9ccfa3",16,"Thompson 1914, p. 130: \\frac{dy}{dx} = -\\frac{3}{(3x-1)^2} + \\frac{4}{(2x+3)^2}"],["blackburn-elements-plane-trigonometry-1863/x-cf907767c0",15,"Blackburn 1863, p. 45: For details of the measurement of a base, and ..."],["blackburn-elements-plane-trigonometry-1863/x-7b534d5cff",15,"Blackburn 1863, p. 47: Where the extent of the earth’s surface surveyed is ..."],["theorem/lateral-area-formula-for-a-cylinder",9,"lateral area formula for a cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-lateral-area-formula-for-a-cylinder"],["concept/method-differentiation",7,"method: differentiation"],["concept/pyramidal-surface",7,"pyramidal surface","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-pyramidal-surface"],["concept/generator",7,"generator","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-generator"],["method/integration-of-polynomials-in-cosines-and-sines-of-multiples-of-x",8,"integration of polynomials in cosines and sines of multiples of x","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-integration-of-polynomials-in-cosines-and-sines-of-multiples-of-x"],["boyden-first-book-in-algebra-1895/ex-50/1",4,"Boyden 1895, Exercise 50 (1)"],["planck-treatise-on-thermodynamics-1903/ch-general-deductions",2,"Planck 1903, General Deductions","../books/planck-treatise-on-thermodynamics-1903/ch/ch-general-deductions/index.html"],["maxwell-elementary-treatise-electricity-1888/eq-9eb4fc0872",16,"Maxwell 1888, scan 63: e(p - P)\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-95c044a5d5",16,"Maxwell 1888, scan 63: ep + EP\\text{.}"],["form/c42d8a0a84",5,"identity: b*x/a**2"],["shape/81ca288e0d",6,"identity: a**N*b*x"],["concept/element-of-a-surface",7,"element of a surface","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-element-of-a-surface"],["hardy-course-of-pure-mathematics-1921/ex-liii/2c",4,"Hardy 1921, Exercise LIII (2c)"],["de-morgan-elementary-illustrations-calculus-1899/x-d6b8ff72bd",15,"De Morgan 1899, p. 108: y and x are no longer independent; for, one ..."],["hardy-course-of-pure-mathematics-1921/eq-c75ad4f187",16,"Hardy 1921, p. 302: \\eta = 4y + (9ay)/x."],["boyden-first-book-in-algebra-1895/ex-28",3,"Boyden 1895, Exercise 28"],["form/380991dd12",5,"integrate: sec(x)**2"],["shape/b38e28abe8",6,"integrate: sec(x)**N"],["hardy-course-of-pure-mathematics-1921/eq-65dbf7f428",16,"Hardy 1921, p. 302: (1 + y_{1}^{2})y_{3} = 3y_{1}y_{2}^{2}"],["form/9232e0559f",5,"solve: (Eq(-a + x, 7141/120), Eq(a + x, 14461/120))"],["thompson-calculus-made-easy-1914/ch-xii",2,"Thompson 1914, ch. XII: Curvature of Curves","../books/thompson-calculus-made-easy-1914/ch/ch-xii/index.html"],["shape/ffda3f24c0",6,"solve: (Eq(-a + x, N), Eq(a + x, N))"],["form/1200f4cd70",5,"integrate: csc(x)**2"],["shape/47c2172336",6,"integrate: csc(x)**N"],["thompson-calculus-made-easy-1914/eq-ee687e7d43",16,"Thompson 1914, p. 210: \\int^{x=x_2}_{x=x_1} y\\, dx = y_2 - y_1"],["hardy-course-of-pure-mathematics-1921/eq-58f087dd73",16,"Hardy 1921, p. 302: a^{3}y = a^{4}x^{2} + a^{2}bxy + (ac - b^{2})y^{2}"],["planck-treatise-on-thermodynamics-1903/x-bf35243829",15,"Planck 1903, p. 105: Our first application of the principle of the entropy ..."],["method/linear-approximation",8,"linear approximation","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-linear-approximation"],["hardy-course-of-pure-mathematics-1921/ex-liii/2e",4,"Hardy 1921, Exercise LIII (2e)"],["form/2d927ee631",5,"integrate: tan(x)*sec(x)"],["de-morgan-elementary-illustrations-calculus-1899/eq-d0eabfc443",16,"De Morgan 1899, p. 62: \\frac{1}{2}gt^{2}"],["concept/nappe",7,"nappe","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-nappe"],["ball-mathematical-recreations-1905/ch-iii",2,"Ball 1905, ch. III: Some Mechanical Questions","../books/ball-mathematical-recreations-1905/ch/ch-iii/index.html"],["wentworth-first-steps-in-algebra-1894/ex-65",3,"Wentworth 1894, Exercise 65"],["thompson-calculus-made-easy-1914/eq-b0bf03b642",16,"Thompson 1914, p. 132: \\frac{dy}{dx} = -\\frac{3}{2x^2\\sqrt{\\dfrac{3}{x} -1}}"],["shape/2d927ee631",6,"integrate: tan(x)*sec(x)"],["concept/method-differentiating-an-inverse-function",7,"method: differentiating an inverse function"],["thompson-calculus-made-easy-1914/eq-4fae0c49d3",16,"Thompson 1914, p. 210: \\text{area~$S$} = b(x_2 - x_1) + \\frac{a}{3}(x_2^3 - x_1^3)"],["thompson-calculus-made-easy-1914/eq-dc790294fa",16,"Thompson 1914, p. 218: \\text{mean~$y$} = \\frac{1}{x_1} \\int^{x=x_1}_{x=0} y · dx"],["form/ea02baad8a",5,"solve: (Eq(5*a + 2*x, 16), Eq(-2*a + 3*x, 5))"],["concept/conical-surface",7,"conical surface","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-conical-surface"],["maxwell-elementary-treatise-electricity-1888/eq-f82769f0de",16,"Maxwell 1888, scan 49: P_s = nP_r"],["thompson-calculus-made-easy-1914/eq-d78ee45a1a",16,"Thompson 1914, p. 217: dA = 2 \\pi r\\, dr"],["thompson-calculus-made-easy-1914/eq-79193fc0d2",16,"Thompson 1914, p. 217: A &= \\pi R^2"],["thompson-calculus-made-easy-1914/eq-8c8d53b6db",16,"Thompson 1914, p. 220: \\tfrac{1}{2} \\int^{\\theta=\\theta_2}_{\\theta=\\theta_1} r^2\\, d\\theta"],["person/cornelius-agrippa",1,"Cornelius Agrippa","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-cornelius-agrippa"],["ball-mathematical-recreations-1905/x-6b715607e6",15,"Ball 1905, scan 162: The reason why such a square is magic can ..."],["hardy-course-of-pure-mathematics-1921/eq-3fae3c1919",16,"Hardy 1921, p. 302: 18\\eta_{2}^{3}T = 9\\eta_{2}^{4}(x - \\xi)^{2} + 6\\eta_{2}^{2}\\eta_{3}(x - \\xi)T + (3\\eta_{2}\\eta_{4} - 4\\eta_{3}^{2})T^{2"],["whitehead-introduction-to-mathematics-1911/x-60126989c9",15,"Whitehead 1911, p. 61: Operations of thought are like cavalry charges in a ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-c845c6f79f",16,"De Morgan 1899, p. 41: dx \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^{2}}"],["thompson-calculus-made-easy-1914/eq-ebb7a56089",16,"Thompson 1914, p. 219: 2\\pi y\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-22117834f2",16,"Hardy 1921, p. 302: T = (y - \\eta) - \\eta_{1}(x - \\xi)"],["thompson-calculus-made-easy-1914/eq-62fc6c1855",16,"Thompson 1914, p. 215: &= b(1-\\epsilon^{-a})"],["hardy-course-of-pure-mathematics-1921/ex-i",3,"Hardy 1921, Exercise I"],["hardy-course-of-pure-mathematics-1921/eq-c8a9afde1e",16,"Hardy 1921, p. 302: x\\frac{\\dd u}{\\dd x} + y\\frac{\\dd u}{\\dd y} + z\\frac{\\dd u}{\\dd z} + \\dots = nu"],["quantity/solar-longitude",11,"solar longitude","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-quantity-solar-longitude"],["slaught-lennes-solid-geometry-1919/x-aeb5293f79",15,"Slaught & Lennes 1919, p. 139: A sphere has a definite area which is less ..."],["thompson-calculus-made-easy-1914/x-7ed6f0f566",15,"Thompson 1914, p. 165: What we have to investigate is the value of ..."],["slaught-lennes-solid-geometry-1919/x-0a2d3fdd25",15,"Slaught & Lennes 1919, p. 139: The student should note that while the statement just ..."],["concept/similar-polyhedron",7,"similar polyhedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-similar-polyhedron"],["thompson-calculus-made-easy-1914/eq-7052ac9d5a",16,"Thompson 1914, p. 135: y + n\\dfrac{y}{n} = 2y"],["thompson-calculus-made-easy-1914/x-20f76f29c9",15,"Thompson 1914, p. 166: But if we regard d \\theta as indefinitely small, ..."],["theorem/equal-circles-on-a-sphere-have-equidistant-planes",9,"equal circles on a sphere have equidistant planes","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-equal-circles-on-a-sphere-have-equidistant-planes"],["theorem/unequally-distant-circles-on-a-sphere",9,"unequally distant circles on a sphere","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-unequally-distant-circles-on-a-sphere"],["form/cea9d06d18",5,"factor: 5*a**2 - 25"],["form/955c632473",5,"solve: Eq(5*x, 20)"],["form/bb8829a1e8",5,"solve: (Eq(b, a + 15), Eq(a + b, 143))"],["whitehead-introduction-to-mathematics-1911/ch-xi",2,"Whitehead 1911, ch. XI: Functions","../books/whitehead-introduction-to-mathematics-1911/ch/ch-xi/index.html"],["boyden-first-book-in-algebra-1895/ex-41",3,"Boyden 1895, Exercise 41"],["shape/48e6ccb5ed",6,"factor: N*a**N + N"],["thompson-calculus-made-easy-1914/eq-aaa5492e0c",16,"Thompson 1914, p. 136: y_n = y_0\\left(1 + \\frac{1}{n}\\right)^n"],["form/7dc2cb7fe5",5,"solve: (Eq(3*a + 2*x, 7), Eq(-5*a + 8*x, 11))"],["form/d647ac3bae",5,"solve: (Eq(7*a + 5*x, 19), Eq(4*a + 7*x, 15))"],["thompson-calculus-made-easy-1914/x-d6e44d38f0",15,"Thompson 1914, p. 168: Now \\cos \\theta=\\sin\\left(\\dfrac{\\pi}{2}-\\theta\\right)."],["concept/pyramid",7,"pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-pyramid"],["thompson-calculus-made-easy-1914/x-6515ac845d",15,"Thompson 1914, p. 171: Passing now from the inverse function to the original ..."],["thompson-calculus-made-easy-1914/x-c67a1ee9bc",15,"Thompson 1914, p. 171: Sines and cosines are the only functions of which ..."],["de-morgan-elementary-illustrations-calculus-1899/x-2e7a78e6e3",15,"De Morgan 1899, p. 111: Here we want a new word, which has not ..."],["concept/stationary-value",7,"stationary value","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-stationary-value"],["hardy-course-of-pure-mathematics-1921/eq-00a79b78fd",16,"Hardy 1921, p. 302: u = x^{n} f(y/x, z/x, \\dots)"],["hardy-course-of-pure-mathematics-1921/eq-2541e4e97b",16,"Hardy 1921, p. 303: xF_{\\xi} + yF_{\\eta} + zF_{\\zeta} = 0"],["blackburn-elements-plane-trigonometry-1863/x-5cd5566dd9",15,"Blackburn 1863, p. 15: In Euclid an angle is not defined as a ..."],["hardy-course-of-pure-mathematics-1921/ex-liii/2f",4,"Hardy 1921, Exercise LIII (2f)"],["concept/polar-distance-of-a-circle",7,"polar distance of a circle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-polar-distance-of-a-circle"],["thompson-calculus-made-easy-1914/x-6de2792c85",15,"Thompson 1914, p. 170: If the frequency, or number of periods per second, ..."],["form/0715de8bb6",5,"integrate: cot(x)*csc(x)"],["de-morgan-elementary-illustrations-calculus-1899/x-04d04dcc26",15,"De Morgan 1899, p. 111: Imagine the diameter of a circle divided into a ..."],["thompson-calculus-made-easy-1914/x-ebc542f1d8",15,"Thompson 1914, p. 171: So we have this curious result that we have ..."],["shape/0715de8bb6",6,"integrate: cot(x)*csc(x)"],["de-morgan-elementary-illustrations-calculus-1899/x-f7b20f15d0",15,"De Morgan 1899, p. 111: We may answer that the notion of time is ..."],["concept/altitude-of-a-pyramid",7,"altitude of a pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-altitude-of-a-pyramid"],["concept/tangent-plane-to-a-sphere",7,"tangent plane to a sphere","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-tangent-plane-to-a-sphere"],["thompson-calculus-made-easy-1914/eq-2243c3cab6",16,"Thompson 1914, p. 148: \\frac{d(\\log_\\epsilon x)}{dx} = x^{-1}"],["thompson-calculus-made-easy-1914/x-6c5cfce66a",15,"Thompson 1914, p. 114: To the left of M the slope is downward, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-93fb8b751b",15,"De Morgan 1899, p. 112: But we, who cannot consider all these perpendiculars at ..."],["blackburn-elements-plane-trigonometry-1863/x-9a5ae2c8af",15,"Blackburn 1863, p. 15: So also in Trigonometry, where angular magnitude in general ..."],["whitehead-introduction-to-mathematics-1911/ch-xvii",2,"Whitehead 1911, ch. XVII: Quantity","../books/whitehead-introduction-to-mathematics-1911/ch/ch-xvii/index.html"],["thompson-calculus-made-easy-1914/eq-8dcff52f3f",16,"Thompson 1914, p. 147: y = \\log_\\epsilon x"],["thompson-calculus-made-easy-1914/eq-904d7931c1",16,"Thompson 1914, p. 146: \\log_\\epsilon a + \\log_\\epsilon b = \\log_\\epsilon ab"],["concept/birectangular-spherical-triangle",7,"birectangular spherical triangle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-birectangular-spherical-triangle"],["concept/base-of-a-pyramid",7,"base of a pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-base-of-a-pyramid"],["hardy-course-of-pure-mathematics-1921/ex-l",3,"Hardy 1921, Exercise L"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi",3,"Hardy 1921, Exercise XXXVI"],["concept/axiom",7,"axiom","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-axiom"],["de-morgan-elementary-illustrations-calculus-1899/x-8141791aec",15,"De Morgan 1899, p. 112: and succession, disguise it as we may, is the ..."],["theorem/midline-of-a-triangle-is-parallel-to-the-third-side",9,"midline of a triangle is parallel to the third side","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-midline-of-a-triangle-is-parallel-to-the-third-side"],["theorem/opposite-sides-of-a-parallelogram-are-equal",9,"opposite sides of a parallelogram are equal","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-opposite-sides-of-a-parallelogram-are-equal"],["concept/dodecahedron",7,"dodecahedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-dodecahedron"],["form/d9437ffae6",5,"integrate: 1/(x*sqrt(x**2 + 2*x + 3))"],["de-morgan-elementary-illustrations-calculus-1899/x-fc2d97d5af",15,"De Morgan 1899, p. 110: who should never take it for granted that because ..."],["de-morgan-elementary-illustrations-calculus-1899/x-27158e53e6",15,"De Morgan 1899, p. 111: The number which represents this line (reference being made ..."],["hardy-course-of-pure-mathematics-1921/x-4890533523",15,"Hardy 1921, p. 345: It is obvious that an absolutely convergent series is ..."],["form/cbce3f24f1",5,"solve: (Eq(5*a + 4*x, 22), Eq(-12*a + 11*x, 9))"],["blackburn-elements-plane-trigonometry-1863/x-fbd11c40e2",15,"Blackburn 1863, p. 15: For instance, if a distance a miles be measured ..."],["blackburn-elements-plane-trigonometry-1863/x-f2a9f1e5f7",15,"Blackburn 1863, p. 16: Then the standard direction may be called the + ..."],["de-morgan-elementary-illustrations-calculus-1899/x-30a6c07b6f",15,"De Morgan 1899, p. 110: This not being the case, it is a cause ..."],["hardy-course-of-pure-mathematics-1921/eq-60dd7b991c",16,"Hardy 1921, p. 303: u_{x}v_{y} - u_{y}v_{x} = 0"],["de-morgan-elementary-illustrations-calculus-1899/x-ae2eea8377",15,"De Morgan 1899, p. 75: For example, let there be a right-angled triangle, whose ..."],["dickson-theory-of-equations-1922/ch-x",2,"Dickson 1922, ch. X: Elimination, Resultants And Discriminants","../books/dickson-theory-of-equations-1922/ch/ch-x/index.html"],["concept/exterior-angle",7,"exterior angle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-exterior-angle"],["shape/c64fbfd311",6,"integrate: (x + 1)*(N*x + N*x**N + 1)**N/(N*x + N*x**N + 1)"],["theorem/symmetrical-spherical-triangles-are-equal-in-area",9,"symmetrical spherical triangles are equal in area","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-symmetrical-spherical-triangles-are-equal-in-area"],["dickson-theory-of-equations-1922/ex-page144",3,"Dickson 1922, Exercise Page144"],["blackburn-elements-plane-trigonometry-1863/x-3566689306",15,"Blackburn 1863, p. 16: Again, with reference to angles, if the hand of ..."],["dickson-theory-of-equations-1922/ex-page147",3,"Dickson 1922, Exercise Page147"],["dickson-theory-of-equations-1922/ex-page150",3,"Dickson 1922, Exercise Page150"],["todhunter-spherical-trigonometry-1886/ch-on-certain-approximate-formul-ae",2,"Todhunter 1886, On certain approximate Formul\\ae","../books/todhunter-spherical-trigonometry-1886/ch/ch-on-certain-approximate-formul-ae/index.html"],["boyden-first-book-in-algebra-1895/ex-1/6",4,"Boyden 1895, Exercise 1 (6)"],["concept/functional-relation",7,"functional relation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-functional-relation"],["concept/regular-pyramid",7,"regular pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-regular-pyramid"],["concept/slant-height",7,"slant height","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-slant-height"],["de-morgan-elementary-illustrations-calculus-1899/ch-algebraical-geometry",2,"De Morgan 1899, Algebraical Geometry","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-algebraical-geometry/index.html"],["concept/contiguous-values",7,"contiguous values","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-contiguous-values"],["dickson-theory-of-equations-1922/ex-page153b",3,"Dickson 1922, Exercise Page153b"],["form/d30c038a20",5,"solve: (Eq(a, 8*x), Eq(a + x, 72))"],["hardy-course-of-pure-mathematics-1921/eq-4c279a68d5",16,"Hardy 1921, p. 303: \\frac{\\dd \\phi}{\\dd u}\\, \\frac{\\dd u}{\\dd x} + \\frac{\\dd \\phi}{\\dd v}\\, \\frac{\\dd v}{\\dd x} = 0"],["form/52e6e78290",5,"solve: (Eq(-c/(a - b) + x/(a + b), 1/(a + b)), Eq(c/(a - b) + x/(a + b), 1/(a - b)))"],["form/9ed84efe4d",5,"identity: 4*c**4/(a**2*b**6)"],["boyden-first-book-in-algebra-1895/ex-29",3,"Boyden 1895, Exercise 29"],["dickson-theory-of-equations-1922/ex-page153",3,"Dickson 1922, Exercise Page153"],["shape/52e6e78290",6,"solve: (Eq(-c/(a - b) + x/(a + b), 1/(a + b)), Eq(c/(a - b) + x/(a + b), 1/(a - b)))"],["de-morgan-elementary-illustrations-calculus-1899/x-a49939cbf2",15,"De Morgan 1899, p. 113: We do not take the increments themselves, but the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-437fc7e689",15,"De Morgan 1899, p. 76: The sun’s longitude is a function of the time; ..."],["de-morgan-elementary-illustrations-calculus-1899/x-d24df0bafe",15,"De Morgan 1899, p. 113: In passing from 1000 to 1003, we have the ..."],["whitehead-introduction-to-mathematics-1911/x-97e361698d",15,"Whitehead 1911, p. 165: The presupposition of periodicity is indeed fundamental to our ..."],["concept/lateral-surface",7,"lateral surface","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-lateral-surface"],["instrument/foot-rule",13,"foot-rule","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-instrument-foot-rule"],["whitehead-introduction-to-mathematics-1911/x-7bf0fc5e88",15,"Whitehead 1911, p. 245: When we have a set of things such as ..."],["de-morgan-elementary-illustrations-calculus-1899/x-cbee4dc0d1",15,"De Morgan 1899, p. 114: But if on adding any interval, however small, to ..."],["concept/similar-cylinders",7,"similar cylinders","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-similar-cylinders"],["form/c37370cd2c",5,"solve: (Eq(x - 200, a/2 + 100), Eq(a - 200, x/3 + 200/3))"],["concept/similar-cones",7,"similar cones","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-similar-cones"],["maxwell-elementary-treatise-electricity-1888/ch-xi",2,"Maxwell 1888, ch. XI: METHODS OF MAINTAINING AN ELECTRIC CURRENT","../books/maxwell-elementary-treatise-electricity-1888/ch/ch-xi/index.html"],["concept/center-of-similitude",7,"center of similitude","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-center-of-similitude"],["dickson-theory-of-equations-1922/eq-070613b1b8",16,"Dickson 1922, p. 86: x^3 - 2x - 5 \\equiv (x-2)^3 + 6(x-2)^2 + 10(x-2) - 1"],["quantity/lateral-area",11,"lateral area","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-quantity-lateral-area"],["method/approximating-the-volume-of-a-sphere-by-pyramids",8,"approximating the volume of a sphere by pyramids","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-approximating-the-volume-of-a-sphere-by-pyramids"],["de-morgan-elementary-illustrations-calculus-1899/x-169285f177",15,"De Morgan 1899, p. 114: The objection becomes of less force as the increment ..."],["concept/resultant",7,"resultant","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-resultant"],["whitehead-introduction-to-mathematics-1911/x-08b76209a5",15,"Whitehead 1911, p. 249: A sense of the flux of time accompanies all ..."],["de-morgan-elementary-illustrations-calculus-1899/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes",2,"De Morgan 1899, On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-on-the-connexion-of-the-signs-of-algebraical-and-the-directions-of-geometrical-magnitudes/index.html"],["hardy-course-of-pure-mathematics-1921/eq-dda9f21a94",16,"Hardy 1921, p. 303: J = \\begin{vmatrix} u_{x} & u_{y}\\\\ v_{x} & v_{y} \\end{vmatrix} = u_{x}v_{y} - u_{y}v_{x} = 0"],["boyden-first-book-in-algebra-1895/ex-1/7",4,"Boyden 1895, Exercise 1 (7)"],["form/d00f105a90",5,"solve: (Eq(a, 2*x), Eq(a + x, 672))"],["boyden-first-book-in-algebra-1895/ex-1/8",4,"Boyden 1895, Exercise 1 (8)"],["de-morgan-elementary-illustrations-calculus-1899/x-1213753f8b",15,"De Morgan 1899, p. 112: Every value of the variable, gives not only a ..."],["de-morgan-elementary-illustrations-calculus-1899/x-5a0f526fd4",15,"De Morgan 1899, p. 115: How well this answers to our previously formed ideas ..."],["dickson-theory-of-equations-1922/eq-d49407fce0",16,"Dickson 1922, p. 87: t^3 + 6.282t^2 + 11.154508t - 0.006153416 = 0"],["hardy-course-of-pure-mathematics-1921/eq-832f7b3485",16,"Hardy 1921, p. 303: J = \\frac{\\dd(u, v)}{\\dd(x, y)}"],["boyden-first-book-in-algebra-1895/ex-40",3,"Boyden 1895, Exercise 40"],["form/ebe731e091",5,"identity: -27*a**3*c**12*(2*a + 3*b)**9/(64*e**6*f**6*(c - d)**6)"],["shape/07f73a7129",6,"solve: (Eq(N + x, N*a + N), Eq(N + a, N*x + N))"],["blackburn-elements-plane-trigonometry-1863/x-d59dc83555",15,"Blackburn 1863, p. 16: In what follows, an angle will be considered as ..."],["method/constructing-a-regular-tetrahedron",8,"constructing a regular tetrahedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-constructing-a-regular-tetrahedron"],["slaught-lennes-solid-geometry-1919/ch-appendix-ii",2,"Slaught & Lennes 1919, ch. Appendix II: PROJECTION OF LINE-SEGMENTS","../books/slaught-lennes-solid-geometry-1919/ch/ch-appendix-ii/index.html"],["concept/corresponding-linear-dimensions",7,"corresponding linear dimensions","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-corresponding-linear-dimensions"],["boyden-first-book-in-algebra-1895/ex-26/30",4,"Boyden 1895, Exercise 26 (30)"],["ball-mathematical-recreations-1905/x-e68a6a66e2",15,"Ball 1905, scan 171: It is unfortunate that no more obvious rule---such, for ..."],["boyden-first-book-in-algebra-1895/ex-26/31",4,"Boyden 1895, Exercise 26 (31)"],["boyden-first-book-in-algebra-1895/x-bc069eab44",15,"Boyden 1895: is indicated by the radical sign and index. When ..."],["cap/cas.series",17,"cas.series"],["todhunter-spherical-trigonometry-1886/ex-iv",3,"Todhunter 1886, Exercise IV"],["quantity/altitude-of-a-spherical-zone",11,"altitude of a spherical zone","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-quantity-altitude-of-a-spherical-zone"],["theorem/volumes-of-similar-tetrahedrons",9,"volumes of similar tetrahedrons","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volumes-of-similar-tetrahedrons"],["hardy-course-of-pure-mathematics-1921/ex-liii/3",4,"Hardy 1921, Exercise LIII (3)"],["boyden-first-book-in-algebra-1895/ex-1/9",4,"Boyden 1895, Exercise 1 (9)"],["ball-mathematical-recreations-1905/x-3d6ea74f75",15,"Ball 1905, scan 159: The majority of the medieval astrologers and physicians were ..."],["hardy-course-of-pure-mathematics-1921/eq-a5accd5bef",16,"Hardy 1921, p. 303: J = \\begin{vmatrix} u_{x} & u_{y} & u_{z}\\\\ v_{x} & v_{y} & v_{z}\\\\ w_{x} & w_{y} & w_{z} \\end{vmatrix} = \\frac{\\dd(u, v"],["thompson-calculus-made-easy-1914/ex-xiii",3,"Thompson 1914, Exercise XIII"],["boyden-first-book-in-algebra-1895/ex-35/10",4,"Boyden 1895, Exercise 35 (10)"],["form/92bda37fbd",5,"solve: (Eq(x/(a + 3), 1/3), Eq((x + 2)/a, 1/2))"],["shape/0b58edb322",6,"solve: (Eq(x/(N + a), N), Eq((N + x)/a, N))"],["planck-treatise-on-thermodynamics-1903/eq-5bc099843f",16,"Planck 1903, p. 49: W = p_{1}V_{1} - p_{2}V_{2},"],["maxwell-elementary-treatise-electricity-1888/eq-184170045b",16,"Maxwell 1888, scan 117: \\text{Electromotive force} = \\text{Current} \\times \\text{Resistance,}"],["ball-mathematical-recreations-1905/x-59133abd89",15,"Ball 1905, scan 207: Thus the cells (x, y) and (9-x, 9-y) are ..."],["concept/truncated-pyramid",7,"truncated pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-truncated-pyramid"],["hardy-course-of-pure-mathematics-1921/eq-715b9fd00e",16,"Hardy 1921, p. 303: abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0"],["concept/oblique-prism",7,"oblique prism","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-oblique-prism"],["whitehead-introduction-to-mathematics-1911/ch-xv",2,"Whitehead 1911, ch. XV: The Differential Calculus","../books/whitehead-introduction-to-mathematics-1911/ch/ch-xv/index.html"],["hardy-course-of-pure-mathematics-1921/eq-abab22f7ee",16,"Hardy 1921, p. 304: \\frac{\\dd(u, v)}{\\dd(x, y)} = \\frac{\\dd(u, v)}{\\dd(\\xi, \\eta)}\\, \\frac{\\dd(\\xi, \\eta)}{\\dd(x, y)}"],["theorem/altitude-of-an-oblique-prism-or-cylinder",9,"altitude of an oblique prism or cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-altitude-of-an-oblique-prism-or-cylinder"],["form/47a211ca07",5,"factor: -64*a**6*b**3*x**3 + 27"],["maxwell-elementary-treatise-electricity-1888/eq-ee127e023d",16,"Maxwell 1888, scan 118: \\text{Heat generated measured in dynamical units}\\\\ = \\text{Square of Current} \\times \\text{Resistance} \\times \\text{Tim"],["shape/05d16d65aa",6,"factor: N*a**N*b**N*x**N + N"],["concept/winning-key-numbers",7,"winning key numbers","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-winning-key-numbers"],["shape/e0d4ee21a8",6,"solve: (Eq((N + x)/(N + a), 1), Eq((a + x)/(N + a), N))"],["hardy-course-of-pure-mathematics-1921/eq-1248d92da4",16,"Hardy 1921, p. 304: f(x) + f(y) = f(xy)"],["de-morgan-elementary-illustrations-calculus-1899/x-044cc972a7",15,"De Morgan 1899, p. 47: The same result may be more simply obtained, by ..."],["hardy-course-of-pure-mathematics-1921/eq-15311498cb",16,"Hardy 1921, p. 304: f(x) + f(y) = f\\left(\\frac{x + y}{1 - xy}\\right)"],["hardy-course-of-pure-mathematics-1921/eq-5cae4be440",16,"Hardy 1921, p. 304: f(x) = \\int_{0}^{x} \\frac{dt}{\\sqrtp{1 - t^{4}}}"],["theorem/dihedral-angle-of-an-oblique-prism",9,"dihedral angle of an oblique prism","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-dihedral-angle-of-an-oblique-prism"],["form/e975e80881",5,"integrate: x**2*cos(x)**2"],["theorem/area-of-the-projection-of-a-plane-segment",9,"area of the projection of a plane-segment","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-area-of-the-projection-of-a-plane-segment"],["shape/5f24ad00fe",6,"integrate: sin(x)**N*cos(N*x)**N"],["shape/d0eefacdd6",6,"integrate: x**N*cos(x)**N"],["concept/circumscribed-prism",7,"circumscribed prism","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-circumscribed-prism"],["quantity/time-constant",11,"time-constant","../books/thompson-calculus-made-easy-1914/terms/index.html#t-quantity-time-constant"],["shape/2b16650e73",6,"identity: N*a**N*c**N*e**N*f**N*(c - d)**N*(N*a + N*b)**N"],["form/0de75cbee3",5,"integrate: cos(x)*cos(2*x)*cos(3*x)"],["dickson-theory-of-equations-1922/eq-7c20e94c81",16,"Dickson 1922, p. 70: \\tfrac{1}{15}f'(x) = x^4 - 5x^2 + 4 = (x^2 - 1)(x^2 - 4)"],["theorem/lateral-area-of-a-regular-pyramid",9,"lateral area of a regular pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-lateral-area-of-a-regular-pyramid"],["planck-treatise-on-thermodynamics-1903/eq-88e72090c5",16,"Planck 1903, p. 52: W = -\\int_{1}^{2} p\\, dV"],["dickson-theory-of-equations-1922/eq-c3a3422621",16,"Dickson 1922, p. 96: d^4 + 9d^3 + 27d^2 + 31d + 6 = 0"],["dickson-theory-of-equations-1922/eq-87ade65e4d",16,"Dickson 1922, p. 90: 6.3r^2 + 11.16196r + 0.000541708 = 0"],["dickson-theory-of-equations-1922/eq-06f2dbb720",16,"Dickson 1922, p. 90: q^3 + 6.3q^2 + 11.23q + 0.061 = 0"],["de-morgan-elementary-illustrations-calculus-1899/x-cf806a9629",15,"De Morgan 1899, p. 115: We have already shown, that when two functions increase ..."],["ball-mathematical-recreations-1905/x-0a01ac4c62",15,"Ball 1905, scan 199: who invented this game---if game is the right term ..."],["dickson-theory-of-equations-1922/eq-9bc7fc81a4",16,"Dickson 1922, p. 96: x - \\sin x = \\tfrac{1}{4} \\pi"],["concept/constant-of-absorption",7,"constant of absorption","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-constant-of-absorption"],["law/newton-s-law-of-cooling",10,"Newton's law of cooling","../books/thompson-calculus-made-easy-1914/terms/index.html#t-law-newton-s-law-of-cooling"],["dickson-theory-of-equations-1922/eq-b09e3bab8b",16,"Dickson 1922, p. 96: \\tfrac{1}{2} r^2(x - \\sin x) = \\tfrac{1}{8} \\pi r^2"],["concept/lever",7,"lever","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-lever"],["dickson-theory-of-equations-1922/eq-711f7fe774",16,"Dickson 1922, p. 97: h = \\frac{-f(a)}{f'(a)} = \\frac{-a + \\sin a + \\tfrac{1}{4} \\pi}{1 - \\cos a}"],["dickson-theory-of-equations-1922/eq-ec8c48a589",16,"Dickson 1922, p. 98: f'(x) = 2 - \\frac{M}{x}"],["theorem/reciprocal-rule-for-derivatives",9,"reciprocal rule for derivatives","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-reciprocal-rule-for-derivatives"],["concept/literal-quadratic-equation",7,"literal quadratic equation","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-literal-quadratic-equation"],["theorem/section-of-a-pyramid-parallel-to-its-base",9,"section of a pyramid parallel to its base","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-section-of-a-pyramid-parallel-to-its-base"],["theorem/limits-of-equal-variables-are-equal",9,"limits of equal variables are equal","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-limits-of-equal-variables-are-equal"],["slaught-lennes-solid-geometry-1919/x-cbda4c3b61",15,"Slaught & Lennes 1919, p. 202: Let a_1, a_2, a_3, \\dotsc be a sequence of ..."],["planck-treatise-on-thermodynamics-1903/x-ceefef0202",15,"Planck 1903, p. 106: Observe, however, that the expressions % [eqn:(64)](64)% for the ..."],["law/electrostatic-attraction-and-repulsion",10,"electrostatic attraction and repulsion","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-law-electrostatic-attraction-and-repulsion"],["form/c3e13cbef4",5,"identity: 96"],["boyden-first-book-in-algebra-1895/ex-1/10",4,"Boyden 1895, Exercise 1 (10)"],["concept/steady-motion",7,"steady motion","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-steady-motion"],["theorem/pyramids-with-equal-altitudes-and-equal-base-areas-have-equal-volumes",9,"pyramids with equal altitudes and equal base areas have equal volumes","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-pyramids-with-equal-altitudes-and-equal-base-areas-have-equal-volumes"],["planck-treatise-on-thermodynamics-1903/eq-02bd55e693",16,"Planck 1903, p. 54: Q = U_{2} - U_{1} + \\int_{1}^{2} p\\, dV"],["hardy-course-of-pure-mathematics-1921/eq-e6d249137d",16,"Hardy 1921, p. 304: f(x) + f(y) = f\\left\\{\\frac{x\\sqrtp{1 - y^{4}} + y\\sqrtp{1 - x^{4}}}{1 + x^{2}y^{2}}\\right\\}"],["form/037c8c5f9d",5,"solve: (Eq(a, x + 2880), Eq(5*x, a))"],["shape/22ab68856d",6,"solve: (Eq(a, N + x), Eq(N*x, a))"],["form/8edca2850e",5,"identity: 16"],["form/6cab9409a6",5,"identity: 36"],["de-morgan-elementary-illustrations-calculus-1899/x-bfdf606489",15,"De Morgan 1899, p. 115: Nevertheless the product \\cos\\theta × \\tan\\theta, of which the ..."],["slaught-lennes-solid-geometry-1919/x-2309e7ddda",15,"Slaught & Lennes 1919, p. 194: The Greeks dealt with the incommensurable cases in an ..."],["concept/three-things-problem",7,"three-things problem","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-three-things-problem"],["form/d1dff33929",5,"solve: (Eq(4*x/(a + 6), 1), Eq(a - x, 15))"],["shape/3c861f149d",6,"solve: (Eq(a, N*b), Eq(c, a + b), Eq(a + b + c, N))"],["theorem/volume-of-a-pyramid",9,"volume of a pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volume-of-a-pyramid"],["shape/5584720264",6,"solve: (Eq(N*x/(N + a), 1), Eq(a - x, N))"],["shape/7f19deb6c0",6,"lcm: (N*x + N*x**N + x**N + 1, N*x + N + a*x + a)"],["person/alexander-macfarlane",1,"Alexander Macfarlane"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-1-9",3,"Macfarlane 1906, Exercise Probs-1-9"],["theorem/volume-of-a-frustum-of-a-pyramid",9,"volume of a frustum of a pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volume-of-a-frustum-of-a-pyramid"],["concept/rational-root",7,"rational root","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-rational-root"],["theorem/volume-of-a-pyramid-lies-between-inscribed-and-circumscribed-prisms",9,"volume of a pyramid lies between inscribed and circumscribed prisms","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volume-of-a-pyramid-lies-between-inscribed-and-circumscribed-prisms"],["hardy-course-of-pure-mathematics-1921/eq-bb8f38cfcf",16,"Hardy 1921, p. 304: f'(x)f'(y)f'(z) \\{f(y) - f(z)\\} \\{f(z) - f(x)\\} \\{f(x) - f(y)\\} = 0"],["boyden-first-book-in-algebra-1895/ex-35/11",4,"Boyden 1895, Exercise 35 (11)"],["dickson-theory-of-equations-1922/ch-i",2,"Dickson 1922, ch. I: Complex Numbers","../books/dickson-theory-of-equations-1922/ch/ch-i/index.html"],["concept/hemisphere",7,"hemisphere","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-hemisphere"],["planck-treatise-on-thermodynamics-1903/eq-c7db2fdfe4",16,"Planck 1903, p. 54: Q = -W"],["planck-treatise-on-thermodynamics-1903/eq-1b72d07003",16,"Planck 1903, p. 54: W = -\\int_{1}^{1} p\\, dV"],["form/a7b9d3bc70",5,"factor: -a**6 + x**3"],["dickson-theory-of-equations-1922/ex-page2",3,"Dickson 1922, Exercise Page2"],["dickson-theory-of-equations-1922/ex-page6",3,"Dickson 1922, Exercise Page6"],["slaught-lennes-solid-geometry-1919/x-a08d3297d2",15,"Slaught & Lennes 1919, p. 80: If a line through the fixed point moves so ..."],["concept/impulse",7,"impulse","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-impulse"],["hardy-course-of-pure-mathematics-1921/ex-liii/4",4,"Hardy 1921, Exercise LIII (4)"],["thompson-calculus-made-easy-1914/x-f56b9be03a",15,"Thompson 1914, p. 135: It is easy to see that if the value ..."],["hardy-course-of-pure-mathematics-1921/ex-liii/5",4,"Hardy 1921, Exercise LIII (5)"],["dickson-theory-of-equations-1922/ex-page10",3,"Dickson 1922, Exercise Page10"],["dickson-theory-of-equations-1922/ex-page9",3,"Dickson 1922, Exercise Page9"],["concept/spin-of-a-ball",7,"spin of a ball","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-spin-of-a-ball"],["planck-treatise-on-thermodynamics-1903/x-2548d13012",15,"Planck 1903, p. 107: In this case the cyclic process results in the ..."],["maxwell-elementary-treatise-electricity-1888/eq-d876ae1035",16,"Maxwell 1888, scan 178: p^2 = \\xp\\dfrac {Q}{B}"],["method/completing-the-square",8,"completing the square","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-method-completing-the-square"],["theorem/square-of-a-difference",9,"square of a difference","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-theorem-square-of-a-difference"],["maxwell-elementary-treatise-electricity-1888/eq-7034dcff4a",16,"Maxwell 1888, scan 178: q^2 = \\xp\\dfrac {Q'}{A}"],["thompson-calculus-made-easy-1914/ch-xi",2,"Thompson 1914, ch. XI: Maxima and Minima","../books/thompson-calculus-made-easy-1914/ch/ch-xi/index.html"],["maxwell-elementary-treatise-electricity-1888/eq-36acb3d76e",16,"Maxwell 1888, scan 178: U_{n + 1} &= U_n - \\frac {Q'}{A} V_n"],["thompson-calculus-made-easy-1914/x-82aaddcf54",15,"Thompson 1914, p. 139: To this mysterious number 2.7182818 etc., the mathematicians have ..."],["boyden-first-book-in-algebra-1895/ex-1/11",4,"Boyden 1895, Exercise 1 (11)"],["slaught-lennes-solid-geometry-1919/x-48340aac51",15,"Slaught & Lennes 1919, p. 82: The lateral area of a regular pyramid is equal ..."],["thompson-calculus-made-easy-1914/eq-97d11fd666",16,"Thompson 1914, p. 93: y = x^2 - 4x + 7."],["thompson-calculus-made-easy-1914/eq-33a88a2752",16,"Thompson 1914, p. 94: y = 3x - x^2"],["thompson-calculus-made-easy-1914/eq-ee70a11391",16,"Thompson 1914, p. 96: \\dfrac{dy}{dx} = 2x - 4"],["concept/method-equating-the-derivative-to-zero",7,"method: equating the derivative to zero"],["thompson-calculus-made-easy-1914/eq-8502897ee8",16,"Thompson 1914, p. 96: 2x - 4 = 0"],["form/44059bc2ec",5,"solve: (Eq(x, 3*a), Eq(x, a + 48))"],["thompson-calculus-made-easy-1914/eq-5ac14c66fc",16,"Thompson 1914, p. 97: \\frac{dy}{dx} = 0"],["shape/271edfdbd3",6,"solve: (Eq(x, N*a), Eq(x, N + a))"],["thompson-calculus-made-easy-1914/eq-0b5060d776",16,"Thompson 1914, p. 98: y = 4x + \\frac{1}{x}."],["thompson-calculus-made-easy-1914/eq-4bf18f4852",16,"Thompson 1914, p. 99: y = nx - x^2"],["thompson-calculus-made-easy-1914/eq-5ecac923e2",16,"Thompson 1914, p. 99: \\dfrac{n}{2} = x"],["thompson-calculus-made-easy-1914/eq-f06c657a66",16,"Thompson 1914, p. 101: x^2 - 4x +3 = 0"],["thompson-calculus-made-easy-1914/eq-ae47a5c558",16,"Thompson 1914, p. 101: y =\\tfrac{1}{3} x^3 - 2x^2 + 3x + 1."],["thompson-calculus-made-easy-1914/eq-ae1cabf995",16,"Thompson 1914, p. 102: (y-b)^2 + (x-a)^2 = r^2."],["thompson-calculus-made-easy-1914/ch-xv",2,"Thompson 1914, ch. XV: How to deal with Sines and Cosines","../books/thompson-calculus-made-easy-1914/ch/ch-xv/index.html"],["thompson-calculus-made-easy-1914/eq-2e6a185a09",16,"Thompson 1914, p. 102: y = \\sqrt{r^2-(x-a)^2} + b."],["hardy-course-of-pure-mathematics-1921/ex-liii/6",4,"Hardy 1921, Exercise LIII (6)"],["slaught-lennes-solid-geometry-1919/x-5133043da9",15,"Slaught & Lennes 1919, p. 88: The volume of any pyramid is one third of ..."],["boyden-first-book-in-algebra-1895/ex-1/12",4,"Boyden 1895, Exercise 1 (12)"],["form/65aacf5c9d",5,"identity: -20*x"],["thompson-calculus-made-easy-1914/eq-a28f2dc210",16,"Thompson 1914, p. 103: \\frac{a-x}{\\sqrt{r^2-(x-a)^2}} = 0."],["slaught-lennes-solid-geometry-1919/x-07e3683e62",15,"Slaught & Lennes 1919, p. 85: A pyramid has a definite volume which is less ..."],["de-morgan-elementary-illustrations-calculus-1899/x-c5547d9bbe",15,"De Morgan 1899, p. 117: But though the two sums increase without limit when ..."],["slaught-lennes-solid-geometry-1919/x-085bb9303a",15,"Slaught & Lennes 1919, p. 89: The volume of a frustum of a pyramid is ..."],["form/906e4d92ee",5,"solve: (Eq(a, 6*x), Eq(a - x, 250))"],["maxwell-elementary-treatise-electricity-1888/x-a4280ba3e0",15,"Maxwell 1888, scan 23: No two different equipotential surfaces can cut one another, ..."],["thompson-calculus-made-easy-1914/eq-9f9aa5bb6b",16,"Thompson 1914, p. 104: 3ax^2 + b = 0"],["thompson-calculus-made-easy-1914/eq-1d793625ae",16,"Thompson 1914, p. 104: \\text{the other side} = \\sqrt{(\\text{diagonal})^2 - x^2}"],["form/c63916bbce",5,"integrate: (a + c*cos(x) + e*sin(x))/(b + d*cos(x) + f*sin(x))"],["hardy-course-of-pure-mathematics-1921/ex-xliv",3,"Hardy 1921, Exercise XLIV"],["shape/c63916bbce",6,"integrate: (a + c*cos(x) + e*sin(x))/(b + d*cos(x) + f*sin(x))"],["thompson-calculus-made-easy-1914/eq-717a946266",16,"Thompson 1914, p. 104: S = x\\sqrt{4R^2 - x^2}"],["thompson-calculus-made-easy-1914/x-cc3aa38f2e",15,"Thompson 1914, p. 136: But this mode of reckoning compound interest once a ..."],["thompson-calculus-made-easy-1914/eq-b65b6703b3",16,"Thompson 1914, p. 105: 4R^2 - 2x^2 = 0"],["thompson-calculus-made-easy-1914/eq-62732c6249",16,"Thompson 1914, p. 105: x = R\\sqrt{2}"],["thompson-calculus-made-easy-1914/eq-500aed3e51",16,"Thompson 1914, p. 105: H = \\sqrt{l^2 - R^2}"],["thompson-calculus-made-easy-1914/eq-607332e8f8",16,"Thompson 1914, p. 105: V = \\pi R^2 × \\dfrac{H}{3} = \\pi R^2 × \\dfrac{\\sqrt{l^2 - R^2}}{3}"],["thompson-calculus-made-easy-1914/eq-26b00246e4",16,"Thompson 1914, p. 105: 2\\pi R(l^2 - R^2) - \\pi R^2 = 0"],["dickson-theory-of-equations-1922/eq-f8a5d1b4f4",16,"Dickson 1922, p. 98: \\log x = M \\log_e x"],["form/5a46546f34",5,"solve: (Eq(a, b + 10*c), Eq(x, 10*b + c), Eq(-a + x, 18), Eq(a + x, 132))"],["thompson-calculus-made-easy-1914/eq-4216dad603",16,"Thompson 1914, p. 105: R = l\\sqrt{\\tfrac{2}{3}}"],["thompson-calculus-made-easy-1914/eq-806df7957f",16,"Thompson 1914, p. 106: y = \\dfrac{x}{4-x} + \\dfrac{4-x}{x}"],["thompson-calculus-made-easy-1914/eq-c11c740a30",16,"Thompson 1914, p. 106: \\dfrac{4}{(4-x)^2} - \\dfrac{4}{x^2} = 0"],["thompson-calculus-made-easy-1914/eq-3f1951f80c",16,"Thompson 1914, p. 106: y = \\sqrt{1+x} + \\sqrt{1-x}"],["slaught-lennes-solid-geometry-1919/x-450ae84868",15,"Slaught & Lennes 1919, p. 85: This process may be repeated by doubling the number ..."],["maxwell-elementary-treatise-electricity-1888/x-fac737735c",15,"Maxwell 1888, scan 36: The total electrification or charge of a body or ..."],["boyden-first-book-in-algebra-1895/ex-31/2",4,"Boyden 1895, Exercise 31 (2)"],["thompson-calculus-made-easy-1914/eq-15deb18216",16,"Thompson 1914, p. 106: \\dfrac{dy}{dx} = \\dfrac{1}{2\\sqrt{1+x}} - \\dfrac{1}{2\\sqrt{1-x}} = 0"],["thompson-calculus-made-easy-1914/eq-800ec7f740",16,"Thompson 1914, p. 106: \\sqrt{1+x} = \\sqrt{1-x}"],["shape/4c2a19890b",6,"solve: (Eq(a, N*c + b), Eq(x, N*b + c), Eq(-a + x, N), Eq(a + x, N))"],["thompson-calculus-made-easy-1914/eq-4ebf478238",16,"Thompson 1914, p. 107: y = \\dfrac{x^2-5}{2x-4}"],["hardy-course-of-pure-mathematics-1921/ex-liii/7",4,"Hardy 1921, Exercise LIII (7)"],["slaught-lennes-solid-geometry-1919/x-7b9e0bdb4e",15,"Slaught & Lennes 1919, p. 81: In this case every face is a triangle, and ..."],["blackburn-elements-plane-trigonometry-1863/x-3caf928c41",15,"Blackburn 1863, p. 50: The four expressions for \\sin (\\theta ± \\phi) and ..."],["slaught-lennes-solid-geometry-1919/x-3641077eee",15,"Slaught & Lennes 1919, p. 94: by rotating a right triangle PMB about one of ..."],["dickson-theory-of-equations-1922/eq-9f4b8e5344",16,"Dickson 1922, p. 86: x^3 - 2x - 5 = 0"],["dickson-theory-of-equations-1922/eq-a06b690dff",16,"Dickson 1922, p. 99: f(x) - f''(x) \\frac{y^2}{1·2} + f''''(x) \\frac{y^4}{1·2·3·4} - \\dotsb = 0"],["form/b36ba21ac0",5,"integrate: 1/(a*cos(x)**2 + 2*b*sin(x)*cos(x) + c*sin(x)**2)"],["boyden-first-book-in-algebra-1895/ch-reduction-of-fractions",2,"Boyden 1895, REDUCTION OF FRACTIONS","../books/boyden-first-book-in-algebra-1895/ch/ch-reduction-of-fractions/index.html"],["form/dd21659c52",5,"identity: 35*a*b"],["shape/8261ded54c",6,"integrate: 1/(N*b*sin(x)*cos(x) + a*cos(x)**N + c*sin(x)**N)"],["form/146f81d4f5",5,"identity: 2*x"],["wentworth-first-steps-in-algebra-1894/eq-53fd2dbd78",16,"Wentworth 1894, p. 8: 10 + (3 + 2) = 10 + 3 + 2"],["hardy-course-of-pure-mathematics-1921/ex-xlv",3,"Hardy 1921, Exercise XLV"],["theorem/difference-of-two-sines",9,"difference of two sines","../books/thompson-calculus-made-easy-1914/terms/index.html#t-theorem-difference-of-two-sines"],["wentworth-first-steps-in-algebra-1894/x-0811c5cd9f",15,"Wentworth 1894, p. 132: An equation which contains the square of the unknown ..."],["blackburn-elements-plane-trigonometry-1863/x-a829afd06a",15,"Blackburn 1863, p. 49: Then \\theta + \\phi is the circular measure of ..."],["ball-mathematical-recreations-1905/x-eea472b2ec",15,"Ball 1905, scan 172: For simplicity I shall apply this method to construct ..."],["ball-mathematical-recreations-1905/x-73112b87b1",15,"Ball 1905, scan 171: Following the analogy of the notation used above, two ..."],["thompson-calculus-made-easy-1914/eq-dbb06339ed",16,"Thompson 1914, p. 107: x^2 - 4x + 5 = 0"],["thompson-calculus-made-easy-1914/eq-402756a770",16,"Thompson 1914, p. 107: x = \\tfrac{5}{2} ± \\sqrt{-1}"],["thompson-calculus-made-easy-1914/eq-86a9dc0637",16,"Thompson 1914, p. 201: \\int x^{-1}\\, dx &&= \\log_\\epsilon x + C."],["thompson-calculus-made-easy-1914/eq-deff2dd9c7",16,"Thompson 1914, p. 201: \\int \\epsilon^x\\, dx &&= \\epsilon ^x + C."],["de-morgan-elementary-illustrations-calculus-1899/x-8450615d69",15,"De Morgan 1899, p. 116: If we take any numbers, such as 1 and ..."],["blackburn-elements-plane-trigonometry-1863/x-44e87e3591",15,"Blackburn 1863, p. 52: So that when, as in the text, R - ..."],["boyden-first-book-in-algebra-1895/ex-26/32",4,"Boyden 1895, Exercise 26 (32)"],["method/resolution-of-a-vector",8,"resolution of a vector","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-method-resolution-of-a-vector"],["blackburn-elements-plane-trigonometry-1863/x-3ffa55dd66",15,"Blackburn 1863, p. 50: Then \\theta - \\phi = circular measure of ACD."],["blackburn-elements-plane-trigonometry-1863/x-6d26e2b2db",15,"Blackburn 1863, p. 51: To shew that, when the difference between R and ..."],["wentworth-first-steps-in-algebra-1894/eq-4548443415",16,"Wentworth 1894, p. 8: 10 + (3 - 2) = 10 + 3 - 2"],["wentworth-first-steps-in-algebra-1894/eq-4df56ad211",16,"Wentworth 1894, p. 9: 10 - (3 + 2) = 10 - 3 - 2"],["thompson-calculus-made-easy-1914/eq-77179615e9",16,"Thompson 1914, p. 107: (y-x^2)^2 = x^5"],["thompson-calculus-made-easy-1914/eq-6d5b7a3bcb",16,"Thompson 1914, p. 107: y = x^2 ± x^{\\efrac{5}{2}}"],["theorem/euclid-euler-theorem",9,"Euclid–Euler theorem","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-euclid-euler-theorem"],["thompson-calculus-made-easy-1914/eq-02ea1873db",16,"Thompson 1914, p. 107: \\dfrac{dy}{dx} = 2x ± \\tfrac{5}{2} x^{\\efrac{3}{2}} = 0"],["thompson-calculus-made-easy-1914/eq-0afbffc3e0",16,"Thompson 1914, p. 201: \\int \\sin x\\, dx &&= -\\cos x + C."],["theorem/fermat-s-theorem-on-sums-of-two-squares",9,"Fermat's theorem on sums of two squares","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-fermat-s-theorem-on-sums-of-two-squares"],["thompson-calculus-made-easy-1914/eq-2f86321752",16,"Thompson 1914, p. 107: 2 ± \\tfrac{5}{2} x^{\\efrac{1}{2}} = 0"],["form/2eabe266f3",5,"identity: -a**3*b**3*c**3"],["thompson-calculus-made-easy-1914/eq-3616554166",16,"Thompson 1914, p. 107: x = \\tfrac{16}{25}"],["thompson-calculus-made-easy-1914/eq-1297fb846b",16,"Thompson 1914, p. 201: \\int \\cos x\\, dx &&= \\sin x + C."],["boyden-first-book-in-algebra-1895/ex-26/33",4,"Boyden 1895, Exercise 26 (33)"],["shape/b626787738",6,"identity: -a**N*b**N*c**N"],["form/a5c15a4591",5,"factor: 64*a*b + 16"],["method/composition-of-successive-vectors",8,"composition of successive vectors","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-method-composition-of-successive-vectors"],["thompson-calculus-made-easy-1914/x-89ce79b44c",15,"Thompson 1914, p. 139: This process of growing proportionately, at every instant, to ..."],["hardy-course-of-pure-mathematics-1921/ex-liv/2",4,"Hardy 1921, Exercise LIV (2)"],["shape/7f53ad7d12",6,"factor: N*a*b + N"],["concept/motion-in-fluids",7,"motion in fluids","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-motion-in-fluids"],["concept/flight-of-birds",7,"flight of birds","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-flight-of-birds"],["thompson-calculus-made-easy-1914/x-f70c47ebb2",15,"Thompson 1914, p. 53: What do we mean by rate? In both these ..."],["thompson-calculus-made-easy-1914/eq-725d6b5fbb",16,"Thompson 1914, p. 109: S = 2(\\pi r^2)+ 2 \\pi r × 2r = 6 \\pi r^2"],["quantity/direction",11,"direction","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-quantity-direction"],["boyden-first-book-in-algebra-1895/ch-operations-upon-fractions",2,"Boyden 1895, OPERATIONS UPON FRACTIONS","../books/boyden-first-book-in-algebra-1895/ch/ch-operations-upon-fractions/index.html"],["boyden-first-book-in-algebra-1895/ex-42",3,"Boyden 1895, Exercise 42"],["boyden-first-book-in-algebra-1895/ex-44",3,"Boyden 1895, Exercise 44"],["wentworth-first-steps-in-algebra-1894/ex-16/6",4,"Wentworth 1894, Exercise 16 (6)"],["thompson-calculus-made-easy-1914/x-5da336fc27",15,"Thompson 1914, p. 53: Ten yards is not the same as 600 yards, ..."],["thompson-calculus-made-easy-1914/x-b94f47f2f7",15,"Thompson 1914, p. 143: The great reason why \\epsilon is regarded of importance ..."],["thompson-calculus-made-easy-1914/x-cde1e65486",15,"Thompson 1914, p. 58: He did not use the notation of the dy ..."],["boyden-first-book-in-algebra-1895/ex-26/34",4,"Boyden 1895, Exercise 26 (34)"],["thompson-calculus-made-easy-1914/x-3b1468536a",15,"Thompson 1914, p. 56: The force necessary to accelerate a mass is proportional ..."],["form/4dbdabdaca",5,"identity: x/5"],["todhunter-spherical-trigonometry-1886/ex-vi",3,"Todhunter 1886, Exercise VI"],["cap/other:geometric_proof",17,"other:geometric_proof"],["thompson-calculus-made-easy-1914/x-c21583f667",15,"Thompson 1914, p. 57: Again, if a force is employed to move something ..."],["thompson-calculus-made-easy-1914/x-5ea5798ed9",15,"Thompson 1914, p. 157: In fact \\epsilon^{-at} serves as a die-away factor for ..."],["thompson-calculus-made-easy-1914/x-6f7a7f37d8",15,"Thompson 1914, p. 145: For the benefit of those who have no tutor ..."],["theorem/lagrange-s-theorem-on-primes",9,"Lagrange's theorem on primes","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-lagrange-s-theorem-on-primes"],["ball-mathematical-recreations-1905/x-f57ba0cde5",15,"Ball 1905, scan 113: The chief cause for this result seems to be ..."],["shape/534dcd288c",6,"factor: a*b + a**N"],["form/2420ad0115",5,"identity: (a**2*b*x - b**3*x**3)/(a*x**2 - b*x**3)"],["thompson-calculus-made-easy-1914/eq-5656f64a41",16,"Thompson 1914, p. 109: V = \\pi r^2 × 2r=2 \\pi r^3"],["thompson-calculus-made-easy-1914/eq-93ef987bb2",16,"Thompson 1914, p. 109: \\frac{dS}{dr} = 12\\pi r"],["wentworth-first-steps-in-algebra-1894/x-d81f756db5",15,"Wentworth 1894, p. 133: The square root of any number is positive or ..."],["thompson-calculus-made-easy-1914/eq-876bb3bc96",16,"Thompson 1914, p. 109: \\frac{dV}{dr}=6 \\pi r^2"],["de-morgan-elementary-illustrations-calculus-1899/x-7d691f8d65",15,"De Morgan 1899, p. 120: Let x have the successive values a, a + ..."],["wentworth-first-steps-in-algebra-1894/x-bcc79a35fd",15,"Wentworth 1894, p. 135: This third term is the square of half the ..."],["thompson-calculus-made-easy-1914/x-9062ffedfa",15,"Thompson 1914, p. 183: The simple reason is that there are a vast ..."],["boyden-first-book-in-algebra-1895/ex-37/23",4,"Boyden 1895, Exercise 37 (23)"],["maxwell-elementary-treatise-electricity-1888/ch-vii",2,"Maxwell 1888, ch. VII: THEORY OF ELECTRICAL IMAGES","../books/maxwell-elementary-treatise-electricity-1888/ch/ch-vii/index.html"],["thompson-calculus-made-easy-1914/ch-xvi",2,"Thompson 1914, ch. XVI: Partial Differentiation","../books/thompson-calculus-made-easy-1914/ch/ch-xvi/index.html"],["whitehead-introduction-to-mathematics-1911/ch-vi",2,"Whitehead 1911, ch. VI: Generalizations of Number","../books/whitehead-introduction-to-mathematics-1911/ch/ch-vi/index.html"],["concept/inverse-points-with-respect-to-a-sphere",7,"inverse points with respect to a sphere","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-inverse-points-with-respect-to-a-sphere"],["boyden-first-book-in-algebra-1895/ex-1/13",4,"Boyden 1895, Exercise 1 (13)"],["concept/centre-of-symmetry",7,"centre of symmetry","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-centre-of-symmetry"],["whitehead-introduction-to-mathematics-1911/ch-i",2,"Whitehead 1911, ch. I: The Abstract Nature of Mathematics","../books/whitehead-introduction-to-mathematics-1911/ch/ch-i/index.html"],["person/alfred-north-whitehead",1,"Alfred North Whitehead"],["wentworth-first-steps-in-algebra-1894/x-3e85a54fb8",15,"Wentworth 1894, p. 136: Since the square root of a negative number cannot ..."],["ball-mathematical-recreations-1905/x-54ef770ccc",15,"Ball 1905, scan 171: we begin by constructing two subsidiary squares, one of ..."],["wentworth-first-steps-in-algebra-1894/x-eb37734410",15,"Wentworth 1894, p. 133: The square root of -5 differs from the square ..."],["wentworth-first-steps-in-algebra-1894/eq-37c8da6320",16,"Wentworth 1894, p. 9: 10 &- (5 - 2) = 10 - 5 + 2"],["wentworth-first-steps-in-algebra-1894/eq-147f653b0d",16,"Wentworth 1894, p. 5: 4a &= a + a + a + a"],["theorem/potential-within-a-closed-equipotential-surface-is-constant",9,"potential within a closed equipotential surface is constant","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-potential-within-a-closed-equipotential-surface-is-constant"],["wentworth-first-steps-in-algebra-1894/x-2b42154605",15,"Wentworth 1894, p. 138: The reason that every root of the equation will ..."],["de-morgan-elementary-illustrations-calculus-1899/x-c86d97de85",15,"De Morgan 1899, p. 121: That is, the integral of \\phi x\\, dx between ..."],["theorem/central-angles-have-the-same-ratio-as-their-intercepted-arcs",9,"central angles have the same ratio as their intercepted arcs","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-central-angles-have-the-same-ratio-as-their-intercepted-arcs"],["theorem/taylor-s-series",9,"Taylor's series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-taylor-s-series"],["thompson-calculus-made-easy-1914/ex-i/5",4,"Thompson 1914, Exercise I (5)"],["ball-mathematical-recreations-1905/x-34bdea7dce",15,"Ball 1905, scan 171: I do not know to whom the modification is ..."],["form/80738ea893",5,"identity: 6*a**2*x*(-a**2*x**2/3 - a*x**3/5 + 5*x**4/6)/5"],["thompson-calculus-made-easy-1914/eq-a38cbd1334",16,"Thompson 1914, p. 201: \\int\\log_\\epsilon x\\, dx &&= x(\\log_\\epsilon x - 1) + C"],["de-morgan-elementary-illustrations-calculus-1899/x-fb07faea49",15,"De Morgan 1899, p. 122: which is said to be the integral of \\phi ..."],["thompson-calculus-made-easy-1914/x-9c0c0e7930",15,"Thompson 1914, p. 177: The first is obtained by supposing y constant, the ..."],["form/771495e761",5,"factor: -3*a**3*b**3*c + 6*a**2*b**2 - 9*a*b**3*c + 3*a*b*c**2"],["shape/fcea689cca",6,"identity: N*a**N*x*(N*a*x**N + N*a**N*x**N + N*x**N)"],["shape/c0bea06ee0",6,"factor: N*a*b*c**N + N*a*b**N*c + N*a**N*b**N*c + N*a**N*b**N"],["boyden-first-book-in-algebra-1895/ex-19/6",4,"Boyden 1895, Exercise 19 (6)"],["wentworth-first-steps-in-algebra-1894/eq-e77a2e57b5",16,"Wentworth 1894, p. 5: a^{4} &= a× a× a× a"],["thompson-calculus-made-easy-1914/eq-9328efa062",16,"Thompson 1914, p. 202: \\int a^x\\, dx &&= \\dfrac{a^x}{\\log_\\epsilon a} + C."],["shape/9288275c4f",6,"identity: (a**N*b*x - b**N*x**N)/(a*x**N - b*x**N)"],["theorem/inscribed-angle-is-measured-by-half-its-intercepted-arc",9,"inscribed angle is measured by half its intercepted arc","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-inscribed-angle-is-measured-by-half-its-intercepted-arc"],["thompson-calculus-made-easy-1914/x-f8bfe8ac67",15,"Thompson 1914, p. 184: If we want to go so far that not ..."],["hardy-course-of-pure-mathematics-1921/eq-42db0aa22a",16,"Hardy 1921, p. 305: \\int_{x_{0}}^{x_{1}} \\frac{dx}{ax^{2} + 2bx + c} = \\frac{1}{\\sqrtp{ac - b^{2}}} \\arctan\\left\\{ \\frac{(x_{1} - x_{0}) \\sq"],["de-morgan-elementary-illustrations-calculus-1899/x-ba4ecf9f65",15,"De Morgan 1899, p. 121: which is the limit arising from supposing x to ..."],["de-morgan-elementary-illustrations-calculus-1899/x-bb2b1399be",15,"De Morgan 1899, p. 121: It is evident that this series bears a great ..."],["wentworth-first-steps-in-algebra-1894/ex-16/7",4,"Wentworth 1894, Exercise 16 (7)"],["person/ahmes",1,"Ahmes","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-ahmes"],["thompson-calculus-made-easy-1914/x-e063224f9e",15,"Thompson 1914, p. 185: For we have seen that differentiating a curve means ..."],["theorem/angle-inscribed-in-a-semicircle-is-a-right-angle",9,"angle inscribed in a semicircle is a right angle","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-angle-inscribed-in-a-semicircle-is-a-right-angle"],["theorem/angle-formed-by-a-tangent-and-a-chord-is-measured-by-half-the-intercepted-arc",9,"angle formed by a tangent and a chord is measured by half the intercepted arc","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-angle-formed-by-a-tangent-and-a-chord-is-measured-by-half-the-intercepted-arc"],["thompson-calculus-made-easy-1914/ch-xvii",2,"Thompson 1914, ch. XVII: Integration","../books/thompson-calculus-made-easy-1914/ch/ch-xvii/index.html"],["theorem/angle-formed-by-two-intersecting-chords-is-measured-by-half-the-sum-of-the-intercepted-arcs",9,"angle formed by two intersecting chords is measured by half the sum of the intercepted arcs","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-angle-formed-by-two-intersecting-chords-is-measured-by-half-the-sum-of-the-intercepted-arcs"],["wentworth-first-steps-in-algebra-1894/ex-16/9",4,"Wentworth 1894, Exercise 16 (9)"],["boyden-first-book-in-algebra-1895/ex-51",3,"Boyden 1895, Exercise 51"],["shape/ff30c2c98d",6,"identity: (N*a**N*x**N + a**N + x**N)**2"],["form/b4a08f8f8a",5,"differentiate: x**(1/3)"],["thompson-calculus-made-easy-1914/eq-0239b587f4",16,"Thompson 1914, p. 49: y = x^5"],["form/57d93277ce",5,"identity: (b/4 - 3*c/(5*a))*(b/4 + 3*c/(5*a))*(b**2/16 + 9*c**2/(25*a))"],["boyden-first-book-in-algebra-1895/ex-52",3,"Boyden 1895, Exercise 52"],["shape/8aaa677352",6,"identity: (N*b + N*c/a)**2*(N*b**N + N*c**N/a)"],["thompson-calculus-made-easy-1914/ex-iv/1b",4,"Thompson 1914, Exercise IV (1b)"],["form/db2a1a17ed",5,"identity: 12*a*b"],["whitehead-introduction-to-mathematics-1911/x-1cc2c688fc",15,"Whitehead 1911, p. 82: Any limitation whatsoever upon the generality of theorems, or ..."],["concept/uniformly-accelerated-motion",7,"uniformly accelerated motion","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-uniformly-accelerated-motion"],["theorem/constant-multiple-rule-for-integration",9,"constant multiple rule for integration","../books/thompson-calculus-made-easy-1914/terms/index.html#t-theorem-constant-multiple-rule-for-integration"],["maxwell-elementary-treatise-electricity-1888/x-9584a2e9d6",15,"Maxwell 1888, scan 77: The boundary between these two regions forms what is ..."],["hardy-course-of-pure-mathematics-1921/ex-liv/3",4,"Hardy 1921, Exercise LIV (3)"],["form/1aeb522bc3",5,"evaluate: 12/35"],["boyden-first-book-in-algebra-1895/ex-53",3,"Boyden 1895, Exercise 53"],["blackburn-elements-plane-trigonometry-1863/eq-48c221fe6c",16,"Blackburn 1863, p. 33: \\log m + \\log n = \\log (m × n)"],["boyden-first-book-in-algebra-1895/ex-54",3,"Boyden 1895, Exercise 54"],["boyden-first-book-in-algebra-1895/ch-equations",2,"Boyden 1895, EQUATIONS","../books/boyden-first-book-in-algebra-1895/ch/ch-equations/index.html"],["thompson-calculus-made-easy-1914/x-854570e8a6",15,"Thompson 1914, p. 196: If a stranger were set down in Trafalgar Square, ..."],["thompson-calculus-made-easy-1914/x-68e963178a",15,"Thompson 1914, p. 196: So, when we work backwards, integrating, the integration will ..."],["theorem/concurrence-of-arcs-through-a-point-of-a-spherical-triangle",9,"concurrence of arcs through a point of a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-concurrence-of-arcs-through-a-point-of-a-spherical-triangle"],["de-morgan-elementary-illustrations-calculus-1899/x-8fdd73711d",15,"De Morgan 1899, p. 123: Thus, since x^{2}, when differentiated, gives 2x, x^{2} is ..."],["theorem/concurrence-of-altitudes-of-a-spherical-triangle",9,"concurrence of altitudes of a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-concurrence-of-altitudes-of-a-spherical-triangle"],["hardy-course-of-pure-mathematics-1921/eq-4de17d44dc",16,"Hardy 1921, p. 305: \\int_{-1}^{1} \\frac{\\sin\\alpha\\, dx}{1 - 2x\\cos\\alpha + x^{2}}"],["method/principle-of-continuity",8,"principle of continuity","../books/wentworth-plane-geometry-1899/terms/index.html#t-method-principle-of-continuity"],["shape/37badf85ae",6,"evaluate: N*b + N*c**N - a"],["form/b2658e75c3",5,"differentiate2: 12*x**2 + 17*x"],["hardy-course-of-pure-mathematics-1921/eq-3b87e537b5",16,"Hardy 1921, p. 305: \\int_{x_{0}}^{x_{1}} \\frac{dx}{y} = \\frac{1}{\\sqrt{a}} \\log \\frac{1 + X\\sqrt{a}}{1 - X\\sqrt{a}}"],["de-morgan-elementary-illustrations-calculus-1899/x-d1488d3b6e",15,"De Morgan 1899, p. 123: the sum of an infinite number of infinitely small ..."],["macfarlane-vector-analysis-quaternions-1906/x-9deb82ee5c",15,"Macfarlane 1906: By a “vector” is meant a quantity which has ..."],["planck-treatise-on-thermodynamics-1903/eq-d2339480c5",16,"Planck 1903, p. 55: Q = dU + p\\, dV"],["concept/convection-current",7,"convection current","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-convection-current"],["theorem/arc-through-midpoints-of-two-sides-of-a-spherical-triangle",9,"arc through midpoints of two sides of a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-arc-through-midpoints-of-two-sides-of-a-spherical-triangle"],["hardy-course-of-pure-mathematics-1921/eq-9b886f3a07",16,"Hardy 1921, p. 305: \\int_{0}^{a} \\frac{dx}{x + \\sqrtp{a^{2} - x^{2}}} = \\tfrac{1}{4}\\pi"],["form/402bc25b18",5,"evaluate: 5*a**2*b**2 - 2*a**2*b + 3*a*b*c at a=-2, b=3, c=-1"],["planck-treatise-on-thermodynamics-1903/eq-8a9cfc9dca",16,"Planck 1903, p. 51: U_{2} - U_{1} = W + Q"],["wentworth-first-steps-in-algebra-1894/ex-1/1",4,"Wentworth 1894, Exercise 1 (1)"],["shape/5ea0c4c6e4",6,"evaluate: N*a*b*c + N*a**N*b + N*a**N*b**N"],["shape/e4eba44b50",6,"differentiate2: N*x + N*x**N"],["de-morgan-elementary-illustrations-calculus-1899/x-0d65fa3132",15,"De Morgan 1899, p. 124: so that mC, or the sum of all the ..."],["shape/7b83515a56",6,"evaluate: x**N/(x + 1)"],["concept/zeno-s-paradoxes",7,"Zeno's paradoxes","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-zeno-s-paradoxes"],["concept/laws-of-indices",7,"laws of indices","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-laws-of-indices"],["hardy-course-of-pure-mathematics-1921/eq-101a259fe1",16,"Hardy 1921, p. 305: \\int_{-1}^{1} \\frac{\\sqrtp{1 - x^{2}}}{a - x}\\, dx = \\pi\\{a - \\sqrtp{a^{2} - 1}\\}"],["hardy-course-of-pure-mathematics-1921/eq-a4b3499791",16,"Hardy 1921, p. 306: \\int_{0}^{1} \\frac{dx}{\\sqrtbr{\\{1 + (p^{2} - 1)x\\}\\{1 - (1 - q^{2}) x\\}}} = \\frac{2\\omega}{(p + q)\\sin\\omega}"],["cap/cas.solve.poly",17,"cas.solve.poly"],["hardy-course-of-pure-mathematics-1921/ex-liv/4",4,"Hardy 1921, Exercise LIV (4)"],["method/solving-a-triangle-from-its-three-angles",8,"solving a triangle from its three angles","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-solving-a-triangle-from-its-three-angles"],["thompson-calculus-made-easy-1914/ch-xviii",2,"Thompson 1914, ch. XVIII: Integrating as the Reverse of Differentiating","../books/thompson-calculus-made-easy-1914/ch/ch-xviii/index.html"],["macfarlane-vector-analysis-quaternions-1906/x-6b1ff61cfd",15,"Macfarlane 1906: Though a vector is represented by a line, its ..."],["concept/concave-polygon",7,"concave polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-concave-polygon"],["todhunter-spherical-trigonometry-1886/x-24696586dc",15,"Todhunter 1886, scan 148: If three arcs be drawn from the angles of ..."],["macfarlane-vector-analysis-quaternions-1906/ex-probs-10-18",3,"Macfarlane 1906, Exercise Probs-10-18"],["hardy-course-of-pure-mathematics-1921/eq-56a9fb91cb",16,"Hardy 1921, p. 306: \\int_{0}^{2\\pi} \\frac{\\sin^{2}\\theta\\, d\\theta}{a - b\\cos\\theta} = \\frac{2\\pi}{b^{2}} \\{a - \\sqrtp{a^{2} - b^{2}}\\}"],["form/7928a524af",5,"solve: Eq(5*x, 60)"],["macfarlane-vector-analysis-quaternions-1906/eq-6af511dc90",16,"Macfarlane 1906: \\sum m_A = \\sum m_R + \\sum\\Bigl\\{m(A - R)\\Bigr\\}"],["maxwell-elementary-treatise-electricity-1888/x-deb06d63a6",15,"Maxwell 1888, scan 80: or the electric force close to the surface of ..."],["macfarlane-vector-analysis-quaternions-1906/ex-probs-10-18/10",4,"Macfarlane 1906, Exercise Probs-10-18 (10)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-1-9/1",4,"Macfarlane 1906, Exercise Probs-1-9 (1)"],["cap/other:related rates (chain rule in t)",17,"other:related rates (chain rule in t)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-10-18/17",4,"Macfarlane 1906, Exercise Probs-10-18 (17)"],["planck-treatise-on-thermodynamics-1903/eq-baf5e6ad97",16,"Planck 1903, p. 55: q = du + p\\, dv"],["thompson-calculus-made-easy-1914/x-ba160de5fd",15,"Thompson 1914, p. 193: So, therefore, when we reverse the process we must ..."],["macfarlane-vector-analysis-quaternions-1906/ex-probs-10-18/18",4,"Macfarlane 1906, Exercise Probs-10-18 (18)"],["person/zeno-of-elea",1,"Zeno of Elea","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-zeno-of-elea"],["thompson-calculus-made-easy-1914/x-4b76f48e9e",15,"Thompson 1914, p. 199: N.B.---Here note this very remarkable fact, that we could ..."],["planck-treatise-on-thermodynamics-1903/eq-f2f1cfc6b3",16,"Planck 1903, p. 55: c = \\frac{q}{d\\theta} = \\frac{du}{d\\theta} + p\\, \\frac{dv}{d\\theta}"],["form/5942c40dcf",5,"solve: (Eq(a, 5*x), Eq(a - x, 48))"],["shape/79dcaa2493",6,"solve: (Eq(a, N*x), Eq(a - x, N))"],["shape/26fb9583ea",6,"solve: Eq(N*x, N)"],["maxwell-elementary-treatise-electricity-1888/x-a65a4c7dec",15,"Maxwell 1888, scan 101: The point B with its imaginary charge is called ..."],["person/ernst-eduard-kummer",1,"Ernst Eduard Kummer","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-ernst-eduard-kummer"],["whitehead-introduction-to-mathematics-1911/x-29affacd83",15,"Whitehead 1911, p. 196: It is easy to verify in the case of ..."],["concept/circum-centre",7,"circum-centre","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-circum-centre"],["concept/in-centre",7,"in-centre","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-in-centre"],["macfarlane-vector-analysis-quaternions-1906/x-8e5c53df39",15,"Macfarlane 1906: The diagonal OC represents in magnitude and direction and ..."],["shape/bf3d3547ba",6,"solve: (Eq(a, N*x), Eq(a + x, N))"],["boyden-first-book-in-algebra-1895/ex-35/12",4,"Boyden 1895, Exercise 35 (12)"],["maxwell-elementary-treatise-electricity-1888/eq-87be207383",16,"Maxwell 1888, scan 209: \\frac{\\gamma^2}{\\beta^2} = 1 + 4\\frac{y - x}{b + c}"],["hardy-course-of-pure-mathematics-1921/eq-cee1bdde4d",16,"Hardy 1921, p. 375: \\int_{a}^{\\xi} \\frac{dx}{x(\\log x)^{s}} = \\frac{(\\log\\xi)^{1-s} - (\\log a)^{1-s}}{1 - s}"],["thompson-calculus-made-easy-1914/x-00a4abfc54",15,"Thompson 1914, p. 197: Hence, when you work the other way and integrate, ..."],["todhunter-spherical-trigonometry-1886/x-165d0ffa96",15,"Todhunter 1886, scan 147: The arc which passes through the middle points of ..."],["whitehead-introduction-to-mathematics-1911/x-fdc7548e5a",15,"Whitehead 1911, p. 9: Now, the first noticeable fact about arithmetic is that ..."],["planck-treatise-on-thermodynamics-1903/eq-15e23bfe52",16,"Planck 1903, p. 56: c_{v} = \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{v}\\Add{,}"],["planck-treatise-on-thermodynamics-1903/eq-f22de43403",16,"Planck 1903, p. 56: c_{v} = \\left(\\frac{\\dd u}{\\dd p}\\right)_{v} \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v}\\Add{.}"],["whitehead-introduction-to-mathematics-1911/x-cf9e730982",15,"Whitehead 1911, p. 213: The importance of the exponential function is that it ..."],["todhunter-spherical-trigonometry-1886/ex-vii",3,"Todhunter 1886, Exercise VII"],["hardy-course-of-pure-mathematics-1921/eq-43c4ec854a",16,"Hardy 1921, p. 375: \\int_{\\DPtypo{}{a}}^{\\xi} \\frac{dx}{x\\log x} = \\log\\log \\xi - \\log\\log a"],["de-morgan-elementary-illustrations-calculus-1899/eq-af284bdd27",16,"De Morgan 1899, p. 63: \\phi t = t^{3}"],["maxwell-elementary-treatise-electricity-1888/ch-ix",2,"Maxwell 1888, ch. IX: THE ELECTRIC CURRENT","../books/maxwell-elementary-treatise-electricity-1888/ch/ch-ix/index.html"],["de-morgan-elementary-illustrations-calculus-1899/eq-43588708dc",16,"De Morgan 1899, p. 64: a + gt"],["todhunter-spherical-trigonometry-1886/ch-circumscribed-and-inscribed-circles",2,"Todhunter 1886, Circumscribed and Inscribed Circles","../books/todhunter-spherical-trigonometry-1886/ch/ch-circumscribed-and-inscribed-circles/index.html"],["de-morgan-elementary-illustrations-calculus-1899/eq-8ad076e259",16,"De Morgan 1899, p. 63: 3t^{2}"],["thompson-calculus-made-easy-1914/ch-xix",2,"Thompson 1914, ch. XIX: On Finding Areas by Integrating","../books/thompson-calculus-made-easy-1914/ch/ch-xix/index.html"],["hardy-course-of-pure-mathematics-1921/ch-iii",2,"Hardy 1921, ch. III: COMPLEX NUMBERS","../books/hardy-course-of-pure-mathematics-1921/ch/ch-iii/index.html"],["maxwell-elementary-treatise-electricity-1888/x-649b610895",15,"Maxwell 1888, scan 164: During the passage of one unit of electricity through ..."],["maxwell-elementary-treatise-electricity-1888/x-447ba6a8fb",15,"Maxwell 1888, scan 166: The electromotive force of an electrochemical apparatus is in ..."],["macfarlane-vector-analysis-quaternions-1906/ex-probs-19-22/19",4,"Macfarlane 1906, Exercise Probs-19-22 (19)"],["todhunter-spherical-trigonometry-1886/eq-6086171e27",16,"Todhunter 1886, scan 148: BQ + CQ = \\pi"],["form/d03ecf7716",5,"evaluate: 200/(6*pi*a/5 + 10)"],["whitehead-introduction-to-mathematics-1911/x-10ea9056ea",15,"Whitehead 1911, p. 214: The curve, which is something like a cocked hat, ..."],["planck-treatise-on-thermodynamics-1903/ch-homogeneous-systems",2,"Planck 1903, Homogeneous Systems","../books/planck-treatise-on-thermodynamics-1903/ch/ch-homogeneous-systems/index.html"],["shape/df3d9fd806",6,"evaluate: N/(pi*N*a + N)"],["form/e2f5f2cf4c",5,"solve: (Eq(a, 3*x), Eq(a + x, 500))"],["form/2b95877a6d",5,"solve: (Eq(a, 11*x), Eq(a - x, 250))"],["thompson-calculus-made-easy-1914/ex-i/9",4,"Thompson 1914, Exercise I (9)"],["boyden-first-book-in-algebra-1895/ex-43",3,"Boyden 1895, Exercise 43"],["form/3156e334c9",5,"factor: a**4/c**4 - b**4/d**4"],["hardy-course-of-pure-mathematics-1921/ex-liv/5",4,"Hardy 1921, Exercise LIV (5)"],["hardy-course-of-pure-mathematics-1921/ex-liv/6",4,"Hardy 1921, Exercise LIV (6)"],["hardy-course-of-pure-mathematics-1921/ex-liv/7",4,"Hardy 1921, Exercise LIV (7)"],["todhunter-spherical-trigonometry-1886/eq-304fee8d60",16,"Todhunter 1886, scan 77: \\text{area of lune } = \\dfrac{A}{2\\pi} 4\\pi r^2 = 2Ar^2."],["maxwell-elementary-treatise-electricity-1888/x-4d4e11145a",15,"Maxwell 1888, scan 168: It does not appear that there is much polarization ..."],["todhunter-spherical-trigonometry-1886/eq-28c67e571a",16,"Todhunter 1886, scan 78: \\text{triangle } ABC=(A+B+C-\\pi)r^2."],["maxwell-elementary-treatise-electricity-1888/x-b572a26865",15,"Maxwell 1888, scan 169: When the current flows through the solution from the ..."],["wentworth-plane-geometry-1899/x-a4d706b429",15,"Wentworth 1899, scan 116: By marking the distinction between quantities measured in opposite ..."],["hardy-course-of-pure-mathematics-1921/eq-708f0fa87e",16,"Hardy 1921, p. 375: \\sum_{n_{0}}^{\\infty} \\frac{1}{n(\\log n)^{s}}"],["boyden-first-book-in-algebra-1895/ex-20/3",4,"Boyden 1895, Exercise 20 (3)"],["wentworth-first-steps-in-algebra-1894/ex-2/17",4,"Wentworth 1894, Exercise 2 (17)"],["maxwell-elementary-treatise-electricity-1888/x-ac8fa0ec8d",15,"Maxwell 1888, scan 117: The whole work done by the external electromotive force ..."],["form/c62afa562e",5,"identity: 3*a*(b + c)"],["shape/8fe604100f",6,"identity: N*a*(b + c)"],["shape/bed354d60c",6,"differentiate: (x**N)**(1/a)"],["wentworth-plane-geometry-1899/x-e63b24a578",15,"Wentworth 1899, scan 116: Here the word sum means the algebraic sum and ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-a9df55201a",16,"De Morgan 1899, p. 41: \\tan VPQ·PQ = VQ"],["concept/convex-plane-curve",7,"convex plane curve","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-convex-plane-curve"],["thompson-calculus-made-easy-1914/x-43c2a987d0",15,"Thompson 1914, p. 213: There are 18 whole squares and four triangles, each ..."],["hardy-course-of-pure-mathematics-1921/ex-liv/8a",4,"Hardy 1921, Exercise LIV (8a)"],["form/915724a8d0",5,"identity: 10*a**5*b**5*c"],["shape/91b29d0403",6,"identity: N*a**N*b**N*c"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-23-27",3,"Macfarlane 1906, Exercise Probs-23-27"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-23-27/23",4,"Macfarlane 1906, Exercise Probs-23-27 (23)"],["concept/inscribed-prism",7,"inscribed prism","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-inscribed-prism"],["concept/right-section",7,"right section","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-right-section"],["concept/element-of-a-cylinder",7,"element of a cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-element-of-a-cylinder"],["form/1016820ba4",5,"identity: 105*a**7*b**5"],["hardy-course-of-pure-mathematics-1921/eq-fe15121998",16,"Hardy 1921, p. 378: e^{x} = 1 + x + \\frac{x^{2}}{2!} + \\dots + \\frac{x^{n-1}}{(n - 1)!} + \\frac{x^{n}}{n!} e^{\\theta x}"],["hardy-course-of-pure-mathematics-1921/eq-e99d881ca5",16,"Hardy 1921, p. 378: x^{n}/n! \\to 0"],["wentworth-plane-geometry-1899/ch-v",2,"Wentworth 1899, ch. V: REGULAR POLYGONS AND CIRCLES","../books/wentworth-plane-geometry-1899/ch/ch-v/index.html"],["wentworth-first-steps-in-algebra-1894/ex-16/12",4,"Wentworth 1894, Exercise 16 (12)"],["theorem/equal-cross-sections-imply-equal-volumes",9,"equal cross-sections imply equal volumes","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-equal-cross-sections-imply-equal-volumes"],["person/cavalieri",1,"Cavalieri","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-person-cavalieri"],["slaught-lennes-solid-geometry-1919/x-e58899fe20",15,"Slaught & Lennes 1919, p. 202: But the limit of this sequence is by definition ..."],["hardy-course-of-pure-mathematics-1921/x-baba544a79",15,"Hardy 1921, p. 140: Later on we shall be able to identify this ..."],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37/28",4,"Macfarlane 1906, Exercise Probs-28-37 (28)"],["shape/21b5a4aecb",6,"identity: N*a**N*(N*a**N*b**N*c + N*a**N*b**N + N*c)/b"],["method/constructing-a-regular-octahedron",8,"constructing a regular octahedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-constructing-a-regular-octahedron"],["slaught-lennes-solid-geometry-1919/x-4b77751d59",15,"Slaught & Lennes 1919, p. 50: A polyhedron is convex if every section of it ..."],["form/a2a2b30cd1",5,"solve: (Eq(x, 3*a), Eq(x, a + 10))"],["form/6a3c4a696e",5,"identity: 6*a**6*b**5*c**7"],["form/180d4fba9f",5,"identity: (x - 4)*(x - 3)"],["slaught-lennes-solid-geometry-1919/x-4df6f1895b",15,"Slaught & Lennes 1919, p. 203: That U = V follows from the fact that ..."],["hardy-course-of-pure-mathematics-1921/ex-liv/8b",4,"Hardy 1921, Exercise LIV (8b)"],["slaught-lennes-solid-geometry-1919/x-29d93f112a",15,"Slaught & Lennes 1919, p. 207: The proof of this general theorem is more difficult ..."],["slaught-lennes-solid-geometry-1919/x-c8f0038d6a",15,"Slaught & Lennes 1919, p. 207: Indeed, this theorem and also that of § [unit:437.]437 ..."],["theorem/de-moivre-s-theorem-for-rational-exponents",9,"De Moivre's theorem for rational exponents","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-de-moivre-s-theorem-for-rational-exponents"],["de-morgan-elementary-illustrations-calculus-1899/x-ee683b9820",15,"De Morgan 1899, p. 126: Hence the curvilinear area MPP'M' is the limit towards ..."],["hardy-course-of-pure-mathematics-1921/ex-lxx",3,"Hardy 1921, Exercise LXX"],["hardy-course-of-pure-mathematics-1921/ex-lxix",3,"Hardy 1921, Exercise LXIX"],["thompson-calculus-made-easy-1914/ch-xx",2,"Thompson 1914, ch. XX: Dodges, Pitfalls, and Triumphs","../books/thompson-calculus-made-easy-1914/ch/ch-xx/index.html"],["hardy-course-of-pure-mathematics-1921/ex-lxviii",3,"Hardy 1921, Exercise LXVIII"],["slaught-lennes-solid-geometry-1919/x-d007cf4a46",15,"Slaught & Lennes 1919, p. 207: Hence, by § [unit:428.]428, V = \\tfrac{1}{3}rS. But by ..."],["hardy-course-of-pure-mathematics-1921/ex-liv/9",4,"Hardy 1921, Exercise LIV (9)"],["hardy-course-of-pure-mathematics-1921/ex-liv/10",4,"Hardy 1921, Exercise LIV (10)"],["hardy-course-of-pure-mathematics-1921/x-2e0e42575a",15,"Hardy 1921, p. 100: There are other special values of n for which ..."],["thompson-calculus-made-easy-1914/eq-e14783d448",16,"Thompson 1914, p. 202: \\int\\cos ax\\, dx &&= \\frac{1}{a} \\sin ax + C"],["thompson-calculus-made-easy-1914/eq-60143ebd06",16,"Thompson 1914, p. 202: \\int\\sin ax\\, dx &&= -\\frac{1}{a} \\cos ax + C."],["shape/b926ae554c",6,"evaluate: N*b + N*c - a**N"],["thompson-calculus-made-easy-1914/eq-8b547a23d3",16,"Thompson 1914, p. 27: \\frac{dy}{dx} = 14x."],["method/subtraction-of-displacements",8,"subtraction of displacements","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-subtraction-of-displacements"],["hardy-course-of-pure-mathematics-1921/ex-liv/11",4,"Hardy 1921, Exercise LIV (11)"],["hardy-course-of-pure-mathematics-1921/eq-71fbae91db",16,"Hardy 1921, p. 378: e^{x} = 1 + x + \\frac{x^{2}}{2!} + \\dots + \\frac{x^{n}}{n!} + \\dots"],["todhunter-spherical-trigonometry-1886/eq-1fc7892b52",16,"Todhunter 1886, scan 135: \\cos^2{TA}+\\cos^2{TB}+\\cos^2{TC}=1"],["wentworth-first-steps-in-algebra-1894/ex-17",3,"Wentworth 1894, Exercise 17"],["whitehead-introduction-to-mathematics-1911/ch-notes",2,"Whitehead 1911, Notes","../books/whitehead-introduction-to-mathematics-1911/ch/ch-notes/index.html"],["theorem/exponential-grows-faster-than-any-power",9,"exponential grows faster than any power","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-exponential-grows-faster-than-any-power"],["wentworth-first-steps-in-algebra-1894/ex-17/1",4,"Wentworth 1894, Exercise 17 (1)"],["hardy-course-of-pure-mathematics-1921/eq-3afad81ab4",16,"Hardy 1921, p. 378: e = 1 + 1 + \\frac{1}{2!} + \\dots + \\frac{1}{n!} + \\dots"],["form/1655c21178",5,"identity: (x + 2)*(x + 7)"],["quantity/eccentricity",11,"eccentricity","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-quantity-eccentricity"],["method/order-of-operations",8,"order of operations","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-method-order-of-operations"],["quantity/momentum",11,"momentum","../books/thompson-calculus-made-easy-1914/terms/index.html#t-quantity-momentum"],["hardy-course-of-pure-mathematics-1921/eq-1967672872",16,"Hardy 1921, p. 378: \\left(1 + 1 + \\frac{1}{2!} + \\dots + \\frac{1}{n!} + \\dots\\right)^{x} = 1 + x + \\frac{x^{2}}{2!} + \\dots + \\frac{x^{n}}{n"],["hardy-course-of-pure-mathematics-1921/eq-1a7c4ebb69",16,"Hardy 1921, p. 379: a^{x} = e^{x\\log a} = 1 + (x\\log a) + \\frac{(x\\log a)^{2}}{2!} + \\dots"],["theorem/change-of-base-of-logarithms",9,"change of base of logarithms","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-change-of-base-of-logarithms"],["cap/other:vector_direction",17,"other:vector_direction"],["hardy-course-of-pure-mathematics-1921/eq-c25c2f3685",16,"Hardy 1921, p. 379: \\left(1 + \\frac{x}{n}\\right)^{n} < E_{n}(x) < \\left(1 - \\frac{x}{n}\\right)^{-n}"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37/29",4,"Macfarlane 1906, Exercise Probs-28-37 (29)"],["concept/three-dimensional-figure",7,"three-dimensional figure","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-three-dimensional-figure"],["boyden-first-book-in-algebra-1895/ex-21",3,"Boyden 1895, Exercise 21"],["thompson-calculus-made-easy-1914/eq-4d0f15a3c6",16,"Thompson 1914, p. 204: \\text{volume} = \\iiint f(x,y,z) · dx · dy · dz."],["hardy-course-of-pure-mathematics-1921/eq-c554128653",16,"Hardy 1921, p. 306: \\int_{0}^{\\pi} \\frac{d\\theta}{a + b\\cos\\theta + c\\sin\\theta} = \\frac{2}{\\sqrtp{a^{2} - b^{2} - c^{2}}} \\arctan \\left\\{\\f"],["concept/prism",7,"prism","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-prism"],["thompson-calculus-made-easy-1914/eq-0d8ec6a754",16,"Thompson 1914, p. 202: \\int\\log_{10} x\\, dx &&= 0.4343x (\\log_\\epsilon x - 1) + C."],["concept/prismatic-surface",7,"prismatic surface","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-prismatic-surface"],["hardy-course-of-pure-mathematics-1921/ex-liv/12",4,"Hardy 1921, Exercise LIV (12)"],["hardy-course-of-pure-mathematics-1921/ex-liv/13a",4,"Hardy 1921, Exercise LIV (13a)"],["concept/weight",7,"weight","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-weight"],["hardy-course-of-pure-mathematics-1921/x-0011ff8d8b",15,"Hardy 1921, p. 77: The required definition is therefore [x, y] [x’, y’] ..."],["hardy-course-of-pure-mathematics-1921/eq-8f0c9b4120",16,"Hardy 1921, p. 306: \\left(\\int_{a}^{b} \\phi\\psi\\, dx\\right)^{2} \\leq \\int_{a}^{b} \\phi^{2}\\, dx \\int_{a}^{b} \\psi^{2}\\, dx"],["whitehead-introduction-to-mathematics-1911/x-17e4c4ec6a",15,"Whitehead 1911, p. 250: In reading these equations it must be noted that ..."],["concept/electric-circuit",7,"electric circuit","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electric-circuit"],["method/transposing-the-terms",8,"transposing the terms","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-method-transposing-the-terms"],["slaught-lennes-solid-geometry-1919/x-3133683442",15,"Slaught & Lennes 1919, p. 62: but no attempt has been made to measure the ..."],["hardy-course-of-pure-mathematics-1921/ex-liv/13d",4,"Hardy 1921, Exercise LIV (13d)"],["concept/curvilinear-figure",7,"curvilinear figure","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-curvilinear-figure"],["method/carnot-cycle",8,"Carnot cycle","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-method-carnot-cycle"],["theorem/opposite-faces-of-a-parallelepiped-are-equal-and-parallel",9,"opposite faces of a parallelepiped are equal and parallel","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-opposite-faces-of-a-parallelepiped-are-equal-and-parallel"],["hardy-course-of-pure-mathematics-1921/ex-liv/14",4,"Hardy 1921, Exercise LIV (14)"],["maxwell-elementary-treatise-electricity-1888/x-a83384a938",15,"Maxwell 1888, scan 210: The battery must never be introduced instead of the ..."],["form/61fa15fa15",5,"identity: a**4*b**2"],["hardy-course-of-pure-mathematics-1921/eq-04df36e3b1",16,"Hardy 1921, p. 306: P_{n}(x) = \\frac{1}{(\\beta - \\alpha)^{n} n!} \\left(\\frac{d}{dx}\\right)^{n} \\{(x - \\alpha)(\\beta - x)\\}^{n}"],["maxwell-elementary-treatise-electricity-1888/x-8d29231383",15,"Maxwell 1888, scan 209: The remaining difference between and will now produce a ..."],["shape/387e2b9271",6,"identity: a**N*b**N"],["planck-treatise-on-thermodynamics-1903/eq-c8504d2ebe",16,"Planck 1903, p. 56: c_{p} = \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{p} + p\\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}\\Add{,}"],["hardy-course-of-pure-mathematics-1921/ex-lix/1",4,"Hardy 1921, Exercise LIX (1)"],["hardy-course-of-pure-mathematics-1921/eq-658918f35b",16,"Hardy 1921, p. 306: \\int_{\\alpha}^{\\beta} P_{n}(x)\\theta(x)\\, dx = 0"],["hardy-course-of-pure-mathematics-1921/eq-8d28dffc10",16,"Hardy 1921, p. 306: \\int_{\\alpha}^{\\beta} P_{m}(x) P_{n}(x)\\, dx = 0"],["hardy-course-of-pure-mathematics-1921/x-d8cfa01763",15,"Hardy 1921, p. 361: This fact is sometimes expressed loosely by saying that ..."],["wentworth-first-steps-in-algebra-1894/eq-332fdb61c5",16,"Wentworth 1894, p. 11: a (b + c) &= ab + ac"],["form/131930a1ef",5,"identity: a**3*b**6"],["hardy-course-of-pure-mathematics-1921/ex-lix/2",4,"Hardy 1921, Exercise LIX (2)"],["hardy-course-of-pure-mathematics-1921/eq-7be7687336",16,"Hardy 1921, p. 307: \\int_{\\alpha}^{\\beta} (Q_{n} - \\kappa P_{n})^{2}\\, dx = 0"],["form/f3d5050d02",5,"identity: -a**9*b**6"],["planck-treatise-on-thermodynamics-1903/eq-7cc98df91a",16,"Planck 1903, p. 105: Q_{1} + Q_{2} + W = 0"],["wentworth-first-steps-in-algebra-1894/x-ffde3bb7bf",15,"Wentworth 1894, p. 19: An equation is a statement in symbols that two ..."],["slaught-lennes-solid-geometry-1919/x-e7b3b477fb",15,"Slaught & Lennes 1919, p. 50: These names are all derived from the Greek and ..."],["hardy-course-of-pure-mathematics-1921/eq-d20dfde0cc",16,"Hardy 1921, p. 379: f(x)f(y) = f(x + y)"],["hardy-course-of-pure-mathematics-1921/ex-lix/3",4,"Hardy 1921, Exercise LIX (3)"],["form/69b6f4a39f",5,"identity: 27*a**6*b**3"],["hardy-course-of-pure-mathematics-1921/ex-lix/4",4,"Hardy 1921, Exercise LIX (4)"],["wentworth-plane-geometry-1899/eq-4e52308087",16,"Wentworth 1899, scan 247: AB = BC"],["de-morgan-elementary-illustrations-calculus-1899/ch-the-same-problem-solved-by-the-principles-of-leibnitz",2,"De Morgan 1899, The Same Problem Solved by the Principles of Leibnitz","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-the-same-problem-solved-by-the-principles-of-leibnitz/index.html"],["theorem/common-potential-after-contact",9,"common potential after contact","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-common-potential-after-contact"],["shape/a56a5d452a",6,"solve: Eq(x, N*a)"],["form/509053f43b",5,"evaluate: 10*a"],["de-morgan-elementary-illustrations-calculus-1899/eq-f9d4f70958",16,"De Morgan 1899, p. 77: \\phi(a + h) = 0"],["wentworth-plane-geometry-1899/eq-989b8e4898",16,"Wentworth 1899, scan 244: \\triangle ACB > \\triangle ADB"],["wentworth-plane-geometry-1899/eq-a7b5b0e974",16,"Wentworth 1899, scan 246: ABCDE > A'B'C'D'E'"],["hardy-course-of-pure-mathematics-1921/x-6454195043",15,"Hardy 1921, p. 367: We take this as our definition of a^{x} when ..."],["boyden-first-book-in-algebra-1895/ex-35/13",4,"Boyden 1895, Exercise 35 (13)"],["wentworth-plane-geometry-1899/x-1224ccd7bd",15,"Wentworth 1899, scan 227: If the number of sides of a regular inscribed ..."],["wentworth-first-steps-in-algebra-1894/x-2304bd5252",15,"Wentworth 1894, p. 19: Thus, a + b = b + a, which ..."],["wentworth-first-steps-in-algebra-1894/x-7645706f2d",15,"Wentworth 1894, p. 22: Any term may be transposed from one side of ..."],["slaught-lennes-solid-geometry-1919/x-2df6f50a5a",15,"Slaught & Lennes 1919, p. 50: The faces, edges, and vertices taken together form the ..."],["wentworth-first-steps-in-algebra-1894/eq-6682317b07",16,"Wentworth 1894, p. 11: a(b - c) &= ab - ac"],["de-morgan-elementary-illustrations-calculus-1899/eq-98878ca9fb",16,"De Morgan 1899, p. 77: \\phi a + \\phi' a\\, h = 0"],["shape/3c55af0a01",6,"factor: x**N - 1"],["hardy-course-of-pure-mathematics-1921/eq-686b64c6d4",16,"Hardy 1921, p. 307: \\int_{0}^{1} \\phi(x)\\, dx = \\tfrac{1}{18}\\{5\\phi(\\alpha) + 8\\phi(\\tfrac{1}{2}) + 5\\phi(\\beta)\\}"],["boyden-first-book-in-algebra-1895/ex-1/14",4,"Boyden 1895, Exercise 1 (14)"],["concept/wave-equation",7,"wave equation","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-wave-equation"],["concept/quadrilateral",7,"quadrilateral","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-quadrilateral"],["wentworth-first-steps-in-algebra-1894/x-073f5c6e2e",15,"Wentworth 1894, p. 22: When the root is substituted for its symbol in ..."],["method/measuring-electric-potential",8,"measuring electric potential","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-measuring-electric-potential"],["maxwell-elementary-treatise-electricity-1888/ch-xii",2,"Maxwell 1888, ch. XII: THE MEASUREMENT OF ELECTRIC RESISTANCE","../books/maxwell-elementary-treatise-electricity-1888/ch/ch-xii/index.html"],["form/0c7a2bd457",5,"solve: (Eq(a, 37*x), Eq(a + x, 4256))"],["form/4fa39afed3",5,"evaluate: a/100"],["wentworth-first-steps-in-algebra-1894/eq-a4c4b75016",16,"Wentworth 1894, p. 19: a + b = b + a"],["ball-mathematical-recreations-1905/x-e9b557869b",15,"Ball 1905, scan 217: Newton’s mathematical career dates from 1665; his reputation, abilities, ..."],["dickson-theory-of-equations-1922/ex-page13",3,"Dickson 1922, Exercise Page13"],["thompson-calculus-made-easy-1914/x-269166d2fd",15,"Thompson 1914, p. 234: He who would attain that facility must work out ..."],["hardy-course-of-pure-mathematics-1921/eq-b285f386eb",16,"Hardy 1921, p. 307: x^{2} - x + \\frac{1}{10} = 0"],["wentworth-plane-geometry-1899/x-8f37984bcb",15,"Wentworth 1899, scan 232: The area of a regular polygon is equal to ..."],["wentworth-plane-geometry-1899/x-289cb9038e",15,"Wentworth 1899, scan 242: \\pi is incommensurable."],["de-morgan-elementary-illustrations-calculus-1899/eq-10f85963ec",16,"De Morgan 1899, p. 77: a - \\dfrac{\\phi a}{\\phi' a}"],["de-morgan-elementary-illustrations-calculus-1899/eq-e1d923dc8e",16,"De Morgan 1899, p. 77: \\phi' x = 2x + 1"],["hardy-course-of-pure-mathematics-1921/eq-974dc8b535",16,"Hardy 1921, p. 307: \\tfrac{1}{4}\\pi = \\int_{0}^{1} \\frac{dx}{1 + x^{2}}"],["cap/core.frac",17,"core.frac"],["form/fdd60920cc",5,"solve: Eq(x + 36, 4*x)"],["wentworth-first-steps-in-algebra-1894/x-582c9d5876",15,"Wentworth 1894, p. 25: Remember that x must not be put for money, ..."],["wentworth-first-steps-in-algebra-1894/x-2e0b7b6eda",15,"Wentworth 1894, p. 26: Now, we know what James had. He had oranges, ..."],["todhunter-spherical-trigonometry-1886/eq-7619587408",16,"Todhunter 1886, scan 75: \\sin^2 s + \\sin^2 (s-a) + \\sin^2 (s-b) + \\sin^2 (s-c) = 2-2 \\cos a \\cos b \\cos c"],["ball-mathematical-recreations-1905/x-7f3ee8ce06",15,"Ball 1905, scan 227: The extraction of roots, the arithmetic of surds, the ..."],["boyden-first-book-in-algebra-1895/ex-35/14",4,"Boyden 1895, Exercise 35 (14)"],["hardy-course-of-pure-mathematics-1921/eq-bfa55a5ab6",16,"Hardy 1921, p. 307: 8.9 < \\int_{3}^{5} \\sqrtp{4 + x^{2}}\\, dx < 9"],["hardy-course-of-pure-mathematics-1921/eq-ce934c2ef1",16,"Hardy 1921, p. 379: f(x)f(-x) = f(0) = 1"],["hardy-course-of-pure-mathematics-1921/eq-b05cdda98b",16,"Hardy 1921, p. 381: \\log(1 + x) = \\int_{0}^{x} \\frac{dt}{1 + t}"],["dickson-theory-of-equations-1922/ex-page15",3,"Dickson 1922, Exercise Page15"],["dickson-theory-of-equations-1922/ex-page17",3,"Dickson 1922, Exercise Page17"],["hardy-course-of-pure-mathematics-1921/ex-lix/5",4,"Hardy 1921, Exercise LIX (5)"],["person/oliver-heaviside",1,"Oliver Heaviside","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-oliver-heaviside"],["dickson-theory-of-equations-1922/ex-page19",3,"Dickson 1922, Exercise Page19"],["boyden-first-book-in-algebra-1895/ex-22",3,"Boyden 1895, Exercise 22"],["hardy-course-of-pure-mathematics-1921/ex-lix/6",4,"Hardy 1921, Exercise LIX (6)"],["theorem/absolute-convergence-of-a-power-series-inside-a-point-of-convergence",9,"absolute convergence of a power series inside a point of convergence","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-absolute-convergence-of-a-power-series-inside-a-point-of-convergence"],["theorem/trichotomy-of-convergence-of-a-power-series",9,"trichotomy of convergence of a power series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-trichotomy-of-convergence-of-a-power-series"],["boyden-first-book-in-algebra-1895/ex-22/1",4,"Boyden 1895, Exercise 22 (1)"],["wentworth-first-steps-in-algebra-1894/eq-b033e067ff",16,"Wentworth 1894, p. 19: 3x + 2 = 8"],["concept/radius-of-convergence",7,"radius of convergence","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-radius-of-convergence"],["boyden-first-book-in-algebra-1895/ex-1/15",4,"Boyden 1895, Exercise 1 (15)"],["hardy-course-of-pure-mathematics-1921/ex-lix/7",4,"Hardy 1921, Exercise LIX (7)"],["hardy-course-of-pure-mathematics-1921/x-a1eceb2a15",15,"Hardy 1921, p. 84: The application of any of the ordinary algebraical operations ..."],["quantity/euler-s-constant",11,"Euler's constant","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-quantity-euler-s-constant"],["form/d1beab7320",5,"solve: (Eq(7*x, a), Eq(a, x + 48))"],["shape/bb2ece34db",6,"solve: Eq(N + x, N*x)"],["shape/c434738b9d",6,"solve: (Eq(N*x, a), Eq(a, N + x))"],["dickson-theory-of-equations-1922/ex-page20",3,"Dickson 1922, Exercise Page20"],["form/1033cc4c79",5,"solve: Eq(12*x/35, 24)"],["form/b54a2ecdb2",5,"factor: -x**3 + 1"],["de-morgan-elementary-illustrations-calculus-1899/x-7582183c36",15,"De Morgan 1899, p. 133: The method so generally followed in our elementary works, ..."],["method/comparing-two-equal-resistances-with-wheatstone-s-bridge",8,"comparing two equal resistances with Wheatstone's bridge","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-comparing-two-equal-resistances-with-wheatstone-s-bridge"],["de-morgan-elementary-illustrations-calculus-1899/ch-calculus-of-finite-differences-successive-differentiation",2,"De Morgan 1899, Calculus of Finite Differences. Successive Differentiation","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-calculus-of-finite-differences-successive-differentiation/index.html"],["method/checking-a-result-by-substitution",8,"checking a result by substitution","../books/thompson-calculus-made-easy-1914/terms/index.html#t-method-checking-a-result-by-substitution"],["form/c05a86a31f",5,"identity: (a + b)**3"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-48-53/49",4,"Macfarlane 1906, Exercise Probs-48-53 (49)"],["cap/other:directed_sine",17,"other:directed_sine"],["form/927263f046",5,"solve: (Eq(-2*a + 8*x, 6), Eq(7*a + 10*x, 36))"],["cap/cas.expand",17,"cas.expand"],["maxwell-elementary-treatise-electricity-1888/x-da0427caf6",15,"Maxwell 1888, scan 206: Of the two resistances, that of the battery and ..."],["method/determining-galvanometer-resistance-by-thomson-s-method",8,"determining galvanometer resistance by Thomson's method","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-determining-galvanometer-resistance-by-thomson-s-method"],["hardy-course-of-pure-mathematics-1921/x-d4bee5e031",15,"Hardy 1921, p. 386: We know that this series is convergent for all ..."],["de-morgan-elementary-illustrations-calculus-1899/x-b76ee1fa7a",15,"De Morgan 1899, p. 25: When any quantity is increased by an increment, which, ..."],["hardy-course-of-pure-mathematics-1921/ex-lix/8",4,"Hardy 1921, Exercise LIX (8)"],["hardy-course-of-pure-mathematics-1921/ex-lxxiv",3,"Hardy 1921, Exercise LXXIV"],["hardy-course-of-pure-mathematics-1921/ex-lxxi",3,"Hardy 1921, Exercise LXXI"],["hardy-course-of-pure-mathematics-1921/ex-lxxii",3,"Hardy 1921, Exercise LXXII"],["dickson-theory-of-equations-1922/ex-page23",3,"Dickson 1922, Exercise Page23"],["concept/parallel-planes",7,"parallel planes","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-parallel-planes"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii",3,"Hardy 1921, Exercise LXXIII"],["hardy-course-of-pure-mathematics-1921/ex-lxxv",3,"Hardy 1921, Exercise LXXV"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi",3,"Hardy 1921, Exercise LXXVI"],["form/2a2b59652d",5,"evaluate: 21*sqrt(3)/40"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-38-43/43",4,"Macfarlane 1906, Exercise Probs-38-43 (43)"],["hardy-course-of-pure-mathematics-1921/x-706456fdde",15,"Hardy 1921, p. 386: Incidentally we have proved that \\exp x is a ..."],["concept/angle-at-the-centre-of-a-regular-polygon",7,"angle at the centre of a regular polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-angle-at-the-centre-of-a-regular-polygon"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-44-47/44",4,"Macfarlane 1906, Exercise Probs-44-47 (44)"],["theorem/sum-of-the-interior-angles-of-a-polygon",9,"sum of the interior angles of a polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-sum-of-the-interior-angles-of-a-polygon"],["boyden-first-book-in-algebra-1895/ex-1/16",4,"Boyden 1895, Exercise 1 (16)"],["form/0d44a7c304",5,"solve: (Eq(x, 7*a), Eq(a + x, 32))"],["hardy-course-of-pure-mathematics-1921/eq-721efac463",16,"Hardy 1921, p. 381: 1/(1 + t) = 1 - t + t^{2} - \\dots + (-1)^{m-1} t^{m-1} + \\frac{(-1)^{m} t^{m}}{1 + t}"],["form/c245d86e67",5,"identity: (-a*b**3 + c**2*d)**2"],["shape/fbc4332921",6,"identity: (-a*b**N + c**N*d)**N"],["cap/other:angle_pair_vector_notation",17,"other:angle_pair_vector_notation"],["boyden-first-book-in-algebra-1895/ex-22/15",4,"Boyden 1895, Exercise 22 (15)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-54-62/54",4,"Macfarlane 1906, Exercise Probs-54-62 (54)"],["concept/magic-square",7,"magic square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-magic-square"],["wentworth-first-steps-in-algebra-1894/ex-17/2",4,"Wentworth 1894, Exercise 17 (2)"],["thompson-calculus-made-easy-1914/x-4d9edf8892",15,"Thompson 1914, p. 250: You don’t teach the rules of syntax to children ..."],["form/1dc5eac63a",5,"identity: (a + 1)**3"],["hardy-course-of-pure-mathematics-1921/eq-bcd0b0feed",16,"Hardy 1921, p. 381: R_{m} = \\int_{0}^{x} \\frac{t^{m}\\, dt}{1 + t}"],["thompson-calculus-made-easy-1914/x-f25e3d5340",15,"Thompson 1914, p. xi: Some calculus-tricks are quite easy. Some are enormously difficult."],["form/a840478272",5,"identity: 3*x**2"],["hardy-course-of-pure-mathematics-1921/eq-43831cc0cb",16,"Hardy 1921, p. 381: R_{m} = (-1)^{m} \\int_{0}^{\\xi} \\frac{u^{m}\\, du}{1 - u}"],["shape/a47ae08828",6,"identity: (a + 1)**N"],["shape/17f8c60261",6,"factor: -a**N*b**N*c**N + d**N*e**N*x**N"],["concept/complementary-numbers",7,"complementary numbers","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-complementary-numbers"],["hardy-course-of-pure-mathematics-1921/ex-lix/9",4,"Hardy 1921, Exercise LIX (9)"],["form/d97808d557",5,"identity: (-a + 1)**4"],["hardy-course-of-pure-mathematics-1921/ex-lix/10",4,"Hardy 1921, Exercise LIX (10)"],["shape/20f2cc5eb1",6,"identity: (-a + 1)**N"],["hardy-course-of-pure-mathematics-1921/eq-fddcdc792f",16,"Hardy 1921, p. 382: 0 < |R_{m}| < \\frac{1}{1 - \\xi} \\int_{0}^{\\xi} u^{m}\\, du"],["wentworth-first-steps-in-algebra-1894/eq-b0c3917b8d",16,"Wentworth 1894, p. 20: ax + b = c"],["hardy-course-of-pure-mathematics-1921/eq-5d580c848f",16,"Hardy 1921, p. 382: \\log(1 + x) = x - \\tfrac{1}{2} x^{2} + \\tfrac{1}{3} x^{3} - \\dots"],["boyden-first-book-in-algebra-1895/ex-22/22",4,"Boyden 1895, Exercise 22 (22)"],["form/ea3334b1dd",5,"identity: (-a**2 + 1)**3"],["ball-mathematical-recreations-1905/x-c2ee0211c2",15,"Ball 1905, scan 19: I shall devote the bulk of this chapter to ..."],["ball-mathematical-recreations-1905/x-e39e588b25",15,"Ball 1905, scan 20: They are given here mainly for their historical---not for ..."],["hardy-course-of-pure-mathematics-1921/eq-2f3d2f7113",16,"Hardy 1921, p. 382: \\log 2 = 1 - \\tfrac{1}{2} + \\tfrac{1}{3} - \\dots"],["thompson-calculus-made-easy-1914/ch-table-of-standard-forms",2,"Thompson 1914, Table of Standard Forms","../books/thompson-calculus-made-easy-1914/ch/ch-table-of-standard-forms/index.html"],["law/laws-of-motion",10,"laws of motion","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-law-laws-of-motion"],["wentworth-first-steps-in-algebra-1894/eq-f9a831ae41",16,"Wentworth 1894, p. 21: x = a - b"],["cap/other:quaternion_rotation",17,"other:quaternion_rotation"],["todhunter-spherical-trigonometry-1886/x-d2580e6a15",15,"Todhunter 1886, scan 18: It will be seen that what are called sides ..."],["macfarlane-vector-analysis-quaternions-1906/ex-probs-54-62/55",4,"Macfarlane 1906, Exercise Probs-54-62 (55)"],["hardy-course-of-pure-mathematics-1921/ex-lix/11",4,"Hardy 1921, Exercise LIX (11)"],["ball-mathematical-recreations-1905/x-c04097d0bb",15,"Ball 1905, scan 28: in arithmetic an integral number is denoted by a ..."],["concept/method-transposing-the-terms",7,"method: transposing the terms"],["concept/oblique-line",7,"oblique line","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-oblique-line"],["shape/49e442e2ae",6,"identity: (-a**N + 1)**N"],["form/b30f4c85e3",5,"factor: 729*a**2*b**10*x**4 - 10000*c**4"],["hardy-course-of-pure-mathematics-1921/ex-lix/12",4,"Hardy 1921, Exercise LIX (12)"],["wentworth-first-steps-in-algebra-1894/eq-92239d6886",16,"Wentworth 1894, p. 21: x = a + b"],["wentworth-first-steps-in-algebra-1894/eq-9d1360e429",16,"Wentworth 1894, p. 77: \\frac{a^{3} + b^{3}}{a + b} = a^{2} - ab + b^{2}"],["wentworth-first-steps-in-algebra-1894/eq-7b490e9a40",16,"Wentworth 1894, p. 78: a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})"],["slaught-lennes-solid-geometry-1919/eq-8276d077d9",16,"Slaught & Lennes 1919, p. 34: EC = ED"],["form/876dfb1c76",5,"identity: (2*a + 3*b**2)**2"],["shape/d7c322b4b9",6,"identity: (N*a + N*b**N)**N"],["ball-mathematical-recreations-1905/x-dab4fc1dc5",15,"Ball 1905, scan 26: Then the sum obtained as the result of this ..."],["de-morgan-elementary-illustrations-calculus-1899/x-65135efea0",15,"De Morgan 1899, p. 62: In this substitute v for nv', and t for ..."],["ball-mathematical-recreations-1905/x-bfe63e6964",15,"Ball 1905, scan 24: Hence, if N is divided by a', the remainder ..."],["concept/determination-of-a-plane",7,"determination of a plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-determination-of-a-plane"],["concept/dihedral-angle",7,"dihedral angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-dihedral-angle"],["concept/half-plane",7,"half-plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-half-plane"],["shape/a3530c911e",6,"factor: N*a**N*b**N*x**N + N*c**N"],["concept/angle-between-line-and-plane",7,"angle between line and plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-angle-between-line-and-plane"],["hardy-course-of-pure-mathematics-1921/ex-lix/13",4,"Hardy 1921, Exercise LIX (13)"],["concept/bisector-of-a-dihedral-angle",7,"bisector of a dihedral angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-bisector-of-a-dihedral-angle"],["whitehead-introduction-to-mathematics-1911/x-0b04c53d06",15,"Whitehead 1911, p. 161: This example brings out the fact that statements about ..."],["ball-mathematical-recreations-1905/x-39bdb8d7de",15,"Ball 1905, scan 30: The result is the man’s age in 1906."],["hardy-course-of-pure-mathematics-1921/eq-22379b1650",16,"Hardy 1921, p. 382: \\arctan x = \\int_{0}^{x} \\frac{dt}{1 + t^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-dc3593a4cc",16,"Hardy 1921, p. 382: \\tfrac{1}{4}\\pi = 1 - \\tfrac{1}{3} + \\tfrac{1}{5} - \\dots"],["concept/right-dihedral-angle",7,"right dihedral angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-right-dihedral-angle"],["concept/mutually-perpendicular-planes",7,"mutually perpendicular planes","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-mutually-perpendicular-planes"],["concept/trihedral-angle",7,"trihedral angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-trihedral-angle"],["shape/6e9f530c44",6,"differentiate: N + a*x**N"],["form/b5adcfed03",5,"evaluate: a**2*b - b*c*(a**2*c - b*d**2) - c**2*d - (a*b + c*d)*(a*c - b*d) at a=2, b=3, c=4, d=0"],["shape/dc5e3f6251",6,"evaluate: a**N*b - b*c*(a**N*c - b*d**N) - c**N*d - (a*b + c*d)*(a*c - b*d)"],["ball-mathematical-recreations-1905/eq-c63f8278d7",16,"Ball 1905, scan 24: A = M(a') + 1"],["hardy-course-of-pure-mathematics-1921/ex-lix/14",4,"Hardy 1921, Exercise LIX (14)"],["de-morgan-elementary-illustrations-calculus-1899/x-43dbb23a83",15,"De Morgan 1899, p. 48: The inaccuracy of this supposition has been already pointed ..."],["concept/prime-to-one-another",7,"prime to one another"],["ball-mathematical-recreations-1905/eq-ec6bf95e11",16,"Ball 1905, scan 24: Aa = M(a') + a"],["ball-mathematical-recreations-1905/eq-687aeb9c19",16,"Ball 1905, scan 24: N = Aa + Bb + Cc + \\dotsb"],["boyden-first-book-in-algebra-1895/ex-31",3,"Boyden 1895, Exercise 31"],["ball-mathematical-recreations-1905/eq-fe0b12eb8d",16,"Ball 1905, scan 24: N-n &= M(a')\\,."],["boyden-first-book-in-algebra-1895/ex-32",3,"Boyden 1895, Exercise 32"],["concept/nasik-square",7,"nasik square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-nasik-square"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-48-53/48",4,"Macfarlane 1906, Exercise Probs-48-53 (48)"],["maxwell-elementary-treatise-electricity-1888/eq-a5e80daab9",16,"Maxwell 1888, scan 216: r = \\alpha T^{\\frac{1}{2}} + \\beta T + \\gamma\\text{,}"],["concept/factored-form",7,"factored form","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-factored-form"],["ball-mathematical-recreations-1905/eq-cb79e0d70c",16,"Ball 1905, scan 24: N &= M(p) + n\\,."],["boyden-first-book-in-algebra-1895/ex-1/17",4,"Boyden 1895, Exercise 1 (17)"],["cap/other:quaternion_product",17,"other:quaternion_product"],["boyden-first-book-in-algebra-1895/ex-33",3,"Boyden 1895, Exercise 33"],["concept/dynamics",7,"dynamics","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-dynamics"],["ball-mathematical-recreations-1905/eq-771c5e1c88",16,"Ball 1905, scan 23: x = e"],["form/f7214c434a",5,"differentiate: sqrt(c*x/d)/(sqrt(pi)*a*b)"],["concept/doubly-magic-square",7,"doubly magic square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-doubly-magic-square"],["shape/cce5c3cf83",6,"differentiate: pi**N*(c*x/d)**N/(a*b)"],["hardy-course-of-pure-mathematics-1921/eq-95310d5613",16,"Hardy 1921, p. 382: \\argtanh x = \\frac{1}{2} \\log\\left(\\dfrac{1 + x}{1 - x}\\right)"],["maxwell-elementary-treatise-electricity-1888/eq-24dc874c31",16,"Maxwell 1888, scan 218: \\rho = \\frac{R_1 - R_2}{{R_1}' - {R_2}'}."],["maxwell-elementary-treatise-electricity-1888/eq-9d52bb38e0",16,"Maxwell 1888, scan 222: R = r \\times 0.8878^t\\text{,}"],["ball-mathematical-recreations-1905/eq-3b62ba80df",16,"Ball 1905, scan 23: y=9-r"],["concept/vector-of-transportation",7,"vector of transportation","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-vector-of-transportation"],["hardy-course-of-pure-mathematics-1921/ex-lix/15",4,"Hardy 1921, Exercise LIX (15)"],["hardy-course-of-pure-mathematics-1921/ex-lix/16",4,"Hardy 1921, Exercise LIX (16)"],["planck-treatise-on-thermodynamics-1903/eq-b561ee08c8",16,"Planck 1903, p. 78: \\ce{\\{H2\\} + $\\tfrac{1}{2}$ \\{O2\\} - (H2O)} = 68,400~\\Unit{cal.}"],["maxwell-elementary-treatise-electricity-1888/eq-2edecd0101",16,"Maxwell 1888, scan 223: E = E_0 + RC\\text{.}"],["ball-mathematical-recreations-1905/eq-f86c8063cd",16,"Ball 1905, scan 23: 9m-y = a-b + 3(c-d)"],["hardy-course-of-pure-mathematics-1921/eq-0269f67e56",16,"Hardy 1921, p. 384: (1 + x)^{m} = 1 + \\binom{m}{1}x + \\binom{m}{2}x^{2} + \\dots"],["hardy-course-of-pure-mathematics-1921/ex-lv/1",4,"Hardy 1921, Exercise LV (1)"],["thompson-calculus-made-easy-1914/ex-ii",3,"Thompson 1914, Exercise II"],["form/e56ec02dfe",5,"identity: -7"],["wentworth-first-steps-in-algebra-1894/eq-ad9a39cd73",16,"Wentworth 1894, p. 41: (+a) × (+b) &= +ab"],["wentworth-first-steps-in-algebra-1894/eq-99af18c2ef",16,"Wentworth 1894, p. 35: +a - a = 0"],["wentworth-first-steps-in-algebra-1894/eq-f0c40bd36a",16,"Wentworth 1894, p. 41: a^{m} × a^{n} = a^{m + n}"],["wentworth-first-steps-in-algebra-1894/eq-ff681e42ac",16,"Wentworth 1894, p. 41: (+a) × (-b) &= -ab"],["slaught-lennes-solid-geometry-1919/eq-c8876d264c",16,"Slaught & Lennes 1919, p. 29: \\angle M-AB-N = \\angle M'-A'B'-N'"],["concept/equal-dihedral-angles",7,"equal dihedral angles"],["hardy-course-of-pure-mathematics-1921/ex-lv/2",4,"Hardy 1921, Exercise LV (2)"],["concept/spherical-segment",7,"spherical segment","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-spherical-segment"],["concept/spherical-cone",7,"spherical cone","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-spherical-cone"],["concept/indicator-diagram",7,"indicator diagram","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-indicator-diagram"],["concept/recurring-series",7,"recurring series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-recurring-series"],["concept/theorem-dihedral-angles-equal-when-plane-angles-are-equal",7,"theorem: dihedral angles equal when plane angles are equal"],["slaught-lennes-solid-geometry-1919/eq-526fca03e6",16,"Slaught & Lennes 1919, p. 29: \\angle CDE = \\angle C'D'E'"],["concept/scale-of-relation",7,"scale of relation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-scale-of-relation"],["maxwell-elementary-treatise-electricity-1888/ch-ii",2,"Maxwell 1888, ch. II: ON THE CHARGES OF ELECTRIFIED BODIES","../books/maxwell-elementary-treatise-electricity-1888/ch/ch-ii/index.html"],["form/204e7786f4",5,"factor: a**5 + x**5"],["shape/9cdd2925e1",6,"factor: a**N + x**N"],["shape/ad973d4708",6,"identity: N"],["hardy-course-of-pure-mathematics-1921/eq-ccff19d66e",16,"Hardy 1921, p. 385: (1 + x)^{m} = e^{m\\log(1+ x)}"],["hardy-course-of-pure-mathematics-1921/ex-lv/3",4,"Hardy 1921, Exercise LV (3)"],["boyden-first-book-in-algebra-1895/ex-23/12",4,"Boyden 1895, Exercise 23 (12)"],["hardy-course-of-pure-mathematics-1921/ex-lxxvii",3,"Hardy 1921, Exercise LXXVII"],["form/e83bbd63e7",5,"identity: 2*a + x**3 - 3*x**2 - 6"],["concept/face-angle",7,"face angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-face-angle"],["shape/71eba71428",6,"identity: N*a + N*x**N + N + x**N"],["hardy-course-of-pure-mathematics-1921/ex-lxxviii",3,"Hardy 1921, Exercise LXXVIII"],["slaught-lennes-solid-geometry-1919/ch-introduction",2,"Slaught & Lennes 1919, ch. INTRODUCTION: ","../books/slaught-lennes-solid-geometry-1919/ch/ch-introduction/index.html"],["slaught-lennes-solid-geometry-1919/eq-5df8500792",16,"Slaught & Lennes 1919, p. 8: ah = bk"],["thompson-calculus-made-easy-1914/eq-174b881fa3",16,"Thompson 1914, p. 32: r = h = \\sqrt{\\dfrac{400}{2\\pi}} = 7.98~\\text{in}."],["todhunter-spherical-trigonometry-1886/ch-geodetical-operations",2,"Todhunter 1886, Geodetical Operations","../books/todhunter-spherical-trigonometry-1886/ch/ch-geodetical-operations/index.html"],["boyden-first-book-in-algebra-1895/ex-20/1",4,"Boyden 1895, Exercise 20 (1)"],["hardy-course-of-pure-mathematics-1921/eq-94a9f9ead4",16,"Hardy 1921, p. 385: D_{x}(1 + x)^{m} = \\{m/(1 + x)\\} e^{m\\log(1 + x)} = m(1 + x)^{m-1}"],["maxwell-elementary-treatise-electricity-1888/eq-d1b8647405",16,"Maxwell 1888, scan 225: ( P + G + Q )\\overline{ x + y} - Gy - Qz &= 0"],["hardy-course-of-pure-mathematics-1921/eq-e363a7251b",16,"Hardy 1921, p. 386: \\exp x = 1 + x + \\frac{x^{2}}{2!} + \\dots"],["slaught-lennes-solid-geometry-1919/eq-f030a1f6e7",16,"Slaught & Lennes 1919, p. 33: \\angle ABD > \\angle ABC"],["hardy-course-of-pure-mathematics-1921/eq-63e9c23aa4",16,"Hardy 1921, p. 386: \\exp x × \\exp y = \\exp(x + y)"],["wentworth-first-steps-in-algebra-1894/eq-19bb597279",16,"Wentworth 1894, p. 41: (-a) × (+b) &= -ab"],["wentworth-first-steps-in-algebra-1894/eq-1a36494be8",16,"Wentworth 1894, p. 41: (-a) × (-b) &= +ab"],["wentworth-first-steps-in-algebra-1894/eq-851fe5e4f4",16,"Wentworth 1894, p. 43: +ab ÷ (+a) = +b"],["wentworth-first-steps-in-algebra-1894/eq-0774b5a15a",16,"Wentworth 1894, p. 43: -ab ÷ (+a) = -b"],["slaught-lennes-solid-geometry-1919/eq-fa0de5ceaa",16,"Slaught & Lennes 1919, p. 8: a = \\tfrac{1}{2} \\cdot 2\\pi r \\cdot r = \\pi r^2."],["maxwell-elementary-treatise-electricity-1888/eq-2ac0395ca5",16,"Maxwell 1888, scan 225: ( R + S + G )y - Sz - G \\overline{x + y} &= 0"],["ball-mathematical-recreations-1905/x-c5ba3d5df6",15,"Ball 1905, scan 171: two numbers which are equidistant from the ends of ..."],["hardy-course-of-pure-mathematics-1921/eq-25bdc4c354",16,"Hardy 1921, p. 386: D_{x} \\exp x = \\exp x"],["form/5616817b02",5,"identity: (a**3 - b**3)/(a + b)"],["shape/e31f9ca13c",6,"identity: (a**N - b**N)/(a + b)"],["wentworth-first-steps-in-algebra-1894/eq-e920c0cf4e",16,"Wentworth 1894, p. 148: \\dfrac{b}{a} = \\dfrac{c}{b} = \\dfrac{d}{c}"],["wentworth-first-steps-in-algebra-1894/ch-xiv",2,"Wentworth 1894, ch. XIV: Geometrical Progression","../books/wentworth-first-steps-in-algebra-1894/ch/ch-xiv/index.html"],["wentworth-first-steps-in-algebra-1894/eq-922e329311",16,"Wentworth 1894, p. 149: l = ar^{n - 1}"],["concept/first-term",7,"first term"],["wentworth-first-steps-in-algebra-1894/eq-435d84bddc",16,"Wentworth 1894, p. 149: G = ± \\sqrt{ab}"],["concept/theorem-acute-angle-with-projection-is-the-least-angle-with-lines-in-the-plane",7,"theorem: acute angle with projection is the least angle with lines in the plane"],["slaught-lennes-solid-geometry-1919/eq-55c2dea1fc",16,"Slaught & Lennes 1919, p. 42: \\angle DFE = \\angle D'F'E'"],["concept/theorem-equal-face-angles-give-equal-opposite-dihedral-angles",7,"theorem: equal face angles give equal opposite dihedral angles"],["thompson-calculus-made-easy-1914/eq-7fe626655f",16,"Thompson 1914, p. 12: \\frac{dy}{dx} = - \\frac{0.11}{1}"],["ball-mathematical-recreations-1905/x-4f9179e7d4",15,"Ball 1905, scan 43: The error in each of the foregoing examples is ..."],["hardy-course-of-pure-mathematics-1921/eq-248d05baef",16,"Hardy 1921, p. 386: \\frac{dy}{dx} = y"],["concept/upper-limit-to-the-roots",7,"upper limit to the roots","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-upper-limit-to-the-roots"],["thompson-calculus-made-easy-1914/eq-475cdd1488",16,"Thompson 1914, p. 11: \\frac{dy}{dx} = \\frac{1}{1.73}"],["thompson-calculus-made-easy-1914/eq-04404005bc",16,"Thompson 1914, p. 13: x^2 + y^2 = l^2"],["concept/lower-limit-to-the-roots",7,"lower limit to the roots","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-lower-limit-to-the-roots"],["hardy-course-of-pure-mathematics-1921/eq-9a497d48d9",16,"Hardy 1921, p. 386: x = \\int_{1}^{y} \\frac{dt}{t}"],["thompson-calculus-made-easy-1914/ch-iii",2,"Thompson 1914, ch. III: On Relative Growings","../books/thompson-calculus-made-easy-1914/ch/ch-iii/index.html"],["theorem/tangent-plane-perpendicular-to-radius",9,"tangent plane perpendicular to radius","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-tangent-plane-perpendicular-to-radius"],["hardy-course-of-pure-mathematics-1921/eq-9929bdda9c",16,"Hardy 1921, p. 387: (\\exp x)^{n} = \\exp nx"],["hardy-course-of-pure-mathematics-1921/ex-lv/4",4,"Hardy 1921, Exercise LV (4)"],["theorem/intersection-of-two-spherical-surfaces-is-a-circle",9,"intersection of two spherical surfaces is a circle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-intersection-of-two-spherical-surfaces-is-a-circle"],["wentworth-first-steps-in-algebra-1894/ex-17/4",4,"Wentworth 1894, Exercise 17 (4)"],["hardy-course-of-pure-mathematics-1921/eq-2a1fccc1b3",16,"Hardy 1921, p. 387: \\exp x \\exp(-x) = 1"],["wentworth-first-steps-in-algebra-1894/eq-d224a759c4",16,"Wentworth 1894, p. 44: a^{m} - a^{n} = a^{m - n}"],["shape/9a5fa823fa",6,"identity: (N*a + 2*N*a**N + N)/(a + a**N - 1)"],["wentworth-first-steps-in-algebra-1894/eq-fb702170bc",16,"Wentworth 1894, p. 43: -ab ÷ (-a) = +b"],["wentworth-first-steps-in-algebra-1894/eq-83a8b029bc",16,"Wentworth 1894, p. 43: +ab ÷ (-a) = -b"],["wentworth-first-steps-in-algebra-1894/eq-d7d115c3f1",16,"Wentworth 1894, p. 79: (x + y)^{2} = x^{2} + 2xy + y^{2}"],["thompson-calculus-made-easy-1914/x-df408bcf42",15,"Thompson 1914, p. 191: But here comes in a curious point. We should ..."],["hardy-course-of-pure-mathematics-1921/eq-fab4a5d4e1",16,"Hardy 1921, p. 387: \\exp x = (\\exp 1)^{x} = e^{x}"],["boyden-first-book-in-algebra-1895/ex-20/4",4,"Boyden 1895, Exercise 20 (4)"],["theorem/shortest-distance-on-a-sphere-follows-a-great-circle-arc",9,"shortest distance on a sphere follows a great circle arc","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-shortest-distance-on-a-sphere-follows-a-great-circle-arc"],["hardy-course-of-pure-mathematics-1921/eq-bae6802363",16,"Hardy 1921, p. 387: e = \\exp 1 = 1 + 1 + \\frac{1}{2!} + \\frac{1}{3!} + \\dots"],["hardy-course-of-pure-mathematics-1921/ex-xxxiii",3,"Hardy 1921, Exercise XXXIII"],["wentworth-first-steps-in-algebra-1894/eq-a28804a446",16,"Wentworth 1894, p. 51: a + (b + c) &= a + b + c"],["wentworth-first-steps-in-algebra-1894/eq-8f84c3b570",16,"Wentworth 1894, p. 51: a + b + c &= a + (b + c)"],["wentworth-first-steps-in-algebra-1894/eq-e10ecaf800",16,"Wentworth 1894, p. 51: a + (b - c) &= a + b - c"],["wentworth-first-steps-in-algebra-1894/eq-e6c973701d",16,"Wentworth 1894, p. 150: s = \\frac{a(r^{n} - 1)}{r - 1}"],["hardy-course-of-pure-mathematics-1921/x-8bf6c132c0",15,"Hardy 1921, p. 1: A fraction r = p/q, where p and q ..."],["wentworth-first-steps-in-algebra-1894/x-e3797b8e51",15,"Wentworth 1894, p. 64: The square of the sum of two numbers is ..."],["hardy-course-of-pure-mathematics-1921/ex-lv/5",4,"Hardy 1921, Exercise LV (5)"],["theorem/parallel-planes-intercept-proportional-segments",9,"parallel planes intercept proportional segments","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-parallel-planes-intercept-proportional-segments"],["theorem/parallel-planes-intercept-equal-segments-on-every-transversal",9,"parallel planes intercept equal segments on every transversal","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-parallel-planes-intercept-equal-segments-on-every-transversal"],["theorem/dihedral-angles-equal-when-plane-angles-are-equal",9,"dihedral angles equal when plane angles are equal","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-dihedral-angles-equal-when-plane-angles-are-equal"],["theorem/locus-of-points-equidistant-from-the-faces-of-a-dihedral-angle",9,"locus of points equidistant from the faces of a dihedral angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-locus-of-points-equidistant-from-the-faces-of-a-dihedral-angle"],["theorem/perpendicular-from-one-of-two-perpendicular-planes-to-their-intersection-is-perpendicular-to-the-other",9,"perpendicular from one of two perpendicular planes to their intersection is perpendicular to the other","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-perpendicular-from-one-of-two-perpendicular-planes-to-their-intersection-is-perpendicular-to-the-other"],["hardy-course-of-pure-mathematics-1921/eq-7412c75fbf",16,"Hardy 1921, p. 395: z = x + iy"],["form/92d22c7fcb",5,"factor: -a**3 - a**2 + x**3 + x**2"],["shape/6c12ab2c79",6,"factor: -2*a**N + 2*x**N"],["hardy-course-of-pure-mathematics-1921/ex-lv/6",4,"Hardy 1921, Exercise LV (6)"],["wentworth-first-steps-in-algebra-1894/x-5186a76e49",15,"Wentworth 1894, p. 64: The square of the difference of two numbers is ..."],["wentworth-first-steps-in-algebra-1894/x-bca6da8048",15,"Wentworth 1894, p. 66: The middle term of each result has for a ..."],["wentworth-first-steps-in-algebra-1894/x-cad1955f3c",15,"Wentworth 1894, p. 66: The intermediate step given above may be omitted, and ..."],["boyden-first-book-in-algebra-1895/ch-algebraic-expressions",2,"Boyden 1895, ALGEBRAIC EXPRESSIONS","../books/boyden-first-book-in-algebra-1895/ch/ch-algebraic-expressions/index.html"],["boyden-first-book-in-algebra-1895/ex-18",3,"Boyden 1895, Exercise 18"],["thompson-calculus-made-easy-1914/ch-viii",2,"Thompson 1914, ch. VIII: When Time Varies","../books/thompson-calculus-made-easy-1914/ch/ch-viii/index.html"],["boyden-first-book-in-algebra-1895/ch-multiplication",2,"Boyden 1895, MULTIPLICATION","../books/boyden-first-book-in-algebra-1895/ch/ch-multiplication/index.html"],["hardy-course-of-pure-mathematics-1921/eq-a14adef8cc",16,"Hardy 1921, p. 395: |z| = \\sqrtp{x^{2} + y^{2}}"],["boyden-first-book-in-algebra-1895/ex-24",3,"Boyden 1895, Exercise 24"],["boyden-first-book-in-algebra-1895/ch-division",2,"Boyden 1895, DIVISION","../books/boyden-first-book-in-algebra-1895/ch/ch-division/index.html"],["boyden-first-book-in-algebra-1895/ex-25",3,"Boyden 1895, Exercise 25"],["hardy-course-of-pure-mathematics-1921/eq-93d92edb58",16,"Hardy 1921, p. 395: \\am z = \\arctan(y/x)"],["hardy-course-of-pure-mathematics-1921/ex-lv/7",4,"Hardy 1921, Exercise LV (7)"],["wentworth-first-steps-in-algebra-1894/x-803c638cbd",15,"Wentworth 1894, p. 67: (-8) + (-7) = -15, (-8)(-7) = +56."],["wentworth-first-steps-in-algebra-1894/x-22060ce047",15,"Wentworth 1894, p. 68: The difference of the squares of two numbers is ..."],["hardy-course-of-pure-mathematics-1921/ex-lv/8",4,"Hardy 1921, Exercise LV (8)"],["concept/probability",7,"probability","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-probability"],["boyden-first-book-in-algebra-1895/ex-30",3,"Boyden 1895, Exercise 30"],["wentworth-first-steps-in-algebra-1894/eq-7932ebc495",16,"Wentworth 1894, p. 79: (x - y)^{2} = x^{2} - 2xy + y^{2}"],["wentworth-first-steps-in-algebra-1894/eq-c55d9182a7",16,"Wentworth 1894, p. 51: a + b - c &= a + (b - c)"],["hardy-course-of-pure-mathematics-1921/x-7629812bb1",15,"Hardy 1921, p. 5: Now it is very easy to see that the ..."],["hardy-course-of-pure-mathematics-1921/x-273bb2c3a3",15,"Hardy 1921, p. 6: But it is easy to see that there is ..."],["wentworth-first-steps-in-algebra-1894/x-26fc7e8d79",15,"Wentworth 1894, p. 33: If we wish to subtract 5 from 2, we ..."],["todhunter-spherical-trigonometry-1886/ex-v",3,"Todhunter 1886, Exercise V"],["maxwell-elementary-treatise-electricity-1888/eq-5518a7e0ae",16,"Maxwell 1888, scan 225: ( Q + S + B )z - Sy - Q \\overline{x + y} &= E"],["wentworth-first-steps-in-algebra-1894/eq-670b148831",16,"Wentworth 1894, p. 51: a - (b + c) &= a - b - c"],["wentworth-first-steps-in-algebra-1894/eq-03e187dbe8",16,"Wentworth 1894, p. 51: a - b - c &= a - (b + c)"],["wentworth-first-steps-in-algebra-1894/ch-ix",2,"Wentworth 1894, ch. IX: Fractions","../books/wentworth-first-steps-in-algebra-1894/ch/ch-ix/index.html"],["todhunter-spherical-trigonometry-1886/ch-solution-of-right-angled-triangles",2,"Todhunter 1886, Solution of Right-angled Triangles","../books/todhunter-spherical-trigonometry-1886/ch/ch-solution-of-right-angled-triangles/index.html"],["thompson-calculus-made-easy-1914/eq-87c041607f",16,"Thompson 1914, p. 14: y &= x \\tan 30°"],["hardy-course-of-pure-mathematics-1921/ex-lv/9",4,"Hardy 1921, Exercise LV (9)"],["method/inserting-arithmetical-means",8,"inserting arithmetical means","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-method-inserting-arithmetical-means"],["theorem/sum-of-an-arithmetical-progression",9,"sum of an arithmetical progression","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-theorem-sum-of-an-arithmetical-progression"],["wentworth-first-steps-in-algebra-1894/x-a314bcd761",15,"Wentworth 1894, p. 142: A series of numbers is said to form an ..."],["wentworth-first-steps-in-algebra-1894/x-72a47cca8d",15,"Wentworth 1894, p. 142: the coefficient of d in each term being always ..."],["wentworth-first-steps-in-algebra-1894/x-f5de054e82",15,"Wentworth 1894, p. 145: it is evident that the series beginning with the ..."],["thompson-calculus-made-easy-1914/eq-5884d32bdf",16,"Thompson 1914, p. 14: y &= \\sqrt{ l^2 - x^2}"],["thompson-calculus-made-easy-1914/eq-5663b53b18",16,"Thompson 1914, p. 14: x^2 + 3 = 2y - 7"],["maxwell-elementary-treatise-electricity-1888/eq-63c64b3508",16,"Maxwell 1888, scan 225: ( P + G + Q )x + ( P + Q )y - Qz &= 0"],["blackburn-elements-plane-trigonometry-1863/eq-ae67d7b780",16,"Blackburn 1863, p. 19: \\theta = \\frac{AB}{AC} = \\frac{AB}{R}"],["hardy-course-of-pure-mathematics-1921/x-d15649689d",15,"Hardy 1921, p. 7: We can therefore divide the rational numbers into two ..."],["maxwell-elementary-treatise-electricity-1888/eq-bd97b7d277",16,"Maxwell 1888, scan 225: -Gx + ( R + S )y - Sz &= 0"],["hardy-course-of-pure-mathematics-1921/eq-8c323ef3cc",16,"Hardy 1921, p. 396: \\log x = \\int_{1}^{x} \\frac{dt}{t}"],["blackburn-elements-plane-trigonometry-1863/eq-61c6a68755",16,"Blackburn 1863, p. 17: \\therefore 1° = 60'"],["blackburn-elements-plane-trigonometry-1863/eq-081328cd8f",16,"Blackburn 1863, p. 17: \\therefore 1' = 60''"],["blackburn-elements-plane-trigonometry-1863/eq-81e5fc28a5",16,"Blackburn 1863, p. 18: 1'' = 60'''"],["blackburn-elements-plane-trigonometry-1863/eq-3570ea0d60",16,"Blackburn 1863, p. 18: 1''' = 60^\\text{iv}"],["blackburn-elements-plane-trigonometry-1863/eq-1777a0cfbf",16,"Blackburn 1863, p. 18: 1^\\text{iv} = 60^\\text{v}"],["blackburn-elements-plane-trigonometry-1863/eq-710183bf7c",16,"Blackburn 1863, p. 19: AB = R\\theta"],["hardy-course-of-pure-mathematics-1921/x-e347f6c98c",15,"Hardy 1921, p. 4: given any rational number r, and any positive integer ..."],["wentworth-first-steps-in-algebra-1894/x-2bb33fd0be",15,"Wentworth 1894, p. 143: Hence, the arithmetical mean of any two numbers is ..."],["wentworth-first-steps-in-algebra-1894/x-be59a84bc4",15,"Wentworth 1894, p. 145: Putting for l its value a + (n - ..."],["wentworth-first-steps-in-algebra-1894/x-2b8527e620",15,"Wentworth 1894, p. 142: If d is positive, the progression is an increasing ..."],["wentworth-first-steps-in-algebra-1894/x-b4bec6cd19",15,"Wentworth 1894, p. 34: In order to subtract a greater number from a ..."],["wentworth-first-steps-in-algebra-1894/x-d48699ca33",15,"Wentworth 1894, p. 35: When no sign stands before a number, the sign ..."],["wentworth-first-steps-in-algebra-1894/x-11cfed4918",15,"Wentworth 1894, p. 35: In the first sense they are signs of operations, ..."],["boyden-first-book-in-algebra-1895/ch-greatest-common-factor",2,"Boyden 1895, GREATEST COMMON FACTOR","../books/boyden-first-book-in-algebra-1895/ch/ch-greatest-common-factor/index.html"],["maxwell-elementary-treatise-electricity-1888/eq-e2e6655f2b",16,"Maxwell 1888, scan 225: -Qx - ( S + Q )y + ( Q + S + B )z &= E"],["hardy-course-of-pure-mathematics-1921/eq-8324d8969c",16,"Hardy 1921, p. 397: \\int_{C} \\{g(x, y)\\, dx + h(x, y)\\, dy\\}"],["hardy-course-of-pure-mathematics-1921/ex-lv/10",4,"Hardy 1921, Exercise LV (10)"],["unit/spherical-degree",12,"spherical degree","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-unit-spherical-degree"],["slaught-lennes-solid-geometry-1919/eq-cf2f6763cf",16,"Slaught & Lennes 1919, p. 82: S = \\tfrac{1}{2} l\\cdot p"],["concept/theorem-lateral-area-of-a-regular-pyramid",7,"theorem: lateral area of a regular pyramid"],["slaught-lennes-solid-geometry-1919/eq-a3f9b2e582",16,"Slaught & Lennes 1919, p. 83: \\dfrac{b'}{b} = \\dfrac{\\overline{PK'}^2}{\\overline{PK}^2}"],["form/ecdf78336b",5,"factor: a**2 + 2*a*x + x**2"],["slaught-lennes-solid-geometry-1919/ch-book-iv",2,"Slaught & Lennes 1919, ch. BOOK IV: Pyramids and Cones","../books/slaught-lennes-solid-geometry-1919/ch/ch-book-iv/index.html"],["shape/aeef1ed4a9",6,"factor: N*a*x + a**N + x**N"],["form/04d7776115",5,"identity: (3*a + 2)/(a**2 + a + 2) - 1"],["shape/93a6b45752",6,"identity: (N*a + N)/(N + a + a**N) - 1"],["theorem/plane-containing-a-perpendicular-line-is-perpendicular-to-the-plane",9,"plane containing a perpendicular line is perpendicular to the plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-plane-containing-a-perpendicular-line-is-perpendicular-to-the-plane"],["hardy-course-of-pure-mathematics-1921/x-15ec1ec725",15,"Hardy 1921, p. 12: It should be observed that we do not obtain ..."],["theorem/plane-perpendicular-to-two-planes-is-perpendicular-to-their-intersection",9,"plane perpendicular to two planes is perpendicular to their intersection","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-plane-perpendicular-to-two-planes-is-perpendicular-to-their-intersection"],["theorem/projection-of-a-line-is-a-straight-line",9,"projection of a line is a straight line","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-projection-of-a-line-is-a-straight-line"],["theorem/trihedral-angles-equal-by-two-face-angles-and-included-dihedral-angle",9,"trihedral angles equal by two face angles and included dihedral angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-trihedral-angles-equal-by-two-face-angles-and-included-dihedral-angle"],["theorem/trihedral-angles-equal-by-a-face-angle-and-two-adjacent-dihedral-angles",9,"trihedral angles equal by a face angle and two adjacent dihedral angles","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-trihedral-angles-equal-by-a-face-angle-and-two-adjacent-dihedral-angles"],["theorem/equal-face-angles-give-equal-opposite-dihedral-angles",9,"equal face angles give equal opposite dihedral angles","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-equal-face-angles-give-equal-opposite-dihedral-angles"],["theorem/construction-postulates",9,"construction postulates","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-construction-postulates"],["theorem/determination-of-a-plane-by-a-line-and-point-two-intersecting-lines-or-two-parallel-lines",9,"determination of a plane by a line and point, two intersecting lines, or two parallel lines","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-determination-of-a-plane-by-a-line-and-point-two-intersecting-lines-or-two-parallel-lines"],["hardy-course-of-pure-mathematics-1921/ex-lv/11",4,"Hardy 1921, Exercise LV (11)"],["maxwell-elementary-treatise-electricity-1888/x-55bff8b0ed",15,"Maxwell 1888, scan 44: Since an electrified system is subject to the law ..."],["form/a82b6fdc85",5,"solve: (Eq(-5*a + 3*x, 15), Eq(3*a + 5*x, 8))"],["thompson-calculus-made-easy-1914/eq-54f6bbf103",16,"Thompson 1914, p. 14: y = \\dfrac{x^2 + 10}{2}"],["thompson-calculus-made-easy-1914/eq-04c4f2ed99",16,"Thompson 1914, p. 15: u = x^2 \\sin \\theta"],["concept/theorem-section-of-a-pyramid-parallel-to-its-base",7,"theorem: section of a pyramid parallel to its base"],["blackburn-elements-plane-trigonometry-1863/eq-7f87858dba",16,"Blackburn 1863, p. 22: \\sin\\theta &= \\sin(2m\\pi + \\theta);"],["concept/theorem-volume-of-a-pyramid",7,"theorem: volume of a pyramid"],["slaught-lennes-solid-geometry-1919/eq-97a9521634",16,"Slaught & Lennes 1919, p. 87: V = \\tfrac{1}{3} bh"],["slaught-lennes-solid-geometry-1919/eq-43931c622a",16,"Slaught & Lennes 1919, p. 89: V = \\tfrac{1}{3} hb + \\tfrac{1}{3} hb' + \\tfrac{1}{3} h \\sqrt{bb'} = \\tfrac{1}{3} h [b + b' + \\sqrt{bb'}]"],["concept/theorem-volume-of-a-frustum-of-a-pyramid",7,"theorem: volume of a frustum of a pyramid"],["blackburn-elements-plane-trigonometry-1863/eq-7bdb332822",16,"Blackburn 1863, p. 22: \\cos\\theta &= \\cos(2m\\pi + \\theta)."],["blackburn-elements-plane-trigonometry-1863/eq-908c8bb301",16,"Blackburn 1863, p. 23: \\sin(-\\theta) &= -\\sin\\theta;"],["maxwell-elementary-treatise-electricity-1888/x-0e4a37480d",15,"Maxwell 1888, scan 40: If the electrified body is moved in such a ..."],["hardy-course-of-pure-mathematics-1921/eq-e56422ab1d",16,"Hardy 1921, p. 1: r = p/q"],["hardy-course-of-pure-mathematics-1921/eq-9b5e363b14",16,"Hardy 1921, p. 1: p/(-q) = (-p)/q"],["maxwell-elementary-treatise-electricity-1888/x-571157f51f",15,"Maxwell 1888, scan 48: There are many such reciprocal relations. They occur in ..."],["hardy-course-of-pure-mathematics-1921/ex-lv/12",4,"Hardy 1921, Exercise LV (12)"],["thompson-calculus-made-easy-1914/x-d06cb0e44a",15,"Thompson 1914, p. 27: So if we take the letter a, or b, ..."],["hardy-course-of-pure-mathematics-1921/eq-cd02b1069d",16,"Hardy 1921, p. 1: (-p)/(-q) = p/q"],["method/constructing-a-plane-perpendicular-to-a-line",8,"constructing a plane perpendicular to a line","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-constructing-a-plane-perpendicular-to-a-line"],["thompson-calculus-made-easy-1914/x-9e4f83dcbe",15,"Thompson 1914, p. 179: Clearly A is maximum when P is maximum."],["maxwell-elementary-treatise-electricity-1888/eq-901085aa7b",16,"Maxwell 1888, scan 225: & = \\frac{E(QR-PS)}{\\Delta}\\text{,}"],["hardy-course-of-pure-mathematics-1921/ex-lv/13",4,"Hardy 1921, Exercise LV (13)"],["wentworth-first-steps-in-algebra-1894/eq-df1a52b47d",16,"Wentworth 1894, p. 51: a - b + c &= a - (b - c)"],["method/constructing-a-line-perpendicular-to-a-plane",8,"constructing a line perpendicular to a plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-constructing-a-line-perpendicular-to-a-plane"],["wentworth-first-steps-in-algebra-1894/eq-267658dc32",16,"Wentworth 1894, p. 120: t = \\frac{a - p}{pr}"],["wentworth-first-steps-in-algebra-1894/eq-65d6f5b071",16,"Wentworth 1894, p. 51: a - (b - c) &= a - b + c"],["wentworth-first-steps-in-algebra-1894/eq-a7eb0e6569",16,"Wentworth 1894, p. 120: r = \\frac{a - p}{pt}"],["wentworth-first-steps-in-algebra-1894/eq-9c8f9bf8f6",16,"Wentworth 1894, p. 53: a(b + c) &= ab + ac"],["maxwell-elementary-treatise-electricity-1888/ch-iii",2,"Maxwell 1888, ch. III: ON ELECTRICAL WORK AND ENERGY","../books/maxwell-elementary-treatise-electricity-1888/ch/ch-iii/index.html"],["wentworth-first-steps-in-algebra-1894/eq-e7d0036322",16,"Wentworth 1894, p. 54: M(m + n + p) = Mm + Mn + Mp"],["wentworth-first-steps-in-algebra-1894/eq-3ab69dec11",16,"Wentworth 1894, p. 57: a(b + c - d) &= ab + ac - ad"],["wentworth-first-steps-in-algebra-1894/eq-8664262bc6",16,"Wentworth 1894, p. 118: x = \\frac{s - d}{2}"],["wentworth-first-steps-in-algebra-1894/eq-6a0a47b52c",16,"Wentworth 1894, p. 119: x = \\frac{ab}{a + b}"],["wentworth-first-steps-in-algebra-1894/eq-2093d86384",16,"Wentworth 1894, p. 150: s = \\frac{a(1 - r^{n})}{1 - r}"],["maxwell-elementary-treatise-electricity-1888/eq-9b98794d09",16,"Maxwell 1888, scan 44: Q = \\tfrac{1}{2}\\sum(EP)"],["thompson-calculus-made-easy-1914/x-624a1e39ea",15,"Thompson 1914, p. 177: The variation when both the radius and the height ..."],["hardy-course-of-pure-mathematics-1921/ex-lv/14",4,"Hardy 1921, Exercise LV (14)"],["form/6c01a906a0",5,"factor: x**2 + 5*x - 6"],["boyden-first-book-in-algebra-1895/ch-modes-of-representing-the-operations",2,"Boyden 1895, MODES OF REPRESENTING THE OPERATIONS","../books/boyden-first-book-in-algebra-1895/ch/ch-modes-of-representing-the-operations/index.html"],["wentworth-first-steps-in-algebra-1894/ch-vi",2,"Wentworth 1894, ch. VI: Multiplication and Division","../books/wentworth-first-steps-in-algebra-1894/ch/ch-vi/index.html"],["wentworth-first-steps-in-algebra-1894/eq-20f9664b9e",16,"Wentworth 1894, p. 68: \\frac{a^{2} - b^{2}}{a + b} = a - b"],["wentworth-first-steps-in-algebra-1894/eq-acd11efd91",16,"Wentworth 1894, p. 68: \\frac{a^{2} - b^{2}}{a - b} = a + b"],["wentworth-first-steps-in-algebra-1894/ch-xi",2,"Wentworth 1894, ch. XI: Simultaneous Equations of the First Degree","../books/wentworth-first-steps-in-algebra-1894/ch/ch-xi/index.html"],["wentworth-first-steps-in-algebra-1894/eq-ccad3db812",16,"Wentworth 1894, p. 122: x + y = 10"],["hardy-course-of-pure-mathematics-1921/x-7c262851b6",15,"Hardy 1921, p. 348: Thus the logarithmic series converges at all points of ..."],["hardy-course-of-pure-mathematics-1921/x-6df0ca4ae2",15,"Hardy 1921, p. 349: It shows that the same function f(z) cannot be ..."],["form/46ebe6e9ed",5,"solve: (Eq(-2*a/7 + 3*x/5, 35), Eq(2*a + x, -63))"],["hardy-course-of-pure-mathematics-1921/eq-89f759d1c2",16,"Hardy 1921, p. 2: A_{0}A_{r}/A_{0}A_{1} = r"],["method/constructing-a-common-perpendicular-to-two-skew-lines",8,"constructing a common perpendicular to two skew lines","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-constructing-a-common-perpendicular-to-two-skew-lines"],["slaught-lennes-solid-geometry-1919/x-2a940e24b4",15,"Slaught & Lennes 1919, p. 28: Two half-planes meeting in a common edge form a ..."],["slaught-lennes-solid-geometry-1919/x-9917bf5de1",15,"Slaught & Lennes 1919, p. 28: angle may be thought of as generated by the ..."],["slaught-lennes-solid-geometry-1919/x-5588179ddd",15,"Slaught & Lennes 1919, p. 32: point on a plane is the foot of the ..."],["slaught-lennes-solid-geometry-1919/x-7f11d71368",15,"Slaught & Lennes 1919, p. 33: The acute angle formed by a straight line with ..."],["maxwell-elementary-treatise-electricity-1888/eq-d803c495fa",16,"Maxwell 1888, scan 225: QR - PS = 0 \\text{, or } \\frac{P}{Q} =\\frac{R}{S}\\text{.}"],["form/a045ea7170",5,"factor: x**6 + 4*x**3 - 77"],["whitehead-introduction-to-mathematics-1911/ch-bibliography",2,"Whitehead 1911, Bibliography","../books/whitehead-introduction-to-mathematics-1911/ch/ch-bibliography/index.html"],["slaught-lennes-solid-geometry-1919/x-8a33e53a3f",15,"Slaught & Lennes 1919, p. 41: The trihedral angles O and O' cannot be made ..."],["slaught-lennes-solid-geometry-1919/x-f5687e2e63",15,"Slaught & Lennes 1919, p. 27: Show that line-segments included between parallel planes and perpendicular ..."],["hardy-course-of-pure-mathematics-1921/x-cf1ed062e5",15,"Hardy 1921, p. 353: A player tossing a coin is to score one ..."],["hardy-course-of-pure-mathematics-1921/eq-2de524e8cd",16,"Hardy 1921, p. 2: AB = -BA"],["slaught-lennes-solid-geometry-1919/ch-book-i",2,"Slaught & Lennes 1919, ch. BOOK I: Properties of the Plane","../books/slaught-lennes-solid-geometry-1919/ch/ch-book-i/index.html"],["hardy-course-of-pure-mathematics-1921/eq-1f52444f80",16,"Hardy 1921, p. 2: A_{0}A_{-s} = -A_{-s}A_{0}"],["hardy-course-of-pure-mathematics-1921/x-4de50457c1",15,"Hardy 1921, p. 170: This function is equal to 1 for all values ..."],["dickson-theory-of-equations-1922/ch-vi",2,"Dickson 1922, ch. VI: Isolation of the Real Roots of a Real Equation","../books/dickson-theory-of-equations-1922/ch/ch-vi/index.html"],["dickson-theory-of-equations-1922/ex-page78",3,"Dickson 1922, Exercise Page78"],["dickson-theory-of-equations-1922/ex-page74",3,"Dickson 1922, Exercise Page74"],["theorem/divisibility-by-nine",9,"divisibility by nine","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-divisibility-by-nine"],["dickson-theory-of-equations-1922/ex-page83",3,"Dickson 1922, Exercise Page83"],["concept/existence-conditions-of-a-triangle",7,"existence conditions of a triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-existence-conditions-of-a-triangle"],["form/f1aba51696",5,"factor: -20*a**2 - a*x + x**2"],["hardy-course-of-pure-mathematics-1921/eq-062e127eca",16,"Hardy 1921, p. 3: A_{0}A_{r} = r · A_{0}A_{1}"],["shape/12481e2c53",6,"solve: (Eq(N*a + N*x, N*a + N*x + N), Eq(N*a + N*x, N*a + N*x + N))"],["dickson-theory-of-equations-1922/ex-page81",3,"Dickson 1922, Exercise Page81"],["dickson-theory-of-equations-1922/eq-e3f521bfae",16,"Dickson 1922, p. 99: f'(x) - f'''(x) \\frac{y^2}{1·2·3} + f^{(5)}(x)\\frac{y^4}{5!} - \\dotsb = 0"],["hardy-course-of-pure-mathematics-1921/eq-aca0a6219f",16,"Hardy 1921, p. 3: A_{0}A_{r} = r"],["dickson-theory-of-equations-1922/ex-page79",3,"Dickson 1922, Exercise Page79"],["hardy-course-of-pure-mathematics-1921/eq-f06381569d",16,"Hardy 1921, p. 3: k · BC > 1"],["ball-mathematical-recreations-1905/ch-i",2,"Ball 1905, ch. I: Some Arithmetical Questions","../books/ball-mathematical-recreations-1905/ch/ch-i/index.html"],["form/6cf6ca677c",5,"factor: -12*a**2 + a*x**2 + x**4"],["dickson-theory-of-equations-1922/ex-page85",3,"Dickson 1922, Exercise Page85"],["shape/1cd0bacc75",6,"factor: N*a**N + a*x**N + x**N"],["hardy-course-of-pure-mathematics-1921/eq-27148deeca",16,"Hardy 1921, p. 7: x^{2} = 1"],["ball-mathematical-recreations-1905/x-6d74caca9f",15,"Ball 1905, scan 21: I may recall the fundamental rule that no trick, ..."],["hardy-course-of-pure-mathematics-1921/eq-ae785d6659",16,"Hardy 1921, p. 5: x^{2} = 2"],["planck-treatise-on-thermodynamics-1903/eq-c61a7f871f",16,"Planck 1903, p. 109: d\\Phi_{0} = -\\frac{Q}{\\theta}"],["form/b085ba92c1",5,"factor: -a**2*b**3*x**3 + a**2 + b**3*x**5 - x**2"],["hardy-course-of-pure-mathematics-1921/eq-a4c4b75016",16,"Hardy 1921, p. 5: a + b = b + a"],["planck-treatise-on-thermodynamics-1903/eq-28f91b322e",16,"Planck 1903, p. 109: d\\Phi_{0} = -\\frac{dU - W}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-b29a20472d",16,"Planck 1903, p. 109: d\\Phi - \\frac{dU - W}{\\theta} \\geq 0"],["hardy-course-of-pure-mathematics-1921/eq-ffef7ae279",16,"Hardy 1921, p. 5: a + (b + c) = (a + b) + c"],["hardy-course-of-pure-mathematics-1921/eq-05e1c6e871",16,"Hardy 1921, p. 5: ab = ba"],["shape/b25ef8e4de",6,"factor: -a**N*b**N*x**N + a**N + b**N*x**N - x**N"],["slaught-lennes-solid-geometry-1919/eq-b0ea85892c",16,"Slaught & Lennes 1919, p. 207: V = \\tfrac{4}{3} \\pi r^3"],["slaught-lennes-solid-geometry-1919/eq-fa944052f1",16,"Slaught & Lennes 1919, p. 207: V = \\tfrac{1}{3}rS"],["slaught-lennes-solid-geometry-1919/eq-6f4168ebad",16,"Slaught & Lennes 1919, p. 207: S = \\tfrac{3}{r} \\cdot \\tfrac{4}{3} \\pi r^3 = 4 \\pi r^2"],["slaught-lennes-solid-geometry-1919/eq-bd67296c88",16,"Slaught & Lennes 1919, p. 203: U = V"],["slaught-lennes-solid-geometry-1919/eq-db5918bc12",16,"Slaught & Lennes 1919, p. 203: p = p'"],["theorem/assumption-on-the-area-of-a-sphere",9,"assumption on the area of a sphere","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-assumption-on-the-area-of-a-sphere"],["hardy-course-of-pure-mathematics-1921/eq-7b1b828802",16,"Hardy 1921, p. 5: a(bc) = (ab)c"],["theorem/angle-and-corresponding-arc-of-polar-triangle-sum-to-180-degrees",9,"angle and corresponding arc of polar triangle sum to 180 degrees","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-angle-and-corresponding-arc-of-polar-triangle-sum-to-180-degrees"],["whitehead-introduction-to-mathematics-1911/ch-viii",2,"Whitehead 1911, ch. VIII: Imaginary Numbers (\\textit{C\\MakeLowercase{ontinued}})","../books/whitehead-introduction-to-mathematics-1911/ch/ch-viii/index.html"],["boyden-first-book-in-algebra-1895/ex-38",3,"Boyden 1895, Exercise 38"],["hardy-course-of-pure-mathematics-1921/eq-b437ed4649",16,"Hardy 1921, p. 5: a(b + c) = ab + ac"],["slaught-lennes-solid-geometry-1919/eq-021b0ad7eb",16,"Slaught & Lennes 1919, p. 203: A = A'"],["hardy-course-of-pure-mathematics-1921/ex-lv/15",4,"Hardy 1921, Exercise LV (15)"],["form/d9422905fd",5,"solve: (Eq(x/(a + 4), 1/4), Eq((x + 1)/a, 1/8))"],["thompson-calculus-made-easy-1914/eq-0fda498405",16,"Thompson 1914, p. 6: x^2 + 2x · dx + (dx)^2"],["boyden-first-book-in-algebra-1895/ex-38/13",4,"Boyden 1895, Exercise 38 (13)"],["thompson-calculus-made-easy-1914/x-670d7d12ef",15,"Thompson 1914, p. 29: If we had begun with y = ax^n, we ..."],["cap/core.limit.num",17,"core.limit.num"],["shape/b200dfb8f0",6,"solve: (Eq(x/(N + a), N), Eq((x + 1)/a, N))"],["form/24fb9b556b",5,"factor: x**3 + 2*x**2 - x - 2"],["thompson-calculus-made-easy-1914/ch-ii",2,"Thompson 1914, ch. II: On Different Degrees of Smallness","../books/thompson-calculus-made-easy-1914/ch/ch-ii/index.html"],["shape/703b061380",6,"factor: N*x**N + N - x + x**N"],["whitehead-introduction-to-mathematics-1911/x-9c4b0f1da9",15,"Whitehead 1911, p. 252: The science has grown to such vast proportions that ..."],["thompson-calculus-made-easy-1914/x-a55e413174",15,"Thompson 1914, p. 231: A beginner is liable to overlook certain points that ..."],["wentworth-plane-geometry-1899/ch-introduction",2,"Wentworth 1899, INTRODUCTION","../books/wentworth-plane-geometry-1899/ch/ch-introduction/index.html"],["hardy-course-of-pure-mathematics-1921/eq-7a49b0b08d",16,"Hardy 1921, p. 5: A_{0}P = x"],["form/7eea74fe46",5,"solve: (Eq(-a/9 + x/9, 4), Eq(a/3 + x/3, 30))"],["maxwell-elementary-treatise-electricity-1888/eq-2015cf8f94",16,"Maxwell 1888, scan 147: a - b - x = 0"],["maxwell-elementary-treatise-electricity-1888/eq-b30a5840ec",16,"Maxwell 1888, scan 154: (\\phi_2 - \\phi_1)(t - t_1)"],["thompson-calculus-made-easy-1914/eq-858d3636f3",16,"Thompson 1914, p. 239: \\ds\\int u dv = uv - \\int v du"],["hardy-course-of-pure-mathematics-1921/eq-21ae12af75",16,"Hardy 1921, p. 6: x = \\sqrt{2}"],["todhunter-spherical-trigonometry-1886/x-591ba3ff2f",15,"Todhunter 1886, scan 96: The degree of closeness with which the measured length ..."],["hardy-course-of-pure-mathematics-1921/eq-7dc3e8e2f7",16,"Hardy 1921, p. 10: x^{q} = n"],["thompson-calculus-made-easy-1914/eq-d7c4ddcc25",16,"Thompson 1914, p. 242: M\\, dx + N\\, dy = 0"],["planck-treatise-on-thermodynamics-1903/eq-274e2abc97",16,"Planck 1903, p. 109: dU - \\theta\\, d\\Phi \\leq W"],["whitehead-introduction-to-mathematics-1911/x-faf95e8c62",15,"Whitehead 1911, p. 252: This elementary course of mathematics is sufficient for some ..."],["hardy-course-of-pure-mathematics-1921/x-8cdff9d343",15,"Hardy 1921, p. 170: The graph of this function consists of the axis ..."],["concept/emulsion",7,"emulsion","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-emulsion"],["hardy-course-of-pure-mathematics-1921/eq-27b0b5fc44",16,"Hardy 1921, p. 10: n^{p/q} = (n^{1/q})^{p}"],["hardy-course-of-pure-mathematics-1921/ex-lvi/1",4,"Hardy 1921, Exercise LVI (1)"],["cap/other:taylor_remainder_bound",17,"other:taylor_remainder_bound"],["wentworth-first-steps-in-algebra-1894/eq-6e032054d2",16,"Wentworth 1894, p. 97: \\frac{a}{b} × \\frac{c}{d} = \\frac{ac}{bd}"],["hardy-course-of-pure-mathematics-1921/eq-0b6271637c",16,"Hardy 1921, p. 10: n^{p/q} n^{-p/q} = 1"],["concept/variation-of-sign",7,"variation of sign","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-variation-of-sign"],["concept/graded-dissociation",7,"graded dissociation","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-graded-dissociation"],["hardy-course-of-pure-mathematics-1921/eq-19ccb4fe51",16,"Hardy 1921, p. 10: n^{r} × n^{s} = n^{r+s}"],["concept/axis-of-symmetry",7,"axis of symmetry","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-axis-of-symmetry"],["law/laws-of-indices-product",10,"laws of indices (product)"],["concept/plane-geometry",7,"plane geometry","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-plane-geometry"],["concept/rhombus",7,"rhombus","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-rhombus"],["concept/geometrical-figure",7,"geometrical figure","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-geometrical-figure"],["theorem/budan-s-theorem",9,"Budan's theorem","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-budan-s-theorem"],["theorem/spherical-triangles-equal-by-two-sides-and-included-angle",9,"spherical triangles equal by two sides and included angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-spherical-triangles-equal-by-two-sides-and-included-angle"],["planck-treatise-on-thermodynamics-1903",0,"Planck, Treatise on Thermodynamics (1903)","../books/planck-treatise-on-thermodynamics-1903/index.html"],["wentworth-first-steps-in-algebra-1894/eq-67a3f12238",16,"Wentworth 1894, p. 132: ax^{2} + bx + c = 0"],["wentworth-first-steps-in-algebra-1894/eq-e6872b1c5d",16,"Wentworth 1894, p. 135: (x + b)^{2} = x^{2} + 2bx + b^{2}"],["wentworth-first-steps-in-algebra-1894/eq-4acc1f6299",16,"Wentworth 1894, p. 135: (x - b)^{2} = x^{2} - 2bx + b^{2}"],["wentworth-first-steps-in-algebra-1894/eq-d6cdd9ede5",16,"Wentworth 1894, p. 135: x^{2} + 2bx = c"],["concept/proposition",7,"proposition","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-proposition"],["hardy-course-of-pure-mathematics-1921/eq-5b63f004b6",16,"Hardy 1921, p. 10: (n^{r})^{s} = n^{rs}"],["wentworth-plane-geometry-1899/x-c4147afd14",15,"Wentworth 1899, scan 65: A concave polygon is a polygon of which two ..."],["law/laws-of-indices-power-of-a-power",10,"laws of indices (power of a power)"],["wentworth-plane-geometry-1899/x-28b87bb4f2",15,"Wentworth 1899, scan 66: Two polygons are equal when they can be divided ..."],["todhunter-spherical-trigonometry-1886/x-04622b30bd",15,"Todhunter 1886, scan 77: A Lune is that portion of the surface of ..."],["wentworth-first-steps-in-algebra-1894/eq-e7134e170d",16,"Wentworth 1894, p. 142: l = a + (n - 1)d"],["wentworth-first-steps-in-algebra-1894/eq-acf2545185",16,"Wentworth 1894, p. 143: A = \\frac{a + b}{2}"],["wentworth-first-steps-in-algebra-1894/eq-37f173344f",16,"Wentworth 1894, p. 144: m + 2 = n"],["concept/number-of-terms",7,"number of terms"],["wentworth-first-steps-in-algebra-1894/eq-8dd2f8b76d",16,"Wentworth 1894, p. 144: l = a + (m + 1)d"],["wentworth-first-steps-in-algebra-1894/eq-250f19306d",16,"Wentworth 1894, p. 144: \\frac{l - a}{m + 1} = d"],["wentworth-first-steps-in-algebra-1894/eq-2e58fe420b",16,"Wentworth 1894, p. 145: 2s = n(a + l)"],["wentworth-first-steps-in-algebra-1894/eq-4b5542aa9c",16,"Wentworth 1894, p. 145: s = \\frac{n}{2}(a + l)"],["wentworth-first-steps-in-algebra-1894/eq-07772edd80",16,"Wentworth 1894, p. 145: s = \\frac{n}{2}\\bigl\\{a + a + (n - 1)d\\bigr\\}"],["wentworth-first-steps-in-algebra-1894/eq-0795807a60",16,"Wentworth 1894, p. 152: 2ab + b^{2} = (2a + b)b"],["blackburn-elements-plane-trigonometry-1863/eq-bcc06dae35",16,"Blackburn 1863, p. 21: \\cos\\theta = \\sin COB = \\frac{BE}{R} = \\frac{OD}{R}."],["maxwell-elementary-treatise-electricity-1888/eq-aa3a898705",16,"Maxwell 1888, scan 45: Q' = Q + \\tfrac{1}{2}\\sum\\{(E' - E)(P' + P)\\}"],["boyden-first-book-in-algebra-1895/ex-20/5",4,"Boyden 1895, Exercise 20 (5)"],["maxwell-elementary-treatise-electricity-1888/eq-0a7bb14a6a",16,"Maxwell 1888, scan 46: \\frac{Q' - Q}{E' - E} = \\tfrac{1}{2}(P' + P)"],["maxwell-elementary-treatise-electricity-1888/eq-db4bfe1cc4",16,"Maxwell 1888, scan 46: \\frac{dQ_e}{dE} = P"],["maxwell-elementary-treatise-electricity-1888/eq-32ac0e2a8f",16,"Maxwell 1888, scan 46: \\sum(EP') = \\sum(E'P)"],["maxwell-elementary-treatise-electricity-1888/eq-8845502cd2",16,"Maxwell 1888, scan 46: \\tfrac{1}{2}\\sum\\{(E' - E)(P' + P)\\} = Q' - Q = \\tfrac{1}{2}\\sum\\{(E' + E)(P' - P)\\}"],["maxwell-elementary-treatise-electricity-1888/eq-dcf42f1f70",16,"Maxwell 1888, scan 47: \\frac{Q' - Q}{P' - P} = \\tfrac{1}{2}(E' + E)"],["hardy-course-of-pure-mathematics-1921/eq-c9267123a9",16,"Hardy 1921, p. 6: x^{n} + p_{1}x^{n-1} + p_{2}x^{n-2} + \\dots + p_{n} = 0"],["theorem/gauss-s-theorem-on-rational-roots-general-form",9,"Gauss's theorem on rational roots (general form)"],["maxwell-elementary-treatise-electricity-1888/eq-291bb8887a",16,"Maxwell 1888, scan 47: E_r{P_r}' + E_s{P_s}' = {E_r}'P_r + {E_s}'P_s"],["maxwell-elementary-treatise-electricity-1888/eq-278c574939",16,"Maxwell 1888, scan 48: E_r{P_r}' = {E_s}'P_s"],["maxwell-elementary-treatise-electricity-1888/eq-33d5ed2a1e",16,"Maxwell 1888, scan 48: \\frac{P_s}{E_r} = \\frac{{P_r}'}{{E_s}'}"],["maxwell-elementary-treatise-electricity-1888/eq-3e510e32cf",16,"Maxwell 1888, scan 48: E_r = {E_s}'"],["maxwell-elementary-treatise-electricity-1888/eq-dee94845a8",16,"Maxwell 1888, scan 48: P_s = {P_r}'"],["maxwell-elementary-treatise-electricity-1888/eq-cff7c360f7",16,"Maxwell 1888, scan 49: 0 = {E_r}'P_r + {E_s}'P_s"],["maxwell-elementary-treatise-electricity-1888/eq-37d36fae7f",16,"Maxwell 1888, scan 49: \\frac{P_s}{P_r} = -\\frac{{E_r}'}{{E_s}'}"],["thompson-calculus-made-easy-1914/eq-d5043f0e8c",16,"Thompson 1914, p. 36: \\dfrac{dy}{dx} &= \\dfrac{du}{dx} + \\dfrac{dv}{dx}"],["hardy-course-of-pure-mathematics-1921/eq-7cbae75279",16,"Hardy 1921, p. 8: y^{2} - x^{2} = (y - x)(y + x)"],["maxwell-elementary-treatise-electricity-1888/eq-43842d678c",16,"Maxwell 1888, scan 50: W = \\tfrac{1}{2}\\sum[E(P - P')]"],["maxwell-elementary-treatise-electricity-1888/eq-d87d9dd652",16,"Maxwell 1888, scan 50: Q' = \\tfrac{1}{2}\\sum(EP')"],["maxwell-elementary-treatise-electricity-1888/eq-3f2c192701",16,"Maxwell 1888, scan 50: W = \\tfrac{1}{2}\\sum[E(P - P_1)]"],["maxwell-elementary-treatise-electricity-1888/eq-08ba10e986",16,"Maxwell 1888, scan 50: \\sum(EP - E_1P_1) = 0"],["maxwell-elementary-treatise-electricity-1888/eq-f9851eafd3",16,"Maxwell 1888, scan 50: W = \\tfrac{1}{2}\\sum[(E_1 - E)P_1]"],["maxwell-elementary-treatise-electricity-1888/eq-1a80a70e12",16,"Maxwell 1888, scan 51: W = \\tfrac{1}{2}\\sum[(E' - E)P]"],["maxwell-elementary-treatise-electricity-1888/eq-283d361b8f",16,"Maxwell 1888, scan 51: W = \\tfrac{1}{2}\\sum(E'P) - \\tfrac{1}{2}\\sum(EP)"],["hardy-course-of-pure-mathematics-1921/eq-53161f6310",16,"Hardy 1921, p. 10: x^{2} = N"],["boyden-first-book-in-algebra-1895/ch-least-common-multiple",2,"Boyden 1895, LEAST COMMON MULTIPLE","../books/boyden-first-book-in-algebra-1895/ch/ch-least-common-multiple/index.html"],["slaught-lennes-solid-geometry-1919/eq-8fe14bf5f3",16,"Slaught & Lennes 1919, p. 53: AF = AE = AB"],["concept/method-constructing-a-regular-octahedron",7,"method: constructing a regular octahedron"],["concept/constant-term",7,"constant term","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-constant-term"],["concept/modular-arithmetic",7,"modular arithmetic","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-modular-arithmetic"],["concept/denary-scale-of-notation",7,"denary scale of notation","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-denary-scale-of-notation"],["theorem/spherical-triangles-equal-by-two-angles-and-included-side",9,"spherical triangles equal by two angles and included side","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-spherical-triangles-equal-by-two-angles-and-included-side"],["blackburn-elements-plane-trigonometry-1863/eq-20d227a15d",16,"Blackburn 1863, p. 12: Ca = Cb = s - c"],["blackburn-elements-plane-trigonometry-1863/eq-cebebab2d5",16,"Blackburn 1863, p. 13: (s - a) \\alpha = r s = \\text{the area}."],["blackburn-elements-plane-trigonometry-1863/eq-4f0b26c1ab",16,"Blackburn 1863, p. 14: s(s-b) : \\Delta :: \\Delta : (s-a)(s-c)"],["blackburn-elements-plane-trigonometry-1863/eq-e6a8b39f8d",16,"Blackburn 1863, p. 14: \\Delta^2 = s(s-a)(s-b)(s-c)"],["concept/theorem-heron-s-formula",7,"theorem: Heron's formula"],["blackburn-elements-plane-trigonometry-1863/eq-8afd4f2418",16,"Blackburn 1863, p. 14: r^2 = \\dfrac{(s-a)(s-b)(s-c)}{s}"],["hardy-course-of-pure-mathematics-1921/eq-1b9860d31e",16,"Hardy 1921, p. 397: \\int_{C} f(z)\\, dz"],["hardy-course-of-pure-mathematics-1921/ex-lvi/2",4,"Hardy 1921, Exercise LVI (2)"],["form/cf5c97e489",5,"solve: Eq(x**2 + 3*x, 18)"],["shape/b548c722c6",6,"solve: Eq(N*x + x**N, N)"],["thompson-calculus-made-easy-1914/eq-03195012b7",16,"Thompson 1914, p. 23: \\frac{dy}{dx} = nx^{(n-1)}"],["boyden-first-book-in-algebra-1895/ex-20/6",4,"Boyden 1895, Exercise 20 (6)"],["wentworth-plane-geometry-1899/x-77bdfd654f",15,"Wentworth 1899, scan 73: If a known truth suggests the required proof, it ..."],["thompson-calculus-made-easy-1914/eq-b535c5eecd",16,"Thompson 1914, p. 24: \\frac{dy}{dx} = -2x^{-3}"],["thompson-calculus-made-easy-1914/eq-57301c9ad4",16,"Thompson 1914, p. 24: \\dfrac{dy}{dx} = \\dfrac{1}{2} x^{-\\efrac{1}{2}}"],["maxwell-elementary-treatise-electricity-1888/x-bbe6c7add1",15,"Maxwell 1888, scan 173: The electrification is here produced between the surfaces of ..."],["maxwell-elementary-treatise-electricity-1888/x-5d18e17163",15,"Maxwell 1888, scan 174: The electromotive force of the machine is the excess ..."],["theorem/complex-curvilinear-integral",9,"complex curvilinear integral"],["hardy-course-of-pure-mathematics-1921/eq-d8c22cdfb0",16,"Hardy 1921, p. 397: \\Log \\zeta = \\int_{C} \\frac{dz}{z}"],["maxwell-elementary-treatise-electricity-1888/eq-7d40d4d662",16,"Maxwell 1888, scan 56: \\tfrac{1}{4}(1 - n)e + (-ne) &= 0\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-2cb1b759fe",16,"Maxwell 1888, scan 56: n &= \\tfrac{1}{5}\\text{,}"],["concept/axis-of-a-circle-on-a-sphere",7,"axis of a circle on a sphere","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-axis-of-a-circle-on-a-sphere"],["ball-mathematical-recreations-1905/x-c62f2686eb",15,"Ball 1905, scan 32: The reason of the rule is obvious, for he ..."],["theorem/two-similar-triangles-may-be-placed-with-a-center-of-similitude",9,"two similar triangles may be placed with a center of similitude","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-two-similar-triangles-may-be-placed-with-a-center-of-similitude"],["ball-mathematical-recreations-1905/x-952424a8ae",15,"Ball 1905, scan 35: Problems like this can be worked out only by ..."],["theorem/volumes-of-similar-polyhedrons",9,"volumes of similar polyhedrons","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volumes-of-similar-polyhedrons"],["ball-mathematical-recreations-1905/x-aef13e9452",15,"Ball 1905, scan 36: Obviously, if A calls 43, then whatever B adds ..."],["theorem/figures-with-a-center-of-similitude-are-similar",9,"figures with a center of similitude are similar","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-figures-with-a-center-of-similitude-are-similar"],["slaught-lennes-solid-geometry-1919/eq-76989eef52",16,"Slaught & Lennes 1919, p. 63: V = l\\cdot w\\cdot h"],["concept/volume-of-a-rectangular-parallelopiped",7,"volume of a rectangular parallelopiped"],["concept/theorem-volume-of-a-rectangular-parallelopiped",7,"theorem: volume of a rectangular parallelopiped"],["slaught-lennes-solid-geometry-1919/eq-36543c5642",16,"Slaught & Lennes 1919, p. 64: \\dfrac{V}{V'} = \\dfrac{a\\cdot b\\cdot c} {a'\\cdot b'\\cdot c'} = \\dfrac{a}{a'}"],["ball-mathematical-recreations-1905/x-752e770190",15,"Ball 1905, scan 39: In other words, if the number of counters taken ..."],["hardy-course-of-pure-mathematics-1921/eq-8bd3c1fa57",16,"Hardy 1921, p. 399: \\zeta = \\rho(\\cos\\phi + i\\sin\\phi)"],["maxwell-elementary-treatise-electricity-1888/x-ffee7d32a9",15,"Maxwell 1888, scan 180: I do not propose it as a useful form ..."],["hardy-course-of-pure-mathematics-1921/ex-lvii/1",4,"Hardy 1921, Exercise LVII (1)"],["planck-treatise-on-thermodynamics-1903/x-a93d34a2eb",15,"Planck 1903, p. 86: As soon as a phenomenon is found to contradict ..."],["slaught-lennes-solid-geometry-1919/eq-b58cf178b8",16,"Slaught & Lennes 1919, p. 75: S = 2 \\pi re"],["slaught-lennes-solid-geometry-1919/eq-e3efb21594",16,"Slaught & Lennes 1919, p. 75: S = 2 \\pi rh"],["slaught-lennes-solid-geometry-1919/eq-9330181188",16,"Slaught & Lennes 1919, p. 76: V = \\pi r_1^2 e"],["slaught-lennes-solid-geometry-1919/eq-53576c3e5b",16,"Slaught & Lennes 1919, p. 76: V = \\pi r_2^2 h"],["slaught-lennes-solid-geometry-1919/eq-aebb1a1c99",16,"Slaught & Lennes 1919, p. 76: V = \\pi r^2h"],["theorem/plane-section-of-a-sphere-is-a-circle",9,"plane section of a sphere is a circle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-plane-section-of-a-sphere-is-a-circle"],["wentworth-plane-geometry-1899/x-06e9780849",15,"Wentworth 1899, scan 67: The sum of the interior angles of a polygon ..."],["planck-treatise-on-thermodynamics-1903/eq-c403953b9f",16,"Planck 1903, p. 109: dU = W"],["planck-treatise-on-thermodynamics-1903/eq-718f9195c2",16,"Planck 1903, p. 109: d\\Phi \\geq 0"],["dickson-theory-of-equations-1922/eq-d2b9c97056",16,"Dickson 1922, p. 99: x^4 - x + 1 - 6x^2 y^2 + y^4 = 0"],["maxwell-elementary-treatise-electricity-1888/eq-e32b66fdcd",16,"Maxwell 1888, scan 64: P_0E_0 + P_1E_1 + P_2E_2 + P_3E_3\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-193f340c7a",16,"Maxwell 1888, scan 63: E = -e"],["maxwell-elementary-treatise-electricity-1888/x-c4f2368c14",15,"Maxwell 1888, scan 183: A great number of the experiments by which Coulomb ..."],["theorem/arc-from-pole-of-great-circle-to-circumference-is-a-quadrant",9,"arc from pole of great circle to circumference is a quadrant","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-arc-from-pole-of-great-circle-to-circumference-is-a-quadrant"],["ball-mathematical-recreations-1905/x-f037a63be4",15,"Ball 1905, scan 44: Now if we put a = d =1 and ..."],["wentworth-plane-geometry-1899/x-061d63fd16",15,"Wentworth 1899, scan 69: A figure is symmetrical with respect to a point ..."],["dickson-theory-of-equations-1922/eq-5e64076267",16,"Dickson 1922, p. 99: 4x^3 - 1 - 4xy^2 = 0"],["dickson-theory-of-equations-1922/eq-c437bf27d5",16,"Dickson 1922, p. 99: y^2 = x^2 - \\frac{1}{4x}"],["thompson-calculus-made-easy-1914/eq-dd661a4283",16,"Thompson 1914, p. 29: \\frac{dy}{dx} = a × 2x."],["thompson-calculus-made-easy-1914/eq-37c50c3c87",16,"Thompson 1914, p. 29: \\dfrac{dy}{dx} = a×nx^{n-1}"],["thompson-calculus-made-easy-1914/eq-5f39ab91bd",16,"Thompson 1914, p. 30: \\frac{dy}{dx} = \\frac{5}{7} x^4."],["thompson-calculus-made-easy-1914/eq-b46eaa9665",16,"Thompson 1914, p. 30: \\frac{dy}{dx} = \\frac{a}{2\\sqrt{x}}."],["thompson-calculus-made-easy-1914/eq-cbd702d5be",16,"Thompson 1914, p. 31: \\frac{dy}{dx} = \\sqrt{\\frac{a+b}{a-b}}."],["thompson-calculus-made-easy-1914/eq-ca747ce947",16,"Thompson 1914, p. 31: V = \\pi r^2 h"],["thompson-calculus-made-easy-1914/eq-dab651fb79",16,"Thompson 1914, p. 31: \\frac{dV}{dr} = 2 \\pi r h."],["thompson-calculus-made-easy-1914/eq-4d74c495a7",16,"Thompson 1914, p. 32: \\dfrac{dV}{dr} = 2\\pi r^2 = 400"],["todhunter-spherical-trigonometry-1886/x-0928c089f4",15,"Todhunter 1886, scan 45: From (4) it follows that \\tan a has the ..."],["dickson-theory-of-equations-1922/eq-feb66f0bea",16,"Dickson 1922, p. 99: -4x^6 + x^2 + \\frac{1}{16} = 0"],["thompson-calculus-made-easy-1914/eq-f5088d8771",16,"Thompson 1914, p. 32: \\dfrac{\\theta}{\\theta_1} = \\left(\\dfrac{t}{t_1}\\right)^4"],["maxwell-elementary-treatise-electricity-1888/eq-a46b771fc2",16,"Maxwell 1888, scan 209: \\frac{c + x}{b - x} = \\frac{\\beta}{\\gamma}"],["thompson-calculus-made-easy-1914/eq-33a36cf519",16,"Thompson 1914, p. 32: \\dfrac{d\\theta}{dt} = \\dfrac{100t^3}{1000^4} = \\dfrac{t^3}{10,000,000,000}."],["method/area-of-a-surface-of-revolution",8,"area of a surface of revolution","../books/thompson-calculus-made-easy-1914/terms/index.html#t-method-area-of-a-surface-of-revolution"],["hardy-course-of-pure-mathematics-1921/x-2e6a3c748c",15,"Hardy 1921, p. 29: then there is a number \\alpha, which has the ..."],["concept/ferry-boat-problems",7,"ferry-boat problems","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-ferry-boat-problems"],["hardy-course-of-pure-mathematics-1921/eq-3b3bf72f35",16,"Hardy 1921, p. 399: \\Log \\zeta = \\log \\rho + i\\phi"],["maxwell-elementary-treatise-electricity-1888/eq-ebea931462",16,"Maxwell 1888, scan 154: (\\phi_2 - \\phi_1) \\theta"],["blackburn-elements-plane-trigonometry-1863/eq-00ded65979",16,"Blackburn 1863, p. 19: \\frac{AB}{R} = \\frac{A'B'}{R'}"],["blackburn-elements-plane-trigonometry-1863/eq-07b37c0fc2",16,"Blackburn 1863, p. 19: \\frac{\\frac{1}{4} \\text{ circumference}}{R} = \\frac{\\pi}{2}"],["blackburn-elements-plane-trigonometry-1863/eq-174c0099f0",16,"Blackburn 1863, p. 19: = \\frac{180°}{\\pi}"],["blackburn-elements-plane-trigonometry-1863/eq-72c1018539",16,"Blackburn 1863, p. 19: \\theta > \\dfrac{\\pi}{2}"],["blackburn-elements-plane-trigonometry-1863/eq-fb2ea2fe2a",16,"Blackburn 1863, p. 19: \\theta > \\pi"],["maxwell-elementary-treatise-electricity-1888/x-808eebec61",15,"Maxwell 1888, scan 186: Besides this, by connecting the guard-ring with a metal ..."],["boyden-first-book-in-algebra-1895/ex-20/7",4,"Boyden 1895, Exercise 20 (7)"],["boyden-first-book-in-algebra-1895/ex-20/8",4,"Boyden 1895, Exercise 20 (8)"],["form/465cba109a",5,"identity: (x - 6)*(x + 5)"],["wentworth-first-steps-in-algebra-1894/ex-17/3",4,"Wentworth 1894, Exercise 17 (3)"],["maxwell-elementary-treatise-electricity-1888/eq-2d9aa78fe6",16,"Maxwell 1888, scan 77: \\sigma = P_A \\sigma _2 - P_B \\sigma _1"],["maxwell-elementary-treatise-electricity-1888/eq-f13d74eff3",16,"Maxwell 1888, scan 77: E_A = P_A q_A - P_B q_{AB}"],["maxwell-elementary-treatise-electricity-1888/eq-aaf3e9d4dd",16,"Maxwell 1888, scan 77: E_B = P_Bq_B - P_A q_{AB}"],["maxwell-elementary-treatise-electricity-1888/eq-bab8621675",16,"Maxwell 1888, scan 77: P_A = P_B \\frac{q_{AB}}{q_A}"],["maxwell-elementary-treatise-electricity-1888/eq-d554a63ec1",16,"Maxwell 1888, scan 77: \\sigma = \\frac{P_B}{q_A}(q_{AB} \\sigma _2 - q_A \\sigma _1)"],["maxwell-elementary-treatise-electricity-1888/x-407d2d5dbf",15,"Maxwell 1888, scan 176: In the ordinary frictional electrical machine the work done ..."],["maxwell-elementary-treatise-electricity-1888/eq-afcdb7bf0e",16,"Maxwell 1888, scan 156: \\xp\\dfrac{H}{\\theta}"],["thompson-calculus-made-easy-1914/eq-a2339bcac3",16,"Thompson 1914, p. 36: \\frac{dy}{dx} &= \\frac{du}{dx} + \\frac{dv}{dx} + \\frac{dw}{dx}"],["thompson-calculus-made-easy-1914/eq-3380a420b0",16,"Thompson 1914, p. 36: \\frac{dy}{dx} &= \\frac{du}{dx} - \\frac{dv}{dx}"],["thompson-calculus-made-easy-1914/eq-e61bb29e2d",16,"Thompson 1914, p. 38: \\dfrac{dy}{dx} = u\\, \\dfrac{dv}{dx} + v\\, \\dfrac{du}{dx}"],["thompson-calculus-made-easy-1914/eq-57dd510b08",16,"Thompson 1914, p. 40: \\dfrac{dy}{dx} &= \\dfrac{v\\, \\dfrac{du}{dx} - u\\, \\dfrac{dv}{dx}}{v^2}"],["thompson-calculus-made-easy-1914/eq-f487d8da66",16,"Thompson 1914, p. 45: V = \\dfrac{H}{3} (A + a + \\sqrt{Aa} )"],["thompson-calculus-made-easy-1914/eq-0f28110e6a",16,"Thompson 1914, p. 45: P = \\left( \\dfrac{40 + t}{140} \\right)^5"],["theorem/dulong-s-formula-for-saturated-steam",9,"Dulong's formula for saturated steam"],["theorem/principal-value-of-the-logarithm",9,"principal value of the logarithm"],["maxwell-elementary-treatise-electricity-1888/eq-6da1b77794",16,"Maxwell 1888, scan 160: QE = H\\text{,}"],["todhunter-spherical-trigonometry-1886/x-668e457509",15,"Todhunter 1886, scan 78: The expression A+B+C-\\pi is called the spherical excess of ..."],["slaught-lennes-solid-geometry-1919/eq-4f6895c1d4",16,"Slaught & Lennes 1919, p. 96: \\dfrac{OF}{MB} = \\dfrac{OP}{MP} = \\dfrac{OG}{MC}"],["slaught-lennes-solid-geometry-1919/eq-208805e749",16,"Slaught & Lennes 1919, p. 98: S = \\tfrac{1}{2}\\cdot 2 \\pi r\\cdot l = \\pi r l"],["slaught-lennes-solid-geometry-1919/eq-ffc7c3fb8c",16,"Slaught & Lennes 1919, p. 99: S = \\tfrac{1}{2} (2\\pi r + 2\\pi r')l = \\pi l(r + r')"],["slaught-lennes-solid-geometry-1919/eq-ebde28790a",16,"Slaught & Lennes 1919, p. 101: V = \\tfrac{1}{3}\\cdot \\pi r^2\\cdot h = \\tfrac{1}{3} \\pi r^2 h"],["slaught-lennes-solid-geometry-1919/eq-e2cdc63f3d",16,"Slaught & Lennes 1919, p. 102: V = \\tfrac{1}{3} h(b + b' + \\sqrt{bb'})"],["slaught-lennes-solid-geometry-1919/eq-d8b2b9c0af",16,"Slaught & Lennes 1919, p. 102: V = \\tfrac{1}{3} \\pi h(r^2 + {r'}^2 + rr')"],["hardy-course-of-pure-mathematics-1921/x-66c4fa7011",15,"Hardy 1921, p. 19: A number of the form ±\\sqrt{a}, where a is ..."],["hardy-course-of-pure-mathematics-1921/eq-164f5639da",16,"Hardy 1921, p. 400: \\log \\zeta = \\log \\rho + i\\phi"],["blackburn-elements-plane-trigonometry-1863/eq-9f3481315a",16,"Blackburn 1863, p. 25: \\tan\\theta = \\frac{AF}{R} \\text{ and } AF = R\\tan\\theta."],["blackburn-elements-plane-trigonometry-1863/eq-889fd6d24e",16,"Blackburn 1863, p. 28: \\tan(-\\theta) = -\\tan\\theta; \\text{ and } \\cot(-\\theta) = -\\cot\\theta."],["blackburn-elements-plane-trigonometry-1863/eq-8ef7675e75",16,"Blackburn 1863, p. 28: \\tan(\\pi - \\theta) = -\\tan\\theta; \\quad \\cot(\\pi - \\theta) = -\\cot\\theta."],["blackburn-elements-plane-trigonometry-1863/eq-e9acdf5af5",16,"Blackburn 1863, p. 26: \\cot\\theta = \\frac{CG}{R} \\text{ and } CG = R\\cot\\theta."],["blackburn-elements-plane-trigonometry-1863/eq-75bf12b854",16,"Blackburn 1863, p. 29: \\sec\\theta = \\frac{OS}{R};\\quad \\cosec\\theta = \\frac{OK}{R};"],["blackburn-elements-plane-trigonometry-1863/eq-1206a8ee44",16,"Blackburn 1863, p. 31: \\versin\\theta = \\frac{AD}{R}; \\text{ and } AD = R\\versin\\theta."],["blackburn-elements-plane-trigonometry-1863/eq-3b226912de",16,"Blackburn 1863, p. 32: R^2 = R^2 \\sin^2\\theta + R^2 \\cos^2\\theta,"],["method/constructing-a-spherical-triangle-from-its-sides",8,"constructing a spherical triangle from its sides","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-constructing-a-spherical-triangle-from-its-sides"],["method/constructing-a-spherical-triangle-from-its-angles",8,"constructing a spherical triangle from its angles","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-constructing-a-spherical-triangle-from-its-angles"],["blackburn-elements-plane-trigonometry-1863/eq-36881400d3",16,"Blackburn 1863, p. 32: \\sin^2\\theta + \\cos^2\\theta = 1\\Add{.}"],["blackburn-elements-plane-trigonometry-1863/eq-6f209ec7e7",16,"Blackburn 1863, p. 32: \\sec^2\\theta = 1 + \\tan^2\\theta\\Add{.}"],["blackburn-elements-plane-trigonometry-1863/eq-5b753037f6",16,"Blackburn 1863, p. 32: \\cosec^2\\theta = 1 + \\cot^2\\theta\\Add{.}"],["blackburn-elements-plane-trigonometry-1863/eq-cf54a66496",16,"Blackburn 1863, p. 32: \\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta}"],["blackburn-elements-plane-trigonometry-1863/eq-472bd8a669",16,"Blackburn 1863, p. 32: \\sec\\theta = \\frac{1}{\\cos\\theta}\\Add{.}"],["blackburn-elements-plane-trigonometry-1863/eq-a3d11a0644",16,"Blackburn 1863, p. 32: \\cotan\\theta = \\frac{\\cos\\theta}{\\sin\\theta} &= \\frac{1}{\\tan\\theta}"],["blackburn-elements-plane-trigonometry-1863/eq-685f1d6298",16,"Blackburn 1863, p. 32: \\cosec\\theta &= \\frac{1}{\\sin\\theta}"],["blackburn-elements-plane-trigonometry-1863/eq-8c0da28a70",16,"Blackburn 1863, p. 32: \\versin\\theta = 1 - \\cos\\theta\\Add{.}"],["blackburn-elements-plane-trigonometry-1863/eq-fea2f16b85",16,"Blackburn 1863, p. 24: \\sin \\frac{\\pi}{4} = \\sin 45° = \\frac{1}{2}\\sqrt{2} = \\cos 45° = \\cos \\frac{\\pi}{4}."],["blackburn-elements-plane-trigonometry-1863/eq-44fcb7bd2e",16,"Blackburn 1863, p. 33: \\log m - \\log n = \\log(m ÷ n)"],["blackburn-elements-plane-trigonometry-1863/eq-401be09736",16,"Blackburn 1863, p. 34: \\log 1 = 0"],["blackburn-elements-plane-trigonometry-1863/eq-cf316b3734",16,"Blackburn 1863, p. 34: \\log n^x = x \\log n"],["blackburn-elements-plane-trigonometry-1863/eq-9535ca31a5",16,"Blackburn 1863, p. 34: \\log n^p = p \\log n"],["blackburn-elements-plane-trigonometry-1863/eq-9f83e647d4",16,"Blackburn 1863, p. 24: \\sin \\frac{\\pi}{6} = \\sin 30° = \\frac{1}{2} = \\cos 60° = \\cos \\frac{\\pi}{3}."],["blackburn-elements-plane-trigonometry-1863/eq-d549076b45",16,"Blackburn 1863, p. 24: \\sin \\frac{\\pi}{3} = \\sin 60° = \\frac{1}{2}\\sqrt{3} = \\cos 30° = \\cos \\frac{\\pi}{6}."],["blackburn-elements-plane-trigonometry-1863/eq-c88ca72782",16,"Blackburn 1863, p. 24: \\sin \\frac{\\pi}{10} = \\sin 18° = \\dfrac{\\sqrt{5} - 1}{4} = \\cos \\frac{4\\pi}{10} = \\cos 72°."],["hardy-course-of-pure-mathematics-1921/x-2a66b15c85",15,"Hardy 1921, p. 21: Two pure quadratic surds are said to be similar ..."],["hardy-course-of-pure-mathematics-1921/x-87db63d560",15,"Hardy 1921, p. 27: It is important to observe that a pair of ..."],["blackburn-elements-plane-trigonometry-1863/x-f8f91f6302",15,"Blackburn 1863, p. 17: The angle of easiest construction is the angle of ..."],["thompson-calculus-made-easy-1914/eq-dca1b357ea",16,"Thompson 1914, p. 133: \\dfrac{dy}{dx} = -\\dfrac{1}{3\\sqrt{(\\theta +5)^4}}"],["thompson-calculus-made-easy-1914/eq-3a66c391c0",16,"Thompson 1914, p. 131: \\frac{dy}{dx} × \\frac{dx}{dy} = 1"],["thompson-calculus-made-easy-1914/eq-73d1546250",16,"Thompson 1914, p. 131: \\frac{dy}{dx} = \\frac{1}{\\ \\dfrac{dx}{dy}\\ }"],["slaught-lennes-solid-geometry-1919/eq-90264bf7ec",16,"Slaught & Lennes 1919, p. 116: PD : PB = PB : PP'"],["concept/method-finding-the-diameter-of-a-sphere",7,"method: finding the diameter of a sphere"],["slaught-lennes-solid-geometry-1919/eq-8f3ff69009",16,"Slaught & Lennes 1919, p. 116: PD × PP' = \\overline{PB}^2"],["slaught-lennes-solid-geometry-1919/eq-cfec2bda51",16,"Slaught & Lennes 1919, p. 122: \\wideparen{AD} + \\wideparen{DC} + \\wideparen{CB} > \\wideparen{AB}"],["concept/theorem-sum-of-two-sides-of-a-spherical-triangle-exceeds-the-third",7,"theorem: sum of two sides of a spherical triangle exceeds the third"],["concept/theorem-shortest-distance-on-a-sphere-follows-a-great-circle-arc",7,"theorem: shortest distance on a sphere follows a great circle arc"],["blackburn-elements-plane-trigonometry-1863/x-cc4998e7e4",15,"Blackburn 1863, p. 18: It is called the unit of circular measure, and ..."],["blackburn-elements-plane-trigonometry-1863/x-420d91dc0b",15,"Blackburn 1863, p. 17: For such reasons perhaps the sexagesimal scale, which has ..."],["blackburn-elements-plane-trigonometry-1863/x-6ab5f52916",15,"Blackburn 1863, p. 19: Two angles are said to be complements, each of ..."],["blackburn-elements-plane-trigonometry-1863/x-5d22d0ea85",15,"Blackburn 1863, p. 17: The 60th part of a degree is a minute, ..."],["wentworth-plane-geometry-1899/ch-i",2,"Wentworth 1899, ch. I: RECTILINEAR FIGURES","../books/wentworth-plane-geometry-1899/ch/ch-i/index.html"],["thompson-calculus-made-easy-1914/eq-2464303fa5",16,"Thompson 1914, p. 50: y = f(x) = x^n"],["thompson-calculus-made-easy-1914/eq-b9ff4f629a",16,"Thompson 1914, p. 50: f'(x) &= nx^{n-1}"],["thompson-calculus-made-easy-1914/eq-ea644f753a",16,"Thompson 1914, p. 50: f''(x) &= n(n-1)x^{n-2}"],["thompson-calculus-made-easy-1914/eq-e6f0a20bb7",16,"Thompson 1914, p. 50: f'''(x) &= n(n-1)(n-2)x^{n-3}"],["thompson-calculus-made-easy-1914/eq-2495612627",16,"Thompson 1914, p. 50: f''''(x) &= n(n-1)(n-2)(n-3)x^{n-4}"],["thompson-calculus-made-easy-1914/eq-95964be744",16,"Thompson 1914, p. 49: y &= f(x)"],["thompson-calculus-made-easy-1914/eq-e7c0e41e3a",16,"Thompson 1914, p. 50: \\frac{dy}{dx} = f'(x)"],["thompson-calculus-made-easy-1914/eq-2238ea562a",16,"Thompson 1914, p. 50: \\frac{d\\left(\\dfrac{dy}{dx}\\right)}{dx} &= f''(x)"],["thompson-calculus-made-easy-1914/eq-a6c46d0e4b",16,"Thompson 1914, p. 50: \\dfrac{d^3y}{dx^3} = f'''(x)"],["wentworth-plane-geometry-1899/x-0f94897358",15,"Wentworth 1899, scan 82: The lines joining the middle points of the sides ..."],["boyden-first-book-in-algebra-1895/ex-20/9",4,"Boyden 1895, Exercise 20 (9)"],["thompson-calculus-made-easy-1914/eq-4c6e72f8ea",16,"Thompson 1914, p. 51: y = f(x) = 7x^4 + 3.5x^3 - \\frac{1}{2}x^2 + x - 2"],["thompson-calculus-made-easy-1914/eq-e45a286b18",16,"Thompson 1914, p. 51: \\frac{dy}{dx} &= f'(x) = 28x^3 + 10.5x^2 - x + 1"],["thompson-calculus-made-easy-1914/eq-119a5c8009",16,"Thompson 1914, p. 51: \\frac{d^2y}{dx^2} &= f''(x) = 84x^2 + 21x - 1"],["thompson-calculus-made-easy-1914/eq-3db345c570",16,"Thompson 1914, p. 51: \\frac{d^3y}{dx^3} &= f'''(x) = 168x + 21"],["thompson-calculus-made-easy-1914/eq-cfe5ebfaff",16,"Thompson 1914, p. 51: \\frac{d^4y}{dx^4} &= f''''(x) = 168"],["thompson-calculus-made-easy-1914/eq-1a2449be3c",16,"Thompson 1914, p. 51: \\frac{d^5y}{dx^5} &= f'''''(x) = 0"],["thompson-calculus-made-easy-1914/eq-ce6bc7cdd3",16,"Thompson 1914, p. 51: y = \\phi(x) = 3x(x^2 - 4)"],["thompson-calculus-made-easy-1914/eq-22d6a162a7",16,"Thompson 1914, p. 51: \\phi'(x) &= \\frac{dy}{dx} = 3\\bigl[x × 2x + (x^2 - 4) × 1\\bigr] = 3(3x^2 - 4)"],["blackburn-elements-plane-trigonometry-1863/eq-85b58e3c73",16,"Blackburn 1863, p. 34: \\log n^{\\tfrac{p}{q}} = \\frac{p}{q} × \\log n"],["blackburn-elements-plane-trigonometry-1863/eq-a3b4ebd3b9",16,"Blackburn 1863, p. 34: \\log_a n = x, \\text{ when } a^x = n"],["form/b2e99696cc",5,"identity: (x - 11)*(x - 2)"],["hardy-course-of-pure-mathematics-1921/ex-iii",3,"Hardy 1921, Exercise III"],["thompson-calculus-made-easy-1914/eq-b9c02bf07e",16,"Thompson 1914, p. 51: \\phi''(x) &= \\frac{d^2y}{dx^2} = 3 × 6x = 18x"],["thompson-calculus-made-easy-1914/eq-d3190b81fa",16,"Thompson 1914, p. 51: \\phi'''(x) &= \\frac{d^3y}{dx^3} = 18"],["thompson-calculus-made-easy-1914/eq-b5c3308c2a",16,"Thompson 1914, p. 51: \\phi''''(x) &= \\frac{d^4y}{dx^4} = 0"],["macfarlane-vector-analysis-quaternions-1906/x-8b08e9b790",15,"Macfarlane 1906: The composition of successive vectors partakes more of the ..."],["hardy-course-of-pure-mathematics-1921/x-37159fb21f",15,"Hardy 1921, p. 26: In the theory of decimals, for instance, we may ..."],["concept/ordered-pair",7,"ordered pair","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-ordered-pair"],["method/multiplying-ordered-pairs",8,"multiplying ordered pairs","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-method-multiplying-ordered-pairs"],["dickson-theory-of-equations-1922/ch-iii",2,"Dickson 1922, ch. III: Constructions with Ruler and Compasses","../books/dickson-theory-of-equations-1922/ch/ch-iii/index.html"],["dickson-theory-of-equations-1922/ex-page40",3,"Dickson 1922, Exercise Page40"],["dickson-theory-of-equations-1922/ex-page44",3,"Dickson 1922, Exercise Page44"],["dickson-theory-of-equations-1922/ex-page30",3,"Dickson 1922, Exercise Page30"],["blackburn-elements-plane-trigonometry-1863/eq-928bc05ad1",16,"Blackburn 1863, p. 35: \\log (10^m × n) &= m + \\log n"],["blackburn-elements-plane-trigonometry-1863/eq-e58183508e",16,"Blackburn 1863, p. 35: \\log (n \\div 10^m) &= -m + \\log n"],["hardy-course-of-pure-mathematics-1921/ex-lvii/2",4,"Hardy 1921, Exercise LVII (2)"],["theorem/volume-of-a-sphere",9,"volume of a sphere","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volume-of-a-sphere"],["concept/quadratic-mean",7,"quadratic mean","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-quadratic-mean"],["concept/line-segment",7,"line segment","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-line-segment"],["method/multiplying-vectors-in-a-plane",8,"multiplying vectors in a plane","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-method-multiplying-vectors-in-a-plane"],["de-morgan-elementary-illustrations-calculus-1899/eq-a6f23902d1",16,"De Morgan 1899, p. 78: u = x^{2} y + 2xy^{3}"],["thompson-calculus-made-easy-1914/eq-2b97326852",16,"Thompson 1914, p. 55: v = \\dfrac{dy}{dt}"],["thompson-calculus-made-easy-1914/eq-030c1614f7",16,"Thompson 1914, p. 55: a = \\dfrac{dv}{dt}"],["thompson-calculus-made-easy-1914/eq-469d8dbfa7",16,"Thompson 1914, p. 56: a = \\frac{d\\left( \\dfrac{dy}{dt} \\right)}{dt}"],["thompson-calculus-made-easy-1914/eq-1392020674",16,"Thompson 1914, p. 56: a = \\dfrac{d^2y}{dt^2}"],["thompson-calculus-made-easy-1914/eq-11d6cc09c6",16,"Thompson 1914, p. 57: f = m \\frac{dv}{dt}"],["thompson-calculus-made-easy-1914/eq-7f03c75e44",16,"Thompson 1914, p. 57: w = f × y"],["thompson-calculus-made-easy-1914/eq-a3d741432a",16,"Thompson 1914, p. 59: v = \\dot{x}"],["thompson-calculus-made-easy-1914/eq-00d5f13566",16,"Thompson 1914, p. 59: a = \\dot{v} = \\ddot{x}"],["thompson-calculus-made-easy-1914/eq-9a49794374",16,"Thompson 1914, p. 59: f = m\\dot{v} = m\\ddot{x}"],["hardy-course-of-pure-mathematics-1921/ex-lvii/3",4,"Hardy 1921, Exercise LVII (3)"],["planck-treatise-on-thermodynamics-1903/x-5fcc9968f4",15,"Planck 1903, p. 119: Therefore, since d\\theta and dv are independent of each ..."],["thompson-calculus-made-easy-1914/eq-d6838a5b6c",16,"Thompson 1914, p. 59: w = x × m \\ddot{x}"],["hardy-course-of-pure-mathematics-1921/ex-lviii/1",4,"Hardy 1921, Exercise LVIII (1)"],["hardy-course-of-pure-mathematics-1921/x-a069a3546c",15,"Hardy 1921, p. 27: He will find interesting examples in ordinary life: policeman ..."],["concept/obtuse-angle",7,"obtuse angle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-obtuse-angle"],["concept/adjacent-angles",7,"adjacent angles","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-adjacent-angles"],["slaught-lennes-solid-geometry-1919/x-4c0bf344db",15,"Slaught & Lennes 1919, p. 135: The number of degrees by which the sum of ..."],["slaught-lennes-solid-geometry-1919/x-cfa0595a1b",15,"Slaught & Lennes 1919, p. 136: The spherical degree differs fundamentally from the units of ..."],["slaught-lennes-solid-geometry-1919/x-f66b2ee7ac",15,"Slaught & Lennes 1919, p. 126: The triangle A'B'C' as thus described is the polar ..."],["slaught-lennes-solid-geometry-1919/eq-255f3085ef",16,"Slaught & Lennes 1919, p. 128: A + a' &= 180\\text{°}"],["slaught-lennes-solid-geometry-1919/eq-e1301e6d39",16,"Slaught & Lennes 1919, p. 129: \\angle A + \\angle B + \\angle C + a' + b' + c' = 6 \\text{ rt.\\ } \\Angles"],["slaught-lennes-solid-geometry-1919/eq-869308b42c",16,"Slaught & Lennes 1919, p. 129: \\angle A + \\angle B + \\angle C < 6 \\text{ rt.\\ } \\Angles"],["slaught-lennes-solid-geometry-1919/eq-f9c89fb8ad",16,"Slaught & Lennes 1919, p. 129: \\angle A + \\angle B + \\angle C > 2 \\text{ rt.\\ } \\Angles"],["slaught-lennes-solid-geometry-1919/eq-c125141b9f",16,"Slaught & Lennes 1919, p. 134: \\text{area } \\triangle ABC = \\text{area } \\triangle A_1B_1C_1"],["slaught-lennes-solid-geometry-1919/eq-fb261286ae",16,"Slaught & Lennes 1919, p. 136: \\triangle ABC = \\angle A + \\angle B + \\angle C - 180"],["slaught-lennes-solid-geometry-1919/eq-4507496565",16,"Slaught & Lennes 1919, p. 140: 2\\pi r × AB = 2\\pi r × 2r = 4\\pi r^2"],["hardy-course-of-pure-mathematics-1921/ex-lviii/2a",4,"Hardy 1921, Exercise LVIII (2a)"],["ball-mathematical-recreations-1905/eq-633ff585e8",16,"Ball 1905, scan 34: (13 - x_1) + (13 - x_2) + \\dotsb + (13 - x_p) + r &= 52"],["ball-mathematical-recreations-1905/eq-011a01715c",16,"Ball 1905, scan 34: x_1 + x_2 + \\dotsb + x_p &= 13p - 52 + r"],["maxwell-elementary-treatise-electricity-1888/eq-662129152a",16,"Maxwell 1888, scan 78: E = -e\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-06590c4a9c",16,"Maxwell 1888, scan 78: s = 4 \\pi r^2"],["maxwell-elementary-treatise-electricity-1888/eq-2a30ac1e48",16,"Maxwell 1888, scan 78: S = 4 \\pi R^2"],["maxwell-elementary-treatise-electricity-1888/eq-c341cd67eb",16,"Maxwell 1888, scan 78: e = s \\sigma"],["maxwell-elementary-treatise-electricity-1888/eq-286b6f320f",16,"Maxwell 1888, scan 78: E = S \\Sigma"],["maxwell-elementary-treatise-electricity-1888/eq-911964209f",16,"Maxwell 1888, scan 78: \\sigma = \\frac{e}{4 \\pi r^2}"],["maxwell-elementary-treatise-electricity-1888/eq-c90f189e3d",16,"Maxwell 1888, scan 78: \\Sigma = \\frac{E}{4 \\pi R^2}"],["maxwell-elementary-treatise-electricity-1888/eq-0756bd782f",16,"Maxwell 1888, scan 78: \\Sigma = \\frac{-e}{4 \\pi R^2}"],["ball-mathematical-recreations-1905/eq-9ef4d4b4e7",16,"Ball 1905, scan 42: \\log(1 + x) = x - \\tfrac{1}{2}x^2 + \\tfrac{1}{3}x^3 - \\dotsb"],["ball-mathematical-recreations-1905/eq-401be09736",16,"Ball 1905, scan 42: \\log 1 = 0"],["concept/vertical-angles",7,"vertical angles","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-vertical-angles"],["maxwell-elementary-treatise-electricity-1888/eq-60050a18a6",16,"Maxwell 1888, scan 80: f = \\frac{ee'}{r^2}"],["maxwell-elementary-treatise-electricity-1888/eq-ac67f5c38b",16,"Maxwell 1888, scan 80: \\mathfrak{E} = \\frac{e}{r'^2}"],["maxwell-elementary-treatise-electricity-1888/eq-d6bcec4059",16,"Maxwell 1888, scan 80: \\mathfrak{E} = \\frac{e}{r^2} = 4 \\pi \\sigma"],["maxwell-elementary-treatise-electricity-1888/eq-7d07c3e678",16,"Maxwell 1888, scan 81: \\mathfrak{E} = 4 \\pi \\sigma"],["maxwell-elementary-treatise-electricity-1888/eq-08b79ca622",16,"Maxwell 1888, scan 80: e = 4 \\pi r^2 \\sigma"],["maxwell-elementary-treatise-electricity-1888/eq-7a66f57d23",16,"Maxwell 1888, scan 81: \\psi' - \\psi = \\overline{BA} \\ldot \\mathfrak{E}"],["maxwell-elementary-treatise-electricity-1888/eq-87a2d55b0f",16,"Maxwell 1888, scan 85: e\\xp\\left(\\dfrac{1}{r} - \\dfrac{1}{b} \\right)"],["maxwell-elementary-treatise-electricity-1888/eq-08c1929c0e",16,"Maxwell 1888, scan 164: (b-a)\\theta+B-A\\text{.}"],["dickson-theory-of-equations-1922/eq-1c592bef2d",16,"Dickson 1922, p. 143: f(x) = a_0x^m + a_1x^{m-1} + \\dotsb + a_m"],["maxwell-elementary-treatise-electricity-1888/eq-f0a0f9c92e",16,"Maxwell 1888, scan 82: \\left(\\frac{1}{b} - \\frac{1}{a}\\right)\\frac{a}{b} < B - A < \\left(\\frac{1}{b} - \\frac{1}{a}\\right) \\frac{a}{b}"],["maxwell-elementary-treatise-electricity-1888/eq-ad365b9501",16,"Maxwell 1888, scan 82: \\left(\\frac{1}{z} - \\frac{1}{a}\\right) \\frac{1}{p} < Z - A < \\left(\\frac{1}{z} - \\frac{1}{y}\\right) p"],["maxwell-elementary-treatise-electricity-1888/eq-c1feeb7495",16,"Maxwell 1888, scan 83: Z - A = \\frac{1}{z} - \\frac{1}{a}"],["maxwell-elementary-treatise-electricity-1888/eq-c1a6d6af48",16,"Maxwell 1888, scan 83: Z = \\frac{1}{z}"],["maxwell-elementary-treatise-electricity-1888/eq-ea99e458f0",16,"Maxwell 1888, scan 83: \\psi = \\frac{e}{r}"],["maxwell-elementary-treatise-electricity-1888/eq-e8a4624014",16,"Maxwell 1888, scan 84: \\psi_a = \\frac{e}{a}"],["maxwell-elementary-treatise-electricity-1888/eq-794cf9f594",16,"Maxwell 1888, scan 84: e = a"],["concept/perigon",7,"perigon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-perigon"],["ball-mathematical-recreations-1905/eq-e94a4ace65",16,"Ball 1905, scan 44: \\sqrt{x-y} = i \\sqrt{y-x}"],["maxwell-elementary-treatise-electricity-1888/eq-85396a49df",16,"Maxwell 1888, scan 85: e\\xp\\left(\\dfrac{1}{a} - \\dfrac{1}{b} \\right)"],["maxwell-elementary-treatise-electricity-1888/eq-f383bf7dda",16,"Maxwell 1888, scan 85: e = \\frac{1}{\\dfrac{1}{a} - \\dfrac{1}{b}} = \\frac{ab}{b - a}"],["maxwell-elementary-treatise-electricity-1888/eq-8b0ab4820e",16,"Maxwell 1888, scan 86: \\sigma = \\frac{A - B}{4 \\pi c}"],["maxwell-elementary-treatise-electricity-1888/eq-06a2f54410",16,"Maxwell 1888, scan 86: \\xp\\dfrac{A-B}{c}"],["maxwell-elementary-treatise-electricity-1888/eq-f45c5c252d",16,"Maxwell 1888, scan 86: e = \\frac{A - B}{4 \\pi c} S"],["maxwell-elementary-treatise-electricity-1888/eq-e9bdb6770a",16,"Maxwell 1888, scan 86: Q = \\tfrac{1}{2}\\{Ae + B(-e)\\} = \\tfrac{1}{2} (A - B)e"],["maxwell-elementary-treatise-electricity-1888/eq-57a933fd37",16,"Maxwell 1888, scan 86: Q = \\frac{2 \\pi}{S} e^2c"],["wentworth-first-steps-in-algebra-1894/ex-2/6",4,"Wentworth 1894, Exercise 2 (6)"],["whitehead-introduction-to-mathematics-1911/x-6963d7998d",15,"Whitehead 1911, p. 102: We came across equations of the form x^{2} = ..."],["ball-mathematical-recreations-1905/eq-dff140c395",16,"Ball 1905, scan 42: a + b &= 2c"],["maxwell-elementary-treatise-electricity-1888/eq-aa849ed6ef",16,"Maxwell 1888, scan 86: Q' = \\frac{2 \\pi}{S} e^2c'"],["maxwell-elementary-treatise-electricity-1888/eq-aef7651eb9",16,"Maxwell 1888, scan 87: Q' - Q = \\frac{2 \\pi}{S} e^2(c' - c)"],["maxwell-elementary-treatise-electricity-1888/eq-35173f0ee6",16,"Maxwell 1888, scan 87: F(c' - c) = \\frac{2 \\pi}{S} e^2(c' - c)"],["maxwell-elementary-treatise-electricity-1888/eq-8c05557d70",16,"Maxwell 1888, scan 87: F = \\frac{2 \\pi}{S} e^2"],["maxwell-elementary-treatise-electricity-1888/eq-dee5661bb7",16,"Maxwell 1888, scan 87: e = \\sqrt\\frac{FS}{2 \\pi}"],["maxwell-elementary-treatise-electricity-1888/eq-4297ebe920",16,"Maxwell 1888, scan 87: A - B = 4 \\pi c \\frac{e}{S} = c \\sqrt{\\frac{8 \\pi F}{S}}"],["maxwell-elementary-treatise-electricity-1888/eq-7a89f1df38",16,"Maxwell 1888, scan 93: V = \\xp\\dfrac{E}{r}"],["theorem/lines-perpendicular-to-the-same-line-in-a-plane-are-parallel",9,"lines perpendicular to the same line in a plane are parallel","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-lines-perpendicular-to-the-same-line-in-a-plane-are-parallel"],["ball-mathematical-recreations-1905/eq-ca50a6b923",16,"Ball 1905, scan 44: a:b = c:d"],["maxwell-elementary-treatise-electricity-1888/eq-58e4dde4eb",16,"Maxwell 1888, scan 93: r = \\xp\\dfrac{E}{V}"],["maxwell-elementary-treatise-electricity-1888/eq-5c7555095c",16,"Maxwell 1888, scan 94: V_1 + V_2 = V"],["law/superposition-of-potentials",10,"superposition of potentials"],["maxwell-elementary-treatise-electricity-1888/eq-199251558f",16,"Maxwell 1888, scan 94: 2 \\pi E(1 - cos \\theta)"],["maxwell-elementary-treatise-electricity-1888/eq-8ea65b18a6",16,"Maxwell 1888, scan 94: E(1 - cos \\theta) = 2 \\Psi"],["maxwell-elementary-treatise-electricity-1888/eq-c240aeff26",16,"Maxwell 1888, scan 94: \\theta = cos^{-1}\\left(1 - 2 \\frac{\\Psi}{E}\\right)"],["maxwell-elementary-treatise-electricity-1888/eq-25d677701f",16,"Maxwell 1888, scan 97: 1 - 2 \\xp\\dfrac{\\Psi}{E}"],["thompson-calculus-made-easy-1914/x-200fc8d0ca",15,"Thompson 1914, p. vii: Advantage has also been taken to enlarge certain parts ..."],["hardy-course-of-pure-mathematics-1921/ex-lviii/2b",4,"Hardy 1921, Exercise LVIII (2b)"],["thompson-calculus-made-easy-1914/eq-bbb20245af",16,"Thompson 1914, p. 68: &= 3x(x^2 + a^2)^{\\efrac{1}{2}}"],["thompson-calculus-made-easy-1914/x-2f58cbb0c8",15,"Thompson 1914, p. xi: The fools who write the textbooks of advanced mathematics---and ..."],["hardy-course-of-pure-mathematics-1921/ex-lviii/3",4,"Hardy 1921, Exercise LVIII (3)"],["thompson-calculus-made-easy-1914/x-d71d640237",15,"Thompson 1914, p. xi: Master these thoroughly, and the rest will follow. What ..."],["whitehead-introduction-to-mathematics-1911/x-fc1c3134d2",15,"Whitehead 1911, p. 102: This is the definition of the meaning of the ..."],["wentworth-plane-geometry-1899/eq-0e95c840cf",16,"Wentworth 1899, scan 234: S:S' = \\pi R^2:\\pi R'^2 = R^2:R'^2"],["maxwell-elementary-treatise-electricity-1888/eq-869922689f",16,"Maxwell 1888, scan 163: JH &= Jh + E"],["maxwell-elementary-treatise-electricity-1888/eq-a78274590c",16,"Maxwell 1888, scan 163: E &= J(H - h)\\text{.}"],["slaught-lennes-solid-geometry-1919/eq-6f6e7b3580",16,"Slaught & Lennes 1919, p. 143: \\tfrac{4}{3} \\pi r^3"],["slaught-lennes-solid-geometry-1919/eq-52f362833c",16,"Slaught & Lennes 1919, p. 143: 4\\pi r^2"],["slaught-lennes-solid-geometry-1919/eq-2a99cddeb3",16,"Slaught & Lennes 1919, p. 147: s = 2\\pi rh"],["slaught-lennes-solid-geometry-1919/eq-d90f6ceff7",16,"Slaught & Lennes 1919, p. 147: v = \\dfrac{r}{3} \\cdot s"],["slaught-lennes-solid-geometry-1919/eq-fd4248d33c",16,"Slaught & Lennes 1919, p. 147: v = \\dfrac{r}{3} \\cdot 2\\pi rh = \\dfrac{2\\pi}{3} r^2h"],["slaught-lennes-solid-geometry-1919/eq-95329cfabd",16,"Slaught & Lennes 1919, p. 147: v = \\dfrac{2\\pi}{3} r^2h"],["slaught-lennes-solid-geometry-1919/eq-dc738616b3",16,"Slaught & Lennes 1919, p. 148: v = \\dfrac{\\pi h}{2} (r_1^2 + r_2^2) + \\dfrac{\\pi}{6} h^3"],["slaught-lennes-solid-geometry-1919/eq-b40f3341f6",16,"Slaught & Lennes 1919, p. 148: v = \\pi h^2 \\left( r - \\dfrac{h}{3} \\right)"],["hardy-course-of-pure-mathematics-1921/ex-lviii/4a",4,"Hardy 1921, Exercise LVIII (4a)"],["slaught-lennes-solid-geometry-1919/eq-53cd9dc361",16,"Slaught & Lennes 1919, p. 148: d &= \\dfrac{r_2^2 - r_1^2 - h^2}{2h}"],["slaught-lennes-solid-geometry-1919/eq-3ede131a14",16,"Slaught & Lennes 1919, p. 148: r^2 = \\dfrac{r_2^4 + r_1^4 + h^4 - 2r_1^2r_2^2 + 2h^2r_2^2 + 2h^2r_1^2}{4h^2}"],["slaught-lennes-solid-geometry-1919/eq-4ccd124ffd",16,"Slaught & Lennes 1919, p. 148: r^2 = r_2^2 + d^2"],["slaught-lennes-solid-geometry-1919/eq-7e50e6461e",16,"Slaught & Lennes 1919, p. 148: r^2 = r_1^2 + (h+d)^2"],["slaught-lennes-solid-geometry-1919/eq-0a2fa5b1db",16,"Slaught & Lennes 1919, p. 152: V = abc"],["slaught-lennes-solid-geometry-1919/eq-67f555ee50",16,"Slaught & Lennes 1919, p. 152: V = hb"],["slaught-lennes-solid-geometry-1919/eq-0529a7d720",16,"Slaught & Lennes 1919, p. 152: S = pe"],["ball-mathematical-recreations-1905/eq-f91984f04f",16,"Ball 1905, scan 44: ad=bc"],["ball-mathematical-recreations-1905/eq-62cd19e4d0",16,"Ball 1905, scan 41: \\phi=(x^2-y^2)/(x^2 + y^2)^2"],["slaught-lennes-solid-geometry-1919/eq-38413ca8c5",16,"Slaught & Lennes 1919, p. 152: S = \\tfrac{1}{2} pl"],["slaught-lennes-solid-geometry-1919/eq-7ff4b23bbc",16,"Slaught & Lennes 1919, p. 152: V = \\tfrac{1}{3} h (b + b' + \\sqrt{bb'})"],["slaught-lennes-solid-geometry-1919/eq-d73b772dd5",16,"Slaught & Lennes 1919, p. 152: S = \\dfrac{a}{720} \\cdot 4 \\pi r^2"],["slaught-lennes-solid-geometry-1919/eq-bd680b3170",16,"Slaught & Lennes 1919, p. 153: V = \\dfrac{2\\pi}{3}r^2h"],["slaught-lennes-solid-geometry-1919/eq-a30389bbc7",16,"Slaught & Lennes 1919, p. 152: S = 4 \\pi r^2"],["ball-mathematical-recreations-1905/eq-917697a68e",16,"Ball 1905, scan 41: \\frac{1}{4}\\pi =-\\frac{1}{4}\\pi"],["thompson-calculus-made-easy-1914/x-f7054e1888",15,"Thompson 1914, p. 207: The secret of solving this problem is to conceive ..."],["concept/one-dimensional-figure",7,"one-dimensional figure","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-one-dimensional-figure"],["concept/falling-bodies",7,"falling bodies","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-falling-bodies"],["concept/skew-lines",7,"skew lines","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-skew-lines"],["thompson-calculus-made-easy-1914/x-6d13a57c77",15,"Thompson 1914, p. 21: Now we know that we may neglect small quantities ..."],["ball-mathematical-recreations-1905/eq-6438c8668e",16,"Ball 1905, scan 33: y < n-m"],["de-morgan-elementary-illustrations-calculus-1899/eq-6dc61d02bb",16,"De Morgan 1899, p. 22: \\phi x + \\phi' x\\, h + \\phi'' x\\, \\frac{h^{2}}{2} + \\phi''' x\\, \\frac{h^{3}}{2·3} + \\etc."],["slaught-lennes-solid-geometry-1919/x-a08831d69a",15,"Slaught & Lennes 1919, p. 1: In plane geometry each figure is restricted so that ..."],["thompson-calculus-made-easy-1914/x-45273359b4",15,"Thompson 1914, p. 20: But, you will say, we neglected a whole unit. ..."],["boyden-first-book-in-algebra-1895/ex-20/10",4,"Boyden 1895, Exercise 20 (10)"],["todhunter-spherical-trigonometry-1886/ch-miscellaneous-propositions",2,"Todhunter 1886, Miscellaneous Propositions","../books/todhunter-spherical-trigonometry-1886/ch/ch-miscellaneous-propositions/index.html"],["planck-treatise-on-thermodynamics-1903/x-6448dd8215",15,"Planck 1903, p. 36: Each one of these experimental results would by itself ..."],["slaught-lennes-solid-geometry-1919/x-635024d72b",15,"Slaught & Lennes 1919, p. 1: A figure, all parts of which lie in one ..."],["slaught-lennes-solid-geometry-1919/x-07981ac5b3",15,"Slaught & Lennes 1919, p. 3: In plane geometry, two lines which do not meet ..."],["slaught-lennes-solid-geometry-1919/x-a6ef59e1f1",15,"Slaught & Lennes 1919, p. 2: In plane geometry, “the locus of all points at ..."],["slaught-lennes-solid-geometry-1919/x-dc448934d2",15,"Slaught & Lennes 1919, p. 2: If all parts of the figure are not required ..."],["slaught-lennes-solid-geometry-1919/x-325976aead",15,"Slaught & Lennes 1919, p. 1: A plane is designated by a single letter in ..."],["slaught-lennes-solid-geometry-1919/x-84f36c2072",15,"Slaught & Lennes 1919, p. 8: The area of a circle is one half the ..."],["slaught-lennes-solid-geometry-1919/x-954cc854e9",15,"Slaught & Lennes 1919, p. 5: Through a point not on a given line only ..."],["ball-mathematical-recreations-1905/eq-d9011b5830",16,"Ball 1905, scan 32: m<12"],["form/62a1369bce",5,"identity: (x - 13)*(x - 1)"],["thompson-calculus-made-easy-1914/eq-88d96171cf",16,"Thompson 1914, p. 68: \\frac{dy}{dx} &= \\frac{dy}{du} × \\frac{du}{dx} = \\frac{1}{2\\sqrt{a+x}}"],["thompson-calculus-made-easy-1914/eq-950a17643c",16,"Thompson 1914, p. 68: \\frac{dy}{dx} &= \\frac{dy}{du}×\\frac{du}{dx} = - \\frac{x}{\\sqrt{(a+x^2)^3}}"],["thompson-calculus-made-easy-1914/eq-691de85314",16,"Thompson 1914, p. 69: \\frac{dy}{dx} &= \\frac{dy}{du} × \\frac{du}{dx} = -\\frac{3x^2}{2\\sqrt{(x^3 - a^2)^3}}"],["thompson-calculus-made-easy-1914/eq-3a83076c90",16,"Thompson 1914, p. 70: \\frac{dy}{dx} &= - \\frac{1}{(1+x)\\sqrt{1-x^2}}"],["thompson-calculus-made-easy-1914/eq-ae836b5c89",16,"Thompson 1914, p. 70: = \\frac{\\sqrt{x}(3+x^2)}{2\\sqrt{(1+x^2)^3}}"],["thompson-calculus-made-easy-1914/eq-0654b7a640",16,"Thompson 1914, p. 71: &= 3\\left(x+\\sqrt{x^2+x+a}\\right)^2 \\left(1 +\\frac{2x+1}{2\\sqrt{x^2+x+a}}\\right)"],["thompson-calculus-made-easy-1914/eq-1d601b014e",16,"Thompson 1914, p. 73: = \\frac{x(3a-4x)}{2b\\sqrt{(a-x)x}}"],["ball-mathematical-recreations-1905/eq-059b1071b2",16,"Ball 1905, scan 33: m < 20"],["de-morgan-elementary-illustrations-calculus-1899/eq-4819483f76",16,"De Morgan 1899, p. 23: \\phi' x\\, h + \\phi'' x\\, \\frac{h^{2}}{2} x + \\phi''' x\\, \\frac{h^{3}}{2·3} + \\etc."],["shape/bca108c142",6,"identity: (N + x)*(x - 1)"],["de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients-of-differential-coefficients",2,"De Morgan 1899, Differential Coefficients of Differential Coefficients","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-differential-coefficients-of-differential-coefficients/index.html"],["todhunter-spherical-trigonometry-1886/ex-xv",3,"Todhunter 1886, Exercise XV"],["thompson-calculus-made-easy-1914/eq-995e34b769",16,"Thompson 1914, p. 73: &= \\frac{3a^2-12ax+8x^2}{4b(a-x)\\sqrt{(a-x)x}}"],["thompson-calculus-made-easy-1914/eq-45dbca0671",16,"Thompson 1914, p. 72: \\frac{d(y^n)}{d(y^5)} = \\frac{ny^{n-1}}{5y^{5-1}} = \\frac{n}{5} y^{n-5}"],["thompson-calculus-made-easy-1914/eq-3914dd9c18",16,"Thompson 1914, p. 74: \\dfrac{dy}{dx} = \\dfrac{dy}{dz} × \\dfrac{dz}{dv} × \\dfrac{dv}{dx}"],["thompson-calculus-made-easy-1914/eq-7d41bcaf3c",16,"Thompson 1914, p. 74: \\frac{dv}{dx} = \\frac{7x(5x-6)}{3\\sqrt[3]{(x-1)^4}}"],["thompson-calculus-made-easy-1914/eq-b4e5b7cb08",16,"Thompson 1914, p. 74: \\frac{dx}{dt} = 3t^2 + \\tfrac{1}{2}"],["thompson-calculus-made-easy-1914/eq-fae65530ab",16,"Thompson 1914, p. 74: \\frac{dt}{d\\theta} = -\\frac{1}{10\\sqrt{\\theta^3}}"],["thompson-calculus-made-easy-1914/eq-5d8083c40b",16,"Thompson 1914, p. 74: \\frac{dv}{d\\theta} = -\\frac{7x(5x-6)(3t^2+\\frac{1}{2})} {30\\sqrt[3]{(x-1)^4} \\sqrt{\\theta^3}}"],["whitehead-introduction-to-mathematics-1911/x-0285cd9d47",15,"Whitehead 1911, p. 108: The product of the two vectors OP and OQ ..."],["hardy-course-of-pure-mathematics-1921/eq-1af659d15b",16,"Hardy 1921, p. 16: -(-\\alpha) = \\alpha"],["wentworth-first-steps-in-algebra-1894/ex-1/2",4,"Wentworth 1894, Exercise 1 (2)"],["thompson-calculus-made-easy-1914/eq-3233f32bc3",16,"Thompson 1914, p. 74: = -\\frac{28}{3x^5\\sqrt{9x^8+7}}"],["thompson-calculus-made-easy-1914/eq-a614f9d5d0",16,"Thompson 1914, p. 75: \\frac{d\\phi}{d\\omega} = \\frac{1}{\\sqrt{2}\\omega^2}"],["thompson-calculus-made-easy-1914/eq-2f9b1c3b73",16,"Thompson 1914, p. 75: \\frac{d\\omega}{d\\theta} = -\\frac{1}{(1+\\theta)\\sqrt{1-\\theta^2}}"],["hardy-course-of-pure-mathematics-1921/eq-1bc81f11ea",16,"Hardy 1921, p. 17: \\gamma = \\alpha + \\beta"],["maxwell-elementary-treatise-electricity-1888/x-937e72bab9",15,"Maxwell 1888, scan 80: Definition.---The electric or electromotive force at a point is ..."],["whitehead-introduction-to-mathematics-1911/x-37178c8146",15,"Whitehead 1911, p. 104: The answer is that it is perfectly indifferent which ..."],["maxwell-elementary-treatise-electricity-1888/x-dfe6e79875",15,"Maxwell 1888, scan 79: Hence the idea of an electrified point is a ..."],["planck-treatise-on-thermodynamics-1903/x-6232a89d50",15,"Planck 1903, p. 120: These two equations lead to an experimental test of ..."],["hardy-course-of-pure-mathematics-1921/eq-d50828eadc",16,"Hardy 1921, p. 17: \\alpha - \\beta = \\alpha + (-\\beta)"],["maxwell-elementary-treatise-electricity-1888/eq-6aa66a3f46",16,"Maxwell 1888, scan 101: \\overline{AP} = m \\overline{BP}"],["maxwell-elementary-treatise-electricity-1888/eq-c0cf6a49d4",16,"Maxwell 1888, scan 101: V &= \\frac{e}{\\overline{AP}} + \\frac{e'}{\\overline{BP}}"],["maxwell-elementary-treatise-electricity-1888/eq-14fc81cfad",16,"Maxwell 1888, scan 102: AD = \\frac{a^2}{c}"],["maxwell-elementary-treatise-electricity-1888/eq-2a169a5a96",16,"Maxwell 1888, scan 102: BE = \\frac{b^2}{c}"],["maxwell-elementary-treatise-electricity-1888/eq-3a3045c436",16,"Maxwell 1888, scan 103: AF = \\frac{a^2}{AE} = \\frac{a^2 c}{c^2 - b^2}"],["maxwell-elementary-treatise-electricity-1888/eq-3de372884b",16,"Maxwell 1888, scan 103: BG = \\frac{b^2}{DB} = \\frac{b^2 c}{c^2 - a^2}"],["maxwell-elementary-treatise-electricity-1888/eq-7f2cc10557",16,"Maxwell 1888, scan 103: q_{aa} &= a + \\frac{a^2 b}{c^2 - b^2} + \\text{\\&c.,}"],["hardy-course-of-pure-mathematics-1921/eq-763bd56411",16,"Hardy 1921, p. 18: (-\\alpha)\\beta = -\\alpha\\beta"],["thompson-calculus-made-easy-1914/x-8778bd96aa",15,"Thompson 1914, p. 213: Now reckon out the area beneath the curve by ..."],["maxwell-elementary-treatise-electricity-1888/eq-795dd232e3",16,"Maxwell 1888, scan 103: q_{ab} &= - \\frac{ab}{c} - \\frac{a^2 b^2}{c (c^2 - a^2 - b^2)} - \\text{\\&c.,}"],["maxwell-elementary-treatise-electricity-1888/eq-27185acbea",16,"Maxwell 1888, scan 103: q_{bb} &= b + \\frac{ab^2}{c^2 -a^2} + \\text{\\&c.,}"],["maxwell-elementary-treatise-electricity-1888/eq-577e5a8e6f",16,"Maxwell 1888, scan 103: E_a = q_{aa} P_a + q_{ab} P_b"],["maxwell-elementary-treatise-electricity-1888/eq-a3d61257ed",16,"Maxwell 1888, scan 103: E_b = q_{ab} P_a + q_{bb} P_b"],["maxwell-elementary-treatise-electricity-1888/eq-eef64e9514",16,"Maxwell 1888, scan 103: &P_a = \\frac{1}{a} E_a + \\frac{1}{c} E_b"],["maxwell-elementary-treatise-electricity-1888/eq-6f45be4433",16,"Maxwell 1888, scan 103: &P_b = \\frac{1}{c} E_a + \\left\\{\\frac{1}{b} - \\frac{a^3}{c^2 (c^2 - a^2)}\\right\\} E_b"],["hardy-course-of-pure-mathematics-1921/eq-6a9fe49173",16,"Hardy 1921, p. 18: \\alpha(-\\beta) = -\\alpha\\beta"],["theorem/area-of-a-circular-sector",9,"area of a circular sector","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-area-of-a-circular-sector"],["de-morgan-elementary-illustrations-calculus-1899/eq-a4c8aad86c",16,"De Morgan 1899, p. 23: \\frac{\\emph{increment of } \\phi x}{\\emph{increment of } x} = \\phi' x + \\phi'' x\\, \\frac{h}{2} x + \\phi''' x\\, \\frac{h^{2"],["maxwell-elementary-treatise-electricity-1888/eq-64bd1bcc1c",16,"Maxwell 1888, scan 103: \\frac{1}{2} (E_a P_a + E_b P_b) = \\frac{1}{2} \\frac{1}{a}\\, E_a^2 + \\frac{1}{c}\\, E_a E_b + \\frac{1}{2} \\left\\{\\frac{1}{"],["maxwell-elementary-treatise-electricity-1888/eq-c5d1ee7bfb",16,"Maxwell 1888, scan 104: R =\\frac{ E_b}{c^2}\\left\\{E_a - E_b\\frac{a^3 (2c^2 - a^2)}{c(c^2 - a^2)^2}\\right\\}"],["maxwell-elementary-treatise-electricity-1888/eq-3bd272abd1",16,"Maxwell 1888, scan 104: E_a \\text{ must be greater than } E_b\\frac{ a^3(2c^2 - a^2)}{c(c^2 - a^2)^2}"],["maxwell-elementary-treatise-electricity-1888/eq-9048db91fa",16,"Maxwell 1888, scan 100: \\xp"],["maxwell-elementary-treatise-electricity-1888/eq-b771914988",16,"Maxwell 1888, scan 105: \\sigma = - \\frac{1}{4\\pi} \\frac{ea}{r^3} (m^2 - 1)"],["maxwell-elementary-treatise-electricity-1888/eq-641dcaed8d",16,"Maxwell 1888, scan 105: 4 \\pi \\sigma = R"],["thompson-calculus-made-easy-1914/x-833928bd2b",15,"Thompson 1914, p. 216: Consider an elementary zone or annulus of the surface ..."],["slaught-lennes-solid-geometry-1919/x-296601393a",15,"Slaught & Lennes 1919, p. 136: This theorem was discovered by Cavalieri."],["hardy-course-of-pure-mathematics-1921/eq-0c2a6c134f",16,"Hardy 1921, p. 18: (-\\alpha)(-\\beta) = \\alpha\\beta"],["thompson-calculus-made-easy-1914/x-e8e575acc5",15,"Thompson 1914, p. 1: (1) d which merely means “a little bit of.”"],["thompson-calculus-made-easy-1914/x-b3750aba2f",15,"Thompson 1914, p. 1: Thus \\ds\\int dx means the sum of all the ..."],["thompson-calculus-made-easy-1914/x-5e1e99e2bb",15,"Thompson 1914, p. 2: The word “integral” simply means “the whole.”"],["theorem/areas-of-similar-segments-are-proportional-to-the-squares-on-their-chords",9,"areas of similar segments are proportional to the squares on their chords","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-areas-of-similar-segments-are-proportional-to-the-squares-on-their-chords"],["wentworth-plane-geometry-1899/eq-ab83981a82",16,"Wentworth 1899, scan 231: \\pi = \\dfrac{C}{2R}"],["wentworth-plane-geometry-1899/eq-3ec41d5145",16,"Wentworth 1899, scan 231: C=2\\pi R"],["wentworth-plane-geometry-1899/eq-cef8e23a78",16,"Wentworth 1899, scan 242: 2\\pi R = C"],["wentworth-plane-geometry-1899/eq-bb8d819cb3",16,"Wentworth 1899, scan 242: \\pi = \\frac{1}{2}C"],["wentworth-plane-geometry-1899/eq-7df4945cbf",16,"Wentworth 1899, scan 242: C = 6.28317"],["concept/similar-curves",7,"similar curves","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-similar-curves"],["maxwell-elementary-treatise-electricity-1888/x-240d7a8fd9",15,"Maxwell 1888, scan 92: But in the vulgar language of the time when ..."],["wentworth-plane-geometry-1899/eq-8dd5758295",16,"Wentworth 1899, scan 242: \\pi = 3.14159"],["hardy-course-of-pure-mathematics-1921/eq-fef2cacbe7",16,"Hardy 1921, p. 18: 1/(-\\alpha) = -(1/\\alpha)"],["wentworth-plane-geometry-1899/eq-19b4e5b2d1",16,"Wentworth 1899, scan 242: \\pi = 3.1416"],["wentworth-plane-geometry-1899/eq-f9f6dce470",16,"Wentworth 1899, scan 242: \\frac{1}{\\pi} = 0.31831"],["hardy-course-of-pure-mathematics-1921/eq-7d46eeeced",16,"Hardy 1921, p. 18: \\alpha/\\beta = \\alpha × (1/\\beta)"],["boyden-first-book-in-algebra-1895/ex-20/11",4,"Boyden 1895, Exercise 20 (11)"],["concept/geodesic",7,"geodesic","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-geodesic"],["theorem/radii-of-inscribed-and-circumscribed-circles-of-doubled-polygons",9,"radii of inscribed and circumscribed circles of doubled polygons","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-radii-of-inscribed-and-circumscribed-circles-of-doubled-polygons"],["method/approximating-pi-by-inscribed-and-circumscribed-polygons",8,"approximating pi by inscribed and circumscribed polygons","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-method-approximating-pi-by-inscribed-and-circumscribed-polygons"],["hardy-course-of-pure-mathematics-1921/eq-498c97fafd",16,"Hardy 1921, p. 19: (\\sqrt{2})^{2} = \\sqrt{2}\\sqrt{2} = 2."],["maxwell-elementary-treatise-electricity-1888/x-d3817b9374",15,"Maxwell 1888, scan 84: Now the electric capacity of a body in a ..."],["person/adrian-metius",1,"Adrian Metius","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-person-adrian-metius"],["blackburn-elements-plane-trigonometry-1863/x-811984a21a",15,"Blackburn 1863, p. 1: A magnitude or ratio, which is fixed in value ..."],["blackburn-elements-plane-trigonometry-1863/x-0a3b94f17c",15,"Blackburn 1863, p. 2: If two variables are at every instant equal their ..."],["hardy-course-of-pure-mathematics-1921/eq-9b363987a0",16,"Hardy 1921, p. 20: x^{2} - 2ax + a^{2} - b = 0"],["boyden-first-book-in-algebra-1895/ex-20/12",4,"Boyden 1895, Exercise 20 (12)"],["blackburn-elements-plane-trigonometry-1863/x-70da0f39c1",15,"Blackburn 1863, p. 3: Then if the number of sides of the polygons ..."],["blackburn-elements-plane-trigonometry-1863/x-3d7208f2fa",15,"Blackburn 1863, p. 5: The area of any circular sector is half the ..."],["blackburn-elements-plane-trigonometry-1863/x-7a9b9b5a41",15,"Blackburn 1863, p. 2: Let a number of points be taken in a ..."],["blackburn-elements-plane-trigonometry-1863/x-1f093d60aa",15,"Blackburn 1863, p. 11: By the method of “continued fractions” it will be ..."],["blackburn-elements-plane-trigonometry-1863/x-7d65261339",15,"Blackburn 1863, p. 11: Of these \\dfrac{22}{7} (=3.14) is the approximation discovered by ..."],["hardy-course-of-pure-mathematics-1921/eq-82071283d8",16,"Hardy 1921, p. 20: ax^{2} + 2bx + c = 0"],["slaught-lennes-solid-geometry-1919/x-3b118a1c5f",15,"Slaught & Lennes 1919, p. 140: The area of a sphere whose radius is r ..."],["blackburn-elements-plane-trigonometry-1863/eq-b60511c0fe",16,"Blackburn 1863, p. 37: a &= c \\sin A"],["hardy-course-of-pure-mathematics-1921/eq-04e14423a8",16,"Hardy 1921, p. 21: \\sqrt{8} = 2\\sqrt{2}"],["blackburn-elements-plane-trigonometry-1863/eq-9fee30e497",16,"Blackburn 1863, p. 37: b &= c \\cos A"],["blackburn-elements-plane-trigonometry-1863/eq-2b468bec3a",16,"Blackburn 1863, p. 37: b &= c \\sin B"],["blackburn-elements-plane-trigonometry-1863/eq-953186424a",16,"Blackburn 1863, p. 37: a &= c \\cos B"],["blackburn-elements-plane-trigonometry-1863/eq-7209e8a3c6",16,"Blackburn 1863, p. 37: a &= b \\tan A"],["blackburn-elements-plane-trigonometry-1863/eq-0749ad0934",16,"Blackburn 1863, p. 37: c &= b \\sec A"],["blackburn-elements-plane-trigonometry-1863/eq-1ab0022818",16,"Blackburn 1863, p. 37: b &= a \\tan B"],["blackburn-elements-plane-trigonometry-1863/eq-ab6adceb96",16,"Blackburn 1863, p. 37: c &= a \\sec B"],["dickson-theory-of-equations-1922/eq-db5d67deec",16,"Dickson 1922, p. 55: x^2 - 6x - 3 = 0"],["maxwell-elementary-treatise-electricity-1888/eq-5d62bfe413",16,"Maxwell 1888, scan 144: E_{12} = CR_{12}\\text{,} \\quad E_{23} = CR_{23}\\text{,} \\quad E_{34} = CR_{34}\\text{.}"],["blackburn-elements-plane-trigonometry-1863/eq-0d0c5585cf",16,"Blackburn 1863, p. 37: \\log a &= \\log c + \\tab\\log \\sin A - 10"],["blackburn-elements-plane-trigonometry-1863/eq-cb9f987031",16,"Blackburn 1863, p. 37: \\log b &= \\log c + \\tab\\log \\cos A - 10"],["blackburn-elements-plane-trigonometry-1863/eq-01855a06d9",16,"Blackburn 1863, p. 37: \\log b &= \\log a + \\tab\\log \\tan B - 10"],["blackburn-elements-plane-trigonometry-1863/eq-4e0bfebbb4",16,"Blackburn 1863, p. 37: \\log c &= \\log a + \\tab\\log \\sec B - 10"],["blackburn-elements-plane-trigonometry-1863/eq-cfdf684cbd",16,"Blackburn 1863, p. 38: \\tab\\log \\sin A &= 10 + \\log a - \\log c"],["blackburn-elements-plane-trigonometry-1863/eq-932924e024",16,"Blackburn 1863, p. 37: B = 90° - A"],["blackburn-elements-plane-trigonometry-1863/eq-c5caf10392",16,"Blackburn 1863, p. 38: AD = AB \\sin B = c \\sin B"],["blackburn-elements-plane-trigonometry-1863/eq-ebbad58551",16,"Blackburn 1863, p. 38: AD = CA \\sin C = b \\sin C"],["maxwell-elementary-treatise-electricity-1888/eq-f5613a74fa",16,"Maxwell 1888, scan 143: E = CR \\text{.}"],["hardy-course-of-pure-mathematics-1921/ex-lviii/4b",4,"Hardy 1921, Exercise LVIII (4b)"],["blackburn-elements-plane-trigonometry-1863/eq-aff341bcef",16,"Blackburn 1863, p. 38: \\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}"],["blackburn-elements-plane-trigonometry-1863/eq-bf2fd41781",16,"Blackburn 1863, p. 39: \\Tab\\log\\sin B = (\\tab\\log\\sin C - \\log c) + \\log b"],["blackburn-elements-plane-trigonometry-1863/eq-faa20b0363",16,"Blackburn 1863, p. 39: A = 180° - (B + C)"],["blackburn-elements-plane-trigonometry-1863/eq-46d9a613a2",16,"Blackburn 1863, p. 39: \\log a = \\tab\\log\\sin A - (\\tab\\log\\sin C - \\log c)"],["blackburn-elements-plane-trigonometry-1863/eq-869d782d87",16,"Blackburn 1863, p. 40: \\tab\\log\\sin B_1 = \\log b + (\\tab\\log\\sin C - \\log c)"],["blackburn-elements-plane-trigonometry-1863/eq-0d7047239a",16,"Blackburn 1863, p. 40: B_2 = 180° - B_1"],["blackburn-elements-plane-trigonometry-1863/eq-bf08a87bf5",16,"Blackburn 1863, p. 40: A_1 = 180° - (B_1 + C)"],["maxwell-elementary-treatise-electricity-1888/eq-c46a61d7b6",16,"Maxwell 1888, scan 144: E &= E_{12} + E_{23} + E_{34}\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-b81327c95f",16,"Maxwell 1888, scan 144: R = R_{12} + R_{23} + R_{34}"],["blackburn-elements-plane-trigonometry-1863/eq-82d3448656",16,"Blackburn 1863, p. 40: A_2 = 180° - (B_2 + C)"],["blackburn-elements-plane-trigonometry-1863/eq-5a54a7477d",16,"Blackburn 1863, p. 40: \\log a_1 = \\tab\\log\\sin A_1 - (\\tab\\log\\sin C - \\log c)"],["blackburn-elements-plane-trigonometry-1863/eq-6716a62e33",16,"Blackburn 1863, p. 40: \\log a_2 = \\tab\\log\\sin A_2 - (\\tab\\log\\sin C - \\log c)"],["blackburn-elements-plane-trigonometry-1863/eq-cfbf497094",16,"Blackburn 1863, p. 40: DCB = A + B = 2BED"],["blackburn-elements-plane-trigonometry-1863/eq-a20fb7a997",16,"Blackburn 1863, p. 40: A = ECF + CFA = ECF + B"],["blackburn-elements-plane-trigonometry-1863/eq-5609d4bce0",16,"Blackburn 1863, p. 40: ECF = A - B = 2FBE"],["blackburn-elements-plane-trigonometry-1863/eq-97f62021ec",16,"Blackburn 1863, p. 41: BED = \\dfrac{1}{2}(A + B)"],["concept/half-angle",7,"half-angle"],["blackburn-elements-plane-trigonometry-1863/eq-514a2b1eae",16,"Blackburn 1863, p. 41: FBE = \\dfrac{1}{2}(A - B)"],["maxwell-elementary-treatise-electricity-1888/eq-c7bc23ee13",16,"Maxwell 1888, scan 144: a - b = R_1C\\text{,} \\quad b - c = R_2C\\text{,}\\quad \\text{and} \\quad a - c = RC\\text{,}"],["hardy-course-of-pure-mathematics-1921/eq-ffc9f70bba",16,"Hardy 1921, p. 22: A + \\sqrt{B} = C + \\sqrt{D}"],["blackburn-elements-plane-trigonometry-1863/eq-3cd5988404",16,"Blackburn 1863, p. 41: \\frac{a+b}{\\cos\\dfrac{1}{2}(A - B)} = \\frac{c}{\\cos\\dfrac{1}{2}(A + B)}"],["blackburn-elements-plane-trigonometry-1863/eq-59ab9f2ca1",16,"Blackburn 1863, p. 41: \\frac{a-b}{\\sin\\dfrac{1}{2}(A - B)} = \\frac{c}{\\sin\\dfrac{1}{2}(A + B)}"],["blackburn-elements-plane-trigonometry-1863/eq-3e319aa0a6",16,"Blackburn 1863, p. 41: \\frac{\\tan\\dfrac{1}{2}(A - B)}{\\tan\\dfrac{1}{2}(A + B)} = \\frac{a - b}{a + b}"],["blackburn-elements-plane-trigonometry-1863/eq-d5062bba0e",16,"Blackburn 1863, p. 42: A + B = 180° - C"],["blackburn-elements-plane-trigonometry-1863/eq-c82095912b",16,"Blackburn 1863, p. 42: \\tab\\log\\tan \\frac{1}{2} (A - B) = \\tab\\log\\tan \\frac{1}{2} (A + B) \\\\ + \\log (a - b) - \\log (a + b)"],["blackburn-elements-plane-trigonometry-1863/eq-923d7fa2af",16,"Blackburn 1863, p. 42: A &= \\frac{1}{2} (A + B) + \\frac{1}{2} (A - B)"],["blackburn-elements-plane-trigonometry-1863/eq-6b23040ab0",16,"Blackburn 1863, p. 42: B &= \\frac{1}{2} (A + B) - \\frac{1}{2} (A - B)"],["maxwell-elementary-treatise-electricity-1888/eq-6eeb6b1493",16,"Maxwell 1888, scan 144: b = \\frac {R_2a + R_1c}{R}\\text{,}"],["thompson-calculus-made-easy-1914/eq-9979996f39",16,"Thompson 1914, p. 244: 2 \\frac{d^2y}{dt^2}\\, \\frac{dy}{dt} = \\frac{d \\left(\\dfrac{dy}{dt}\\right)^2}{dt}"],["blackburn-elements-plane-trigonometry-1863/eq-a76e6b28a5",16,"Blackburn 1863, p. 42: \\log c = \\tab\\log\\sin C - (\\tab\\log\\sin A - \\log a)"],["blackburn-elements-plane-trigonometry-1863/eq-690664468e",16,"Blackburn 1863, p. 42: \\log c = \\tab\\log\\cos \\frac{1}{2} (A + B) - \\tab\\log\\cos \\frac{1}{2} (A - B) + \\log (a + b)"],["blackburn-elements-plane-trigonometry-1863/eq-94001971fb",16,"Blackburn 1863, p. 42: \\log c = \\tab\\log\\sin \\frac{1}{2} (A + B) - \\tab\\log\\sin \\frac{1}{2} (A - B) + \\log (a - b)"],["blackburn-elements-plane-trigonometry-1863/eq-abc3e60521",16,"Blackburn 1863, p. 43: a + b + c = 2s"],["blackburn-elements-plane-trigonometry-1863/eq-4e6d6434e1",16,"Blackburn 1863, p. 43: r = (s - a) \\tan \\frac{1}{2} A"],["blackburn-elements-plane-trigonometry-1863/eq-50392e3c58",16,"Blackburn 1863, p. 43: r^2 = \\frac{(s - a)(s - b)(s - c)}{s}"],["maxwell-elementary-treatise-electricity-1888/eq-acfc970331",16,"Maxwell 1888, scan 145: E = C_1R_1 = C_2R_2 = C_3R_3 = CR\\text{,}"],["hardy-course-of-pure-mathematics-1921/eq-df46acf064",16,"Hardy 1921, p. 22: A - \\sqrt{B} = C - \\sqrt{D}"],["blackburn-elements-plane-trigonometry-1863/eq-bea37ab88f",16,"Blackburn 1863, p. 43: \\log r = \\frac{1}{2} \\bigl\\{ \\log (s - a) + \\log (s - b) + \\log (s - c) - \\log s \\bigr\\}"],["blackburn-elements-plane-trigonometry-1863/eq-1e2ca23c5a",16,"Blackburn 1863, p. 44: \\tab\\log\\tan \\frac{1}{2} A &= 10 + \\log r - \\log (s - a)"],["blackburn-elements-plane-trigonometry-1863/eq-17dbaaaa58",16,"Blackburn 1863, p. 44: \\tab\\log\\tan \\frac{1}{2} B &= 10 + \\log r - \\log (s - b)"],["blackburn-elements-plane-trigonometry-1863/eq-79b401ec53",16,"Blackburn 1863, p. 44: \\tab\\log\\tan \\frac{1}{2} C &= 10 + \\log r - \\log (s - c)"],["maxwell-elementary-treatise-electricity-1888/x-228b4a1c97",15,"Maxwell 1888, scan 91: This circle is an example of a line of ..."],["maxwell-elementary-treatise-electricity-1888/x-7fb3c4dd6e",15,"Maxwell 1888, scan 85: The smaller the distance between the surfaces and the ..."],["hardy-course-of-pure-mathematics-1921/eq-7431d6e597",16,"Hardy 1921, p. 24: z = \\sqrtp[3]{4 + \\sqrt{15}} + \\sqrtp[3]{4 - \\sqrt{15}}"],["boyden-first-book-in-algebra-1895/ex-20/13",4,"Boyden 1895, Exercise 20 (13)"],["form/b1ce9ff245",5,"identity: (x - 15)*(x + 2)"],["hardy-course-of-pure-mathematics-1921/ex-lviii/4c",4,"Hardy 1921, Exercise LVIII (4c)"],["hardy-course-of-pure-mathematics-1921/eq-c928f7a75d",16,"Hardy 1921, p. 24: z^{3} = 3z + 8"],["person/george-albert-wentworth",1,"George Albert Wentworth"],["thompson-calculus-made-easy-1914/eq-9a63a2d72f",16,"Thompson 1914, p. 84: y = ax+b"],["hardy-course-of-pure-mathematics-1921/eq-2c5f1a9e62",16,"Hardy 1921, p. 25: x^{5} = x + 16\\DPtypo{.}{,}"],["de-morgan-elementary-illustrations-calculus-1899/eq-2f07fe1e72",16,"De Morgan 1899, p. 23: h\\left(\\phi'' x\\, \\frac{1}{2} + \\phi''' x\\, \\frac{h}{2·3} + \\etc.\\right)"],["wentworth-first-steps-in-algebra-1894/ex-1/3",4,"Wentworth 1894, Exercise 1 (3)"],["concept/lowering-of-vapour-pressure",7,"lowering of vapour pressure","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-lowering-of-vapour-pressure"],["wentworth-plane-geometry-1899/ch-ii",2,"Wentworth 1899, ch. II: THE CIRCLE","../books/wentworth-plane-geometry-1899/ch/ch-ii/index.html"],["de-morgan-elementary-illustrations-calculus-1899/x-322ef48dbb",15,"De Morgan 1899, p. 54: we can, at the end of every time, assign ..."],["form/6f53557c16",5,"identity: 13"],["hardy-course-of-pure-mathematics-1921/ex-lviii/5a",4,"Hardy 1921, Exercise LVIII (5a)"],["hardy-course-of-pure-mathematics-1921/ex-lviii/5b",4,"Hardy 1921, Exercise LVIII (5b)"],["hardy-course-of-pure-mathematics-1921/ex-lviii/6a",4,"Hardy 1921, Exercise LVIII (6a)"],["hardy-course-of-pure-mathematics-1921/eq-a30bd5fbde",16,"Hardy 1921, p. 25: \\pi^{5} = \\pi + n"],["thompson-calculus-made-easy-1914/eq-ab730b882d",16,"Thompson 1914, p. 84: \\dfrac{dy}{dx} = a"],["concept/heterostatic-instrument",7,"heterostatic instrument","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-heterostatic-instrument"],["de-morgan-elementary-illustrations-calculus-1899/eq-812bb93f4b",16,"De Morgan 1899, p. 24: kh^{n} : lh^{n+1} + mh^{n+2} + \\etc.,\\ ::\\ k : lh + mh^{2} + \\etc.,"],["wentworth-plane-geometry-1899/eq-ad4a6ec4d9",16,"Wentworth 1899, scan 170: \\overline{AC}^2=AB × AF"],["hardy-course-of-pure-mathematics-1921/eq-2acdfb1442",16,"Hardy 1921, p. 29: \\beta \\leq x \\leq \\gamma"],["maxwell-elementary-treatise-electricity-1888/x-b5c74a1dd0",15,"Maxwell 1888, scan 21: Whatever produces or tends to produce a transfer of ..."],["maxwell-elementary-treatise-electricity-1888/x-352310df7e",15,"Maxwell 1888, scan 22: The electromotive force from any point, along a path ..."],["maxwell-elementary-treatise-electricity-1888/x-692b6bf64f",15,"Maxwell 1888, scan 20: Hence bodies when electrified in the same way are ..."],["maxwell-elementary-treatise-electricity-1888/x-6e763f8234",15,"Maxwell 1888, scan 19: Bodies may therefore be divided into two classes: conductors, ..."],["maxwell-elementary-treatise-electricity-1888/x-ca597c7300",15,"Maxwell 1888, scan 17: The shreds of paper will move, the lighter ones ..."],["thompson-calculus-made-easy-1914/eq-fa903cb6b7",16,"Thompson 1914, p. 85: y = ax^2 + b"],["hardy-course-of-pure-mathematics-1921/eq-96cede1da4",16,"Hardy 1921, p. 401: \\log |\\zeta| + i\\am \\zeta = \\log \\rho + i(2k\\pi + \\phi)"],["maxwell-elementary-treatise-electricity-1888/x-85a1e2e16c",15,"Maxwell 1888, scan 24: We must also remember that temperature corresponds to a ..."],["maxwell-elementary-treatise-electricity-1888/eq-0cf23e8fdd",16,"Maxwell 1888, scan 209: \\frac{c + y}{b - y} = \\frac{\\gamma}{\\beta}"],["hardy-course-of-pure-mathematics-1921/ex-lviii/6b",4,"Hardy 1921, Exercise LVIII (6b)"],["maxwell-elementary-treatise-electricity-1888/eq-e44bdd033f",16,"Maxwell 1888, scan 209: \\frac{\\gamma ^2}{\\beta ^2} = 1 + \\frac{(b+c)(y-x)}{(c+x)(b-y)}"],["maxwell-elementary-treatise-electricity-1888/eq-9e6da24093",16,"Maxwell 1888, scan 145: \\frac{1}{R} = \\frac{1}{R_1} + \\frac{1}{R_2} + \\frac{1}{R_3}\\text{.}"],["boyden-first-book-in-algebra-1895/ex-20/14",4,"Boyden 1895, Exercise 20 (14)"],["form/a5a7c821f3",5,"identity: (x + 2)*(x + 16)"],["concept/point-of-accumulation",7,"point of accumulation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-point-of-accumulation"],["maxwell-elementary-treatise-electricity-1888/eq-0cdb2617b2",16,"Maxwell 1888, scan 145: C = C_1 + C_2 + C_3\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-4961708d1f",16,"Maxwell 1888, scan 145: C_1 = C\\frac{R}{R_1}\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-baf87a6468",16,"Maxwell 1888, scan 146: R = \\frac{l\\rho}{s}\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-176abcfeb8",16,"Maxwell 1888, scan 146: R = \\frac{l^2r}{m}\\text{.}"],["de-morgan-elementary-illustrations-calculus-1899/x-42179164f4",15,"De Morgan 1899, p. 57: That is, at the end of four seconds a ..."],["hardy-course-of-pure-mathematics-1921/eq-0db684eb56",16,"Hardy 1921, p. 402: \\Log z_{1} z_{2} = \\Log z_{1} + \\Log z_{2}"],["planck-treatise-on-thermodynamics-1903/ch-molecular-weight",2,"Planck 1903, Molecular Weight","../books/planck-treatise-on-thermodynamics-1903/ch/ch-molecular-weight/index.html"],["slaught-lennes-solid-geometry-1919/eq-8802f8e67d",16,"Slaught & Lennes 1919, p. 160: S = 2\\pi r(r + h)"],["slaught-lennes-solid-geometry-1919/eq-d540dbf5a7",16,"Slaught & Lennes 1919, p. 160: V = \\pi {r'}^2h'"],["slaught-lennes-solid-geometry-1919/eq-d4cfef6e4b",16,"Slaught & Lennes 1919, p. 160: \\dfrac{r}{r'} = \\dfrac{h}{h'}"],["slaught-lennes-solid-geometry-1919/eq-4b7ffe76a7",16,"Slaught & Lennes 1919, p. 160: \\dfrac{s}{s'} = \\dfrac{S}{S'} = \\dfrac{r^2}{r'^2} = \\dfrac{h^2}{h'^2}"],["thompson-calculus-made-easy-1914/eq-5025b30a18",16,"Thompson 1914, p. 116: C = aP + \\dfrac{b}{c+P} + d"],["thompson-calculus-made-easy-1914/eq-f4179b8d63",16,"Thompson 1914, p. 116: \\dfrac{dC}{dP} = a - \\frac{b}{(c+P)^2} = 0"],["hardy-course-of-pure-mathematics-1921/ex-lx/1",4,"Hardy 1921, Exercise LX (1)"],["slaught-lennes-solid-geometry-1919/eq-346882baa5",16,"Slaught & Lennes 1919, p. 160: \\dfrac{s}{s'} = \\dfrac{S}{S'} = \\dfrac{r^2}{r'^2} = \\dfrac{h^2}{h'^2} \\text{ and } \\dfrac{V}{V'} = \\dfrac{r^3}{r'^3} = \\"],["slaught-lennes-solid-geometry-1919/eq-96e1a345b5",16,"Slaught & Lennes 1919, p. 163: \\dfrac{V}{V'} = \\dfrac{PA \\cdot PB \\cdot PC}{P'A' \\cdot P'B' \\cdot P'C'}"],["concept/equal-trihedral-angles",7,"equal trihedral angles"],["slaught-lennes-solid-geometry-1919/eq-6aaab052f2",16,"Slaught & Lennes 1919, p. 164: \\dfrac{V}{V'} = \\dfrac{\\overline{PA}^3}{\\overline{P'A'}^3}"],["slaught-lennes-solid-geometry-1919/eq-5aa83460ab",16,"Slaught & Lennes 1919, p. 165: OA : OA' = OB : OB'"],["thompson-calculus-made-easy-1914/eq-6206efcc63",16,"Thompson 1914, p. 116: P = ±\\sqrt{\\dfrac{b}{a}} - c"],["thompson-calculus-made-easy-1914/eq-dd77c51d00",16,"Thompson 1914, p. 117: C = N\\left(\\frac{C_l}{t} + \\frac{EPC_e}{1000}\\right)"],["thompson-calculus-made-easy-1914/eq-4fa343c1bf",16,"Thompson 1914, p. 117: t = mE^n"],["concept/common-ratio",7,"common ratio","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-common-ratio"],["slaught-lennes-solid-geometry-1919/eq-dc1d03ed3b",16,"Slaught & Lennes 1919, p. 170: \\dfrac{V}{V'} = \\dfrac{\\overline{AB}^3}{\\overline{A'B'}^3}"],["slaught-lennes-solid-geometry-1919/eq-ccda6ac7b8",16,"Slaught & Lennes 1919, p. 172: \\dfrac{\\text{area } ABC}{\\text{area } A'B'C'} = \\dfrac{m^2}{n^2}"],["concept/corresponding-cross-sections",7,"corresponding cross-sections"],["slaught-lennes-solid-geometry-1919/eq-5430f639bf",16,"Slaught & Lennes 1919, p. 172: \\dfrac{\\text{vol.\\ } ABCD}{\\text{vol.\\ } A'B'C'D'} = \\dfrac{m^3}{n^3}"],["slaught-lennes-solid-geometry-1919/eq-d0175894ee",16,"Slaught & Lennes 1919, p. 174: w = kh^3"],["theorem/nth-term-of-a-geometrical-progression",9,"nth term of a geometrical progression","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-theorem-nth-term-of-a-geometrical-progression"],["theorem/arithmetic-geometric-mean-inequality",9,"arithmetic–geometric mean inequality","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-arithmetic-geometric-mean-inequality"],["blackburn-elements-plane-trigonometry-1863/eq-043253d284",16,"Blackburn 1863, p. 48: A'B'= AB \\cos BAC"],["wentworth-first-steps-in-algebra-1894/x-0e19f333c9",15,"Wentworth 1894, p. 149: In formula (1), put 5 for n, 3 for ..."],["hardy-course-of-pure-mathematics-1921/eq-8e827e52a7",16,"Hardy 1921, p. 402: \\log z_{1}z_{2} = \\log z_{1} + \\log z_{2}"],["thompson-calculus-made-easy-1914/x-bf4acef86e",15,"Thompson 1914, p. 3: Obviously 1 minute is a very small quantity of ..."],["law/law-of-definite-proportions",10,"law of definite proportions","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-law-law-of-definite-proportions"],["thompson-calculus-made-easy-1914/eq-f5f5920db7",16,"Thompson 1914, p. 117: \\frac{PC_e}{1000} - \\frac{nC_l}{m} E^{-(n+1)} = 0"],["form/b82d4eb7bf",5,"identity: (x**2 - 5)*(x**2 + 7)"],["law/law-of-multiple-proportions",10,"law of multiple proportions","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-law-law-of-multiple-proportions"],["boyden-first-book-in-algebra-1895/ex-20/15",4,"Boyden 1895, Exercise 20 (15)"],["shape/aa379e81bd",6,"identity: (N + x**N)**2"],["boyden-first-book-in-algebra-1895/ex-20/16",4,"Boyden 1895, Exercise 20 (16)"],["form/8bce2f2b74",5,"identity: (x - 9)*(x + 9)"],["blackburn-elements-plane-trigonometry-1863/eq-aec7cee42c",16,"Blackburn 1863, p. 48: A'D' = A'B' + B'C' + C'D'"],["person/g-tarry",1,"G. Tarry","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-g-tarry"],["wentworth-first-steps-in-algebra-1894/x-e120fc3933",15,"Wentworth 1894, p. 149: the geometrical mean of any two numbers is the ..."],["wentworth-first-steps-in-algebra-1894/x-0fbd766b52",15,"Wentworth 1894, p. 149: Therefore the series is 3, ±6, 12, ±24, \\dots."],["dickson-theory-of-equations-1922/eq-6b933c787c",16,"Dickson 1922, p. 55: y = x^2 - 6x - 3"],["wentworth-first-steps-in-algebra-1894/x-ca64b2c4cc",15,"Wentworth 1894, p. 151: If a blacksmith uses seven nails in putting a ..."],["concept/line-parallel-to-a-plane",7,"line parallel to a plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-line-parallel-to-a-plane"],["dickson-theory-of-equations-1922/eq-1901272b31",16,"Dickson 1922, p. 55: y = 8x^4 - 14x^3 - 9x^2 + 11x - 2"],["hardy-course-of-pure-mathematics-1921/eq-c9403a896c",16,"Hardy 1921, p. 402: \\Log z^{m} = m\\Log z"],["theorem/two-distinct-planes-cannot-meet-in-one-point-only",9,"two distinct planes cannot meet in one point only","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-two-distinct-planes-cannot-meet-in-one-point-only"],["theorem/oblique-lines-at-equal-distances-from-the-foot-are-equal",9,"oblique lines at equal distances from the foot are equal","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-oblique-lines-at-equal-distances-from-the-foot-are-equal"],["theorem/perpendicular-is-the-shortest-distance-from-a-point-to-a-plane",9,"perpendicular is the shortest distance from a point to a plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-perpendicular-is-the-shortest-distance-from-a-point-to-a-plane"],["theorem/lines-perpendicular-to-the-same-plane-are-parallel",9,"lines perpendicular to the same plane are parallel","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-lines-perpendicular-to-the-same-plane-are-parallel"],["theorem/planes-perpendicular-to-the-same-line-are-parallel",9,"planes perpendicular to the same line are parallel","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-planes-perpendicular-to-the-same-line-are-parallel"],["theorem/plane-parallel-to-two-intersecting-lines-is-parallel-to-their-plane",9,"plane parallel to two intersecting lines is parallel to their plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-plane-parallel-to-two-intersecting-lines-is-parallel-to-their-plane"],["theorem/through-a-point-not-in-a-plane-there-is-one-parallel-plane",9,"through a point not in a plane there is one parallel plane","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-through-a-point-not-in-a-plane-there-is-one-parallel-plane"],["theorem/through-one-of-two-skew-lines-there-is-one-plane-parallel-to-the-other",9,"through one of two skew lines there is one plane parallel to the other","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-through-one-of-two-skew-lines-there-is-one-plane-parallel-to-the-other"],["theorem/angles-whose-sides-are-parallel-are-equal",9,"angles whose sides are parallel are equal","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-angles-whose-sides-are-parallel-are-equal"],["slaught-lennes-solid-geometry-1919/eq-cdcc0d802d",16,"Slaught & Lennes 1919, p. 176: \\dfrac{p}{l} = \\text{cosine}\\, \\angle BAE"],["slaught-lennes-solid-geometry-1919/x-beb86089f3",15,"Slaught & Lennes 1919, p. 9: If a line or a plane contains a point, ..."],["slaught-lennes-solid-geometry-1919/x-4d6f7b6163",15,"Slaught & Lennes 1919, p. 9: While two points determine a straight line it is ..."],["slaught-lennes-solid-geometry-1919/x-9a1a9d95a3",15,"Slaught & Lennes 1919, p. 9: We, therefore, say that three non-collinear points determine a ..."],["slaught-lennes-solid-geometry-1919/x-955bbba1bc",15,"Slaught & Lennes 1919, p. 10: In the figure, PA is perpendicular to the plane ..."],["slaught-lennes-solid-geometry-1919/x-175d5645cb",15,"Slaught & Lennes 1919, p. 17: The perpendicular is the shortest distance from a point ..."],["slaught-lennes-solid-geometry-1919/x-91e91c5753",15,"Slaught & Lennes 1919, p. 24: Note that the four vertices of a quadrilateral in ..."],["slaught-lennes-solid-geometry-1919/x-60f72a7a4e",15,"Slaught & Lennes 1919, p. 20: Any line, as l_1, in either of two parallel ..."],["boyden-first-book-in-algebra-1895/ex-20/17",4,"Boyden 1895, Exercise 20 (17)"],["dickson-theory-of-equations-1922/eq-b7dd6bc71a",16,"Dickson 1922, p. 24: a_0 x^n + \\dotsb + a_{n-1}x + a_n = 0"],["hardy-course-of-pure-mathematics-1921/x-685e60e2ee",15,"Hardy 1921, p. 30: In this case we shall say that \\xi is ..."],["theorem/triangle-area-as-semi-perimeter-times-inradius",9,"triangle area as semi-perimeter times inradius","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-triangle-area-as-semi-perimeter-times-inradius"],["theorem/triangle-area-as-excircle-radius-times-tangent",9,"triangle area as excircle radius times tangent","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-triangle-area-as-excircle-radius-times-tangent"],["theorem/triangle-area-as-a-mean-proportional",9,"triangle area as a mean proportional","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-triangle-area-as-a-mean-proportional"],["hardy-course-of-pure-mathematics-1921/x-ef2b31d0f1",15,"Hardy 1921, p. 29: On the other hand, if S consists of all ..."],["blackburn-elements-plane-trigonometry-1863/x-ced77fd644",15,"Blackburn 1863, p. 11: A triangle is equal to the rectangle contained by ..."],["blackburn-elements-plane-trigonometry-1863/x-01975c8f91",15,"Blackburn 1863, p. 12: The two tangents from each angle to the inscribed ..."],["blackburn-elements-plane-trigonometry-1863/x-8505dbbfc9",15,"Blackburn 1863, p. 12: Let two of the sides of the triangle ABC ..."],["blackburn-elements-plane-trigonometry-1863/x-83c76e1a90",15,"Blackburn 1863, p. 12: This word is often spelled “escribed” improperly."],["blackburn-elements-plane-trigonometry-1863/x-d38af63a9f",15,"Blackburn 1863, p. 14: This most useful proposition was known to the Greeks ..."],["blackburn-elements-plane-trigonometry-1863/x-e56315ac0d",15,"Blackburn 1863, p. 14: whence the area can be calculated in square units ..."],["blackburn-elements-plane-trigonometry-1863/x-a4e77e2c7a",15,"Blackburn 1863, p. 12: Then, numerically, the Area = rs."],["thompson-calculus-made-easy-1914/x-a70480b0be",15,"Thompson 1914, p. 243: There is no one rule for discovering such an ..."],["hardy-course-of-pure-mathematics-1921/eq-808f2a413b",16,"Hardy 1921, p. 402: \\Log (1/z) = -\\Log z"],["form/a7a1808c3e",5,"identity: (x**2 - 16)*(x**2 - 2)"],["slaught-lennes-solid-geometry-1919/eq-3f9f666c54",16,"Slaught & Lennes 1919, p. 176: \\sin A = \\dfrac{a}{c},\\quad \\cos A = \\dfrac{b}{c},\\quad \\tan A = \\dfrac{a}{b}"],["slaught-lennes-solid-geometry-1919/eq-8994ca1c43",16,"Slaught & Lennes 1919, p. 179: p = l \\cos A"],["concept/theorem-projection-of-a-line-on-a-line",7,"theorem: projection of a line on a line"],["slaught-lennes-solid-geometry-1919/eq-642039cf78",16,"Slaught & Lennes 1919, p. 181: BE &= AB \\cdot \\cos D"],["hardy-course-of-pure-mathematics-1921/x-017b6bb384",15,"Hardy 1921, p. 30: If a set S contains infinitely many points, and ..."],["hardy-course-of-pure-mathematics-1921/ex-lx/2",4,"Hardy 1921, Exercise LX (2)"],["concept/theorem-altitude-of-an-oblique-prism-or-cylinder",7,"theorem: altitude of an oblique prism or cylinder"],["slaught-lennes-solid-geometry-1919/eq-b1d96f5e7f",16,"Slaught & Lennes 1919, p. 184: c &= b \\cos \\angle 1"],["thompson-calculus-made-easy-1914/eq-6d8c567678",16,"Thompson 1914, p. 117: E = \\sqrt[n+1]{\\frac{1000 × nC_l}{mPC_e}}"],["quantity/quantity",11,"quantity","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-quantity-quantity"],["concept/closed-plane-curve",7,"closed plane curve","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-closed-plane-curve"],["hardy-course-of-pure-mathematics-1921/x-ef8f497277",15,"Hardy 1921, p. 35: As a corollary, if a + b\\sqrt[3]{2} + c\\sqrt[3]{4} ..."],["hardy-course-of-pure-mathematics-1921/ex-lx/3",4,"Hardy 1921, Exercise LX (3)"],["slaught-lennes-solid-geometry-1919/eq-d55b598482",16,"Slaught & Lennes 1919, p. 184: S' &= S \\cos \\angle 1"],["todhunter-spherical-trigonometry-1886/ex-xi",3,"Todhunter 1886, Exercise XI"],["concept/small-variation",7,"small variation","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-small-variation"],["concept/inequality",7,"inequality","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-inequality"],["dickson-theory-of-equations-1922/eq-2896e2d772",16,"Dickson 1922, p. 25: f(x) \\equiv x^4 - 9x^3 + 24x^2 - 23x + 15 = 0"],["hardy-course-of-pure-mathematics-1921/ex-lx/4",4,"Hardy 1921, Exercise LX (4)"],["todhunter-spherical-trigonometry-1886/ch-on-small-variations-in-the-parts-of-a-spherical-triangle",2,"Todhunter 1886, On small variations in the parts of a Spherical Triangle","../books/todhunter-spherical-trigonometry-1886/ch/ch-on-small-variations-in-the-parts-of-a-spherical-triangle/index.html"],["planck-treatise-on-thermodynamics-1903/x-0e0d97db1c",15,"Planck 1903, p. 89: It increases or decreases according as heat is absorbed ..."],["person/de-fonteney",1,"De Fonteney","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-de-fonteney"],["ball-mathematical-recreations-1905/x-fc784b8214",15,"Ball 1905, scan 73: Whoever first gets three (or any other assigned number) ..."],["concept/dominant-letter",7,"dominant letter","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-dominant-letter"],["hardy-course-of-pure-mathematics-1921/eq-4a15684938",16,"Hardy 1921, p. 407: \\Log e^{\\zeta} = (1 + 2m\\pi i)\\zeta + 2n\\pi i"],["planck-treatise-on-thermodynamics-1903/x-b8f37c7ff9",15,"Planck 1903, p. 87: Not a single really rational proof of the second ..."],["hardy-course-of-pure-mathematics-1921/eq-e9b0e2db1d",16,"Hardy 1921, p. 403: z = \\exp \\zeta"],["theorem/irrationality-of-e",9,"irrationality of e","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-irrationality-of-e"],["wentworth-first-steps-in-algebra-1894/ex-1/8",4,"Wentworth 1894, Exercise 1 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- 9y + 1 = 0"],["wentworth-first-steps-in-algebra-1894/x-bd47485243",15,"Wentworth 1894, p. 8: If a man has 10 dollars and afterwards collects ..."],["wentworth-first-steps-in-algebra-1894/x-60a21c22c3",15,"Wentworth 1894, p. 9: If a man has 10 dollars consisting of 2 ..."],["wentworth-first-steps-in-algebra-1894/x-9ce5e9ece4",15,"Wentworth 1894, p. 13: In finding the value of a compound expression the ..."],["concept/cumulative-voting",7,"cumulative voting","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cumulative-voting"],["concept/exploration-problem",7,"exploration problem","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-exploration-problem"],["concept/bachet-s-weights-problem",7,"Bachet's weights problem","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-bachet-s-weights-problem"],["thompson-calculus-made-easy-1914/x-29fbd257d8",15,"Thompson 1914, p. 184: The remainder needed will always be equal to the ..."],["hardy-course-of-pure-mathematics-1921/ex-lxi/1",4,"Hardy 1921, Exercise LXI (1)"],["maxwell-elementary-treatise-electricity-1888/x-38968ba637",15,"Maxwell 1888, scan 36: For if the process of electrification is conducted within ..."],["maxwell-elementary-treatise-electricity-1888/x-6e5681eae9",15,"Maxwell 1888, scan 36: If an electrified body or system of bodies be ..."],["maxwell-elementary-treatise-electricity-1888/x-34226c5841",15,"Maxwell 1888, scan 33: The electrification of the gold leaves when tested is ..."],["thompson-calculus-made-easy-1914/eq-4e08db8e31",16,"Thompson 1914, p. 118: \\frac{d^2C}{dE^2} = (n + 1) \\frac{nC_l}{m} E^{-(n+2)}"],["slaught-lennes-solid-geometry-1919/eq-a2416ba81b",16,"Slaught & Lennes 1919, p. 185: \\pi ab"],["concept/theorem-area-of-an-ellipse",7,"theorem: area of an ellipse"],["slaught-lennes-solid-geometry-1919/eq-9c25319856",16,"Slaught & Lennes 1919, p. 186: ABCD = A'BCD' \\cos \\angle 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18^{\\text{th}} term! ..."],["theorem/fermat-s-last-theorem",9,"Fermat's Last Theorem","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-fermat-s-last-theorem"],["ball-mathematical-recreations-1905/x-45b0ac6493",15,"Ball 1905, scan 76: If more than two colours are used, the problems ..."],["whitehead-introduction-to-mathematics-1911/x-6bd5d1afd0",15,"Whitehead 1911, p. 17: Thus, as here used, any implies some and some ..."],["dickson-theory-of-equations-1922/eq-f90c60584a",16,"Dickson 1922, p. 27: c_0 x^n + c_1 x^{n-1} + \\dotsb + c_{n-1} x + c_n = 0"],["concept/trisection-of-an-angle",7,"trisection of an angle","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-trisection-of-an-angle"],["dickson-theory-of-equations-1922/eq-175d468f2c",16,"Dickson 1922, p. 26: f(x) \\equiv (x-d)Q(x)"],["hardy-course-of-pure-mathematics-1921/eq-4e4c372aed",16,"Hardy 1921, p. 404: \\exp \\zeta = e^{\\zeta}"],["dickson-theory-of-equations-1922/eq-fdeb715f60",16,"Dickson 1922, p. 26: f(m) = (m-d) q"],["todhunter-spherical-trigonometry-1886/eq-906f30a7ce",16,"Todhunter 1886, scan 80: E=A+B+C-\\pi"],["concept/diagonal-term-of-a-determinant",7,"diagonal term of a determinant","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-diagonal-term-of-a-determinant"],["thompson-calculus-made-easy-1914/x-654cc5f7c1",15,"Thompson 1914, p. 189: Clearly, at any point P of the curve, the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-87df7086eb",15,"De Morgan 1899, p. 4: while the magnitudes diminish, we may not assume either ..."],["whitehead-introduction-to-mathematics-1911/eq-cb7bdbb482",16,"Whitehead 1911, p. 117: x + y = 1"],["hardy-course-of-pure-mathematics-1921/ex-lxi/2b",4,"Hardy 1921, Exercise LXI (2b)"],["maxwell-elementary-treatise-electricity-1888/eq-2ce575ef51",16,"Maxwell 1888, scan 177: z = -QU"],["concept/instrument-carrier",7,"instrument: carrier"],["concept/quantity-coefficient-of-induction",7,"quantity: coefficient of induction"],["whitehead-introduction-to-mathematics-1911/eq-f74fcbc030",16,"Whitehead 1911, p. 117: x - y = 1"],["dickson-theory-of-equations-1922/x-4f17717f9e",15,"Dickson 1922, p. 71: But in the contrary case, narrower limits are necessary, ..."],["todhunter-spherical-trigonometry-1886/eq-e3c9cf0aee",16,"Todhunter 1886, scan 79: \\text{area of polygon} = \\Bigl\\{\\Sigma - (n-2)\\pi \\Bigr\\} r^2."],["method/horner-s-method",8,"Horner's method","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-horner-s-method"],["hardy-course-of-pure-mathematics-1921/eq-b725935cc4",16,"Hardy 1921, p. 404: \\exp (\\xi + i\\eta) = e^{\\xi} (\\cos\\eta + i\\sin\\eta)"],["maxwell-elementary-treatise-electricity-1888/x-5fe580a502",15,"Maxwell 1888, scan 52: Finally, when we contemplate the region occupied by the ..."],["theorem/line-meets-circumference-in-at-most-two-points",9,"line meets circumference in at most two points","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-line-meets-circumference-in-at-most-two-points"],["method/regula-falsi",8,"regula falsi","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-regula-falsi"],["dickson-theory-of-equations-1922/x-f6648aee9d",15,"Dickson 1922, p. 72: The number of positive real roots of an equation ..."],["method/solving-simultaneous-equations-by-determinants",8,"solving simultaneous equations by determinants","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-solving-simultaneous-equations-by-determinants"],["concept/transformed-equation",7,"transformed equation","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-transformed-equation"],["dickson-theory-of-equations-1922/eq-6758a72b9a",16,"Dickson 1922, p. 56: 8x^4 - 14x^3 - 9x^2 + 11x - 2 = 0"],["maxwell-elementary-treatise-electricity-1888/x-059299b19c",15,"Maxwell 1888, scan 53: Since, therefore, the force which acts on the ball ..."],["maxwell-elementary-treatise-electricity-1888/x-3190a27d3d",15,"Maxwell 1888, scan 54: This shews that there has been an actual transference ..."],["maxwell-elementary-treatise-electricity-1888/x-5ec99e2ae0",15,"Maxwell 1888, scan 55: The force with which they tend to separate is ..."],["maxwell-elementary-treatise-electricity-1888/x-6e9055148e",15,"Maxwell 1888, scan 56: The fact that the electromotive force at a point ..."],["maxwell-elementary-treatise-electricity-1888/x-6bf5d06fa9",15,"Maxwell 1888, scan 58: All these points lie on a certain surface, which ..."],["maxwell-elementary-treatise-electricity-1888/x-b2c0669ac6",15,"Maxwell 1888, scan 60: A line of force in every part of its ..."],["theorem/equal-central-angles-intercept-equal-arcs",9,"equal central angles intercept equal 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f(x)}{h}"],["concept/difference-quotient",7,"difference quotient"],["wentworth-first-steps-in-algebra-1894/x-fee9735e18",15,"Wentworth 1894, p. 41: From these four cases it follows that in finding ..."],["theorem/tangents-from-an-external-point-are-equal",9,"tangents from an external point are equal","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-tangents-from-an-external-point-are-equal"],["concept/constant-factor",7,"constant factor","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-constant-factor"],["wentworth-plane-geometry-1899/x-1ea73a8cd8",15,"Wentworth 1899, scan 84: A circle is a portion of a plane bounded ..."],["wentworth-plane-geometry-1899/x-63260e02da",15,"Wentworth 1899, scan 101: Two quantities of the same kind that cannot both ..."],["hardy-course-of-pure-mathematics-1921/ex-lxi/3",4,"Hardy 1921, Exercise LXI (3)"],["dickson-theory-of-equations-1922/x-c0efe78486",15,"Dickson 1922, p. 72: Descartes’ rule will be derived in §73 as a ..."],["thompson-calculus-made-easy-1914/x-039416112f",15,"Thompson 1914, p. 189: But, as in the previous case, this requires the ..."],["person/ferdinand-gotthold-max-eisenstein",1,"Ferdinand Gotthold Max Eisenstein","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-ferdinand-gotthold-max-eisenstein"],["wentworth-plane-geometry-1899/x-06e6bf1984",15,"Wentworth 1899, scan 88: If, however, a theorem is in fact a group ..."],["hardy-course-of-pure-mathematics-1921/ex-lxi/4",4,"Hardy 1921, Exercise LXI (4)"],["wentworth-plane-geometry-1899/x-9d64287eef",15,"Wentworth 1899, scan 101: To measure a quantity of any kind is to ..."],["wentworth-plane-geometry-1899/x-5a487c1b47",15,"Wentworth 1899, scan 95: A straight line perpendicular to a radius at its ..."],["theorem/limit-of-a-rational-function",9,"limit of a rational 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Leibnitz","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-rational-explanation-of-the-language-of-leibnitz/index.html"],["wentworth-first-steps-in-algebra-1894/x-7ffed2b24a",15,"Wentworth 1894, p. 130: The interest for one year is \\dfrac{y}{100} of the ..."],["concept/solution",7,"solution","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-solution"],["hardy-course-of-pure-mathematics-1921/ex-lxii/1",4,"Hardy 1921, Exercise LXII (1)"],["concept/initial-value",7,"initial value","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-initial-value"],["ball-mathematical-recreations-1905/x-1ba47bcbe6",15,"Ball 1905, scan 48: Thus only four weights are required, namely, 1 lb., ..."],["method/numerical-solution-of-a-spherical-triangle",8,"numerical solution of a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-numerical-solution-of-a-spherical-triangle"],["hardy-course-of-pure-mathematics-1921/eq-d853ad2904",16,"Hardy 1921, p. 405: a^{\\zeta} = \\exp (\\zeta\\Log a)"],["hardy-course-of-pure-mathematics-1921/eq-66e4f1cbc8",16,"Hardy 1921, p. 406: |a^{\\zeta}| = e^{\\xi\\log \\sigma - \\eta(\\psi+2m\\pi)}"],["quantity/frequency",11,"frequency","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-quantity-frequency"],["boyden-first-book-in-algebra-1895/ex-20/18",4,"Boyden 1895, Exercise 20 (18)"],["hardy-course-of-pure-mathematics-1921/eq-6d645de494",16,"Hardy 1921, p. 408: \\zeta = \\Log_{e} z = \\frac{\\log |z| + (\\am z + 2m\\pi)i}{1 + 2n\\pi i}"],["hardy-course-of-pure-mathematics-1921/x-f81287ae67",15,"Hardy 1921, p. 123: Such an assertion is palpably absurd when made of ..."],["whitehead-introduction-to-mathematics-1911/eq-c1cb2f2c8f",16,"Whitehead 1911, p. 118: ax^{2} + 2hxy + by^{2} + 2gx + 2fy = c"],["concept/cissoid-of-diocles",7,"cissoid of Diocles","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cissoid-of-diocles"],["hardy-course-of-pure-mathematics-1921/eq-8a98e6d22d",16,"Hardy 1921, p. 407: e^{\\zeta} = e^{\\xi-2m\\pi\\eta} \\{\\cos(\\eta + 2m\\pi\\xi) + i\\sin(\\eta + 2m\\pi\\xi)\\}"],["theorem/discriminant-of-a-quartic-equals-that-of-its-resolvent-cubic",9,"discriminant of a quartic equals that of its resolvent cubic","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-discriminant-of-a-quartic-equals-that-of-its-resolvent-cubic"],["whitehead-introduction-to-mathematics-1911/eq-d45bc5789e",16,"Whitehead 1911, p. 124: y - x = 0"],["concept/bisector",7,"bisector"],["slaught-lennes-solid-geometry-1919/x-4dd8dca176",15,"Slaught & Lennes 1919, p. 53: From these propositions it follows that each of the ..."],["slaught-lennes-solid-geometry-1919/x-f13cc37fe2",15,"Slaught & Lennes 1919, p. 53: The sum of the face angles of a polyhedral ..."],["slaught-lennes-solid-geometry-1919/x-fdeabe6260",15,"Slaught & Lennes 1919, p. 53: At the center E of an equilateral triangle ABC ..."],["form/e66d5d210c",5,"identity: (x**3 - 10)*(x**3 + 12)"],["whitehead-introduction-to-mathematics-1911/eq-40afa4cdfd",16,"Whitehead 1911, p. 124: y + x = 0"],["whitehead-introduction-to-mathematics-1911/eq-7b8ab7c0b7",16,"Whitehead 1911, p. 124: ax + by = 0"],["dickson-theory-of-equations-1922/ex-page124",3,"Dickson 1922, Exercise Page124"],["dickson-theory-of-equations-1922/ex-page125",3,"Dickson 1922, Exercise Page125"],["whitehead-introduction-to-mathematics-1911/eq-f3e2110c91",16,"Whitehead 1911, p. 124: y - x = 1"],["whitehead-introduction-to-mathematics-1911/eq-e1db74d023",16,"Whitehead 1911, p. 124: y + x = 1"],["dickson-theory-of-equations-1922/eq-c409af3e74",16,"Dickson 1922, p. 58: f(x) = a_0 x^n + a_1 x^{n-1} + \\dotsb + a_{n-1} x + a_n"],["ball-mathematical-recreations-1905/x-db41f96b5a",15,"Ball 1905, scan 49: To determine the arrangement of the weights to weigh ..."],["thompson-calculus-made-easy-1914/x-157b910651",15,"Thompson 1914, p. 235: Now the mere inspection of this relation tells us ..."],["thompson-calculus-made-easy-1914/x-2cc9b13449",15,"Thompson 1914, p. 235: As both y and dy occur in the equation ..."],["thompson-calculus-made-easy-1914/x-bb959d8a5c",15,"Thompson 1914, p. 238: Now, as it stands, the left side is not ..."],["blackburn-elements-plane-trigonometry-1863/x-8f81ec33e7",15,"Blackburn 1863, p. 24: Since the perpendicular from the centre of a circle ..."],["blackburn-elements-plane-trigonometry-1863/x-fdb188cb69",15,"Blackburn 1863, p. 31: The seven ratios defined above are altogether independent of ..."],["thompson-calculus-made-easy-1914/x-209af684c9",15,"Thompson 1914, p. 247: In this case the simplified equation represents the propagation ..."],["hardy-course-of-pure-mathematics-1921/ex-lxii/2",4,"Hardy 1921, Exercise LXII (2)"],["wentworth-first-steps-in-algebra-1894/x-03905975e9",15,"Wentworth 1894, p. 126: To find x in problem 26, add the equations; ..."],["dickson-theory-of-equations-1922/eq-8dc4c6294c",16,"Dickson 1922, p. 143: g(x) = \\;b_0x^n + \\;b_1x^{n-1} + \\dotsb + \\, b_n"],["blackburn-elements-plane-trigonometry-1863/x-527feb98f6",15,"Blackburn 1863, p. 32: From these equations all the functions can be found, ..."],["dickson-theory-of-equations-1922/eq-576fe1c9bf",16,"Dickson 1922, p. 144: R(f, g) = a_0^n g(\\alpha_1)g(\\alpha_2) \\dotsm g(\\alpha_m)"],["hardy-course-of-pure-mathematics-1921/ex-lxii/3",4,"Hardy 1921, Exercise LXII (3)"],["concept/complementary-angles",7,"complementary angles","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-complementary-angles"],["ball-mathematical-recreations-1905/eq-24e48116c8",16,"Ball 1905, scan 46: (m + n) ! / m! n !"],["blackburn-elements-plane-trigonometry-1863/x-5c3bea5050",15,"Blackburn 1863, p. 20: When one magnitude or ratio is so connected with ..."],["blackburn-elements-plane-trigonometry-1863/x-eef9e960ab",15,"Blackburn 1863, p. 24: The side of the hexagon inscribed is = R, ..."],["blackburn-elements-plane-trigonometry-1863/x-9c7c821295",15,"Blackburn 1863, p. 23: It should be observed that, while the number \\theta ..."],["blackburn-elements-plane-trigonometry-1863/x-4ab36fab10",15,"Blackburn 1863, p. 25: If AF be measured towards T, it is to ..."],["dickson-theory-of-equations-1922/x-b37e967bc0",15,"Dickson 1922, p. 46: The expression A + B for a root was ..."],["ball-mathematical-recreations-1905/eq-6eb1898e08",16,"Ball 1905, scan 47: (52!)/(13!)^4"],["dickson-theory-of-equations-1922/x-234ee0c0f6",15,"Dickson 1922, p. 46: The pairs of values of z whose product is ..."],["ball-mathematical-recreations-1905/eq-eeb3450d00",16,"Ball 1905, scan 47: pr/(ns + r)"],["dickson-theory-of-equations-1922/x-e5ac8e5fc0",15,"Dickson 1922, p. 47: This expression (and not P itself) is called the ..."],["concept/minority-representation",7,"minority representation"],["dickson-theory-of-equations-1922/eq-f99c612e6b",16,"Dickson 1922, p. 145: F = \\begin{vmatrix} a_0 & a_1 & a_2 & a_3 & 0 \\\\ 0 & a_0 & a_1 & a_2 & a_3 \\\\ b_0 & b_1 & b_2 & 0 & 0 \\\\ 0 & b_0 & b_1 &"],["dickson-theory-of-equations-1922/eq-b511a4cb80",16,"Dickson 1922, p. 147: \\beta f + \\alpha g \\equiv 0"],["theorem/sylvester-determinant-cubic-and-quadratic",9,"Sylvester determinant (cubic and quadratic)"],["dickson-theory-of-equations-1922/eq-34898b9e96",16,"Dickson 1922, p. 146: b_0^3z^2 + kz + F = 0"],["dickson-theory-of-equations-1922/eq-26bf9eb023",16,"Dickson 1922, p. 146: F = b_0^3 f(\\beta_1) f(\\beta_2)"],["ball-mathematical-recreations-1905/eq-c5e702b7eb",16,"Ball 1905, scan 47: na/(n + 1)"],["concept/vertex",7,"vertex","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-vertex"],["hardy-course-of-pure-mathematics-1921/ex-lxii/4",4,"Hardy 1921, Exercise LXII 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..."],["blackburn-elements-plane-trigonometry-1863/x-70ac2f58fe",15,"Blackburn 1863, p. 47: It is convenient to take one direction of the ..."],["blackburn-elements-plane-trigonometry-1863/x-f881adea3e",15,"Blackburn 1863, p. 48: If BAC be acute, \\cos BAC is +, and ..."],["blackburn-elements-plane-trigonometry-1863/x-9275af9721",15,"Blackburn 1863, p. 48: The projection of a broken line is the algebraic ..."],["blackburn-elements-plane-trigonometry-1863/x-112f3e089e",15,"Blackburn 1863, p. 48: both in the case where all these projections are ..."],["method/testing-a-stationary-value-by-neighbouring-values",8,"testing a stationary value by neighbouring values","../books/thompson-calculus-made-easy-1914/terms/index.html#t-method-testing-a-stationary-value-by-neighbouring-values"],["ball-mathematical-recreations-1905/eq-ada03c56fe",16,"Ball 1905, scan 54: x^n + y^n = z^n"],["concept/curved-surface",7,"curved 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a_{2}^{p}"],["form/9b6f93a6bf",5,"identity: (x - 1/2)*(x - 1/4)"],["hardy-course-of-pure-mathematics-1921/eq-9cd67eb446",16,"Hardy 1921, p. 32: \\frac{a_{1}^{p+q} + a_{2}^{p+q}}{2} \\geq \\left(\\frac{a_{1}^{p} + a_{2}^{p}}{2}\\right) \\left(\\frac{a_{1}^{q} + a_{2}^{q}}"],["concept/guarini-s-knights-problem",7,"Guarini's knights problem","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-guarini-s-knights-problem"],["hardy-course-of-pure-mathematics-1921/eq-31a6e6bc73",16,"Hardy 1921, p. 32: \\frac{a_{1}^{p} + a_{2}^{p}}{2} \\geq \\left(\\frac{a_{1} + a_{2}}{2}\\right)^{p}"],["concept/acute-angle",7,"acute angle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-acute-angle"],["de-morgan-elementary-illustrations-calculus-1899/eq-7b1bf0f4df",16,"De Morgan 1899, p. 50: BP + Pa : Pa :: a^{2} + b^{2} : b^{2}"],["concept/straight-angle",7,"straight angle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-straight-angle"],["hardy-course-of-pure-mathematics-1921/ex-lxii/7",4,"Hardy 1921, Exercise LXII (7)"],["de-morgan-elementary-illustrations-calculus-1899/eq-121859064c",16,"De Morgan 1899, p. 50: \\PadTo[r]{BP + Pa}{Bb} : A'a :: \\PadTo[l]{a^{2} + b^{2}}{a^{2}} : b^{2}"],["wentworth-plane-geometry-1899/x-ba6d055e6c",15,"Wentworth 1899, scan 65: A convex polygon is a polygon of which no ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-5ef55225ff",16,"De Morgan 1899, p. 50: a^{2} + b^{2} = l^{2}"],["law/pythagorean-relation",10,"Pythagorean relation"],["hardy-course-of-pure-mathematics-1921/ex-lxii/8",4,"Hardy 1921, Exercise LXII (8)"],["hardy-course-of-pure-mathematics-1921/eq-6e73c613f3",16,"Hardy 1921, p. 32: n \\tsum{a^{p+q}} \\geq \\tsum a^{p} \\tsum a^{q}"],["hardy-course-of-pure-mathematics-1921/eq-463b0de2d3",16,"Hardy 1921, p. 32: \\left(\\tsum a^{p}\\right)/n \\geq \\left\\{\\left(\\tsum a\\right)/n\\right\\}^{p}"],["concept/secant",7,"secant","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-secant"],["person/johann-benedict-listing",1,"Johann Benedict Listing","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-johann-benedict-listing"],["thompson-calculus-made-easy-1914/eq-1d81cdfd2e",16,"Thompson 1914, p. 192: y = \\frac{1}{n + 1} x^{n+1} + C"],["thompson-calculus-made-easy-1914/eq-890c8e50c1",16,"Thompson 1914, p. 199: \\int x^n\\, dx = \\dfrac{1}{n+1} x^{n+1}"],["thompson-calculus-made-easy-1914/eq-781cc487d5",16,"Thompson 1914, p. 192: \\frac{dy}{dx} = anx^{n-1}"],["hardy-course-of-pure-mathematics-1921/eq-3dfc7516a9",16,"Hardy 1921, p. 32: a_{r}' + a_{s}' - a_{r} - a_{s} = (a_{r} - G)(a_{s} - G)/G"],["hardy-course-of-pure-mathematics-1921/eq-22583f6776",16,"Hardy 1921, p. 33: \\left(\\tsum a_{r} b_{r}\\right)^{2} = \\tsum a_{r}^{2} \\tsum a_{s}^{2} - \\tsum (a_{r} b_{s} - a_{s} b_{r})^{2}"],["hardy-course-of-pure-mathematics-1921/eq-880a76a6f3",16,"Hardy 1921, p. 33: \\left(\\tsum a_{r} b_{r}\\right)^{2} \\leq \\tsum a_{r}^{2} \\tsum b_{r}^{2}"],["boyden-first-book-in-algebra-1895/ex-20/20",4,"Boyden 1895, Exercise 20 (20)"],["ball-mathematical-recreations-1905/x-8893d4f7c9",15,"Ball 1905, scan 90: Since there is only one cell on the board ..."],["hardy-course-of-pure-mathematics-1921/eq-d29a40bc99",16,"Hardy 1921, p. 407: a^{\\zeta} × b^{\\zeta} = (ab)^{\\zeta}"],["hardy-course-of-pure-mathematics-1921/eq-912bd9770e",16,"Hardy 1921, p. 407: a^{\\zeta} × a^{\\zeta'} = a^{\\zeta+\\zeta'}"],["thompson-calculus-made-easy-1914/eq-c2d6e017d5",16,"Thompson 1914, p. 197: y = \\frac{1}{n+1} x^{n+1} + bx + C"],["concept/cotangent",7,"cotangent","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-cotangent"],["form/33af07cb01",5,"identity: (x + 1/6)*(x + 1/3)"],["hardy-course-of-pure-mathematics-1921/eq-6501297543",16,"Hardy 1921, p. 36: a_{0}x^{n} + a_{1}x^{n-1} + \\dots + a_{n} = 0"],["ball-mathematical-recreations-1905/x-f57be59906",15,"Ball 1905, scan 92: Next suppose that the end A is twisted once ..."],["ball-mathematical-recreations-1905/x-61954ae79c",15,"Ball 1905, scan 89: Hence to interchange all the pieces will require 15 ..."],["hardy-course-of-pure-mathematics-1921/ex-lxii/9",4,"Hardy 1921, Exercise LXII (9)"],["wentworth-plane-geometry-1899/x-b32e5c0e6f",15,"Wentworth 1899, scan 66: And, except in the case of triangles, two polygons ..."],["thompson-calculus-made-easy-1914/eq-1a024152aa",16,"Thompson 1914, p. 201: \\int x^{-1}\\, dx = \\log_\\epsilon x + C"],["wentworth-plane-geometry-1899/x-837d29ebcf",15,"Wentworth 1899, scan 69: Two points are said to be symmetrical with respect ..."],["ball-mathematical-recreations-1905/eq-a5f1024497",16,"Ball 1905, scan 67: 5 \\times 13 - 8^2 = 1"],["whitehead-introduction-to-mathematics-1911/ch-vii",2,"Whitehead 1911, ch. VII: Imaginary Numbers","../books/whitehead-introduction-to-mathematics-1911/ch/ch-vii/index.html"],["ball-mathematical-recreations-1905/eq-3de0f1b3c4",16,"Ball 1905, scan 65: \\frac{1}{2}\\pi ab"],["planck-treatise-on-thermodynamics-1903/x-81f8f59d12",15,"Planck 1903, p. 82: A process which can in no way be completely ..."],["form/5e0b1fc436",5,"factor: -a**3*x + x**7"],["boyden-first-book-in-algebra-1895/ex-35/19",4,"Boyden 1895, Exercise 35 (19)"],["ball-mathematical-recreations-1905/ch-ii",2,"Ball 1905, ch. II: Some Geometrical Questions","../books/ball-mathematical-recreations-1905/ch/ch-ii/index.html"],["wentworth-plane-geometry-1899/x-facebe6baf",15,"Wentworth 1899, scan 67: In general, each angle of an equiangular polygon of ..."],["boyden-first-book-in-algebra-1895/ex-35/20",4,"Boyden 1895, Exercise 35 (20)"],["shape/80d8e193d0",6,"identity: N*a*(N*a + b)"],["theorem/polar-triangle-relation-is-symmetric",9,"polar triangle relation is symmetric","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-polar-triangle-relation-is-symmetric"],["ball-mathematical-recreations-1905/eq-aea449f253",16,"Ball 1905, scan 72: h &=1 + p_1 + 2p_2 + \\dotsb\\,"],["concept/summit",7,"summit"],["ball-mathematical-recreations-1905/eq-9e217b298e",16,"Ball 1905, scan 72: d &=1 + f_1 + 2f_2 + \\dotsb\\,"],["method/adding-ordered-pairs",8,"adding ordered pairs","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-method-adding-ordered-pairs"],["thompson-calculus-made-easy-1914/eq-d3f3ce7879",16,"Thompson 1914, p. 201: \\int \\frac{1}{x+a}\\, dx = \\log_\\epsilon (x+a) + C"],["thompson-calculus-made-easy-1914/eq-0b52d71969",16,"Thompson 1914, p. 201: \\int \\epsilon^x\\, dx = \\epsilon ^x + C"],["thompson-calculus-made-easy-1914/eq-728aebef93",16,"Thompson 1914, p. 201: \\int \\epsilon^{-x}\\, dx = -\\epsilon^{-x} + C"],["todhunter-spherical-trigonometry-1886/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry",2,"Todhunter 1886, On the connexion of Formul\\ae\\ in Plane and Spherical Trigonometry","../books/todhunter-spherical-trigonometry-1886/ch/ch-on-the-connexion-of-formul-ae-in-plane-and-spherical-trigonometry/index.html"],["hardy-course-of-pure-mathematics-1921/ex-xcii",3,"Hardy 1921, Exercise XCII"],["theorem/inverse-tangent-series",9,"inverse tangent 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II: FUNCTIONS OF REAL VARIABLES","../books/hardy-course-of-pure-mathematics-1921/ch/ch-ii/index.html"],["shape/c94ba868ff",6,"identity: (N - x)**2"],["wentworth-first-steps-in-algebra-1894/ex-17/9",4,"Wentworth 1894, Exercise 17 (9)"],["shape/ab4be38af4",6,"integrate: 1/(x**N + 1)"],["thompson-calculus-made-easy-1914/x-724c67dad5",15,"Thompson 1914, p. 35: If you have any doubt whether this is right, ..."],["hardy-course-of-pure-mathematics-1921/ex-lxiii/3b",4,"Hardy 1921, Exercise LXIII (3b)"],["hardy-course-of-pure-mathematics-1921/eq-784473b83c",16,"Hardy 1921, p. 42: Ax + By + C = 0"],["concept/perfect-system",7,"perfect system","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-perfect-system"],["concept/order-of-a-magic-square",7,"order of a magic square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-order-of-a-magic-square"],["person/plato",1,"Plato","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-plato"],["concept/inverse-hyperbolic-function",7,"inverse hyperbolic function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-inverse-hyperbolic-function"],["whitehead-introduction-to-mathematics-1911/x-f937a75ca7",15,"Whitehead 1911, p. 44: The state of a body unacted on by force ..."],["quantity/magic-sum",11,"magic sum","../books/ball-mathematical-recreations-1905/terms/index.html#t-quantity-magic-sum"],["hardy-course-of-pure-mathematics-1921/eq-76d3aa6695",16,"Hardy 1921, p. 42: (x - \\alpha)^{2} + (y - \\beta)^{2} = \\rho^{2}"],["hardy-course-of-pure-mathematics-1921/x-7a2d6df5ac",15,"Hardy 1921, p. 382: If x = 1, we obtain the formula 14= ..."],["hardy-course-of-pure-mathematics-1921/ex-lxiii/4a",4,"Hardy 1921, Exercise LXIII (4a)"],["planck-treatise-on-thermodynamics-1903/x-52ea77912e",15,"Planck 1903, p. 38: it is in no way possible, either by mechanical, ..."],["wentworth-first-steps-in-algebra-1894/x-55a6255f82",15,"Wentworth 1894, p. 81: Since the product is +12, the two numbers are ..."],["form/dc0fe59c7f",5,"identity: -9*x"],["hardy-course-of-pure-mathematics-1921/eq-d0e17f1921",16,"Hardy 1921, p. 43: x^{2} + y^{2} + 2Gx + 2Fy + C = 0"],["form/138086ce31",5,"integrate: 1/(x**2 - x + 1)"],["form/4413cd37dc",5,"identity: 5*x"],["shape/273216dba9",6,"integrate: 1/(-x + x**N + 1)"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/4b",4,"Hardy 1921, Exercise LXIII (4b)"],["de-morgan-elementary-illustrations-calculus-1899/eq-a45a1f8223",16,"De Morgan 1899, p. 52: dp = dq"],["form/69af96f310",5,"integrate: 1/(x**2 + x + 1)"],["de-morgan-elementary-illustrations-calculus-1899/eq-e5e8aba377",16,"De Morgan 1899, p. 52: dp + \\mu &: (p - dp) \\sin d\\theta &&:: a &&: b"],["de-morgan-elementary-illustrations-calculus-1899/eq-ed2f7564f5",16,"De Morgan 1899, p. 52: q \\sin d\\theta &: \\PadTo{(p - dp) \\sin d\\theta}{dq + \\nu} &&:: a - da &&: b + db"],["shape/ebf90a8e48",6,"integrate: 1/(x + x**N + 1)"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/5",4,"Hardy 1921, Exercise LXIII (5)"],["hardy-course-of-pure-mathematics-1921/eq-de413e8886",16,"Hardy 1921, p. 43: Ax^{2} + 2Hxy + By^{2} + 2Gx + 2Fy + C = 0"],["form/741b2e0572",5,"integrate: 1/(x**2 + 2*x*cos(a) + 1)"],["hardy-course-of-pure-mathematics-1921/eq-17269f34a7",16,"Hardy 1921, p. 43: x = r\\cos\\theta"],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv",3,"Hardy 1921, Exercise LXXXIV"],["hardy-course-of-pure-mathematics-1921/x-14b53d7fa6",15,"Hardy 1921, p. 383: This series may be used to calculate \\log 2, ..."],["hardy-course-of-pure-mathematics-1921/eq-8028194b49",16,"Hardy 1921, p. 43: y = r\\sin\\theta"],["concept/arithmetical-fallacy",7,"arithmetical fallacy","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-arithmetical-fallacy"],["hardy-course-of-pure-mathematics-1921/ex-lxxxv",3,"Hardy 1921, Exercise LXXXV"],["shape/7ac6ba07d3",6,"integrate: 1/(N*x*cos(a) + x**N + 1)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxvi",3,"Hardy 1921, Exercise LXXXVI"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/6a",4,"Hardy 1921, Exercise LXIII (6a)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxiii",3,"Hardy 1921, Exercise LXXXIII"],["hardy-course-of-pure-mathematics-1921/ex-lxxxii",3,"Hardy 1921, Exercise LXXXII"],["ball-mathematical-recreations-1905/x-ee523a62af",15,"Ball 1905, scan 45: To the above examples I may add the following ..."],["form/008499e3ee",5,"integrate: sqrt(-x**2 + 1)"],["hardy-course-of-pure-mathematics-1921/eq-60c6b3a6d5",16,"Hardy 1921, p. 43: r = \\sqrtp{x^{2} + y^{2}}"],["shape/282ea60f0b",6,"integrate: (-x**N + 1)**N"],["hardy-course-of-pure-mathematics-1921/eq-7fcb54123a",16,"Hardy 1921, p. 43: r\\cos(\\theta - \\alpha) = p"],["hardy-course-of-pure-mathematics-1921/eq-8fa23504a1",16,"Hardy 1921, p. 43: r = 2a\\cos\\theta"],["de-morgan-elementary-illustrations-calculus-1899/x-2ba6b11533",15,"De Morgan 1899, p. 11: Since (a + h)^{2} = a^{2} + 2ah + ..."],["hardy-course-of-pure-mathematics-1921/eq-a76400b775",16,"Hardy 1921, p. 43: r^{2} + c^{2} - 2rc\\cos(\\theta - \\alpha) = A^{2}"],["hardy-course-of-pure-mathematics-1921/eq-02f4b9b50c",16,"Hardy 1921, p. 43: l/r = 1 - e\\cos\\theta"],["person/bachet",1,"Bachet","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-bachet"],["concept/unit-tube-of-induction",7,"unit tube of induction","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-unit-tube-of-induction"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/6b",4,"Hardy 1921, Exercise LXIII (6b)"],["hardy-course-of-pure-mathematics-1921/x-149528ce78",15,"Hardy 1921, p. 386: We shall now give an outline of a method ..."],["theorem/areas-of-parallel-cross-sections-of-a-cone-are-proportional-to-the-squares-of-their-distances-from-the-vertex",9,"areas of parallel cross-sections of a cone are proportional to the squares of their distances from the vertex","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-areas-of-parallel-cross-sections-of-a-cone-are-proportional-to-the-squares-of-their-distances-from-the-vertex"],["method/approximating-the-lateral-area-of-a-cone-by-inscribed-and-circumscribed-pyramids",8,"approximating the lateral area of a cone by inscribed and circumscribed pyramids","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-approximating-the-lateral-area-of-a-cone-by-inscribed-and-circumscribed-pyramids"],["concept/inscribed-pyramid",7,"inscribed pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-inscribed-pyramid"],["concept/isohydric-solution",7,"isohydric solution","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-isohydric-solution"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/7",4,"Hardy 1921, Exercise LXIII (7)"],["hardy-course-of-pure-mathematics-1921/eq-817c6600eb",16,"Hardy 1921, p. 43: -l/r = 1 - e\\cos\\theta"],["form/0833efd67e",5,"integrate: 1/(a + b*cos(x))"],["hardy-course-of-pure-mathematics-1921/eq-abda9c0403",16,"Hardy 1921, p. 44: a_{0}x^{m} + a_{1}x^{m-1} + \\dots + a_{m}"],["whitehead-introduction-to-mathematics-1911/ch-ix",2,"Whitehead 1911, ch. IX: Coordinate Geometry","../books/whitehead-introduction-to-mathematics-1911/ch/ch-ix/index.html"],["de-morgan-elementary-illustrations-calculus-1899/x-2d73479cca",15,"De Morgan 1899, p. 77: Let a + h be the real value, in ..."],["hardy-course-of-pure-mathematics-1921/eq-54adebf06c",16,"Hardy 1921, p. 47: R(x) = \\frac{P(x)}{Q(x)}"],["hardy-course-of-pure-mathematics-1921/eq-2230f48d01",16,"Hardy 1921, p. 41: y = f(x)"],["shape/0833efd67e",6,"integrate: 1/(a + b*cos(x))"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/8",4,"Hardy 1921, Exercise LXIII (8)"],["form/1aff8fea5f",5,"integrate: 1/(a**2*cos(x)**2 + b**2*sin(x)**2)"],["shape/f906c3b08b",6,"integrate: 1/(a**N*cos(x)**N + b**N*sin(x)**N)"],["de-morgan-elementary-illustrations-calculus-1899/x-c838ba3ea6",15,"De Morgan 1899, p. 14: Again, it would be said (1) that if AB ..."],["ball-mathematical-recreations-1905/x-899711b864",15,"Ball 1905, scan 158: I confine my account to such magic squares, that ..."],["hardy-course-of-pure-mathematics-1921/eq-2d3137ab63",16,"Hardy 1921, p. 46: y - \\{(ac - b^{2})/a\\} = a\\{x + (b/a)\\}^{2}"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/9a",4,"Hardy 1921, Exercise LXIII (9a)"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/9b",4,"Hardy 1921, Exercise LXIII (9b)"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/9c",4,"Hardy 1921, Exercise LXIII (9c)"],["method/measuring-the-potential-at-a-point-in-air",8,"measuring the potential at a point in air","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-measuring-the-potential-at-a-point-in-air"],["experiment/faraday-s-cube-experiment",14,"Faraday's cube experiment","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-experiment-faraday-s-cube-experiment"],["hardy-course-of-pure-mathematics-1921/eq-b450dfe11a",16,"Hardy 1921, p. 45: (-x)^{m} = x^{m}"],["boyden-first-book-in-algebra-1895/ex-20/26",4,"Boyden 1895, Exercise 20 (26)"],["concept/circumscribed-pyramid",7,"circumscribed pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-circumscribed-pyramid"],["theorem/lateral-area-of-a-right-circular-cone",9,"lateral area of a right circular cone","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-lateral-area-of-a-right-circular-cone"],["theorem/lateral-area-of-a-frustum-of-a-cone",9,"lateral area of a frustum of a cone","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-lateral-area-of-a-frustum-of-a-cone"],["concept/frustum-of-a-cone",7,"frustum of a cone","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-frustum-of-a-cone"],["form/f87f628284",5,"identity: (-x + 3)*(-x + 5)"],["maxwell-elementary-treatise-electricity-1888/x-218d927df8",15,"Maxwell 1888, scan 61: If we consider a portion of an electrified surface ..."],["maxwell-elementary-treatise-electricity-1888/x-da33720093",15,"Maxwell 1888, scan 62: A system of lines of force forming a tubular ..."],["hardy-course-of-pure-mathematics-1921/ex-lxiii/10a",4,"Hardy 1921, Exercise LXIII (10a)"],["maxwell-elementary-treatise-electricity-1888/x-0bb1f8ee2d",15,"Maxwell 1888, scan 67: The similarity which constitutes the analogy is not between ..."],["method/approximating-air-potential-by-repeated-contact-with-a-discharged-sphere",8,"approximating air potential by repeated contact with a discharged sphere","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-approximating-air-potential-by-repeated-contact-with-a-discharged-sphere"],["hardy-course-of-pure-mathematics-1921/eq-be2f5bde5f",16,"Hardy 1921, p. 45: (-x)^{m} = -x^{m}"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/10b",4,"Hardy 1921, Exercise LXIII (10b)"],["maxwell-elementary-treatise-electricity-1888/x-199a25977a",15,"Maxwell 1888, scan 63: Hence in this simple case the number of cells ..."],["maxwell-elementary-treatise-electricity-1888/x-6c5a5458a8",15,"Maxwell 1888, scan 73: It follows from (2) that the potential at the ..."],["theorem/volume-of-a-cone",9,"volume of a cone","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volume-of-a-cone"],["method/approximating-the-volume-of-a-cone-by-inscribed-and-circumscribed-prisms",8,"approximating the volume of a cone by inscribed and circumscribed prisms","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-approximating-the-volume-of-a-cone-by-inscribed-and-circumscribed-prisms"],["theorem/volume-of-a-frustum-of-a-cone",9,"volume of a frustum of a cone","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volume-of-a-frustum-of-a-cone"],["concept/dodecagon",7,"dodecagon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-dodecagon"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/10c",4,"Hardy 1921, Exercise LXIII (10c)"],["hardy-course-of-pure-mathematics-1921/eq-2477242342",16,"Hardy 1921, p. 39: W = Ap_{0}"],["hardy-course-of-pure-mathematics-1921/eq-4537d415bf",16,"Hardy 1921, p. 39: pv = a"],["dickson-theory-of-equations-1922/eq-b11e2ec77a",16,"Dickson 1922, p. 128: x^n + c_1x^{n-1} + c_2x^{n-2} + \\dotsb + c_n = 0"],["dickson-theory-of-equations-1922/eq-fb80718189",16,"Dickson 1922, p. 130: x^n - E_1 x^{n-1} + E_2 x^{n-2} - \\dotsb + (-1)^n E_n = 0"],["concept/function-of-a-complex-variable",7,"function of a complex variable","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-function-of-a-complex-variable"],["hardy-course-of-pure-mathematics-1921/eq-d2b6447cd4",16,"Hardy 1921, p. 40: \\left(p + \\frac{\\alpha}{v^{2}}\\right)(v - \\beta) = \\gamma"],["hardy-course-of-pure-mathematics-1921/eq-f82c697c2a",16,"Hardy 1921, p. 40: h = \\tfrac{1}{2}g(2n\\tau - t)^{2}"],["theorem/sum-of-the-angles-of-a-polygon",9,"sum of the angles of a polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-sum-of-the-angles-of-a-polygon"],["slaught-lennes-solid-geometry-1919/x-e08370fde1",15,"Slaught & Lennes 1919, p. 97: The circumscribed pyramids all have the same slant height ..."],["slaught-lennes-solid-geometry-1919/x-adb53ded8c",15,"Slaught & Lennes 1919, p. 97: Evidently either of these processes may be repeated indefinitely ..."],["concept/inflection-point",7,"inflection point","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-inflection-point"],["dickson-theory-of-equations-1922/x-91d521a554",15,"Dickson 1922, p. 86: Hence -1 is the remainder obtained when the given ..."],["maxwell-elementary-treatise-electricity-1888/x-2504f2dea7",15,"Maxwell 1888, scan 191: In order to determine large differences of potential in ..."],["theorem/sum-of-the-exterior-angles-of-a-polygon",9,"sum of the exterior angles of a polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-sum-of-the-exterior-angles-of-a-polygon"],["concept/rebound",7,"rebound"],["method/synthetic-method",8,"synthetic method","../books/wentworth-plane-geometry-1899/terms/index.html#t-method-synthetic-method"],["hardy-course-of-pure-mathematics-1921/ex-lxiii/10d",4,"Hardy 1921, Exercise LXIII (10d)"],["method/analytic-method",8,"analytic method","../books/wentworth-plane-geometry-1899/terms/index.html#t-method-analytic-method"],["slaught-lennes-solid-geometry-1919/x-03e9b17713",15,"Slaught & Lennes 1919, p. 98: In the case of a cone which is not ..."],["dickson-theory-of-equations-1922/eq-3ba551ef85",16,"Dickson 1922, p. 129: rx_1^2 + rx_2^2 + sx_1 + sx_2 \\equiv r(E_1^2 - 2E_2) + sE_1"],["hardy-course-of-pure-mathematics-1921/ex-misc-viii",3,"Hardy 1921, Exercise Misc-VIII"],["hardy-course-of-pure-mathematics-1921/eq-b6fa0df6d8",16,"Hardy 1921, p. 409: \\sin\\zeta = -\\tfrac{1}{2}i \\{\\exp (i\\zeta) - \\exp (-i\\zeta)\\}"],["hardy-course-of-pure-mathematics-1921/ex-lxiv/1",4,"Hardy 1921, Exercise LXIV (1)"],["hardy-course-of-pure-mathematics-1921/eq-9021b427be",16,"Hardy 1921, p. 409: \\tan \\zeta = \\frac{\\sin \\zeta}{\\cos \\zeta}"],["hardy-course-of-pure-mathematics-1921/eq-b22a4f7b65",16,"Hardy 1921, p. 409: \\cot \\zeta = \\frac{\\cos \\zeta}{\\sin \\zeta}"],["hardy-course-of-pure-mathematics-1921/eq-f1650ef6ad",16,"Hardy 1921, p. 409: \\sec \\zeta = \\frac{1}{\\cos \\zeta}"],["slaught-lennes-solid-geometry-1919/x-a14ee2911e",15,"Slaught & Lennes 1919, p. 102: The theorem of § [unit:272.]272 holds for any cone ..."],["thompson-calculus-made-easy-1914/eq-13dd054127",16,"Thompson 1914, p. 202: \\int\\log_{10} x\\, dx = 0.4343x (\\log_\\epsilon x - 1) + C"],["hardy-course-of-pure-mathematics-1921/eq-4265396288",16,"Hardy 1921, p. 409: \\cosec \\zeta = \\frac{1}{\\sin \\zeta}"],["hardy-course-of-pure-mathematics-1921/eq-2d612aa7bd",16,"Hardy 1921, p. 409: \\cos \\zeta = \\tfrac{1}{2} \\{t + (1/t)\\}"],["hardy-course-of-pure-mathematics-1921/eq-eba14f6f25",16,"Hardy 1921, p. 409: \\sin \\zeta = -\\tfrac{1}{2}i \\{t - (1/t)\\}"],["concept/path-of-integration",7,"path of integration","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-path-of-integration"],["boyden-first-book-in-algebra-1895/ex-20/27",4,"Boyden 1895, Exercise 20 (27)"],["slaught-lennes-solid-geometry-1919/x-bac58dce38",15,"Slaught & Lennes 1919, p. 106: His work on the circle, cone, cylinder, and sphere ..."],["maxwell-elementary-treatise-electricity-1888/x-14c680b43f",15,"Maxwell 1888, scan 191: If the conductor is not large compared with the ..."],["maxwell-elementary-treatise-electricity-1888/x-922f36ed8a",15,"Maxwell 1888, scan 192: Since the sphere is not electrified it will be ..."],["thompson-calculus-made-easy-1914/eq-b6cf0444ce",16,"Thompson 1914, p. 202: \\int\\sin ax\\, dx = -\\frac{1}{a} \\cos ax + C"],["thompson-calculus-made-easy-1914/eq-84cb0d8fc5",16,"Thompson 1914, p. 202: \\int a^x\\, dx = \\dfrac{a^x}{\\log_\\epsilon a} + C"],["thompson-calculus-made-easy-1914/eq-3d8c55f5e0",16,"Thompson 1914, p. 202: \\int\\cos ax\\, dx = \\frac{1}{a} \\sin ax + C"],["thompson-calculus-made-easy-1914/eq-b69529f8de",16,"Thompson 1914, p. 202: \\cos 2\\theta = \\cos^2\\theta - \\sin^2\\theta"],["form/24d3bcf146",5,"identity: (-x + 6)*(x + 7)"],["thompson-calculus-made-easy-1914/eq-cdfa4fe8ed",16,"Thompson 1914, p. 155: y = b\\epsilon^{ax}"],["thompson-calculus-made-easy-1914/eq-31a51320f7",16,"Thompson 1914, p. 155: \\log_\\epsilon \\frac{y}{b}=ax"],["thompson-calculus-made-easy-1914/eq-352f515c18",16,"Thompson 1914, p. 156: y=b\\epsilon^{-ax}"],["concept/constant-of-decrement",7,"constant of decrement","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-constant-of-decrement"],["thompson-calculus-made-easy-1914/eq-8e257b8aaa",16,"Thompson 1914, p. 156: \\theta_t=\\theta_0 \\epsilon^{-at}"],["thompson-calculus-made-easy-1914/eq-30be9bfc0e",16,"Thompson 1914, p. 157: Q_t=Q_0 \\epsilon^{-at}"],["concept/radioactive-decay",7,"radioactive decay"],["dickson-theory-of-equations-1922/eq-f34f9cb43a",16,"Dickson 1922, p. 134: f(x) \\equiv (x - \\alpha_1)(x - \\alpha_2) \\dotsm (x - \\alpha_n)"],["shape/d4c9bda8ea",6,"identity: (N - x)*(N + x)"],["concept/absorption-of-light",7,"absorption of light"],["maxwell-elementary-treatise-electricity-1888/x-10ac65d8a8",15,"Maxwell 1888, scan 193: If there be a hollow nearly surrounded by the ..."],["hardy-course-of-pure-mathematics-1921/ex-lxiv/2",4,"Hardy 1921, Exercise LXIV (2)"],["todhunter-spherical-trigonometry-1886/ex-viii",3,"Todhunter 1886, Exercise VIII"],["form/0db5863348",5,"integrate: x**2"],["wentworth-plane-geometry-1899/ex-ii-2",3,"Wentworth 1899, Exercise II.2"],["thompson-calculus-made-easy-1914/x-9296a72814",15,"Thompson 1914, p. 49: Begin with a concrete case."],["boyden-first-book-in-algebra-1895/ex-20/28",4,"Boyden 1895, Exercise 20 (28)"],["thompson-calculus-made-easy-1914/x-4aad470c43",15,"Thompson 1914, p. 49: This is called the “derived function” of x."],["ball-mathematical-recreations-1905/x-9a01ff5a8d",15,"Ball 1905, scan 158: If the integers are the consecutive numbers from 1 ..."],["planck-treatise-on-thermodynamics-1903/eq-2681fe33b6",16,"Planck 1903, p. 56: c_{p} = \\left[\\left(\\frac{\\dd u}{\\dd v}\\right)_{p} + p\\right]\\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}\\Add{.}"],["form/05805be3c8",5,"identity: (-x + 11)*(x + 3)"],["boyden-first-book-in-algebra-1895/ex-20/29",4,"Boyden 1895, Exercise 20 (29)"],["shape/b2f1f8c768",6,"integrate: x**N"],["hardy-course-of-pure-mathematics-1921/ex-lxiv/3a",4,"Hardy 1921, Exercise LXIV (3a)"],["thompson-calculus-made-easy-1914/x-5684fabdfa",15,"Thompson 1914, p. 49: So the statement y=f(x) merely tells us that y ..."],["form/36767d5788",5,"integrate: x"],["shape/36767d5788",6,"integrate: x"],["hardy-course-of-pure-mathematics-1921/eq-8f5d09ead1",16,"Hardy 1921, p. 409: \\cos^{2} \\zeta + \\sin^{2} \\zeta = \\tfrac{1}{4}[\\{t + (1/t)\\}^{2} - \\{t - (1/t)\\}^{2}] = 1"],["planck-treatise-on-thermodynamics-1903/eq-5f422f0ea1",16,"Planck 1903, p. 56: c_{p} = c_{v} + \\left[\\left(\\frac{\\dd u}{\\dd v}\\right)_{\\theta} + p\\right]\\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}"],["hardy-course-of-pure-mathematics-1921/eq-399708c107",16,"Hardy 1921, p. 410: \\sin (\\zeta + \\zeta') = \\sin\\zeta \\cos\\zeta' + \\cos\\zeta \\sin\\zeta'"],["theorem/lhuilier-s-theorem",9,"Lhuilier's theorem","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-lhuilier-s-theorem"],["dickson-theory-of-equations-1922/ch-v",2,"Dickson 1922, ch. V: The Graph of an Equation","../books/dickson-theory-of-equations-1922/ch/ch-v/index.html"],["hardy-course-of-pure-mathematics-1921/ex-lxiv/3b",4,"Hardy 1921, Exercise LXIV (3b)"],["dickson-theory-of-equations-1922/ex-page55",3,"Dickson 1922, Exercise Page55"],["hardy-course-of-pure-mathematics-1921/x-edc6c1a4fa",15,"Hardy 1921, p. 40: ‘The largest prime factor of \\frac{11}{3} or of \\sqrt{2} ..."],["todhunter-spherical-trigonometry-1886/x-b87ec29ae0",15,"Todhunter 1886, scan 77: Hence since the whole surface of a sphere may ..."],["concept/centre-of-a-circle",7,"centre of a circle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-centre-of-a-circle"],["boyden-first-book-in-algebra-1895/ex-20/30",4,"Boyden 1895, Exercise 20 (30)"],["todhunter-spherical-trigonometry-1886/x-d8d1b13964",15,"Todhunter 1886, scan 79: the area of a spherical triangle is the same ..."],["concept/many-valued-function",7,"many-valued function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-many-valued-function"],["theorem/exponential-function-is-one-valued",9,"exponential function is one-valued","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-exponential-function-is-one-valued"],["ball-mathematical-recreations-1905/x-991736fa82",15,"Ball 1905, scan 158: The formation of these squares is an old amusement, ..."],["dickson-theory-of-equations-1922/ex-page69",3,"Dickson 1922, Exercise Page69"],["hardy-course-of-pure-mathematics-1921/ex-lxiv/4a",4,"Hardy 1921, Exercise LXIV (4a)"],["shape/3ef9109ded",6,"integrate: cos(a*x)"],["hardy-course-of-pure-mathematics-1921/eq-57fbf9c8a9",16,"Hardy 1921, p. 410: \\cosh\\zeta = \\tfrac{1}{2} \\{\\exp \\zeta + \\exp (-\\zeta)\\}"],["hardy-course-of-pure-mathematics-1921/eq-de6211a0a0",16,"Hardy 1921, p. 410: \\cos(\\zeta + \\tfrac{1}{2}\\pi) = -\\sin\\zeta"],["ball-mathematical-recreations-1905/x-7b204a6df8",15,"Ball 1905, scan 159: Magic squares of an odd order were constructed in ..."],["hardy-course-of-pure-mathematics-1921/ex-lxiv/4b",4,"Hardy 1921, Exercise LXIV (4b)"],["thompson-calculus-made-easy-1914/x-78d83b4ce5",15,"Thompson 1914, p. 251: There are amongst young engineers a number on whose ..."],["hardy-course-of-pure-mathematics-1921/eq-2dbdc7923d",16,"Hardy 1921, p. 410: \\sin(\\zeta + \\tfrac{1}{2}\\pi) = \\cos\\zeta"],["dickson-theory-of-equations-1922/eq-d24e436492",16,"Dickson 1922, p. 135: f'(x) \\equiv \\frac{f(x)}{x - \\alpha_1} + \\frac{f(x)}{x - \\alpha_2} + \\dotsb + \\frac{f(x)}{x - \\alpha_n}"],["dickson-theory-of-equations-1922/eq-08c8145195",16,"Dickson 1922, p. 134: s_k = \\Sigma \\alpha_1^k"],["concept/sum-of-like-powers-of-the-roots",7,"sum of like powers of the roots"],["hardy-course-of-pure-mathematics-1921/eq-dd4c6df1c5",16,"Hardy 1921, p. 410: \\sinh\\zeta = \\tfrac{1}{2} \\{\\exp \\zeta - \\exp (-\\zeta)\\}"],["concept/perpendicular-bisector",7,"perpendicular 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fallacy","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-geometrical-fallacy"],["hardy-course-of-pure-mathematics-1921/x-ade63c1449",15,"Hardy 1921, p. 47: Thus the function x/x is equal to 1 if ..."],["concept/logarithm-to-any-base",7,"logarithm to any base","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-logarithm-to-any-base"],["dickson-theory-of-equations-1922/eq-9583c8ffe0",16,"Dickson 1922, p. 135: s_1 = -c_1"],["theorem/half-side-formulae-for-a-spherical-triangle",9,"half-side formulae for a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-half-side-formulae-for-a-spherical-triangle"],["hardy-course-of-pure-mathematics-1921/x-b2ddb5f5f7",15,"Hardy 1921, p. 47: A rational function is the quotient of one polynomial ..."],["dickson-theory-of-equations-1922/eq-27a706ae34",16,"Dickson 1922, p. 135: s_2 = c_1^2 - 2c_2"],["dickson-theory-of-equations-1922/eq-c37d7f0687",16,"Dickson 1922, p. 135: s_k + c_1 s_{k-1} + c_2 s_{k-2} + \\dotsb + c_{k-1} s_1 + kc_k = 0"],["theorem/newton-s-identities",9,"Newton's identities"],["method/plotting-a-curve",8,"plotting a curve","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-plotting-a-curve"],["hardy-course-of-pure-mathematics-1921/ex-lxix/1",4,"Hardy 1921, Exercise LXIX (1)"],["hardy-course-of-pure-mathematics-1921/x-04b516938c",15,"Hardy 1921, p. 42: We call the aggregate of all these points the ..."],["theorem/limit-of-a-power-of-a-complex-number",9,"limit of a power of a complex number","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-of-a-power-of-a-complex-number"],["todhunter-spherical-trigonometry-1886/x-aee12a2ff4",15,"Todhunter 1886, scan 38: It should be observed that the two triangles in ..."],["hardy-course-of-pure-mathematics-1921/eq-c91df48771",16,"Hardy 1921, p. 410: \\cos i\\zeta = \\cosh 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1914, p. 252: \\sec^2 x& \\tan x & -\\log_\\epsilon \\cos x + ..."],["ball-mathematical-recreations-1905/eq-afeb08ebcd",16,"Ball 1905, scan 81: y = 2n - 1"],["todhunter-spherical-trigonometry-1886/x-3a3e44fecb",15,"Todhunter 1886, scan 35: The formul (4), (5), (6), (7) may be put ..."],["ball-mathematical-recreations-1905/eq-d5a055f8da",16,"Ball 1905, scan 75: x=\\pm a"],["ball-mathematical-recreations-1905/eq-5c6bb2edf4",16,"Ball 1905, scan 75: y=\\pm x"],["concept/element-of-a-determinant",7,"element of a determinant","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-element-of-a-determinant"],["theorem/ratio-test-for-convergence",9,"ratio test for convergence","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-ratio-test-for-convergence"],["hardy-course-of-pure-mathematics-1921/eq-c7b8d250ed",16,"Hardy 1921, p. 410: \\cosh 2\\zeta = \\cosh^{2} \\zeta + \\sinh^{2} \\zeta"],["de-morgan-elementary-illustrations-calculus-1899/ch-accelerated-motion",2,"De Morgan 1899, Accelerated Motion","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-accelerated-motion/index.html"],["hardy-course-of-pure-mathematics-1921/eq-0c0df7759e",16,"Hardy 1921, p. 411: \\zeta = 2k\\pi ± \\arccos a"],["concept/dissection-proof",7,"dissection proof","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-dissection-proof"],["hardy-course-of-pure-mathematics-1921/x-6694cdb76a",15,"Hardy 1921, p. 335: The explanation is to be found in a closer ..."],["hardy-course-of-pure-mathematics-1921/x-d2f08dc4eb",15,"Hardy 1921, p. 341: This example shows that the condition that \\phi_{n} should ..."],["hardy-course-of-pure-mathematics-1921/eq-3b352f7d86",16,"Hardy 1921, p. 413: \\int \\frac{dx}{x^{2} + \\alpha} = \\frac{1}{\\sqrt{\\alpha}} \\arctan \\frac{x}{\\sqrt{\\alpha}}"],["theorem/power-rule-for-differentiation",9,"power rule for differentiation","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-power-rule-for-differentiation"],["hardy-course-of-pure-mathematics-1921/eq-5c2d2a2d7d",16,"Hardy 1921, p. 413: \\int \\frac{dx}{x^{2} + \\alpha} = \\frac{1}{2\\sqrtp{-\\alpha}} \\log \\left|\\frac{x - \\sqrtp{-\\alpha}}{x + \\sqrtp{-\\alpha}}\\r"],["wentworth-first-steps-in-algebra-1894/x-7790d96b4a",15,"Wentworth 1894, p. 53: To multiply a polynomial by a monomial, therefore, Theorem ..."],["wentworth-first-steps-in-algebra-1894/x-0c005b237b",15,"Wentworth 1894, p. 59: Arrange both the dividend and divisor in ascending or ..."],["wentworth-first-steps-in-algebra-1894/x-ef422e8f06",15,"Wentworth 1894, p. 60: It is of fundamental importance to arrange the dividend ..."],["wentworth-first-steps-in-algebra-1894/x-6bc3cb3111",15,"Wentworth 1894, p. 56: The pupil should observe that, with a view to ..."],["wentworth-first-steps-in-algebra-1894/x-090911c9f7",15,"Wentworth 1894, p. 60: The pupil will notice that by this process we ..."],["boyden-first-book-in-algebra-1895/ex-20/31",4,"Boyden 1895, Exercise 20 (31)"],["form/2b9e967831",5,"solve: Eq(6*x + 15, 141)"],["wentworth-first-steps-in-algebra-1894/x-5e911c2a47",15,"Wentworth 1894, p. 59: The first term of the dividend is an; that ..."],["concept/line-of-slope",7,"line of slope","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-line-of-slope"],["method/solving-a-right-angled-triangle-logarithmically",8,"solving a right-angled triangle logarithmically","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-method-solving-a-right-angled-triangle-logarithmically"],["theorem/tangent-formula-for-two-sides-and-the-included-angle",9,"tangent formula for two sides and the included angle","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-tangent-formula-for-two-sides-and-the-included-angle"],["blackburn-elements-plane-trigonometry-1863/x-0f54b14a5e",15,"Blackburn 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Tartaglia","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-niccol-tartaglia"],["dickson-theory-of-equations-1922/x-6830ce5423",15,"Dickson 1922, p. 55: To find geometrically the real roots of a real ..."],["concept/vector-analysis",7,"vector analysis","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-vector-analysis"],["macfarlane-vector-analysis-quaternions-1906/ch-introduction",2,"Macfarlane 1906, Introduction","../books/macfarlane-vector-analysis-quaternions-1906/ch/ch-introduction/index.html"],["concept/decimal-fraction",7,"decimal fraction","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-decimal-fraction"],["ball-mathematical-recreations-1905/x-b14842756c",15,"Ball 1905, scan 257: The insertion of Plato’s name is an obvious anachronism."],["concept/place-value",7,"place value","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-place-value"],["concept/zero",7,"zero","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-zero"],["wentworth-plane-geometry-1899/eq-d30f217eec",16,"Wentworth 1899, scan 181: \\dfrac{AH}{m} = \\dfrac{HK}{n} = \\dfrac{KB}{p}"],["hardy-course-of-pure-mathematics-1921/x-4b20c68d95",15,"Hardy 1921, p. 338: Dirichlet’s Theorem ([§]169) shows that the terms of a ..."],["boyden-first-book-in-algebra-1895/ex-21/1",4,"Boyden 1895, Exercise 21 (1)"],["dickson-theory-of-equations-1922/x-41a01ff24a",15,"Dickson 1922, p. 56: A point (like M or M' in Fig. 14) ..."],["concept/space-analysis",7,"space analysis","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-space-analysis"],["hardy-course-of-pure-mathematics-1921/ex-lxix/3",4,"Hardy 1921, Exercise LXIX (3)"],["hardy-course-of-pure-mathematics-1921/x-eac8dc28f5",15,"Hardy 1921, p. 406: We conclude that a^{\\zeta} is infinitely many-valued unless \\zeta ..."],["dickson-theory-of-equations-1922/x-a93a96ffb4",15,"Dickson 1922, p. 57: We call 3x^2 + 8x the derivative of x^3 ..."],["dickson-theory-of-equations-1922/x-aaaf2f808e",15,"Dickson 1922, p. 59: Thus the derivative of a_0 x^n is na_0 x^{n-1}, ..."],["concept/isosceles-trapezoid",7,"isosceles trapezoid","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-isosceles-trapezoid"],["theorem/steadily-increasing-function-limit-theorem",9,"steadily increasing function limit theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-steadily-increasing-function-limit-theorem"],["hardy-course-of-pure-mathematics-1921/x-27df2b0342",15,"Hardy 1921, p. 131: The function \\phi(n) will be said to increase steadily ..."],["hardy-course-of-pure-mathematics-1921/ex-xcvii",3,"Hardy 1921, Exercise XCVII"],["dickson-theory-of-equations-1922/x-7e0fc26cea",15,"Dickson 1922, p. 56: The true curve between two points below the x-axis ..."],["ball-mathematical-recreations-1905/x-e8b9f29785",15,"Ball 1905, scan 75: Thus at present it is not possible to say ..."],["whitehead-introduction-to-mathematics-1911/eq-0ce27a921e",16,"Whitehead 1911, p. 195: n × (n - 1) × (n - 2) × (n - 3) × \\dots × 4 × 3 × 2 × 1\\Add{,}"],["concept/cylindrical-surface",7,"cylindrical surface","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-cylindrical-surface"],["hardy-course-of-pure-mathematics-1921/ex-xcv",3,"Hardy 1921, Exercise XCV"],["quantity/absolute-gas-constant",11,"absolute gas constant","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-absolute-gas-constant"],["boyden-first-book-in-algebra-1895/ex-21/2",4,"Boyden 1895, Exercise 21 (2)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi",3,"Hardy 1921, Exercise XCVI"],["dickson-theory-of-equations-1922/x-6e76e36cf4",15,"Dickson 1922, p. 59: This formula (8) is known as Taylor’s theorem for ..."],["wentworth-plane-geometry-1899/x-a71066095f",15,"Wentworth 1899, scan 139: Prove that the locus of the vertex of a ..."],["ball-mathematical-recreations-1905/x-04351237fc",15,"Ball 1905, scan 77: Take any face of the cube K: it has ..."],["ball-mathematical-recreations-1905/x-0638355da0",15,"Ball 1905, scan 80: The construction and the initial arrangement ensure that at ..."],["de-morgan-elementary-illustrations-calculus-1899/ch-nature-of-integration",2,"De Morgan 1899, Nature of Integration","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-nature-of-integration/index.html"],["hardy-course-of-pure-mathematics-1921/x-74107c82f2",15,"Hardy 1921, p. 61: We can in this case at once obtain a ..."],["concept/cartesian-analysis",7,"Cartesian analysis","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-cartesian-analysis"],["person/thomas-harriot",1,"Thomas Harriot","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-thomas-harriot"],["person/bhaskara",1,"Bhaskara","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-bhaskara"],["ball-mathematical-recreations-1905/x-e5a8255474",15,"Ball 1905, scan 81: Let y denote the number of passages from one ..."],["todhunter-spherical-trigonometry-1886/x-052b8e5459",15,"Todhunter 1886, scan 56: The solution of oblique-angled triangles may be made in ..."],["ball-mathematical-recreations-1905/x-03cf7a3e99",15,"Ball 1905, scan 84: Thus the problem is reduced to finding the way ..."],["wentworth-plane-geometry-1899/x-269be65811",15,"Wentworth 1899, scan 138: The required point is the intersection of the given ..."],["wentworth-plane-geometry-1899/x-3e5d1fc1f4",15,"Wentworth 1899, scan 143: Make use of the point which forms with P ..."],["boyden-first-book-in-algebra-1895/ex-21/3",4,"Boyden 1895, Exercise 21 (3)"],["whitehead-introduction-to-mathematics-1911/x-4fd6c8e340",15,"Whitehead 1911, p. 65: Similarly the important way of writing the equation x ..."],["form/5f0955799d",5,"identity: a**8*b**2"],["person/walter-william-rouse-ball",1,"Walter William Rouse Ball"],["todhunter-spherical-trigonometry-1886/x-0797e7599a",15,"Todhunter 1886, scan 57: these determine \\tfrac{1}{2}(A + B) and \\tfrac{1}{2}(A - B), ..."],["ball-mathematical-recreations-1905/eq-bc9efdd699",16,"Ball 1905, scan 98: a + a/n + a/n^2 + a/n^3 + \\dotsb"],["concept/oblique-cylinder",7,"oblique cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-oblique-cylinder"],["concept/battery",7,"battery","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-battery"],["whitehead-introduction-to-mathematics-1911/eq-f25ee20249",16,"Whitehead 1911, p. 200: s_{n} = u_{1} + u_{2} + u_{3} + \\dots + u_{n}"],["concept/axis-of-a-cylinder",7,"axis of a cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-axis-of-a-cylinder"],["boyden-first-book-in-algebra-1895/ex-21/4",4,"Boyden 1895, Exercise 21 (4)"],["maxwell-elementary-treatise-electricity-1888/eq-def5e68130",16,"Maxwell 1888, scan 200: E = IR = I_1( R + r_1 ) = I_2( R + r_2 )"],["maxwell-elementary-treatise-electricity-1888/eq-3d0f752e76",16,"Maxwell 1888, scan 200: \\frac{r_1}{r_2} = \\frac{(I-I_1)I_2}{(I-I_2)I_1}"],["method/approximating-the-area-and-volume-of-a-cylinder-by-prisms",8,"approximating the area and volume of a cylinder by prisms","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-approximating-the-area-and-volume-of-a-cylinder-by-prisms"],["ball-mathematical-recreations-1905/eq-51a1332c94",16,"Ball 1905, scan 98: an/(n-1)"],["theorem/bases-of-a-cylinder-are-equal",9,"bases of a cylinder are equal","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-bases-of-a-cylinder-are-equal"],["theorem/longitudinal-section-of-a-cylinder-is-a-parallelogram",9,"longitudinal section of a cylinder is a parallelogram","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-longitudinal-section-of-a-cylinder-is-a-parallelogram"],["slaught-lennes-solid-geometry-1919/x-fde93cb217",15,"Slaught & Lennes 1919, p. 74: Indeed, in these cases approximate measurement only is possible, ..."],["quantity/perimeter",11,"perimeter","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-quantity-perimeter"],["whitehead-introduction-to-mathematics-1911/eq-6d9d81f711",16,"Whitehead 1911, p. 202: \\tfrac{1}{9} = .1 + \\tfrac{1}{90}"],["whitehead-introduction-to-mathematics-1911/eq-02b51eb846",16,"Whitehead 1911, p. 206: s_{n} = 1 + x + x^{2} + x^{3} + \\dots + x^{n}"],["de-morgan-elementary-illustrations-calculus-1899/ch-inverse-functions",2,"De Morgan 1899, Inverse Functions","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-inverse-functions/index.html"],["slaught-lennes-solid-geometry-1919/x-e7a92791bb",15,"Slaught & Lennes 1919, p. 75: A cylinder has a definite lateral area and a ..."],["slaught-lennes-solid-geometry-1919/x-55a9bd14b8",15,"Slaught & Lennes 1919, p. 76: The lateral area of a cylinder can be computed ..."],["slaught-lennes-solid-geometry-1919/x-76d6cdfa3a",15,"Slaught & Lennes 1919, p. 70: The moving line is the generator, and the generator ..."],["slaught-lennes-solid-geometry-1919/x-625bba13fc",15,"Slaught & Lennes 1919, p. 76: In this case r = r_1 = r_2, and ..."],["maxwell-elementary-treatise-electricity-1888/eq-c58b07ec95",16,"Maxwell 1888, scan 201: \\delta = mI_1 - nI_2"],["maxwell-elementary-treatise-electricity-1888/eq-af9963254a",16,"Maxwell 1888, scan 201: C-D = I_1(A+\\alpha) = I_2(B+\\beta) = E-Ir"],["maxwell-elementary-treatise-electricity-1888/eq-81bcfd42ef",16,"Maxwell 1888, scan 181: (C'+c')U+C'V'=aV"],["concept/instrument-regenerator",7,"instrument: regenerator"],["maxwell-elementary-treatise-electricity-1888/eq-5ca8187e13",16,"Maxwell 1888, scan 181: C'V'=aV"],["maxwell-elementary-treatise-electricity-1888/eq-6f9f640298",16,"Maxwell 1888, scan 182: C'V'=aV\\text{,\\quad and\\quad}CV = a'V'"],["maxwell-elementary-treatise-electricity-1888/eq-9b4afbeb40",16,"Maxwell 1888, scan 201: I_1 + I_2 = I"],["maxwell-elementary-treatise-electricity-1888/eq-adea4180e5",16,"Maxwell 1888, scan 188: \\alpha = 0.220635 (R' - R)"],["todhunter-spherical-trigonometry-1886/x-6f3e9017bd",15,"Todhunter 1886, scan 58: Thus, in the present case, there is no real ..."],["whitehead-introduction-to-mathematics-1911/eq-38b8fe1c89",16,"Whitehead 1911, p. 206: s_{n} = \\frac{1 - x^{n+1}}{1 - x}"],["maxwell-elementary-treatise-electricity-1888/eq-e0a7f361ef",16,"Maxwell 1888, scan 183: Fa \\cos \\tfrac{1}{2} \\theta = M ( \\theta - \\phi )"],["concept/instrument-coulomb-s-torsion-balance",7,"instrument: Coulomb's torsion balance"],["concept/law-coulomb-s-law",7,"law: Coulomb's law"],["maxwell-elementary-treatise-electricity-1888/eq-4f3e2e80ef",16,"Maxwell 1888, scan 183: M = \\frac{4\\pi^2I}{T^2}"],["maxwell-elementary-treatise-electricity-1888/eq-cf7bf3881e",16,"Maxwell 1888, scan 184: =\\frac{EE_1}{r^2}"],["maxwell-elementary-treatise-electricity-1888/eq-60e4a6043b",16,"Maxwell 1888, scan 184: \\frac{EE_1aa_1 \\sin \\theta}{ r^3}"],["maxwell-elementary-treatise-electricity-1888/eq-c6b8c744d0",16,"Maxwell 1888, scan 184: = EE_1 \\frac{aa_1\\sin\\theta}{b^3 \\left\\{ 1 - 2 \\dfrac{aa_1}{b^2} \\cos \\theta + \\dfrac{a^2{a_1}^2}{b^4} \\right\\}^\\frac{3}"],["maxwell-elementary-treatise-electricity-1888/eq-c59a6c1945",16,"Maxwell 1888, scan 201: I_1 = E\\,\\frac {B + \\beta}{D}"],["maxwell-elementary-treatise-electricity-1888/eq-cec2a5607e",16,"Maxwell 1888, scan 201: D = (A + \\alpha)(B + \\beta) + r(A + \\alpha + B + \\beta)"],["maxwell-elementary-treatise-electricity-1888/eq-36d75498d7",16,"Maxwell 1888, scan 185: EE_1 aa_1\\sin\\theta\\left\\{\\frac{1}{r^3}-\\frac{1}{b^3}\\right\\}=M(\\theta-\\phi)"],["maxwell-elementary-treatise-electricity-1888/eq-6d6590beb4",16,"Maxwell 1888, scan 188: Wg &= \\frac{V^2A}{8 \\pi D^2}"],["concept/instrument-attracted-disk-electrometer",7,"instrument: attracted disk electrometer"],["maxwell-elementary-treatise-electricity-1888/eq-3b301c71ac",16,"Maxwell 1888, scan 188: V = D\\, \\sqrt{\\frac{8 \\pi gW}{A}}"],["maxwell-elementary-treatise-electricity-1888/eq-0c4daf0b3a",16,"Maxwell 1888, scan 188: A = \\tfrac{1}{2} \\pi (R^2 + R'^2)"],["maxwell-elementary-treatise-electricity-1888/eq-e65f588127",16,"Maxwell 1888, scan 188: V = 4D\\, \\sqrt{ \\frac{gW}{R^2 + R'^2}}"],["maxwell-elementary-treatise-electricity-1888/eq-288eb0bcb0",16,"Maxwell 1888, scan 188: \\alpha = B \\frac{log_e2}{\\pi}"],["maxwell-elementary-treatise-electricity-1888/eq-253b85f181",16,"Maxwell 1888, scan 201: \\delta =\\frac{E}{D} \\{m(B + \\beta) - n(A + \\alpha)\\}"],["maxwell-elementary-treatise-electricity-1888/eq-7481ab0506",16,"Maxwell 1888, scan 188: Q &= V \\left\\{ \\frac{R^2 + R'^2}{8D} - \\frac{R'^2 - R^2}{8D} \\frac{\\alpha}{D + \\alpha} \\right\\}"],["maxwell-elementary-treatise-electricity-1888/eq-0310e4cdfb",16,"Maxwell 1888, scan 188: Q = V \\left\\{ \\frac{R^2 + R'^2}{8D} - \\frac{R'^2 - R^2}{8D} \\frac{\\alpha}{D + \\alpha} + \\frac{R+R'}{D}(D'-D) \\log_e \\fra"],["maxwell-elementary-treatise-electricity-1888/eq-9b0a6c7b47",16,"Maxwell 1888, scan 188: A = \\tfrac{1}{2} \\pi \\left\\{R^2 + R'^2 -(R'^2 - R^2) \\frac{\\alpha}{D + \\alpha} + 8 (R + R')(D' - D) \\log_e \\frac{4 \\pi ("],["maxwell-elementary-treatise-electricity-1888/eq-16a09aaac1",16,"Maxwell 1888, scan 189: V - V' = (D - D') \\sqrt{ \\frac{8 \\pi g W}{A}}"],["hardy-course-of-pure-mathematics-1921/ex-lxv/1a",4,"Hardy 1921, Exercise LXV (1a)"],["whitehead-introduction-to-mathematics-1911/eq-b6e6e99deb",16,"Whitehead 1911, p. 207: \\frac{1}{1 - x} = 1 + x + x^{2} + \\dots + x^{n} + \\dots"],["theorem/sign-of-surface-charge-follows-potential",9,"sign of surface charge follows potential","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-sign-of-surface-charge-follows-potential"],["hardy-course-of-pure-mathematics-1921/ex-lxv/1b",4,"Hardy 1921, Exercise LXV (1b)"],["maxwell-elementary-treatise-electricity-1888/eq-9d3003fc2f",16,"Maxwell 1888, scan 202: n(A' - A) = \\frac{D}{E} \\delta - \\frac{D'}{E'} \\delta'"],["de-morgan-elementary-illustrations-calculus-1899/ch-on-functions",2,"De Morgan 1899, On Functions","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-on-functions/index.html"],["theorem/sum-of-the-geometric-series",9,"sum of the geometric series"],["whitehead-introduction-to-mathematics-1911/eq-e5c7020535",16,"Whitehead 1911, p. 211: \\exp x = 1 + x + \\frac{x^{2}}{2!} + \\frac{x^{3}}{3!} + \\dots + \\frac{x^{n}}{n!} + \\dots"],["hardy-course-of-pure-mathematics-1921/ex-lxv/1c",4,"Hardy 1921, Exercise LXV (1c)"],["concept/electric-induction",7,"electric induction","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electric-induction"],["boyden-first-book-in-algebra-1895/ex-21/6",4,"Boyden 1895, Exercise 21 (6)"],["theorem/connection-between-the-logarithm-and-the-inverse-circular-functions",9,"connection between the logarithm and the inverse circular functions","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-connection-between-the-logarithm-and-the-inverse-circular-functions"],["form/7c93871921",5,"identity: 49*a**2*b**4*c**6"],["boyden-first-book-in-algebra-1895/ex-21/5",4,"Boyden 1895, Exercise 21 (5)"],["boyden-first-book-in-algebra-1895/ex-21/7",4,"Boyden 1895, Exercise 21 (7)"],["form/5f02f758a7",5,"identity: a**5*b**5*c**10"],["shape/2e444a793e",6,"identity: a**N*b**N*c**N"],["hardy-course-of-pure-mathematics-1921/ex-lxv/2",4,"Hardy 1921, Exercise LXV (2)"],["maxwell-elementary-treatise-electricity-1888/x-796308f33a",15,"Maxwell 1888, scan 76: Under the action of the inductor B part of ..."],["maxwell-elementary-treatise-electricity-1888/eq-782a494c92",16,"Maxwell 1888, scan 203: (m + n)(B - A) = \\frac{D}{E} \\delta - \\frac{D'}{ E} \\delta'"],["maxwell-elementary-treatise-electricity-1888/eq-f97cd3de6d",16,"Maxwell 1888, scan 203: B - A = \\frac{1}{2nE} (A + \\alpha)(A + \\alpha + 2r)(\\delta - \\delta')"],["whitehead-introduction-to-mathematics-1911/eq-f517f9b38a",16,"Whitehead 1911, p. 211: (\\exp x) × (\\exp y) = \\exp(x + y)"],["concept/conditionally-convergent-series",7,"conditionally convergent series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-conditionally-convergent-series"],["hardy-course-of-pure-mathematics-1921/eq-3c39bc4ad8",16,"Hardy 1921, p. 50: y = \\frac{\\sqrtp{1 + x} - \\sqrtp[3]{1 - x}} {\\sqrtp{1 + x} + \\sqrtp[3]{1 - x}}"],["hardy-course-of-pure-mathematics-1921/ex-lxv/3",4,"Hardy 1921, Exercise LXV (3)"],["whitehead-introduction-to-mathematics-1911/eq-5718dd3c31",16,"Whitehead 1911, p. 212: \\sin(x + y) = \\sin x \\cos y + \\cos x \\sin y"],["theorem/addition-theorem-for-the-sine",9,"addition theorem for the sine"],["whitehead-introduction-to-mathematics-1911/eq-5c5cb37b34",16,"Whitehead 1911, p. 212: \\cos(x + y) = \\cos x \\cos y - \\sin x \\sin y"],["concept/motion-of-fluids",7,"motion of fluids","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-motion-of-fluids"],["maxwell-elementary-treatise-electricity-1888/eq-93fa00a148",16,"Maxwell 1888, scan 203: \\alpha = \\tfrac{1}{3}(A + r) \\left\\{2 \\sqrt{1 - \\frac{3}{4}\\frac{r^2}{(A + r)^2}}-1\\right\\}"],["theorem/addition-theorem-for-the-cosine",9,"addition theorem for the cosine"],["theorem/power-series-for-the-exponential-function",9,"power series for the exponential function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-power-series-for-the-exponential-function"],["theorem/power-series-for-the-sine-and-cosine",9,"power series for the sine and cosine","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-power-series-for-the-sine-and-cosine"],["boyden-first-book-in-algebra-1895/ex-21/8",4,"Boyden 1895, Exercise 21 (8)"],["form/af7c2f23a0",5,"identity: a**4*b**8*c**4"],["hardy-course-of-pure-mathematics-1921/x-94e9b92345",15,"Hardy 1921, p. 39: Let y = 0 whatever be the value of ..."],["hardy-course-of-pure-mathematics-1921/eq-6612408ed2",16,"Hardy 1921, p. 50: \\left(\\frac{1 + y}{1 - y}\\right)^{6} = \\frac{(1 + x)^{3}}{(1 - x)^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-1ee0d81ec5",16,"Hardy 1921, p. 50: y = \\sqrt{x} + \\sqrtp{x + \\sqrt{x}}"],["hardy-course-of-pure-mathematics-1921/eq-f2fd082c65",16,"Hardy 1921, p. 50: y^{4} - (4y^{2} + 4y + 1)x = 0"],["maxwell-elementary-treatise-electricity-1888/eq-b09442fa97",16,"Maxwell 1888, scan 203: \\alpha = \\tfrac{1}{3} A"],["maxwell-elementary-treatise-electricity-1888/eq-6795c0715e",16,"Maxwell 1888, scan 203: B - A = \\frac{8}{9}\\frac{A^2}{nE}(\\delta - \\delta')"],["hardy-course-of-pure-mathematics-1921/eq-e37735b278",16,"Hardy 1921, p. 50: y^{m} + R_{1}y^{m-1} + \\dots + R_{m} = 0"],["ball-mathematical-recreations-1905/x-a87f55e0e6",15,"Ball 1905, scan 125: Would that English writers were in the habit of ..."],["todhunter-spherical-trigonometry-1886/ex-xiii",3,"Todhunter 1886, Exercise XIII"],["method/factoring-by-taking-out-a-common-factor",8,"factoring by taking out a common factor","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-method-factoring-by-taking-out-a-common-factor"],["hardy-course-of-pure-mathematics-1921/eq-74b6524873",16,"Hardy 1921, p. 51: y = \\tfrac{1}{2}\\{-R_{1} ± \\sqrtp{R_{1}^{2} - 4R_{2}}\\}"],["hardy-course-of-pure-mathematics-1921/ex-lxv/4",4,"Hardy 1921, Exercise LXV (4)"],["todhunter-spherical-trigonometry-1886/ch-polyhedrons",2,"Todhunter 1886, Polyhedrons","../books/todhunter-spherical-trigonometry-1886/ch/ch-polyhedrons/index.html"],["hardy-course-of-pure-mathematics-1921/eq-44f1beed63",16,"Hardy 1921, p. 58: f(x) = \\phi(x)"],["maxwell-elementary-treatise-electricity-1888/eq-fdf305cdc2",16,"Maxwell 1888, scan 178: V_{n + 1} &= V_n - \\frac {Q}{B} U_n"],["concept/sine",7,"sine","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-sine"],["hardy-course-of-pure-mathematics-1921/eq-9ce4847511",16,"Hardy 1921, p. 60: f(x, y) = 0"],["hardy-course-of-pure-mathematics-1921/ex-lxv/5",4,"Hardy 1921, Exercise LXV (5)"],["whitehead-introduction-to-mathematics-1911/eq-f283bbb29c",16,"Whitehead 1911, p. 212: \\cos x = 1 - \\frac{x^{2}}{2!} + \\frac{x^{4}}{4!} - \\frac{x^{6}}{6!} + \\text{etc.} \\dots"],["hardy-course-of-pure-mathematics-1921/eq-50cad42c65",16,"Hardy 1921, p. 60: x = f(t)"],["hardy-course-of-pure-mathematics-1921/ex-lxv/6",4,"Hardy 1921, Exercise LXV (6)"],["maxwell-elementary-treatise-electricity-1888/eq-f29f90f57e",16,"Maxwell 1888, scan 185: EE_1 aa_1\\sin\\theta\\left\\{\\frac{1}{r^3}-\\frac{1}{b^3}\\right\\}=M(\\theta-\\phi)\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-8ff385375c",16,"Maxwell 1888, scan 188: V = 4D\\, \\sqrt{ \\frac{gW}{R^2 + R'^2}}."],["hardy-course-of-pure-mathematics-1921/ex-lxv/7",4,"Hardy 1921, Exercise LXV (7)"],["whitehead-introduction-to-mathematics-1911/eq-7fe52db00b",16,"Whitehead 1911, p. 212: \\sin x = x - \\frac{x^{3}}{3!} + \\frac{x^{5}}{5!} - \\frac{x^{7}}{7!} + \\text{etc.} \\dots"],["maxwell-elementary-treatise-electricity-1888/eq-cfb3423057",16,"Maxwell 1888, scan 204: O = \\frac{B \\gamma + C \\beta}{ \\beta + \\gamma}"],["hardy-course-of-pure-mathematics-1921/ex-lxv/8",4,"Hardy 1921, Exercise LXV (8)"],["whitehead-introduction-to-mathematics-1911/eq-eefb986697",16,"Whitehead 1911, p. 214: y = \\exp(-x^{2})"],["hardy-course-of-pure-mathematics-1921/ex-lxv/9",4,"Hardy 1921, Exercise LXV (9)"],["whitehead-introduction-to-mathematics-1911/eq-c7d8305ef1",16,"Whitehead 1911, p. 215: y = \\exp(-cx) × \\sin \\frac{2\\pi x}{p}"],["wentworth-plane-geometry-1899/eq-d3ca1ab9b7",16,"Wentworth 1899, scan 185: AG:AB = AB:AF"],["maxwell-elementary-treatise-electricity-1888/eq-47524b189e",16,"Maxwell 1888, scan 203: \\Delta = \\frac{mE}{A + \\alpha + r} = \\frac{3}{4}\\frac{nE}{A}"],["maxwell-elementary-treatise-electricity-1888/eq-12fd21e886",16,"Maxwell 1888, scan 203: \\frac{B - A}{A} = \\frac{2}{3}\\frac{\\delta - \\delta'}{\\Delta}"],["concept/van-der-waals-equation",7,"van der Waals' equation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-van-der-waals-equation"],["todhunter-spherical-trigonometry-1886/x-b0bb9a841e",15,"Todhunter 1886, scan 60: In this case, since B is found from its ..."],["hardy-course-of-pure-mathematics-1921/eq-45d8b989b6",16,"Hardy 1921, p. 61: x = a\\cos t"],["planck-treatise-on-thermodynamics-1903/eq-08b35d6a70",16,"Planck 1903, p. 72: -Q = U_{1} - U_{2} + 1.97 \\theta (n_{1} - n_{2})~\\Unit{cal.}"],["hardy-course-of-pure-mathematics-1921/ex-lxv/10",4,"Hardy 1921, Exercise LXV (10)"],["hardy-course-of-pure-mathematics-1921/eq-d896ff8fc8",16,"Hardy 1921, p. 61: y = a\\sin t"],["maxwell-elementary-treatise-electricity-1888/eq-155e956d4f",16,"Maxwell 1888, scan 204: A = \\frac{Bb + Cc}{b + c}"],["maxwell-elementary-treatise-electricity-1888/eq-5f7a73da2f",16,"Maxwell 1888, scan 204: b \\beta = c \\gamma"],["hardy-course-of-pure-mathematics-1921/ex-lxv/11",4,"Hardy 1921, Exercise LXV (11)"],["hardy-course-of-pure-mathematics-1921/ex-lxv/12",4,"Hardy 1921, Exercise LXV (12)"],["hardy-course-of-pure-mathematics-1921/x-668b828cc5",15,"Hardy 1921, p. 418: We shall in fact prove rather more than this, ..."],["quantity/density",11,"density","../books/ball-mathematical-recreations-1905/terms/index.html#t-quantity-density"],["theorem/cosine-of-the-sum-of-two-angles",9,"cosine of the sum of two angles","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-cosine-of-the-sum-of-two-angles"],["theorem/sine-of-the-difference-of-two-angles",9,"sine of the difference of two angles","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-sine-of-the-difference-of-two-angles"],["theorem/limiting-radius-of-inscribed-and-circumscribed-circles",9,"limiting radius of inscribed and circumscribed circles","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-limiting-radius-of-inscribed-and-circumscribed-circles"],["hardy-course-of-pure-mathematics-1921/eq-b21e5a7237",16,"Hardy 1921, p. 61: x^{2} + y^{2} = a^{2}"],["de-morgan-elementary-illustrations-calculus-1899/ch-differential-and-integral-calculus",2,"De Morgan 1899, Differential and Integral Calculus","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-differential-and-integral-calculus/index.html"],["de-morgan-elementary-illustrations-calculus-1899/x-480a0d560f",15,"De Morgan 1899, p. 67: It has been shown that by taking \\dfrac{1}{x} sufficiently ..."],["planck-treatise-on-thermodynamics-1903/eq-89526f2f3b",16,"Planck 1903, p. 72: -Q = \\ce{\\{H2\\} + $\\tfrac{1}{2}$ \\{O2\\}} - \\ce{(H2O)} + 860~\\Unit{cal.}"],["dickson-theory-of-equations-1922/eq-7ec82acf8c",16,"Dickson 1922, p. 29: x^2 - ax + b = 0"],["de-morgan-elementary-illustrations-calculus-1899/x-bb9a140e04",15,"De Morgan 1899, p. 1: The Differential and Integral Calculus, or, as it was ..."],["concept/hexahedron",7,"hexahedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-hexahedron"],["concept/physical-change",7,"physical change","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-physical-change"],["hardy-course-of-pure-mathematics-1921/ex-lxv/13",4,"Hardy 1921, Exercise LXV (13)"],["concept/integral-expression",7,"integral expression","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-integral-expression"],["wentworth-first-steps-in-algebra-1894/x-828d6e999f",15,"Wentworth 1894, p. 74: Take the square root of the first term and ..."],["wentworth-first-steps-in-algebra-1894/x-1253379b54",15,"Wentworth 1894, p. 72: The first two terms of ac + ad + ..."],["wentworth-first-steps-in-algebra-1894/x-5e9b8437ff",15,"Wentworth 1894, p. 73: Here the last two terms, -bc - bd, being ..."],["hardy-course-of-pure-mathematics-1921/eq-a4a6b68c9d",16,"Hardy 1921, p. 59: z = f(x, y)"],["planck-treatise-on-thermodynamics-1903/eq-26b5709b36",16,"Planck 1903, p. 72: (U + p_{0} V)_{2} - (U + p_{0} V)_{1} = Q"],["hardy-course-of-pure-mathematics-1921/eq-201f6cafe8",16,"Hardy 1921, p. 59: Ax + By + Cz + D = 0"],["hardy-course-of-pure-mathematics-1921/ex-lxv/14",4,"Hardy 1921, Exercise LXV (14)"],["maxwell-elementary-treatise-electricity-1888/x-4858de6d8d",15,"Maxwell 1888, scan 98: This method has the great advantage of being intelligible ..."],["planck-treatise-on-thermodynamics-1903/eq-29babe90e0",16,"Planck 1903, p. 76: (U_{2} + p_{0} V_{2})_{\\theta} - (U_{1} + p_{0} V_{1})_{\\theta} = Q_{\\theta}"],["de-morgan-elementary-illustrations-calculus-1899/x-b8d6e0b2d5",15,"De Morgan 1899, p. 67: Hence \\dfrac{(x + 1)^{m}}{x^{m}} = 1 + \\dfrac{mx^{m-1} + ..."],["planck-treatise-on-thermodynamics-1903/eq-bab8e54140",16,"Planck 1903, p. 73: \\ce{(NaHCO3 $\\aq$) + (NaHO $\\aq$) - (Na2CO3 $\\aq$)} = 9200~\\Unit{cal.}"],["hardy-course-of-pure-mathematics-1921/ex-lxv/15",4,"Hardy 1921, Exercise LXV (15)"],["de-morgan-elementary-illustrations-calculus-1899/x-51f3f733bf",15,"De Morgan 1899, p. 1: It is matter of common observation, that any one ..."],["hardy-course-of-pure-mathematics-1921/eq-1ef11af9ed",16,"Hardy 1921, p. 59: (x - \\alpha)^{2} + (y - \\beta)^{2} + (z - \\gamma)^{2} = \\rho^{2}"],["de-morgan-elementary-illustrations-calculus-1899/x-3f72ea146b",15,"De Morgan 1899, p. 1: The reason of this may be, that it is ..."],["maxwell-elementary-treatise-electricity-1888/x-9744a43ca8",15,"Maxwell 1888, scan 106: It can only exist when A is insulated, and ..."],["maxwell-elementary-treatise-electricity-1888/x-7eddc05411",15,"Maxwell 1888, scan 106: the negative electrification is more concentrated than the positive, ..."],["planck-treatise-on-thermodynamics-1903/eq-ba24371b22",16,"Planck 1903, p. 88: du = c_{v}\\, d\\theta"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/1",4,"Hardy 1921, Exercise LXVI (1)"],["de-morgan-elementary-illustrations-calculus-1899/x-9fb6bbbde5",15,"De Morgan 1899, p. 122: Hence results a new branch of the inquiry, the ..."],["hardy-course-of-pure-mathematics-1921/eq-f77b1bb2b3",16,"Hardy 1921, p. 59: x^{2} + y^{2} + z^{2} + 2Fx + 2Gy + 2Hz + C = 0"],["hardy-course-of-pure-mathematics-1921/eq-5f3ca93a44",16,"Hardy 1921, p. 59: F^{2} + G^{2} + H^{2} - C > 0"],["concept/inclination-of-two-lines",7,"inclination of two lines","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-inclination-of-two-lines"],["concept/symmetric-function",7,"symmetric function","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-symmetric-function"],["concept/sigma-function",7,"sigma function","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-sigma-function"],["boyden-first-book-in-algebra-1895/ex-21/9",4,"Boyden 1895, Exercise 21 (9)"],["form/437dccb6fa",5,"identity: -125*a**9*b**12*c**3"],["boyden-first-book-in-algebra-1895/ex-57/4",4,"Boyden 1895, Exercise 57 (4)"],["dickson-theory-of-equations-1922/ex-page142",3,"Dickson 1922, Exercise Page142"],["hardy-course-of-pure-mathematics-1921/eq-4b86569ba4",16,"Hardy 1921, p. 62: f(x, y, z) = 0"],["theorem/sphere-may-be-circumscribed-about-a-tetrahedron",9,"sphere may be circumscribed about a tetrahedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-sphere-may-be-circumscribed-about-a-tetrahedron"],["method/inscribing-a-sphere-in-a-tetrahedron",8,"inscribing a sphere in a tetrahedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-inscribing-a-sphere-in-a-tetrahedron"],["method/finding-the-diameter-of-a-sphere",8,"finding the diameter of a sphere","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-method-finding-the-diameter-of-a-sphere"],["slaught-lennes-solid-geometry-1919/x-0539f9fbee",15,"Slaught & Lennes 1919, p. 113: A plane tangent to a sphere is perpendicular to ..."],["slaught-lennes-solid-geometry-1919/x-2c01e4d9ce",15,"Slaught & Lennes 1919, p. 122: The shortest distance on a sphere between two of ..."],["boyden-first-book-in-algebra-1895/ex-21/10",4,"Boyden 1895, Exercise 21 (10)"],["theorem/sum-of-two-sides-of-a-spherical-triangle-exceeds-the-third",9,"sum of two sides of a spherical triangle exceeds the third","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-sum-of-two-sides-of-a-spherical-triangle-exceeds-the-third"],["concept/polyhedron-inscribed-in-a-sphere",7,"polyhedron inscribed in a sphere","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-polyhedron-inscribed-in-a-sphere"],["slaught-lennes-solid-geometry-1919/x-18d3c962f5",15,"Slaught & Lennes 1919, p. 111: It follows from the preceding theorem and corollaries that, ..."],["slaught-lennes-solid-geometry-1919/x-cf99d5fa51",15,"Slaught & Lennes 1919, p. 116: PB and BD being known, we may now compute ..."],["de-morgan-elementary-illustrations-calculus-1899/x-1792c4c84d",15,"De Morgan 1899, p. 122: For whatever differential coefficient \\psi x gives, C + ..."],["de-morgan-elementary-illustrations-calculus-1899/x-d024a19551",15,"De Morgan 1899, p. 2: We would not, nevertheless, that the student should imagine ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvi/2",4,"Hardy 1921, Exercise LXVI (2)"],["hardy-course-of-pure-mathematics-1921/eq-2331a6c30c",16,"Hardy 1921, p. 60: x^{2} + y^{2} + 2Gx + 2Fy+ C = 0"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/3a",4,"Hardy 1921, Exercise LXVI (3a)"],["concept/problem",7,"problem","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-problem"],["method/reducing-a-fraction-to-an-integral-or-mixed-expression",8,"reducing a fraction to an integral or mixed expression","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-method-reducing-a-fraction-to-an-integral-or-mixed-expression"],["hardy-course-of-pure-mathematics-1921/eq-d4ca52196d",16,"Hardy 1921, p. 60: y = -F + \\sqrtp{F^{2} - x^{2} - 2Gx - C}"],["theorem/exponential-limit",9,"exponential limit","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-exponential-limit"],["form/a7a9559860",5,"identity: 121*a**10*b**24*c**8"],["boyden-first-book-in-algebra-1895/ex-21/11",4,"Boyden 1895, Exercise 21 (11)"],["ball-mathematical-recreations-1905/x-af1df12a95",15,"Ball 1905, scan 108: Hence the forces that act on the machine and ..."],["hardy-course-of-pure-mathematics-1921/eq-c7f06c7575",16,"Hardy 1921, p. 60: x^{5} + y^{5} - ay = 0"],["whitehead-introduction-to-mathematics-1911/ch-x",2,"Whitehead 1911, ch. X: Conic Sections","../books/whitehead-introduction-to-mathematics-1911/ch/ch-x/index.html"],["form/3376dcfa4d",5,"identity: a**2*b**6*c**4/4"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/3b",4,"Hardy 1921, Exercise LXVI (3b)"],["de-morgan-elementary-illustrations-calculus-1899/x-cd3d38155a",15,"De Morgan 1899, p. 61: Thus, an impulse which changes the velocity from 50 ..."],["ball-mathematical-recreations-1905/x-ada2b6c92e",15,"Ball 1905, scan 110: But as the velocity of the boat increases, a ..."],["hardy-course-of-pure-mathematics-1921/eq-8ef1f63d15",16,"Hardy 1921, p. 52: y^{5} - y - x = 0"],["ball-mathematical-recreations-1905/x-7caff6710f",15,"Ball 1905, scan 111: Hence, if the wind makes the same angle \\alpha ..."],["ball-mathematical-recreations-1905/x-9641f26bef",15,"Ball 1905, scan 113: Hence the usual movements of the crew in the ..."],["boyden-first-book-in-algebra-1895/ex-21/12",4,"Boyden 1895, Exercise 21 (12)"],["dickson-theory-of-equations-1922/eq-188bc48bb7",16,"Dickson 1922, p. 135: s_n + c_1 s_{n-1} + c_2 s_{n-2} + \\dotsb + c_{n-1} s_1 + nc_n = 0"],["de-morgan-elementary-illustrations-calculus-1899/x-979d409a70",15,"De Morgan 1899, p. 61: It is said to act uniformly, when the velocity ..."],["de-morgan-elementary-illustrations-calculus-1899/x-9fc5b1f99e",15,"De Morgan 1899, p. 122: the value of an integral is not to be ..."],["de-morgan-elementary-illustrations-calculus-1899/x-8d4c9bca98",15,"De Morgan 1899, p. 124: bringing before him modes of speech, which, taken quite ..."],["thompson-calculus-made-easy-1914/x-3c1a98925f",15,"Thompson 1914, p. 77: This tangent to the curve has evidently the same ..."],["thompson-calculus-made-easy-1914/x-375950e9b3",15,"Thompson 1914, p. 81: The characteristic of a minimum is that y must ..."],["form/ea0479ba7d",5,"identity: a**2*b**6*c**2/9"],["boyden-first-book-in-algebra-1895/ex-21/13",4,"Boyden 1895, Exercise 21 (13)"],["form/b73c8ace26",5,"identity: 225*a**12*b**2*c**4"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/4",4,"Hardy 1921, Exercise LXVI (4)"],["boyden-first-book-in-algebra-1895/ex-21/14",4,"Boyden 1895, Exercise 21 (14)"],["wentworth-first-steps-in-algebra-1894/x-71c2354a6b",15,"Wentworth 1894, p. 152: Since the square of a + b is a^{2} ..."],["de-morgan-elementary-illustrations-calculus-1899/x-35f3dad49a",15,"De Morgan 1899, p. 122: if \\phi x be the differential coefficient of \\psi ..."],["planck-treatise-on-thermodynamics-1903/eq-9d5d6f0cda",16,"Planck 1903, p. 73: \\ce{(CO2 $\\aq$) + 2(NaHO $\\aq$) - (Na2CO3 $\\aq$)} = 20,200~\\Unit{cal.}"],["planck-treatise-on-thermodynamics-1903/eq-4997d20cab",16,"Planck 1903, p. 73: \\ce{(CO2 $\\aq$) + (NaHO $\\aq$) - (NaHCO3 $\\aq$)} = 11,000~\\Unit{cal.}"],["hardy-course-of-pure-mathematics-1921/eq-a12ab9760f",16,"Hardy 1921, p. 414: \\arctan \\left(\\frac{x}{\\alpha}\\right) = \\frac{1}{2i} \\log\\left(\\frac{x - i\\alpha}{x + i\\alpha}\\right) + C"],["hardy-course-of-pure-mathematics-1921/eq-a94084b540",16,"Hardy 1921, p. 414: \\arctan x = \\frac{1}{2i} \\Log\\left(\\frac{1 + ix}{1 - ix}\\right)"],["boyden-first-book-in-algebra-1895/ex-21/15",4,"Boyden 1895, Exercise 21 (15)"],["form/0c7a01d02d",5,"identity: a**36*b**8*c**16*d**8"],["shape/aa8ba091d2",6,"identity: a**N*b**N*c**N*d**N"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/5",4,"Hardy 1921, Exercise LXVI (5)"],["wentworth-first-steps-in-algebra-1894/x-45040d228f",15,"Wentworth 1894, p. 153: The same method will apply to longer expressions, if ..."],["theorem/product-of-determinants",9,"product of determinants","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-product-of-determinants"],["de-morgan-elementary-illustrations-calculus-1899/x-0d27742c57",15,"De Morgan 1899, p. 20: This never happens in the developments which we shall ..."],["todhunter-spherical-trigonometry-1886/x-288619f49d",15,"Todhunter 1886, scan 72: A circle which touches one side of a triangle ..."],["theorem/derivative-of-the-complex-exponential",9,"derivative of the complex exponential","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-derivative-of-the-complex-exponential"],["theorem/derivative-of-a-complex-power",9,"derivative of a complex power","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-derivative-of-a-complex-power"],["wentworth-first-steps-in-algebra-1894/x-00c54e6160",15,"Wentworth 1894, p. 157: Since the cube of a + b is a^{3} ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvi/6",4,"Hardy 1921, Exercise LXVI (6)"],["wentworth-first-steps-in-algebra-1894/x-6a7ca4662a",15,"Wentworth 1894, p. 154: If, therefore, an integral square number is divided into ..."],["wentworth-first-steps-in-algebra-1894/x-e263c7501e",15,"Wentworth 1894, p. 155: If the square root of a number has decimal ..."],["wentworth-first-steps-in-algebra-1894/x-268182ecbb",15,"Wentworth 1894, p. 155: If a number contain an odd number of decimal ..."],["hardy-course-of-pure-mathematics-1921/eq-2da0fde5dd",16,"Hardy 1921, p. 415: f'(y) = \\lim_{k \\to 0} \\frac{f(y + k) - f(y)}{k} = if(y)"],["planck-treatise-on-thermodynamics-1903/ch-gaseous-system",2,"Planck 1903, Gaseous System","../books/planck-treatise-on-thermodynamics-1903/ch/ch-gaseous-system/index.html"],["hardy-course-of-pure-mathematics-1921/eq-93d72b1554",16,"Hardy 1921, p. 414: \\exp z = 1 + z + \\frac{z^{2}}{2!} + \\dots"],["hardy-course-of-pure-mathematics-1921/eq-06ee5ba57a",16,"Hardy 1921, p. 415: F(z) F(h) = F(z + h)"],["wentworth-first-steps-in-algebra-1894/x-60e075fd53",15,"Wentworth 1894, p. 153: It will be noticed that each successive trial-divisor may ..."],["thompson-calculus-made-easy-1914/eq-8f31be2d78",16,"Thompson 1914, p. 182: \\ds\\int dy = y"],["thompson-calculus-made-easy-1914/eq-b8f451fe18",16,"Thompson 1914, p. 182: \\ds\\int dx = x"],["thompson-calculus-made-easy-1914/eq-8195fe4b70",16,"Thompson 1914, p. 187: \\dfrac{y}{x} = a"],["hardy-course-of-pure-mathematics-1921/eq-e058ba38ee",16,"Hardy 1921, p. 415: f(y) = \\cos Y + i \\sin Y"],["theorem/discriminant-of-a-conic",9,"discriminant of a conic","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-theorem-discriminant-of-a-conic"],["hardy-course-of-pure-mathematics-1921/eq-4bc326b487",16,"Hardy 1921, p. 416: F(iy) = \\cos y + i\\sin y"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii",3,"Hardy 1921, Exercise XXXVII"],["dickson-theory-of-equations-1922/ex-page59",3,"Dickson 1922, Exercise Page59"],["concept/concurrent-lines",7,"concurrent lines","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-concurrent-lines"],["hardy-course-of-pure-mathematics-1921/ex-xxxviii",3,"Hardy 1921, Exercise XXXVIII"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/7",4,"Hardy 1921, Exercise LXVI (7)"],["planck-treatise-on-thermodynamics-1903/eq-58f8c0a837",16,"Planck 1903, p. 74: \\ce{(SnCl2 . 2HCl $\\aq$) + (H2O2 $\\aq$) - (SnCl4 $\\aq$)} = 88,800~\\Unit{cal.}"],["hardy-course-of-pure-mathematics-1921/eq-82757cef79",16,"Hardy 1921, p. 416: F(x + iy) = F(x) F(iy) = \\exp x(\\cos y + i\\sin y) = \\exp(x + iy)"],["planck-treatise-on-thermodynamics-1903/eq-0902be4d32",16,"Planck 1903, p. 74: \\ce{(SnCl2 . 2HCl $\\aq$) + $\\tfrac{1}{2}$ \\{O2\\} - (SnCl4 $\\aq$)} = 65,700~\\Unit{cal.}"],["hardy-course-of-pure-mathematics-1921/eq-426d320adc",16,"Hardy 1921, p. 416: \\cos z = 1 - \\frac{z^{2}}{2!} + \\frac{z^{4}}{4!} - \\dots"],["planck-treatise-on-thermodynamics-1903/eq-e828309c86",16,"Planck 1903, p. 74: \\ce{(H2O2 $\\aq$) - $\\tfrac{1}{2}$ \\{O2\\} - ($\\aq$)} = 23,100~\\Unit{cal.}"],["planck-treatise-on-thermodynamics-1903/eq-a1879ab372",16,"Planck 1903, p. 74: \\ce{[C] + \\{O2\\} - \\{CO2\\}} = 97,000~\\Unit{cal.}"],["planck-treatise-on-thermodynamics-1903/eq-075c9d643e",16,"Planck 1903, p. 74: \\ce{\\{CO\\} + $\\tfrac{1}{2}$ \\{O2\\} - \\{CO2\\}} = 68,000~\\Unit{cal.}"],["person/gabriel-cramer",1,"Gabriel Cramer","../books/dickson-theory-of-equations-1922/terms/index.html#t-person-gabriel-cramer"],["hardy-course-of-pure-mathematics-1921/eq-34403013fa",16,"Hardy 1921, p. 416: \\sin z = z - \\frac{z^{3}}{3!} + \\frac{z^{5}}{5!} - \\dots"],["hardy-course-of-pure-mathematics-1921/ex-xxi",3,"Hardy 1921, Exercise XXI"],["concept/denominator",7,"denominator","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-denominator"],["person/aristotle",1,"Aristotle","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-aristotle"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/8",4,"Hardy 1921, Exercise LXVI (8)"],["person/galileo-galilei",1,"Galileo Galilei","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-galileo-galilei"],["whitehead-introduction-to-mathematics-1911/x-4adae97035",15,"Whitehead 1911, p. 52: For example, a velocity requires for its definition the ..."],["planck-treatise-on-thermodynamics-1903/eq-aaafaa01bc",16,"Planck 1903, p. 74: \\ce{[C] + $\\tfrac{1}{2}$ \\{O2\\} - \\{CO\\}} = 29,000~\\Unit{cal.}"],["planck-treatise-on-thermodynamics-1903/eq-e87e0b12cb",16,"Planck 1903, p. 75: \\ce{[S] + \\{O2\\} - \\{SO2\\}} = 71,100~\\Unit{cal.}"],["hardy-course-of-pure-mathematics-1921/eq-9ffde08d95",16,"Hardy 1921, p. 417: \\log(1 + z) = z - \\tfrac{1}{2} z^{2} + \\tfrac{1}{3} z^{3} - \\dots"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/9",4,"Hardy 1921, Exercise LXVI (9)"],["planck-treatise-on-thermodynamics-1903/eq-d35bef4ef9",16,"Planck 1903, p. 75: \\ce{\\{CS2\\} + 3\\{O2\\} - \\{CO2\\} - 2\\{SO2\\}} = 265,100~\\Unit{cal.}"],["theorem/trigonometric-identity",9,"trigonometric identity","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-trigonometric-identity"],["planck-treatise-on-thermodynamics-1903/eq-17b4d2a2ad",16,"Planck 1903, p. 75: \\ce{\\{CS2\\} - (CS2)} = 6400~\\Unit{cal.}"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/10",4,"Hardy 1921, Exercise LXVI (10)"],["concept/stereographic-projection",7,"stereographic projection","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-stereographic-projection"],["concept/mercator-s-projection",7,"Mercator's projection","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-mercator-s-projection"],["dickson-theory-of-equations-1922/ex-page62",3,"Dickson 1922, Exercise Page62"],["dickson-theory-of-equations-1922/ex-page64",3,"Dickson 1922, Exercise Page64"],["dickson-theory-of-equations-1922/ex-page66",3,"Dickson 1922, Exercise Page66"],["ball-mathematical-recreations-1905/x-d549719745",15,"Ball 1905, scan 99: Thus, if a number of dominoes or draughts are ..."],["ball-mathematical-recreations-1905/x-6c415a86c0",15,"Ball 1905, scan 102: In other words, if in order to cause a ..."],["ball-mathematical-recreations-1905/x-a23522339c",15,"Ball 1905, scan 96: To establish this, Zeno argued that when Achilles had ..."],["ball-mathematical-recreations-1905/x-c18bc7c6df",15,"Ball 1905, scan 95: So naturalists observe, a flea hath smaller fleas that ..."],["ball-mathematical-recreations-1905/x-76e1dc0b30",15,"Ball 1905, scan 102: Montucla says that in his time it was not ..."],["theorem/reduction-of-a-rational-function-to-x-yi",9,"reduction of a rational function to X + Yi","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-reduction-of-a-rational-function-to-x-yi"],["theorem/conjugation-of-real-rational-functions",9,"conjugation of real rational functions","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-conjugation-of-real-rational-functions"],["hardy-course-of-pure-mathematics-1921/eq-4da4219942",16,"Hardy 1921, p. 419: \\log \\left(\\frac{1}{1 - z}\\right) = -\\log(1 - z) = z + \\tfrac{1}{2} z^{2} + \\tfrac{1}{3} z^{3} + \\dots"],["dickson-theory-of-equations-1922/ex-page68",3,"Dickson 1922, Exercise Page68"],["maxwell-elementary-treatise-electricity-1888/eq-b068590bb6",16,"Maxwell 1888, scan 209: \\gamma = \\beta\\left(1 + 2 \\frac{y - x}{b + c}\\right)"],["maxwell-elementary-treatise-electricity-1888/eq-d69896bcb5",16,"Maxwell 1888, scan 210: c=\\sqrt{a\\alpha}"],["maxwell-elementary-treatise-electricity-1888/eq-ba61ae24b2",16,"Maxwell 1888, scan 210: b=\\sqrt{a\\gamma\\frac{\\alpha + \\gamma}{a + \\gamma}}"],["maxwell-elementary-treatise-electricity-1888/eq-41c016ab5c",16,"Maxwell 1888, scan 210: \\beta = \\sqrt{\\alpha \\gamma \\frac{a + \\gamma }{\\alpha + \\gamma}}"],["maxwell-elementary-treatise-electricity-1888/eq-b59162bc3d",16,"Maxwell 1888, scan 211: b = \\frac{c\\gamma}{\\beta}"],["maxwell-elementary-treatise-electricity-1888/eq-6bff70ac2b",16,"Maxwell 1888, scan 212: y = \\frac{E\\alpha}{b\\alpha + c(b+\\alpha+\\gamma)}"],["method/rationalizing-a-complex-denominator",8,"rationalizing a complex denominator","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-rationalizing-a-complex-denominator"],["hardy-course-of-pure-mathematics-1921/x-8ee021cb9a",15,"Hardy 1921, p. 85: [fig:24]Fig. 24 is usually known as Argand’s diagram."],["hardy-course-of-pure-mathematics-1921/eq-75f34e8be4",16,"Hardy 1921, p. 418: \\log(1 + z) = \\int_{C} \\frac{du}{u}"],["maxwell-elementary-treatise-electricity-1888/eq-d1de377c70",16,"Maxwell 1888, scan 212: \\frac{y_0-y_1}{y} = \\frac{\\alpha}{\\gamma}\\frac{c\\gamma-a\\alpha}{(c+\\alpha )(\\alpha+\\gamma)}"],["concept/sensibility-of-a-method",7,"sensibility of a method"],["maxwell-elementary-treatise-electricity-1888/eq-e0107d09ec",16,"Maxwell 1888, scan 212: c\\gamma=a\\alpha"],["maxwell-elementary-treatise-electricity-1888/eq-ef872efefc",16,"Maxwell 1888, scan 213: x_1 = y \\left(1 + \\frac{b}{\\alpha + \\gamma} \\right)"],["maxwell-elementary-treatise-electricity-1888/eq-f6931ea895",16,"Maxwell 1888, scan 213: x_0 = y \\left(1 + \\frac{b}{\\gamma} + \\frac{\\alpha c}{\\gamma (\\alpha + c)} \\right)"],["maxwell-elementary-treatise-electricity-1888/eq-571717dfb6",16,"Maxwell 1888, scan 213: a = \\frac{c \\gamma}{\\alpha}"],["maxwell-elementary-treatise-electricity-1888/eq-f3315449ec",16,"Maxwell 1888, scan 213: E = y \\left( b + c + \\frac{c}{\\alpha} ( b + \\gamma ) \\right)"],["hardy-course-of-pure-mathematics-1921/x-27beec6920",15,"Hardy 1921, p. 85: When y = 0 we say that z is ..."],["dickson-theory-of-equations-1922/ex-page70",3,"Dickson 1922, Exercise Page70"],["maxwell-elementary-treatise-electricity-1888/eq-e6023668b0",16,"Maxwell 1888, scan 213: \\frac{y_0 - y_1}{y} = \\frac{\\beta}{\\gamma} \\frac{c \\gamma - b \\beta}{(c + \\beta)(\\beta + \\gamma)}"],["maxwell-elementary-treatise-electricity-1888/eq-9df34973ed",16,"Maxwell 1888, scan 214: E_1 = R_1C"],["maxwell-elementary-treatise-electricity-1888/eq-e96101c536",16,"Maxwell 1888, scan 214: E_2 = R_2C"],["maxwell-elementary-treatise-electricity-1888/eq-f0bd12c32a",16,"Maxwell 1888, scan 214: E_1 : E_2 :: R_1 : R_2"],["hardy-course-of-pure-mathematics-1921/x-09cc278e74",15,"Hardy 1921, p. 85: Two numbers x + yi, x - yi which ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvi/11",4,"Hardy 1921, Exercise LXVI (11)"],["dickson-theory-of-equations-1922/ex-page133",3,"Dickson 1922, Exercise Page133"],["dickson-theory-of-equations-1922/ex-page136",3,"Dickson 1922, Exercise Page136"],["dickson-theory-of-equations-1922/ex-page129",3,"Dickson 1922, Exercise Page129"],["dickson-theory-of-equations-1922/ex-page140",3,"Dickson 1922, Exercise Page140"],["whitehead-introduction-to-mathematics-1911/x-f538b06879",15,"Whitehead 1911, p. 55: If the steamer were still, in one minute he ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvi/12a",4,"Hardy 1921, Exercise LXVI (12a)"],["concept/simple-harmonic-motion",7,"simple harmonic motion","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-simple-harmonic-motion"],["wentworth-first-steps-in-algebra-1894/ex-15/11",4,"Wentworth 1894, Exercise 15 (11)"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/12b",4,"Hardy 1921, Exercise LXVI (12b)"],["thompson-calculus-made-easy-1914/eq-aef0c5588d",16,"Thompson 1914, p. 187: y = ax + C."],["thompson-calculus-made-easy-1914/eq-a9755c7adf",16,"Thompson 1914, p. 187: \\frac{dy}{dx} = ax."],["form/0c99fbb649",5,"identity: 0"],["dickson-theory-of-equations-1922/ex-page141",3,"Dickson 1922, Exercise Page141"],["concept/curve",7,"curve","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-curve"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/13",4,"Hardy 1921, Exercise LXVI (13)"],["wentworth-plane-geometry-1899/ex-v-1",3,"Wentworth 1899, Exercise V.1"],["hardy-course-of-pure-mathematics-1921/eq-7e3caa33a7",16,"Hardy 1921, p. 420: \\arctan z = z - \\tfrac{1}{3}z^{3} + \\tfrac{1}{5}z^{5} - \\dots"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/14",4,"Hardy 1921, Exercise LXVI (14)"],["thompson-calculus-made-easy-1914/eq-9a206fc1ba",16,"Thompson 1914, p. 187: \\frac{dy}{dx} = \\tfrac{1}{5} x"],["thompson-calculus-made-easy-1914/eq-de1c65827e",16,"Thompson 1914, p. 189: \\ds\\int \\tfrac{1}{5} x\\, dx = \\tfrac{1}{10} x^2"],["thompson-calculus-made-easy-1914/eq-d13e1cec80",16,"Thompson 1914, p. 189: y=\\frac{1}{10}x^2"],["thompson-calculus-made-easy-1914/eq-5ba5d309bf",16,"Thompson 1914, p. 190: y = \\tfrac{1}{10}x^2 + C."],["concept/associated-triangles",7,"associated triangles","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-associated-triangles"],["wentworth-plane-geometry-1899/ex-misc",3,"Wentworth 1899, Exercise Misc"],["wentworth-plane-geometry-1899/x-6754238415",15,"Wentworth 1899, scan 243: Of all triangles having two given sides, that in ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvi/15",4,"Hardy 1921, Exercise LXVI (15)"],["form/e3c603a03c",5,"identity: -3*a**3*b**3*x**3"],["maxwell-elementary-treatise-electricity-1888/x-3898cbdd03",15,"Maxwell 1888, scan 107: the amount of the charge is the greater the ..."],["planck-treatise-on-thermodynamics-1903/x-e5e16f18ad",15,"Planck 1903, p. 150: No off-hand statement can be made with regard to ..."],["method/carnot-s-method",8,"Carnot's method","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-carnot-s-method"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/16",4,"Hardy 1921, Exercise LXVI (16)"],["hardy-course-of-pure-mathematics-1921/x-29370dd380",15,"Hardy 1921, p. 429: If f(z) is a function of the complex variable ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvi/17",4,"Hardy 1921, Exercise LXVI (17)"],["concept/right-circular-cone",7,"right circular cone","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-right-circular-cone"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/18",4,"Hardy 1921, Exercise LXVI (18)"],["planck-treatise-on-thermodynamics-1903/eq-8024063bd3",16,"Planck 1903, p. 75: \\ce{[C] + 2[S] - (CS2)} = - 19,500~\\Unit{cal.}"],["wentworth-first-steps-in-algebra-1894/ex-1/10",4,"Wentworth 1894, Exercise 1 (10)"],["planck-treatise-on-thermodynamics-1903/x-a6ad026515",15,"Planck 1903, p. 158: A geometrical representation may facilitate a general survey of ..."],["hardy-course-of-pure-mathematics-1921/x-6dbc417b07",15,"Hardy 1921, p. 428: In this map parallels of latitude and longitude are ..."],["wentworth-first-steps-in-algebra-1894/ex-1/11",4,"Wentworth 1894, Exercise 1 (11)"],["de-morgan-elementary-illustrations-calculus-1899/ch-differential-coefficients",2,"De Morgan 1899, Differential Coefficients","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-differential-coefficients/index.html"],["wentworth-first-steps-in-algebra-1894/ex-1/12",4,"Wentworth 1894, Exercise 1 (12)"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/19",4,"Hardy 1921, Exercise LXVI (19)"],["person/alessandro-volta",1,"Alessandro Volta","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-alessandro-volta"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/20",4,"Hardy 1921, Exercise LXVI (20)"],["de-morgan-elementary-illustrations-calculus-1899/x-2e20ace9d0",15,"De Morgan 1899, p. 62: Hence the limit to which we approximate by diminishing ..."],["boyden-first-book-in-algebra-1895/ex-21/16",4,"Boyden 1895, Exercise 21 (16)"],["maxwell-elementary-treatise-electricity-1888/eq-6904475e4b",16,"Maxwell 1888, scan 188: Wg &= \\frac{V^2A}{8 \\pi D^2}\\text{,}"],["form/23758ac750",5,"identity: -a**10*b**5*c**40*d**5*e**15"],["shape/2540758bdf",6,"identity: -a**N*b**N*c**N*d**N*e**N"],["boyden-first-book-in-algebra-1895/ex-21/17",4,"Boyden 1895, Exercise 21 (17)"],["form/0332ecbf9e",5,"identity: 4*a**4*b**2*c**8/9"],["hardy-course-of-pure-mathematics-1921/ex-lxvi/21",4,"Hardy 1921, Exercise LXVI (21)"],["maxwell-elementary-treatise-electricity-1888/x-d6dc070f66",15,"Maxwell 1888, scan 108: In this case the capacity of the inner conductor ..."],["hardy-course-of-pure-mathematics-1921/eq-becab479c4",16,"Hardy 1921, p. 419: \\cos\\theta - \\tfrac{1}{2} \\cos 2\\theta + \\tfrac{1}{3} \\cos 3\\theta - \\dots = \\tfrac{1}{2} \\log(4\\cos^{2} \\tfrac{1}{2}\\th"],["planck-treatise-on-thermodynamics-1903/x-52efad7777",15,"Planck 1903, p. 151: Watt assumed this to be the case for steam."],["de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-or-proportion-of-two-magnitudes",2,"De Morgan 1899, On the Ratio or Proportion of Two Magnitudes","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-on-the-ratio-or-proportion-of-two-magnitudes/index.html"],["maxwell-elementary-treatise-electricity-1888/eq-1c10bf3661",16,"Maxwell 1888, scan 216: r &= 0.039369T^{\\frac{1}{2}} + 0.00216407T - 0.2413\\text{,}"],["dickson-theory-of-equations-1922/ch-appendix",2,"Dickson 1922, Appendix","../books/dickson-theory-of-equations-1922/ch/ch-appendix/index.html"],["todhunter-spherical-trigonometry-1886/x-adaa97ce20",15,"Todhunter 1886, scan 130: Thus for a regular tetrahedron we have 144\\hspace{3pt}V^2=2a^6."],["maxwell-elementary-treatise-electricity-1888/eq-0c2541ef19",16,"Maxwell 1888, scan 216: r = \\alpha T^{\\frac{1}{2}} + \\beta T + \\gamma"],["boyden-first-book-in-algebra-1895/ex-21/18",4,"Boyden 1895, Exercise 21 (18)"],["maxwell-elementary-treatise-electricity-1888/eq-c70d5f336b",16,"Maxwell 1888, scan 216: r &= 0.026577T^{\\frac{1}{2}} + 0.0031443T - 0.22751\\text{,}"],["form/7c53bc6d5f",5,"identity: 25*a**2*b**4*c**6/36"],["hardy-course-of-pure-mathematics-1921/ex-lxvii/1",4,"Hardy 1921, Exercise LXVII (1)"],["maxwell-elementary-treatise-electricity-1888/ch-xiii",2,"Maxwell 1888, ch. XIII: ON THE ELECTRIC RESISTANCE OF SUBSTANCES","../books/maxwell-elementary-treatise-electricity-1888/ch/ch-xiii/index.html"],["concept/metal",7,"metal","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-metal"],["cap/other:series_convergence",17,"other:series_convergence"],["maxwell-elementary-treatise-electricity-1888/eq-45c9d1f410",16,"Maxwell 1888, scan 216: r &= 0.072545T^{\\frac{1}{2}} + 0.0038133T - 1.23971\\text{.}"],["concept/commensurable-magnitudes",7,"commensurable magnitudes","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-commensurable-magnitudes"],["wentworth-first-steps-in-algebra-1894/ex-21/5",4,"Wentworth 1894, Exercise 21 (5)"],["hardy-course-of-pure-mathematics-1921/ex-lxvii/2",4,"Hardy 1921, Exercise LXVII (2)"],["hardy-course-of-pure-mathematics-1921/eq-4b6ee51cca",16,"Hardy 1921, p. 419: \\sin\\theta - \\tfrac{1}{2} \\sin 2\\theta + \\tfrac{1}{3} \\sin 3\\theta - \\dots = \\tfrac{1}{2} \\theta"],["de-morgan-elementary-illustrations-calculus-1899/x-fd7a039cc2",15,"De Morgan 1899, p. 22: Let there be any function of x, which we ..."],["maxwell-elementary-treatise-electricity-1888/eq-6fa9caabb2",16,"Maxwell 1888, scan 218: \\rho = \\frac{R_1 - R_2}{{R_1}' - {R_2}'}"],["de-morgan-elementary-illustrations-calculus-1899/ch-partial-and-total-differentials",2,"De Morgan 1899, Partial and Total Differentials","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-partial-and-total-differentials/index.html"],["form/83ec872261",5,"identity: -9*a*b*c*x"],["shape/290f74450d",6,"identity: N*a*b*c*x"],["macfarlane-vector-analysis-quaternions-1906/ch-coaxial-quaternions",2,"Macfarlane 1906, Coaxial Quaternions","../books/macfarlane-vector-analysis-quaternions-1906/ch/ch-coaxial-quaternions/index.html"],["hardy-course-of-pure-mathematics-1921/x-5376ee36cb",15,"Hardy 1921, p. 51: a function y = f(x) will be said to ..."],["hardy-course-of-pure-mathematics-1921/x-0c64f122e2",15,"Hardy 1921, p. 54: A function is said to be periodic, with period ..."],["todhunter-spherical-trigonometry-1886/x-2b0ef355a6",15,"Todhunter 1886, scan 98: This formula is called General Roy’s rule, as it ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvii/3",4,"Hardy 1921, Exercise LXVII (3)"],["hardy-course-of-pure-mathematics-1921/x-81ae2b2397",15,"Hardy 1921, p. 62: The locus is the surface formed by drawing lines ..."],["concept/diagonal",7,"diagonal","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-diagonal"],["wentworth-first-steps-in-algebra-1894/x-b53aea0c98",15,"Wentworth 1894, p. 85: Resolve each expression into its simplest factors. Find the ..."],["wentworth-first-steps-in-algebra-1894/x-88e592b989",15,"Wentworth 1894, p. 85: We cannot apply the terms greatest and least to ..."],["wentworth-first-steps-in-algebra-1894/x-162a035eff",15,"Wentworth 1894, p. 87: The L. C. M. must evidently contain each factor ..."],["wentworth-first-steps-in-algebra-1894/x-4b677a7bee",15,"Wentworth 1894, p. 85: The highest common factor in Algebra corresponds to the ..."],["wentworth-first-steps-in-algebra-1894/x-0b96e8bb4a",15,"Wentworth 1894, p. 84: The highest common factor of two or more integral ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvii/4",4,"Hardy 1921, Exercise LXVII (4)"],["hardy-course-of-pure-mathematics-1921/ex-lxvii/5",4,"Hardy 1921, Exercise LXVII (5)"],["wentworth-plane-geometry-1899/ex-iv-2",3,"Wentworth 1899, Exercise IV.2"],["planck-treatise-on-thermodynamics-1903/eq-c2de726761",16,"Planck 1903, p. 75: \\ce{\\{CH4\\} + 2\\{O2\\} - \\{CO2\\} - 2(H2O)} &= 211,900~\\Unit{cal.}"],["planck-treatise-on-thermodynamics-1903/eq-69179b653b",16,"Planck 1903, p. 75: \\ce{\\{H2\\} + $\\tfrac{1}{2}$ \\{O2\\} - (H2O)} &= \\Z68,400~\\Unit{cal.}"],["hardy-course-of-pure-mathematics-1921/ex-lxvii/6",4,"Hardy 1921, Exercise LXVII (6)"],["hardy-course-of-pure-mathematics-1921/x-270a15857d",15,"Hardy 1921, p. 58: The abscissae of its intersections with the axis of ..."],["hardy-course-of-pure-mathematics-1921/eq-31beedaa72",16,"Hardy 1921, p. 421: \\log(1 + hz) = hz - \\tfrac{1}{2}(hz)^{2} + \\tfrac{1}{3}(hz)^{3} - \\dots"],["hardy-course-of-pure-mathematics-1921/x-4f075be07b",15,"Hardy 1921, p. 71: Common sense at once suggests that we should define ..."],["form/8a3cb85bbf",5,"evaluate: -2*a - 2*b at a=4, b=-2, c=-3"],["hardy-course-of-pure-mathematics-1921/ex-lxvii/7",4,"Hardy 1921, Exercise LXVII (7)"],["form/19f31105b8",5,"identity: -a*(3*a - b)"],["shape/1f2981b2c9",6,"identity: -a*(N*a - b)"],["form/7113e7b76c",5,"identity: -3*a*(2*a**2 - 3*a*b)"],["quantity/self-induction",11,"self-induction","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-quantity-self-induction"],["hardy-course-of-pure-mathematics-1921/ex-lxvii/8",4,"Hardy 1921, Exercise LXVII (8)"],["concept/pawn-interchange-puzzle",7,"pawn interchange puzzle","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-pawn-interchange-puzzle"],["concept/solubility-product",7,"solubility product","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-solubility-product"],["concept/coaxial-quaternions",7,"coaxial quaternions","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-coaxial-quaternions"],["planck-treatise-on-thermodynamics-1903/eq-fc1d6acce4",16,"Planck 1903, p. 75: \\ce{[C] + 2\\{H2\\} - \\{CH4\\}} = 21,900~\\Unit{cal.}"],["de-morgan-elementary-illustrations-calculus-1899/x-b16d51ec5b",15,"De Morgan 1899, p. 86: (2) y = a^{x}. Here dy = a^{x}\\log a\\, ..."],["macfarlane-vector-analysis-quaternions-1906/ex-probs-19-22",3,"Macfarlane 1906, Exercise Probs-19-22"],["form/92e86e5872",5,"identity: 5*b*(2*a**2 + 3*a*b)"],["shape/d8981bafdd",6,"identity: N*b*(N*a*b + N*a**N)"],["concept/form-factor",7,"form factor","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-form-factor"],["concept/paradromic-ring",7,"paradromic ring","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-paradromic-ring"],["concept/one-sided-surface",7,"one-sided surface","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-one-sided-surface"],["theorem/sine-addition-formula",9,"sine addition formula","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-sine-addition-formula"],["concept/area-of-a-surface-of-revolution",7,"area of a surface of revolution","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-area-of-a-surface-of-revolution"],["de-morgan-elementary-illustrations-calculus-1899/x-0ddc2aa34e",15,"De Morgan 1899, p. 86: y = \\log x (the Naperian logarithm). Here dy ..."],["concept/error-of-measurement",7,"error of measurement","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-error-of-measurement"],["hardy-course-of-pure-mathematics-1921/ex-lxvii/9",4,"Hardy 1921, Exercise LXVII (9)"],["thompson-calculus-made-easy-1914/x-f639b4580b",15,"Thompson 1914, p. 98: It does not of itself discriminate; it finds for ..."],["thompson-calculus-made-easy-1914/x-6a2d721ab9",15,"Thompson 1914, p. 108: It is necessary therefore always to check by taking ..."],["thompson-calculus-made-easy-1914/x-172de47127",15,"Thompson 1914, p. 99: Let the number to be cut into two parts ..."],["thompson-calculus-made-easy-1914/x-e0d3323226",15,"Thompson 1914, p. 99: This is a very useful rule, and applies to ..."],["de-morgan-elementary-illustrations-calculus-1899/x-a982f55088",15,"De Morgan 1899, p. 2: Thus, the ratio of the diagonal of a square ..."],["method/finding-an-area-in-polar-coordinates",8,"finding an area in polar coordinates","../books/thompson-calculus-made-easy-1914/terms/index.html#t-method-finding-an-area-in-polar-coordinates"],["hardy-course-of-pure-mathematics-1921/eq-c585cceb31",16,"Hardy 1921, p. 421: \\frac{\\log(1 + hz)}{h} = z + \\phi(h, z)"],["ball-mathematical-recreations-1905/x-c2ad4fe5c5",15,"Ball 1905, scan 86: Suppose the pieces to be arranged originally in circular ..."],["ball-mathematical-recreations-1905/x-0fd6b516f4",15,"Ball 1905, scan 91: The solution is tolerably obvious. First, move the pieces ..."],["hardy-course-of-pure-mathematics-1921/eq-b2270fa7eb",16,"Hardy 1921, p. 421: \\lim_{h\\to 0} \\frac{\\log(1 + hz)}{h} = z"],["de-morgan-elementary-illustrations-calculus-1899/x-c06a89fe72",15,"De Morgan 1899, p. 3: In estimating the approach to, or departure from equality, ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvii/10",4,"Hardy 1921, Exercise LXVII (10)"],["shape/89b4bdb771",6,"identity: -b**N*(N*a*b - a**N - b**N)"],["thompson-calculus-made-easy-1914/x-e7983ea4f0",15,"Thompson 1914, p. 209: Here we have the clue as to what to ..."],["thompson-calculus-made-easy-1914/x-328e83d165",15,"Thompson 1914, p. 214: N.B.---Notice that in dealing with definite integrals the constant ..."],["ball-mathematical-recreations-1905/x-17c84553e9",15,"Ball 1905, scan 92: If any of my readers think that these results ..."],["theorem/descartes-rule-of-signs",9,"Descartes' rule of signs","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-descartes-rule-of-signs"],["hardy-course-of-pure-mathematics-1921/ex-lxvii/11",4,"Hardy 1921, Exercise LXVII (11)"],["de-morgan-elementary-illustrations-calculus-1899/x-2c44cc7b3f",15,"De Morgan 1899, p. 58: But if d\\theta be taken sufficiently small, \\sin d\\theta, ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvii/12",4,"Hardy 1921, Exercise LXVII (12)"],["form/8bc175f713",5,"identity: 5*a**2*b**2*(a**3*b + 3*a**2*b**2 - 4*a*b**3)"],["shape/ab5fee81b8",6,"identity: N*a**N*b**N*(N*a*b**N + N*a**N*b**N + a**N*b)"],["theorem/sturm-s-theorem",9,"Sturm's theorem","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-sturm-s-theorem"],["concept/sturm-s-functions",7,"Sturm's functions","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-sturm-s-functions"],["hardy-course-of-pure-mathematics-1921/ex-lxvii/13",4,"Hardy 1921, Exercise LXVII (13)"],["ball-mathematical-recreations-1905/eq-d725ea4f54",16,"Ball 1905, scan 111: W\\!AR = \\theta"],["de-morgan-elementary-illustrations-calculus-1899/x-50dc29dafd",15,"De Morgan 1899, p. 59: If the angle so described be always increased by ..."],["ball-mathematical-recreations-1905/eq-af74ae2ee4",16,"Ball 1905, scan 111: BAS = \\alpha"],["ball-mathematical-recreations-1905/eq-67a2299902",16,"Ball 1905, scan 111: W\\!AL=\\theta + \\alpha"],["thompson-calculus-made-easy-1914/x-dadf41b10d",15,"Thompson 1914, p. 208: That is all very well; but a little thought ..."],["form/d1a8b9ada1",5,"solve: Eq(x, 8*a)"],["boyden-first-book-in-algebra-1895/ex-21/20a",4,"Boyden 1895, Exercise 21 (20a)"],["theorem/area-of-a-trapezoid",9,"area of a trapezoid","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-area-of-a-trapezoid"],["boyden-first-book-in-algebra-1895/ex-21/19",4,"Boyden 1895, Exercise 21 (19)"],["wentworth-first-steps-in-algebra-1894/ex-22",3,"Wentworth 1894, Exercise 22"],["ball-mathematical-recreations-1905/eq-f5c8e2442a",16,"Ball 1905, scan 111: v \\sin \\alpha = u \\sin (\\theta + \\alpha)"],["thompson-calculus-made-easy-1914/x-f3abe7ab01",15,"Thompson 1914, p. 222: By “quadratic mean” is denoted the square root of ..."],["ball-mathematical-recreations-1905/eq-0d3c3a6367",16,"Ball 1905, scan 111: v>u"],["concept/projective-geometry",7,"projective geometry","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-projective-geometry"],["concept/pencil-of-lines",7,"pencil of lines","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-pencil-of-lines"],["ball-mathematical-recreations-1905/eq-fb7ad75362",16,"Ball 1905, scan 111: \\sin (\\theta + \\alpha)>\\allowbreak\\sin \\alpha"],["law/kepler-s-laws-of-planetary-motion",10,"Kepler's laws of planetary motion","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-law-kepler-s-laws-of-planetary-motion"],["thompson-calculus-made-easy-1914/x-16f1f2ccf1",15,"Thompson 1914, p. 219: Instead of a strip of area, we consider a ..."],["dickson-theory-of-equations-1922/x-c7df95e232",15,"Dickson 1922, p. 128: A rational function of the independent variables x_1, x_2, ..."],["whitehead-introduction-to-mathematics-1911/x-a85f7f4f18",15,"Whitehead 1911, p. 142: If ab - h^{2} is a positive number, the ..."],["ball-mathematical-recreations-1905/eq-280fc5f950",16,"Ball 1905, scan 111: \\theta + \\alpha = \\frac{1}{2}\\pi"],["whitehead-introduction-to-mathematics-1911/x-9b7da76a5d",15,"Whitehead 1911, p. 135: The characteristic property of a focus, S, and its ..."],["whitehead-introduction-to-mathematics-1911/x-71345d269b",15,"Whitehead 1911, p. 138: The orbits of the planets are ellipses, the sun ..."],["whitehead-introduction-to-mathematics-1911/x-7016b1d6ae",15,"Whitehead 1911, p. 129: There is a certain type of mathematician who is ..."],["whitehead-introduction-to-mathematics-1911/x-8c2e41f55e",15,"Whitehead 1911, p. 138: Novel ideas are more apt to spring from an ..."],["hardy-course-of-pure-mathematics-1921/ex-lxvii/14",4,"Hardy 1921, Exercise LXVII (14)"],["dickson-theory-of-equations-1922/eq-e672c0143d",16,"Dickson 1922, p. 136: s_k + c_1 s_{k-1} + c_2 s_{k-2} + \\dotsb + c_n s_{k-n} = 0"],["ball-mathematical-recreations-1905/eq-0ef6b17d24",16,"Ball 1905, scan 111: v = u \\cosec \\alpha"],["ball-mathematical-recreations-1905/eq-dc2ec1c628",16,"Ball 1905, scan 111: v \\sin \\alpha = u \\sin \\phi"],["planck-treatise-on-thermodynamics-1903/eq-d3d3dd2e52",16,"Planck 1903, p. 57: (c_{p} - c_{v})\\, \\frac{\\dd^{2} \\theta}{\\dd p\\, \\dd v} + \\frac{\\dd c_{p}}{\\dd p} · \\frac{\\dd \\theta}{\\dd v} - \\frac{\\dd "],["de-morgan-elementary-illustrations-calculus-1899/x-7d99b88a66",15,"De Morgan 1899, p. 3: For example, if a geometrical figure, two of whose ..."],["hardy-course-of-pure-mathematics-1921/ex-lxviii/1",4,"Hardy 1921, Exercise LXVIII (1)"],["planck-treatise-on-thermodynamics-1903/eq-bab4a8c510",16,"Planck 1903, p. 57: p = \\frac{R}{m} · \\frac{\\theta}{v}"],["dickson-theory-of-equations-1922/ex-page115",3,"Dickson 1922, Exercise Page115"],["dickson-theory-of-equations-1922/ex-page119",3,"Dickson 1922, Exercise Page119"],["dickson-theory-of-equations-1922/ex-page120",3,"Dickson 1922, Exercise Page120"],["dickson-theory-of-equations-1922/ex-page121",3,"Dickson 1922, Exercise Page121"],["ball-mathematical-recreations-1905/eq-653c731ed4",16,"Ball 1905, scan 111: \\phi = \\text{angle } W\\!AS = \\pi - \\theta - \\alpha"],["ball-mathematical-recreations-1905/eq-253a2ddc86",16,"Ball 1905, scan 111: v = u \\sin \\phi \\cosec \\alpha"],["ball-mathematical-recreations-1905/eq-6d58cd09ed",16,"Ball 1905, scan 111: w =\\allowbreak v \\cos BAW =\\allowbreak v \\cos (\\alpha + \\phi) =\\allowbreak u \\sin \\phi \\cosec \\alpha \\cos (\\alpha + \\phi"],["ball-mathematical-recreations-1905/eq-262775e3ed",16,"Ball 1905, scan 111: \\phi = \\frac{1}{4}\\pi - \\frac{1}{2}\\alpha"],["ball-mathematical-recreations-1905/eq-4b65af3edc",16,"Ball 1905, scan 111: w = \\frac{1}{2}u (\\cosec\\alpha - 1)"],["thompson-calculus-made-easy-1914/x-abee18b9b0",15,"Thompson 1914, p. 192: Clearly, in dealing with powers of x, the rule ..."],["thompson-calculus-made-easy-1914/x-b7a2b507ad",15,"Thompson 1914, p. 193: Thus, if \\dfrac{dy}{dx} = 4x^2, the reverse process gives ..."],["de-morgan-elementary-illustrations-calculus-1899/x-35f5f1eb56",15,"De Morgan 1899, p. 3: Thus, twenty miles would be a material error in ..."],["ball-mathematical-recreations-1905/eq-1e7ba35e99",16,"Ball 1905, scan 111: \\sin\\alpha< \\frac{1}{3}"],["hardy-course-of-pure-mathematics-1921/ex-lxviii/2",4,"Hardy 1921, Exercise LXVIII (2)"],["dickson-theory-of-equations-1922/eq-48e43dbe19",16,"Dickson 1922, p. 136: x^2 + px + q \\equiv (x - \\alpha)(x - \\beta)"],["thompson-calculus-made-easy-1914/eq-f36105247b",16,"Thompson 1914, p. 226: \\int u\\, dx = ux - \\int x\\, du + C."],["theorem/integration-by-parts-formula",9,"integration by parts formula"],["concept/method-integration-by-parts",7,"method: integration by parts"],["thompson-calculus-made-easy-1914/eq-4d9549004f",16,"Thompson 1914, p. 226: d(ux) = u\\, dx + x\\, du,"],["concept/theorem-product-rule-for-differentiation",7,"theorem: product rule for differentiation"],["thompson-calculus-made-easy-1914/eq-c8faa15c3e",16,"Thompson 1914, p. 229: \\int \\sqrt{1-x^2}\\, dx = \\frac{x \\sqrt{1-x^2}}{2} + \\tfrac{1}{2} \\arcsin x +C."],["dickson-theory-of-equations-1922/eq-e95fa6bad7",16,"Dickson 1922, p. 137: 1 + py + qy^2 \\equiv (1 - \\alpha y)(1 - \\beta y)"],["dickson-theory-of-equations-1922/eq-01e3548216",16,"Dickson 1922, p. 137: \\frac{-p - 2qy}{1 + py + qy^2} \\equiv \\frac{\\alpha}{1 - \\alpha y} + \\frac{\\beta}{1 - \\beta y}"],["dickson-theory-of-equations-1922/eq-b033c9c0af",16,"Dickson 1922, p. 137: \\frac{1}{1 - r} \\equiv 1 + r + r^2 + \\dotsb + r^{k-1} + \\frac{r^k}{1-r}"],["theorem/heron-s-formula",9,"Heron's formula","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-heron-s-formula"],["ball-mathematical-recreations-1905/eq-aefcc06c6e",16,"Ball 1905, scan 113: p = \\Pi\\alpha^{-v^2}"],["wentworth-plane-geometry-1899/ex-ii-1",3,"Wentworth 1899, Exercise II.1"],["de-morgan-elementary-illustrations-calculus-1899/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines",2,"De Morgan 1899, A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-a-geometrical-illustration-limit-of-the-intersections-of-two-coinciding-straight-lines/index.html"],["thompson-calculus-made-easy-1914/eq-d4aa9340cf",16,"Thompson 1914, p. 232: \\frac{1}{a^2-x^2} = \\frac{1}{2a(a+x)} + \\frac{1}{2a(a-x)},"],["thompson-calculus-made-easy-1914/eq-f44292469d",16,"Thompson 1914, p. 232: \\dfrac{dy}{dx} = \\dfrac{1}{a^2-x^2}"],["wentworth-plane-geometry-1899",0,"Wentworth, Plane Geometry (1899)","../books/wentworth-plane-geometry-1899/index.html"],["method/area-of-an-irregular-polygon",8,"area of an irregular polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-method-area-of-an-irregular-polygon"],["dickson-theory-of-equations-1922/eq-1206e991cb",16,"Dickson 1922, p. 137: \\frac{\\alpha}{1 - \\alpha y} + \\frac{\\beta}{1 - \\beta y} = s_1 + s_2 y + \\dotsb + s_k y^{k-1} + \\frac{\\phi y^k}{1 + py + "],["de-morgan-elementary-illustrations-calculus-1899/x-ccb6b238ba",15,"De Morgan 1899, p. 4: In future, when we talk of an approach towards ..."],["theorem/de-moivre-s-theorem",9,"de Moivre's theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-de-moivre-s-theorem"],["de-morgan-elementary-illustrations-calculus-1899/x-fbded606c7",15,"De Morgan 1899, p. 2: This is only saying that P can be taken ..."],["hardy-course-of-pure-mathematics-1921/ex-lxviii/3",4,"Hardy 1921, Exercise LXVIII (3)"],["hardy-course-of-pure-mathematics-1921/ex-lxviii/4",4,"Hardy 1921, Exercise LXVIII (4)"],["planck-treatise-on-thermodynamics-1903/x-16706281af",15,"Planck 1903, p. 25: Thus Avogadro’s law enables us to give in quite ..."],["maxwell-elementary-treatise-electricity-1888/x-aa5597dfca",15,"Maxwell 1888, scan 114: There will thus be a transference of positive electricity ..."],["hardy-course-of-pure-mathematics-1921/ex-lxx/1",4,"Hardy 1921, Exercise LXX (1)"],["maxwell-elementary-treatise-electricity-1888/x-1c83b24150",15,"Maxwell 1888, scan 117: The ratio of the numerical value of the electromotive ..."],["maxwell-elementary-treatise-electricity-1888/x-840f8469df",15,"Maxwell 1888, scan 118: This operation, in which a compound body is decomposed ..."],["maxwell-elementary-treatise-electricity-1888/x-47a8589b5b",15,"Maxwell 1888, scan 122: According to the theory of molecular motion of which ..."],["hardy-course-of-pure-mathematics-1921/ex-lxx/2",4,"Hardy 1921, Exercise LXX (2)"],["hardy-course-of-pure-mathematics-1921/ex-lxx/3",4,"Hardy 1921, Exercise LXX (3)"],["concept/commutativity-of-operations",7,"commutativity of operations","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-commutativity-of-operations"],["concept/limit-operation",7,"limit operation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-limit-operation"],["concept/double-limit-problem",7,"double limit problem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-double-limit-problem"],["wentworth-first-steps-in-algebra-1894/x-aa3a64158f",15,"Wentworth 1894, p. 89: The introduction of the same factor into the dividend ..."],["wentworth-first-steps-in-algebra-1894/x-865d75c450",15,"Wentworth 1894, p. 92: The dividing line between the terms of a fraction ..."],["wentworth-first-steps-in-algebra-1894/x-0e0d7ec149",15,"Wentworth 1894, p. 92: If, therefore, a minus sign precedes the dividing line, ..."],["wentworth-first-steps-in-algebra-1894/x-73b11d75ed",15,"Wentworth 1894, p. 91: By division, \\dfrac{x^{3} - 1}{x + 1} = x^{2} ..."],["wentworth-first-steps-in-algebra-1894/x-84d25a0a14",15,"Wentworth 1894, p. 98: The reciprocal of a fraction, therefore, is the fraction ..."],["wentworth-first-steps-in-algebra-1894/x-5fd066f232",15,"Wentworth 1894, p. 98: Every mixed expression should first be reduced to a ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxi/1",4,"Hardy 1921, Exercise LXXI (1)"],["concept/simple-interest",7,"simple interest","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-simple-interest"],["theorem/necessary-condition-for-a-maximum-or-minimum",9,"necessary condition for a maximum or minimum","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-necessary-condition-for-a-maximum-or-minimum"],["hardy-course-of-pure-mathematics-1921/eq-e9359426a0",16,"Hardy 1921, p. 421: \\lim_{n\\to \\infty} n\\log \\left(1 + \\frac{z}{n}\\right) = z"],["whitehead-introduction-to-mathematics-1911/x-e28df660a5",15,"Whitehead 1911, p. 147: The essential point is that when x is given, ..."],["whitehead-introduction-to-mathematics-1911/x-b37fafbee5",15,"Whitehead 1911, p. 154: For the value of the function on the negative ..."],["cap/other:convergence-proof",17,"other:convergence-proof"],["hardy-course-of-pure-mathematics-1921/ex-lxxi/2",4,"Hardy 1921, Exercise LXXI (2)"],["form/49e10b541e",5,"identity: 3*a*(b - c)"],["hardy-course-of-pure-mathematics-1921/ex-lxxi/3",4,"Hardy 1921, Exercise LXXI (3)"],["whitehead-introduction-to-mathematics-1911/x-c7272cdcb3",15,"Whitehead 1911, p. 155: The whole difference between the older and the newer ..."],["whitehead-introduction-to-mathematics-1911/x-d4362625e3",15,"Whitehead 1911, p. 155: This is exactly the sort of definition which satisfied ..."],["whitehead-introduction-to-mathematics-1911/x-b304c4aa3a",15,"Whitehead 1911, p. 152: A man who, trusting that the mean height of ..."],["quantity/pi",11,"pi","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-quantity-pi"],["hardy-course-of-pure-mathematics-1921/ex-lxxi/4",4,"Hardy 1921, Exercise LXXI (4)"],["hardy-course-of-pure-mathematics-1921/x-903e114164",15,"Hardy 1921, p. 441: The preceding examples suggest that there are three possibilities ..."],["form/e14fbbdb0b",5,"identity: 5*a*(b**2 - c**2)"],["shape/1dff4dcaaf",6,"identity: N*a*(b**N - c**N)"],["de-morgan-elementary-illustrations-calculus-1899/eq-6b2b7a8ffa",16,"De Morgan 1899, p. 102: y = \\phi x"],["hardy-course-of-pure-mathematics-1921/ex-lxxi/5",4,"Hardy 1921, Exercise LXXI (5)"],["thompson-calculus-made-easy-1914/x-1e595e4f58",15,"Thompson 1914, p. 112: Clearly it means the rate (per unit of length ..."],["thompson-calculus-made-easy-1914/x-13c5ffbb80",15,"Thompson 1914, p. 113: If \\dfrac{d^2y}{dx^2} comes out positive, then you know that ..."],["theorem/angles-inscribed-in-the-same-segment-are-equal",9,"angles inscribed in the same segment are equal","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-angles-inscribed-in-the-same-segment-are-equal"],["theorem/chord-angle-measured-by-half-the-sum-of-intercepted-arcs",9,"chord angle measured by half the sum of intercepted arcs","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-chord-angle-measured-by-half-the-sum-of-intercepted-arcs"],["theorem/tangent-chord-angle-measured-by-half-its-intercepted-arc",9,"tangent-chord angle measured by half its intercepted arc","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-tangent-chord-angle-measured-by-half-its-intercepted-arc"],["wentworth-first-steps-in-algebra-1894/ex-30/1",4,"Wentworth 1894, Exercise 30 (1)"],["form/9e3cacf833",5,"identity: (x**3 + 1)/(x + 1)"],["shape/e589ae8d12",6,"identity: (x**N + 1)/(x + 1)"],["hardy-course-of-pure-mathematics-1921/ex-lxxi/6",4,"Hardy 1921, Exercise LXXI (6)"],["todhunter-spherical-trigonometry-1886/x-281c33fdb8",15,"Todhunter 1886, scan 85: If the sides of the triangle are small compared ..."],["todhunter-spherical-trigonometry-1886/x-8b5df53b01",15,"Todhunter 1886, scan 86: This gives the circular measure of \\theta"],["hardy-course-of-pure-mathematics-1921/x-dbadc785e2",15,"Hardy 1921, p. 442: Of course, in an exact science like pure mathematics, ..."],["hardy-course-of-pure-mathematics-1921/x-e1eced374f",15,"Hardy 1921, p. 442: Detailed investigations of a large number of important double ..."],["theorem/external-angle-measured-by-half-the-difference-of-intercepted-arcs",9,"external angle measured by half the difference of intercepted arcs","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-external-angle-measured-by-half-the-difference-of-intercepted-arcs"],["dickson-theory-of-equations-1922/x-cbdf831b14",15,"Dickson 1922, p. 128: In general, if t is a rational function of ..."],["boyden-first-book-in-algebra-1895/ex-57/5",4,"Boyden 1895, Exercise 57 (5)"],["form/74bf522917",5,"solve: Eq(23*x, x**2 + 120)"],["shape/f608983e93",6,"solve: Eq(N*x, N + x**N)"],["form/cec43f1127",5,"identity: (125*a**3 + 8*x**3)/(5*a + 2*x)"],["wentworth-first-steps-in-algebra-1894/ex-16/1",4,"Wentworth 1894, Exercise 16 (1)"],["hardy-course-of-pure-mathematics-1921/ex-lxxii/1",4,"Hardy 1921, Exercise LXXII (1)"],["form/863724803b",5,"identity: 30*a**5"],["thompson-calculus-made-easy-1914/ch-xxi",2,"Thompson 1914, ch. XXI: Finding some Solutions","../books/thompson-calculus-made-easy-1914/ch/ch-xxi/index.html"],["thompson-calculus-made-easy-1914/eq-7d76322ac5",16,"Thompson 1914, p. 234: ay + b \\frac{dy}{dx} = 0"],["planck-treatise-on-thermodynamics-1903/eq-5ac7f3372a",16,"Planck 1903, p. 57: \\theta = \\frac{m}{R}\\, pv"],["hardy-course-of-pure-mathematics-1921/ex-lxxii/2",4,"Hardy 1921, Exercise LXXII (2)"],["wentworth-first-steps-in-algebra-1894/ex-16/2",4,"Wentworth 1894, Exercise 16 (2)"],["form/260bf3a464",5,"identity: 40*a**4*b**3"],["concept/spherical-sector",7,"spherical sector","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-spherical-sector"],["concept/circumscribed-polyhedron",7,"circumscribed polyhedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-circumscribed-polyhedron"],["theorem/area-of-a-spherical-zone",9,"area of a spherical zone","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-area-of-a-spherical-zone"],["theorem/volume-of-a-spherical-cone",9,"volume of a spherical cone","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volume-of-a-spherical-cone"],["theorem/volume-of-a-spherical-sector",9,"volume of a spherical sector","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volume-of-a-spherical-sector"],["theorem/volume-of-a-spherical-segment",9,"volume of a spherical segment","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volume-of-a-spherical-segment"],["slaught-lennes-solid-geometry-1919/x-c2d0e9e24e",15,"Slaught & Lennes 1919, p. 144: The volume of a sphere whose radius is r ..."],["slaught-lennes-solid-geometry-1919/x-44e906cf71",15,"Slaught & Lennes 1919, p. 143: The sphere is covered with a network of spherical ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-3e196bc390",16,"De Morgan 1899, p. 102: x = \\psi y"],["form/943fb91680",5,"identity: (x**9 + 27)/(x**3 + 3)"],["slaught-lennes-solid-geometry-1919/x-0288b4429d",15,"Slaught & Lennes 1919, p. 143: Hence, their combined volume is \\frac{1}{3} r × (\\text{area ..."],["slaught-lennes-solid-geometry-1919/x-978643a2b7",15,"Slaught & Lennes 1919, p. 143: This is of course obvious at a glance, though ..."],["slaught-lennes-solid-geometry-1919/x-d5c5b6f3dd",15,"Slaught & Lennes 1919, p. 142: The total area of the sphere is 4\\pi × ..."],["slaught-lennes-solid-geometry-1919/x-4b7ec6ba4d",15,"Slaught & Lennes 1919, p. 159: He was one of three commissioners who introduced the ..."],["slaught-lennes-solid-geometry-1919/x-a9b3b090d7",15,"Slaught & Lennes 1919, p. 146: The portion of a sphere included between two parallel ..."],["slaught-lennes-solid-geometry-1919/x-f00b67f93e",15,"Slaught & Lennes 1919, p. 146: The perpendicular distance between the planes is the altitude ..."],["thompson-calculus-made-easy-1914/eq-d6838ccf1d",16,"Thompson 1914, p. 232: y = \\dfrac{1}{2a} \\log_\\epsilon \\dfrac{a+x}{a-x} + C?"],["wentworth-plane-geometry-1899/x-961e532717",15,"Wentworth 1899, scan 243: Thus, the diameter of a circle is the maximum ..."],["thompson-calculus-made-easy-1914/eq-84169f93af",16,"Thompson 1914, p. 236: y = C \\epsilon^{-\\efrac{a}{b} x}"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/1",4,"Hardy 1921, Exercise LXXIII (1)"],["thompson-calculus-made-easy-1914/eq-9d17621a84",16,"Thompson 1914, p. 230: u=\\dfrac{1}{a} \\arctan \\dfrac{u}{a}"],["wentworth-plane-geometry-1899/x-6404f08d5c",15,"Wentworth 1899, scan 109: In the same circle or in equal circles, two ..."],["concept/closed-contour",7,"closed contour","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-closed-contour"],["wentworth-plane-geometry-1899/x-61192f61ef",15,"Wentworth 1899, scan 111: An inscribed angle is measured by half the arc ..."],["wentworth-plane-geometry-1899/x-453bf214bf",15,"Wentworth 1899, scan 113: An angle formed by two chords intersecting within the ..."],["cap/other:improper_integral_convergence",17,"other:improper_integral_convergence"],["method/nested-squares-argument",8,"nested squares argument","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-nested-squares-argument"],["dickson-theory-of-equations-1922/x-a56f1131a1",15,"Dickson 1922, p. 115: If D \\neq 0, the unique values of x_1, ..."],["hardy-course-of-pure-mathematics-1921/x-b595546ace",15,"Hardy 1921, p. 67: Starting from a unit length we can construct any ..."],["hardy-course-of-pure-mathematics-1921/x-0c03675fe2",15,"Hardy 1921, p. 66: Show that if x is a rational function of ..."],["hardy-course-of-pure-mathematics-1921/x-2b2bbd30db",15,"Hardy 1921, p. 400: We shall call this particular value of \\Log \\zeta ..."],["hardy-course-of-pure-mathematics-1921/eq-2625fde525",16,"Hardy 1921, p. 421: \\lim_{n\\to \\infty} \\left(1 + \\frac{z}{n}\\right)^{n} = \\lim_{n\\to \\infty} \\exp\\left\\{n\\log\\left(1 + \\frac{z}{n}\\right)\\ri"],["wentworth-plane-geometry-1899/x-816fd5bcb1",15,"Wentworth 1899, scan 112: An angle inscribed in a semicircle is a right ..."],["dickson-theory-of-equations-1922/x-f32340740c",15,"Dickson 1922, p. 128: In particular, \\Sigma \\alpha = \\alpha + \\beta + ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-2f78e0fd4d",16,"De Morgan 1899, p. 102: y = x^{2}"],["hardy-course-of-pure-mathematics-1921/eq-de3802a6e7",16,"Hardy 1921, p. 65: y = f(x) = (ax + b)/(cx - a)"],["de-morgan-elementary-illustrations-calculus-1899/eq-62914ba88a",16,"De Morgan 1899, p. 102: x = y^{\\efrac{1}{2}}"],["wentworth-first-steps-in-algebra-1894/ex-31",3,"Wentworth 1894, Exercise 31"],["de-morgan-elementary-illustrations-calculus-1899/eq-402a2a290e",16,"De Morgan 1899, p. 103: x = \\psi(\\phi x)"],["de-morgan-elementary-illustrations-calculus-1899/eq-65dc05be67",16,"De Morgan 1899, p. 103: \\dfrac{dy}{dx} = \\phi' x"],["concept/chess-piece-placement-problem",7,"chess-piece placement problem","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-chess-piece-placement-problem"],["concept/kirkman-s-schoolgirl-problem",7,"Kirkman's schoolgirl problem","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-kirkman-s-schoolgirl-problem"],["concept/triplet",7,"triplet","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-triplet"],["wentworth-plane-geometry-1899/ch-iii",2,"Wentworth 1899, ch. III: PROPORTION\\@. SIMILAR POLYGONS","../books/wentworth-plane-geometry-1899/ch/ch-iii/index.html"],["de-morgan-elementary-illustrations-calculus-1899/eq-6edf8a1461",16,"De Morgan 1899, p. 103: \\dfrac{dx}{dy} = \\psi' y"],["de-morgan-elementary-illustrations-calculus-1899/eq-205c1efb5d",16,"De Morgan 1899, p. 104: \\frac{dy}{dx} = \\phi' x = \\frac{1}{\\psi' y} = \\frac{1}{p}"],["todhunter-spherical-trigonometry-1886/ch-numerical-solution-of-spherical-triangles",2,"Todhunter 1886, Numerical Solution of Spherical Triangles","../books/todhunter-spherical-trigonometry-1886/ch/ch-numerical-solution-of-spherical-triangles/index.html"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/2",4,"Hardy 1921, Exercise LXXIII (2)"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/3",4,"Hardy 1921, Exercise LXXIII (3)"],["cap/other:oscillation_behaviour",17,"other:oscillation_behaviour"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/4",4,"Hardy 1921, Exercise LXXIII (4)"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/5",4,"Hardy 1921, Exercise LXXIII (5)"],["wentworth-first-steps-in-algebra-1894/ex-71",3,"Wentworth 1894, Exercise 71"],["wentworth-first-steps-in-algebra-1894/ex-71/1",4,"Wentworth 1894, Exercise 71 (1)"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/6",4,"Hardy 1921, Exercise LXXIII (6)"],["hardy-course-of-pure-mathematics-1921/eq-f0a2799f96",16,"Hardy 1921, p. 65: f(x) = f(-x)"],["maxwell-elementary-treatise-electricity-1888/x-1e2fb6ba36",15,"Maxwell 1888, scan 128: This analogy is so complete that we may make ..."],["maxwell-elementary-treatise-electricity-1888/x-5aab6e6aa7",15,"Maxwell 1888, scan 129: The excess of water in the tube D may ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/7",4,"Hardy 1921, Exercise LXXIII (7)"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/8",4,"Hardy 1921, Exercise LXXIII (8)"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/9",4,"Hardy 1921, Exercise LXXIII (9)"],["maxwell-elementary-treatise-electricity-1888/x-80838f83da",15,"Maxwell 1888, scan 131: The electric strength of a dielectric medium depends on ..."],["maxwell-elementary-treatise-electricity-1888/x-f9974b7304",15,"Maxwell 1888, scan 139: The light of the spark or other discharge is ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/10a",4,"Hardy 1921, Exercise LXXIII (10a)"],["concept/complementary-suffixes",7,"complementary suffixes","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-complementary-suffixes"],["method/frost-s-method",8,"Frost's method","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-frost-s-method"],["person/t-p-kirkman",1,"T. P. Kirkman","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-t-p-kirkman"],["ball-mathematical-recreations-1905/x-c1eee91df8",15,"Ball 1905, scan 136: The first of these is to place eight queens ..."],["thompson-calculus-made-easy-1914/eq-39e7722f6c",16,"Thompson 1914, p. 236: y = y_0 \\epsilon^{-\\efrac{a}{b} x}"],["cap/other:abel_theorem_integral_test",17,"other:abel_theorem_integral_test"],["form/f97e6cb625",5,"solve: Eq(5*x**2 - 2, 3*x**2 + 6)"],["wentworth-first-steps-in-algebra-1894/ex-71/2",4,"Wentworth 1894, Exercise 71 (2)"],["hardy-course-of-pure-mathematics-1921/eq-1642a5ac72",16,"Hardy 1921, p. 65: f(x) = -f(-x)"],["thompson-calculus-made-easy-1914/eq-fab79c6135",16,"Thompson 1914, p. 236: ay + b \\frac{dy}{dx} = g"],["dickson-theory-of-equations-1922/eq-051f7a6f18",16,"Dickson 1922, p. 29: \\left(x - \\frac{a}{2}\\right)^2 + \\left(y - \\frac{b+1}{2}\\right)^2 = \\frac{a^2 + (b-1)^2}{4}"],["ball-mathematical-recreations-1905/x-e33cdf3a22",15,"Ball 1905, scan 139: Let a stand for a_1, or a_2, b for ..."],["ball-mathematical-recreations-1905/x-165be835b9",15,"Ball 1905, scan 139: Thus there are seven possible triads, such as abc, ..."],["dickson-theory-of-equations-1922/eq-c0b1cf9c5b",16,"Dickson 1922, p. 31: y = mx + b"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/10b",4,"Hardy 1921, Exercise LXXIII (10b)"],["form/a218c14825",5,"solve: Eq(3*x**2 + 1, 2*x**2 + 10)"],["hardy-course-of-pure-mathematics-1921/eq-56c058468e",16,"Hardy 1921, p. 421: \\frac{d}{dt} \\{\\log(1 + tz)\\} = \\frac{z}{1 + tz}"],["ball-mathematical-recreations-1905/x-cb308bebfa",15,"Ball 1905, scan 139: The suffixes 1 and 2 are called complementary."],["shape/09b7dc90ea",6,"solve: Eq(N*x**N + 1, N*x**N + N)"],["thompson-calculus-made-easy-1914/eq-82126f0eaa",16,"Thompson 1914, p. 237: y = \\frac{g}{a} + C\\epsilon^{-\\efrac{a}{b}x}"],["hardy-course-of-pure-mathematics-1921/eq-8803562418",16,"Hardy 1921, p. 65: f(x) = \\frac{1}{2}\\{f(x) + f(-x)\\} + \\frac{1}{2}\\{f(x) - f(-x)\\}"],["method/solving-a-spherical-triangle-from-its-three-sides",8,"solving a spherical triangle from its three sides","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-solving-a-spherical-triangle-from-its-three-sides"],["theorem/volumes-of-similar-solids",9,"volumes of similar solids","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volumes-of-similar-solids"],["theorem/area-and-volume-ratios-of-similar-solids",9,"area and volume ratios of similar solids","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-area-and-volume-ratios-of-similar-solids"],["slaught-lennes-solid-geometry-1919/x-adcc3e958f",15,"Slaught & Lennes 1919, p. 172: The fact that the ratio of the areas of ..."],["dickson-theory-of-equations-1922/x-89f475ee0a",15,"Dickson 1922, p. 129: (\\Sigma \\alpha)^2 = \\Sigma \\alpha^2 + 2\\Sigma \\alpha\\beta, whence ..."],["form/a304da9762",5,"identity: (x - 7)*(x + 6)"],["thompson-calculus-made-easy-1914/eq-8b42feb43b",16,"Thompson 1914, p. 238: y = \\frac{g}{a} (1-\\epsilon^{-\\efrac{a}{b} x})"],["ball-mathematical-recreations-1905/x-d25436e792",15,"Ball 1905, scan 137: It has been asserted that, if k = 2, ..."],["hardy-course-of-pure-mathematics-1921/eq-b35c08fed8",16,"Hardy 1921, p. 65: x^{3} + px + q = 0"],["thompson-calculus-made-easy-1914/eq-854385e821",16,"Thompson 1914, p. 238: y_{\\text{max.}} = \\dfrac{g}{a}"],["hardy-course-of-pure-mathematics-1921/eq-2f78e0fd4d",16,"Hardy 1921, p. 65: y = x^{2}"],["hardy-course-of-pure-mathematics-1921/ex-lxxiii/11",4,"Hardy 1921, Exercise LXXIII (11)"],["hardy-course-of-pure-mathematics-1921/eq-9232e96e9c",16,"Hardy 1921, p. 65: x^{2} + y^{2} + (p - 1)y + qx = 0"],["cap/other:series_integral_comparison",17,"other:series_integral_comparison"],["hardy-course-of-pure-mathematics-1921/ex-xxix",3,"Hardy 1921, Exercise XXIX"],["de-morgan-elementary-illustrations-calculus-1899/ch-the-drawing-of-a-tangent-to-a-curve",2,"De Morgan 1899, The Drawing of a Tangent to a Curve","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-the-drawing-of-a-tangent-to-a-curve/index.html"],["hardy-course-of-pure-mathematics-1921/ch-iv",2,"Hardy 1921, ch. IV: LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE","../books/hardy-course-of-pure-mathematics-1921/ch/ch-iv/index.html"],["hardy-course-of-pure-mathematics-1921/eq-0e632d93da",16,"Hardy 1921, p. 65: x^{4} + nx^{3} + px^{2} + qx + r = 0"],["hardy-course-of-pure-mathematics-1921/eq-6533b30b96",16,"Hardy 1921, p. 65: x^{2} = y - \\frac{1}{2}nx"],["de-morgan-elementary-illustrations-calculus-1899/ch-the-method-of-fluxions",2,"De Morgan 1899, The Method of Fluxions","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-the-method-of-fluxions/index.html"],["hardy-course-of-pure-mathematics-1921/eq-dc184fa7c3",16,"Hardy 1921, p. 65: x^{2} + y^{2} + (\\tfrac{1}{8}n^{2} - \\tfrac{1}{2}pn + \\tfrac{1}{2}n + q)x + (p - 1 - \\tfrac{1}{4}n^{2})y + r = 0"],["hardy-course-of-pure-mathematics-1921/eq-378b9fd1a7",16,"Hardy 1921, p. 65: x^{m} + ax^{2} + bx + c = 0"],["hardy-course-of-pure-mathematics-1921/eq-cabd31f31f",16,"Hardy 1921, p. 65: y = x^{m}"],["hardy-course-of-pure-mathematics-1921/ex-lxxiv/1",4,"Hardy 1921, Exercise LXXIV (1)"],["hardy-course-of-pure-mathematics-1921/eq-9eeedd6c6e",16,"Hardy 1921, p. 65: y = -ax^{2} - bx - c"],["hardy-course-of-pure-mathematics-1921/eq-3260c9af14",16,"Hardy 1921, p. 66: 2x = (2n + 1)\\pi(1 - \\cos x)"],["concept/least-upper-bound",7,"least upper bound","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-least-upper-bound"],["hardy-course-of-pure-mathematics-1921/ex-xxx",3,"Hardy 1921, Exercise XXX"],["thompson-calculus-made-easy-1914/eq-0b93ffafcf",16,"Thompson 1914, p. 238: y = y_{\\text{max.}}(1-\\epsilon^{-\\efrac{a}{b} x})"],["thompson-calculus-made-easy-1914/x-db0f6ece47",15,"Thompson 1914, p. 122: If we perform many additions of two or more ..."],["thompson-calculus-made-easy-1914/x-da3f3e33af",15,"Thompson 1914, p. 124: If we make x=1, we get 4 = (A ..."],["hardy-course-of-pure-mathematics-1921/eq-071bea3fcb",16,"Hardy 1921, p. 66: \\frac{2}{3}x\\sin x = 1"],["boyden-first-book-in-algebra-1895/ex-57/6",4,"Boyden 1895, Exercise 57 (6)"],["hardy-course-of-pure-mathematics-1921/eq-a58d711d19",16,"Hardy 1921, p. 66: \\cot x + x - \\frac{3}{2}\\pi = 0"],["hardy-course-of-pure-mathematics-1921/ex-lxxiv/2",4,"Hardy 1921, Exercise LXXIV (2)"],["form/6807cd45df",5,"solve: Eq(187, x**2 + 6*x)"],["shape/eba5413da8",6,"solve: Eq(N, N*x + x**N)"],["shape/ef2e9f4c62",6,"factor: N*x + N*x**N"],["thompson-calculus-made-easy-1914/x-1f18d6359c",15,"Thompson 1914, p. 126: Since the given fraction and the fraction found by ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxiv/3",4,"Hardy 1921, Exercise LXXIV (3)"],["thompson-calculus-made-easy-1914/eq-7a9b9d76b2",16,"Thompson 1914, p. 241: \\tan \\phi = \\dfrac{2 \\pi n b}{ a}"],["hardy-course-of-pure-mathematics-1921/eq-4c7228ac31",16,"Hardy 1921, p. 66: x^{2} + \\sin^{2} x = 1"],["wentworth-first-steps-in-algebra-1894/ex-33/3",4,"Wentworth 1894, Exercise 33 (3)"],["cap/other:integral_test",17,"other:integral_test"],["hardy-course-of-pure-mathematics-1921/ex-lxxiv/4",4,"Hardy 1921, Exercise LXXIV (4)"],["form/c21cc91f7e",5,"factor: -21*x**3 + 7*x**2 + 14*x"],["hardy-course-of-pure-mathematics-1921/eq-582b07c65b",16,"Hardy 1921, p. 66: \\tan x = 2x/(1 + x^{2})"],["form/c0ce56512e",5,"factor: -10*a**2*x**2 + 15*a*x**2 + 5"],["shape/0a0b1ddca6",6,"factor: N*a*x**N + N*a**N*x**N + N"],["wentworth-first-steps-in-algebra-1894/ex-31/9",4,"Wentworth 1894, Exercise 31 (9)"],["hardy-course-of-pure-mathematics-1921/eq-82f488c8fd",16,"Hardy 1921, p. 66: \\sin x - x + \\frac{1}{6}x^{3} = 0"],["thompson-calculus-made-easy-1914/eq-794581565a",16,"Thompson 1914, p. 241: \\sin \\phi = \\frac{2 \\pi nb}{\\sqrt{a^2 + 4 \\pi^2 n^2 b^2}}"],["form/e5e15d3cae",5,"integrate: 1/(sqrt(x)*(x + 1))"],["hardy-course-of-pure-mathematics-1921/eq-1050dbc1f4",16,"Hardy 1921, p. 66: (1 - \\cos x)\\tan\\alpha - x + \\sin x = 0"],["theorem/sign-persistence-of-a-continuous-function",9,"sign persistence of a continuous function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-sign-persistence-of-a-continuous-function"],["form/66bd0339f1",5,"factor: 9*a**2 - x**2"],["shape/452b203469",6,"integrate: x**N/(x + 1)"],["boyden-first-book-in-algebra-1895/ex-21/20b",4,"Boyden 1895, Exercise 21 (20b)"],["boyden-first-book-in-algebra-1895/ex-28/12",4,"Boyden 1895, Exercise 28 (12)"],["hardy-course-of-pure-mathematics-1921/ex-lxxiv/5",4,"Hardy 1921, Exercise LXXIV (5)"],["form/e7a4328639",5,"integrate: sqrt(x)/(x + 1)**2"],["wentworth-first-steps-in-algebra-1894/ex-32",3,"Wentworth 1894, Exercise 32"],["hardy-course-of-pure-mathematics-1921/eq-0918f60e32",16,"Hardy 1921, p. 422: (\\psi' + i\\chi') \\exp(\\psi + i\\chi) = \\phi' \\exp\\phi"],["hardy-course-of-pure-mathematics-1921/eq-0d08286ed7",16,"Hardy 1921, p. 66: \\alpha\\frac{(x - b)(x - c)}{(a - b)(a - c)} + \\beta \\frac{(x - c)(x - a)}{(b - c)(b - a)} + \\gamma\\frac{(x - a)(x - b)}{"],["thompson-calculus-made-easy-1914/eq-b454b9e594",16,"Thompson 1914, p. 241: \\cos \\phi = \\frac{a}{\\sqrt{a^2 + 4 \\pi^2 n^2 b^2}}"],["thompson-calculus-made-easy-1914/eq-d92c6b1133",16,"Thompson 1914, p. 241: y = g \\frac{\\sin(2 \\pi nt - \\phi)}{\\sqrt{a^2 + 4 \\pi^2 n^2 b^2}}"],["thompson-calculus-made-easy-1914/eq-bec276802f",16,"Thompson 1914, p. 239: \\int u dv = uv - \\int v du"],["thompson-calculus-made-easy-1914/eq-cba570ff5a",16,"Thompson 1914, p. 242: \\frac{dM}{dy} = \\frac{dN}{dx}"],["shape/c1efb540fb",6,"integrate: x**N*(x + 1)**N"],["hardy-course-of-pure-mathematics-1921/eq-e51d8f7636",16,"Hardy 1921, p. 66: Axy + Bx + Cy + D = 0"],["hardy-course-of-pure-mathematics-1921/eq-db2303e548",16,"Hardy 1921, p. 66: \\cos\\tfrac{1}{2}\\pi x = 1 - \\frac{x^{2}}{x + (x - 1)\\bigsqrtp{\\dfrac{2 - x}{3}}}"],["wentworth-plane-geometry-1899/ch-iv",2,"Wentworth 1899, ch. IV: AREAS OF POLYGONS","../books/wentworth-plane-geometry-1899/ch/ch-iv/index.html"],["hardy-course-of-pure-mathematics-1921/ex-lxxiv/6",4,"Hardy 1921, Exercise LXXIV (6)"],["wentworth-plane-geometry-1899/ex-iv-1",3,"Wentworth 1899, Exercise IV.1"],["thompson-calculus-made-easy-1914/eq-d0800304b6",16,"Thompson 1914, p. 242: \\frac{\\partial U}{\\partial x} = M"],["thompson-calculus-made-easy-1914/eq-29b73a251f",16,"Thompson 1914, p. 242: \\frac{\\partial U}{\\partial y} = N"],["thompson-calculus-made-easy-1914/eq-a064e51410",16,"Thompson 1914, p. 243: w = 2x^3y"],["todhunter-spherical-trigonometry-1886/x-3e33e11a3c",15,"Todhunter 1886, scan 97: If the three angles of a plane triangle be ..."],["todhunter-spherical-trigonometry-1886/x-4c5122419c",15,"Todhunter 1886, scan 99: Now in modern observations h will not exceed the ..."],["thompson-calculus-made-easy-1914/eq-a49f4cfb28",16,"Thompson 1914, p. 243: U = x^2 + 2x^3y + C"],["wentworth-plane-geometry-1899/x-7e930eced8",15,"Wentworth 1899, scan 245: Hence, every vertex lies on the circumference; that is, ..."],["cap/other:limit_argument",17,"other:limit_argument"],["hardy-course-of-pure-mathematics-1921/eq-113e30761d",16,"Hardy 1921, p. 66: z = [x] + [y]"],["shape/6955a60e01",6,"factor: N*a**N - x**N"],["form/dd6dda8a45",5,"factor: -x**2 + 25"],["hardy-course-of-pure-mathematics-1921/ex-lxxix",3,"Hardy 1921, Exercise LXXIX"],["wentworth-first-steps-in-algebra-1894/ex-33",3,"Wentworth 1894, Exercise 33"],["thompson-calculus-made-easy-1914/eq-e0c86d815f",16,"Thompson 1914, p. 244: \\left(\\frac{dy}{dt}\\right)^2 + n^2 (y^2-C^2) = 0"],["thompson-calculus-made-easy-1914/eq-c403af7a35",16,"Thompson 1914, p. 244: \\frac{1}{\\sqrt{C^2 - y^2}} = \\frac{d (\\arcsin \\dfrac{y}{C})}{dy}"],["wentworth-first-steps-in-algebra-1894/ex-33/1",4,"Wentworth 1894, Exercise 33 (1)"],["thompson-calculus-made-easy-1914/eq-e69472fe2b",16,"Thompson 1914, p. 245: y = A \\sin nt + B \\cos nt"],["hardy-course-of-pure-mathematics-1921/eq-37ca9f0534",16,"Hardy 1921, p. 66: z = x + y - [x] - [y]"],["hardy-course-of-pure-mathematics-1921/ex-lxxix/1",4,"Hardy 1921, Exercise LXXIX (1)"],["form/750de27778",5,"factor: -x**2 + 4"],["shape/7f285668fd",6,"factor: N - x**N"],["form/259d144150",5,"factor: -x**2 + 9"],["concept/literal-equation",7,"literal equation","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-literal-equation"],["concept/rule",7,"rule","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-rule"],["concept/combined-rate",7,"combined rate","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-concept-combined-rate"],["hardy-course-of-pure-mathematics-1921/eq-84976b1937",16,"Hardy 1921, p. 66: z = \\sin x + \\sin y"],["thompson-calculus-made-easy-1914/eq-3df345ca31",16,"Thompson 1914, p. 244: y = C \\sin (nt + C_1)"],["hardy-course-of-pure-mathematics-1921/eq-a6f558ff3d",16,"Hardy 1921, p. 66: z = \\sin x\\sin y"],["hardy-course-of-pure-mathematics-1921/x-2d11f38873",15,"Hardy 1921, p. 435: Hence, if \\am Z is changed when z describes ..."],["wentworth-first-steps-in-algebra-1894/x-1a53d6ab7a",15,"Wentworth 1894, p. 103: If a fraction is preceded by a minus sign, ..."],["wentworth-first-steps-in-algebra-1894/x-31b78d1183",15,"Wentworth 1894, p. 103: Multiply each term by the L. C. M. of ..."],["wentworth-first-steps-in-algebra-1894/x-536d994939",15,"Wentworth 1894, p. 107: If the denominators contain both simple and compound expressions, ..."],["wentworth-first-steps-in-algebra-1894/x-3b627f7889",15,"Wentworth 1894, p. 108: Literal equations are equations in which some or all ..."],["wentworth-first-steps-in-algebra-1894/x-6892527ec1",15,"Wentworth 1894, p. 112: By working 2 days each they build \\frac{1}{12} + ..."],["wentworth-first-steps-in-algebra-1894/x-04362a925d",15,"Wentworth 1894, p. 118: Such an expression is called a formula, and the ..."],["wentworth-first-steps-in-algebra-1894/x-a9e5a04d3b",15,"Wentworth 1894, p. 115: A hare takes 4 leaps to a greyhound’s 3; ..."],["hardy-course-of-pure-mathematics-1921/eq-271eb4c1ae",16,"Hardy 1921, p. 66: z = \\sin xy"],["concept/ratio-of-similitude",7,"ratio of similitude","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-ratio-of-similitude"],["concept/corresponding-parts",7,"corresponding parts","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-corresponding-parts"],["thompson-calculus-made-easy-1914/eq-17748598d4",16,"Thompson 1914, p. 245: \\frac{dw}{dy} = \\frac{1}{\\sqrt{ y^2 + c^2}}"],["instrument/pantograph",13,"pantograph","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-instrument-pantograph"],["theorem/ratios-of-similar-cylinders",9,"ratios of similar cylinders","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-ratios-of-similar-cylinders"],["theorem/ratios-of-similar-cones",9,"ratios of similar cones","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-ratios-of-similar-cones"],["theorem/two-tetrahedrons-are-similar-if-three-faces-are-similar-and-similarly-placed",9,"two tetrahedrons are similar if three faces are similar and similarly placed","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-two-tetrahedrons-are-similar-if-three-faces-are-similar-and-similarly-placed"],["theorem/volumes-of-tetrahedrons-with-equal-trihedral-angles",9,"volumes of tetrahedrons with equal trihedral angles","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volumes-of-tetrahedrons-with-equal-trihedral-angles"],["form/1b40c7b254",5,"factor: 81*a**2*x**2 - 1"],["shape/935a384f2d",6,"factor: N*a**N*x**N - 1"],["slaught-lennes-solid-geometry-1919/x-f80a05cef5",15,"Slaught & Lennes 1919, p. 165: Any two figures which have a center of similitude ..."],["slaught-lennes-solid-geometry-1919/x-6b3219a64c",15,"Slaught & Lennes 1919, p. 170: This proposition may be rendered evident by noticing that ..."],["slaught-lennes-solid-geometry-1919/x-6f4a53fc18",15,"Slaught & Lennes 1919, p. 171: The essential property of all such contrivances is that ..."],["slaught-lennes-solid-geometry-1919/x-3d5b568d5a",15,"Slaught & Lennes 1919, p. 170: Note that the ratio of similitude of two similar ..."],["slaught-lennes-solid-geometry-1919/x-72a4d0b1b5",15,"Slaught & Lennes 1919, p. 172: Thus the ratio of the weights of two similar ..."],["hardy-course-of-pure-mathematics-1921/eq-2f45892ce9",16,"Hardy 1921, p. 66: z = \\sin(x^{2} + y^{2})"],["dickson-theory-of-equations-1922/eq-6fac5553c3",16,"Dickson 1922, p. 31: x = \\frac{b' - b}{m - m'}"],["thompson-calculus-made-easy-1914/eq-ce738cf544",16,"Thompson 1914, p. 245: y + \\sqrt{y^2 + c^2} = C \\epsilon^{nx}"],["boyden-first-book-in-algebra-1895/ex-22/2",4,"Boyden 1895, Exercise 22 (2)"],["theorem/quadratic-formula-setting",9,"quadratic formula setting"],["hardy-course-of-pure-mathematics-1921/ex-lxxix/2",4,"Hardy 1921, Exercise LXXIX (2)"],["thompson-calculus-made-easy-1914/eq-1cf4c92cec",16,"Thompson 1914, p. 245: -y + \\sqrt{y^2 + c^2} = \\dfrac{c^2}{C} \\epsilon^{-nx}"],["thompson-calculus-made-easy-1914/eq-092c804380",16,"Thompson 1914, p. 246: y = \\frac{1}{2} C \\epsilon^{nx} - \\frac{1}{2}\\, \\frac{c^2}{C} \\epsilon^{-nx}"],["form/8639dc2974",5,"identity: (a + b)**4"],["hardy-course-of-pure-mathematics-1921/ex-lxxix/3",4,"Hardy 1921, Exercise LXXIX (3)"],["form/729572cab6",5,"factor: 49*a**2*x**2 - 4"],["thompson-calculus-made-easy-1914/eq-44a0772ea2",16,"Thompson 1914, p. 246: y = A \\epsilon^{nx} + B \\epsilon^{-nx}"],["thompson-calculus-made-easy-1914/eq-bda730c0db",16,"Thompson 1914, p. 246: b \\frac{d^2y}{dt^2} + a \\frac{dy}{dt} + gy = 0"],["hardy-course-of-pure-mathematics-1921/ex-lxxix/4",4,"Hardy 1921, Exercise LXXIX (4)"],["thompson-calculus-made-easy-1914/eq-7260ac1080",16,"Thompson 1914, p. 135: y + n\\dfrac{y}{n} = 2y."],["thompson-calculus-made-easy-1914/eq-fcc5d9aabb",16,"Thompson 1914, p. 246: m = \\frac{a}{2b}"],["thompson-calculus-made-easy-1914/eq-c15187bd6d",16,"Thompson 1914, p. 136: y_n = y_0\\left(1 + \\frac{1}{n}\\right)^n."],["thompson-calculus-made-easy-1914/eq-592273159a",16,"Thompson 1914, p. 137: y_n = £100 \\left( 1 + \\tfrac{1}{100} \\right)^{100};"],["wentworth-first-steps-in-algebra-1894/ex-1/13",4,"Wentworth 1894, Exercise 1 (13)"],["wentworth-first-steps-in-algebra-1894/ex-1/15",4,"Wentworth 1894, Exercise 1 (15)"],["form/2bb251b029",5,"identity: 4"],["form/ca3821e75d",5,"factor: 9*a**6*x**8 - 16*b**10"],["hardy-course-of-pure-mathematics-1921/eq-5474628d33",16,"Hardy 1921, p. 67: (x - \\alpha)^{2} + (y - \\beta)^{2} = \\rho ^{2}"],["hardy-course-of-pure-mathematics-1921/eq-df5488c548",16,"Hardy 1921, p. 67: x^{2} + y^{2} + 2gx + 2fy + c = 0"],["hardy-course-of-pure-mathematics-1921/eq-eba7197e71",16,"Hardy 1921, p. 67: x^{2} - 34x + 190 = 0"],["hardy-course-of-pure-mathematics-1921/ex-lxxix/5",4,"Hardy 1921, Exercise LXXIX (5)"],["thompson-calculus-made-easy-1914/eq-3fafd9025a",16,"Thompson 1914, p. 137: y_n = £100 \\left( 1 + \\tfrac{1}{1000} \\right)^{1000};"],["thompson-calculus-made-easy-1914/eq-cf563fbd9f",16,"Thompson 1914, p. 137: y_n = £100 \\left( 1 + \\tfrac{1}{10,000} \\right)^{10,000};"],["concept/magnetic-action-of-an-electric-current",7,"magnetic action of an electric current","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-magnetic-action-of-an-electric-current"],["boyden-first-book-in-algebra-1895/ex-22/3",4,"Boyden 1895, Exercise 22 (3)"],["hardy-course-of-pure-mathematics-1921/ex-lxxix/6",4,"Hardy 1921, Exercise LXXIX (6)"],["thompson-calculus-made-easy-1914/eq-0cdd5fd386",16,"Thompson 1914, p. 246: n = \\sqrt{\\frac{a^2}{4b^2}} - \\frac{g}{b}"],["hardy-course-of-pure-mathematics-1921/eq-acd8c8e842",16,"Hardy 1921, p. 68: AM/R = \\tfrac{13}{25}\\sqrt{146}"],["maxwell-elementary-treatise-electricity-1888/x-ab0c29f4b6",15,"Maxwell 1888, scan 140: But there are methods by which the difference of ..."],["thompson-calculus-made-easy-1914/eq-1b70be9b9d",16,"Thompson 1914, p. 246: y = (\\epsilon^{-mt})(A \\epsilon^{nt} + B \\epsilon^{-nt})"],["de-morgan-elementary-illustrations-calculus-1899/eq-088f9597aa",16,"De Morgan 1899, p. 105: u = \\frac{1}{p}"],["hardy-course-of-pure-mathematics-1921/ex-lxxv/1",4,"Hardy 1921, Exercise LXXV (1)"],["maxwell-elementary-treatise-electricity-1888/x-3d25fe9cde",15,"Maxwell 1888, scan 142: Thus we see that the electric current has a ..."],["thompson-calculus-made-easy-1914/eq-f472af2c45",16,"Thompson 1914, p. 247: \\frac{d^2y}{dt^2} = a^2 \\frac{d^2y}{dx^2}"],["hardy-course-of-pure-mathematics-1921/eq-6211e961ac",16,"Hardy 1921, p. 68: y^{2} = 4x"],["hardy-course-of-pure-mathematics-1921/eq-2470c718f3",16,"Hardy 1921, p. 68: x^{2} = 2y"],["de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratio-of-magnitudes-that-vanish-together",2,"De Morgan 1899, On the Ratio of Magnitudes that Vanish Together","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-on-the-ratio-of-magnitudes-that-vanish-together/index.html"],["hardy-course-of-pure-mathematics-1921/ex-lxxv/2",4,"Hardy 1921, Exercise LXXV (2)"],["law/kirchhoff-s-laws",10,"Kirchhoff's laws","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-law-kirchhoff-s-laws"],["hardy-course-of-pure-mathematics-1921/eq-ac6c1fd12c",16,"Hardy 1921, p. 68: SQ = \\sqrt[3]{2}"],["thompson-calculus-made-easy-1914/eq-0ba207cf50",16,"Thompson 1914, p. 238: ay+b\\frac{dy}{dt} = g · \\sin 2\\pi nt"],["thompson-calculus-made-easy-1914/eq-e1d1db5dde",16,"Thompson 1914, p. 241: y = g \\left\\{\\frac{ a · \\sin 2 \\pi n t - 2 \\pi n b · \\cos 2 \\pi nt}{ a^2 + 4 \\pi^2 n^2 b^2}\\right\\}"],["concept/magnitudes-vanishing-together",7,"magnitudes vanishing together","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-magnitudes-vanishing-together"],["hardy-course-of-pure-mathematics-1921/ex-lxxv/3",4,"Hardy 1921, Exercise LXXV (3)"],["hardy-course-of-pure-mathematics-1921/ex-lxxv/4",4,"Hardy 1921, Exercise LXXV (4)"],["wentworth-first-steps-in-algebra-1894/ex-34",3,"Wentworth 1894, Exercise 34"],["hardy-course-of-pure-mathematics-1921/eq-baf19c7993",16,"Hardy 1921, p. 68: (x^{2} + y^{2})x - y^{2} = 0"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/1",4,"Hardy 1921, Exercise LXXVI (1)"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/2",4,"Hardy 1921, Exercise LXXVI (2)"],["hardy-course-of-pure-mathematics-1921/eq-7c307f868c",16,"Hardy 1921, p. 68: AQ = \\sqrt[3]{2}"],["shape/3d1ae48bd4",6,"factor: c**N - (a + b)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/4",4,"Wentworth 1894, Exercise 34 (4)"],["form/23e0905832",5,"factor: c**2 - (a - b)**2"],["shape/14db7858b3",6,"factor: c**N - (a - b)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/5",4,"Wentworth 1894, Exercise 34 (5)"],["form/18a7a09565",5,"factor: -4*c**2 + (a + b)**2"],["shape/f7a5d7daee",6,"factor: N*c**N + (a + b)**N"],["thompson-calculus-made-easy-1914/eq-7387993e71",16,"Thompson 1914, p. 247: y = F(x+at) + f(x-at)"],["quantity/magnitude",11,"magnitude","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-quantity-magnitude"],["whitehead-introduction-to-mathematics-1911/x-da44a725ea",15,"Whitehead 1911, p. 46: But according to the Newtonian law, apart from some ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/3",4,"Hardy 1921, Exercise LXXVI (3)"],["thompson-calculus-made-easy-1914/eq-6376bae709",16,"Thompson 1914, p. 248: a = \\sqrt{\\frac{k}{m}}"],["maxwell-elementary-treatise-electricity-1888/x-7736342d61",15,"Maxwell 1888, scan 195: In the electromagnetic system a resistance is a quantity ..."],["ball-mathematical-recreations-1905/eq-5bf540796b",16,"Ball 1905, scan 125: 2^{64}-1"],["ball-mathematical-recreations-1905/eq-78c46b5717",16,"Ball 1905, scan 125: 2^n-1"],["ball-mathematical-recreations-1905/eq-c0b66baf82",16,"Ball 1905, scan 124: 2x +1"],["ball-mathematical-recreations-1905/eq-fbc382aaa8",16,"Ball 1905, scan 127: \\frac{1}{3}(2^{n+1}-1)"],["todhunter-spherical-trigonometry-1886/x-7f35ebec37",15,"Todhunter 1886, scan 103: It is sometimes important to know what amount of ..."],["todhunter-spherical-trigonometry-1886/x-ab9a5994ca",15,"Todhunter 1886, scan 103: Suppose C and c to remain constant."],["ball-mathematical-recreations-1905/eq-de4fad9a9b",16,"Ball 1905, scan 127: \\frac{1}{3}(2^{n+1}-2)"],["boyden-first-book-in-algebra-1895/ex-22/4",4,"Boyden 1895, Exercise 22 (4)"],["form/9b6ccc5228",5,"identity: (a - b)**3"],["shape/e82dc0e1a8",6,"identity: (a - b)**N"],["boyden-first-book-in-algebra-1895/ex-22/5",4,"Boyden 1895, Exercise 22 (5)"],["form/af66fb64af",5,"identity: (a + b)**2"],["dickson-theory-of-equations-1922/eq-7ec99d9584",16,"Dickson 1922, p. 31: (x - c)^2 + (y - d)^2 = r^2"],["thompson-calculus-made-easy-1914/eq-baedea5c2f",16,"Thompson 1914, p. 248: m \\frac{d^2y}{dt^2} = k\\, \\frac{d^2y}{dx^2}"],["ball-mathematical-recreations-1905/eq-3df447e0eb",16,"Ball 1905, scan 128: \\tfrac{1}{3}(2^{2n+2} - 1)"],["ball-mathematical-recreations-1905/eq-9ad40513e9",16,"Ball 1905, scan 128: \\tfrac{1}{3}(2^{2n+1}-2)"],["dickson-theory-of-equations-1922/eq-eb38515661",16,"Dickson 1922, p. 31: y = \\frac{mb' - m'b}{m - m'}"],["ball-mathematical-recreations-1905",0,"Ball, Mathematical Recreations and Essays (1905)","../books/ball-mathematical-recreations-1905/index.html"],["concept/type-of-order",7,"type of order","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-type-of-order"],["dickson-theory-of-equations-1922/eq-fcfbab8685",16,"Dickson 1922, p. 30: \\tfrac{1}{2}(a ± \\sqrt{a^2 -4b)}"],["dickson-theory-of-equations-1922/eq-1042fbdf3f",16,"Dickson 1922, p. 32: s = \\tfrac{1}{2}\\sqrt{10 - 2\\sqrt{5}}"],["ball-mathematical-recreations-1905/ch-viii",2,"Ball 1905, ch. VIII: Three Geometrical Problems","../books/ball-mathematical-recreations-1905/ch/ch-viii/index.html"],["concept/principal-value-of-logarithm",7,"principal value of logarithm"],["concept/ratio-decreasing-without-limit",7,"ratio decreasing without limit","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-ratio-decreasing-without-limit"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/4a",4,"Hardy 1921, Exercise LXXVI (4a)"],["whitehead-introduction-to-mathematics-1911/x-31c243ad28",15,"Whitehead 1911, p. 43: Galileo affirmed that they would fall in the same ..."],["whitehead-introduction-to-mathematics-1911/x-8ae0b01c36",15,"Whitehead 1911, p. 194: The general mathematical idea of a series is that ..."],["whitehead-introduction-to-mathematics-1911",0,"Whitehead, An Introduction to Mathematics (1911)","../books/whitehead-introduction-to-mathematics-1911/index.html"],["maxwell-elementary-treatise-electricity-1888/x-ebcf565c74",15,"Maxwell 1888, scan 202: The merit of the method consists in the fact ..."],["whitehead-introduction-to-mathematics-1911/ch-xiii",2,"Whitehead 1911, ch. XIII: Trigonometry","../books/whitehead-introduction-to-mathematics-1911/ch/ch-xiii/index.html"],["whitehead-introduction-to-mathematics-1911/x-faba375778",15,"Whitehead 1911, p. 197: But, if the series has an infinite number of ..."],["whitehead-introduction-to-mathematics-1911/x-d3a408d718",15,"Whitehead 1911, p. 195: When the number of things considered is finite, the ..."],["whitehead-introduction-to-mathematics-1911/x-981a1254c4",15,"Whitehead 1911, p. 202: This decimal is merely a way of symbolizing the ..."],["ball-mathematical-recreations-1905/eq-060de4ea59",16,"Ball 1905, scan 128: 2^{2n}"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/4b",4,"Hardy 1921, Exercise LXXVI (4b)"],["ball-mathematical-recreations-1905/eq-0f2d319ac4",16,"Ball 1905, scan 128: 2^{2n-1}-1"],["wentworth-plane-geometry-1899/eq-f2b1d13523",16,"Wentworth 1899, scan 101: \\dfrac{a}{b}"],["boyden-first-book-in-algebra-1895/ex-22/6",4,"Boyden 1895, Exercise 22 (6)"],["form/f784b96214",5,"identity: (a - b)**2"],["person/hipparchus",1,"Hipparchus","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-hipparchus"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/4c",4,"Hardy 1921, Exercise LXXVI (4c)"],["concept/reciprocal",7,"reciprocal","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-reciprocal"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/4d",4,"Hardy 1921, Exercise LXXVI (4d)"],["ball-mathematical-recreations-1905/eq-ac74f8ab5b",16,"Ball 1905, scan 128: 4x"],["planck-treatise-on-thermodynamics-1903/x-ebd4106d72",15,"Planck 1903, p. 113: This leads to the proposition that chemical reactions, in ..."],["concept/recursion",7,"recursion"],["ball-mathematical-recreations-1905/eq-b6431073bf",16,"Ball 1905, scan 128: 1 + 4 + 4^2 + \\dotsb + 4^n"],["thompson-calculus-made-easy-1914/x-aa0c2a83de",15,"Thompson 1914, p. 145: Another reason why \\epsilon is important is because it ..."],["ball-mathematical-recreations-1905/eq-47454ee531",16,"Ball 1905, scan 130: (1+2^1 + 2^2 + \\ldots + 2^{2n})"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/4e",4,"Hardy 1921, Exercise LXXVI (4e)"],["planck-treatise-on-thermodynamics-1903/x-7a946b39ce",15,"Planck 1903, p. 117: Among all the states of the system which can ..."],["ball-mathematical-recreations-1905/eq-46183fb45e",16,"Ball 1905, scan 130: (2 + 2^3 + \\ldots + 2^{2n-1})"],["theorem/arctangent-series",9,"arctangent series","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-arctangent-series"],["person/max-thiesen",1,"Max Thiesen","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-max-thiesen"],["ball-mathematical-recreations-1905/eq-f8df9b3cef",16,"Ball 1905, scan 123: mn-3"],["slaught-lennes-solid-geometry-1919/x-456a72a179",15,"Slaught & Lennes 1919, p. 179: The length of the projection of a line-segment upon ..."],["slaught-lennes-solid-geometry-1919/x-daf88d8f44",15,"Slaught & Lennes 1919, p. 176: Likewise we define the sine of an acute angle ..."],["slaught-lennes-solid-geometry-1919/x-0b217327b2",15,"Slaught & Lennes 1919, p. 183: The area of the projection of a plane-segment on ..."],["slaught-lennes-solid-geometry-1919/x-89b21a1191",15,"Slaught & Lennes 1919, p. 181: The altitude of an oblique prism or cylinder is ..."],["slaught-lennes-solid-geometry-1919/x-03d29ae0d6",15,"Slaught & Lennes 1919, p. 185: Note that when a and b are equal, the ..."],["wentworth-plane-geometry-1899/x-09627a50d3",15,"Wentworth 1899, scan 193: The area of a surface is the number of ..."],["slaught-lennes-solid-geometry-1919/x-8c87fb6de6",15,"Slaught & Lennes 1919, p. 186: The comparatively low temperature of the earth’s surface near ..."],["slaught-lennes-solid-geometry-1919/x-239be3bc6b",15,"Slaught & Lennes 1919, p. 179: We assume that it lies halfway between these numbers. ..."],["maxwell-elementary-treatise-electricity-1888/x-4fda1f8c87",15,"Maxwell 1888, scan 204: The conductors BC and OA are then said to ..."],["maxwell-elementary-treatise-electricity-1888/x-fd3639a2ff",15,"Maxwell 1888, scan 199: This method, founded on the binary scale, is that ..."],["maxwell-elementary-treatise-electricity-1888/x-5e90638488",15,"Maxwell 1888, scan 195: It is sometimes referred to as the B.A. unit, ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/4f",4,"Hardy 1921, Exercise LXXVI (4f)"],["wentworth-first-steps-in-algebra-1894/ex-36",3,"Wentworth 1894, Exercise 36"],["theorem/infinitesimal-arc-coincides-with-its-chord",9,"infinitesimal arc coincides with its chord","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-infinitesimal-arc-coincides-with-its-chord"],["planck-treatise-on-thermodynamics-1903/x-715292d9b8",15,"Planck 1903, p. 110: has been called by H. v. Helmholtz the free ..."],["ball-mathematical-recreations-1905/eq-51aeea8ffa",16,"Ball 1905, scan 121: n-1"],["theorem/machin-s-formula-for-pi",9,"Machin's formula for pi","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-machin-s-formula-for-pi"],["hardy-course-of-pure-mathematics-1921/x-8c1e20e2f5",15,"Hardy 1921, p. 402: The equation \\Log z^{m} = m\\Log z, where m ..."],["hardy-course-of-pure-mathematics-1921/x-6c705a6dbf",15,"Hardy 1921, p. 395: The definition, although perfectly legitimate, is futile because it ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/4g",4,"Hardy 1921, Exercise LXXVI (4g)"],["todhunter-spherical-trigonometry-1886",0,"Todhunter, Spherical Trigonometry, for the Use of Colleges and Schools (1886)","../books/todhunter-spherical-trigonometry-1886/index.html"],["law/second-law-of-motion",10,"second law of motion","../books/ball-mathematical-recreations-1905/terms/index.html#t-law-second-law-of-motion"],["concept/scale-of-a-map",7,"scale of a map","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-scale-of-a-map"],["law/third-law-of-motion",10,"third law of motion","../books/ball-mathematical-recreations-1905/terms/index.html#t-law-third-law-of-motion"],["concept/inertia",7,"inertia","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-inertia"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/5",4,"Hardy 1921, Exercise LXXVI (5)"],["shape/14586c0cb7",6,"factor: x**N + 1"],["shape/c618474772",6,"identity: N*a**N*x"],["de-morgan-elementary-illustrations-calculus-1899/x-2f0539dd99",15,"De Morgan 1899, p. 39: All which must be interpreted to mean that, the ..."],["form/92fee1cf3f",5,"factor: 4*x**2 - 4*x + 1"],["de-morgan-elementary-illustrations-calculus-1899/ch-illustration-of-the-rules-for-differentiation",2,"De Morgan 1899, Illustration of the Rules for Differentiation","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-illustration-of-the-rules-for-differentiation/index.html"],["form/d7fd4d2160",5,"identity: 12*a**4*x"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/6",4,"Hardy 1921, Exercise LXXVI (6)"],["hardy-course-of-pure-mathematics-1921/x-e0bb7c160b",15,"Hardy 1921, p. 406: This is a further generalisation of De Moivre’s Theorem"],["quantity/principal",11,"principal","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-quantity-principal"],["quantity/interest",11,"interest","../books/thompson-calculus-made-easy-1914/terms/index.html#t-quantity-interest"],["quantity/amount",11,"amount","../books/wentworth-first-steps-in-algebra-1894/terms/index.html#t-quantity-amount"],["wentworth-plane-geometry-1899/x-337ff1d460",15,"Wentworth 1899, scan 193: Two rectangles having equal altitudes are to each other ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/7",4,"Hardy 1921, Exercise LXXVI (7)"],["theorem/sum-of-the-first-x-natural-numbers",9,"sum of the first x natural numbers","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-sum-of-the-first-x-natural-numbers"],["shape/b8a110b96b",6,"identity: N*a*b*x"],["wentworth-first-steps-in-algebra-1894/ex-17/26",4,"Wentworth 1894, Exercise 17 (26)"],["form/6ffb93c824",5,"identity: 1/(a*b*x)"],["wentworth-plane-geometry-1899/eq-3bad41cdf3",16,"Wentworth 1899, scan 104: \\frac{3}{10} + \\frac{3}{100} + \\frac{3}{1000} + \\cdots"],["wentworth-first-steps-in-algebra-1894/x-659de2c6fc",15,"Wentworth 1894, p. 122: Independent equations involving the same unknown numbers are called ..."],["wentworth-first-steps-in-algebra-1894/x-9e99d9ed7d",15,"Wentworth 1894, p. 122: If we have two unknown numbers and but one ..."],["wentworth-first-steps-in-algebra-1894/x-8d45059f42",15,"Wentworth 1894, p. 122: If we have two unknown numbers, and two independent ..."],["wentworth-first-steps-in-algebra-1894/x-3a98fc05f3",15,"Wentworth 1894, p. 124: Multiply the equations by such numbers as will make ..."],["wentworth-first-steps-in-algebra-1894/x-331d88883d",15,"Wentworth 1894, p. 124: It is generally best to select the letter to ..."],["wentworth-first-steps-in-algebra-1894/x-8b2d75a1be",15,"Wentworth 1894, p. 129: The expression 64 means 60 + 4, that is, ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/8",4,"Hardy 1921, Exercise LXXVI (8)"],["maxwell-elementary-treatise-electricity-1888/x-1005974b5c",15,"Maxwell 1888, scan 149: Any ordinate such as 0°P, 1°Q, &c., is called ..."],["maxwell-elementary-treatise-electricity-1888/x-3d54dc9631",15,"Maxwell 1888, scan 148: The electromotive force from one point of a conductor ..."],["concept/thermal-effect-of-an-electric-current",7,"thermal effect of an electric current","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-thermal-effect-of-an-electric-current"],["theorem/composition-and-division-of-proportions",9,"composition and division of proportions","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-composition-and-division-of-proportions"],["wentworth-plane-geometry-1899/x-6e3c52d657",15,"Wentworth 1899, scan 144: A proportion is an expression of equality between two ..."],["planck-treatise-on-thermodynamics-1903/x-16659e19ff",15,"Planck 1903, p. 132: It is still very uncertain whether the molecules of ..."],["form/243ebebc85",5,"identity: 2*a**2 + 2*b**2"],["shape/b1f4cd2669",6,"identity: N*a**N + N*b**N"],["maxwell-elementary-treatise-electricity-1888/x-0663c7e389",15,"Maxwell 1888, scan 153: We may express both the Peltier and the Thomson ..."],["maxwell-elementary-treatise-electricity-1888/x-f349248a81",15,"Maxwell 1888, scan 160: It is manifest that in a heterogeneous circuit we ..."],["maxwell-elementary-treatise-electricity-1888/x-c68116957a",15,"Maxwell 1888, scan 155: In this treatise we have avoided making any assumption ..."],["wentworth-plane-geometry-1899/eq-6d370a6549",16,"Wentworth 1899, scan 102: \\sqrt{2} = 1.41421356\\cdots"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/9",4,"Hardy 1921, Exercise LXXVI (9)"],["form/0a283c65e7",5,"identity: -a**3 - a**2*b - 2*a*b**2 - 2*b**3"],["shape/71f83f78cd",6,"identity: N*a*b**N + N*b**N - a**N*b - a**N"],["concept/centrifugal-force",7,"centrifugal force","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-centrifugal-force"],["todhunter-spherical-trigonometry-1886/x-e7eb828cda",15,"Todhunter 1886, scan 159: We shall give in this Chapter examples of the ..."],["todhunter-spherical-trigonometry-1886/x-c0d4e3b27c",15,"Todhunter 1886, scan 159: We shall first take right-angled triangles, and then oblique-angled ..."],["wentworth-plane-geometry-1899/eq-5597d2b614",16,"Wentworth 1899, scan 105: \\dfrac{x}{k} = \\dfrac{1}{k} × x"],["todhunter-spherical-trigonometry-1886/x-bc35865994",15,"Todhunter 1886, scan 161: Here \\tan c is negative; and therefore \\tan b ..."],["todhunter-spherical-trigonometry-1886/x-4bc5a22075",15,"Todhunter 1886, scan 160: The numerical value of \\cos c is the same ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/10",4,"Hardy 1921, Exercise LXXVI (10)"],["todhunter-spherical-trigonometry-1886/x-ebdb84046b",15,"Todhunter 1886, scan 160: Here \\cos c is negative; and therefore \\cot B ..."],["wentworth-first-steps-in-algebra-1894/ex-37",3,"Wentworth 1894, Exercise 37"],["shape/067716de7a",6,"factor: N*x + N*x**N + 1"],["wentworth-plane-geometry-1899/eq-202e463938",16,"Wentworth 1899, scan 104: kx=0"],["wentworth-plane-geometry-1899/eq-cef083168f",16,"Wentworth 1899, scan 105: kx = ka"],["hardy-course-of-pure-mathematics-1921/eq-38834fd8b4",16,"Hardy 1921, p. 308: u_{m} + u_{m+1} + \\dots + u_{n} = \\sum_{m}^{n} \\phi(\\nu)"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/11",4,"Hardy 1921, Exercise LXXVI (11)"],["wentworth-plane-geometry-1899/eq-3113b2ca4c",16,"Wentworth 1899, scan 106: xy = ab"],["wentworth-first-steps-in-algebra-1894/ex-37/7",4,"Wentworth 1894, Exercise 37 (7)"],["wentworth-plane-geometry-1899/eq-5a9c3a850f",16,"Wentworth 1899, scan 106: x^n = a^n"],["wentworth-first-steps-in-algebra-1894/ex-38",3,"Wentworth 1894, Exercise 38"],["concept/antecedent-of-a-proportion",7,"antecedent of a proportion","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-antecedent-of-a-proportion"],["person/ptolemy",1,"Ptolemy","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-person-ptolemy"],["hardy-course-of-pure-mathematics-1921/eq-902f750389",16,"Hardy 1921, p. 309: u_{0} + u_{1} + \\dots + u_{n} < K"],["theorem/product-of-extremes-equals-product-of-means",9,"product of extremes equals product of means","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-product-of-extremes-equals-product-of-means"],["theorem/equal-products-give-a-proportion",9,"equal products give a proportion","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-equal-products-give-a-proportion"],["theorem/mean-proportional-is-square-root-of-product",9,"mean proportional is square root of product","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-mean-proportional-is-square-root-of-product"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/12",4,"Hardy 1921, Exercise LXXVI (12)"],["form/683a3fc176",5,"factor: x**2 - 9*x + 20"],["shape/84e36ece73",6,"identity: N*a + 3*a**N"],["form/b404ff74f3",5,"identity: -4*a**2*b + 2*a*b**2 + 2*b**3"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/13",4,"Hardy 1921, Exercise LXXVI (13)"],["shape/7b5b1e9527",6,"identity: (-a + x)*(-b + x)"],["form/f04374aac4",5,"factor: x**2 - 5*x - 14"],["wentworth-first-steps-in-algebra-1894/ex-38/14",4,"Wentworth 1894, Exercise 38 (14)"],["maxwell-elementary-treatise-electricity-1888/x-c40af54fd8",15,"Maxwell 1888, scan 212: In this method of measuring the resistance of the ..."],["maxwell-elementary-treatise-electricity-1888/x-5b4c48bbab",15,"Maxwell 1888, scan 214: This method, in which, at the time of the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-16792ebada",15,"De Morgan 1899, p. 4: For example, let a point A move on a ..."],["form/60b6273335",5,"identity: (a + x)*(a**2 - a*x + x**2)"],["wentworth-first-steps-in-algebra-1894/ex-22/20",4,"Wentworth 1894, Exercise 22 (20)"],["shape/b2efdbaaf2",6,"identity: (a + x)*(-a*x + a**N + x**N)"],["hardy-course-of-pure-mathematics-1921/eq-e4d99a205a",16,"Hardy 1921, p. 309: v_{n} \\leq Ku_{n}"],["de-morgan-elementary-illustrations-calculus-1899/x-f596991fe5",15,"De Morgan 1899, p. 4: But while the magnitudes diminish, we may not assume ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/14",4,"Hardy 1921, Exercise LXXVI (14)"],["method/finding-a-limit-by-rejecting-negligible-terms",8,"finding a limit by rejecting negligible terms","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-finding-a-limit-by-rejecting-negligible-terms"],["method/dividing-numerator-and-denominator-by-the-highest-power",8,"dividing numerator and denominator by the highest power","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-dividing-numerator-and-denominator-by-the-highest-power"],["wentworth-first-steps-in-algebra-1894/ex-1",3,"Wentworth 1894, Exercise 1"],["wentworth-first-steps-in-algebra-1894/ex-38/15",4,"Wentworth 1894, Exercise 38 (15)"],["form/f25e165913",5,"factor: x**2 - x - 20"],["form/c54dfdb5c6",5,"identity: 14"],["form/d687261146",5,"identity: 10"],["hardy-course-of-pure-mathematics-1921/x-8919c874d7",15,"Hardy 1921, p. 99: That these n roots are in reality all distinct ..."],["hardy-course-of-pure-mathematics-1921/eq-3e788ab1b7",16,"Hardy 1921, p. 309: \\sum v_{n} \\leq K \\sum u_{n}"],["de-morgan-elementary-illustrations-calculus-1899/x-0e1b247626",15,"De Morgan 1899, p. 5: The first possible case is that the ratio of ..."],["thompson-calculus-made-easy-1914/x-118ce46b78",15,"Thompson 1914, p. 227: Write u = w, and for \\sin w · ..."],["wentworth-plane-geometry-1899/eq-65ddeae06e",16,"Wentworth 1899, scan 105: d+d'+d''+\\cdots < nd"],["wentworth-plane-geometry-1899/eq-946022f27e",16,"Wentworth 1899, scan 107: \\dfrac{a}{b} = r"],["wentworth-plane-geometry-1899/eq-05728c061c",16,"Wentworth 1899, scan 98: AB = AC"],["planck-treatise-on-thermodynamics-1903/eq-bc59b0790b",16,"Planck 1903, p. 57: c_{p} - c_{v} + p\\, \\frac{\\dd c_{p}}{\\dd p} - v\\, \\frac{\\dd c_{v}}{\\dd v} = \\frac{R}{m}"],["de-morgan-elementary-illustrations-calculus-1899/x-a48a0caa11",15,"De Morgan 1899, p. 6: The second possible case is that in which the ..."],["thompson-calculus-made-easy-1914/x-f2c13dfa15",15,"Thompson 1914, p. 233: There are whole treatises, such as Boole’s Differential Equations, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-c489b9d7d2",15,"De Morgan 1899, p. 6: The difference between this case and the last is, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-07fb32e5bf",15,"De Morgan 1899, p. 7: Therefore (1) \\dfrac{M}{N} continually increases; (2) may be brought ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/15",4,"Hardy 1921, Exercise LXXVI (15)"],["wentworth-first-steps-in-algebra-1894/ex-23",3,"Wentworth 1894, Exercise 23"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/16",4,"Hardy 1921, Exercise LXXVI (16)"],["hardy-course-of-pure-mathematics-1921/eq-e1b471951c",16,"Hardy 1921, p. 310: v_{n} \\leq Kr^{n}"],["de-morgan-elementary-illustrations-calculus-1899/x-601d97c476",15,"De Morgan 1899, p. 4: In introducing the notion of time, we consult only ..."],["boyden-first-book-in-algebra-1895/ex-22/7",4,"Boyden 1895, Exercise 22 (7)"],["hardy-course-of-pure-mathematics-1921/x-82afc18dbf",15,"Hardy 1921, p. 102: This is the analytical equivalent of the geometrical theorem ..."],["hardy-course-of-pure-mathematics-1921/eq-f8d4257289",16,"Hardy 1921, p. 310: v_{n}^{1/n} \\leq r"],["quantity/circumference",11,"circumference","../books/ball-mathematical-recreations-1905/terms/index.html#t-quantity-circumference"],["hardy-course-of-pure-mathematics-1921/eq-8827d24aca",16,"Hardy 1921, p. 310: v_{n}^{1/n} \\geq 1"],["hardy-course-of-pure-mathematics-1921/eq-6b719c60a1",16,"Hardy 1921, p. 310: v_{n+1}/v_{n} \\leq r"],["form/8c6d6d52a4",5,"identity: (a**2 + b**2)**3"],["shape/5e19ae7326",6,"identity: (a**N + b**N)**N"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/17",4,"Hardy 1921, Exercise LXXVI (17)"],["concept/magic-bottle",7,"magic bottle","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-magic-bottle"],["hardy-course-of-pure-mathematics-1921/x-7e6d37dd96",15,"Hardy 1921, p. 102: [The amplitudes have not necessarily their principal values.]"],["planck-treatise-on-thermodynamics-1903/x-1cae37aaab",15,"Planck 1903, p. 119: For the present, besides M, let \\theta and v ..."],["planck-treatise-on-thermodynamics-1903/x-93cca2f397",15,"Planck 1903, p. 119: Then the pressure p, the specific energy u = ..."],["hardy-course-of-pure-mathematics-1921/eq-7ef7861f04",16,"Hardy 1921, p. 310: v_{n+1}/v_{n} \\geq r \\geq 1"],["boyden-first-book-in-algebra-1895/ex-22/8",4,"Boyden 1895, Exercise 22 (8)"],["form/08e555506a",5,"identity: (a**3 - b**2)**2"],["shape/7bc748d7ab",6,"identity: (a**N - b**N)**N"],["form/5343058820",5,"factor: a**2*x**2 - 19*a*b*x + 48*b**2"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/18",4,"Hardy 1921, Exercise LXXVI (18)"],["shape/4f4dbf1eef",6,"factor: N*a*b*x + N*b**N + a**N*x**N"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/19",4,"Hardy 1921, Exercise LXXVI (19)"],["de-morgan-elementary-illustrations-calculus-1899/x-ab8b2a35fe",15,"De Morgan 1899, p. 74: The following is a recapitulation of the principal results ..."],["de-morgan-elementary-illustrations-calculus-1899/x-371bf9eadc",15,"De Morgan 1899, p. 74: That if in the equation y = \\phi(x), the ..."],["hardy-course-of-pure-mathematics-1921/eq-da7feb4b8f",16,"Hardy 1921, p. 311: v_{n+1}/v_{n} \\to l"],["hardy-course-of-pure-mathematics-1921/eq-6ec870b328",16,"Hardy 1921, p. 314: u_{0} v_{0} + (u_{1} v_{0} + u_{0} v_{1}) + (u_{2} v_{0} + u_{1} v_{1} + u_{0} v_{2}) + \\dots"],["boyden-first-book-in-algebra-1895/ex-22/9",4,"Boyden 1895, Exercise 22 (9)"],["hardy-course-of-pure-mathematics-1921/eq-ca4138217a",16,"Hardy 1921, p. 314: (u_{0} + u_{1} + \\dots + u_{n})(v_{0} + v_{1} + \\dots + v_{n})"],["form/0caadb6f43",5,"identity: (a**2 - b**2)**4"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/20",4,"Hardy 1921, Exercise LXXVI (20)"],["form/7c994e8b9a",5,"factor: a**2*x**2 + 15*a*b*x + 44*b**2"],["form/6006fd18a2",5,"factor: 36*a**2 - 13*a*x + x**2"],["concept/continuous-quantity",7,"continuous quantity","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-continuous-quantity"],["concept/decreasing-without-limit",7,"decreasing without limit","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-decreasing-without-limit"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/21",4,"Hardy 1921, Exercise LXXVI (21)"],["thompson-calculus-made-easy-1914/eq-a07b8ba3b1",16,"Thompson 1914, p. 141: \\epsilon = 1 + 1 + \\dfrac{1}{2!} + \\dfrac{1}{3!} + \\dfrac{1}{4!} + \\text{etc}.\\ldots"],["planck-treatise-on-thermodynamics-1903/ch-proof",2,"Planck 1903, Proof","../books/planck-treatise-on-thermodynamics-1903/ch/ch-proof/index.html"],["boyden-first-book-in-algebra-1895/ex-22/10",4,"Boyden 1895, Exercise 22 (10)"],["form/c5d4842d99",5,"identity: (a**2 + b**4)**3"],["wentworth-first-steps-in-algebra-1894/ex-1/16",4,"Wentworth 1894, Exercise 1 (16)"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/22",4,"Hardy 1921, Exercise LXXVI (22)"],["quantity/temperature-coefficient-of-resistance",11,"temperature coefficient of resistance","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-quantity-temperature-coefficient-of-resistance"],["maxwell-elementary-treatise-electricity-1888/x-ad7bee4e88",15,"Maxwell 1888, scan 215: In all these substances conduction takes place without any ..."],["maxwell-elementary-treatise-electricity-1888/x-9f3ec35fcd",15,"Maxwell 1888, scan 215: The resistance of this class of bodies is enormous ..."],["boyden-first-book-in-algebra-1895/ex-22/11",4,"Boyden 1895, Exercise 22 (11)"],["concept/alloy",7,"alloy","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-alloy"],["form/227fee97ce",5,"identity: (a**2*b + c)**2"],["shape/2ee459fdd8",6,"identity: (a**N*b + c)**N"],["maxwell-elementary-treatise-electricity-1888/x-68a8313e38",15,"Maxwell 1888, scan 218: The measurement of the electric resistance of electrolytes is ..."],["maxwell-elementary-treatise-electricity-1888/x-961c9e9dca",15,"Maxwell 1888, scan 221: These phenomena seem to be due to a condition ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/23",4,"Hardy 1921, Exercise LXXVI (23)"],["maxwell-elementary-treatise-electricity-1888/x-b51d67aae3",15,"Maxwell 1888, scan 217: Hence ordinary resistance coils are made of German silver, ..."],["planck-treatise-on-thermodynamics-1903/eq-0169318754",16,"Planck 1903, p. 57: \\left(\\frac{\\dd u}{\\dd v}\\right)_{\\theta} = 0"],["planck-treatise-on-thermodynamics-1903/eq-127731aa97",16,"Planck 1903, p. 58: du = \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{v} d\\theta + \\left(\\frac{\\dd u}{\\dd v}\\right)_{\\theta} dv"],["dickson-theory-of-equations-1922/eq-3734d44d89",16,"Dickson 1922, p. 32: x^3 + \\alpha x^2 + \\beta x + \\gamma = 0"],["person/max-planck",1,"Max Planck"],["person/karl-weierstrass",1,"Karl Weierstrass","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-karl-weierstrass"],["unit/foot-pound",12,"foot-pound","../books/ball-mathematical-recreations-1905/terms/index.html#t-unit-foot-pound"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/24",4,"Hardy 1921, Exercise LXXVI (24)"],["thompson-calculus-made-easy-1914/x-f365c78b0a",15,"Thompson 1914, p. 242: Now the test of the matter is this. If ..."],["form/3d27ab1d4c",5,"factor: 48*a**2*b**2 + 49*a*b*x + x**2"],["thompson-calculus-made-easy-1914/eq-de3b335fe5",16,"Thompson 1914, p. 143: \\epsilon^x = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\text{etc.}\\dots"],["thompson-calculus-made-easy-1914/eq-c3958c1cc4",16,"Thompson 1914, p. 145: \\epsilon^x = 1 + \\dfrac{x}{1} + \\dfrac{x^2}{1·2} + \\dfrac{x^3}{1· 2· 3} + \\dfrac{x^4}{1· 2· 3· 4} + \\text{etc}."],["thompson-calculus-made-easy-1914/eq-760a737dd7",16,"Thompson 1914, p. 147: y = \\log_\\epsilon x."],["thompson-calculus-made-easy-1914/eq-dd5d2e0991",16,"Thompson 1914, p. 148: \\frac{d(\\log_\\epsilon x)}{dx} = x^{-1}."],["thompson-calculus-made-easy-1914/eq-a7ed6dda75",16,"Thompson 1914, p. 146: \\log_\\epsilon a + \\log_\\epsilon b = \\log_\\epsilon ab."],["thompson-calculus-made-easy-1914/eq-648e8f4ccf",16,"Thompson 1914, p. 146: n × \\log_\\epsilon a = \\log_\\epsilon a^n."],["thompson-calculus-made-easy-1914/x-5d5e1cd3c5",15,"Thompson 1914, p. 245: The only function we know that has this property ..."],["ball-mathematical-recreations-1905/x-166dc7e372",15,"Ball 1905, scan 267: We may say that the \\pi-calculators who used the ..."],["planck-treatise-on-thermodynamics-1903/eq-2525e532da",16,"Planck 1903, p. 58: du = \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{v} d\\theta"],["concept/arc-of-a-circle",7,"arc of a circle","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-arc-of-a-circle"],["boyden-first-book-in-algebra-1895/ex-22/12",4,"Boyden 1895, Exercise 22 (12)"],["planck-treatise-on-thermodynamics-1903/eq-676a63c785",16,"Planck 1903, p. 58: du = c_{v} · d\\theta"],["thompson-calculus-made-easy-1914/x-624276693a",15,"Thompson 1914, p. 252: [-12pt]0pt32ptnx^{n-1} & x^n & \\dfrac{1}{n+1} x^{n+1} + C"],["person/william-kingdon-clifford",1,"William Kingdon Clifford","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-william-kingdon-clifford"],["hardy-course-of-pure-mathematics-1921/ex-lxxvi/25",4,"Hardy 1921, Exercise LXXVI (25)"],["person/jean-tienne-montucla",1,"Jean-Étienne Montucla","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-jean-tienne-montucla"],["form/93cda8311c",5,"identity: (a**2 - b**3*c)**3"],["form/3964479804",5,"identity: (a**2*b - c)**4"],["planck-treatise-on-thermodynamics-1903/eq-bef2f09b64",16,"Planck 1903, p. 58: c_{p} = c_{v} + p \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}"],["planck-treatise-on-thermodynamics-1903/eq-545f113302",16,"Planck 1903, p. 58: c_{p} = c_{v} + \\frac{R}{m}"],["shape/4ed85f3995",6,"identity: (a**N*b - c)**N"],["boyden-first-book-in-algebra-1895/ex-22/13",4,"Boyden 1895, Exercise 22 (13)"],["shape/ef7da15a18",6,"identity: (a**N - b**N*c)**N"],["hardy-course-of-pure-mathematics-1921/ex-lxxvii/1",4,"Hardy 1921, Exercise LXXVII (1)"],["person/gergonne",1,"Gergonne","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-gergonne"],["hardy-course-of-pure-mathematics-1921/eq-b7b07b154f",16,"Hardy 1921, p. 318: u_{n} = \\phi(n)"],["ball-mathematical-recreations-1905/x-24569f54ec",15,"Ball 1905, scan 277: “Only prove to me that it is impossible,” said ..."],["form/77c9441cbd",5,"factor: -243*a**2*b**2 - 18*a*b*x + x**2"],["planck-treatise-on-thermodynamics-1903/eq-f0138a7b70",16,"Planck 1903, p. 58: mc_{p} - mc_{v} = R"],["cap/other:general_principle_of_convergence",17,"other:general_principle_of_convergence"],["law/mayer-relation",10,"Mayer relation"],["planck-treatise-on-thermodynamics-1903/eq-117de13736",16,"Planck 1903, p. 59: mc_{p} - mc_{v} = \\frac{R}{J} = \\frac{826 · 10^{5}}{419 · 10^{5}} = 1.971\\Add{.}"],["hardy-course-of-pure-mathematics-1921/eq-9362cab691",16,"Hardy 1921, p. 316: u_{n+1} \\leq u_{n}"],["form/c2c58b9bbb",5,"factor: a**2*x**2 - a*b*x - 182*b**2"],["concept/linear-relation",7,"linear relation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-linear-relation"],["hardy-course-of-pure-mathematics-1921/eq-6ab53686eb",16,"Hardy 1921, p. 318: \\phi(\\nu - 1) \\geq \\phi(x) \\geq \\phi(\\nu)"],["de-morgan-elementary-illustrations-calculus-1899/ch-determination-of-curvilinear-areas-the-parabola",2,"De Morgan 1899, Determination of Curvilinear Areas. The Parabola","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-determination-of-curvilinear-areas-the-parabola/index.html"],["thompson-calculus-made-easy-1914/eq-3c037289f5",16,"Thompson 1914, p. 156: p=\\epsilon^{-a}"],["thompson-calculus-made-easy-1914/eq-6ca4e37ce5",16,"Thompson 1914, p. 156: y=bp^x;"],["thompson-calculus-made-easy-1914/eq-a0a5823e09",16,"Thompson 1914, p. 155: y = b\\epsilon^{ax}."],["thompson-calculus-made-easy-1914/eq-526bf8e68a",16,"Thompson 1914, p. 155: \\log_\\epsilon \\frac{y}{b}=ax,"],["wentworth-first-steps-in-algebra-1894/ex-39",3,"Wentworth 1894, Exercise 39"],["hardy-course-of-pure-mathematics-1921/ex-lxxvii/2",4,"Hardy 1921, Exercise LXXVII (2)"],["cap/other:comparison_test",17,"other:comparison_test"],["hardy-course-of-pure-mathematics-1921/eq-03cac8156c",16,"Hardy 1921, p. 316: \\lim nu_{n} = 0"],["ball-mathematical-recreations-1905/x-7cdcfe8e69",15,"Ball 1905, scan 98: Probably the meaning of the law is best expressed ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvii/3",4,"Hardy 1921, Exercise LXXVII (3)"],["wentworth-first-steps-in-algebra-1894/ex-1/17",4,"Wentworth 1894, Exercise 1 (17)"],["hardy-course-of-pure-mathematics-1921/ex-lxxvii/4",4,"Hardy 1921, Exercise LXXVII (4)"],["thompson-calculus-made-easy-1914/eq-b9c6d80f8e",16,"Thompson 1914, p. 156: y=b\\epsilon^{-ax}."],["thompson-calculus-made-easy-1914/eq-66efe184f8",16,"Thompson 1914, p. 156: \\theta_t=\\theta_0 \\epsilon^{-at};"],["form/172b75ecd2",5,"identity: 1"],["form/0332464b80",5,"factor: 3*a**3*x + 3*a**2*x**2 - 2*a*x**3"],["shape/03ed626580",6,"factor: N*a*x**N + N*a**N*x + N*a**N*x**N"],["ball-mathematical-recreations-1905/x-60f2bb43cc",15,"Ball 1905, scan 101: A given agent in a given time can do ..."],["ball-mathematical-recreations-1905/x-4369f8bda4",15,"Ball 1905, scan 95: The assertion was that if Achilles ran ten times ..."],["ball-mathematical-recreations-1905/x-1b8bf86333",15,"Ball 1905, scan 102: Such a bottle is made of thin glass or ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxvii/5",4,"Hardy 1921, Exercise LXXVII (5)"],["hardy-course-of-pure-mathematics-1921/eq-29aa7fb076",16,"Hardy 1921, p. 318: v_{\\nu} = \\phi(\\nu - 1) - \\int_{\\nu-1}^{\\nu} \\phi(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-e24481e948",16,"Hardy 1921, p. 318: 0 \\leq v_{\\nu} \\leq \\phi(\\nu - 1) - \\phi(\\nu)"],["boyden-first-book-in-algebra-1895/ex-22/14",4,"Boyden 1895, Exercise 22 (14)"],["boyden-first-book-in-algebra-1895/ex-22/16",4,"Boyden 1895, Exercise 22 (16)"],["boyden-first-book-in-algebra-1895/ex-22/17",4,"Boyden 1895, Exercise 22 (17)"],["ball-mathematical-recreations-1905/eq-edcd10b5b0",16,"Ball 1905, scan 137: \\frac{1}{2}(n-1)(n-2) (n^2 + 3n-2)"],["cap/other:subseries",17,"other:subseries"],["ball-mathematical-recreations-1905/eq-28b1069961",16,"Ball 1905, scan 137: \\frac{1}{6}(n-1) (n-2) (n^4+ 3n^3-20n^2-30n + 132)"],["hardy-course-of-pure-mathematics-1921/ex-lxxvii/6",4,"Hardy 1921, Exercise LXXVII (6)"],["ball-mathematical-recreations-1905/eq-ccca12397a",16,"Ball 1905, scan 146: x_1=\\frac{1}{2}(2p+x_0+1)"],["form/8de632c48e",5,"identity: (a**2 - 1)**4"],["ball-mathematical-recreations-1905/ch-iv",2,"Ball 1905, ch. IV: Some Miscellaneous Questions","../books/ball-mathematical-recreations-1905/ch/ch-iv/index.html"],["shape/776df485d2",6,"identity: (a**N - 1)**N"],["boyden-first-book-in-algebra-1895/ex-22/18",4,"Boyden 1895, Exercise 22 (18)"],["form/1a71259b99",5,"identity: (a**3 + 1)**3"],["cap/other:triangle_inequality",17,"other:triangle_inequality"],["form/f7302f1f61",5,"factor: x**2 - 2*x - 3"],["wentworth-first-steps-in-algebra-1894/ex-39/11",4,"Wentworth 1894, Exercise 39 (11)"],["theorem/area-of-a-parabolic-segment",9,"area of a parabolic segment","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-area-of-a-parabolic-segment"],["unit/degree-of-angle",12,"degree of angle","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-unit-degree-of-angle"],["concept/antiderivative",7,"antiderivative","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-antiderivative"],["unit/minute-of-arc",12,"minute of arc","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-unit-minute-of-arc"],["ball-mathematical-recreations-1905/eq-3299123e29",16,"Ball 1905, scan 146: x_1=\\frac{1}{2}(2p-x_0 + 2)"],["ball-mathematical-recreations-1905/eq-d68fe73cf6",16,"Ball 1905, scan 146: 2^{m+1}x_m=(4p+1)(2^{m-1} \\pm 2^{m-2} \\pm \\dotsb \\pm 2 \\pm 1) \\pm 2x_0 + 2^m \\pm 1"],["shape/2ff461b55d",6,"identity: (a**N + 1)**N"],["boyden-first-book-in-algebra-1895/ex-22/19",4,"Boyden 1895, Exercise 22 (19)"],["de-morgan-elementary-illustrations-calculus-1899/eq-66984acf53",16,"De Morgan 1899, p. 105: \\frac{du}{dp} = -\\frac{1}{p^{2}}"],["de-morgan-elementary-illustrations-calculus-1899/eq-fb02532e85",16,"De Morgan 1899, p. 105: p = \\psi' y"],["thompson-calculus-made-easy-1914/eq-e9b04ad5f3",16,"Thompson 1914, p. 157: Q_t=Q_0 \\epsilon^{-at},"],["ball-mathematical-recreations-1905/eq-233d9f8903",16,"Ball 1905, scan 141: 3 (2m+1)(3m+1)"],["todhunter-spherical-trigonometry-1886/x-adc430ec0a",15,"Todhunter 1886, scan 89: The importance of Legendre’s Theorem in the application of ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxviii/1",4,"Hardy 1921, Exercise LXXVIII (1)"],["todhunter-spherical-trigonometry-1886/x-fbcf2881ce",15,"Todhunter 1886, scan 88: It will be seen that in the above approximation ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-1bbb3cb130",16,"De Morgan 1899, p. 105: \\frac{dp}{dy} = \\psi'' y"],["de-morgan-elementary-illustrations-calculus-1899/eq-3aa4580c3c",16,"De Morgan 1899, p. 104: \\dfrac{d^{2} y}{dx^{2}} = \\phi'' x"],["concept/parallel-lines",7,"parallel lines","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-parallel-lines"],["form/543bbb6f90",5,"identity: (a*b - 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1) \\log v = \\const"],["planck-treatise-on-thermodynamics-1903/x-90e705d634",15,"Planck 1903, p. 86: It is impossible to construct an engine which will ..."],["ball-mathematical-recreations-1905/eq-9b37697379",16,"Ball 1905, scan 144: z=91"],["ball-mathematical-recreations-1905/eq-80856b6001",16,"Ball 1905, scan 144: y=84"],["ball-mathematical-recreations-1905/eq-130477cf9b",16,"Ball 1905, scan 144: z=28"],["boyden-first-book-in-algebra-1895/ex-22/20",4,"Boyden 1895, Exercise 22 (20)"],["form/1a5d12980d",5,"identity: (a**2*b - 3)**2"],["ball-mathematical-recreations-1905/eq-0d4542329f",16,"Ball 1905, scan 145: y=7"],["planck-treatise-on-thermodynamics-1903/eq-2bf7733693",16,"Planck 1903, p. 60: -\\gamma \\log \\theta + (\\gamma - 1) \\log p = \\const"],["ball-mathematical-recreations-1905/eq-f2fad6b3f4",16,"Ball 1905, scan 145: y=155"],["planck-treatise-on-thermodynamics-1903/eq-776e64e106",16,"Planck 1903, p. 60: \\log p + \\gamma \\log v = \\const"],["de-morgan-elementary-illustrations-calculus-1899/x-3e04ffee72",15,"De Morgan 1899, p. 9: Let \\theta = 1°; then \\sin\\theta = .0174524 and ..."],["shape/030969362f",6,"identity: (N + a**N*b)**N"],["hardy-course-of-pure-mathematics-1921/ex-lxxviii/4",4,"Hardy 1921, Exercise LXXVIII (4)"],["planck-treatise-on-thermodynamics-1903/x-8a2fb8b5ad",15,"Planck 1903, p. 89: The entropy of the gas, therefore, remains constant during ..."],["form/63df134772",5,"factor: x**2 - 19*x + 88"],["ball-mathematical-recreations-1905/eq-e0619e6433",16,"Ball 1905, scan 145: y=66"],["planck-treatise-on-thermodynamics-1903/eq-a90d199fed",16,"Planck 1903, p. 60: pv^{\\gamma} = \\const"],["ball-mathematical-recreations-1905/eq-b6223fc5e7",16,"Ball 1905, scan 145: y=140"],["de-morgan-elementary-illustrations-calculus-1899/x-243749ab47",15,"De Morgan 1899, p. 9: Thus if \\angle BOA = 1°, BM ÷ MA ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxviii/5",4,"Hardy 1921, Exercise LXXVIII (5)"],["law/adiabatic-law",10,"adiabatic law"],["form/710a5e839a",5,"factor: -10*a**2 + 3*a*x + x**2"],["planck-treatise-on-thermodynamics-1903/eq-46f3e29ccf",16,"Planck 1903, p. 61: \\frac{p}{\\rho^{\\gamma}} = \\const"],["ball-mathematical-recreations-1905/eq-24df0367f6",16,"Ball 1905, scan 140: (455)^7"],["form/8a9531d521",5,"factor: -16*x**2 + 1"],["ball-mathematical-recreations-1905/eq-e554d25a59",16,"Ball 1905, scan 140: 15567,552000"],["hardy-course-of-pure-mathematics-1921/eq-c17ffdc2e6",16,"Hardy 1921, p. 318: v_{2} + v_{3} + \\dots + v_{n} \\leq \\phi(1) - \\phi(n) \\leq \\phi(1)"],["hardy-course-of-pure-mathematics-1921/eq-5b960b7ae6",16,"Hardy 1921, p. 318: \\Phi(\\xi) = \\int_{1}^{\\xi} \\phi(x)\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/ch-application-of-the-theorem-for-total-differentials-to-the-determination-of-total-resultant-errors",2,"De Morgan 1899, Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-application-of-the-theorem-for-total-differentials-to-the-determination-of-total-resultant-errors/index.html"],["hardy-course-of-pure-mathematics-1921/eq-8b11aaf72a",16,"Hardy 1921, p. 318: \\sum_{1}^{n-1} \\phi(\\nu) - \\int_{1}^{n} \\phi(x)\\, dx"],["boyden-first-book-in-algebra-1895/ex-22/21",4,"Boyden 1895, Exercise 22 (21)"],["concept/same-affection",7,"same affection","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-same-affection"],["wentworth-plane-geometry-1899/ex-iii-1",3,"Wentworth 1899, Exercise III.1"],["wentworth-plane-geometry-1899/ex-iii-2",3,"Wentworth 1899, Exercise III.2"],["wentworth-plane-geometry-1899/ex-iii-3",3,"Wentworth 1899, Exercise III.3"],["planck-treatise-on-thermodynamics-1903/eq-e964dbc77d",16,"Planck 1903, p. 61: \\frac{dp}{d\\rho} = \\frac{\\gamma p}{\\rho} = \\gamma pv"],["planck-treatise-on-thermodynamics-1903/eq-d1a107c4a1",16,"Planck 1903, p. 61: \\frac{dp}{d\\rho} = \\gamma \\frac{R}{m} \\theta"],["hardy-course-of-pure-mathematics-1921/eq-a0ac0f607d",16,"Hardy 1921, p. 78: x + yi = x' + y'i"],["theorem/pythagorean-theorem",9,"pythagorean theorem","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-pythagorean-theorem"],["hardy-course-of-pure-mathematics-1921/eq-1ee947d43d",16,"Hardy 1921, p. 78: (x + yi) + (x' + y'i) = (x + x') + (y + y')i"],["concept/mutually-equilateral-polygons",7,"mutually equilateral polygons","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-mutually-equilateral-polygons"],["wentworth-plane-geometry-1899/x-8670d14d59",15,"Wentworth 1899, scan 171: The sum of the squares of the two legs ..."],["concept/extreme-and-mean-ratio",7,"extreme and mean ratio","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-extreme-and-mean-ratio"],["hardy-course-of-pure-mathematics-1921/eq-1ccade3fb5",16,"Hardy 1921, p. 78: (x + yi) (x' + y'i) = xx' - yy' + (xy' + yx')i"],["boyden-first-book-in-algebra-1895/ex-22/23",4,"Boyden 1895, Exercise 22 (23)"],["hardy-course-of-pure-mathematics-1921/eq-b0736d8105",16,"Hardy 1921, p. 319: v_{\\nu} < \\phi(\\nu - 1) - \\phi(\\nu)"],["hardy-course-of-pure-mathematics-1921/eq-1c6ad6e2cc",16,"Hardy 1921, p. 81: (x + yi)i = -y + xi"],["planck-treatise-on-thermodynamics-1903/eq-3bb3ecfc81",16,"Planck 1903, p. 64: W = -\\frac{R}{m} \\left(\\theta_{2} \\log \\frac{v_{2}'}{v_{2}} + \\theta_{1} \\log \\frac{v_{1}'}{v_{1}}\\right)"],["hardy-course-of-pure-mathematics-1921/eq-9326a473e9",16,"Hardy 1921, p. 80: (x + yi)(x - yi) = x^{2} + y^{2}"],["concept/constant",7,"constant","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-constant"],["hardy-course-of-pure-mathematics-1921/ex-lxxviii/6",4,"Hardy 1921, Exercise LXXVIII (6)"],["hardy-course-of-pure-mathematics-1921/eq-1decb5173d",16,"Hardy 1921, p. 79: (x + yi) (x' + y'i) = (x' + y'i) (x + yi)"],["planck-treatise-on-thermodynamics-1903/eq-6f9c23c2cc",16,"Planck 1903, p. 61: \\gamma = \\frac{m}{R\\theta} · \\frac{dp}{d\\rho}"],["hardy-course-of-pure-mathematics-1921/eq-c766573983",16,"Hardy 1921, p. 80: i^{2} = ii = (0 + 1i) (0 + 1i) = (0 · 0 - 1 · 1) + (0 · 1 + 1 · 0)i = -1"],["planck-treatise-on-thermodynamics-1903/eq-e47eff52a0",16,"Planck 1903, p. 61: \\sqrt{\\dfrac{dp}{d\\rho}}"],["planck-treatise-on-thermodynamics-1903/eq-c3be53c89a",16,"Planck 1903, p. 61: \\gamma = \\frac{28.8}{826 · 10^{5}} · \\frac{33280^{2}}{273} = 1.41"],["hardy-course-of-pure-mathematics-1921/eq-89f9c4faec",16,"Hardy 1921, p. 81: az^{2} + 2bz + c = 0"],["wentworth-plane-geometry-1899/x-0030f8cea3",15,"Wentworth 1899, scan 184: A straight line is divided in extreme and mean ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxviii/7",4,"Hardy 1921, Exercise LXXVIII (7)"],["hardy-course-of-pure-mathematics-1921/eq-9e3e7eda9a",16,"Hardy 1921, p. 82: \\{z + (b/a)\\}^{2} = -(ac - b^{2})/a^{2}"],["hardy-course-of-pure-mathematics-1921/eq-c4c76bdaf0",16,"Hardy 1921, p. 84: \\alpha + \\beta = -(2b/a)"],["wentworth-first-steps-in-algebra-1894/ex-3",3,"Wentworth 1894, Exercise 3"],["wentworth-first-steps-in-algebra-1894/ex-3/1",4,"Wentworth 1894, Exercise 3 (1)"],["hardy-course-of-pure-mathematics-1921/eq-8e53532f9f",16,"Hardy 1921, p. 84: \\alpha\\beta = (c/a)"],["wentworth-plane-geometry-1899/x-782272bdfb",15,"Wentworth 1899, scan 169: The perpendicular is the mean proportional between the segments ..."],["de-morgan-elementary-illustrations-calculus-1899/x-b788dc34d7",15,"De Morgan 1899, p. 85: The effect of all the errors will then be ..."],["hardy-course-of-pure-mathematics-1921/eq-e963136739",16,"Hardy 1921, p. 84: \\alpha + \\beta + \\gamma = -(3b/a)"],["hardy-course-of-pure-mathematics-1921/ex-lxxx",3,"Hardy 1921, Exercise LXXX"],["boyden-first-book-in-algebra-1895/ex-22/24",4,"Boyden 1895, Exercise 22 (24)"],["shape/faa2e47107",6,"identity: (N*a*b - c**N*d)**N"],["hardy-course-of-pure-mathematics-1921/ex-lxxx/1",4,"Hardy 1921, Exercise LXXX (1)"],["shape/1da1af9198",6,"evaluate: N*a*b"],["form/38cc227322",5,"evaluate: 4*a**2*b at a=7, b=5, c=3"],["shape/3d816cb99a",6,"evaluate: N*a**N*b"],["hardy-course-of-pure-mathematics-1921/eq-7f09de9701",16,"Hardy 1921, p. 84: \\beta\\gamma + \\gamma\\alpha + \\alpha\\beta = (3c/a)"],["hardy-course-of-pure-mathematics-1921/eq-f4e9f4227b",16,"Hardy 1921, p. 84: \\alpha\\beta\\gamma = -(d/a)"],["hardy-course-of-pure-mathematics-1921/ex-lxxx/2",4,"Hardy 1921, Exercise LXXX (2)"],["todhunter-spherical-trigonometry-1886/eq-7b4d659cac",16,"Todhunter 1886, scan 145: \\cot \\tfrac{1}{2}E = \\cot \\tfrac{1}{2}\\theta \\cot \\tfrac{1}{2}c \\operatorname{cosec} \\phi + \\cot \\phi"],["hardy-course-of-pure-mathematics-1921/ex-lxxx/3",4,"Hardy 1921, Exercise LXXX (3)"],["form/a5ddb935c7",5,"evaluate: a*b*c/3 at a=7, b=5, c=3"],["shape/d2d534ada3",6,"evaluate: N*a*b*c"],["form/1d3369eaa6",5,"evaluate: a*b**2*c/5 at a=7, b=5, c=3"],["shape/cdf3018418",6,"evaluate: N*a*b**N*c"],["hardy-course-of-pure-mathematics-1921/eq-86a9738d5f",16,"Hardy 1921, p. 83: f(z) = A(z - a_{1}) (z - a_{2}) \\dots (z - a_{n})"],["dickson-theory-of-equations-1922/eq-7035756e71",16,"Dickson 1922, p. 33: x_1 = \\frac{a + b \\sqrt{k}}{c + d \\sqrt{k}}"],["wentworth-first-steps-in-algebra-1894/ex-40",3,"Wentworth 1894, Exercise 40"],["dickson-theory-of-equations-1922/eq-ec9001ece3",16,"Dickson 1922, p. 33: x_1 = e + f\\sqrt{k}"],["thompson-calculus-made-easy-1914/eq-cf999a08ae",16,"Thompson 1914, p. 160: C = \\dfrac{E}{R}\\left\\{1 - \\epsilon^{-\\efrac{Rt}{L}}\\right\\}"],["dickson-theory-of-equations-1922/eq-cd2d133997",16,"Dickson 1922, p. 33: (e + f \\sqrt{k})^3 + \\alpha(e + f \\sqrt{k})^2 + \\beta(e + f \\sqrt{k}) + \\gamma = A + B\\sqrt{k}"],["dickson-theory-of-equations-1922/eq-b508460d24",16,"Dickson 1922, p. 34: x_2 = e - f \\sqrt{k}"],["dickson-theory-of-equations-1922/eq-a977661c4a",16,"Dickson 1922, p. 34: x_3 = -\\alpha - x_1 - x_2 = -\\alpha - 2e"],["dickson-theory-of-equations-1922/eq-2b55e8aed3",16,"Dickson 1922, p. 34: x_3 = g + h \\sqrt{s}"],["de-morgan-elementary-illustrations-calculus-1899/x-af0e988ca6",15,"De Morgan 1899, p. 10: The arc BA always lies between BA and BN ..."],["planck-treatise-on-thermodynamics-1903/x-664fa2f591",15,"Planck 1903, p. 65: Carnot’s cycle, performed with a perfect gas, thus affords ..."],["hardy-course-of-pure-mathematics-1921/eq-b0f858a963",16,"Hardy 1921, p. 79: x' \\xi - y' \\eta = x"],["planck-treatise-on-thermodynamics-1903/eq-b82c946383",16,"Planck 1903, p. 64: \\frac{v_{2}'}{v_{2}} = \\frac{v_{1}'}{v_{1}}"],["hardy-course-of-pure-mathematics-1921/eq-3771af24ef",16,"Hardy 1921, p. 79: x' \\eta + y' \\xi = y"],["hardy-course-of-pure-mathematics-1921/eq-036215f1b7",16,"Hardy 1921, p. 79: \\xi = \\frac{xx' + yy'}{x'^{2} + y'^{2}}"],["maxwell-elementary-treatise-electricity-1888/eq-99cd191488",16,"Maxwell 1888, scan 178: pU_{n + 1} + qV_{n + 1} &= \\left(pU_n + qV_n \\right) \\left( 1 - pq \\right) = \\left(pU_0 + qV_0 \\right) \\left( 1 - pq \\ri"],["maxwell-elementary-treatise-electricity-1888/eq-43b029a35d",16,"Maxwell 1888, scan 178: U_n &= U_0 \\left\\{ ( 1 - pq )^n + ( 1 + pq )^n \\right\\} + \\frac{q}{p} V_0 \\left\\{ ( 1 - pq )^n - ( 1 + pq )^n \\right\\}"],["maxwell-elementary-treatise-electricity-1888/eq-25ff1430f1",16,"Maxwell 1888, scan 178: V_n &= \\frac{p}{q} U_0 \\left\\{ ( 1 - pq )^n - ( 1 + pq )^n \\right\\} + V_0 \\left\\{ ( 1 - pq )^n + ( 1 + pq )^n \\right\\}"],["maxwell-elementary-treatise-electricity-1888/eq-824047d55e",16,"Maxwell 1888, scan 188: V &= D\\, \\sqrt{\\frac{8 \\pi gW}{A}}\\text{.}"],["hardy-course-of-pure-mathematics-1921/eq-fef6643a35",16,"Hardy 1921, p. 79: \\eta = \\frac{yx' - xy'}{x'^{2} + y'^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-8dc5d1d858",16,"Hardy 1921, p. 77: x = \\rho\\cos\\theta"],["hardy-course-of-pure-mathematics-1921/ch-appendix-iv",2,"Hardy 1921, ch. Appendix IV: The infinite in analysis and geometry","../books/hardy-course-of-pure-mathematics-1921/ch/ch-appendix-iv/index.html"],["maxwell-elementary-treatise-electricity-1888/eq-66eb24d4a1",16,"Maxwell 1888, scan 181: C'V'=aV\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-cdbc851e26",16,"Maxwell 1888, scan 181: (C'+c')U+C'V'=aV\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-1ff057ea96",16,"Maxwell 1888, scan 182: -\\frac{A'V'+BV}{A'+a'}\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-d8e2fdaef0",16,"Maxwell 1888, scan 182: C'V'=aV\\text{,\\quad and\\quad}CV = a'V'\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-15304f59ef",16,"Maxwell 1888, scan 183: Fa \\cos \\tfrac{1}{2} \\theta = M ( \\theta - \\phi )\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-b91787a4c5",16,"Maxwell 1888, scan 183: M = \\frac{4\\pi^2I}{T^2}\\text{.}"],["planck-treatise-on-thermodynamics-1903/eq-832337b05e",16,"Planck 1903, p. 64: W = -\\frac{R}{m} (\\theta_{2} - \\theta_{1}) \\log \\frac{v_{1}'}{v_{1}}"],["maxwell-elementary-treatise-electricity-1888/eq-8295cebfea",16,"Maxwell 1888, scan 189: V - V' = (D - D') \\sqrt{ \\frac{8 \\pi g W}{A}}\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-9af0520009",16,"Maxwell 1888, scan 189: v = (D - D') \\sqrt{ \\frac{8 \\pi gW}{A}}\\text{.}"],["concept/small-electromotive-force-measured",7,"small electromotive force measured"],["maxwell-elementary-treatise-electricity-1888/eq-6955ac08ac",16,"Maxwell 1888, scan 188: Q &= V \\left\\{ \\frac{R^2 + R'^2}{8D} - \\frac{R'^2 - R^2}{8D} \\frac{\\alpha}{D + \\alpha} \\right\\}\\text{,}"],["planck-treatise-on-thermodynamics-1903/x-68f69cf0c7",15,"Planck 1903, p. 259: It follows, then, that when two equally diluted solutions ..."],["planck-treatise-on-thermodynamics-1903/eq-8d8d8bffe2",16,"Planck 1903, p. 64: Q = Q_{1} + Q_{2}= -W"],["dickson-theory-of-equations-1922/eq-4ee77a2fcc",16,"Dickson 1922, p. 34: g - h \\sqrt{s} = e ± f \\sqrt{k}"],["hardy-course-of-pure-mathematics-1921/ex-lxxx/4",4,"Hardy 1921, Exercise LXXX (4)"],["law/first-law-of-motion",10,"first law of motion","../books/ball-mathematical-recreations-1905/terms/index.html#t-law-first-law-of-motion"],["hardy-course-of-pure-mathematics-1921/eq-94fb865b8e",16,"Hardy 1921, p. 77: y = \\rho\\sin\\theta"],["hardy-course-of-pure-mathematics-1921/eq-8a69df811b",16,"Hardy 1921, p. 73: [x, y] = [x, 0] + [0, y]"],["hardy-course-of-pure-mathematics-1921/eq-52cb06e032",16,"Hardy 1921, p. 72: [x, y] + [x', y'] = [x + x', y + y']"],["dickson-theory-of-equations-1922/eq-26c685639e",16,"Dickson 1922, p. 34: \\cos A = 4 \\cos^3 \\frac{A}{3} - 3 \\cos \\frac{A}{3}"],["theorem/triple-angle-formula-for-cosine",9,"triple-angle formula for cosine"],["dickson-theory-of-equations-1922/eq-6492bf3ef7",16,"Dickson 1922, p. 35: x^3 - 3x = 2\\cos A"],["dickson-theory-of-equations-1922/eq-b79aff2c45",16,"Dickson 1922, p. 35: \\cos A = -\\frac{1}{2}"],["hardy-course-of-pure-mathematics-1921/eq-06fe54cab8",16,"Hardy 1921, p. 73: [x, y] - [x', y'] = [x, y] + (-[x', y'])"],["concept/common-cartesian-geometry",7,"common Cartesian geometry","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-common-cartesian-geometry"],["person/archimedes",1,"Archimedes","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-archimedes"],["concept/pendulum",7,"pendulum","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-pendulum"],["hardy-course-of-pure-mathematics-1921/eq-eed30b77f0",16,"Hardy 1921, p. 71: \\alpha[x, y] = [\\alpha x, \\alpha y]"],["hardy-course-of-pure-mathematics-1921/eq-c0d8ad3630",16,"Hardy 1921, p. 73: [0, 0] = 0"],["hardy-course-of-pure-mathematics-1921/eq-cca858b0d3",16,"Hardy 1921, p. 77: [x, y] [x', y'] = [xx' - yy', xy' + yx']"],["concept/real-homogeneous-cartesian-geometry",7,"real homogeneous Cartesian geometry","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-real-homogeneous-cartesian-geometry"],["hardy-course-of-pure-mathematics-1921/ex-lxxx/5",4,"Hardy 1921, Exercise LXXX (5)"],["de-morgan-elementary-illustrations-calculus-1899/x-eb12b3705c",15,"De Morgan 1899, p. 126: These are the altitudes of a set of parallelograms, ..."],["hardy-course-of-pure-mathematics-1921/eq-84ee7a0c92",16,"Hardy 1921, p. 76: [x, 0] [x', y'] = [xx', xy']"],["todhunter-spherical-trigonometry-1886/eq-d4633f403d",16,"Todhunter 1886, scan 145: \\cos \\theta \\cot \\tfrac{1}{2}c \\sin \\tfrac{1}{2} E + \\sin \\theta \\cos \\left(\\phi - \\tfrac{1}{2}E + \\dfrac{\\pi}{2}\\right)"],["hardy-course-of-pure-mathematics-1921/eq-244c1cdb94",16,"Hardy 1921, p. 318: \\phi(1) + \\phi(2) + \\dots"],["dickson-theory-of-equations-1922/eq-c220d80036",16,"Dickson 1922, p. 35: x^3 - 3x + 1 = 0"],["whitehead-introduction-to-mathematics-1911/ch-xiv",2,"Whitehead 1911, ch. XIV: Series","../books/whitehead-introduction-to-mathematics-1911/ch/ch-xiv/index.html"],["hardy-course-of-pure-mathematics-1921/eq-9d9f45da8e",16,"Hardy 1921, p. 319: \\Phi(\\xi) = \\int_{1}^{\\xi} \\frac{dx}{x^{s}} = \\frac{\\xi^{1-s} - 1}{1 - s}"],["todhunter-spherical-trigonometry-1886/eq-c2afe48dd1",16,"Todhunter 1886, scan 145: \\beta = \\tfrac{1}{2} E - \\frac{\\pi}{2}"],["concept/series-of-n-s",7,"series of n^(-s)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxi",3,"Hardy 1921, Exercise LXXXI"],["hardy-course-of-pure-mathematics-1921/ex-lxxxi/1",4,"Hardy 1921, Exercise LXXXI (1)"],["form/9812e43864",5,"solve: Eq(3*x - 24, x)"],["concept/point",7,"point","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-point"],["hardy-course-of-pure-mathematics-1921/ex-lxxxi/2",4,"Hardy 1921, Exercise LXXXI (2)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxi/3",4,"Hardy 1921, Exercise LXXXI (3)"],["concept/triad",7,"triad","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-triad"],["de-morgan-elementary-illustrations-calculus-1899/x-c9c24e93e7",15,"De Morgan 1899, p. 127: If we take the function cx^{n}, c being independent ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxi/4",4,"Hardy 1921, Exercise LXXXI (4)"],["hardy-course-of-pure-mathematics-1921/eq-ae97425246",16,"Hardy 1921, p. 319: \\Phi(\\xi) \\to \\frac{1}{(s - 1)} = l"],["wentworth-first-steps-in-algebra-1894/ex-10/10",4,"Wentworth 1894, Exercise 10 (10)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxi/5",4,"Hardy 1921, Exercise LXXXI (5)"],["shape/a1e7ea10ee",6,"solve: Eq(N*x + N, x)"],["de-morgan-elementary-illustrations-calculus-1899/x-5fdd8b024c",15,"De Morgan 1899, p. 127: Here y = p^{\\efrac{1}{2}} x^{\\efrac{1}{2}}, and we must find ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxi/6",4,"Hardy 1921, Exercise LXXXI (6)"],["concept/consequent-of-a-proportion",7,"consequent of a proportion","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-consequent-of-a-proportion"],["concept/means-of-a-proportion",7,"means of a proportion","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-means-of-a-proportion"],["concept/extremes-of-a-proportion",7,"extremes of a proportion","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-extremes-of-a-proportion"],["boyden-first-book-in-algebra-1895/ex-28/13",4,"Boyden 1895, Exercise 28 (13)"],["concept/tends-to-infinity",7,"tends to infinity","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-tends-to-infinity"],["shape/e289f05592",6,"solve: Eq(N + a, N - x)"],["boyden-first-book-in-algebra-1895/ex-30/15",4,"Boyden 1895, Exercise 30 (15)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxi/7",4,"Hardy 1921, Exercise LXXXI (7)"],["whitehead-introduction-to-mathematics-1911/x-29bd387c8d",15,"Whitehead 1911, p. 220: This idea is immediately presented to us by the ..."],["concept/continued-proportion",7,"continued proportion","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-continued-proportion"],["whitehead-introduction-to-mathematics-1911/x-9227aa7012",15,"Whitehead 1911, p. 220: The really inspiring reflection suggested by the history of ..."],["hardy-course-of-pure-mathematics-1921/eq-567e7850a0",16,"Hardy 1921, p. 319: \\Phi(\\xi) = \\int_{1}^{\\xi} \\frac{dx}{x}"],["hardy-course-of-pure-mathematics-1921/eq-03dd00067e",16,"Hardy 1921, p. 320: \\ds\\Phi(\\xi) > n\\int_{1}^{2} \\frac{du}{u}"],["form/29aacab503",5,"evaluate: sqrt(2173945)/250"],["boyden-first-book-in-algebra-1895/ex-30/16",4,"Boyden 1895, Exercise 30 (16)"],["form/a6994b9289",5,"evaluate: sqrt(7)"],["concept/relatively-prime",7,"relatively prime","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-relatively-prime"],["hardy-course-of-pure-mathematics-1921/eq-fe79c967e6",16,"Hardy 1921, p. 320: \\int_{2^{r}}^{2^{r+1}} \\frac{dx}{x} = \\int_{1}^{2} \\frac{du}{u}"],["hardy-course-of-pure-mathematics-1921/ex-lxxxi/8",4,"Hardy 1921, Exercise LXXXI (8)"],["dickson-theory-of-equations-1922/eq-5e538f39c6",16,"Dickson 1922, p. 35: x = 2 \\cos B"],["hardy-course-of-pure-mathematics-1921/eq-d2ced67269",16,"Hardy 1921, p. 423: \\phi^{(n)}(t) = m(m - 1) \\dots (m - n + 1)z^{n} (1 + tz)^{m-n}"],["wentworth-first-steps-in-algebra-1894/ex-1/4",4,"Wentworth 1894, Exercise 1 (4)"],["form/dd515b3ec6",5,"solve: (Eq(b, 4*a), Eq(a + b, 210))"],["form/f9f3974600",5,"identity: 11"],["dickson-theory-of-equations-1922/eq-67e891a912",16,"Dickson 1922, p. 35: \\cos 3B = \\cos 4B"],["dickson-theory-of-equations-1922/eq-5c388569d0",16,"Dickson 1922, p. 35: 2(4 \\cos^3 B - 3 \\cos B) = x^3 - 3x"],["shape/fd481e43b3",6,"evaluate: N**N"],["concept/third-proportional",7,"third proportional","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-third-proportional"],["hardy-course-of-pure-mathematics-1921/ex-lxxxi/9",4,"Hardy 1921, Exercise LXXXI (9)"],["hardy-course-of-pure-mathematics-1921/x-499174b4ed",15,"Hardy 1921, p. 445: The object of this brief note is to point ..."],["concept/conjugate-arcs",7,"conjugate arcs","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-conjugate-arcs"],["concept/semicircle",7,"semicircle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-semicircle"],["dickson-theory-of-equations-1922/eq-b01e00317d",16,"Dickson 1922, p. 35: 4(2 \\cos^2 B - 1)^2 - 2 = (x^2 - 2)^2 - 2"],["dickson-theory-of-equations-1922/eq-70f9c72cd5",16,"Dickson 1922, p. 35: 0 = x^4 - 4x^2 + 2 - (x^3 - 3x) = (x - 2)(x^3 + x^2 - 2x - 1)"],["dickson-theory-of-equations-1922/eq-e329073739",16,"Dickson 1922, p. 35: x^3 + x^2 - 2x - 1 = 0"],["form/ad853fde36",5,"solve: (Eq(a + 10, 2*b + 20), Eq(a + b, 40))"],["theorem/basic-proportionality-theorem",9,"basic proportionality theorem","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-basic-proportionality-theorem"],["shape/ed8210ad16",6,"solve: (Eq(N + a, N*b + N), Eq(a + b, N))"],["de-morgan-elementary-illustrations-calculus-1899/x-326537879a",15,"De Morgan 1899, p. 126: Hence, y being the ordinate, the area included between ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxii/1",4,"Hardy 1921, Exercise LXXXII (1)"],["concept/cylinder",7,"cylinder","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-cylinder"],["hardy-course-of-pure-mathematics-1921/x-1256770113",15,"Hardy 1921, p. 445: In what may be called ‘common Cartesian geometry’, a ..."],["theorem/alternation-of-proportions",9,"alternation of proportions","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-alternation-of-proportions"],["theorem/inversion-of-proportions",9,"inversion of proportions","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-inversion-of-proportions"],["hardy-course-of-pure-mathematics-1921/ex-lxxxii/2",4,"Hardy 1921, Exercise LXXXII (2)"],["de-morgan-elementary-illustrations-calculus-1899/x-92631a44d1",15,"De Morgan 1899, p. 127: at a time when such a step was one ..."],["boyden-first-book-in-algebra-1895/ex-22/25",4,"Boyden 1895, Exercise 22 (25)"],["theorem/relations-between-roots-and-coefficients",9,"relations between roots and coefficients","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-relations-between-roots-and-coefficients"],["macfarlane-vector-analysis-quaternions-1906/eq-97ce7fee53",16,"Macfarlane 1906: R = \\frac{\\sum mA}{\\sum m},\\quad\\text{or}\\quad R \\sum m = \\sum mA"],["macfarlane-vector-analysis-quaternions-1906/eq-742cb68077",16,"Macfarlane 1906: x &= \\frac{\\sum (ma)}{\\sum m}"],["macfarlane-vector-analysis-quaternions-1906/eq-fe0c3aa6c6",16,"Macfarlane 1906: y = \\frac{\\sum (mb)}{\\sum m}"],["macfarlane-vector-analysis-quaternions-1906/eq-367be9dbee",16,"Macfarlane 1906: z = \\frac{\\sum (mc)}{\\sum m}"],["macfarlane-vector-analysis-quaternions-1906/eq-3c7a4696f2",16,"Macfarlane 1906: A &= a_1j + b_1j+c_1k;"],["macfarlane-vector-analysis-quaternions-1906/eq-41d6ed4d1f",16,"Macfarlane 1906: F_A &= F_R + \\mathrm{V}(A-R)F"],["hardy-course-of-pure-mathematics-1921/ex-lxxxii/3",4,"Hardy 1921, Exercise LXXXII (3)"],["method/solving-an-equation-graphically",8,"solving an equation graphically","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-solving-an-equation-graphically"],["concept/parallelogram",7,"parallelogram","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-parallelogram"],["hardy-course-of-pure-mathematics-1921/ex-lxxxii/4",4,"Hardy 1921, Exercise LXXXII (4)"],["form/771e8b4242",5,"solve: Eq(-x + 60, -2*x + 100)"],["hardy-course-of-pure-mathematics-1921/eq-c5aaff6190",16,"Hardy 1921, p. 423: \\frac{\\phi^{n}(0)}{n!} = \\binom{m}{n} z^{n}"],["hardy-course-of-pure-mathematics-1921/eq-7a0455f910",16,"Hardy 1921, p. 423: \\phi(1) = \\phi(0) + \\phi'(0) + \\frac{\\phi''(0)}{2!} + \\dots + \\frac{\\phi^{(n-1)}(0)}{(n - 1)!} + R_{n}"],["theorem/taylor-s-theorem-with-remainder",9,"Taylor's theorem with remainder"],["hardy-course-of-pure-mathematics-1921/ex-lxxxiii/1",4,"Hardy 1921, Exercise LXXXIII (1)"],["hardy-course-of-pure-mathematics-1921/x-7941a2e75f",15,"Hardy 1921, p. 64: It is entirely built up of straight lines; but ..."],["hardy-course-of-pure-mathematics-1921/x-090ac09035",15,"Hardy 1921, p. 56: The reader may possibly regard this as an unreasonable ..."],["hardy-course-of-pure-mathematics-1921/x-0a3d21532d",15,"Hardy 1921, p. 51: This function is defined for all values of x ..."],["macfarlane-vector-analysis-quaternions-1906/eq-5697f9bc09",16,"Macfarlane 1906: \\sum \\left(F_A\\right) &= \\sum \\left(F_R\\right) + \\sum \\mathrm{V}\\left(A - R\\right)F"],["macfarlane-vector-analysis-quaternions-1906/eq-2c679ce9e9",16,"Macfarlane 1906: \\mathrm{V} R \\sum F &= \\sum \\mathrm{V} A F"],["shape/b6d7fcf4b7",6,"solve: Eq(N - x, N*x + N)"],["hardy-course-of-pure-mathematics-1921/ch-v",2,"Hardy 1921, ch. V: LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS","../books/hardy-course-of-pure-mathematics-1921/ch/ch-v/index.html"],["hardy-course-of-pure-mathematics-1921/ex-lxxxiii/2",4,"Hardy 1921, Exercise LXXXIII (2)"],["hardy-course-of-pure-mathematics-1921/x-b971722392",15,"Hardy 1921, p. 83: We conclude that a quadratic equation with real coefficients ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-v",3,"Hardy 1921, Exercise Misc-V"],["form/7a937fc7f6",5,"solve: Eq(-x + 100, x + 20)"],["concept/melting-point",7,"melting point","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-melting-point"],["shape/c30669ec03",6,"solve: Eq(N - x, N + x)"],["concept/base-line",7,"base line","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-base-line"],["form/9d64258b47",5,"solve: Eq(x + 50, 3*x + 30)"],["shape/47574556c7",6,"solve: Eq(N + x, N*x + N)"],["person/thomas-andrews",1,"Thomas Andrews","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-thomas-andrews"],["hardy-course-of-pure-mathematics-1921/x-1623dda3d7",15,"Hardy 1921, p. 80: One most important property of real numbers is that ..."],["hardy-course-of-pure-mathematics-1921/x-0a87af19ed",15,"Hardy 1921, p. 84: All such theorems as these are true whether a, ..."],["form/6d7ffe79f6",5,"identity: (-3*a**2*b + 4*c*d**3)**4"],["hardy-course-of-pure-mathematics-1921/eq-8e25ded8b4",16,"Hardy 1921, p. 423: R_{n} = \\frac{1}{(n - 1)!}\\int_{0}^{1} (1 - t)^{n-1} \\phi^{(n)}(t)\\, dt"],["form/0adc8e4811",5,"identity: (a/2 - b)**2"],["shape/2a774f8a3a",6,"identity: (N*a**N*b + N*c*d**N)**N"],["concept/chordal-triangle",7,"chordal triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-chordal-triangle"],["boyden-first-book-in-algebra-1895/ex-22/26",4,"Boyden 1895, Exercise 22 (26)"],["shape/53cd771105",6,"identity: (N*a - 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(28)"],["todhunter-spherical-trigonometry-1886/x-7c4fa9c288",15,"Todhunter 1886, scan 95: At various points of the country suitable stations are ..."],["de-morgan-elementary-illustrations-calculus-1899/ch-applications-of-the-theorem-for-implicit-differentiation",2,"De Morgan 1899, Applications of the Theorem for Implicit Differentiation","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-applications-of-the-theorem-for-implicit-differentiation/index.html"],["concept/sailing-faster-than-the-wind",7,"sailing faster than the wind","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-sailing-faster-than-the-wind"],["concept/relative-motion",7,"relative motion","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-relative-motion"],["concept/proportion",7,"proportion","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-proportion"],["todhunter-spherical-trigonometry-1886/x-e24a68a1ab",15,"Todhunter 1886, scan 101: thus in observing three angles, we suppose that in ..."],["form/3d739e9b8c",5,"identity: (a**2 - 3)**4"],["todhunter-spherical-trigonometry-1886/x-235471c74c",15,"Todhunter 1886, scan 97: The three methods which we have indicated were all ..."],["shape/67955403d3",6,"identity: (N + a**N)**N"],["wentworth-plane-geometry-1899/x-5a42042395",15,"Wentworth 1899, scan 144: Thus, in the proportion a:b = b:c; b is ..."],["concept/isolated-system",7,"isolated system","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-isolated-system"],["concept/kinematics",7,"kinematics","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-kinematics"],["concept/boat-moved-by-a-rope",7,"boat moved by a rope","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-boat-moved-by-a-rope"],["theorem/existence-of-a-continuous-inverse-function",9,"existence of a continuous inverse 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centres","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-line-of-centres"],["dickson-theory-of-equations-1922/eq-5481ae5491",16,"Dickson 1922, p. 36: \\frac{1}{R} = \\cos\\frac{2\\pi}{7} -i \\sin\\frac{2\\pi}{7}"],["ball-mathematical-recreations-1905/x-f676ad7d4a",15,"Ball 1905, scan 108: For example, if the side of a cube is ..."],["ball-mathematical-recreations-1905/x-148e1fbd2f",15,"Ball 1905, scan 107: Of course it is only in isolated systems that ..."],["ball-mathematical-recreations-1905/x-0b33997537",15,"Ball 1905, scan 101: Montucla calculated the mass of the earth and, assuming ..."],["ball-mathematical-recreations-1905/x-a54bfde0a1",15,"Ball 1905, scan 108: Again, if the linear dimensions of a man of ..."],["planck-treatise-on-thermodynamics-1903/x-caaea4eaef",15,"Planck 1903, p. 162: If, as a rough approximation, we assume this same ..."],["dickson-theory-of-equations-1922/eq-ff10398641",16,"Dickson 1922, p. 36: R + \\frac{1}{R} = 2\\cos\\frac{2\\pi}{7}"],["de-morgan-elementary-illustrations-calculus-1899/x-309884c9fd",15,"De Morgan 1899, p. 12: Nothing is either small or great in itself, these ..."],["todhunter-spherical-trigonometry-1886/x-e210bfbeed",15,"Todhunter 1886, scan 138: A sphere is described about a regular polyhedron; from ..."],["ball-mathematical-recreations-1905/x-a51b2fc84e",15,"Ball 1905, scan 113: Thus, if a current of air is moving in ..."],["ball-mathematical-recreations-1905/x-ad73525262",15,"Ball 1905, scan 114: If anyone blows steadily through the tube so formed, ..."],["concept/cosecant",7,"cosecant","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cosecant"],["theorem/tangent-perpendicular-to-radius",9,"tangent perpendicular to radius","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-tangent-perpendicular-to-radius"],["de-morgan-elementary-illustrations-calculus-1899/x-892fd7cd5f",15,"De Morgan 1899, p. 13: Here all dispute about a standard of smallness is ..."],["ball-mathematical-recreations-1905/eq-1b9074e714",16,"Ball 1905, scan 149: \\frac{1}{2}n(n+1)"],["concept/homologous-sides",7,"homologous sides","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-homologous-sides"],["todhunter-spherical-trigonometry-1886/x-c39907c4c1",15,"Todhunter 1886, scan 136: We leave to the student the exercise of shewing ..."],["ball-mathematical-recreations-1905/eq-f55dd01c56",16,"Ball 1905, scan 152: m^m"],["wentworth-plane-geometry-1899/x-d48594ea2f",15,"Wentworth 1899, scan 84: A tangent is a straight line of unlimited length ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv/2",4,"Hardy 1921, Exercise LXXXIV (2)"],["de-morgan-elementary-illustrations-calculus-1899/x-e029409be3",15,"De Morgan 1899, p. 13: The desire of combining simplicity with the appearance of ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv/3",4,"Hardy 1921, Exercise LXXXIV (3)"],["method/anstice-s-method",8,"Anstice's method","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-anstice-s-method"],["form/d37ef4dca0",5,"identity: 9*a**7*b**9*c**9"],["dickson-theory-of-equations-1922/eq-48711b4d4d",16,"Dickson 1922, p. 36: \\left(y^3 + \\frac{1}{y^3}\\right) + \\left(y^2 + \\frac{1}{y^2}\\right) + \\left(y + \\frac{1}{y }\\right) + 1=0"],["maxwell-elementary-treatise-electricity-1888/eq-02ef363089",16,"Maxwell 1888, scan 208: G = 2n(1-n)R\\text{,}"],["form/4e968b2960",5,"identity: -10*a**2"],["dickson-theory-of-equations-1922/eq-7ded9bbf7a",16,"Dickson 1922, p. 36: y^6 + y^5 + y^4 + y^3 + y^2 + y + 1 = 0"],["dickson-theory-of-equations-1922/eq-012ec9707c",16,"Dickson 1922, p. 36: y+ \\frac{1}{y} = x"],["boyden-first-book-in-algebra-1895/ex-22/29",4,"Boyden 1895, Exercise 22 (29)"],["person/monge",1,"Monge","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-monge"],["theorem/kirchhoff-s-formula",9,"Kirchhoff's formula","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-theorem-kirchhoff-s-formula"],["dickson-theory-of-equations-1922/eq-d4c872d09a",16,"Dickson 1922, p. 36: y^2 + \\frac{1}{y^2} = x^2 - 2"],["method/equating-real-and-imaginary-parts",8,"equating real and imaginary parts","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-equating-real-and-imaginary-parts"],["concept/inflection-tangent",7,"inflection tangent","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-inflection-tangent"],["hardy-course-of-pure-mathematics-1921/x-b9b5b5b17d",15,"Hardy 1921, p. 195: If \\phi(x) = 1/q when x = p/q, and ..."],["form/918fd9237c",5,"solve: Eq(x, 20*a**2 - b)"],["dickson-theory-of-equations-1922/eq-a64f147991",16,"Dickson 1922, p. 36: y^3 + \\frac{1}{y^3} = x^3 - 3x"],["de-morgan-elementary-illustrations-calculus-1899/x-ebd6cb0f5e",15,"De Morgan 1899, p. 13: If a be increased by h, a^{2} is increased ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv/4",4,"Hardy 1921, Exercise LXXXIV (4)"],["de-morgan-elementary-illustrations-calculus-1899/x-738869f2e0",15,"De Morgan 1899, p. 14: This should be considered as an abbreviation of the ..."],["concept/common-ion",7,"common ion","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-common-ion"],["dickson-theory-of-equations-1922/eq-afbce9cf36",16,"Dickson 1922, p. 37: x_1 = R + \\frac{1}{R} = R + R^6"],["dickson-theory-of-equations-1922/eq-18ac9e662d",16,"Dickson 1922, p. 37: x_2 = R^2 + \\frac{1}{R^2} = R^2 + R^5"],["method/method-of-indivisibles",8,"method of indivisibles","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-method-of-indivisibles"],["dickson-theory-of-equations-1922/eq-7556512d7c",16,"Dickson 1922, p. 37: x_3 = R^3 + \\frac{1}{R^3} = R^3 + R^4"],["ball-mathematical-recreations-1905/x-ab6f0b9c3b",15,"Ball 1905, scan 149: For example, if we are told that in figure ..."],["concept/leg-of-a-right-triangle",7,"leg of a right triangle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-leg-of-a-right-triangle"],["hardy-course-of-pure-mathematics-1921/eq-73b6dda4b1",16,"Hardy 1921, p. 422: (1 + z)^{m} = \\exp\\{m\\log(1 + z)\\}"],["concept/sign-of-a-polynomial-at-infinity",7,"sign of a polynomial at infinity","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-sign-of-a-polynomial-at-infinity"],["dickson-theory-of-equations-1922/x-fb53dcb8e9",15,"Dickson 1922, p. 56: The purpose of the example was, however, not to ..."],["dickson-theory-of-equations-1922/eq-afd1a67805",16,"Dickson 1922, p. 37: x_1 + x_2 + x_3 = R + R^2 + \\dotsb + R^6 = -1"],["dickson-theory-of-equations-1922/eq-69b49ea4e1",16,"Dickson 1922, p. 37: x_1 x_2 + x_1 x_3 + x_2 x_3 = 2(R + R^2 + \\dotsb +R^6) = -2"],["ball-mathematical-recreations-1905/eq-39afb562ae",16,"Ball 1905, scan 152: n=km^{m-1}-jm^{m-2} + \\dotsb + bm - a + 1"],["shape/d30753a921",6,"solve: Eq(x, N*a**N - b)"],["dickson-theory-of-equations-1922/eq-56eefb0d98",16,"Dickson 1922, p. 37: x_1 x_2 x_3 = 2 + R + R^2 + \\dotsb + R^6 = 1"],["dickson-theory-of-equations-1922/ex-page89",3,"Dickson 1922, Exercise Page89"],["form/a8dc265f3c",5,"solve: Eq(7*x/2, 84)"],["dickson-theory-of-equations-1922/eq-0b11f3e3fa",16,"Dickson 1922, p. 37: y^n f\\left(\\frac{1}{y}\\right) \\equiv ±f(y)"],["dickson-theory-of-equations-1922/ex-page100",3,"Dickson 1922, Exercise Page100"],["dickson-theory-of-equations-1922/ex-page94",3,"Dickson 1922, Exercise Page94"],["dickson-theory-of-equations-1922/ex-page96",3,"Dickson 1922, Exercise Page96"],["dickson-theory-of-equations-1922/ex-page98",3,"Dickson 1922, Exercise Page98"],["dickson-theory-of-equations-1922/ex-page99",3,"Dickson 1922, Exercise Page99"],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv/5",4,"Hardy 1921, Exercise LXXXIV (5)"],["dickson-theory-of-equations-1922",0,"Dickson, First Course in the Theory of Equations (1922)","../books/dickson-theory-of-equations-1922/index.html"],["dickson-theory-of-equations-1922/eq-2691c26fd1",16,"Dickson 1922, p. 37: c^2 = 1"],["dickson-theory-of-equations-1922/eq-dd15919ba8",16,"Dickson 1922, p. 38: f(y) \\equiv y^n ± 1 + p_1(y^{n-1} ± y) + p_2 (y^{n-2} ± y^2) + \\dotsb"],["dickson-theory-of-equations-1922/eq-036d6e2de2",16,"Dickson 1922, p. 38: y^{n-1} Q \\left(\\frac{1}{y}\\right) \\equiv Q(y)"],["dickson-theory-of-equations-1922/ch-vii",2,"Dickson 1922, ch. VII: Solution of Numerical Equations","../books/dickson-theory-of-equations-1922/ch/ch-vii/index.html"],["ball-mathematical-recreations-1905/eq-ae3f8b6617",16,"Ball 1905, scan 152: n=km^{m-1}-jm^{m-2} + \\dotsb - bm + a"],["ball-mathematical-recreations-1905/eq-c9450cf8e7",16,"Ball 1905, scan 153: 64d-16c + 4b-a + 1"],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv/6",4,"Hardy 1921, Exercise LXXXIV (6)"],["theorem/binomial-theorem-general-form",9,"binomial theorem (general form)"],["de-morgan-elementary-illustrations-calculus-1899/x-64fef63503",15,"De Morgan 1899, p. 74: \\phi' x is the limit of \\dfrac{dy}{dx}, or the ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv/7",4,"Hardy 1921, Exercise LXXXIV (7)"],["theorem/expansion-of-a-function-by-increments",9,"expansion of a function by increments","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-expansion-of-a-function-by-increments"],["form/8514cdf9d2",5,"solve: (Eq(a - x, 104), Eq(a + x, 2044))"],["shape/e0419646b4",6,"solve: (Eq(a - x, N), Eq(a + x, N))"],["ball-mathematical-recreations-1905/eq-c8fcb70de5",16,"Ball 1905, scan 153: 64-16c + 4b-a + 1"],["ball-mathematical-recreations-1905/eq-a1784b9886",16,"Ball 1905, scan 154: n = 9c-3b + a"],["theorem/napier-s-rules-of-circular-parts",9,"Napier's Rules of Circular Parts","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-napier-s-rules-of-circular-parts"],["ball-mathematical-recreations-1905/eq-2e8aaa9ca4",16,"Ball 1905, scan 153: 9-3b + a"],["maxwell-elementary-treatise-electricity-1888/eq-68850259e4",16,"Maxwell 1888, scan 207: \\delta = C \\sqrt{G} \\xi\\text{,}"],["concept/proportionality",7,"proportionality"],["de-morgan-elementary-illustrations-calculus-1899/x-5d0244ec86",15,"De Morgan 1899, p. 74: That in every case which occurs in practice, dx ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv/8a",4,"Hardy 1921, Exercise LXXXIV (8a)"],["de-morgan-elementary-illustrations-calculus-1899/x-ee49dc297a",15,"De Morgan 1899, p. 74: It is called the differential coefficient of y."],["form/1c15345973",5,"differentiate: log(log(x))"],["ball-mathematical-recreations-1905/eq-136f91922a",16,"Ball 1905, scan 154: 9(c-1) + (8-3b + a) + 1"],["concept/tangent-circles",7,"tangent circles","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-tangent-circles"],["concept/subtangent",7,"subtangent","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-subtangent"],["ball-mathematical-recreations-1905/eq-a1166c63be",16,"Ball 1905, scan 154: n = 9c-3b + a= 14"],["shape/1c15345973",6,"differentiate: log(log(x))"],["concept/middle",7,"middle"],["person/william-george-horner",1,"William George Horner","../books/dickson-theory-of-equations-1922/terms/index.html#t-person-william-george-horner"],["shape/b240c5ff55",6,"solve: (Eq(x, N*a), Eq(x, N*a + N))"],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv/8b",4,"Hardy 1921, Exercise LXXXIV (8b)"],["form/073ccd27bc",5,"differentiate: log(log(log(x)))"],["shape/073ccd27bc",6,"differentiate: log(log(log(x)))"],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv/9a",4,"Hardy 1921, Exercise LXXXIV (9a)"],["form/cbc22adb57",5,"solve: (Eq(x, 4*a), Eq(x, 3*a + 6))"],["form/92d763e961",5,"solve: (Eq(x, 2*a), Eq(a + 30, 2*x - 60))"],["ball-mathematical-recreations-1905/eq-55ced50757",16,"Ball 1905, scan 155: 3p = 9c-3b"],["concept/incommensurable-ratio",7,"incommensurable ratio","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-incommensurable-ratio"],["concept/numerical-measure",7,"numerical measure","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-numerical-measure"],["ball-mathematical-recreations-1905/eq-eff498fcee",16,"Ball 1905, scan 155: p = 3c-b"],["dickson-theory-of-equations-1922/eq-a8012a68aa",16,"Dickson 1922, p. 38: y^{2t} + 1 + c_1 (y^{2t-1} + y) + c_2 (y^{2t-2} + y^2) + \\dotsb + c_{t-1} (y^{t+1} + y^{t-1}) + c_t y^t = 0"],["de-morgan-elementary-illustrations-calculus-1899/x-07d0c83f06",15,"De Morgan 1899, p. 14: An expression may be a function of more quantities ..."],["dickson-theory-of-equations-1922/eq-79adf0e5ff",16,"Dickson 1922, p. 38: y^k + \\frac{1}{y^k} = x \\left(y^{k-1} + \\frac{1}{y^{k-1}}\\right) - \\left(y^{k-2} + \\frac{1}{y^{k-2}}\\right)"],["dickson-theory-of-equations-1922/eq-776c7f347f",16,"Dickson 1922, p. 38: n = 2t+1"],["form/77c56ec49f",5,"differentiate: log(x)**a"],["method/laplace-s-development",8,"Laplace's development","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-laplace-s-development"],["shape/77c56ec49f",6,"differentiate: log(x)**a"],["theorem/gergonne-s-equation",9,"Gergonne's equation"],["de-morgan-elementary-illustrations-calculus-1899/x-9d0eb71507",15,"De Morgan 1899, p. 15: Thus in 1, the length of the radius OB ..."],["ball-mathematical-recreations-1905/eq-b4377910ed",16,"Ball 1905, scan 155: 9z + 3y + x = n-1"],["de-morgan-elementary-illustrations-calculus-1899/x-6d33ed0759",15,"De Morgan 1899, p. 15: And, as in algebra we reason on numbers by ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxiv/9b",4,"Hardy 1921, Exercise LXXXIV (9b)"],["hardy-course-of-pure-mathematics-1921/eq-412cfdb012",16,"Hardy 1921, p. 422: \\left|\\frac{a_{n+1}}{a_{n}}\\right| = \\left|\\frac{m - n}{n + 1}\\right| \\to 1"],["wentworth-plane-geometry-1899/x-a9f1c88e27",15,"Wentworth 1899, scan 101: But by taking the unit sufficiently small, an approximate ..."],["form/b880aa77e0",5,"differentiate: log(log(x))**a"],["shape/b880aa77e0",6,"differentiate: log(log(x))**a"],["de-morgan-elementary-illustrations-calculus-1899/x-65b25baf5b",15,"De Morgan 1899, p. 15: Here it must be borne in mind that \\phi ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxix/1",4,"Hardy 1921, Exercise LXXXIX (1)"],["concept/fifteen-puzzle",7,"fifteen puzzle","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-fifteen-puzzle"],["concept/cyclical-permutation",7,"cyclical permutation","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cyclical-permutation"],["concept/tower-of-hanoi",7,"Tower of Hanoi","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-tower-of-hanoi"],["concept/chinese-rings",7,"Chinese rings","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-chinese-rings"],["ball-mathematical-recreations-1905/x-2431185198",15,"Ball 1905, scan 121: Now a cyclical permutation of n letters is equivalent ..."],["wentworth-first-steps-in-algebra-1894/ex-1/18",4,"Wentworth 1894, Exercise 1 (18)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxix/2",4,"Hardy 1921, Exercise LXXXIX (2)"],["ball-mathematical-recreations-1905/x-f11c553f38",15,"Ball 1905, scan 124: Hence, if it requires x transfers of simple discs ..."],["wentworth-first-steps-in-algebra-1894/ex-16/8",4,"Wentworth 1894, Exercise 16 (8)"],["dickson-theory-of-equations-1922/eq-bf244731b0",16,"Dickson 1922, p. 39: x (x^3-3x) - (x^2-2) = x^4 - 4x^2 + 2"],["dickson-theory-of-equations-1922/eq-d27c7b37fb",16,"Dickson 1922, p. 40: R^8 = R"],["form/79c66ad900",5,"solve: (Eq(a, 2*x), Eq(2*a + x/4, 51/4))"],["wentworth-first-steps-in-algebra-1894/ex-1/19",4,"Wentworth 1894, Exercise 1 (19)"],["form/595a63889b",5,"identity: -a**4*b**3"],["dickson-theory-of-equations-1922/eq-4d3eaab8e1",16,"Dickson 1922, p. 39: R = \\cos\\frac{ 2\\pi}{9} + i \\sin\\frac{ 2\\pi}{9}"],["shape/aab7b018ca",6,"identity: -a**N*b**N"],["dickson-theory-of-equations-1922/eq-9b36a1ffcb",16,"Dickson 1922, p. 39: \\frac{y^9 - 1}{y^3 - 1} = y^6 + y^3 + 1 = 0"],["dickson-theory-of-equations-1922/eq-739a438991",16,"Dickson 1922, p. 41: z_1 = R + R^2 + R^4"],["hardy-course-of-pure-mathematics-1921/ex-lxxxix/3",4,"Hardy 1921, Exercise LXXXIX (3)"],["hardy-course-of-pure-mathematics-1921/x-7b382ad15e",15,"Hardy 1921, p. 363: We define e as the number whose logarithm is ..."],["hardy-course-of-pure-mathematics-1921/x-a0edd4c0b1",15,"Hardy 1921, p. 363: Since \\log x is an increasing function of x, ..."],["hardy-course-of-pure-mathematics-1921/eq-8954d0489f",16,"Hardy 1921, p. 422: \\frac{d}{dt}(1 + tz)^{m} = mz(1 + tz)^{m-1}"],["wentworth-first-steps-in-algebra-1894/ex-1/20",4,"Wentworth 1894, Exercise 1 (20)"],["wentworth-first-steps-in-algebra-1894/ex-17/5",4,"Wentworth 1894, Exercise 17 (5)"],["person/j-thomsen",1,"J. Thomsen","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-j-thomsen"],["dickson-theory-of-equations-1922/eq-147b3b4862",16,"Dickson 1922, p. 41: z_2 = R^3 + R^6 + R^5"],["dickson-theory-of-equations-1922/eq-c151e5de6f",16,"Dickson 1922, p. 41: z_1 z_2 = 3 + R + \\dotsb + R^6 = 2"],["dickson-theory-of-equations-1922/eq-3db1099072",16,"Dickson 1922, p. 41: z^2 + z + 2 = 0"],["ball-mathematical-recreations-1905/x-0803ab2792",15,"Ball 1905, scan 125: Proceeding in this way we see that with a ..."],["concept/major-axis",7,"major axis","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-major-axis"],["concept/discriminant-of-a-conic",7,"discriminant of a conic","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-discriminant-of-a-conic"],["wentworth-plane-geometry-1899/eq-fe99585815",16,"Wentworth 1899, scan 198: \\triangle{}ABC=\\frac{1}{2}a × b"],["wentworth-plane-geometry-1899/eq-6dd9773ac6",16,"Wentworth 1899, scan 194: \\dfrac{\\rect AF}{\\rect AC} = \\dfrac{AE}{AB}"],["wentworth-plane-geometry-1899/eq-f10a3c8214",16,"Wentworth 1899, scan 196: \\dfrac{R}{U} = \\dfrac{a × b}{1 × 1}"],["wentworth-plane-geometry-1899/eq-5a61a291b1",16,"Wentworth 1899, scan 197: \\Par AEFD = a × b"],["wentworth-plane-geometry-1899/eq-109140c2f8",16,"Wentworth 1899, scan 199: ABCH=\\frac{1}{2}a(b+b')"],["concept/specific-gravity",7,"specific gravity","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-specific-gravity"],["hardy-course-of-pure-mathematics-1921/eq-ec86ce3556",16,"Hardy 1921, p. 433: Z = P(z) = \\alpha_{0} z^{n} + \\alpha_{1} z^{n-1} + \\dots + \\alpha_{n}"],["dickson-theory-of-equations-1922/eq-a5b3b0f054",16,"Dickson 1922, p. 41: w^3 - z_1w^2 + z_2w - 1 = 0"],["dickson-theory-of-equations-1922/eq-2fb1f035ce",16,"Dickson 1922, p. 41: \\frac{R^{17} - 1}{R - 1} = R^{16} + R^{15} + \\dotsb + R + 1 = 0"],["dickson-theory-of-equations-1922/eq-11b5480f09",16,"Dickson 1922, p. 41: y_1 + y_2 = -1"],["dickson-theory-of-equations-1922/eq-31bfd5a67c",16,"Dickson 1922, p. 41: y_1 y_2 = 4(R + \\dotsb + R^{16}) = -4"],["dickson-theory-of-equations-1922/eq-b7e47d7070",16,"Dickson 1922, p. 41: y^2 + y - 4 = 0"],["dickson-theory-of-equations-1922/eq-a46a7513f5",16,"Dickson 1922, p. 41: R + R^9 + R^{13} + R^{15} + R^{16} + R^8 + R^4 + R^2"],["wentworth-plane-geometry-1899/eq-6a08926d90",16,"Wentworth 1899, scan 200: \\dfrac{\\triangle ABC} {\\triangle ADE} = \\dfrac{AB × AC} {AD × AE}"],["method/interpolating-in-logarithmic-tables",8,"interpolating in logarithmic tables","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-method-interpolating-in-logarithmic-tables"],["ball-mathematical-recreations-1905/x-cc3482bea4",15,"Ball 1905, scan 129: Denote the rings which are on the bar by ..."],["ball-mathematical-recreations-1905/x-fa71bd947a",15,"Ball 1905, scan 132: In the case of an ordinary chess-board the determinant ..."],["boyden-first-book-in-algebra-1895/ex-29/1",4,"Boyden 1895, Exercise 29 (1)"],["concept/circular-cylinder",7,"circular cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-circular-cylinder"],["hardy-course-of-pure-mathematics-1921/eq-ac0e59712b",16,"Hardy 1921, p. 436: |Z| = |P(x + iy)|"],["concept/directrix",7,"directrix","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-directrix"],["de-morgan-elementary-illustrations-calculus-1899/x-9c37135fd4",15,"De Morgan 1899, p. 15: Thus, if \\phi x = x + x^{2}, \\phi ..."],["de-morgan-elementary-illustrations-calculus-1899/x-fc14f3b25e",15,"De Morgan 1899, p. 15: It may be easily conceived that this notion is ..."],["quantity/real-part-of-a-complex-number",11,"real part of a complex number","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-quantity-real-part-of-a-complex-number"],["todhunter-spherical-trigonometry-1886/x-bf9c3ef821",15,"Todhunter 1886, scan 42: The solution of spherical triangles is the process by ..."],["boyden-first-book-in-algebra-1895/ex-29/2",4,"Boyden 1895, Exercise 29 (2)"],["todhunter-spherical-trigonometry-1886/x-fe3829ed82",15,"Todhunter 1886, scan 44: These six formul comprise ten equations; and thus we ..."],["todhunter-spherical-trigonometry-1886/x-2d5f3913eb",15,"Todhunter 1886, scan 45: the side opposite the right angle is called the ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxix/4",4,"Hardy 1921, Exercise LXXXIX (4)"],["quantity/amplitude-of-a-complex-number",11,"amplitude of a complex number","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-quantity-amplitude-of-a-complex-number"],["todhunter-spherical-trigonometry-1886/x-3134c57182",15,"Todhunter 1886, scan 45: Napier was also the inventor of Logarithms, and the ..."],["todhunter-spherical-trigonometry-1886/x-9b98fafd49",15,"Todhunter 1886, scan 50: We do not give them, because we are convinced ..."],["person/menaechmus",1,"Menaechmus","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-menaechmus"],["hardy-course-of-pure-mathematics-1921/eq-42539b078a",16,"Hardy 1921, p. 435: |P(x + iy) - P(x_{0} + iy_{0})| < \\tfrac{1}{2}\\rho"],["hardy-course-of-pure-mathematics-1921/eq-361981b4fc",16,"Hardy 1921, p. 435: P(x_{0} + iy_{0}) = 0"],["person/nicolaus-copernicus",1,"Nicolaus Copernicus","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-nicolaus-copernicus"],["hardy-course-of-pure-mathematics-1921/eq-2e9a31a790",16,"Hardy 1921, p. 435: P(x + iy) = a + \\phi"],["dickson-theory-of-equations-1922/x-68501f81c6",15,"Dickson 1922, p. 145: Given without proof by Sylvester, Philosophical Magazine, 1840, p. ..."],["concept/board-of-mathematical-studies",7,"Board of Mathematical Studies","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-board-of-mathematical-studies"],["person/pierre-simon-laplace",1,"Pierre-Simon Laplace","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-pierre-simon-laplace"],["hardy-course-of-pure-mathematics-1921/eq-94cd082b71",16,"Hardy 1921, p. 435: \\delta_{m} = \\delta_{1}/2^{m-1}"],["whitehead-introduction-to-mathematics-1911/x-93dacdce5f",15,"Whitehead 1911, p. 129: Nothing illustrates better the gain in power which is ..."],["hardy-course-of-pure-mathematics-1921/eq-c0bd5e2e79",16,"Hardy 1921, p. 437: z = z_{0} + \\zeta"],["whitehead-introduction-to-mathematics-1911/x-a24d43a195",15,"Whitehead 1911, p. 144: This fact is worth noting; for it is characteristic ..."],["hardy-course-of-pure-mathematics-1921/eq-8696acd0cb",16,"Hardy 1921, p. 437: |\\zeta| = \\rho"],["concept/partial-sum",7,"partial sum","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-partial-sum"],["todhunter-spherical-trigonometry-1886/x-e73455694d",15,"Todhunter 1886, scan 52: There are limitations of the data in order to ..."],["hardy-course-of-pure-mathematics-1921/eq-7a49549e21",16,"Hardy 1921, p. 437: P(z) = P(z_{0}) + A_{1}\\zeta + A_{2}\\zeta^{2} + \\dots + A_{n}\\zeta^{n}"],["wentworth-first-steps-in-algebra-1894/ex-1/21",4,"Wentworth 1894, Exercise 1 (21)"],["wentworth-first-steps-in-algebra-1894/ex-2/8",4,"Wentworth 1894, Exercise 2 (8)"],["form/9d56b10402",5,"identity: 4*a + 4*b"],["todhunter-spherical-trigonometry-1886/x-69ea8cfea0",15,"Todhunter 1886, scan 52: Thus if one triangle exists with the given parts, ..."],["dickson-theory-of-equations-1922/eq-aa219b353a",16,"Dickson 1922, p. 41: R^3 + R^{10} + R^5 + R^{11} + R^{14} + R^7 + R^{12} + R^6"],["maxwell-elementary-treatise-electricity-1888/eq-95cfd2348c",16,"Maxwell 1888, scan 200: E = IR = I_1( R + r_1 ) = I_2( R + r_2 )\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-87d335477c",16,"Maxwell 1888, scan 200: \\frac{r_1}{r_2} = \\frac{(I-I_1)I_2}{(I-I_2)I_1}\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-72d35a878e",16,"Maxwell 1888, scan 201: \\delta = mI_1 - nI_2\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-e500ab6f04",16,"Maxwell 1888, scan 201: \\delta =\\frac{E}{D} \\{m(B + \\beta) - n(A + \\alpha)\\}\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-ce8d135990",16,"Maxwell 1888, scan 201: D = (A + \\alpha)(B + \\beta) + r(A + \\alpha + B + \\beta)\\text{.}"],["de-morgan-elementary-illustrations-calculus-1899/ch-rules-for-differentiation",2,"De Morgan 1899, Rules for Differentiation","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-rules-for-differentiation/index.html"],["maxwell-elementary-treatise-electricity-1888/eq-83478d15e6",16,"Maxwell 1888, scan 203: B - A = \\frac{1}{2nE} (A + \\alpha)(A + \\alpha + 2r)(\\delta - \\delta')\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-535fbd3eaa",16,"Maxwell 1888, scan 203: \\alpha = \\tfrac{1}{3} A\\text{;}"],["concept/optimal-design",7,"optimal design"],["maxwell-elementary-treatise-electricity-1888/eq-9ba64bdcaa",16,"Maxwell 1888, scan 203: \\frac{B - A}{A} = \\frac{2}{3}\\frac{\\delta - \\delta'}{\\Delta}\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-e630cd921f",16,"Maxwell 1888, scan 203: \\Delta = \\frac{mE}{A + \\alpha + r} = \\frac{3}{4}\\frac{nE}{A}\\text{ if }r = 0\\text{ and }\\alpha = \\frac{1}{3} A\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-dd955426fe",16,"Maxwell 1888, scan 204: b \\beta = c \\gamma\\text{,}"],["law/wheatstone-s-bridge-balance-condition",10,"Wheatstone's bridge balance condition"],["whitehead-introduction-to-mathematics-1911/eq-60fcddf9e0",16,"Whitehead 1911, p. 136: \\dfrac{SP}{PN}"],["maxwell-elementary-treatise-electricity-1888/eq-1ed5ae6afb",16,"Maxwell 1888, scan 204: O = \\frac{B \\gamma + C \\beta}{ \\beta + \\gamma}\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-a8d2c270f2",16,"Maxwell 1888, scan 205: \\xi = \\frac{E}{D}(b\\beta - c\\gamma)\\text{,}"],["maxwell-elementary-treatise-electricity-1888/eq-f75a775690",16,"Maxwell 1888, scan 205: D = abc + bc(\\beta+\\gamma)+ca(\\gamma+\\alpha)+ab(\\alpha+\\beta)+(a+b+c)(\\beta\\gamma+\\gamma\\alpha+\\alpha\\beta)"],["maxwell-elementary-treatise-electricity-1888/eq-5abcd1a512",16,"Maxwell 1888, scan 207: G = n(1 - n)(R + S)\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-7bc463f847",16,"Maxwell 1888, scan 208: B =\\frac{RS}{R + S}\\text{.}"],["maxwell-elementary-treatise-electricity-1888/eq-9bd792f824",16,"Maxwell 1888, scan 208: S^2 = \\frac{BR}{B + R}\\left(R +\\frac{G}{n(1 - n)}\\right)\\text{.}"],["theorem/triangle-inequality",9,"triangle inequality","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-triangle-inequality"],["concept/composition-of-functions",7,"composition of functions","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-composition-of-functions"],["planck-treatise-on-thermodynamics-1903/eq-883aae3b4c",16,"Planck 1903, p. 64: Q = Q_{1} + Q_{2} = \\frac{R}{m} (\\theta_{2} - \\theta_{1}) \\log \\frac{v_{1}'}{v_{1}}"],["concept/higher-order-difference",7,"higher-order difference","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-higher-order-difference"],["hardy-course-of-pure-mathematics-1921/eq-2fe6ce0428",16,"Hardy 1921, p. 437: |A_{k}| = \\mu"],["hardy-course-of-pure-mathematics-1921/ex-lxxxix/5",4,"Hardy 1921, Exercise LXXXIX (5)"],["hardy-course-of-pure-mathematics-1921/eq-5d7a79bef0",16,"Hardy 1921, p. 437: |A_{k+1}|\\rho + |A_{k+2}|\\rho^{2} + \\dots + |A_{n}|\\rho^{n-k} < \\tfrac{1}{2}\\mu"],["hardy-course-of-pure-mathematics-1921/eq-ec63881b2a",16,"Hardy 1921, p. 437: |P(z) - P(z_{0}) - A_{k}\\zeta^{k}| < \\tfrac{1}{2}\\mu\\rho^{k}"],["concept/ratio-of-successive-terms",7,"ratio of successive terms","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-ratio-of-successive-terms"],["dickson-theory-of-equations-1922/eq-a09c5c1387",16,"Dickson 1922, p. 41: z_2 = R^9 + R^{15} + R^8 + R^2"],["theorem/ratio-test",9,"ratio test","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-ratio-test"],["method/comparison-of-series",8,"comparison of series","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-comparison-of-series"],["boyden-first-book-in-algebra-1895/ex-22/30",4,"Boyden 1895, Exercise 22 (30)"],["planck-treatise-on-thermodynamics-1903/x-adb7b613ae",15,"Planck 1903, p. 144: This shows that the external work forms only a ..."],["form/312343abe5",5,"solve: (Eq(a, 100*d), Eq(x, a - b), Eq(b, c**2))"],["shape/cd0189c622",6,"solve: (Eq(a, N*d), Eq(x, a - b), Eq(b, c**N))"],["dickson-theory-of-equations-1922/eq-ca6a3c5cbf",16,"Dickson 1922, p. 41: z_1 = R + R^{13} + R^{16} + R^4"],["dickson-theory-of-equations-1922/eq-f3ffcaf514",16,"Dickson 1922, p. 41: w_1 = R^3 + R^5 + R^{14} + R^{12}"],["dickson-theory-of-equations-1922/eq-412c876665",16,"Dickson 1922, p. 41: w_2 = R^{10} + R^{11} + R^7 + R^6"],["de-morgan-elementary-illustrations-calculus-1899/x-cc1e79c25f",15,"De Morgan 1899, p. 17: A series is said to be convergent when the ..."],["boyden-first-book-in-algebra-1895/ex-35/22",4,"Boyden 1895, Exercise 35 (22)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxix/6",4,"Hardy 1921, Exercise LXXXIX (6)"],["wentworth-first-steps-in-algebra-1894/ex-1/5",4,"Wentworth 1894, Exercise 1 (5)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxix/7",4,"Hardy 1921, Exercise LXXXIX (7)"],["wentworth-first-steps-in-algebra-1894/ex-1/6",4,"Wentworth 1894, Exercise 1 (6)"],["form/fce8979ed1",5,"identity: 7"],["shape/ee7ba66e52",6,"factor: -x + 1"],["form/bbadfb4fa2",5,"identity: 4*a - 4*b"],["hardy-course-of-pure-mathematics-1921/x-67e21d5585",15,"Hardy 1921, p. 85: we call z the complex variable."],["hardy-course-of-pure-mathematics-1921/x-186e0e32dd",15,"Hardy 1921, p. 85: It must be observed that \\theta or \\am z ..."],["hardy-course-of-pure-mathematics-1921/eq-d269f8b6ff",16,"Hardy 1921, p. 437: |P(z)| < |P(z_{0} + A_{k}\\zeta^{k}| + \\tfrac{1}{2}\\mu\\rho^{k}"],["wentworth-first-steps-in-algebra-1894/ex-15/1",4,"Wentworth 1894, Exercise 15 (1)"],["wentworth-first-steps-in-algebra-1894/ex-1/7",4,"Wentworth 1894, Exercise 1 (7)"],["form/fb7ed69801",5,"identity: 40*x"],["dickson-theory-of-equations-1922/eq-17846b7dfe",16,"Dickson 1922, p. 41: z_1 + z_2 = y_1"],["form/6830501430",5,"identity: 9"],["wentworth-first-steps-in-algebra-1894/ex-2/9",4,"Wentworth 1894, Exercise 2 (9)"],["wentworth-first-steps-in-algebra-1894/ex-2/10",4,"Wentworth 1894, Exercise 2 (10)"],["form/277f031e68",5,"factor: x**6 - x**5 + x**3 - x**2 - x + 1"],["de-morgan-elementary-illustrations-calculus-1899/x-615126b778",15,"De Morgan 1899, p. 17: A series cannot be convergent, unless its separate terms ..."],["hardy-course-of-pure-mathematics-1921/x-b58dfc300a",15,"Hardy 1921, p. 87: Hence De Moivre’s Theorem holds for all integral values ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxix/8",4,"Hardy 1921, Exercise LXXXIX (8)"],["dickson-theory-of-equations-1922/x-e7c12b3a70",15,"Dickson 1922, p. 157: This becomes intuitive geometrically. The portion of the graph ..."],["planck-treatise-on-thermodynamics-1903/x-514d3c7197",15,"Planck 1903, p. 105: This time, the system operated upon may be of ..."],["dickson-theory-of-equations-1922/x-96c7d33a88",15,"Dickson 1922, p. 155: has a complex real or imaginary root."],["de-morgan-elementary-illustrations-calculus-1899/x-925739d155",15,"De Morgan 1899, p. 127: A line is considered as the sum of an ..."],["hardy-course-of-pure-mathematics-1921/x-da1a8aed37",15,"Hardy 1921, p. 96: The four points are said to be harmonic or ..."],["dickson-theory-of-equations-1922/x-7d58af1318",15,"Dickson 1922, p. 158: In other words, if |f(z)|\\leqq P, the point representing ..."],["hardy-course-of-pure-mathematics-1921/x-d0410175ef",15,"Hardy 1921, p. 86: in this notation, suggested by Profs. Harkness and Morley, ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxv/1",4,"Hardy 1921, Exercise LXXXV (1)"],["form/74db6d16cf",5,"identity: 6"],["concept/trigonometry",7,"trigonometry","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-trigonometry"],["form/d935ac9a57",5,"identity: 2"],["cap/cas.ode",17,"cas.ode"],["de-morgan-elementary-illustrations-calculus-1899/x-89de2a30cd",15,"De Morgan 1899, p. 86: At the risk of being tedious to some readers, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-eae3e7b2b7",15,"De Morgan 1899, p. 87: Let x = 1000, whence y = \\text{common log}~ ..."],["hardy-course-of-pure-mathematics-1921/eq-506edf4e9f",16,"Hardy 1921, p. 85: x = \\Real(z)"],["de-morgan-elementary-illustrations-calculus-1899/ch-implicit-functions",2,"De Morgan 1899, Implicit Functions","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-implicit-functions/index.html"],["hardy-course-of-pure-mathematics-1921/eq-a4ee600775",16,"Hardy 1921, p. 85: y = \\Imag(z)"],["hardy-course-of-pure-mathematics-1921/eq-c5d0f21779",16,"Hardy 1921, p. 85: r = |z|"],["hardy-course-of-pure-mathematics-1921/eq-00f2388873",16,"Hardy 1921, p. 85: \\theta = \\am z"],["de-morgan-elementary-illustrations-calculus-1899/x-0280f41905",15,"De Morgan 1899, p. 18: And the terms of a series may at first ..."],["hardy-course-of-pure-mathematics-1921/eq-189b44fa98",16,"Hardy 1921, p. 86: 1/z = (\\cos\\theta - i\\sin\\theta)/r"],["de-morgan-elementary-illustrations-calculus-1899/x-480d9916a3",15,"De Morgan 1899, p. 128: If twenty points be taken on a straight line, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-bb0d03fa7d",15,"De Morgan 1899, p. 128: We would therefore recommend to the student not to ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxv/2",4,"Hardy 1921, Exercise LXXXV (2)"],["de-morgan-elementary-illustrations-calculus-1899/x-7874592d87",15,"De Morgan 1899, p. 129: Though a point is not treated as a length, ..."],["hardy-course-of-pure-mathematics-1921/eq-134704b2b1",16,"Hardy 1921, p. 86: (\\cos\\theta + i\\sin\\theta)^{n} = \\cos n\\theta + i\\sin n\\theta"],["hardy-course-of-pure-mathematics-1921/eq-6d44966a68",16,"Hardy 1921, p. 86: r(\\cos\\theta + i\\sin\\theta) × \\rho(\\cos\\phi + i\\sin\\phi) = r\\rho\\{\\cos(\\theta + \\phi) + i\\sin(\\theta + \\phi)\\}"],["concept/circle",7,"circle","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-circle"],["de-morgan-elementary-illustrations-calculus-1899/x-400324c8c0",15,"De Morgan 1899, p. 130: A cubic inch of gold weighs more than a ..."],["hardy-course-of-pure-mathematics-1921/eq-43f5747fb0",16,"Hardy 1921, p. 96: \\frac{(z_{1} - z_{3}) (z_{2} - z_{4})}{(z_{1} - z_{4}) (z_{2} - z_{3})} = -1"],["concept/polar-triangle",7,"polar triangle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-polar-triangle"],["de-morgan-elementary-illustrations-calculus-1899/x-a3d7b4ad13",15,"De Morgan 1899, p. 129: But if the weight of every two cubic inches ..."],["hardy-course-of-pure-mathematics-1921/eq-4f5efeb54f",16,"Hardy 1921, p. 88: a_{0}z^{n} + a_{1}z^{n-1} + \\dots + a_{n} = 0"],["concept/inscribed-circle",7,"inscribed circle","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-inscribed-circle"],["hardy-course-of-pure-mathematics-1921/eq-1d6ca04d59",16,"Hardy 1921, p. 91: z^{2} + 2(b + Bi)z + (c + Ci) = 0"],["de-morgan-elementary-illustrations-calculus-1899/x-905861baaa",15,"De Morgan 1899, p. 132: The integral of bwx^{2}\\, dx is \\frac{1}{3}bwx^{3}, which taken ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxv/3",4,"Hardy 1921, Exercise LXXXV (3)"],["whitehead-introduction-to-mathematics-1911/eq-b07cecfdaa",16,"Whitehead 1911, p. 141: ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0"],["whitehead-introduction-to-mathematics-1911/eq-8de156b5fc",16,"Whitehead 1911, p. 142: ab - h^{2} = 0"],["hardy-course-of-pure-mathematics-1921/eq-0ed0d6b5c2",16,"Hardy 1921, p. 91: x^{2} - y^{2} + 2(bx - By) + c = 0"],["hardy-course-of-pure-mathematics-1921/eq-61945d8237",16,"Hardy 1921, p. 91: 2xy + 2(by + Bx) + C = 0"],["form/2129e3f7ef",5,"identity: 39*x"],["todhunter-spherical-trigonometry-1886/x-87f70aac01",15,"Todhunter 1886, scan 145: It may be presumed from symmetry that the pole ..."],["hardy-course-of-pure-mathematics-1921/eq-658d69bb6f",16,"Hardy 1921, p. 91: \\xi^{2} - \\eta^{2} = h"],["hardy-course-of-pure-mathematics-1921/eq-0b636aa994",16,"Hardy 1921, p. 91: 2\\xi\\eta = k"],["hardy-course-of-pure-mathematics-1921/ex-lxxxvi/1",4,"Hardy 1921, Exercise LXXXVI (1)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxvi/2",4,"Hardy 1921, Exercise LXXXVI (2)"],["hardy-course-of-pure-mathematics-1921/eq-ae30aef37d",16,"Hardy 1921, p. 92: \\xi^{2} + \\eta^{2} = \\sqrtp{h^{2} + k^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-cf5461d31f",16,"Hardy 1921, p. 92: c + Ci = (b + Bi)^{2}"],["whitehead-introduction-to-mathematics-1911/eq-c315b5ac1f",16,"Whitehead 1911, p. 142: a(x^{2} + y^{2}) + 2gx + 2fy + c = 0"],["planck-treatise-on-thermodynamics-1903/ch-dilute-solutions",2,"Planck 1903, Dilute Solutions","../books/planck-treatise-on-thermodynamics-1903/ch/ch-dilute-solutions/index.html"],["boyden-first-book-in-algebra-1895/ex-22/31",4,"Boyden 1895, Exercise 22 (31)"],["hardy-course-of-pure-mathematics-1921/eq-e0da529494",16,"Hardy 1921, p. 92: C^{2} - 4bBC + 4cB^{2} = 0"],["form/48bb364729",5,"solve: Eq(x, a*(-b + 25))"],["quantity/solubility",11,"solubility","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-solubility"],["hardy-course-of-pure-mathematics-1921/eq-9a48920231",16,"Hardy 1921, p. 92: C^{2} - 4bBC - 4b^{2}c = 0"],["hardy-course-of-pure-mathematics-1921/eq-9c4f225594",16,"Hardy 1921, p. 92: z^{3} + 3Hz + G = 0"],["todhunter-spherical-trigonometry-1886/x-3ce611b5bc",15,"Todhunter 1886, scan 150: the volume of a tetrahedron is one sixth of ..."],["hardy-course-of-pure-mathematics-1921/eq-d42a93ef3b",16,"Hardy 1921, p. 92: \\sigma^{3} + 27\\lambda\\mu^{2}\\sigma - 27\\mu^{3}\\rho = 0"],["shape/7222e1a6bf",6,"solve: Eq(x, a*(N - b))"],["wentworth-first-steps-in-algebra-1894/ex-2/12",4,"Wentworth 1894, Exercise 2 (12)"],["form/5334fd87ad",5,"identity: 3*a*b + 3*c"],["shape/eef5f21885",6,"identity: N*a*b + N*c"],["boyden-first-book-in-algebra-1895/ex-22/32",4,"Boyden 1895, Exercise 22 (32)"],["hardy-course-of-pure-mathematics-1921/eq-cbec8d0654",16,"Hardy 1921, p. 92: \\rho^{3} - 27\\lambda\\mu^{2}\\rho - 27\\mu^{3}\\sigma = 0"],["hardy-course-of-pure-mathematics-1921/eq-cd50db4a27",16,"Hardy 1921, p. 92: 2x(x^{2} + y^{2}) = G"],["whitehead-introduction-to-mathematics-1911/eq-4e9953e6f0",16,"Whitehead 1911, p. 142: (dx + ey)^{2} + 2gx + 2fy + c = 0"],["hardy-course-of-pure-mathematics-1921/eq-0466cea82e",16,"Hardy 1921, p. 92: y^{2} - 3x^{2} = 3H"],["form/797af1303c",5,"evaluate: 7*a/(11*b - 3*c) + 11*b/(8*b - 7*c) - 10*c/(7*a - 5*b) at a=5, x=4, y=3"],["dickson-theory-of-equations-1922/eq-a1dcae5abc",16,"Dickson 1922, p. 41: z_1 z_2 = w_1 w_2 = -1"],["dickson-theory-of-equations-1922/eq-dddb132073",16,"Dickson 1922, p. 41: w^2 - y_2 w - 1 = 0"],["shape/edd308062e",6,"evaluate: N*a/(N*b + N*c) + N*b/(N*b + N*c) + N*c/(N*a + N*b)"],["dickson-theory-of-equations-1922/eq-483c5e4f1c",16,"Dickson 1922, p. 41: z^2 - y_1 z - 1 = 0"],["hardy-course-of-pure-mathematics-1921/eq-d9cd998e61",16,"Hardy 1921, p. 91: \\alpha z + \\beta = 0"],["de-morgan-elementary-illustrations-calculus-1899/x-195fa2b09d",15,"De Morgan 1899, p. 88: We have already had occasion to notice these successive ..."],["hardy-course-of-pure-mathematics-1921/eq-cb17542466",16,"Hardy 1921, p. 91: z = -(\\beta/\\alpha)"],["concept/phase-rule",7,"phase rule","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-phase-rule"],["dickson-theory-of-equations-1922/eq-dfc34646c5",16,"Dickson 1922, p. 42: v_1 = R + R^{16}"],["dickson-theory-of-equations-1922/eq-28d7fcac2f",16,"Dickson 1922, p. 42: v_1v_2 = w_1"],["dickson-theory-of-equations-1922/eq-d470d87629",16,"Dickson 1922, p. 42: v_2 = R^{13} + R^4"],["dickson-theory-of-equations-1922/eq-de0ee14927",16,"Dickson 1922, p. 42: v_1 + v_2 = z_1"],["dickson-theory-of-equations-1922/eq-87a089ce8d",16,"Dickson 1922, p. 42: v^2 - z_1v + w_1 = 0"],["hardy-course-of-pure-mathematics-1921/eq-2212fcdb6d",16,"Hardy 1921, p. 91: aB - bA = 0"],["hardy-course-of-pure-mathematics-1921/eq-531183b9c6",16,"Hardy 1921, p. 93: 8\\alpha^{3} + 6\\alpha H - G = 0"],["hardy-course-of-pure-mathematics-1921/ex-lxxxvi/3",4,"Hardy 1921, Exercise LXXXVI (3)"],["hardy-course-of-pure-mathematics-1921/eq-5a17fadeba",16,"Hardy 1921, p. 93: \\left|\\frac{z - b}{z - a}\\right| = \\lambda"],["todhunter-spherical-trigonometry-1886/x-37e4e98cdc",15,"Todhunter 1886, scan 164: since \\sin C is greater than \\sin A we ..."],["wentworth-plane-geometry-1899/eq-6fc133375f",16,"Wentworth 1899, scan 142: EG = FC = \\frac{1}{2}DC"],["wentworth-plane-geometry-1899/eq-b9723b04a2",16,"Wentworth 1899, scan 141: \\angle BAC = \\angle BCA = 45°"],["planck-treatise-on-thermodynamics-1903/eq-eef029c36a",16,"Planck 1903, p. 65: Q_{2} = \\frac{R}{m} \\theta_{2} \\log \\frac{v_{2}'}{v_{2}} = \\frac{R}{m} \\theta_{2} \\log \\frac{v_{1}'}{v_{1}}"],["hardy-course-of-pure-mathematics-1921/eq-4cfeef3415",16,"Hardy 1921, p. 94: z = Z + a"],["boyden-first-book-in-algebra-1895/ex-29/3",4,"Boyden 1895, Exercise 29 (3)"],["hardy-course-of-pure-mathematics-1921/eq-e92f1e1ec8",16,"Hardy 1921, p. 94: z = \\rho Z"],["hardy-course-of-pure-mathematics-1921/eq-6d2590a78e",16,"Hardy 1921, p. 94: z = (\\cos\\phi + i \\sin\\phi)Z"],["hardy-course-of-pure-mathematics-1921/eq-76516ed5b4",16,"Hardy 1921, p. 94: z = aZ + b"],["hardy-course-of-pure-mathematics-1921/eq-681396e93f",16,"Hardy 1921, p. 95: z = 1/Z"],["wentworth-first-steps-in-algebra-1894/ex-17/6",4,"Wentworth 1894, Exercise 17 (6)"],["hardy-course-of-pure-mathematics-1921/ex-lxvii",3,"Hardy 1921, Exercise LXVII"],["hardy-course-of-pure-mathematics-1921/ch-viii",2,"Hardy 1921, ch. VIII: THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS","../books/hardy-course-of-pure-mathematics-1921/ch/ch-viii/index.html"],["hardy-course-of-pure-mathematics-1921/eq-9913521d71",16,"Hardy 1921, p. 95: z = \\frac{aZ + b}{cZ + d}"],["unit/unit-of-surface",12,"unit of surface","../books/wentworth-plane-geometry-1899/terms/index.html#t-unit-unit-of-surface"],["theorem/area-of-a-rectangle",9,"area of a rectangle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-area-of-a-rectangle"],["hardy-course-of-pure-mathematics-1921/eq-bef13d8959",16,"Hardy 1921, p. 95: Z = \\frac{dz - b}{cz - a}"],["hardy-course-of-pure-mathematics-1921/eq-56ead0336f",16,"Hardy 1921, p. 96: ac' + a'c - 2bb' = 0"],["planck-treatise-on-thermodynamics-1903/x-08e4f44d03",15,"Planck 1903, p. 242: the concentration of the dissolved gas is proportional to ..."],["concept/trapezoid",7,"trapezoid","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-trapezoid"],["hardy-course-of-pure-mathematics-1921/eq-ddc33ee789",16,"Hardy 1921, p. 90: \\cot\\omega = \\cot A + \\cot B + \\cot C"],["hardy-course-of-pure-mathematics-1921/eq-36b14306f2",16,"Hardy 1921, p. 90: z = \\frac{1}{3}(\\alpha + \\beta + \\gamma)"],["hardy-course-of-pure-mathematics-1921/eq-70a1d5bef8",16,"Hardy 1921, p. 98: z = c + \\rho\\left(\\frac{1 + ti}{1 - ti}\\right)"],["concept/complex-function-of-a-real-variable",7,"complex function of a real variable"],["hardy-course-of-pure-mathematics-1921/eq-c9b7338322",16,"Hardy 1921, p. 98: z = a + 2bt + ct^{2}"],["de-morgan-elementary-illustrations-calculus-1899/x-8fa413d97d",15,"De Morgan 1899, p. 18: It may be shown (1) that if the terms ..."],["hardy-course-of-pure-mathematics-1921/eq-4b3a7c5b4e",16,"Hardy 1921, p. 90: \\am\\left(\\frac{a - b}{c - d}\\right) = ±\\tfrac{1}{2} \\pi"],["hardy-course-of-pure-mathematics-1921/eq-fcf350bf3a",16,"Hardy 1921, p. 97: \\tan\\phi_{m+n} &= \\tan\\phi_{m} \\sec\\phi_{n} &&+ \\sec\\phi_{m} \\tan\\phi_{n}"],["de-morgan-elementary-illustrations-calculus-1899/x-5cd6f38aa4",15,"De Morgan 1899, p. 19: But since \\dfrac{l}{k} is less than unity, the first ..."],["concept/order-of-a-node",7,"order of a node","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-order-of-a-node"],["de-morgan-elementary-illustrations-calculus-1899/x-82dbee3fec",15,"De Morgan 1899, p. 19: (2) The second theorem on the divergence of series ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxvi/4",4,"Hardy 1921, Exercise LXXXVI (4)"],["hardy-course-of-pure-mathematics-1921/x-4a1d889a7e",15,"Hardy 1921, p. 120: A function can only tend to +\\infty or to ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxvi/5",4,"Hardy 1921, Exercise LXXXVI (5)"],["planck-treatise-on-thermodynamics-1903/eq-511cc78739",16,"Planck 1903, p. 65: Q_{1} = \\frac{R}{m} \\theta_{1} \\log \\frac{v_{1}}{v_{1}'} = -\\frac{R}{m} \\theta_{1} \\log \\frac{v_{1}'}{v_{1}}"],["hardy-course-of-pure-mathematics-1921/ex-lxxxviii/1",4,"Hardy 1921, Exercise LXXXVIII (1)"],["hardy-course-of-pure-mathematics-1921/eq-978ca4b7f2",16,"Hardy 1921, p. 97: \\sec\\phi_{m+n} &= \\sec\\phi_{m} \\sec\\phi_{n} &&+ \\tan\\phi_{m} \\tan\\phi_{n}"],["hardy-course-of-pure-mathematics-1921/eq-97314614df",16,"Hardy 1921, p. 97: \\tan\\phi_{m} + \\sec\\phi_{m} = (\\tan\\phi_{1} + \\sec\\phi_{1})^{m}"],["concept/odd-node",7,"odd node","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-odd-node"],["concept/even-node",7,"even node","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-even-node"],["concept/free-end",7,"free end","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-free-end"],["cap/other:series_convergence_test",17,"other:series_convergence_test"],["de-morgan-elementary-illustrations-calculus-1899/ch-infinite-series",2,"De Morgan 1899, Infinite Series","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-infinite-series/index.html"],["hardy-course-of-pure-mathematics-1921/ex-lxxxviii/2",4,"Hardy 1921, Exercise LXXXVIII (2)"],["person/augustus-de-morgan",1,"Augustus De Morgan"],["ball-mathematical-recreations-1905/eq-4aa5141752",16,"Ball 1905, scan 167: N = \\frac{1}{2}n (n^2 + 1)"],["concept/geometric-series",7,"geometric series","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-geometric-series"],["boyden-first-book-in-algebra-1895/ex-22/33",4,"Boyden 1895, Exercise 22 (33)"],["dickson-theory-of-equations-1922/ex-page106",3,"Dickson 1922, Exercise Page106"],["dickson-theory-of-equations-1922/ex-page102",3,"Dickson 1922, Exercise Page102"],["ball-mathematical-recreations-1905/eq-92583284c1",16,"Ball 1905, scan 162: \\frac{1}{2}n(n-1)"],["dickson-theory-of-equations-1922/ex-page104",3,"Dickson 1922, Exercise Page104"],["dickson-theory-of-equations-1922/ch-viii",2,"Dickson 1922, ch. VIII: Determinants; Systems of Linear Equations","../books/dickson-theory-of-equations-1922/ch/ch-viii/index.html"],["ball-mathematical-recreations-1905/eq-4df55d172a",16,"Ball 1905, scan 162: \\frac{1}{2}(n+1)"],["hardy-course-of-pure-mathematics-1921/x-4a9cbc7dbd",15,"Hardy 1921, p. 111: Large is in fact a word which, standing by ..."],["concept/permutation",7,"permutation","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-permutation"],["concept/power",7,"power","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-power"],["wentworth-first-steps-in-algebra-1894/ex-17/7",4,"Wentworth 1894, Exercise 17 (7)"],["form/dae5f63be9",5,"identity: -4*x"],["concept/vertically-related-cells",7,"vertically related cells"],["ball-mathematical-recreations-1905/eq-8dab8c6118",16,"Ball 1905, scan 168: n(n-2x + 1)"],["ball-mathematical-recreations-1905/eq-d1ad0234a4",16,"Ball 1905, scan 168: n-2y+1"],["concept/horizontally-related-cells",7,"horizontally related cells"],["concept/route",7,"route","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-route"],["theorem/odd-nodes-of-a-closed-network-are-even-in-number",9,"odd nodes of a closed network are even in number","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-odd-nodes-of-a-closed-network-are-even-in-number"],["dickson-theory-of-equations-1922/eq-2e0afa26fe",16,"Dickson 1922, p. 42: 2 \\cos \\frac{6\\pi}{17} + 2 \\cos \\frac{10\\pi}{17} + 2 \\cos \\frac{12\\pi}{17} + 2 \\cos \\frac{14\\pi}{17} < 0"],["ball-mathematical-recreations-1905/eq-029f4b08dd",16,"Ball 1905, scan 168: N-\\frac{1}{2}n^2(n-2x+1)"],["concept/magic-square-of-an-even-order",7,"magic square of an even order"],["ball-mathematical-recreations-1905/eq-f1a81f0618",16,"Ball 1905, scan 168: N + \\frac{1}{2}n^2(n-2x + 1)"],["dickson-theory-of-equations-1922/eq-5f1212c0c5",16,"Dickson 1922, p. 42: \\rho^2 - v_1\\rho + 1 = 0"],["dickson-theory-of-equations-1922/eq-57a1d6f8a9",16,"Dickson 1922, p. 43: v_1 = 2 \\cos\\frac{2\\pi}{17}"],["ball-mathematical-recreations-1905/eq-8a8eda536a",16,"Ball 1905, scan 168: N-\\frac{1}{2}n (n-2y + 1)"],["concept/tangent",7,"tangent","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-tangent"],["dickson-theory-of-equations-1922/eq-31020f0b49",16,"Dickson 1922, p. 42: 2 \\cos \\frac{6\\pi}{17} + 2 \\cos \\frac{10\\pi}{17} = 2 \\cos \\frac{6\\pi}{17} - 2 \\cos \\frac{7\\pi}{17}"],["de-morgan-elementary-illustrations-calculus-1899/x-c1b9c1ff0e",15,"De Morgan 1899, p. 15: If in \\phi x, any function of x, the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-2dbf6596cc",15,"De Morgan 1899, p. 16: It will happen, however, in many functions, that one ..."],["ball-mathematical-recreations-1905/ch-ix",2,"Ball 1905, ch. IX: Mersenne's Numbers","../books/ball-mathematical-recreations-1905/ch/ch-ix/index.html"],["de-morgan-elementary-illustrations-calculus-1899/x-3fef7f36fe",15,"De Morgan 1899, p. 16: As the notion of a series which has no ..."],["concept/mersenne-number",7,"Mersenne number","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-mersenne-number"],["hardy-course-of-pure-mathematics-1921/x-897d29f1de",15,"Hardy 1921, p. 414: where -1 \\leq x \\leq 1: each of these ..."],["hardy-course-of-pure-mathematics-1921/x-4ac2970531",15,"Hardy 1921, p. 409: It is evident that \\cos \\zeta and \\sec \\zeta ..."],["ball-mathematical-recreations-1905/eq-aa6c93d3b5",16,"Ball 1905, scan 168: N + \\frac{1}{2}n(n-2y +1)"],["wentworth-first-steps-in-algebra-1894/ex-2/13",4,"Wentworth 1894, Exercise 2 (13)"],["method/card-shuffling",8,"card 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..."],["concept/maze",7,"maze","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-maze"],["concept/tree",7,"tree","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-tree"],["concept/ramification-base",7,"ramification base","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-ramification-base"],["ball-mathematical-recreations-1905/x-e64763a29b",15,"Ball 1905, scan 185: It is evident that the question will not be ..."],["concept/quadratic-residue",7,"quadratic residue","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-quadratic-residue"],["hardy-course-of-pure-mathematics-1921/x-737e38d1c4",15,"Hardy 1921, p. 414: the question is suggested whether, now that we have ..."],["hardy-course-of-pure-mathematics-1921/eq-e4579af174",16,"Hardy 1921, p. 436: Z = a_{0} z^{n} \\left(1 + \\frac{a_{1}}{a_{0}z} + \\frac{a_{2}}{a_{0} z^{2}} + \\dots + \\frac{a_{n}}{a_{0} z^{n}}\\right)"],["de-morgan-elementary-illustrations-calculus-1899/x-a19e4a097a",15,"De Morgan 1899, p. 16: Thus it generally happens that x^{2} - 10x + ..."],["hardy-course-of-pure-mathematics-1921/eq-6e4dfa7c71",16,"Hardy 1921, p. 98: z^{n} = a"],["hardy-course-of-pure-mathematics-1921/ex-lxxxviii/4",4,"Hardy 1921, Exercise LXXXVIII (4)"],["cap/other:integral_test_theorem",17,"other:integral_test_theorem"],["hardy-course-of-pure-mathematics-1921/eq-d5f877f9e7",16,"Hardy 1921, p. 99: r^{n} = \\rho"],["hardy-course-of-pure-mathematics-1921/eq-ed4965a3df",16,"Hardy 1921, p. 98: a = \\rho(\\cos\\phi + i\\sin\\phi)"],["hardy-course-of-pure-mathematics-1921/eq-05b4f5629a",16,"Hardy 1921, p. 99: z = r(\\cos\\theta + i\\sin\\theta)"],["concept/congruence",7,"congruence","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-congruence"],["person/alfred-pringsheim",1,"Alfred 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..."],["hardy-course-of-pure-mathematics-1921/eq-c1f6796b61",16,"Hardy 1921, p. 321: \\lim_{x \\to \\infty} \\int_{1}^{x} \\phi(t)\\, dt = l"],["concept/average",7,"average","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-average"],["concept/calculus-of-finite-differences",7,"calculus of finite differences","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-calculus-of-finite-differences"],["ball-mathematical-recreations-1905/eq-dcb09d0eb7",16,"Ball 1905, scan 175: \\frac{1}{2}(n^2 + 1)"],["hardy-course-of-pure-mathematics-1921/x-932601a847",15,"Hardy 1921, p. 4: From these considerations the reader might be tempted to ..."],["ball-mathematical-recreations-1905/eq-1b5b81af19",16,"Ball 1905, scan 175: \\frac{1}{2}(n-2)\\{(n-2)^2+1\\}"],["ball-mathematical-recreations-1905/eq-c30e82734d",16,"Ball 1905, scan 175: \\frac{1}{2}\\{(n-2)^2+1\\}"],["wentworth-first-steps-in-algebra-1894/ex-20/2",4,"Wentworth 1894, Exercise 20 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Seelhoff","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-p-h-h-seelhoff"],["person/a-j-c-cunningham",1,"A.J.C. Cunningham","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-a-j-c-cunningham"],["concept/field-of-a-variable",7,"field of a variable","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-field-of-a-variable"],["concept/continuous-real-variable",7,"continuous real variable","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-continuous-real-variable"],["wentworth-first-steps-in-algebra-1894/ex-21",3,"Wentworth 1894, Exercise 21"],["wentworth-first-steps-in-algebra-1894/ex-21/1",4,"Wentworth 1894, Exercise 21 (1)"],["form/4f263c1242",5,"identity: a*(a + 7)"],["shape/2d7c5cd99e",6,"identity: a*(N + a)"],["hardy-course-of-pure-mathematics-1921/ex-lxxxviii/6",4,"Hardy 1921, Exercise LXXXVIII (6)"],["wentworth-plane-geometry-1899/x-7525f46e3b",15,"Wentworth 1899, scan 231: The ratio of the circumference of a circle to ..."],["hardy-course-of-pure-mathematics-1921/x-b8e8d4ab94",15,"Hardy 1921, p. 15: We have to show that these ideas can be ..."],["hardy-course-of-pure-mathematics-1921/x-3eb76b5594",15,"Hardy 1921, p. 24: We add a few further examples to show how ..."],["ball-mathematical-recreations-1905/x-871ceb45ae",15,"Ball 1905, scan 280: It is evident that, if p is not a ..."],["hardy-course-of-pure-mathematics-1921/x-bce88ff2da",15,"Hardy 1921, p. 98: These definitions do not prejudge the question as to ..."],["theorem/area-of-a-circle",9,"area of a circle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-area-of-a-circle"],["boyden-first-book-in-algebra-1895/ex-23/5",4,"Boyden 1895, Exercise 23 (5)"],["ball-mathematical-recreations-1905/x-286c137789",15,"Ball 1905, scan 292: We know by Fermat’s Theorem that if x + ..."],["ball-mathematical-recreations-1905/x-f471bbedfe",15,"Ball 1905, scan 292: The number N when expressed in the binary scale, ..."],["hardy-course-of-pure-mathematics-1921/eq-f0fbadf928",16,"Hardy 1921, p. 323: \\int_{a}^{\\infty} \\phi(x)\\, dx = \\int_{a}^{b} \\phi(x)\\, dx + \\int_{b}^{\\infty}\\phi(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/ex-lxxxviii/7",4,"Hardy 1921, Exercise LXXXVIII (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxiii",3,"Hardy 1921, Exercise XXIII"],["hardy-course-of-pure-mathematics-1921/ex-xxiv",3,"Hardy 1921, Exercise XXIV"],["hardy-course-of-pure-mathematics-1921/eq-3735dd186f",16,"Hardy 1921, p. 323: \\int_{a}^{x} \\phi(t)\\, dt < K"],["hardy-course-of-pure-mathematics-1921/ex-xxv",3,"Hardy 1921, Exercise XXV"],["ball-mathematical-recreations-1905/x-c6fd53b2ee",15,"Ball 1905, scan 286: but the riddle as to how it was discovered ..."],["hardy-course-of-pure-mathematics-1921/x-b44afed958",15,"Hardy 1921, p. 25: It can in fact be proved (though the proof ..."],["ball-mathematical-recreations-1905/x-e9f313a3a4",15,"Ball 1905, scan 280: Qui vndecim alios repererit, nouerit se analysim omnem, quae ..."],["hardy-course-of-pure-mathematics-1921/x-c80c693751",15,"Hardy 1921, p. 26: And this number \\pi is no isolated or exceptional ..."],["hardy-course-of-pure-mathematics-1921/x-7f3db1c9ae",15,"Hardy 1921, p. 315: The tests derived from comparison with it are therefore ..."],["form/02956b9dbc",5,"solve: (Eq(4*x, 2*a + 10), Eq(a, x + 8))"],["shape/f945e541bf",6,"solve: (Eq(N*x, N*a + N), Eq(a, N + x))"],["hardy-course-of-pure-mathematics-1921/x-f89a92df3b",15,"Hardy 1921, p. 120: Take for instance the case of k > 0. ..."],["ball-mathematical-recreations-1905/eq-06b0159307",16,"Ball 1905, scan 198: n^{n-2}"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii",3,"Hardy 1921, Exercise Misc-II"],["ball-mathematical-recreations-1905/ch-vi",2,"Ball 1905, ch. VI: Unicursal Problems","../books/ball-mathematical-recreations-1905/ch/ch-vi/index.html"],["ball-mathematical-recreations-1905/eq-872dc18479",16,"Ball 1905, scan 198: (1-x)^{-1} (1-x^2)^{-A_1} (1-x^3)^{-A_2} \\dotsm & = 1 + A_1 x + A_2 x^2 + A_3 x^3 + \\dotsb\\, ,"],["hardy-course-of-pure-mathematics-1921/x-7188788249",15,"Hardy 1921, p. 28: This conclusion is of very great importance; for it ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/1",4,"Hardy 1921, Exercise Misc-II (1)"],["hardy-course-of-pure-mathematics-1921/eq-a78410e544",16,"Hardy 1921, p. 436: Z = a_{0} z^{n} (1 + \\rho)"],["boyden-first-book-in-algebra-1895/ex-23/6",4,"Boyden 1895, Exercise 23 (6)"],["de-morgan-elementary-illustrations-calculus-1899/ch-concluding-remarks-on-the-study-of-the-calculus",2,"De Morgan 1899, Concluding Remarks on the Study of the Calculus","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-concluding-remarks-on-the-study-of-the-calculus/index.html"],["form/7144ada5ef",5,"identity: 2*x**3 + 4*x**2 + 10"],["shape/c04eafe678",6,"identity: 2*N*x**N + N"],["boyden-first-book-in-algebra-1895/ex-23/7",4,"Boyden 1895, Exercise 23 (7)"],["form/fe10db7b18",5,"identity: a**2/2 - 4*a*x/3 + x**2/3"],["de-morgan-elementary-illustrations-calculus-1899/x-b9cb535ed4",15,"De Morgan 1899, p. 110: Hence \\dfrac{dy}{dx} (meaning the limit) is -\\dfrac{1}{x^{2}}, which will ..."],["concept/algebraical-number",7,"algebraical number","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-algebraical-number"],["shape/317afcee9d",6,"identity: N*a*x + N*a**N + N*x**N"],["hardy-course-of-pure-mathematics-1921/x-9f027d6270",15,"Hardy 1921, p. 310: This hardly requires proof, for v_{n}^{1/n} \\geq 1 involves ..."],["ball-mathematical-recreations-1905/x-5490b84afd",15,"Ball 1905, scan 191: a chess-board, divided as usual by straight lines into ..."],["hardy-course-of-pure-mathematics-1921/x-a37d1aa8c8",15,"Hardy 1921, p. 312: If 0 < a < b < 1, then ..."],["planck-treatise-on-thermodynamics-1903/eq-fc7b0cf825",16,"Planck 1903, p. 65: Q_{1} : Q_{2}: W = (-\\theta_{1}): \\theta_{2} : (\\theta_{1} - \\theta_{2})"],["planck-treatise-on-thermodynamics-1903/eq-378b1bef73",16,"Planck 1903, p. 62: Q + W = 0"],["concept/rectilinear-figure",7,"rectilinear figure","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-rectilinear-figure"],["ball-mathematical-recreations-1905/eq-3b06e11cd2",16,"Ball 1905, scan 198: (1-x)^{-1} (1-x^2)^{-B_2} (1-x^3)^{-B_3} \\dotsm & = 1 + x + 2B_2 x^2 + 2B_3 x^3 + \\dotsb\\,."],["hardy-course-of-pure-mathematics-1921/eq-cc20ee237d",16,"Hardy 1921, p. 323: \\int_{a}^{\\infty} \\psi(x)\\, dx \\leq K\\int_{a}^{\\infty} \\phi(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-460e8debee",16,"Hardy 1921, p. 324: \\lim x^{s}\\phi(x) = l"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/2",4,"Hardy 1921, Exercise Misc-II (2)"],["whitehead-introduction-to-mathematics-1911/eq-98e73ff08d",16,"Whitehead 1911, p. 145: s = 20 × t"],["whitehead-introduction-to-mathematics-1911/eq-77e6d1fe89",16,"Whitehead 1911, p. 145: y = x + 1"],["whitehead-introduction-to-mathematics-1911/eq-2f78e0fd4d",16,"Whitehead 1911, p. 146: y = x^{2}"],["concept/augmented-matrix",7,"augmented matrix","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-augmented-matrix"],["hardy-course-of-pure-mathematics-1921/eq-6d4b5b8a02",16,"Hardy 1921, p. 107: y = \\phi(n)"],["hardy-course-of-pure-mathematics-1921/eq-72131843bc",16,"Hardy 1921, p. 107: y = \\phi(x) + \\sin x\\pi"],["hardy-course-of-pure-mathematics-1921/eq-47a7dd4bea",16,"Hardy 1921, p. 107: \\sin n\\pi = 0"],["hardy-course-of-pure-mathematics-1921/eq-0edd12453e",16,"Hardy 1921, p. 108: (-1)^{n} = \\cos n\\pi"],["whitehead-introduction-to-mathematics-1911/eq-481c31abbb",16,"Whitehead 1911, p. 146: y = 2x^{2} + 3x + 1"],["hardy-course-of-pure-mathematics-1921/eq-18d8020916",16,"Hardy 1921, p. 108: y = 1 · 2 \\dots n = n!"],["hardy-course-of-pure-mathematics-1921/eq-2bdd5b47a5",16,"Hardy 1921, p. 114: \\lim_{n\\to\\infty} \\frac{1}{n} = 0"],["whitehead-introduction-to-mathematics-1911/eq-93048ab716",16,"Whitehead 1911, p. 145: y = x"],["hardy-course-of-pure-mathematics-1921/eq-73ee399921",16,"Hardy 1921, p. 115: \\lim_{n\\to\\infty} \\left(1 - \\frac{1}{n}\\right) = 1"],["hardy-course-of-pure-mathematics-1921/eq-55586002c3",16,"Hardy 1921, p. 114: 1 - \\phi(n) = 1/n"],["concept/transcendental-number",7,"transcendental number","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-transcendental-number"],["hardy-course-of-pure-mathematics-1921/ex-xxvii",3,"Hardy 1921, Exercise XXVII"],["hardy-course-of-pure-mathematics-1921/x-7e69424422",15,"Hardy 1921, p. 30: Let us suppose that (ii) is true. Then any ..."],["boyden-first-book-in-algebra-1895/ex-37/24",4,"Boyden 1895, Exercise 37 (24)"],["hardy-course-of-pure-mathematics-1921/eq-7bc3ec4605",16,"Hardy 1921, p. 115: n^2 \\to \\infty"],["hardy-course-of-pure-mathematics-1921/eq-c8c35752ba",16,"Hardy 1921, p. 115: -n^{2} \\to -\\infty"],["whitehead-introduction-to-mathematics-1911/eq-5112129174",16,"Whitehead 1911, p. 146: y = \\log x"],["whitehead-introduction-to-mathematics-1911/eq-2230f48d01",16,"Whitehead 1911, p. 147: y = f(x)"],["whitehead-introduction-to-mathematics-1911/eq-c9972e28a3",16,"Whitehead 1911, p. 148: f(1) = 0"],["concept/decimal",7,"decimal","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-decimal"],["planck-treatise-on-thermodynamics-1903/eq-a0752067de",16,"Planck 1903, p. 62: Q = Q_{1} + Q_{2}"],["concept/indirect-function",7,"indirect function","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-indirect-function"],["hardy-course-of-pure-mathematics-1921/eq-df25b25d7d",16,"Hardy 1921, p. 116: \\lim_{n \\to \\infty} \\phi(n) = l"],["whitehead-introduction-to-mathematics-1911/eq-44daafd179",16,"Whitehead 1911, p. 155: f(x) = 1"],["hardy-course-of-pure-mathematics-1921/eq-cb6034f95d",16,"Hardy 1921, p. 116: |\\phi(n) - l| < \\DELTA"],["hardy-course-of-pure-mathematics-1921/x-e4948bad0e",15,"Hardy 1921, p. 30: The general theory of sets of points is of ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/3",4,"Hardy 1921, Exercise Misc-II (3)"],["concept/hamiltonian-cycle",7,"Hamiltonian cycle","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-hamiltonian-cycle"],["de-morgan-elementary-illustrations-calculus-1899/x-cce5c84c3b",15,"De Morgan 1899, p. 96: Generally, the complete differential coefficient of z with respect ..."],["de-morgan-elementary-illustrations-calculus-1899/x-a0f72cc48f",15,"De Morgan 1899, p. 100: Find the differential coefficient belonging to each of the ..."],["hardy-course-of-pure-mathematics-1921/eq-fbecf63fcf",16,"Hardy 1921, p. 117: \\phi(n) > \\Delta"],["hardy-course-of-pure-mathematics-1921/eq-e64e78d1e0",16,"Hardy 1921, p. 118: \\phi(n) \\to +\\infty"],["hardy-course-of-pure-mathematics-1921/eq-1d674baf2e",16,"Hardy 1921, p. 116: n_{0} = 1 + [1/\\DELTA]"],["hardy-course-of-pure-mathematics-1921/eq-2859d478d3",16,"Hardy 1921, p. 113: 1/n < \\DELTA"],["de-morgan-elementary-illustrations-calculus-1899/x-4d168900a3",15,"De Morgan 1899, p. 99: In saying that z is a function of x ..."],["dickson-theory-of-equations-1922/eq-55a3a70e08",16,"Dickson 1922, p. 44: \\cos LOP = OL = \\cos\\frac{2\\pi}{17}"],["dickson-theory-of-equations-1922/eq-c2b1e0e29b",16,"Dickson 1922, p. 32: \\tfrac{1}{2}\\sqrt{10 - 2\\sqrt{5}}"],["dickson-theory-of-equations-1922/eq-9d40885121",16,"Dickson 1922, p. 33: st = \\sqrt{5}"],["hardy-course-of-pure-mathematics-1921/eq-8356ab3815",16,"Hardy 1921, p. 111: 1000\\{1 + (-1)^{n}\\}/n < 1"],["hardy-course-of-pure-mathematics-1921/eq-c0d5acee59",16,"Hardy 1921, p. 324: \\int_{0}^{\\xi} \\phi(x)\\, dx < \\sum_{0}^{\\infty} \\frac{1}{(n + 1)^{2}}"],["form/9b8ff69635",5,"factor: 45*a**4 - 14*a**2*x**2 + x**4"],["shape/3dd54026d5",6,"factor: N*a**N*x**N + N*a**N + x**N"],["de-morgan-elementary-illustrations-calculus-1899/x-dee9e1b6ba",15,"De Morgan 1899, p. 132: Thus, if he has the area of a curve ..."],["de-morgan-elementary-illustrations-calculus-1899/x-b9f5c8ee73",15,"De Morgan 1899, p. 133: let him remark that if an approximate solution only ..."],["method/pairs-of-cards-trick",8,"pairs of cards trick","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-pairs-of-cards-trick"],["concept/isoperimetric-polygon",7,"isoperimetric polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-isoperimetric-polygon"],["method/three-pile-problem",8,"three-pile problem","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-three-pile-problem"],["theorem/gergonne-s-theorem",9,"Gergonne's theorem","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-gergonne-s-theorem"],["wentworth-plane-geometry-1899/x-d0aa9ea5a4",15,"Wentworth 1899, scan 243: Among geometrical magnitudes which satisfy given conditions, the greatest ..."],["concept/mouse-trap-game",7,"mouse trap game","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-mouse-trap-game"],["de-morgan-elementary-illustrations-calculus-1899/x-3210994472",15,"De Morgan 1899, p. 133: The mathematical method begins with the same principle, investigating ..."],["law/dalton-s-law",10,"Dalton's law","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-law-dalton-s-law"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/4",4,"Hardy 1921, Exercise Misc-II (4)"],["concept/treize-game",7,"treize game","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-treize-game"],["de-morgan-elementary-illustrations-calculus-1899/x-3dc3432fd0",15,"De Morgan 1899, p. 133: This limit is shown to be that of which ..."],["person/steen",1,"Steen","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-steen"],["ball-mathematical-recreations-1905/x-09005ce4a5",15,"Ball 1905, scan 149: This depends on the fact that the number of ..."],["boyden-first-book-in-algebra-1895/ex-23/8",4,"Boyden 1895, Exercise 23 (8)"],["whitehead-introduction-to-mathematics-1911/eq-c4759de494",16,"Whitehead 1911, p. 155: f(x) = 2"],["whitehead-introduction-to-mathematics-1911/eq-495dcc2ca8",16,"Whitehead 1911, p. 153: p = \\dfrac{1}{v}"],["whitehead-introduction-to-mathematics-1911/eq-cb7891b045",16,"Whitehead 1911, p. 153: y = \\dfrac{1}{x}"],["dickson-theory-of-equations-1922/eq-812d01f5d5",16,"Dickson 1922, p. 33: t = \\tfrac{1}{2} \\sqrt{10 + 2\\sqrt{5}}"],["planck-treatise-on-thermodynamics-1903/eq-c1d65fe52e",16,"Planck 1903, p. 88: q = c_{v}\\, d\\theta + \\frac{R}{m} · \\frac{\\theta}{v}\\, dv"],["planck-treatise-on-thermodynamics-1903/eq-ad3d941ab7",16,"Planck 1903, p. 88: \\phi = c_{v} \\log \\theta + \\frac{R}{m} \\log v + \\const"],["dickson-theory-of-equations-1922/eq-21a5ab2501",16,"Dickson 1922, p. 59: f'(x) = na_0 x^{n-1} + (n-1)a_1 x^{n-2} + \\dotsb + 2a_{n-2} x + a_{n-1}"],["dickson-theory-of-equations-1922/eq-693f81b1fb",16,"Dickson 1922, p. 59: f''(x) = n(n-1)a_0 x^{n-2} + (n-1)(n-2)a_1 x^{n-3} + \\dotsb + 2a_{n-2}"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/5",4,"Hardy 1921, Exercise Misc-II (5)"],["ball-mathematical-recreations-1905/x-671f35d47a",15,"Ball 1905, scan 153: The reason is that after the first deal you ..."],["person/mile-clapeyron",1,"Émile Clapeyron","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-mile-clapeyron"],["ball-mathematical-recreations-1905/x-889d7fc46b",15,"Ball 1905, scan 155: Hence, if n-1 is expressed in the ternary scale ..."],["ball-mathematical-recreations-1905/x-6eab8eab08",15,"Ball 1905, scan 156: If the kth card has the number k on ..."],["ball-mathematical-recreations-1905/x-80b7a79e49",15,"Ball 1905, scan 151: I believe that these arrangements by sentences are known, ..."],["form/ec9d3fa983",5,"identity: 48*a**6*b**7*x**7"],["whitehead-introduction-to-mathematics-1911/x-258d7f085e",15,"Whitehead 1911, p. 59: By relieving the brain of all unnecessary work, a ..."],["todhunter-spherical-trigonometry-1886/ch-solution-of-oblique-angled-triangles",2,"Todhunter 1886, Solution of Oblique-Angled Triangles","../books/todhunter-spherical-trigonometry-1886/ch/ch-solution-of-oblique-angled-triangles/index.html"],["shape/ec5b0448fb",6,"identity: N*a**N*b**N*x**N"],["hardy-course-of-pure-mathematics-1921/x-cbd8bf0652",15,"Hardy 1921, p. 32: It is clear that we may repeat this argument ..."],["concept/gamma-function",7,"gamma function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-gamma-function"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/6i",4,"Hardy 1921, Exercise Misc-II (6(i))"],["boyden-first-book-in-algebra-1895/ex-23/9",4,"Boyden 1895, Exercise 23 (9)"],["form/e66ae46bc2",5,"identity: 16*a**8*b**12*x**4/81"],["boyden-first-book-in-algebra-1895/ex-23/10",4,"Boyden 1895, Exercise 23 (10)"],["form/e4bb988797",5,"identity: (x - 8)*(x - 5) + (x - 2)*(x + 7)"],["de-morgan-elementary-illustrations-calculus-1899/eq-cf5c482601",16,"De Morgan 1899, p. 44: \\phi' x\\, dx + \\phi'' x\\, \\dfrac{(dx)^{2}}{2}"],["todhunter-spherical-trigonometry-1886/eq-8d3e533101",16,"Todhunter 1886, scan 69: \\tan r = \\tan\\dfrac{A}{2} \\sin (s-a)"],["shape/486445e3d4",6,"identity: 2*(N + x)**2"],["de-morgan-elementary-illustrations-calculus-1899",0,"Morgan, Elementary Illustrations of the Differential and Integral Calculus (1899)","../books/de-morgan-elementary-illustrations-calculus-1899/index.html"],["whitehead-introduction-to-mathematics-1911/x-0b50831bd9",15,"Whitehead 1911, p. 69: Thus, by now varying a, b, and c, we ..."],["hardy-course-of-pure-mathematics-1921/eq-59c507e61e",16,"Hardy 1921, p. 436: \\frac{|a_{1}|}{|a_{0}| R} + \\frac{|a_{2}|}{|a_{0}| R^{2}} + \\dots + \\frac{|a_{n}|}{|a_{0}| R^{n}} < \\delta"],["todhunter-spherical-trigonometry-1886/eq-6332f4dec3",16,"Todhunter 1886, scan 56: \\cos A = \\dfrac{\\cos a - \\cos b \\cos c}{\\sin b \\sin c}"],["hardy-course-of-pure-mathematics-1921/eq-5f1a29f57f",16,"Hardy 1921, p. 438: f'(z) = f(z) \\left(\\frac{1}{z - z_{1}} + \\frac{1}{z - z_{2}} + \\frac{1}{z - z_{3}}\\right)"],["whitehead-introduction-to-mathematics-1911/x-ddcfc37f89",15,"Whitehead 1911, p. 64: This service is performed by 0, the symbol for ..."],["whitehead-introduction-to-mathematics-1911/x-30c30062b1",15,"Whitehead 1911, p. 62: Mathematicians have chosen to make their symbolism more concise ..."],["whitehead-introduction-to-mathematics-1911/x-2f7b6ac605",15,"Whitehead 1911, p. 62: Thus when we substitute 2 for x and 3 ..."],["concept/method-finding-the-angular-radius-of-the-inscribed-circle-of-a-spherical-triangle",7,"method: finding the angular radius of the inscribed circle of a spherical triangle"],["concept/integer",7,"integer","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-integer"],["concept/power-series",7,"power series","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-power-series"],["concept/coefficient",7,"coefficient","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-coefficient"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/6ii",4,"Hardy 1921, Exercise Misc-II (6(ii))"],["hardy-course-of-pure-mathematics-1921/eq-b52333897a",16,"Hardy 1921, p. 326: \\int_{a}^{\\xi} \\phi(x)\\, dx = \\int_{b}^{\\tau} \\phi\\{f(t)\\}f'(t)\\, dt"],["hardy-course-of-pure-mathematics-1921/eq-e3813bf344",16,"Hardy 1921, p. 439: -1 < x \\leq 1"],["hardy-course-of-pure-mathematics-1921/eq-9552626931",16,"Hardy 1921, p. 439: \\log(1 + x) = x - \\tfrac{1}{2}x^{2} + \\tfrac{1}{3}x^{3} - \\dots"],["hardy-course-of-pure-mathematics-1921/x-46761d54c6",15,"Hardy 1921, p. 199: The existence of a derived function \\phi'(x) for all ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/7",4,"Hardy 1921, Exercise Misc-II (7)"],["theorem/inverse-function-rule",9,"inverse function rule","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-inverse-function-rule"],["hardy-course-of-pure-mathematics-1921/eq-60dc235b0d",16,"Hardy 1921, p. 331: \\int_{a}^{\\infty} \\phi(x)\\, dx = \\int_{b}^{c} \\phi\\{f(t)\\}f'(t)\\, dt"],["hardy-course-of-pure-mathematics-1921/eq-a840f7a8bb",16,"Hardy 1921, p. 439: 1/(1 + t) = 1 - t + t^{2} - \\dots"],["wentworth-plane-geometry-1899/eq-6d98059ba0",16,"Wentworth 1899, scan 110: \\dfrac{\\angle A'C'B'}{\\angle ACB} = \\dfrac{\\arc A'B'}{\\arc AB}"],["wentworth-plane-geometry-1899/eq-a58cb94bbe",16,"Wentworth 1899, scan 137: \\angle ECF = 90° + \\frac{1}{2}\\angle ACB"],["hardy-course-of-pure-mathematics-1921/eq-4f42c1d0ed",16,"Hardy 1921, p. 439: \\int_{0}^{x} \\frac{dt}{1 + t} = \\int_{0}^{x} dt - \\int_{0}^{x} t\\, dt + \\int_{0}^{x} t^{2}\\, dt - \\dots"],["hardy-course-of-pure-mathematics-1921/x-765646ea2d",15,"Hardy 1921, p. 118: On the other hand we cannot as a rule ..."],["boyden-first-book-in-algebra-1895/ex-23/11",4,"Boyden 1895, Exercise 23 (11)"],["hardy-course-of-pure-mathematics-1921/x-466a35aef5",15,"Hardy 1921, p. 201: The reader should observe that this method cannot be ..."],["form/ac05105603",5,"identity: (2*a*x**2 - 3*b*c)**3"],["hardy-course-of-pure-mathematics-1921/eq-7cb5f84a9a",16,"Hardy 1921, p. 439: D_{x} \\left(1 + x + \\frac{x^{2}}{2!} + \\dots\\right) = D_{x}1 + D_{x}x + D_{x} \\frac{x^{2}}{2!} + \\dots"],["hardy-course-of-pure-mathematics-1921/eq-bb81891aa4",16,"Hardy 1921, p. 440: \\lim_{x\\to\\xi} \\left(1 + x + \\frac{x^{2}}{2!} + \\dots\\right) = 1 + \\xi + \\frac{\\xi^{2}}{2!} + \\dots = \\lim_{x\\to\\xi} 1 +"],["wentworth-plane-geometry-1899/eq-84fc1b0f14",16,"Wentworth 1899, scan 137: \\angle E+\\angle F+\\frac{1}{2}\\angle ACB = 90°"],["shape/2c880f9d50",6,"identity: (N*a*x**N + N*b*c)**N"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/8",4,"Hardy 1921, Exercise Misc-II (8)"],["hardy-course-of-pure-mathematics-1921/x-aa7b6c644a",15,"Hardy 1921, p. 107: Were the problem however merely that of finding some ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/9",4,"Hardy 1921, Exercise Misc-II (9)"],["method/vandermonde-s-method-for-the-knight-s-tour",8,"Vandermonde's method for the knight's tour","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-vandermonde-s-method-for-the-knight-s-tour"],["method/roget-s-method-for-the-knight-s-tour",8,"Roget's method for the knight's tour","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-roget-s-method-for-the-knight-s-tour"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/10",4,"Hardy 1921, Exercise Misc-II (10)"],["method/warnsdorff-s-rule",8,"Warnsdorff's rule","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-warnsdorff-s-rule"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/11a",4,"Hardy 1921, Exercise Misc-II (11a)"],["form/71d8128200",5,"solve: Eq(csc(x) + sec(x), 2*sqrt(2))"],["hardy-course-of-pure-mathematics-1921/x-43efba2dd7",15,"Hardy 1921, p. 108: For mathematicians have succeeded in discovering a function (the ..."],["hardy-course-of-pure-mathematics-1921/eq-9cde6cbf00",16,"Hardy 1921, p. 328: \\int_{a}^{\\xi} f(x)\\phi'(x)\\, dx = f(\\xi)\\phi(\\xi) - f(a)\\phi(a) - \\int_{a}^{\\xi} f'(x)\\phi(x)\\, dx"],["shape/43d21b619b",6,"solve: Eq(csc(x) + sec(x), N*N**N)"],["hardy-course-of-pure-mathematics-1921/eq-975d2fe3d7",16,"Hardy 1921, p. 440: \\lim_{x\\to 0} \\{\\lim_{y\\to 0} (x + y)\\} = \\lim_{x\\to 0} x = 0"],["theorem/zero-factorial",9,"zero factorial"],["wentworth-plane-geometry-1899/eq-3e18ee3c7e",16,"Wentworth 1899, scan 137: \\angle E+\\angle F = 90° - \\frac{1}{2}\\angle ACB"],["dickson-theory-of-equations-1922/eq-966d1ba68d",16,"Dickson 1922, p. 59: 0! = 1"],["dickson-theory-of-equations-1922/eq-647b90521e",16,"Dickson 1922, p. 62: y = f(\\alpha) + f'(\\alpha)(x-\\alpha)"],["dickson-theory-of-equations-1922/eq-8dd67f90f9",16,"Dickson 1922, p. 62: y-\\beta = s(x -\\alpha)"],["method/sylvester-s-dialytic-method",8,"Sylvester's dialytic method","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-sylvester-s-dialytic-method"],["method/b-zout-s-method-of-elimination",8,"Bézout's method of elimination","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-b-zout-s-method-of-elimination"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/11b",4,"Hardy 1921, Exercise Misc-II (11b)"],["ball-mathematical-recreations-1905/eq-33996dabdb",16,"Ball 1905, scan 200: lr^2l = rlr"],["ball-mathematical-recreations-1905/eq-21de75d737",16,"Ball 1905, scan 200: rl^2r = lrl"],["person/william-rowan-hamilton",1,"William Rowan Hamilton","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-william-rowan-hamilton"],["unit/dyne",12,"dyne","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-unit-dyne"],["dickson-theory-of-equations-1922/eq-2961943cc5",16,"Dickson 1922, p. 63: \\tan\\theta=f'(\\alpha)"],["dickson-theory-of-equations-1922/eq-1ec24a4431",16,"Dickson 1922, p. 63: X = x\\cos\\theta"],["dickson-theory-of-equations-1922/eq-0244af2213",16,"Dickson 1922, p. 63: Y = f'(\\alpha)X + f''(\\alpha)\\frac{X^2}{2} + \\dotsb"],["ball-mathematical-recreations-1905/eq-b5119ee40d",16,"Ball 1905, scan 200: lr^3l=r^2"],["ball-mathematical-recreations-1905/eq-9e020c4c96",16,"Ball 1905, scan 200: rl^3r=l^2"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/12",4,"Hardy 1921, Exercise Misc-II (12)"],["hardy-course-of-pure-mathematics-1921/eq-e1867ae145",16,"Hardy 1921, p. 440: \\lim_{y\\to 0} \\{\\lim_{x\\to 0} (x + y)\\} = \\lim_{y\\to 0} y = 0"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/13",4,"Hardy 1921, Exercise Misc-II (13)"],["hardy-course-of-pure-mathematics-1921/eq-673073db4a",16,"Hardy 1921, p. 441: \\lim_{x\\to 0} \\left(\\lim_{y\\to 0} \\frac{x - y}{x + y}\\right) &= \\lim_{x\\to 0} \\frac{x}{x} &&= \\lim_{x\\to 0} 1 = 1"],["ball-mathematical-recreations-1905/eq-8f858dc55b",16,"Ball 1905, scan 200: l^5=1"],["ball-mathematical-recreations-1905/eq-a6ff5d8f22",16,"Ball 1905, scan 200: r^5=1"],["ball-mathematical-recreations-1905/x-565585b221",15,"Ball 1905, scan 199: in the determination of a route along the edges ..."],["hardy-course-of-pure-mathematics-1921/eq-8f955076f9",16,"Hardy 1921, p. 441: \\lim_{y\\to 0} \\left(\\lim_{x\\to 0} \\frac{x - y}{x + y}\\right) &= \\lim_{y\\to 0}\\frac{-y}{y} &&= \\lim_{y\\to 0} (-1) = -1"],["hardy-course-of-pure-mathematics-1921/eq-7ac4f47ec0",16,"Hardy 1921, p. 441: \\lim_{x\\to 1} \\left\\{\\sum_{1}^{\\infty} \\frac{(-1)^{n}}{n}x^{n}\\right\\} &= \\lim_{x\\to 1}\\log(1 + x) &&= \\log 2"],["ball-mathematical-recreations-1905/eq-96fe6230f9",16,"Ball 1905, scan 200: \\{r^3l^3(rl)^2\\}^2=1"],["ball-mathematical-recreations-1905/eq-d89825597f",16,"Ball 1905, scan 200: \\{l^3r^3(lr)^2\\}^2=1"],["hardy-course-of-pure-mathematics-1921/eq-55c47f213c",16,"Hardy 1921, p. 330: \\int_{a}^{A} (x - a)^{-s}\\, dx = \\lim_{\\epsilon\\to +0} \\int_{a+\\epsilon}^{A} (x - a)^{-s}\\, dx"],["person/tienne-b-zout",1,"Étienne Bézout","../books/dickson-theory-of-equations-1922/terms/index.html#t-person-tienne-b-zout"],["hardy-course-of-pure-mathematics-1921/eq-ed12986fe8",16,"Hardy 1921, p. 441: \\sum_{1}^{\\infty} \\left\\{\\lim_{x\\to 1} \\frac{(-1)^{n}}{n}x^{n}\\right\\} &= \\quad \\sum_{1}^{\\infty} \\frac{(-1)^{n}}{n} &&="],["dickson-theory-of-equations-1922/x-27eb0767d3",15,"Dickson 1922, p. 143: We call R the resultant (or eliminant) of the ..."],["dickson-theory-of-equations-1922/x-697ac048f9",15,"Dickson 1922, p. 143: Methods of elimination which seem plausible often yield not ..."],["dickson-theory-of-equations-1922/x-f7317aa938",15,"Dickson 1922, p. 145: Multiply the first equation by x and the second ..."],["dickson-theory-of-equations-1922/x-7a46dec5b9",15,"Dickson 1922, p. 152: Evidently D is unaltered by the interchange of any ..."],["concept/broken-diagonal",7,"broken diagonal","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-broken-diagonal"],["hardy-course-of-pure-mathematics-1921/x-ed3216b934",15,"Hardy 1921, p. 149: This number M we call the upper bound of ..."],["quantity/pressure-coefficient",11,"pressure coefficient","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-pressure-coefficient"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/141",4,"Hardy 1921, Exercise Misc-II (14(1))"],["ball-mathematical-recreations-1905/x-c7be2bfba8",15,"Ball 1905, scan 208: The rule has not been proved to be true, ..."],["concept/magic-square-cell",7,"magic square cell","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-magic-square-cell"],["concept/solubility",7,"solubility","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-solubility"],["hardy-course-of-pure-mathematics-1921/eq-5612e88b46",16,"Hardy 1921, p. 441: \\lim_{x\\to 1} \\left\\{\\sum_{1}^{\\infty} (x^{n} - x^{n+1})\\right\\} &= \\lim_{x\\to 1} \\{(1 - x) + (x - x^{2}) + \\dots\\}"],["hardy-course-of-pure-mathematics-1921/x-bd0b0bc969",15,"Hardy 1921, p. 147: the function which is equal to 1/(1 - x) ..."],["concept/natural-logarithm",7,"natural logarithm","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-natural-logarithm"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/142",4,"Hardy 1921, Exercise Misc-II (14(2))"],["concept/general-power",7,"general power","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-general-power"],["hardy-course-of-pure-mathematics-1921/eq-b99bf33261",16,"Hardy 1921, p. 441: \\lim_{x\\to 1} \\{(1 - x) + (x - x^{2}) + \\dots\\} = \\lim_{x\\to 1} 1 = 1"],["hardy-course-of-pure-mathematics-1921/eq-5ea86a1826",16,"Hardy 1921, p. 441: \\sum_{1}^{\\infty} \\left\\{\\lim_{x\\to 1} (x^{n} - x^{n+1})\\right\\} &= \\sum_{1}^{\\infty} (1 - 1) = 0 + 0 + 0 + \\dots = 0"],["concept/complementary-rows-and-columns",7,"complementary rows and columns","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-complementary-rows-and-columns"],["boyden-first-book-in-algebra-1895/ex-35/23",4,"Boyden 1895, Exercise 35 (23)"],["method/cross-interchange",8,"cross interchange","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-cross-interchange"],["method/first-method-for-even-magic-squares",8,"First method for even magic squares","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-first-method-for-even-magic-squares"],["method/de-la-loub-re-s-method",8,"De la Loubère's method","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-de-la-loub-re-s-method"],["method/bachet-s-method-for-odd-magic-squares",8,"Bachet's method for odd magic squares","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-bachet-s-method-for-odd-magic-squares"],["method/de-la-hire-s-method-for-odd-magic-squares",8,"De la Hire's method for odd magic squares","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-de-la-hire-s-method-for-odd-magic-squares"],["boyden-first-book-in-algebra-1895/ex-35/24",4,"Boyden 1895, Exercise 35 (24)"],["boyden-first-book-in-algebra-1895/ex-37/25",4,"Boyden 1895, Exercise 37 (25)"],["method/moon-s-method-for-the-knight-s-tour",8,"Moon's method for the knight's tour","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-moon-s-method-for-the-knight-s-tour"],["dickson-theory-of-equations-1922/eq-ad9fce4e94",16,"Dickson 1922, p. 63: y = cx^m + dx^{m+1} + \\dotsb"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/143",4,"Hardy 1921, Exercise Misc-II (14(3))"],["hardy-course-of-pure-mathematics-1921/eq-fef21e88d9",16,"Hardy 1921, p. 330: \\int_{1/A}^{\\eta} y^{s-2}\\, dy = \\int_{1/\\eta}^{A} x^{-s}\\, dx"],["quantity/coefficient-of-elasticity",11,"coefficient of elasticity","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-coefficient-of-elasticity"],["form/002a6a02f7",5,"factor: x**5 - 23*x**4 + 132*x**3"],["shape/2c51445e88",6,"factor: 2*N*x**N + x**N"],["method/jaenisch-s-method-for-the-knight-s-tour",8,"Jaenisch's method for the knight's tour","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-jaenisch-s-method-for-the-knight-s-tour"],["concept/closed-circuit",7,"closed circuit","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-closed-circuit"],["dickson-theory-of-equations-1922/eq-7d5977b7db",16,"Dickson 1922, p. 63: c = \\frac{f^{(m)}(\\alpha)\\cos^m \\theta}{m!}"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/144",4,"Hardy 1921, Exercise Misc-II (14(4))"],["hardy-course-of-pure-mathematics-1921/eq-357ca4d096",16,"Hardy 1921, p. 326: \\int_{a}^{\\infty} \\phi(x)\\, dx = \\lim_{\\tau\\to c} \\int_{b}^{\\tau} \\phi\\{f(t)\\}f'(t)\\, dt"],["hardy-course-of-pure-mathematics-1921/eq-628a00c9db",16,"Hardy 1921, p. 443: y = y(x) = \\arctan x = \\ds\\int_{0}^{x} \\frac{dt}{1 + t^{2}}"],["boyden-first-book-in-algebra-1895/ex-29/5",4,"Boyden 1895, Exercise 29 (5)"],["hardy-course-of-pure-mathematics-1921/eq-43716da505",16,"Hardy 1921, p. 443: x = x(y) = \\tan y"],["boyden-first-book-in-algebra-1895/ex-29/4",4,"Boyden 1895, Exercise 29 (4)"],["planck-treatise-on-thermodynamics-1903/x-671bfa87ee",15,"Planck 1903, p. 259: Conversely, the dissociation of the sodium acetate increases on ..."],["boyden-first-book-in-algebra-1895/ex-29/6",4,"Boyden 1895, Exercise 29 (6)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/145",4,"Hardy 1921, Exercise Misc-II (14(5))"],["person/de-la-hire",1,"De la Hire","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-de-la-hire"],["hardy-course-of-pure-mathematics-1921/eq-b02e3eacc3",16,"Hardy 1921, p. 443: \\frac{1}{2}\\pi = \\ds\\int_{0}^{\\infty} \\frac{dt}{1 + t^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-40ff3920ff",16,"Hardy 1921, p. 443: \\cos y = \\dfrac{1}{\\sqrt{1 + x^{2}}}"],["hardy-course-of-pure-mathematics-1921/eq-ccb9ff2d51",16,"Hardy 1921, p. 443: \\sin y = \\dfrac{x}{\\sqrt{1 + x^{2}}}"],["boyden-first-book-in-algebra-1895/ex-29/7",4,"Boyden 1895, Exercise 29 (7)"],["planck-treatise-on-thermodynamics-1903/x-802b149728",15,"Planck 1903, p. 33: This, in general, varies with temperature, but very slowly ..."],["person/claude-gaspar-bachet-de-m-ziriac",1,"Claude Gaspar Bachet de Méziriac","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-claude-gaspar-bachet-de-m-ziriac"],["person/moschopulus",1,"Moschopulus","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-moschopulus"],["ball-mathematical-recreations-1905/x-79860dc816",15,"Ball 1905, scan 164: The cells filled by the same number form a ..."],["hardy-course-of-pure-mathematics-1921/x-78e5a2beba",15,"Hardy 1921, p. 421: Let z be any complex number, and h a ..."],["hardy-course-of-pure-mathematics-1921/x-2dcbcf23ea",15,"Hardy 1921, p. 421: _h0 (1 + hz)h = z."],["ball-mathematical-recreations-1905/ch-x",2,"Ball 1905, ch. X: Astrology","../books/ball-mathematical-recreations-1905/ch/ch-x/index.html"],["concept/astrology",7,"astrology","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-astrology"],["concept/corner-cell",7,"corner cell","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-corner-cell"],["method/natal-astrology",8,"natal astrology","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-natal-astrology"],["method/horary-astrology",8,"horary astrology","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-horary-astrology"],["de-morgan-elementary-illustrations-calculus-1899/x-b272c3af95",15,"De Morgan 1899, p. 107: In this case y is said to be implicitly ..."],["method/verification",8,"verification","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-verification"],["hardy-course-of-pure-mathematics-1921/eq-d775accb12",16,"Hardy 1921, p. 120: \\phi(n) = n^{k}"],["boyden-first-book-in-algebra-1895/ex-23/13",4,"Boyden 1895, Exercise 23 (13)"],["de-morgan-elementary-illustrations-calculus-1899/x-7c6e08f887",15,"De Morgan 1899, p. 107: For example, in x^{2} - xy + y^{2} = ..."],["hardy-course-of-pure-mathematics-1921/eq-973f1edbbe",16,"Hardy 1921, p. 120: \\lim n^{k} = 0"],["hardy-course-of-pure-mathematics-1921/eq-127b8f88bd",16,"Hardy 1921, p. 120: \\lim n^{k} = 1"],["concept/scheme-of-nativity",7,"scheme of nativity","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-scheme-of-nativity"],["hardy-course-of-pure-mathematics-1921/eq-b3174e71c5",16,"Hardy 1921, p. 120: \\phi(n) = p_{n}"],["concept/right-circular-cylinder",7,"right circular cylinder","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-right-circular-cylinder"],["hardy-course-of-pure-mathematics-1921/eq-ff0d7d4f3f",16,"Hardy 1921, p. 120: \\phi(n) > n"],["de-morgan-elementary-illustrations-calculus-1899/x-0ffd6d930f",15,"De Morgan 1899, p. 21: It appears, then, that the development of \\phi(x + ..."],["concept/trigonometric-form-of-a-complex-number",7,"trigonometric form of a complex number","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-trigonometric-form-of-a-complex-number"],["hardy-course-of-pure-mathematics-1921/eq-3c8eada3cb",16,"Hardy 1921, p. 121: \\phi(n) = [\\alpha n]"],["hardy-course-of-pure-mathematics-1921/eq-e1a4954646",16,"Hardy 1921, p. 121: \\phi(n) = 0\\quad (0 \\leq n < 1 / \\alpha)"],["form/51d2b2b94b",5,"identity: (-a**3 + 3*a**2*x - 3*a*x**2 + x**3)*(a**3 + 3*a**2*x + 3*a*x**2 + x**3)"],["concept/astrological-planet",7,"astrological planet","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-astrological-planet"],["shape/89185f1a85",6,"identity: (N*a*x**N + N*a**N*x - a**N + x**N)*(N*a*x**N + N*a**N*x + a**N + x**N)"],["concept/differential",7,"differential","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-differential"],["hardy-course-of-pure-mathematics-1921/eq-e87e9add02",16,"Hardy 1921, p. 121: \\phi(n) = 1/\\{n - (-1)^{n}\\}"],["hardy-course-of-pure-mathematics-1921/eq-e3a9531bea",16,"Hardy 1921, p. 121: |\\phi(n)| < 1/n"],["de-morgan-elementary-illustrations-calculus-1899/x-038f54f7d7",15,"De Morgan 1899, p. 107: Here, though we know that y must be a ..."],["hardy-course-of-pure-mathematics-1921/eq-07c4b568f1",16,"Hardy 1921, p. 123: \\lim \\sin n\\theta\\pi = l"],["hardy-course-of-pure-mathematics-1921/eq-5fb6e131f2",16,"Hardy 1921, p. 122: \\phi(n) = (-1)^{n}"],["hardy-course-of-pure-mathematics-1921/eq-a927e81010",16,"Hardy 1921, p. 122: \\phi(n) = (-1)^{n} + (1/n)"],["hardy-course-of-pure-mathematics-1921/eq-f1fd6f67c1",16,"Hardy 1921, p. 122: \\phi(n) = (-1)^{n}n"],["hardy-course-of-pure-mathematics-1921/eq-a7eb5fcecd",16,"Hardy 1921, p. 123: \\cos(n + \\tfrac{1}{2})\\theta\\pi = \\cos n\\theta\\pi \\cos\\tfrac{1}{2}\\theta\\pi - \\sin n\\theta\\pi \\sin\\tfrac{1}{2}\\theta\\pi"],["boyden-first-book-in-algebra-1895/ex-23/14",4,"Boyden 1895, Exercise 23 (14)"],["de-morgan-elementary-illustrations-calculus-1899/x-0fd8000cae",15,"De Morgan 1899, p. 21: The following relation exists between \\phi x, \\phi' x, ..."],["hardy-course-of-pure-mathematics-1921/eq-9b023ec2f4",16,"Hardy 1921, p. 123: \\cos(n - \\tfrac{1}{2})\\theta\\pi = \\cos n\\theta\\pi \\cos\\tfrac{1}{2}\\theta\\pi + \\sin n\\theta\\pi \\sin\\tfrac{1}{2}\\theta\\pi"],["hardy-course-of-pure-mathematics-1921/eq-18a198b539",16,"Hardy 1921, p. 121: \\sin(np\\pi/q) = (-1)^{ap}\\sin(bp\\pi/q)"],["de-morgan-elementary-illustrations-calculus-1899/x-92f138648e",15,"De Morgan 1899, p. 20: In this case, by theorem (1b), the series is ..."],["hardy-course-of-pure-mathematics-1921/eq-8720e04a30",16,"Hardy 1921, p. 126: \\lim\\{\\phi(n) + \\psi(n)\\} = a + b"],["hardy-course-of-pure-mathematics-1921/eq-32b3ecf1c6",16,"Hardy 1921, p. 126: |\\phi(n) + \\psi(n) - a - b| < \\DELTA"],["hardy-course-of-pure-mathematics-1921/eq-b274e3d7c2",16,"Hardy 1921, p. 126: |\\phi(n) + \\psi(n) - a - b| \\leq |\\phi(n) - a| + |\\psi(n) - b|"],["de-morgan-elementary-illustrations-calculus-1899/x-ce9b504082",15,"De Morgan 1899, p. 21: When \\phi x = a^{x}, \\phi' x = ka^{x}, ..."],["concept/proper-algebraic-fraction",7,"proper algebraic fraction","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-proper-algebraic-fraction"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/15",4,"Hardy 1921, Exercise Misc-II (15)"],["form/f23c7619e9",5,"identity: (x - 1)*(x + 1)*(x**2 + 1)"],["shape/27672bc48c",6,"identity: (x - 1)*(x + 1)*(x**N + 1)"],["hardy-course-of-pure-mathematics-1921/eq-4d3331cb1b",16,"Hardy 1921, p. 128: \\phi(n)\\psi(n) = ab + a\\psi_{1}(n) + b\\phi_{1}(n) + \\phi_{1}(n)\\psi_{1}(n)"],["concept/re-entrant-route",7,"re-entrant route","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-re-entrant-route"],["method/euler-s-method-for-the-knight-s-tour",8,"Euler's method for the knight's tour","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-euler-s-method-for-the-knight-s-tour"],["de-morgan-elementary-illustrations-calculus-1899/x-a8d74ead19",15,"De Morgan 1899, p. 22: Again, if \\phi x = \\log x, \\phi' x ..."],["ball-mathematical-recreations-1905/x-1045ef21b2",15,"Ball 1905, scan 297: This sphere was divided into twelve equal spaces by ..."],["ball-mathematical-recreations-1905/x-41e9151b50",15,"Ball 1905, scan 302: It must be remembered also that the rules were ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/16",4,"Hardy 1921, Exercise Misc-II (16)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/17",4,"Hardy 1921, Exercise Misc-II (17)"],["hardy-course-of-pure-mathematics-1921/eq-5ee96e8e96",16,"Hardy 1921, p. 128: \\lim\\phi(n)\\psi(n) = ab"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/18",4,"Hardy 1921, Exercise Misc-II (18)"],["ball-mathematical-recreations-1905/x-41398b5e33",15,"Ball 1905, scan 302: This is the less necessary as the rules---especially as ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/19",4,"Hardy 1921, Exercise Misc-II (19)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/20",4,"Hardy 1921, Exercise Misc-II (20)"],["planck-treatise-on-thermodynamics-1903/x-5ba60dbc81",15,"Planck 1903, p. 35: A gas at 0° and atmospheric pressure can be ..."],["hardy-course-of-pure-mathematics-1921/eq-c1f934c5de",16,"Hardy 1921, p. 129: \\lim k\\phi(n) = ka"],["hardy-course-of-pure-mathematics-1921/eq-d51186254d",16,"Hardy 1921, p. 130: \\left|\\frac{1}{\\phi(n)} - \\frac{1}{a}\\right| = \\frac{|\\phi_{1}(n)|}{|a| |a + \\phi_{1}(n)|}"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/21",4,"Hardy 1921, Exercise Misc-II (21)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/22",4,"Hardy 1921, Exercise Misc-II (22)"],["hardy-course-of-pure-mathematics-1921/eq-78033223a3",16,"Hardy 1921, p. 129: \\lim\\frac{1}{\\phi(n)} = \\frac{1}{a}"],["boyden-first-book-in-algebra-1895/ex-29/8",4,"Boyden 1895, Exercise 29 (8)"],["hardy-course-of-pure-mathematics-1921/eq-2ceb827003",16,"Hardy 1921, p. 130: \\lim\\frac{\\phi(n)}{\\psi(n)} = \\frac{a}{b}"],["boyden-first-book-in-algebra-1895/ex-29/9",4,"Boyden 1895, Exercise 29 (9)"],["form/56051c8c9b",5,"identity: 3"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/23",4,"Hardy 1921, Exercise Misc-II (23)"],["hardy-course-of-pure-mathematics-1921/eq-ae75aafe52",16,"Hardy 1921, p. 130: \\lim R\\{\\phi(n), \\psi(n), \\chi(n), \\dots\\} = R(a, b, c, \\dots)"],["hardy-course-of-pure-mathematics-1921/eq-b1ffb49130",16,"Hardy 1921, p. 130: S(n) = \\frac{a_{0}n^{p} + a_{1}n^{p-1} + \\dots + a_{p}} {b_{0}n^{q} + b_{1}n^{q-1} + \\dots + b_{q}}"],["concept/thermodynamic-equilibrium",7,"thermodynamic equilibrium","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-thermodynamic-equilibrium"],["boyden-first-book-in-algebra-1895/ex-23/15",4,"Boyden 1895, Exercise 23 (15)"],["form/fb9e58b6a8",5,"identity: -a**4 + x**4"],["shape/09a6b72e84",6,"identity: -a**N + x**N"],["form/750d81c6ac",5,"identity: -40*x"],["hardy-course-of-pure-mathematics-1921/eq-dc8abe6dd2",16,"Hardy 1921, p. 131: \\lim S(n) = 0\\quad (p < q)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/24",4,"Hardy 1921, Exercise Misc-II (24)"],["hardy-course-of-pure-mathematics-1921/eq-f8b5d14936",16,"Hardy 1921, p. 131: \\lim S(n) = a_{0}/b_{0}\\quad (p = q)"],["hardy-course-of-pure-mathematics-1921/x-8954d0489f",15,"Hardy 1921, p. 422: ddt(1 + tz)^m = mz(1 + tz)^m-1"],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/25",4,"Hardy 1921, Exercise Misc-II (25)"],["hardy-course-of-pure-mathematics-1921/x-ea7e29707b",15,"Hardy 1921, p. 423: for all values of m, real or complex, and ..."],["wentworth-first-steps-in-algebra-1894/ex-1/14",4,"Wentworth 1894, Exercise 1 (14)"],["form/a7623c7409",5,"identity: 8"],["concept/squaring-the-circle",7,"squaring the circle","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-squaring-the-circle"],["ball-mathematical-recreations-1905/ch-vii",2,"Ball 1905, ch. VII: The Mathematical Tripos","../books/ball-mathematical-recreations-1905/ch/ch-vii/index.html"],["hardy-course-of-pure-mathematics-1921/eq-54a817f896",16,"Hardy 1921, p. 334: J = \\int_{1}^{7} (x^{2} - 6x + 13)\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-a4d67b61ac",16,"Hardy 1921, p. 334: J = 48"],["concept/decimation-problem",7,"decimation problem","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-decimation-problem"],["method/frequency-analysis",8,"frequency analysis","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-frequency-analysis"],["hardy-course-of-pure-mathematics-1921/eq-98ddd3ffc5",16,"Hardy 1921, p. 334: y = x^{2} - 6x + 13"],["hardy-course-of-pure-mathematics-1921/eq-a6868aa802",16,"Hardy 1921, p. 334: x = 3 ± \\sqrtp{y - 4}"],["hardy-course-of-pure-mathematics-1921/eq-3a1c3fab01",16,"Hardy 1921, p. 335: J = \\int_{1}^{7} y\\, dx = \\int_{8}^{4} \\left\\{-\\frac{y}{2\\sqrtp{y - 4}}\\right\\} dy + \\int_{4}^{20} \\frac{y}{2\\sqrtp{y - "],["hardy-course-of-pure-mathematics-1921/eq-76c13f95dc",16,"Hardy 1921, p. 335: \\int_{0}^{\\pi} dx = \\pi"],["hardy-course-of-pure-mathematics-1921/eq-9f90ceb50b",16,"Hardy 1921, p. 335: x = \\arcsin y"],["concept/surd",7,"surd","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-surd"],["law/distribution-law",10,"distribution law","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-law-distribution-law"],["planck-treatise-on-thermodynamics-1903/x-4b6def4bd2",15,"Planck 1903, p. 171: The total mass M, the volume V, and the ..."],["quantity/osmotic-pressure",11,"osmotic pressure","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-osmotic-pressure"],["ball-mathematical-recreations-1905/x-2bb1ab2b9b",15,"Ball 1905, scan 318: The defect of the method is that the broken ..."],["hardy-course-of-pure-mathematics-1921/eq-bdcc88f13c",16,"Hardy 1921, p. 335: dx/dy = 1/\\sqrtp{1 - y^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-99f13dd4e4",16,"Hardy 1921, p. 335: dx/dy = -1/\\sqrtp{1 - y^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-99c0850d64",16,"Hardy 1921, p. 336: |u_{n}| = \\alpha_{n}"],["hardy-course-of-pure-mathematics-1921/x-bc81274a88",15,"Hardy 1921, p. 120: This number is evidently not divisible by any of ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-ii/26",4,"Hardy 1921, Exercise Misc-II (26)"],["hardy-course-of-pure-mathematics-1921/x-b15fb71f23",15,"Hardy 1921, p. 120: There are however, as was first shown by Euclid, ..."],["hardy-course-of-pure-mathematics-1921/x-dd0b922e67",15,"Hardy 1921, p. 65: If f(x) = f(-x) for all values of x, ..."],["planck-treatise-on-thermodynamics-1903/x-7d907c885c",15,"Planck 1903, p. 37: Latent heat, as in the case of specific heat, ..."],["todhunter-spherical-trigonometry-1886/eq-4a22df4026",16,"Todhunter 1886, scan 57: \\cos a = \\dfrac{\\cos A + \\cos B \\cos C}{\\sin B \\sin C}"],["concept/variable",7,"variable","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-variable"],["concept/subsidiary-square",7,"subsidiary square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-subsidiary-square"],["law/law-of-dissociation-of-an-electrolyte",10,"law of dissociation of an electrolyte","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-law-law-of-dissociation-of-an-electrolyte"],["theorem/van-der-waals-equation",9,"van der Waals' equation","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-theorem-van-der-waals-equation"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/1",4,"Hardy 1921, Exercise Misc-III (1)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/2",4,"Hardy 1921, Exercise Misc-III (2)"],["concept/even-magic-square",7,"even magic square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-even-magic-square"],["shape/172b75ecd2",6,"identity: 1"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/3",4,"Hardy 1921, Exercise Misc-III (3)"],["concept/equation-of-the-second-degree",7,"equation of the second degree","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-equation-of-the-second-degree"],["concept/parabola",7,"parabola","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-parabola"],["concept/origin",7,"origin","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-origin"],["wentworth-first-steps-in-algebra-1894/ex-20/1",4,"Wentworth 1894, Exercise 20 (1)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/4",4,"Hardy 1921, Exercise Misc-III (4)"],["form/9fdd95356d",5,"identity: a"],["hardy-course-of-pure-mathematics-1921/ex-xlvi",3,"Hardy 1921, Exercise XLVI"],["shape/9fdd95356d",6,"identity: a"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/5",4,"Hardy 1921, Exercise Misc-III (5)"],["hardy-course-of-pure-mathematics-1921/eq-02be3723c4",16,"Hardy 1921, p. 336: u_{n} = v_{n} - w_{n}"],["hardy-course-of-pure-mathematics-1921/eq-5b57b6a962",16,"Hardy 1921, p. 336: \\alpha_{n} = v_{n} + w_{n}"],["dickson-theory-of-equations-1922/eq-39c6da23aa",16,"Dickson 1922, p. 65: f(x) = x^3 - 3lx + q"],["hardy-course-of-pure-mathematics-1921/eq-0ab8704e5e",16,"Hardy 1921, p. 338: \\tsum u'_{n} = \\tsum v'_{n} - \\tsum w'_{n} = \\tsum v_{n} - \\tsum w_{n} = \\tsum u_{n}"],["todhunter-spherical-trigonometry-1886/eq-05abc0038d",16,"Todhunter 1886, scan 57: \\tan \\tfrac{1}{2}(A + B) = \\dfrac{\\cos\\tfrac{1}{2}(a - b)}{\\cos\\tfrac{1}{2}(a + b)}\\cot\\tfrac{1}{2}C"],["todhunter-spherical-trigonometry-1886/eq-33b84efeb1",16,"Todhunter 1886, scan 57: \\tan \\tfrac{1}{2}(A - B) = \\dfrac{\\sin\\tfrac{1}{2}(a - b)}{\\sin\\tfrac{1}{2}(a + b)}\\cot\\tfrac{1}{2}C"],["dickson-theory-of-equations-1922/eq-32a16d1e96",16,"Dickson 1922, p. 65: q^2 = 4l^3"],["dickson-theory-of-equations-1922/eq-d30219e428",16,"Dickson 1922, p. 65: q^2 < 4l^3"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/6",4,"Hardy 1921, Exercise Misc-III (6)"],["concept/bordered-magic-square",7,"bordered magic square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-bordered-magic-square"],["method/constructing-an-even-magic-square",8,"constructing an even magic square","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-constructing-an-even-magic-square"],["form/960ea66f8d",5,"evaluate: 2**(2/3)*251**(1/3)/20"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/7",4,"Hardy 1921, Exercise Misc-III (7)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/8",4,"Hardy 1921, Exercise Misc-III (8)"],["concept/composite-magic-square",7,"composite magic square","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-composite-magic-square"],["concept/magic-pencil",7,"magic pencil","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-magic-pencil"],["boyden-first-book-in-algebra-1895/ex-30/17",4,"Boyden 1895, Exercise 30 (17)"],["shape/af14cef8d4",6,"identity: -a + b"],["ball-mathematical-recreations-1905/x-a379ac0fd8",15,"Ball 1905, scan 172: The square so formed is necessarily magic in rows, ..."],["ball-mathematical-recreations-1905/x-385446ac83",15,"Ball 1905, scan 173: In the case of a singly even square, that ..."],["ball-mathematical-recreations-1905/x-85743adf58",15,"Ball 1905, scan 178: By reciprocating the figures composed of the points on ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/9",4,"Hardy 1921, Exercise Misc-III (9)"],["hardy-course-of-pure-mathematics-1921/eq-4f3e49adf5",16,"Hardy 1921, p. 338: \\sum_{0}^{N} u_{n} = \\sum_{0}^{N} v_{n} - \\sum_{0}^{N} w_{n}"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/10",4,"Hardy 1921, Exercise Misc-III (10)"],["ball-mathematical-recreations-1905/x-c1750611a8",15,"Ball 1905, scan 217: Probably the science (as distinct from the art) of ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/11",4,"Hardy 1921, Exercise Misc-III (11)"],["ball-mathematical-recreations-1905/x-8cdf2e27d7",15,"Ball 1905, scan 176: I believe that with a little patience a magic ..."],["hardy-course-of-pure-mathematics-1921/x-bcc8264b79",15,"Hardy 1921, p. 39: Boyle’s law, however, only gives a reasonable approximation to ..."],["concept/mechanics",7,"mechanics","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-mechanics"],["hardy-course-of-pure-mathematics-1921/eq-ceb2c24679",16,"Hardy 1921, p. 339: |v_{0}| + |v_{1}| + \\dots + |v_{n}| < |u_{0}| + \\dots + |u_{n}|"],["hardy-course-of-pure-mathematics-1921/x-bbc7621d31",15,"Hardy 1921, p. 41: Let y be defined as the height in inches ..."],["concept/change-of-energy",7,"change of energy","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-change-of-energy"],["hardy-course-of-pure-mathematics-1921/eq-d7dec9985a",16,"Hardy 1921, p. 340: s_{2n} - s_{2n-2} = -(\\phi_{2n-1} - \\phi_{2n}) \\leq 0"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/12",4,"Hardy 1921, Exercise Misc-III (12)"],["planck-treatise-on-thermodynamics-1903/eq-acad5749c4",16,"Planck 1903, p. 43: U_{2} - U_{1} = Q + W\\Add{.}"],["hardy-course-of-pure-mathematics-1921/eq-425f5305f1",16,"Hardy 1921, p. 340: s_{n} = \\phi_{0} - \\phi_{1} + \\phi_{2} - \\dots + (-1)^{n}\\phi_{n}"],["hardy-course-of-pure-mathematics-1921/eq-cfbb063153",16,"Hardy 1921, p. 340: s_{2n+1} - s_{2n-1} = \\phi_{2n} - \\phi_{2n+1}\\geq 0"],["hardy-course-of-pure-mathematics-1921/eq-8da8affd87",16,"Hardy 1921, p. 340: \\lim (s_{2n+1} - s_{2n}) = \\lim (-1)^{2n+1} \\phi_{2n+1} = 0"],["ball-mathematical-recreations-1905/eq-a82c0c42c2",16,"Ball 1905, scan 255: x^3 = 2a^3"],["todhunter-spherical-trigonometry-1886/eq-ffe0de22f5",16,"Todhunter 1886, scan 57: \\cos c = \\cos a\\, \\cos b + \\sin a\\, \\sin b\\, \\cos C"],["planck-treatise-on-thermodynamics-1903/eq-015a3f5d2c",16,"Planck 1903, p. 43: U_{2} - U_{1} = U_{2}"],["theorem/duplication-of-the-cube-condition",9,"duplication of the cube condition"],["planck-treatise-on-thermodynamics-1903/eq-ed3cf157b4",16,"Planck 1903, p. 44: Q + W = 0\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-b9dc081efd",16,"Planck 1903, p. 44: U = \\const"],["concept/normal-state-zero-of-energy",7,"normal state (zero of energy)"],["planck-treatise-on-thermodynamics-1903/eq-833599be42",16,"Planck 1903, p. 43: U_{1} = 0"],["concept/zero-energy",7,"zero energy"],["ball-mathematical-recreations-1905/eq-4c54602da6",16,"Ball 1905, scan 255: 4x^3=3x-a"],["todhunter-spherical-trigonometry-1886/eq-3d19927f42",16,"Todhunter 1886, scan 57: \\sin c = \\dfrac{\\sin a\\, \\sin C}{\\sin A}"],["todhunter-spherical-trigonometry-1886/eq-d925511c1c",16,"Todhunter 1886, scan 57: \\cos c = \\cos b\\, (\\cos a + \\sin a \\tan b \\cos C)"],["todhunter-spherical-trigonometry-1886/eq-255143d90f",16,"Todhunter 1886, scan 58: \\tan \\theta = \\tan b\\, \\cos C"],["ball-mathematical-recreations-1905/eq-43b9e85cc4",16,"Ball 1905, scan 255: x^2 + y^2 + ax + by + c = 0"],["hardy-course-of-pure-mathematics-1921/x-fcd5f47abf",15,"Hardy 1921, p. 44: The reader has no doubt some notion as to ..."],["ball-mathematical-recreations-1905/eq-deb71e7d9f",16,"Ball 1905, scan 255: \\alpha x + \\beta y +\\gamma = 0"],["hardy-course-of-pure-mathematics-1921/ex-misc-vii",3,"Hardy 1921, Exercise Misc-VII"],["todhunter-spherical-trigonometry-1886/eq-4908843390",16,"Todhunter 1886, scan 58: \\cos c = \\cos b\\, (\\cos a + \\sin a\\, \\tan \\theta) = \\dfrac{\\cos b\\, \\cos (a - \\theta)}{\\cos \\theta}"],["todhunter-spherical-trigonometry-1886/eq-f02f3d8708",16,"Todhunter 1886, scan 58: \\tan CD = \\tan b \\cos C"],["ball-mathematical-recreations-1905/eq-8c22cd2f71",16,"Ball 1905, scan 257: x: a = \\sqrt[3]{2}: 1"],["ball-mathematical-recreations-1905/eq-25cb39eb5f",16,"Ball 1905, scan 258: a: x = x: y = y: 2a"],["ball-mathematical-recreations-1905/eq-415022ec5c",16,"Ball 1905, scan 259: r = 2a \\sin\\theta"],["ball-mathematical-recreations-1905/eq-aa240f72ea",16,"Ball 1905, scan 259: r \\sin\\theta = 2a \\cos\\phi"],["concept/binomial-coefficient",7,"binomial coefficient","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-binomial-coefficient"],["de-morgan-elementary-illustrations-calculus-1899/ch-taylor-s-theorem-derived-functions",2,"De Morgan 1899, Taylor's Theorem. Derived Functions","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-taylor-s-theorem-derived-functions/index.html"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/13",4,"Hardy 1921, Exercise Misc-III (13)"],["concept/saturation-point",7,"saturation point","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-saturation-point"],["de-morgan-elementary-illustrations-calculus-1899/eq-b5720c9ca6",16,"De Morgan 1899, p. 20: (x + h)^{n} = x^{n} + nx^{n-1}h + n(n - 1)x^{n-2} \\frac{h^{2}}{2} + n(n - 1)(n - 2)x^{n-3} \\frac{h^{3}}{2·3}"],["ball-mathematical-recreations-1905/eq-9bde780238",16,"Ball 1905, scan 259: \\sin\\theta \\cos\\phi = \\frac{1}{2}"],["ball-mathematical-recreations-1905/eq-4e4d29a1d9",16,"Ball 1905, scan 259: \\sin^3\\theta = \\frac{1}{2}"],["ball-mathematical-recreations-1905/eq-63c24ab59e",16,"Ball 1905, scan 259: (r\\sin\\theta)^3=2a^3"],["de-morgan-elementary-illustrations-calculus-1899/eq-2041e1ecc9",16,"De Morgan 1899, p. 20: \\log(x + h) = \\log x + \\frac{1}{x}\\, h - \\frac{1}{x^{2}}\\, \\frac{h^{2}}{2} + \\frac{2}{x^{3}}\\, \\frac{h^{3}}{2·3}"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/14",4,"Hardy 1921, Exercise Misc-III (14)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/15a",4,"Hardy 1921, Exercise Misc-III (15a)"],["wentworth-first-steps-in-algebra-1894/ex-15/8",4,"Wentworth 1894, Exercise 15 (8)"],["form/e9c3bb9514",5,"identity: -66*x"],["de-morgan-elementary-illustrations-calculus-1899/eq-17c3a81421",16,"De Morgan 1899, p. 20: \\cos(x + h) = \\cos x - \\sin x\\, h - \\cos x\\, \\frac{h^{2}}{2} + \\sin x\\, \\frac{h^{3}}{2·3}"],["todhunter-spherical-trigonometry-1886/eq-0d82b9577d",16,"Todhunter 1886, scan 58: \\tan AD = \\tan C \\sin CD"],["ball-mathematical-recreations-1905/eq-d31d808c30",16,"Ball 1905, scan 259: PC : PB = PB : PA = PA : PD"],["form/733b2127d8",5,"solve: (Eq(a, 5*b), Eq(a + b, 60))"],["concept/graph-of-a-function",7,"graph of a function","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-graph-of-a-function"],["ball-mathematical-recreations-1905/eq-d2e13dbfaa",16,"Ball 1905, scan 259: y^2 = 2ax"],["de-morgan-elementary-illustrations-calculus-1899/x-721f28bb80",15,"De Morgan 1899, p. 112: value of the variable, is, if we may use ..."],["concept/standard-form",7,"standard form","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-standard-form"],["boyden-first-book-in-algebra-1895/ex-31/1",4,"Boyden 1895, Exercise 31 (1)"],["concept/latus-rectum",7,"latus rectum"],["ball-mathematical-recreations-1905/eq-0085ee58ba",16,"Ball 1905, scan 259: x^2=ay"],["ball-mathematical-recreations-1905/eq-165535994a",16,"Ball 1905, scan 260: x^3 = 2l^3"],["ball-mathematical-recreations-1905/eq-591644ece6",16,"Ball 1905, scan 260: y^3 = 4l^3"],["ball-mathematical-recreations-1905/eq-894d31e6df",16,"Ball 1905, scan 260: x^2 = ly"],["concept/parametric-representation-of-a-curve",7,"parametric representation of a curve","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-parametric-representation-of-a-curve"],["ball-mathematical-recreations-1905/eq-e87187e2c1",16,"Ball 1905, scan 260: xy = 2l^2"],["form/537f91c99b",5,"solve: Eq(a**3 - 3*a*x + x**3 + 1, 0)"],["ball-mathematical-recreations-1905/eq-eade4e93a0",16,"Ball 1905, scan 260: l : x = x : y = y : 2l"],["de-morgan-elementary-illustrations-calculus-1899/x-cf79c267e2",15,"De Morgan 1899, p. 114: In the same way, if a point is moving, ..."],["ball-mathematical-recreations-1905/eq-e185414129",16,"Ball 1905, scan 260: OA : Bb = Bb : Aa = Aa : OB"],["ball-mathematical-recreations-1905/eq-30020875e1",16,"Ball 1905, scan 261: a : x = x : y = y : b"],["ball-mathematical-recreations-1905/eq-df5830cdf9",16,"Ball 1905, scan 261: x^2 + y^2 = ay + bx"],["ball-mathematical-recreations-1905/eq-3113b2ca4c",16,"Ball 1905, scan 261: xy = ab"],["ball-mathematical-recreations-1905/eq-fe03761827",16,"Ball 1905, scan 262: BC : OD = OD : CE = CE : OA"],["shape/007f83ab39",6,"solve: Eq(N*a*x + a**N + x**N + 1, 0)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/15b",4,"Hardy 1921, Exercise Misc-III (15b)"],["form/83c1eb84e9",5,"solve: Eq(a**5 + 5*a**2*x - 5*a*x**3 + x**5 + 1, 0)"],["ball-mathematical-recreations-1905/eq-7838adc7db",16,"Ball 1905, scan 261: AB : GC = GC : GA = GA : CD"],["shape/b986c607e4",6,"solve: Eq(N*a*x**N + N*a**N*x + a**N + x**N + 1, 0)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/16",4,"Hardy 1921, Exercise Misc-III (16)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/17",4,"Hardy 1921, Exercise Misc-III (17)"],["ball-mathematical-recreations-1905/eq-01c528dab8",16,"Ball 1905, scan 262: QR = 2\\dotm OP"],["form/c90b44651d",5,"solve: (Eq(a, 8*b), Eq(a + b, 90))"],["form/a047a20a8b",5,"solve: (Eq(a - b, 7), Eq(a + b, 53))"],["concept/degree-of-freedom",7,"degree of freedom","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-degree-of-freedom"],["ball-mathematical-recreations-1905/eq-6858febed1",16,"Ball 1905, scan 262: \\tan^{-1} (b/x)=\\frac{1}{3}\\tan^{-1}(b/a)"],["de-morgan-elementary-illustrations-calculus-1899/eq-6e24c4adb9",16,"De Morgan 1899, p. 21: \\phi'' x = -\\sin x"],["de-morgan-elementary-illustrations-calculus-1899/eq-575dcb219a",16,"De Morgan 1899, p. 21: \\phi' x = ka^{x}"],["de-morgan-elementary-illustrations-calculus-1899/eq-a4a82055ae",16,"De Morgan 1899, p. 21: \\phi(x + h) = \\phi x + \\phi' x\\, h + \\phi''x\\, \\frac{h^{2}}{2} + \\phi''' x\\, \\frac{h^{3}}{2·3} + \\etc."],["de-morgan-elementary-illustrations-calculus-1899/eq-197e96a074",16,"De Morgan 1899, p. 21: \\phi'(x + h) = ka^{x+h} = k(a^{x} + ka^{x}\\, h + \\etc.)"],["de-morgan-elementary-illustrations-calculus-1899/eq-fe73ebca9e",16,"De Morgan 1899, p. 21: \\phi' x = nx^{n-1}"],["de-morgan-elementary-illustrations-calculus-1899/eq-cd1a255f5a",16,"De Morgan 1899, p. 21: \\phi'' x = n(n - 1)x^{n-2}"],["de-morgan-elementary-illustrations-calculus-1899/eq-9c0c495ec8",16,"De Morgan 1899, p. 21: \\phi' x = \\cos x"],["ball-mathematical-recreations-1905/eq-69a51df3ae",16,"Ball 1905, scan 262: AOR=\\frac{1}{3}AOB"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/18",4,"Hardy 1921, Exercise Misc-III (18)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/19",4,"Hardy 1921, Exercise Misc-III (19)"],["ball-mathematical-recreations-1905/eq-73317d3d10",16,"Ball 1905, scan 262: (x-a)^2 + (y-b)^2 = 4(a^2 + b^2)"],["de-morgan-elementary-illustrations-calculus-1899/eq-a7dac87ce5",16,"De Morgan 1899, p. 22: \\dfrac{1}{x + h} = \\dfrac{1}{x} - \\dfrac{h}{x^{2}} + \\etc."],["de-morgan-elementary-illustrations-calculus-1899/eq-8d2205e664",16,"De Morgan 1899, p. 22: \\phi''(x + h) = -\\dfrac{1}{(x + h)^{2}} = -(x + h)^{-2}"],["todhunter-spherical-trigonometry-1886/eq-07bf8dd39e",16,"Todhunter 1886, scan 58: \\tan ABD \\sin DB = \\tan C \\sin \\theta"],["todhunter-spherical-trigonometry-1886/eq-5d2551fa89",16,"Todhunter 1886, scan 59: \\tan \\tfrac{1}{2} (a + b) = \\dfrac{\\cos \\tfrac{1}{2} (A - B)}{\\cos \\tfrac{1}{2} (A + B)} \\tan \\tfrac{1}{2} c"],["concept/ruled-surface",7,"ruled surface","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-ruled-surface"],["ball-mathematical-recreations-1905/eq-00fbf51215",16,"Ball 1905, scan 263: y^2 = \\frac{1}{4}x"],["ball-mathematical-recreations-1905/eq-597fd3eac3",16,"Ball 1905, scan 262: PR = x-a"],["concept/contour-line",7,"contour-line","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-contour-line"],["ball-mathematical-recreations-1905/eq-15165553aa",16,"Ball 1905, scan 263: AOE = \\frac{1}{3} AOB"],["ball-mathematical-recreations-1905/eq-79897189e9",16,"Ball 1905, scan 263: SOP = \\frac{1}{3}SOA"],["ball-mathematical-recreations-1905/eq-e1f367ca65",16,"Ball 1905, scan 263: x^2 + y^2 - \\frac{13}{4}x + 4ay = 0"],["ball-mathematical-recreations-1905/eq-8530060ca9",16,"Ball 1905, scan 263: 4y^3 = 3y - a"],["ball-mathematical-recreations-1905/eq-d4ca16ff6f",16,"Ball 1905, scan 264: AH = 2 \\dotm HL"],["ball-mathematical-recreations-1905/eq-c34f53f3c6",16,"Ball 1905, scan 264: AP : PM = AH : HL = 2 : 1"],["concept/arithmetical-progression",7,"arithmetical progression","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-arithmetical-progression"],["theorem/abel-s-summation-identity",9,"Abel's summation identity"],["hardy-course-of-pure-mathematics-1921/x-e381016ac0",15,"Hardy 1921, p. 51: We are therefore led to give the following definition: ..."],["hardy-course-of-pure-mathematics-1921/eq-0ea90775f0",16,"Hardy 1921, p. 340: s_{1} = \\phi_{0} - \\phi_{1}"],["ball-mathematical-recreations-1905/eq-ce01061e16",16,"Ball 1905, scan 264: AP = 2 \\dotm PM = PQ"],["ball-mathematical-recreations-1905/eq-31d746161e",16,"Ball 1905, scan 264: AP = PQ = QR"],["hardy-course-of-pure-mathematics-1921/eq-dd4ebb813b",16,"Hardy 1921, p. 342: a_{0}\\phi_{0} + a_{1}\\phi_{1} + \\dots + a_{n}\\phi_{n} = s_{0}(\\phi_{0} - \\phi_{1}) + s_{1}(\\phi_{1} - \\phi_{2}) + \\dots "],["hardy-course-of-pure-mathematics-1921/eq-82b8dce2d0",16,"Hardy 1921, p. 342: s_{n} = a_{0} + a_{1} + \\dots + a_{n}"],["hardy-course-of-pure-mathematics-1921/eq-35b8ac461a",16,"Hardy 1921, p. 342: |s_{\\nu}| < K"],["hardy-course-of-pure-mathematics-1921/x-57a19e9bf8",15,"Hardy 1921, p. 51: For it is known that in general such an ..."],["concept/right-angle",7,"right angle","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-right-angle"],["hardy-course-of-pure-mathematics-1921/x-0432dabffd",15,"Hardy 1921, p. 52: All functions of x which are not rational or ..."],["hardy-course-of-pure-mathematics-1921/x-0689cee364",15,"Hardy 1921, p. 54: It is easy to see that no periodic function ..."],["hardy-course-of-pure-mathematics-1921/x-f2750f2e8a",15,"Hardy 1921, p. 53: It oscillates up and down, the rapidity of the ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/20",4,"Hardy 1921, Exercise Misc-III (20)"],["concept/displacement",7,"displacement","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-displacement"],["method/multiplication-of-displacements-by-numbers",8,"multiplication of displacements by numbers","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-multiplication-of-displacements-by-numbers"],["de-morgan-elementary-illustrations-calculus-1899/x-29a944d5ea",15,"De Morgan 1899, p. 116: Generally, if A diminishes without limit at the same ..."],["concept/sign-of-the-derivative",7,"sign of the derivative","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-sign-of-the-derivative"],["de-morgan-elementary-illustrations-calculus-1899/x-23a2644d2e",15,"De Morgan 1899, p. 119: There is as yet no general agreement on this ..."],["method/multiplication-of-displacements",8,"multiplication of displacements","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-multiplication-of-displacements"],["method/multiplication-of-complex-numbers",8,"multiplication of complex numbers","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-multiplication-of-complex-numbers"],["hardy-course-of-pure-mathematics-1921/eq-b92057290c",16,"Hardy 1921, p. 343: s_{m, \\nu} = a_{m} + a_{m+1} + \\dots + a_{\\nu}"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/21",4,"Hardy 1921, Exercise Misc-III (21)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/22",4,"Hardy 1921, Exercise Misc-III (22)"],["dickson-theory-of-equations-1922/eq-52fe7e096e",16,"Dickson 1922, p. 53: k_1 k_2 k_3 = -\\frac{r}{8}"],["dickson-theory-of-equations-1922/eq-412d4651ba",16,"Dickson 1922, p. 53: k_1 + k_2 + k_3"],["form/e48798cdc7",5,"solve: Eq(31*x - 40, -9*x + 40)"],["shape/79c3713a5d",6,"solve: True"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/23",4,"Hardy 1921, Exercise Misc-III (23)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/24",4,"Hardy 1921, Exercise Misc-III (24)"],["theorem/necessary-condition-for-convergence",9,"necessary condition for convergence","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-necessary-condition-for-convergence"],["concept/imaginary-unit",7,"imaginary unit","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-imaginary-unit"],["theorem/comparison-test",9,"comparison test","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-comparison-test"],["theorem/limit-of-a-power",9,"limit of a power","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-of-a-power"],["method/indirect-method",8,"indirect method","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-indirect-method"],["boyden-first-book-in-algebra-1895/ex-31/3",4,"Boyden 1895, Exercise 31 (3)"],["form/09b62c9e82",5,"factor: -2*a**2 + 2*a"],["shape/7bde7e7637",6,"factor: N*a + N*a**N"],["form/e7b5bc117c",5,"solve: Eq(7*x, 63)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/25",4,"Hardy 1921, Exercise Misc-III (25)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/26",4,"Hardy 1921, Exercise Misc-III (26)"],["method/multiplication-by-i",8,"multiplication by i","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-multiplication-by-i"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/27",4,"Hardy 1921, Exercise Misc-III (27)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/28",4,"Hardy 1921, Exercise Misc-III (28)"],["hardy-course-of-pure-mathematics-1921/eq-523398b605",16,"Hardy 1921, p. 343: |a_{m}\\phi_{m} + a_{m+1}\\phi_{m+1} + \\dots + a_{n}\\phi_{n}| < \\DELTA \\phi_{m} \\leq \\DELTA \\phi_{1}"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/29",4,"Hardy 1921, Exercise Misc-III (29)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/30",4,"Hardy 1921, Exercise Misc-III (30)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/31",4,"Hardy 1921, Exercise Misc-III (31)"],["todhunter-spherical-trigonometry-1886/eq-a0542d9004",16,"Todhunter 1886, scan 59: \\tan \\tfrac{1}{2} (a - b) = \\dfrac{\\sin \\tfrac{1}{2} (A - B)}{\\sin \\tfrac{1}{2} (A + B)} \\tan \\tfrac{1}{2} c"],["todhunter-spherical-trigonometry-1886/eq-005a749ed8",16,"Todhunter 1886, scan 59: \\sin C = \\dfrac{\\sin A \\sin c}{\\sin a}"],["form/94f0ea3303",5,"solve: Eq(8*x - 64, 6*x - 36)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iii/32",4,"Hardy 1921, Exercise Misc-III (32)"],["theorem/zero-product-property",9,"zero-product property","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-zero-product-property"],["concept/conjugate-complex-numbers",7,"conjugate complex numbers","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-conjugate-complex-numbers"],["hardy-course-of-pure-mathematics-1921/x-f565b2eda4",15,"Hardy 1921, p. 132: That is to say, while there are in general ..."],["dickson-theory-of-equations-1922/eq-6b7c2caf8a",16,"Dickson 1922, p. 48: R = -\\Delta/108"],["dickson-theory-of-equations-1922/eq-8b024973a0",16,"Dickson 1922, p. 66: D = f(a+h) - f(a)"],["planck-treatise-on-thermodynamics-1903/x-80526b2ea4",15,"Planck 1903, p. 232: The influence of the temperature on K, and therewith ..."],["dickson-theory-of-equations-1922/eq-25e92bf677",16,"Dickson 1922, p. 66: F = a_1 h + a_2 h^2 + \\dotsb + a_n h^n"],["theorem/factor-theorem",9,"factor theorem","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-factor-theorem"],["concept/quadratic-equation",7,"quadratic equation","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-quadratic-equation"],["hardy-course-of-pure-mathematics-1921/eq-a6443df9f5",16,"Hardy 1921, p. 343: z = \\Cis\\theta"],["hardy-course-of-pure-mathematics-1921/x-7c99adc882",15,"Hardy 1921, p. 132: We divide the real numbers \\xi into two classes ..."],["de-morgan-elementary-illustrations-calculus-1899/ch-an-illustration-from-dynamics-velocity-acceleration-etc",2,"De Morgan 1899, An Illustration from Dynamics: Velocity, Acceleration, etc","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-an-illustration-from-dynamics-velocity-acceleration-etc/index.html"],["de-morgan-elementary-illustrations-calculus-1899/eq-c696d116d9",16,"De Morgan 1899, p. 56: a = 16\\frac{1}{12}"],["hardy-course-of-pure-mathematics-1921/eq-1c0930c267",16,"Hardy 1921, p. 343: |s_{n} + it_{n}| = \\left|\\frac{1 - z^{n}}{1 - z}\\right| \\leq \\frac{1 + |z^{n}|}{|1 - z|} \\leq \\frac{2}{|1 - z|}"],["concept/mathematical-tripos",7,"Mathematical Tripos","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-mathematical-tripos"],["law/commutative-law",10,"commutative law","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-law-commutative-law"],["boyden-first-book-in-algebra-1895/ex-36/1",4,"Boyden 1895, Exercise 36 (1)"],["hardy-course-of-pure-mathematics-1921/eq-25f7810ef4",16,"Hardy 1921, p. 344: \\tsum u_{n} = \\tsum (v_{n} + iw_{n})"],["dickson-theory-of-equations-1922/eq-166d10edce",16,"Dickson 1922, p. 67: k < \\frac{p}{p + g}"],["dickson-theory-of-equations-1922/eq-2f7718e15f",16,"Dickson 1922, p. 68: f(x) = x^n (a_0 + \\phi)"],["hardy-course-of-pure-mathematics-1921/eq-583d0c481d",16,"Hardy 1921, p. 344: u_{n} = v_{n} + iw_{n}"],["hardy-course-of-pure-mathematics-1921/eq-a56ee4ff4f",16,"Hardy 1921, p. 344: |u_{n}| = \\sqrtp{v_{n}^{2} + w_{n}^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-a2093ee5ce",16,"Hardy 1921, p. 344: |u_{n}| = \\sqrtp{v_{n}^{2} + w_{n}^{2}} \\leq |v_{n}| + |w_{n}|"],["hardy-course-of-pure-mathematics-1921/eq-d6d17a64c8",16,"Hardy 1921, p. 345: |v_{n}| \\leq \\sqrtp{v_{n}^{2} + w_{n}^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-c06eef33f4",16,"Hardy 1921, p. 345: |u_{n+1}|/|u_{n}| = |z|/(n + 1) \\to 0"],["hardy-course-of-pure-mathematics-1921/eq-3f2fc7bc64",16,"Hardy 1921, p. 345: |u_{n+1}|/|u_{n}| = (n + 1)|z|"],["hardy-course-of-pure-mathematics-1921/ex-misc-iv",3,"Hardy 1921, Exercise Misc-IV"],["hardy-course-of-pure-mathematics-1921/ex-misc-iv/1",4,"Hardy 1921, Exercise Misc-IV (1)"],["hardy-course-of-pure-mathematics-1921/x-695f599de5",15,"Hardy 1921, p. 143: It is perhaps hardly necessary to point out that ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-iv/2",4,"Hardy 1921, Exercise Misc-IV (2)"],["shape/e1ca3cac71",6,"solve: Eq(N*x, N*x + N)"],["hardy-course-of-pure-mathematics-1921/ex-misc-iv/3",4,"Hardy 1921, Exercise Misc-IV (3)"],["hardy-course-of-pure-mathematics-1921/eq-ae58b7fbd0",16,"Hardy 1921, p. 341: \\lim \\left[\\frac{1}{2n + 1} - \\frac{1}{2n + 2} + \\frac{1}{2n + 3} - \\dots + \\frac{1}{4n - 1} - \\frac{1}{4n}\\right] = 0"],["form/ca595a00c4",5,"solve: Eq(30*x, 1200)"],["form/9ff69bf00d",5,"solve: Eq(x/10, x/12 + 250)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/1",4,"Hardy 1921, Exercise Misc-IX (1)"],["hardy-course-of-pure-mathematics-1921/eq-8d95712e4a",16,"Hardy 1921, p. 443: \\tan(y + \\pi) &= &&\\tan y"],["hardy-course-of-pure-mathematics-1921/x-758de01772",15,"Hardy 1921, p. 221: There is no derivative for x = 0, and ..."],["hardy-course-of-pure-mathematics-1921/x-b6904b26af",15,"Hardy 1921, p. 221: The reader, if he considers what the question means ..."],["form/e10426d5e3",5,"solve: (Eq(a, 3*x), Eq(10*a + 25*x, 330))"],["hardy-course-of-pure-mathematics-1921/eq-d50680d562",16,"Hardy 1921, p. 443: \\cos(y + \\pi) &= -&&\\cos y"],["planck-treatise-on-thermodynamics-1903/eq-705171867c",16,"Planck 1903, p. 113: dF \\leq W - \\Phi\\, d\\theta"],["planck-treatise-on-thermodynamics-1903/eq-2d45e60f0c",16,"Planck 1903, p. 113: U = Mu = M(c_{v} \\theta + \\const)"],["hardy-course-of-pure-mathematics-1921/eq-6979a154b1",16,"Hardy 1921, p. 443: \\sin(y + \\pi) &= -&&\\sin y"],["hardy-course-of-pure-mathematics-1921/eq-f9d9ba64c3",16,"Hardy 1921, p. 444: \\tan (y_{1} + y_{2}) = \\dfrac{\\tan y_{1} + \\tan y_{2}}{1 - \\tan y_{1}\\tan y_{2}}"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/2",4,"Hardy 1921, Exercise Misc-IX (2)"],["hardy-course-of-pure-mathematics-1921/x-f750079d88",15,"Hardy 1921, p. 215: It will then follow by the principle of mathematical ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/3",4,"Hardy 1921, Exercise Misc-IX (3)"],["form/cfe9eb5358",5,"identity: x**2"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/4",4,"Hardy 1921, Exercise Misc-IX (4)"],["shape/f3a90dd3b2",6,"identity: x**N"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/5",4,"Hardy 1921, Exercise Misc-IX (5)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/6",4,"Hardy 1921, Exercise Misc-IX (6)"],["cap/core.prog",17,"core.prog"],["de-morgan-elementary-illustrations-calculus-1899/ch-simple-harmonic-motion",2,"De Morgan 1899, Simple Harmonic Motion","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-simple-harmonic-motion/index.html"],["form/c030a2b351",5,"identity: -21*x**3"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/7",4,"Hardy 1921, Exercise Misc-IX (7)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/8",4,"Hardy 1921, Exercise Misc-IX (8)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/9",4,"Hardy 1921, Exercise Misc-IX (9)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/10",4,"Hardy 1921, Exercise Misc-IX (10)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/11",4,"Hardy 1921, Exercise Misc-IX (11)"],["boyden-first-book-in-algebra-1895/ex-10/1",4,"Boyden 1895, Exercise 10 (1)"],["todhunter-spherical-trigonometry-1886/eq-0d2af6669e",16,"Todhunter 1886, scan 59: \\cos C = -\\cos A \\cos B + \\sin A \\sin B \\cos c"],["form/5f49278990",5,"evaluate: a - b at a=-4, b=-5"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/12",4,"Hardy 1921, Exercise Misc-IX (12)"],["hardy-course-of-pure-mathematics-1921/eq-cbbc63a009",16,"Hardy 1921, p. 444: \\cos(y_{1} + y_{2}) = ±(\\cos y_{1}\\cos y_{2} - \\sin y_{1}\\sin y_{2})"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/13",4,"Hardy 1921, Exercise Misc-IX (13)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/14",4,"Hardy 1921, Exercise Misc-IX (14)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/15",4,"Hardy 1921, Exercise Misc-IX (15)"],["todhunter-spherical-trigonometry-1886/eq-6042a75488",16,"Todhunter 1886, scan 59: \\cos C = \\cos B (-\\cos A + \\cot \\phi \\sin A) = \\dfrac{\\cos B \\sin (A-\\phi)}{\\sin \\phi}"],["todhunter-spherical-trigonometry-1886/eq-9c59fb0e7c",16,"Todhunter 1886, scan 59: \\cot \\phi = \\tan B\\, \\cos c"],["form/e069ea1315",5,"identity: -12*a**5*b**5*c**5*d**5"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/16",4,"Hardy 1921, Exercise Misc-IX (16)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/17",4,"Hardy 1921, Exercise Misc-IX (17)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/18",4,"Hardy 1921, Exercise Misc-IX (18)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/19",4,"Hardy 1921, Exercise Misc-IX (19)"],["concept/k-nigsberg-bridge-problem",7,"Königsberg bridge problem","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-k-nigsberg-bridge-problem"],["concept/node",7,"node","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-node"],["form/9728232d5b",5,"evaluate: 4*a**2 - 2*b**2 - c**2 at a=-2, b=3, c=-1"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/20",4,"Hardy 1921, Exercise Misc-IX (20)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/21",4,"Hardy 1921, Exercise Misc-IX (21)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/22",4,"Hardy 1921, Exercise Misc-IX (22)"],["hardy-course-of-pure-mathematics-1921/eq-2f20c83664",16,"Hardy 1921, p. 444: \\cos x = 1 - \\frac{x^{2}}{2!} + \\frac{x^{4}}{4!} - \\dots"],["todhunter-spherical-trigonometry-1886/eq-fd242a7122",16,"Todhunter 1886, scan 59: \\cos c = \\cot B \\cot DAB"],["todhunter-spherical-trigonometry-1886/eq-fda1c375c3",16,"Todhunter 1886, scan 59: \\cos AD \\sin CAD = \\cos C"],["todhunter-spherical-trigonometry-1886/eq-7d137a01ab",16,"Todhunter 1886, scan 59: \\cos AD \\sin BAD = \\cos B"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/23",4,"Hardy 1921, Exercise Misc-IX (23)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/24",4,"Hardy 1921, Exercise Misc-IX (24)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/25",4,"Hardy 1921, Exercise Misc-IX (25)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/26",4,"Hardy 1921, Exercise Misc-IX (26)"],["todhunter-spherical-trigonometry-1886/eq-fe7db1381d",16,"Todhunter 1886, scan 59: \\dfrac{\\cos C}{\\sin CAD} = \\dfrac{\\cos B}{\\sin BAD}"],["todhunter-spherical-trigonometry-1886/eq-7f5190e61a",16,"Todhunter 1886, scan 60: \\tan b \\cos CAD = \\tan c \\cos \\phi"],["form/c193ba1ab7",5,"identity: 3*a**3*x**3/5"],["form/7cc6c77f2e",5,"identity: -3*x/2"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/27",4,"Hardy 1921, Exercise Misc-IX (27)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/28",4,"Hardy 1921, Exercise Misc-IX (28)"],["concept/edge",7,"edge","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-edge"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/29",4,"Hardy 1921, Exercise Misc-IX (29)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/30",4,"Hardy 1921, Exercise Misc-IX (30)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/31",4,"Hardy 1921, Exercise Misc-IX (31)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/32",4,"Hardy 1921, Exercise Misc-IX (32)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/33",4,"Hardy 1921, Exercise Misc-IX (33)"],["law/associative-law",10,"associative law","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-law-associative-law"],["concept/closed-network",7,"closed network","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-closed-network"],["theorem/unicursal-route-for-a-figure-with-no-odd-node",9,"unicursal route for a figure with no odd node","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-unicursal-route-for-a-figure-with-no-odd-node"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/34",4,"Hardy 1921, Exercise Misc-IX (34)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/35",4,"Hardy 1921, Exercise Misc-IX (35)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/36",4,"Hardy 1921, Exercise Misc-IX (36)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/37",4,"Hardy 1921, Exercise Misc-IX (37)"],["boyden-first-book-in-algebra-1895/ex-10/2",4,"Boyden 1895, Exercise 10 (2)"],["form/0aac3ed84f",5,"identity: 9*a**2 - 2*a + 6"],["shape/b60a19f620",6,"identity: N*a + N*a**N + N"],["todhunter-spherical-trigonometry-1886/eq-138053619c",16,"Todhunter 1886, scan 60: \\sin B = \\frac{\\sin b}{\\sin a} \\sin A"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/38",4,"Hardy 1921, Exercise Misc-IX (38)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/39",4,"Hardy 1921, Exercise Misc-IX (39)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/40",4,"Hardy 1921, Exercise Misc-IX (40)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/41a",4,"Hardy 1921, Exercise Misc-IX (41a)"],["form/432ce6da42",5,"identity: 8*a*b + 3*a*c"],["de-morgan-elementary-illustrations-calculus-1899/eq-3f89993684",16,"De Morgan 1899, p. 31: x^{2} + y^{2} = r^{2}"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/42",4,"Hardy 1921, Exercise Misc-IX (42)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/43",4,"Hardy 1921, Exercise Misc-IX (43)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/44",4,"Hardy 1921, Exercise Misc-IX (44)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/45",4,"Hardy 1921, Exercise Misc-IX (45)"],["concept/real-number",7,"real number","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-real-number"],["form/4730837005",5,"identity: 3*a**3 + 7*a**2 + 2"],["shape/cf36898a56",6,"identity: 2*N*a**N + N"],["hardy-course-of-pure-mathematics-1921/x-39d2ad48e1",15,"Hardy 1921, p. 70: If cricket were a mathematical science, it would be ..."],["boyden-first-book-in-algebra-1895/ex-10/3",4,"Boyden 1895, Exercise 10 (3)"],["boyden-first-book-in-algebra-1895/ex-10/4",4,"Boyden 1895, Exercise 10 (4)"],["boyden-first-book-in-algebra-1895/ex-10/5",4,"Boyden 1895, Exercise 10 (5)"],["wentworth-first-steps-in-algebra-1894/ex-18/20",4,"Wentworth 1894, Exercise 18 (20)"],["theorem/unicursal-route-for-a-figure-with-two-odd-nodes",9,"unicursal route for a figure with two odd nodes","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-unicursal-route-for-a-figure-with-two-odd-nodes"],["theorem/more-than-two-odd-nodes-forbid-a-single-route",9,"more than two odd nodes forbid a single route","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-more-than-two-odd-nodes-forbid-a-single-route"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/46",4,"Hardy 1921, Exercise Misc-IX (46)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/47",4,"Hardy 1921, Exercise Misc-IX (47)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/48",4,"Hardy 1921, Exercise Misc-IX (48)"],["hardy-course-of-pure-mathematics-1921/x-788c198ca3",15,"Hardy 1921, p. 69: To specify a displacement completely three things are needed, ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/49",4,"Hardy 1921, Exercise Misc-IX (49)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/50",4,"Hardy 1921, Exercise Misc-IX (50)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/51",4,"Hardy 1921, Exercise Misc-IX (51)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/52",4,"Hardy 1921, Exercise Misc-IX (52)"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/53",4,"Hardy 1921, Exercise Misc-IX (53)"],["hardy-course-of-pure-mathematics-1921/x-76c3235201",15,"Hardy 1921, p. 75: In the first place our definition would be futile. ..."],["theorem/2n-odd-nodes-need-n-routes",9,"2n odd nodes need n routes","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-2n-odd-nodes-need-n-routes"],["hardy-course-of-pure-mathematics-1921/ex-misc-ix/54",4,"Hardy 1921, Exercise Misc-IX (54)"],["hardy-course-of-pure-mathematics-1921/x-8d4ba67256",15,"Hardy 1921, p. 78: For the present the reader must regard x + ..."],["hardy-course-of-pure-mathematics-1921/x-d9601b2804",15,"Hardy 1921, p. 83: We can only attach a meaning to 3 - ..."],["method/maze-traversal-rule",8,"maze traversal rule","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-maze-traversal-rule"],["method/re-coupling-of-hooks",8,"re-coupling of hooks","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-re-coupling-of-hooks"],["method/wall-following",8,"wall-following","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-wall-following"],["ball-mathematical-recreations-1905/x-a64c2d32a7",15,"Ball 1905, scan 187: Since a node of the nth order is one ..."],["form/47f91d94ea",5,"identity: 4*a**3 - 3*a**2*b - 4*a*b**2 + 7*b**3"],["shape/9ed18fb581",6,"identity: N*a*b**N + N*a**N*b + N*a**N + N*b**N"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/1",4,"Hardy 1921, Exercise Misc-V (1)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/2",4,"Hardy 1921, Exercise Misc-V (2)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/3",4,"Hardy 1921, Exercise Misc-V (3)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/4",4,"Hardy 1921, Exercise Misc-V (4)"],["hardy-course-of-pure-mathematics-1921/eq-d9f7f836c4",16,"Hardy 1921, p. 341: \\lim \\left(\\frac{1}{2n + 2} + \\frac{1}{2n + 4} + \\dots + \\frac{1}{4n}\\right) = \\tfrac{1}{2} \\lim \\frac{1}{n} \\sum_{r=1}^"],["form/3f6b973876",5,"identity: 2*a + b"],["person/edgar-allan-poe",1,"Edgar Allan Poe","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-edgar-allan-poe"],["shape/782ea820cb",6,"identity: N*a + b"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/5",4,"Hardy 1921, Exercise Misc-V (5)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/6",4,"Hardy 1921, Exercise Misc-V (6)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/7",4,"Hardy 1921, Exercise Misc-V (7)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/8",4,"Hardy 1921, Exercise Misc-V (8)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/9",4,"Hardy 1921, Exercise Misc-V (9)"],["boyden-first-book-in-algebra-1895/ex-10/6",4,"Boyden 1895, Exercise 10 (6)"],["hardy-course-of-pure-mathematics-1921/eq-2ab1b94b1a",16,"Hardy 1921, p. 341: \\lim t_{3n} = s + \\tfrac{1}{2} \\int_{1}^{2} \\frac{dx}{x}"],["theorem/equation-of-the-circle",9,"equation of the circle"],["form/18d5bbd933",5,"identity: 2*a + 3*b + c"],["shape/f6e2d07868",6,"identity: N*a + N*b + c"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/10",4,"Hardy 1921, Exercise Misc-V (10)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/11",4,"Hardy 1921, Exercise Misc-V (11)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/12",4,"Hardy 1921, Exercise Misc-V (12)"],["concept/imaginary-straight-line",7,"imaginary straight line","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-imaginary-straight-line"],["concept/fixed-point-of-a-transformation",7,"fixed point of a transformation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-fixed-point-of-a-transformation"],["wentworth-first-steps-in-algebra-1894/ex-20/16",4,"Wentworth 1894, Exercise 20 (16)"],["ball-mathematical-recreations-1905/x-75911dbe1b",15,"Ball 1905, scan 187: The number of hooks at each node is even, ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-v/13",4,"Hardy 1921, Exercise Misc-V (13)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/14",4,"Hardy 1921, Exercise Misc-V (14)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/15",4,"Hardy 1921, Exercise Misc-V (15)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/16",4,"Hardy 1921, Exercise Misc-V (16)"],["hardy-course-of-pure-mathematics-1921/x-81cadd824e",15,"Hardy 1921, p. 98: Thus we define \\sqrt[n]{a} or a^{1/n}, where n is ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-v/17",4,"Hardy 1921, Exercise Misc-V (17)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/18",4,"Hardy 1921, Exercise Misc-V (18)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/19",4,"Hardy 1921, Exercise Misc-V (19)"],["boyden-first-book-in-algebra-1895/ex-10/7",4,"Boyden 1895, Exercise 10 (7)"],["de-morgan-elementary-illustrations-calculus-1899/eq-f425d5cb6f",16,"De Morgan 1899, p. 32: ay + bx = ab"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/20",4,"Hardy 1921, Exercise Misc-V (20)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/21",4,"Hardy 1921, Exercise Misc-V (21)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/22",4,"Hardy 1921, Exercise Misc-V (22)"],["hardy-course-of-pure-mathematics-1921/ex-xlvii",3,"Hardy 1921, Exercise XLVII"],["hardy-course-of-pure-mathematics-1921/ex-xlviii",3,"Hardy 1921, Exercise XLVIII"],["de-morgan-elementary-illustrations-calculus-1899/eq-2e1ac207a9",16,"De Morgan 1899, p. 32: ay - bx = ab"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/23",4,"Hardy 1921, Exercise Misc-V (23)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/24",4,"Hardy 1921, Exercise Misc-V (24)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/25",4,"Hardy 1921, Exercise Misc-V (25)"],["boyden-first-book-in-algebra-1895/ex-10/8",4,"Boyden 1895, Exercise 10 (8)"],["ball-mathematical-recreations-1905/x-eed573a30e",15,"Ball 1905, scan 192: Of course to walk twice over every path in ..."],["form/20c3b5aa95",5,"identity: -3*a*c**2*(-a*b**2 + 3*a*b*c**2 - a*c**4)"],["hardy-course-of-pure-mathematics-1921/ex-misc-v/26",4,"Hardy 1921, Exercise Misc-V (26)"],["hardy-course-of-pure-mathematics-1921/x-f0bdc3bfa6",15,"Hardy 1921, p. 99: The only possible value of r is \\sqrt[n]{\\rho}, the ..."],["theorem/argument-principle",9,"argument principle","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-argument-principle"],["thompson-calculus-made-easy-1914/x-d4ef91cf5f",15,"Thompson 1914, p. 3: Nowadays we call these small quantities of the second ..."],["hardy-course-of-pure-mathematics-1921/x-86af3bc017",15,"Hardy 1921, p. 99: These numbers are called the nth roots of unity; ..."],["ball-mathematical-recreations-1905/x-cca5528aa6",15,"Ball 1905, scan 190: said to have been originally traced in the sand ..."],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/1",4,"Hardy 1921, Exercise Misc-VI (1)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/2",4,"Hardy 1921, Exercise Misc-VI (2)"],["thompson-calculus-made-easy-1914/ex-i",3,"Thompson 1914, Exercise I"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/3",4,"Hardy 1921, Exercise Misc-VI (3)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/4",4,"Hardy 1921, Exercise Misc-VI (4)"],["hardy-course-of-pure-mathematics-1921/x-6a76ef8834",15,"Hardy 1921, p. 101: Raising each of these expressions to the power p ..."],["thompson-calculus-made-easy-1914/ch-iv",2,"Thompson 1914, ch. 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a**2*b**2 + a*b)/(a*b)"],["shape/6b41f3a8b0",6,"identity: -1"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/39",4,"Hardy 1921, Exercise Misc-VI (39)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/40",4,"Hardy 1921, Exercise Misc-VI (40)"],["boyden-first-book-in-algebra-1895/ex-10/12",4,"Boyden 1895, Exercise 10 (12)"],["boyden-first-book-in-algebra-1895/ex-4",3,"Boyden 1895, Exercise 4"],["boyden-first-book-in-algebra-1895/ex-1",3,"Boyden 1895, Exercise 1"],["person/bernhard-riemann",1,"Bernhard Riemann","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-bernhard-riemann"],["person/nikolai-lobachevsky",1,"Nikolai Lobachevsky","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-nikolai-lobachevsky"],["person/j-nos-bolyai",1,"János Bolyai","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-j-nos-bolyai"],["boyden-first-book-in-algebra-1895/ex-5",3,"Boyden 1895, Exercise 5"],["form/3b838845e9",5,"identity: (3*a**3*b**3 + 3*a**2*b - 3*a*b**2)/(3*a*b)"],["shape/4bb5a14b20",6,"identity: N*(N*a*b**N + N*a**N*b + N*a**N*b**N)/(a*b)"],["boyden-first-book-in-algebra-1895",0,"Boyden, A First Book in Algebra (1895)","../books/boyden-first-book-in-algebra-1895/index.html"],["boyden-first-book-in-algebra-1895/ch-preface",2,"Boyden 1895, PREFACE","../books/boyden-first-book-in-algebra-1895/ch/ch-preface/index.html"],["boyden-first-book-in-algebra-1895/ex-6",3,"Boyden 1895, Exercise 6"],["boyden-first-book-in-algebra-1895/ex-2",3,"Boyden 1895, Exercise 2"],["person/wallace-clarke-boyden",1,"Wallace Clarke Boyden"],["boyden-first-book-in-algebra-1895/ex-3",3,"Boyden 1895, Exercise 3"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/41",4,"Hardy 1921, Exercise Misc-VI (41)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/42",4,"Hardy 1921, Exercise Misc-VI (42)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/43",4,"Hardy 1921, Exercise Misc-VI (43)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/44",4,"Hardy 1921, Exercise Misc-VI (44)"],["wentworth-first-steps-in-algebra-1894/ex-24/1",4,"Wentworth 1894, Exercise 24 (1)"],["form/5a1ab2280e",5,"identity: (x**2 + 15*x + 56)/(x + 7)"],["shape/ad43095326",6,"identity: (N*x + N + x**N)/(N + x)"],["boyden-first-book-in-algebra-1895/ex-7",3,"Boyden 1895, Exercise 7"],["boyden-first-book-in-algebra-1895/ex-8",3,"Boyden 1895, Exercise 8"],["boyden-first-book-in-algebra-1895/ex-9",3,"Boyden 1895, Exercise 9"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/45",4,"Hardy 1921, Exercise Misc-VI (45)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/46",4,"Hardy 1921, Exercise Misc-VI (46)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/47",4,"Hardy 1921, Exercise Misc-VI (47)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/48",4,"Hardy 1921, Exercise Misc-VI (48)"],["theorem/triangle-inequality-for-complex-numbers",9,"triangle inequality for complex numbers","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-triangle-inequality-for-complex-numbers"],["boyden-first-book-in-algebra-1895/ex-10/13",4,"Boyden 1895, Exercise 10 (13)"],["dickson-theory-of-equations-1922/eq-db7be62b8a",16,"Dickson 1922, p. 68: \\phi = \\frac{a_1}{x} + \\frac{a_2}{x^2} + \\dotsb + \\frac{a_n}{x^n}"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/49",4,"Hardy 1921, Exercise Misc-VI (49)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi/50",4,"Hardy 1921, Exercise Misc-VI (50)"],["thompson-calculus-made-easy-1914/x-dc72085496",15,"Thompson 1914, p. 18: Now remember that the fundamental notion about the calculus ..."],["form/a85d54133c",5,"identity: (-a**3*x**3 + 1)/(-a*x + 1)"],["shape/76b12b59e6",6,"identity: (-a**N*x**N + 1)/(-a*x + 1)"],["hardy-course-of-pure-mathematics-1921/ex-misc-vii/none",4,"Hardy 1921, Exercise Misc-VII (None)"],["de-morgan-elementary-illustrations-calculus-1899/eq-54038ee0b9",16,"De Morgan 1899, p. 27: \\dfrac{dy}{dx} = 2x + dx"],["wentworth-first-steps-in-algebra-1894/ex-24/16",4,"Wentworth 1894, Exercise 24 (16)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/1",4,"Hardy 1921, Exercise Misc-X (1)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/2",4,"Hardy 1921, Exercise Misc-X (2)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/3",4,"Hardy 1921, Exercise Misc-X (3)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/4",4,"Hardy 1921, Exercise Misc-X (4)"],["form/553da4c04f",5,"identity: (x**4 + 6*x**3 + 13*x**2 + 12*x + 4)/(x**2 + 3*x + 2)"],["de-morgan-elementary-illustrations-calculus-1899/eq-4df3fc90aa",16,"De Morgan 1899, p. 27: \\dfrac{dy}{dx} = 2x"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/5",4,"Hardy 1921, Exercise Misc-X (5)"],["concept/translation",7,"translation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-translation"],["hardy-course-of-pure-mathematics-1921/x-ccc31e2d74",15,"Hardy 1921, p. 143: The commonest example of an infinite geometric series is ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-c0e0693e66",16,"De Morgan 1899, p. 28: dy = 2x\\, dx + (dx)^{2}"],["hardy-course-of-pure-mathematics-1921/x-f13b200ee5",15,"Hardy 1921, p. 144: Thus every proper fraction can be expressed as a ..."],["concept/transformation",7,"transformation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-transformation"],["wentworth-first-steps-in-algebra-1894/ex-24/26",4,"Wentworth 1894, Exercise 24 (26)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/6",4,"Hardy 1921, Exercise Misc-X (6)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/7",4,"Hardy 1921, Exercise Misc-X (7)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/8",4,"Hardy 1921, Exercise Misc-X (8)"],["form/fcb53cca98",5,"identity: (a**2 - b**2 + c**2 + x**2)*(a**2 + b**2 - c**2 + x**2)"],["ball-mathematical-recreations-1905/eq-f308c3e8d0",16,"Ball 1905, scan 269: 6336/2017\\frac14 <\\pi<14688/4673\\frac12"],["ball-mathematical-recreations-1905/eq-0526c3a1a6",16,"Ball 1905, scan 269: \\sin\\theta < \\theta < \\tan\\theta"],["method/reduction-formula",8,"reduction formula","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-reduction-formula"],["shape/298d09d9bb",6,"identity: (a**N - b**N + c**N + x**N)*(a**N + b**N - c**N + x**N)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/9",4,"Hardy 1921, Exercise Misc-X (9)"],["ball-mathematical-recreations-1905/eq-1dcafac6c4",16,"Ball 1905, scan 269: \\theta= \\pi/96"],["concept/magnification",7,"magnification","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-magnification"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/10",4,"Hardy 1921, Exercise Misc-X (10)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/11",4,"Hardy 1921, Exercise Misc-X (11)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/12",4,"Hardy 1921, Exercise Misc-X (12)"],["cap/core.limit",17,"core.limit"],["form/fc47885ddd",5,"identity: (a + b + x)*(a**2 - a*b - a*x + b**2 - b*x + x**2)"],["todhunter-spherical-trigonometry-1886/eq-cd8bdfda20",16,"Todhunter 1886, scan 148: DQ = \\tfrac{1}{2} (BQ + CQ) = \\tfrac{1}{2} \\pi"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/13",4,"Hardy 1921, Exercise Misc-X (13)"],["ball-mathematical-recreations-1905/eq-d5e6f70c02",16,"Ball 1905, scan 269: \\pi = 3^{\\circ} 8' 30''"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/14",4,"Hardy 1921, Exercise Misc-X (14)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/15",4,"Hardy 1921, Exercise Misc-X (15)"],["todhunter-spherical-trigonometry-1886/eq-3350ce60f9",16,"Todhunter 1886, scan 60: \\tan \\tfrac{1}{2} c = \\dfrac{\\cos \\tfrac{1}{2} (A + B)}{\\cos \\tfrac{1}{2} (A - B)} \\tan \\tfrac{1}{2} (a + b)"],["form/454caadf27",5,"identity: 3*x**2 + 24*x - 12"],["shape/d73ab1e07f",6,"identity: N*x + N*x**N + N"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/16",4,"Hardy 1921, Exercise Misc-X (16)"],["concept/rotation",7,"rotation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-rotation"],["ball-mathematical-recreations-1905/eq-86cf3cf683",16,"Ball 1905, scan 269: \\pi = 3 + \\frac8{60} + \\frac{30}{3600} =\\allowbreak 3\\frac{17}{120} =\\allowbreak 3.141\\dot6"],["todhunter-spherical-trigonometry-1886/eq-b5bdefe289",16,"Todhunter 1886, scan 60: \\cot a\\, \\sin b = \\cos b\\, \\cos C + \\sin C\\, \\cot A"],["todhunter-spherical-trigonometry-1886/eq-534695d56e",16,"Todhunter 1886, scan 60: \\cos (C - \\phi) = \\cos \\phi \\cot a \\tan b"],["todhunter-spherical-trigonometry-1886/eq-a37e877cc1",16,"Todhunter 1886, scan 61: \\cos a = \\cos b \\cos c + \\sin b \\sin c \\cos A"],["todhunter-spherical-trigonometry-1886/eq-faf7877fe3",16,"Todhunter 1886, scan 61: \\tan\\theta = \\tan b \\cos A"],["ball-mathematical-recreations-1905/eq-0b59413ddd",16,"Ball 1905, scan 270: b^2=\\frac{1}{2}-\\frac{1}{2}(1-a^2)^{\\frac{1}{2}}"],["form/9ac7007bef",5,"identity: a*x**3 - b*x**2 + b*x - c*x**2 - c*x + x**3"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/17",4,"Hardy 1921, Exercise Misc-X (17)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/18",4,"Hardy 1921, Exercise Misc-X (18)"],["ball-mathematical-recreations-1905/eq-bc44af1f24",16,"Ball 1905, scan 272: 2 \\sin^2\\frac{1}{2}\\theta = 1-\\cos \\theta"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/19",4,"Hardy 1921, Exercise Misc-X (19)"],["ball-mathematical-recreations-1905/eq-f9900663f0",16,"Ball 1905, scan 272: \\frac{2}{\\pi} = \\frac{\\surd 2}{2} \\frac{\\surd (2+\\surd 2)}{2} \\frac{\\surd \\{2+\\surd (2+\\surd 2)\\}}{2} \\dotsm\\;"],["theorem/vieta-s-product",9,"Vieta's product"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/20",4,"Hardy 1921, Exercise Misc-X (20)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/21",4,"Hardy 1921, Exercise Misc-X (21)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/22",4,"Hardy 1921, Exercise Misc-X (22)"],["planck-treatise-on-thermodynamics-1903/eq-bc97b30821",16,"Planck 1903, p. 232: \\log K = \\log a - \\frac{b}{\\theta} + (\\nu_{1} + \\nu_{2} + \\dots) \\log \\frac{\\theta}{p}."],["planck-treatise-on-thermodynamics-1903/eq-8b39b434b7",16,"Planck 1903, p. 235: 2\\, \\frac{\\dd \\log c_{1}}{\\dd \\theta} = \\frac{1}{R} · \\frac{L}{\\theta^{2}}."],["form/8257b980f3",5,"identity: (-2*a + b)**2"],["shape/150b93ae49",6,"identity: (N*a + b)**N"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/23",4,"Hardy 1921, Exercise Misc-X (23)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/24",4,"Hardy 1921, Exercise Misc-X (24)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/25",4,"Hardy 1921, Exercise Misc-X (25)"],["concept/coaxal-circles",7,"coaxal circles","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-coaxal-circles"],["ball-mathematical-recreations-1905/eq-5da534babd",16,"Ball 1905, scan 272: 1-\\cos A = 2 \\sin^2\\frac{1}{2}A"],["concept/dependent-variable",7,"dependent variable","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-dependent-variable"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/26",4,"Hardy 1921, Exercise Misc-X (26)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/27",4,"Hardy 1921, Exercise Misc-X (27)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/28",4,"Hardy 1921, Exercise Misc-X (28)"],["todhunter-spherical-trigonometry-1886/eq-701895c36e",16,"Todhunter 1886, scan 61: \\cos(c - \\theta) = \\dfrac{\\cos a \\cos\\theta}{\\cos b}"],["form/f7bcf9609b",5,"identity: (x - 3)*(x + 7)"],["todhunter-spherical-trigonometry-1886/eq-f07fb02d95",16,"Todhunter 1886, scan 61: \\cos b = \\cot A \\cot ACD"],["form/0188a06610",5,"identity: (x - 4)*(x - 2)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/29",4,"Hardy 1921, Exercise Misc-X (29)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/30",4,"Hardy 1921, Exercise Misc-X (30)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/31i",4,"Hardy 1921, Exercise Misc-X (31i)"],["ball-mathematical-recreations-1905/eq-c76ccfe2f2",16,"Ball 1905, scan 273: 3 \\sin\\theta /(2 + \\cos\\theta) < \\theta < (2 \\sin\\frac{1}{3}\\theta + \\tan\\frac{1}{3}\\theta)"],["planck-treatise-on-thermodynamics-1903/eq-d3c907de54",16,"Planck 1903, p. 235: -\\log c_{0} + \\log c_{1} + \\log c_{2} = K"],["planck-treatise-on-thermodynamics-1903/eq-ba98912c9e",16,"Planck 1903, p. 235: L = \\frac{4045000}{\\theta}"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/31ii",4,"Hardy 1921, Exercise Misc-X (31ii)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/32",4,"Hardy 1921, Exercise Misc-X (32)"],["form/d44408e573",5,"identity: (-x + 3)*(2*x + 3)"],["dickson-theory-of-equations-1922/eq-9192ea45fc",16,"Dickson 1922, p. 141: \\Sigma x_1^2x_2x_3x_4 = E_1E_4 - 5E_5"],["dickson-theory-of-equations-1922/eq-ef9257bddf",16,"Dickson 1922, p. 133: \\Sigma \\frac{\\alpha^2 + \\beta^2}{\\alpha + \\beta} = \\frac{2q^2 - 2p^2q + 4pr}{pq - r}"],["dickson-theory-of-equations-1922/eq-4c32a6fa99",16,"Dickson 1922, p. 149: (a_0b_1) = a_0b_1 - a_1b_0"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/33",4,"Hardy 1921, Exercise Misc-X (33)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/34",4,"Hardy 1921, Exercise Misc-X (34)"],["theorem/snell-s-theorem",9,"Snell's theorem"],["concept/pentadecagon",7,"pentadecagon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-pentadecagon"],["concept/pentagon",7,"pentagon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-pentagon"],["form/abd7eb96b8",5,"identity: (-3*a**2*x + a*x**2)*(-a**2*x + a*x**2)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/35",4,"Hardy 1921, Exercise Misc-X (35)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/36",4,"Hardy 1921, Exercise Misc-X (36)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/37",4,"Hardy 1921, Exercise Misc-X (37)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/38",4,"Hardy 1921, Exercise Misc-X (38)"],["ball-mathematical-recreations-1905/eq-21b4497145",16,"Ball 1905, scan 274: \\frac{\\pi}{2}=\\frac{2\\dotm 2\\dotm 4\\dotm 4\\dotm 6\\dotm 6\\dotsm} {1\\dotm 3\\dotm 3\\dotm 5\\dotm 5\\dotm 7\\dotm 7\\dotsm}"],["theorem/wallis-s-product",9,"Wallis's product"],["boyden-first-book-in-algebra-1895/ex-29/10",4,"Boyden 1895, Exercise 29 (10)"],["hardy-course-of-pure-mathematics-1921/ex-misc-x/39",4,"Hardy 1921, Exercise Misc-X (39)"],["ball-mathematical-recreations-1905/eq-90497d531d",16,"Ball 1905, scan 274: \\frac{\\pi}{4}=1+\\frac{1^2}{2} \\genfrac{}{}{0pt}{}{}{+} \\frac{3^2}{2} \\genfrac{}{}{0pt}{}{}{+} \\frac{5^2}{2} \\genfrac{}{}"],["todhunter-spherical-trigonometry-1886/eq-a74d6af74a",16,"Todhunter 1886, scan 98: s=Er^2"],["todhunter-spherical-trigonometry-1886/eq-9463a7c87c",16,"Todhunter 1886, scan 98: E=\\frac{n\\pi}{180\\centerdot 60\\centerdot 60}"],["hardy-course-of-pure-mathematics-1921/ex-v/1",4,"Hardy 1921, Exercise V (1)"],["shape/f9376053f4",6,"identity: (N*x**N - a**N)/(N*x - a)"],["hardy-course-of-pure-mathematics-1921/ex-v/2",4,"Hardy 1921, Exercise V (2)"],["hardy-course-of-pure-mathematics-1921/ex-v/3",4,"Hardy 1921, Exercise V (3)"],["hardy-course-of-pure-mathematics-1921/ex-v/4",4,"Hardy 1921, Exercise V (4)"],["hardy-course-of-pure-mathematics-1921/ex-v/5",4,"Hardy 1921, Exercise V (5)"],["todhunter-spherical-trigonometry-1886/eq-67dfd61a4c",16,"Todhunter 1886, scan 98: = \\frac{n}{206265}\\ \\text{ approximately;}"],["todhunter-spherical-trigonometry-1886/eq-fa88a49409",16,"Todhunter 1886, scan 98: s=\\frac{nr^2}{206265}\\,"],["theorem/brouncker-s-series",9,"Brouncker's series"],["form/a1b039bce4",5,"identity: (-a**2 + 49*x**2)/(-a + 7*x)"],["hardy-course-of-pure-mathematics-1921/ex-v/6",4,"Hardy 1921, Exercise V (6)"],["hardy-course-of-pure-mathematics-1921/ex-v/7",4,"Hardy 1921, Exercise V (7)"],["hardy-course-of-pure-mathematics-1921/ex-v/8",4,"Hardy 1921, Exercise V (8)"],["hardy-course-of-pure-mathematics-1921/ex-v/9",4,"Hardy 1921, Exercise V (9)"],["ball-mathematical-recreations-1905/eq-93416b2378",16,"Ball 1905, scan 275: \\theta = \\tan\\theta - \\frac{1}{3}\\tan^3\\theta + \\frac{1}{5}\\tan^5\\theta - \\dotsb"],["theorem/general-roy-s-rule-intermediate-form",9,"General Roy's rule (intermediate form)"],["todhunter-spherical-trigonometry-1886/eq-26ec02c88b",16,"Todhunter 1886, scan 98: \\frac{\\pi r}{180}=365155"],["todhunter-spherical-trigonometry-1886/eq-3cb795b6e7",16,"Todhunter 1886, scan 98: \\log n = \\log s - 9.326774"],["hardy-course-of-pure-mathematics-1921/ex-v/10",4,"Hardy 1921, Exercise V (10)"],["todhunter-spherical-trigonometry-1886/eq-5f9a147c3f",16,"Todhunter 1886, scan 62: \\sin b = \\dfrac{\\sin B \\sin a}{\\sin A}"],["form/f2cc6652ec",5,"identity: ((-7*a + 5*x)**2 - 1)/(-7*a + 5*x - 1)"],["hardy-course-of-pure-mathematics-1921/ex-vi/1",4,"Hardy 1921, Exercise VI (1)"],["hardy-course-of-pure-mathematics-1921/ex-vi/2",4,"Hardy 1921, Exercise VI (2)"],["hardy-course-of-pure-mathematics-1921/ex-vi/3",4,"Hardy 1921, Exercise VI (3)"],["hardy-course-of-pure-mathematics-1921/ex-vi/4",4,"Hardy 1921, Exercise VI (4)"],["concept/cross-ratio",7,"cross ratio","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-cross-ratio"],["method/mathematical-induction",8,"mathematical induction","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-mathematical-induction"],["ball-mathematical-recreations-1905/eq-6fb399cf50",16,"Ball 1905, scan 275: \\tfrac{1}{4}\\pi = 4 \\tan^{-1}\\tfrac{1}{5} -\\tan^{-1}\\tfrac{1}{239}"],["hardy-course-of-pure-mathematics-1921/ex-vi/5",4,"Hardy 1921, Exercise VI (5)"],["hardy-course-of-pure-mathematics-1921/x-7f28a22a45",15,"Hardy 1921, p. 85: A line originally lying along OX will, if turned ..."],["hardy-course-of-pure-mathematics-1921/ex-vi/6",4,"Hardy 1921, Exercise VI (6)"],["shape/29a3abe08c",6,"identity: (N*x**N + 1)/(N*x + 1)"],["hardy-course-of-pure-mathematics-1921/ex-vi/7",4,"Hardy 1921, Exercise VI (7)"],["hardy-course-of-pure-mathematics-1921/ex-vi/8",4,"Hardy 1921, Exercise VI (8)"],["form/602ae80c0b",5,"identity: (-8*x**3 + 1)/(-2*x + 1)"],["hardy-course-of-pure-mathematics-1921/ch-appendix-iii",2,"Hardy 1921, ch. Appendix III: The circular functions","../books/hardy-course-of-pure-mathematics-1921/ch/ch-appendix-iii/index.html"],["hardy-course-of-pure-mathematics-1921/ex-vii/1",4,"Hardy 1921, Exercise VII (1)"],["ball-mathematical-recreations-1905/eq-4e6e9e4e4d",16,"Ball 1905, scan 275: \\frac{1}{4}\\pi =\\allowbreak \\tan^{-1}\\frac{1}{2} +\\allowbreak \\tan^{-1}\\frac{1}{3}"],["shape/1bf82fa706",6,"identity: (N + a**N*x**N)/(N + a*x)"],["cap/other:geometric_construction",17,"other:geometric_construction"],["hardy-course-of-pure-mathematics-1921/ex-vii/2",4,"Hardy 1921, Exercise VII (2)"],["hardy-course-of-pure-mathematics-1921/ex-vii/3",4,"Hardy 1921, Exercise VII (3)"],["ball-mathematical-recreations-1905/eq-55e1596d7e",16,"Ball 1905, scan 275: \\frac{1}{4}\\pi = 5 \\tan^{-1}\\frac{1}{7} + 2\\tan^{-1}\\frac{3}{79}"],["form/b1529a9dca",5,"identity: (a**3*x**3 - 8)/(a*x - 2)"],["hardy-course-of-pure-mathematics-1921/ex-vii/4",4,"Hardy 1921, Exercise VII (4)"],["ball-mathematical-recreations-1905/eq-808b1f2f78",16,"Ball 1905, scan 276: \\frac{1}{4}\\pi=4\\tan^{-1}\\frac{1}{5} - \\tan^{-1}\\frac{1}{70} + \\tan^{-1}\\frac{1}{99}"],["ball-mathematical-recreations-1905/eq-533530bbd6",16,"Ball 1905, scan 276: \\frac{1}{4}\\pi= \\tan^{-1}\\frac{1}{2} +\\tan^{-1}\\frac{1}{5} + \\tan^{-1}\\frac{1}{8}"],["boyden-first-book-in-algebra-1895/ex-31/4",4,"Boyden 1895, Exercise 31 (4)"],["hardy-course-of-pure-mathematics-1921/ex-viii/1",4,"Hardy 1921, Exercise VIII (1)"],["hardy-course-of-pure-mathematics-1921/ex-viii/2",4,"Hardy 1921, Exercise VIII (2)"],["hardy-course-of-pure-mathematics-1921/ex-viii/3",4,"Hardy 1921, Exercise VIII (3)"],["hardy-course-of-pure-mathematics-1921/ex-viii/4",4,"Hardy 1921, Exercise VIII (4)"],["hardy-course-of-pure-mathematics-1921/x-ecf6eadca9",15,"Hardy 1921, p. 85: It will be observed that the sum 2x of ..."],["form/d95cc45cca",5,"factor: -225*a**4 + 15*a**2"],["de-morgan-elementary-illustrations-calculus-1899/eq-d2b3c906d5",16,"De Morgan 1899, p. 101: z = ab"],["hardy-course-of-pure-mathematics-1921/ex-viii/5",4,"Hardy 1921, Exercise VIII (5)"],["hardy-course-of-pure-mathematics-1921/x-a71919951f",15,"Hardy 1921, p. 86: Thus the modulus of the reciprocal of z is ..."],["hardy-course-of-pure-mathematics-1921/x-814012f45b",15,"Hardy 1921, p. 87: The length OU is the modulus of the sum ..."],["boyden-first-book-in-algebra-1895/ex-29/11",4,"Boyden 1895, Exercise 29 (11)"],["wentworth-first-steps-in-algebra-1894/ex-2/21",4,"Wentworth 1894, Exercise 2 (21)"],["form/dbdca9cf23",5,"identity: 5*a**2*(b**2 - c)"],["ball-mathematical-recreations-1905/eq-315a02ca7e",16,"Ball 1905, scan 276: \\frac{1}{4}\\pi = 2\\tan^{-1}\\frac{1}{3} + \\tan^{-1}\\frac{1}{7}"],["ball-mathematical-recreations-1905/eq-b4b2eccd75",16,"Ball 1905, scan 277: \\frac{\\pi}{6} = \\frac{1}{2} + \\frac{1}{2}\\dotm \\frac{1}{3\\dotm 2^3} + \\frac{1\\dotm 3}{2\\dotm 4} \\dotm \\frac{1}{5\\dotm 2^"],["concept/mean-solar-day",7,"mean solar day","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-mean-solar-day"],["hardy-course-of-pure-mathematics-1921/ex-viii/6",4,"Hardy 1921, Exercise VIII (6)"],["hardy-course-of-pure-mathematics-1921/ex-viii/7",4,"Hardy 1921, Exercise VIII (7)"],["hardy-course-of-pure-mathematics-1921/ex-viii/8",4,"Hardy 1921, Exercise VIII (8)"],["form/203deaf6c1",5,"identity: 1/(a + sqrt(b))"],["shape/95a2a4b9d4",6,"identity: 1/(a + b**N)"],["boyden-first-book-in-algebra-1895/ex-10/14",4,"Boyden 1895, Exercise 10 (14)"],["ball-mathematical-recreations-1905/eq-2255548018",16,"Ball 1905, scan 277: \\frac{\\pi}{4} = 2 + 22\\tan^{-1}\\frac{1}{28} + \\tan^{-1}\\frac{1}{443} - 5\\tan^{-1}\\frac{1}{1393} - 10\\tan^{-1}\\frac{1}{11"],["hardy-course-of-pure-mathematics-1921/ex-viii/9",4,"Hardy 1921, Exercise VIII (9)"],["hardy-course-of-pure-mathematics-1921/ex-viii/10",4,"Hardy 1921, Exercise VIII (10)"],["hardy-course-of-pure-mathematics-1921/ex-viii/11",4,"Hardy 1921, Exercise VIII (11)"],["hardy-course-of-pure-mathematics-1921/ex-viii/12",4,"Hardy 1921, Exercise VIII (12)"],["concept/functional-interpolation",7,"functional interpolation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-functional-interpolation"],["shape/a2bd275ad8",6,"factor: 2*N*a**N"],["ball-mathematical-recreations-1905/eq-8f20de365f",16,"Ball 1905, scan 275: \\tan^{-1}\\tfrac{1}{5}"],["hardy-course-of-pure-mathematics-1921/ex-xc/1",4,"Hardy 1921, Exercise XC (1)"],["hardy-course-of-pure-mathematics-1921/ex-xc/2",4,"Hardy 1921, Exercise XC (2)"],["hardy-course-of-pure-mathematics-1921/ex-xc/3",4,"Hardy 1921, Exercise XC (3)"],["hardy-course-of-pure-mathematics-1921/ex-xc/4",4,"Hardy 1921, Exercise XC (4)"],["concept/set",7,"set","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-set"],["hardy-course-of-pure-mathematics-1921/ex-xc/5",4,"Hardy 1921, Exercise XC (5)"],["hardy-course-of-pure-mathematics-1921/ex-xc/6",4,"Hardy 1921, Exercise XC (6)"],["hardy-course-of-pure-mathematics-1921/ex-xc/7",4,"Hardy 1921, Exercise XC (7)"],["concept/finite-set",7,"finite set","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-finite-set"],["form/e75e51e99b",5,"identity: (8*a**3*b**3*x**3 + 27)/(2*a*b*x + 3)"],["todhunter-spherical-trigonometry-1886/eq-5bd06f71ac",16,"Todhunter 1886, scan 100: \\delta A + \\delta B + \\delta C = 0"],["hardy-course-of-pure-mathematics-1921/ex-xc/8",4,"Hardy 1921, Exercise XC (8)"],["hardy-course-of-pure-mathematics-1921/ex-xc/9",4,"Hardy 1921, Exercise XC (9)"],["hardy-course-of-pure-mathematics-1921/ex-xc/10",4,"Hardy 1921, Exercise XC (10)"],["concept/threshold-index",7,"threshold index","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-threshold-index"],["form/380fbbe246",5,"identity: (-a**5 + x**5)/(-a + x)"],["form/e68848f435",5,"identity: (a**5 + x**5)/(a + x)"],["ball-mathematical-recreations-1905/eq-b4b20813d9",16,"Ball 1905, scan 271: \\frac{3}{4}(\\sqrt{3} + \\sqrt{6})"],["instrument/watch",13,"watch","../books/ball-mathematical-recreations-1905/terms/index.html#t-instrument-watch"],["hardy-course-of-pure-mathematics-1921/ex-xc/11",4,"Hardy 1921, Exercise XC (11)"],["hardy-course-of-pure-mathematics-1921/ex-xc/12",4,"Hardy 1921, Exercise XC (12)"],["hardy-course-of-pure-mathematics-1921/ex-xc/13",4,"Hardy 1921, Exercise XC (13)"],["hardy-course-of-pure-mathematics-1921/ex-xc/14",4,"Hardy 1921, Exercise XC (14)"],["boyden-first-book-in-algebra-1895/ex-10/15",4,"Boyden 1895, Exercise 10 (15)"],["form/2849303678",5,"factor: -10*a**3*x**3 + 5*a**2*x**2"],["ball-mathematical-recreations-1905/eq-bbb167059b",16,"Ball 1905, scan 277: 2l/\\pi a"],["hardy-course-of-pure-mathematics-1921/ex-xc/15",4,"Hardy 1921, Exercise XC (15)"],["hardy-course-of-pure-mathematics-1921/x-51855c17eb",15,"Hardy 1921, p. 111: It is a truism that in common life a ..."],["hardy-course-of-pure-mathematics-1921/x-e970eb4063",15,"Hardy 1921, p. 112: The reader cannot too strongly impress upon himself that ..."],["hardy-course-of-pure-mathematics-1921/x-9b6b1ed7fd",15,"Hardy 1921, p. 114: The reader should imagine himself confronted by an opponent ..."],["todhunter-spherical-trigonometry-1886/eq-c49c01a15d",16,"Todhunter 1886, scan 100: \\dfrac{a \\sin (C + \\delta C)}{\\sin (A + \\delta A)}"],["todhunter-spherical-trigonometry-1886/eq-72fb5d1823",16,"Todhunter 1886, scan 100: \\sin (C + \\delta C) = \\sin C + \\delta C \\cos C"],["hardy-course-of-pure-mathematics-1921/ex-xci/1",4,"Hardy 1921, Exercise XCI (1)"],["hardy-course-of-pure-mathematics-1921/x-c70f879e42",15,"Hardy 1921, p. 119: The reader cannot impress these facts too strongly on ..."],["hardy-course-of-pure-mathematics-1921/ex-xci/2",4,"Hardy 1921, Exercise XCI (2)"],["hardy-course-of-pure-mathematics-1921/ex-xci/3",4,"Hardy 1921, Exercise XCI (3)"],["hardy-course-of-pure-mathematics-1921/ex-xci/4",4,"Hardy 1921, Exercise XCI (4)"],["hardy-course-of-pure-mathematics-1921/x-5c1faddaeb",15,"Hardy 1921, p. 109: If n is any positive integer, such as 1000, ..."],["form/371167f06b",5,"factor: -a*c + a*x + b*c - b*x"],["shape/371167f06b",6,"factor: -a*c + a*x + b*c - b*x"],["hardy-course-of-pure-mathematics-1921/ex-misc-vi",3,"Hardy 1921, Exercise Misc-VI"],["hardy-course-of-pure-mathematics-1921/ex-xci/5",4,"Hardy 1921, Exercise XCI (5)"],["hardy-course-of-pure-mathematics-1921/ex-xci/6",4,"Hardy 1921, Exercise XCI (6)"],["hardy-course-of-pure-mathematics-1921/ex-xci/7",4,"Hardy 1921, Exercise XCI (7)"],["hardy-course-of-pure-mathematics-1921/ex-xci/8",4,"Hardy 1921, Exercise XCI (8)"],["wentworth-first-steps-in-algebra-1894/ex-32/8",4,"Wentworth 1894, Exercise 32 (8)"],["todhunter-spherical-trigonometry-1886/eq-65524eacc4",16,"Todhunter 1886, scan 62: \\cos A= -\\cos B\\cos C + \\sin B\\sin C\\cos a"],["hardy-course-of-pure-mathematics-1921/ex-xci/9",4,"Hardy 1921, Exercise XCI (9)"],["ball-mathematical-recreations-1905/eq-d6fa02cc07",16,"Ball 1905, scan 278: 6/\\pi^2 = 154/250"],["dickson-theory-of-equations-1922/eq-8706881e8c",16,"Dickson 1922, p. 151: (a_0b_1)(a_2b_3) - (a_0b_2)(a_1b_3) + (a_0b_3)(a_1b_2)"],["hardy-course-of-pure-mathematics-1921/ex-xci/10",4,"Hardy 1921, Exercise XCI (10)"],["hardy-course-of-pure-mathematics-1921/ex-xci/11",4,"Hardy 1921, Exercise XCI (11)"],["dickson-theory-of-equations-1922/eq-5bef72c8bf",16,"Dickson 1922, p. 151: F=(a_0b_3)R"],["hardy-course-of-pure-mathematics-1921/ex-xci/12",4,"Hardy 1921, Exercise XCI (12)"],["form/606abfb105",5,"factor: -a**2 + 25*x**2"],["hardy-course-of-pure-mathematics-1921/ex-xci/13",4,"Hardy 1921, Exercise XCI (13)"],["hardy-course-of-pure-mathematics-1921/ex-xci/14",4,"Hardy 1921, Exercise XCI (14)"],["hardy-course-of-pure-mathematics-1921/ex-xci/15",4,"Hardy 1921, Exercise XCI (15)"],["hardy-course-of-pure-mathematics-1921/ex-xci/16",4,"Hardy 1921, Exercise XCI (16)"],["ball-mathematical-recreations-1905/eq-3fcabcbcc4",16,"Ball 1905, scan 278: \\pi=3.12"],["planck-treatise-on-thermodynamics-1903/eq-1188b1d409",16,"Planck 1903, p. 239: c_{1} + c_{3} + c_{4} = c"],["planck-treatise-on-thermodynamics-1903/eq-469d1a024a",16,"Planck 1903, p. 231: \\frac{\\dd \\varphi_{0}}{\\dd \\theta} = \\frac{u_{0} + pv_{0}}{\\theta^{2}};\\quad"],["form/630258822e",5,"factor: 144*a**2*x**2 - 1"],["form/28abba9498",5,"factor: 100*a**2*b**4*x**6 - 1"],["hardy-course-of-pure-mathematics-1921/ex-xci/17",4,"Hardy 1921, Exercise XCI (17)"],["hardy-course-of-pure-mathematics-1921/ex-xci/18",4,"Hardy 1921, Exercise XCI (18)"],["todhunter-spherical-trigonometry-1886/eq-d7f2b836fe",16,"Todhunter 1886, scan 62: \\sin(C - \\phi)=\\dfrac{\\cos A \\sin \\phi}{\\cos B}"],["todhunter-spherical-trigonometry-1886/eq-492b807b28",16,"Todhunter 1886, scan 63: \\cot A \\sin B = \\cot a \\sin c - \\cos c \\cos B"],["form/8c7bb4884b",5,"evaluate: 90000"],["hardy-course-of-pure-mathematics-1921/ex-xcii/1",4,"Hardy 1921, Exercise XCII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xcii/2",4,"Hardy 1921, Exercise XCII (2)"],["hardy-course-of-pure-mathematics-1921/x-a4ee3819e0",15,"Hardy 1921, p. 118: We may obviously alter the values of \\phi(n) for ..."],["todhunter-spherical-trigonometry-1886/eq-2041131c9f",16,"Todhunter 1886, scan 63: \\cot \\theta = \\dfrac{\\cot a}{\\cos B}"],["form/2e085129a5",5,"factor: 4*c**2 - (a - b)**2"],["shape/f03c5c92d3",6,"factor: N*c**N - (a - b)**N"],["hardy-course-of-pure-mathematics-1921/ex-xcii/3",4,"Hardy 1921, Exercise XCII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xcii/4",4,"Hardy 1921, Exercise XCII (4)"],["hardy-course-of-pure-mathematics-1921/ex-xcii/5",4,"Hardy 1921, Exercise XCII (5)"],["concept/left-right-operation-notation",7,"left-right operation notation","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-left-right-operation-notation"],["todhunter-spherical-trigonometry-1886/eq-4568a482ee",16,"Todhunter 1886, scan 63: \\sin (c - \\theta) = \\cot A \\tan B \\sin \\theta"],["todhunter-spherical-trigonometry-1886/eq-d04c5b16d7",16,"Todhunter 1886, scan 63: \\cot \\tfrac{1}{2}C = \\tan A \\cos a"],["hardy-course-of-pure-mathematics-1921/ex-xcii/6",4,"Hardy 1921, Exercise XCII (6)"],["hardy-course-of-pure-mathematics-1921/ex-xcii/7a",4,"Hardy 1921, Exercise XCII (7a)"],["hardy-course-of-pure-mathematics-1921/ex-xcii/7b",4,"Hardy 1921, Exercise XCII (7b)"],["hardy-course-of-pure-mathematics-1921/ex-xcii/7c",4,"Hardy 1921, Exercise XCII (7c)"],["method/de-moivre-s-method-for-the-knight-s-tour",8,"De Moivre's method for the knight's tour","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-de-moivre-s-method-for-the-knight-s-tour"],["concept/complementary-cell",7,"complementary cell","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-complementary-cell"],["concept/knight-s-move",7,"knight's move","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-knight-s-move"],["hardy-course-of-pure-mathematics-1921/ex-xciii/1",4,"Hardy 1921, Exercise XCIII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xciii/2",4,"Hardy 1921, Exercise XCIII (2)"],["wentworth-first-steps-in-algebra-1894/ex-40/1",4,"Wentworth 1894, Exercise 40 (1)"],["theorem/knight-s-move-alternates-colour",9,"knight's move alternates colour","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-knight-s-move-alternates-colour"],["person/abraham-de-moivre",1,"Abraham de Moivre","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-abraham-de-moivre"],["person/alexandre-th-ophile-vandermonde",1,"Alexandre-Théophile Vandermonde","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-alexandre-th-ophile-vandermonde"],["hardy-course-of-pure-mathematics-1921/ex-xciii/3",4,"Hardy 1921, Exercise XCIII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xciii/4",4,"Hardy 1921, Exercise XCIII (4)"],["hardy-course-of-pure-mathematics-1921/ex-xciii/5",4,"Hardy 1921, Exercise XCIII (5)"],["person/peter-mark-roget",1,"Peter Mark Roget","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-peter-mark-roget"],["ball-mathematical-recreations-1905/x-f5a360f271",15,"Ball 1905, scan 202: On a board containing an even number of cells ..."],["boyden-first-book-in-algebra-1895/ex-12/1",4,"Boyden 1895, Exercise 12 (1)"],["hardy-course-of-pure-mathematics-1921/ex-xciii/6",4,"Hardy 1921, Exercise XCIII (6)"],["ball-mathematical-recreations-1905/x-ba25f8db36",15,"Ball 1905, scan 208: His rule is that the knight must be moved ..."],["ball-mathematical-recreations-1905/x-26b151acfa",15,"Ball 1905, scan 200: if we approach a point by one edge, the ..."],["ball-mathematical-recreations-1905/x-942f6faff4",15,"Ball 1905, scan 201: It is convenient to make a mark or to ..."],["concept/finite-oscillation",7,"finite oscillation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-finite-oscillation"],["concept/infinite-oscillation",7,"infinite oscillation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-infinite-oscillation"],["boyden-first-book-in-algebra-1895/ex-12/2",4,"Boyden 1895, Exercise 12 (2)"],["hardy-course-of-pure-mathematics-1921/ex-xciii/7",4,"Hardy 1921, Exercise XCIII (7)"],["hardy-course-of-pure-mathematics-1921/ex-xciii/8",4,"Hardy 1921, Exercise XCIII (8)"],["hardy-course-of-pure-mathematics-1921/ex-xciii/9",4,"Hardy 1921, Exercise XCIII (9)"],["hardy-course-of-pure-mathematics-1921/ex-xciii/10",4,"Hardy 1921, Exercise XCIII (10)"],["hardy-course-of-pure-mathematics-1921/ex-xciii/11",4,"Hardy 1921, Exercise XCIII (11)"],["hardy-course-of-pure-mathematics-1921/eq-1f005cb506",16,"Hardy 1921, p. 143: u_{n} = r^{n-1}"],["hardy-course-of-pure-mathematics-1921/eq-7b248dcb7e",16,"Hardy 1921, p. 143: s_{n} = 1 + r + r^{2} + \\dots + r^{n-1} = (1 - r^{n})/(1 - r)"],["form/194e3b34c3",5,"factor: -a**3 + 125*x**3"],["form/d31bff7045",5,"factor: x**3 - 343"],["hardy-course-of-pure-mathematics-1921/ex-xciii/12",4,"Hardy 1921, Exercise XCIII (12)"],["hardy-course-of-pure-mathematics-1921/eq-cbaf5acf74",16,"Hardy 1921, p. 143: s_{n} = 1 + 1 + \\dots + 1 = n"],["hardy-course-of-pure-mathematics-1921/eq-ba6e8675c9",16,"Hardy 1921, p. 143: s_{n} \\to +\\infty"],["hardy-course-of-pure-mathematics-1921/eq-ec8a22f1f1",16,"Hardy 1921, p. 143: s_{n} \\geq n"],["form/b532de51b6",5,"factor: -64*a**6 + 27*x**3"],["hardy-course-of-pure-mathematics-1921/ex-xciv/1",4,"Hardy 1921, Exercise XCIV (1)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/2",4,"Hardy 1921, Exercise XCIV (2)"],["wentworth-first-steps-in-algebra-1894/ex-35/17",4,"Wentworth 1894, Exercise 35 (17)"],["hardy-course-of-pure-mathematics-1921/eq-a9482dbff2",16,"Hardy 1921, p. 147: \\lim_{n \\to \\infty} \\phi_{n}(x)"],["concept/prime-number",7,"prime number","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-prime-number"],["hardy-course-of-pure-mathematics-1921/eq-677cebf27f",16,"Hardy 1921, p. 143: 1/(1 - r)"],["hardy-course-of-pure-mathematics-1921/eq-6f69fdb020",16,"Hardy 1921, p. 147: f(x) = \\lim_{n \\to \\infty} n(\\sqrt[n]{x} - 1)"],["hardy-course-of-pure-mathematics-1921/eq-d61b44e8ef",16,"Hardy 1921, p. 147: u_{1}(x) + u_{2}(x) + \\dots = \\lim_{n \\to \\infty}\\{u_{1}(x) + u_{2}(x) + \\dots + u_{n}(x)\\}"],["hardy-course-of-pure-mathematics-1921/eq-960dd58fe7",16,"Hardy 1921, p. 149: s \\leq K"],["hardy-course-of-pure-mathematics-1921/ex-xciv/3",4,"Hardy 1921, Exercise XCIV (3)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/4",4,"Hardy 1921, Exercise XCIV (4)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/5",4,"Hardy 1921, Exercise XCIV (5)"],["hardy-course-of-pure-mathematics-1921/eq-87b3725757",16,"Hardy 1921, p. 149: s \\geq k"],["concept/cardioid",7,"cardioid","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-cardioid"],["form/a24a60c47b",5,"factor: a**3 + 8*x**3"],["hardy-course-of-pure-mathematics-1921/ex-xciv/6",4,"Hardy 1921, Exercise XCIV (6)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/7",4,"Hardy 1921, Exercise XCIV (7)"],["form/433994060c",5,"hcf: (330, 546)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/8",4,"Hardy 1921, Exercise XCIV (8)"],["shape/43afa90f98",6,"hcf: (N, N)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/9",4,"Hardy 1921, Exercise XCIV (9)"],["wentworth-first-steps-in-algebra-1894/ex-40/2",4,"Wentworth 1894, Exercise 40 (2)"],["cap/cas.pgcd",17,"cas.pgcd"],["form/27884093e9",5,"hcf: (20*a**3, 15*a**4)"],["form/140721155f",5,"factor: x**3 + 343"],["shape/3dca7cf967",6,"hcf: (N*a**N, N*a**N)"],["todhunter-spherical-trigonometry-1886/eq-c44f373df0",16,"Todhunter 1886, scan 63: \\tan \\tfrac{1}{2}c = \\tan a \\cos A"],["hardy-course-of-pure-mathematics-1921/ex-xciv/10",4,"Hardy 1921, Exercise XCIV (10)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/11",4,"Hardy 1921, Exercise XCIV (11)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/12",4,"Hardy 1921, Exercise XCIV (12)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/13",4,"Hardy 1921, Exercise XCIV (13)"],["hardy-course-of-pure-mathematics-1921/eq-3fccc8b42b",16,"Hardy 1921, p. 149: k \\leq m \\leq M \\leq K"],["hardy-course-of-pure-mathematics-1921/eq-caede7193f",16,"Hardy 1921, p. 150: \\phi(n) \\leq M"],["boyden-first-book-in-algebra-1895/ex-12/3",4,"Boyden 1895, Exercise 12 (3)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/14",4,"Hardy 1921, Exercise XCIV (14)"],["todhunter-spherical-trigonometry-1886/eq-7a48453b8b",16,"Todhunter 1886, scan 64: \\sin B = \\dfrac{\\sin b \\sin A}{\\sin a}"],["todhunter-spherical-trigonometry-1886/eq-1f53d9bd64",16,"Todhunter 1886, scan 64: \\beta' = \\pi - \\beta"],["form/b4617af100",5,"factor: 9*a**2 + 6*a*x + x**2"],["de-morgan-elementary-illustrations-calculus-1899/eq-210c626c47",16,"De Morgan 1899, p. 32: bx - ay = ab"],["hardy-course-of-pure-mathematics-1921/ex-xciv/15",4,"Hardy 1921, Exercise XCIV (15)"],["hardy-course-of-pure-mathematics-1921/ex-xciv/16",4,"Hardy 1921, Exercise XCIV (16)"],["concept/circular-segment",7,"circular segment","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-circular-segment"],["form/4e9eb43986",5,"factor: 9*a**2 - 12*a*x + 4*x**2"],["shape/6ae9a493a9",6,"factor: N*a*x + N*a**N + N*x**N"],["hardy-course-of-pure-mathematics-1921/ex-xcv/1",4,"Hardy 1921, Exercise XCV (1)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/2",4,"Hardy 1921, Exercise XCV (2)"],["person/euclid",1,"Euclid","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-euclid"],["hardy-course-of-pure-mathematics-1921/eq-711dd0b189",16,"Hardy 1921, p. 150: \\phi(n) > M - \\DELTA"],["de-morgan-elementary-illustrations-calculus-1899/eq-be98437520",16,"De Morgan 1899, p. 33: x^{2} + b^{2}\\, \\frac{(a - x)^{2}}{a^{2}} = r^{2}"],["hardy-course-of-pure-mathematics-1921/eq-9a4a59c1b7",16,"Hardy 1921, p. 150: \\phi(n) \\geq m"],["hardy-course-of-pure-mathematics-1921/ex-xcv/3",4,"Hardy 1921, Exercise XCV (3)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/4",4,"Hardy 1921, Exercise XCV (4)"],["concept/compact-series",7,"compact series","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-compact-series"],["hardy-course-of-pure-mathematics-1921/ex-xcv/5",4,"Hardy 1921, Exercise XCV (5)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/6",4,"Hardy 1921, Exercise XCV (6)"],["form/34e9f279f3",5,"hcf: (a**2 + 3*a*b - 4*b**2, a**2 + 7*a*b + 12*b**2)"],["cap/cas.factor",17,"cas.factor"],["shape/de35536a08",6,"hcf: (N*a*b + N*b**N + a**N, N*a*b + N*b**N + a**N)"],["form/306c977993",5,"factor: x**2 + 5*x + 6"],["wentworth-first-steps-in-algebra-1894/ex-40/23",4,"Wentworth 1894, Exercise 40 (23)"],["form/a6a4f84061",5,"hcf: (a**3 - 8*b**3, a**2 + 2*a*b + 4*b**2)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/7",4,"Hardy 1921, Exercise XCV (7)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/8",4,"Hardy 1921, Exercise XCV (8)"],["hardy-course-of-pure-mathematics-1921/eq-965ea2bc61",16,"Hardy 1921, p. 150: \\phi(n) < m + \\DELTA"],["form/1027347de1",5,"solve: Eq(cos(x), a)"],["shape/1027347de1",6,"solve: Eq(cos(x), a)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/9",4,"Hardy 1921, Exercise XCV (9)"],["form/d7ddc85c75",5,"solve: Eq(sin(x), a)"],["shape/d7ddc85c75",6,"solve: Eq(sin(x), a)"],["hardy-course-of-pure-mathematics-1921/eq-52c51d96ab",16,"Hardy 1921, p. 150: \\phi(n) \\leq k"],["form/e3ebfbb636",5,"factor: x**2 - 9*x + 18"],["form/e19fb0799b",5,"factor: x**2 + 9*x + 18"],["hardy-course-of-pure-mathematics-1921/ex-xcv/10",4,"Hardy 1921, Exercise XCV (10)"],["form/31cf648cdb",5,"solve: Eq(cos(x), a*c + b)"],["shape/0a3ea17640",6,"hcf: (N*b**N + a**N, N*a*b + N*b**N + a**N)"],["shape/31cf648cdb",6,"solve: Eq(cos(x), a*c + b)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/11",4,"Hardy 1921, Exercise XCV (11)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/12",4,"Hardy 1921, Exercise XCV (12)"],["hardy-course-of-pure-mathematics-1921/eq-bc8d51a5ec",16,"Hardy 1921, p. 150: m \\leq \\lambda \\leq \\Lambda \\leq M"],["hardy-course-of-pure-mathematics-1921/eq-a57de6f0b8",16,"Hardy 1921, p. 150: \\Lambda = \\limsup \\phi(n)"],["boyden-first-book-in-algebra-1895/ex-12/4",4,"Boyden 1895, Exercise 12 (4)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/13",4,"Hardy 1921, Exercise XCV (13)"],["wentworth-first-steps-in-algebra-1894/ex-40/24",4,"Wentworth 1894, Exercise 40 (24)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/14",4,"Hardy 1921, Exercise XCV (14)"],["form/a789a28a8f",5,"solve: Eq(tan(x), a)"],["form/612781f386",5,"hcf: (a**2 - 4*a + 4, a**3 - 2*a**2 - a + 2)"],["shape/a789a28a8f",6,"solve: Eq(tan(x), a)"],["shape/8ed9b84164",6,"hcf: (N*a + N + a**N, N*a**N + N - a + a**N)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/15",4,"Hardy 1921, Exercise XCV (15)"],["whitehead-introduction-to-mathematics-1911/x-7771622743",15,"Whitehead 1911, p. 86: If a balance at the bank is positive, an ..."],["form/e3a83a57c6",5,"factor: x**2 - 10*x + 9"],["hardy-course-of-pure-mathematics-1921/ex-xcv/16",4,"Hardy 1921, Exercise XCV (16)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/17",4,"Hardy 1921, Exercise XCV (17)"],["de-morgan-elementary-illustrations-calculus-1899/eq-43e4701644",16,"De Morgan 1899, p. 33: (a^{2} + b^{2}) x^{2} - 2ab^{2}x + a^{2}(b^{2} - r^{2}) = 0"],["hardy-course-of-pure-mathematics-1921/ex-xcv/18",4,"Hardy 1921, Exercise XCV (18)"],["hardy-course-of-pure-mathematics-1921/ex-xcv/19",4,"Hardy 1921, Exercise XCV (19)"],["de-morgan-elementary-illustrations-calculus-1899/eq-a74922b914",16,"De Morgan 1899, p. 33: (a^{2} + b^{2}) y^{2} - 2a^{2}by + b^{2}(a^{2} - r^{2}) = 0"],["form/b4073b1966",5,"factor: -4*a**2 + 3*a*x + x**2"],["hardy-course-of-pure-mathematics-1921/ex-xcv/20",4,"Hardy 1921, Exercise XCV (20)"],["wentworth-first-steps-in-algebra-1894/ex-41/1",4,"Wentworth 1894, Exercise 41 (1)"],["form/fb6a14338d",5,"lcm: (9*a*b**3, 6*a**2*b)"],["hardy-course-of-pure-mathematics-1921/eq-0c4e905fb8",16,"Hardy 1921, p. 150: \\lambda = \\liminf \\phi(n)"],["shape/dbd8b22b8e",6,"lcm: (N*a*b**N, N*a**N*b)"],["wentworth-first-steps-in-algebra-1894/ex-41/2",4,"Wentworth 1894, Exercise 41 (2)"],["hardy-course-of-pure-mathematics-1921/ex-x/1",4,"Hardy 1921, Exercise X (1)"],["hardy-course-of-pure-mathematics-1921/ex-x/2",4,"Hardy 1921, Exercise X (2)"],["hardy-course-of-pure-mathematics-1921/ex-x/3",4,"Hardy 1921, Exercise X (3)"],["planck-treatise-on-thermodynamics-1903/eq-eab16f482f",16,"Planck 1903, p. 231: \\frac{\\dd \\varphi_{0}}{\\dd p} = -\\frac{v_{0}}{\\theta}"],["hardy-course-of-pure-mathematics-1921/eq-3187caff78",16,"Hardy 1921, p. 150: \\phi(n) < \\Lambda + \\DELTA"],["hardy-course-of-pure-mathematics-1921/ex-x/4",4,"Hardy 1921, Exercise X (4)"],["hardy-course-of-pure-mathematics-1921/ex-x/5",4,"Hardy 1921, Exercise X (5)"],["hardy-course-of-pure-mathematics-1921/eq-dc448812b0",16,"Hardy 1921, p. 150: \\phi(n) > \\Lambda - \\DELTA"],["hardy-course-of-pure-mathematics-1921/eq-75d736a5a6",16,"Hardy 1921, p. 151: l - \\DELTA < \\phi(n) < l + \\DELTA"],["form/9c7995ef64",5,"factor: 8*a**3 + x**3"],["form/81f6de2989",5,"factor: -4*a**2 - 2*a + x**2 + x"],["hardy-course-of-pure-mathematics-1921/ex-x/6",4,"Hardy 1921, Exercise X (6)"],["hardy-course-of-pure-mathematics-1921/ex-x/7",4,"Hardy 1921, Exercise X (7)"],["form/4781bb3af4",5,"lcm: (3*a*b*c**2, 2*a**2*b*c**3)"],["shape/d75ad05492",6,"lcm: (N*a*b*c**N, N*a**N*b*c**N)"],["wentworth-first-steps-in-algebra-1894/ex-41/3",4,"Wentworth 1894, Exercise 41 (3)"],["de-morgan-elementary-illustrations-calculus-1899/x-f17ab51c11",15,"De Morgan 1899, p. 18: We have introduced the new terms, \\dfrac{b}{a}, \\dfrac{c}{b}, etc., ..."],["form/545d5e7581",5,"lcm: (10*a*b**3, 4*a**3*b)"],["hardy-course-of-pure-mathematics-1921/ex-x/8",4,"Hardy 1921, Exercise X (8)"],["wentworth-first-steps-in-algebra-1894/ex-42",3,"Wentworth 1894, Exercise 42"],["hardy-course-of-pure-mathematics-1921/eq-9047e0caec",16,"Hardy 1921, p. 151: \\Lambda - \\lambda \\leq 2\\DELTA"],["hardy-course-of-pure-mathematics-1921/eq-6a87c80af3",16,"Hardy 1921, p. 152: |\\phi(n_{2}) - \\phi(n_{1})| < \\DELTA"],["form/420bcba9c9",5,"factor: -(-a + x)**2 + 1"],["shape/f122b8bcc6",6,"factor: -(-a + x)**N + 1"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/1",4,"Hardy 1921, Exercise XCVI (1)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/2",4,"Hardy 1921, Exercise XCVI (2)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/3",4,"Hardy 1921, Exercise XCVI (3)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/4",4,"Hardy 1921, Exercise XCVI (4)"],["hardy-course-of-pure-mathematics-1921/eq-f01d262e0d",16,"Hardy 1921, p. 153: |u_{n_{1}+1} + u_{n_{1}+2} + \\dots + u_{n_{2}}| < \\DELTA"],["hardy-course-of-pure-mathematics-1921/eq-7ea4a8d07b",16,"Hardy 1921, p. 152: \\phi(n_{1}) - \\DELTA < \\phi(n_{2}) < \\phi(n_{1}) + \\DELTA"],["form/97a5765ea6",5,"factor: 4*x**4 - (3*x - 1)**2"],["concept/unbounded-function",7,"unbounded function"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/5",4,"Hardy 1921, Exercise XCVI (5)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/6",4,"Hardy 1921, Exercise XCVI (6)"],["wentworth-first-steps-in-algebra-1894/ex-42/1",4,"Wentworth 1894, Exercise 42 (1)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/7",4,"Hardy 1921, Exercise XCVI (7)"],["form/08657ebb8f",5,"identity: 1/(3*a)"],["shape/02d31a76cb",6,"identity: N/a"],["wentworth-first-steps-in-algebra-1894/ex-42/2",4,"Wentworth 1894, Exercise 42 (2)"],["theorem/power-of-a-point",9,"power of a point","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-power-of-a-point"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/8",4,"Hardy 1921, Exercise XCVI (8)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/9",4,"Hardy 1921, Exercise XCVI (9)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/10",4,"Hardy 1921, Exercise XCVI (10)"],["hardy-course-of-pure-mathematics-1921/eq-b7dea93b92",16,"Hardy 1921, p. 153: \\rho(n) + i\\sigma(n)"],["boyden-first-book-in-algebra-1895/ex-12/5",4,"Boyden 1895, Exercise 12 (5)"],["hardy-course-of-pure-mathematics-1921/eq-50b8e2840a",16,"Hardy 1921, p. 153: \\lim\\phi(n) = l"],["hardy-course-of-pure-mathematics-1921/eq-2aa3eb113b",16,"Hardy 1921, p. 153: l = r + is"],["boyden-first-book-in-algebra-1895/ex-12/6",4,"Boyden 1895, Exercise 12 (6)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/11",4,"Hardy 1921, Exercise XCVI (11)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/12",4,"Hardy 1921, Exercise XCVI (12)"],["hardy-course-of-pure-mathematics-1921/ex-xcvi/13",4,"Hardy 1921, Exercise XCVI (13)"],["hardy-course-of-pure-mathematics-1921/eq-670b9c0fcf",16,"Hardy 1921, p. 153: s_{n} = u_{1} + u_{2} + \\dots + u_{n}"],["de-morgan-elementary-illustrations-calculus-1899/eq-9e1036605a",16,"De Morgan 1899, p. 33: x = a\\, \\frac{b^{2} ± \\sqrt{(a^{2} + b^{2})r^{2} - a^{2}b^{2}}}{a^{2} + b^{2}}"],["hardy-course-of-pure-mathematics-1921/ex-xcvii/1",4,"Hardy 1921, Exercise XCVII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xcvii/2",4,"Hardy 1921, Exercise XCVII (2)"],["form/a8f19d0794",5,"identity: 4*a/(5*b)"],["hardy-course-of-pure-mathematics-1921/ex-xcvii/3a",4,"Hardy 1921, Exercise XCVII (3a)"],["shape/7301cba403",6,"identity: N*a/b"],["wentworth-first-steps-in-algebra-1894/ex-42/3",4,"Wentworth 1894, Exercise 42 (3)"],["form/663243aa20",5,"identity: 3*a/(4*b**2)"],["dickson-theory-of-equations-1922/eq-cbdb6ee0b7",16,"Dickson 1922, p. 158: G(x_1,y_1) \\leqq G(x,y)"],["hardy-course-of-pure-mathematics-1921/ex-xcvii/3b",4,"Hardy 1921, Exercise XCVII (3b)"],["hardy-course-of-pure-mathematics-1921/ex-xcvii/3c",4,"Hardy 1921, Exercise XCVII (3c)"],["hardy-course-of-pure-mathematics-1921/eq-8c7c184fea",16,"Hardy 1921, p. 154: (v_{1} + v_{2} + \\dots + v_{n}) + i(w_{1} + w_{2} + \\dots + w_{n})"],["hardy-course-of-pure-mathematics-1921/x-3c0919c195",15,"Hardy 1921, p. 123: It should be observed that in this case \\phi(2k ..."],["hardy-course-of-pure-mathematics-1921/x-f78022741b",15,"Hardy 1921, p. 259: A formula such as this is called a formula ..."],["form/7259fa05e5",5,"evaluate: 3*a*b*c*d*e at a=5, b=2, c=0, x=1, y=3"],["hardy-course-of-pure-mathematics-1921/ex-xcvii/3d",4,"Hardy 1921, Exercise XCVII (3d)"],["hardy-course-of-pure-mathematics-1921/ex-xcvii/4a",4,"Hardy 1921, Exercise XCVII (4a)"],["hardy-course-of-pure-mathematics-1921/ex-xcvii/4b",4,"Hardy 1921, Exercise XCVII (4b)"],["hardy-course-of-pure-mathematics-1921/x-c1b58b7453",15,"Hardy 1921, p. 259: Put x + \\frac{1}{2}p = t, q - \\frac{1}{4}p^{2} ..."],["wentworth-first-steps-in-algebra-1894/ex-40/3",4,"Wentworth 1894, Exercise 40 (3)"],["hardy-course-of-pure-mathematics-1921/ex-xcvii/4c",4,"Hardy 1921, Exercise XCVII (4c)"],["wentworth-first-steps-in-algebra-1894/ex-40/8",4,"Wentworth 1894, Exercise 40 (8)"],["planck-treatise-on-thermodynamics-1903/eq-28059cbaac",16,"Planck 1903, p. 110: d(U - \\theta\\Phi) \\leq W"],["hardy-course-of-pure-mathematics-1921/ex-xcviii/1",4,"Hardy 1921, Exercise XCVIII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xcviii/2",4,"Hardy 1921, Exercise XCVIII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xcviii/3",4,"Hardy 1921, Exercise XCVIII (3)"],["de-morgan-elementary-illustrations-calculus-1899/eq-b5508a74e5",16,"De Morgan 1899, p. 4: \\dfrac{x + a}{x} = 1 + \\dfrac{a}{x}"],["de-morgan-elementary-illustrations-calculus-1899/eq-0eb46ee9ff",16,"De Morgan 1899, p. 4: \\dfrac{x + m + a}{x + m} = 1 + \\dfrac{a}{x + m}"],["hardy-course-of-pure-mathematics-1921/ex-xcviii/4",4,"Hardy 1921, Exercise XCVIII (4)"],["hardy-course-of-pure-mathematics-1921/ex-xcviii/5",4,"Hardy 1921, Exercise XCVIII (5)"],["hardy-course-of-pure-mathematics-1921/ex-xcviii/6",4,"Hardy 1921, Exercise XCVIII (6)"],["de-morgan-elementary-illustrations-calculus-1899/eq-3b422fe3af",16,"De Morgan 1899, p. 33: y = b\\, \\frac{a^{2} \\mp \\sqrt{(a^{2} + b^{2})r^{2} - a^{2}b^{2}}}{a^{2} + b^{2}}"],["de-morgan-elementary-illustrations-calculus-1899/eq-8cdd831823",16,"De Morgan 1899, p. 33: (a^{2} + b^{2})r^{2} > a^{2}b^{2}"],["form/bb575770d3",5,"hcf: (a**2 - 3*a*b + 2*b**2, a**2 - 2*a*b + b**2)"],["hardy-course-of-pure-mathematics-1921/ex-xi/1",4,"Hardy 1921, Exercise XI (1)"],["hardy-course-of-pure-mathematics-1921/ex-xi/2",4,"Hardy 1921, Exercise XI (2)"],["hardy-course-of-pure-mathematics-1921/ex-xi/3",4,"Hardy 1921, Exercise XI (3)"],["cap/cas.complete_square",17,"cas.complete_square"],["hardy-course-of-pure-mathematics-1921/ex-xi/4",4,"Hardy 1921, Exercise XI (4)"],["person/pythagoras",1,"Pythagoras","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-pythagoras"],["de-morgan-elementary-illustrations-calculus-1899/eq-3da072c933",16,"De Morgan 1899, p. 34: (a^{2} + b^{2})r^{2} = a^{2}b^{2}"],["de-morgan-elementary-illustrations-calculus-1899/eq-afa4a026ac",16,"De Morgan 1899, p. 34: (a^{2} + b^{2})r^{2} < a^{2}b^{2}"],["form/1ae8a147ae",5,"hcf: (6*a**2 - 5*a + 1, 12*a**2 - 7*a + 1)"],["hardy-course-of-pure-mathematics-1921/ex-xii/1",4,"Hardy 1921, Exercise XII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xii/2",4,"Hardy 1921, Exercise XII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xii/3",4,"Hardy 1921, Exercise XII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xii/4",4,"Hardy 1921, Exercise XII (4)"],["hardy-course-of-pure-mathematics-1921/ex-xii/5",4,"Hardy 1921, Exercise XII (5)"],["de-morgan-elementary-illustrations-calculus-1899/x-d46087aa6d",15,"De Morgan 1899, p. 1: Differential and Integral Calculus, or, as it was formerly ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-a9e3feda6d",16,"De Morgan 1899, p. 35: (-x)^{2} = x^{2}"],["form/220f5cbca2",5,"lcm: (15*a**2*b**4, 6*a**3*b**3)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/1",4,"Hardy 1921, Exercise XIII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/2a",4,"Hardy 1921, Exercise XIII (2a)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/2b",4,"Hardy 1921, Exercise XIII (2b)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/2c",4,"Hardy 1921, Exercise XIII (2c)"],["whitehead-introduction-to-mathematics-1911/x-12f728cdb4",15,"Whitehead 1911, p. 8: The reason for this failure of the science to ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-3f3222b0c3",16,"De Morgan 1899, p. 31: OE = a"],["form/f3376b00f6",5,"lcm: (a**2 - 1, a**2 + a)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/2d",4,"Hardy 1921, Exercise XIII (2d)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/2e",4,"Hardy 1921, Exercise XIII (2e)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/3a",4,"Hardy 1921, Exercise XIII (3a)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/3b",4,"Hardy 1921, Exercise XIII (3b)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/3c",4,"Hardy 1921, Exercise XIII (3c)"],["dickson-theory-of-equations-1922/eq-b365c1e9da",16,"Dickson 1922, p. 158: |f(z)|^2 = G(x,y)"],["wentworth-first-steps-in-algebra-1894/ex-41/14",4,"Wentworth 1894, Exercise 41 (14)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/4a",4,"Hardy 1921, Exercise XIII (4a)"],["hardy-course-of-pure-mathematics-1921/ex-xiii/4b",4,"Hardy 1921, Exercise XIII (4b)"],["wentworth-first-steps-in-algebra-1894/ex-42/4",4,"Wentworth 1894, Exercise 42 (4)"],["whitehead-introduction-to-mathematics-1911/x-10051c0cf1",15,"Whitehead 1911, p. 9: Thus we write down as the leading characteristic of ..."],["form/32dfda954c",5,"lcm: ((a - b)**2, (a + b)**2, a**2 - b**2)"],["hardy-course-of-pure-mathematics-1921/ex-xiv/1",4,"Hardy 1921, Exercise XIV (1)"],["hardy-course-of-pure-mathematics-1921/ex-xiv/2",4,"Hardy 1921, Exercise XIV (2)"],["form/344afcf63a",5,"identity: a**2/(2*b*c)"],["hardy-course-of-pure-mathematics-1921/ex-xiv/3a",4,"Hardy 1921, Exercise XIV (3a)"],["form/7634e7f126",5,"solve: Eq(-a**2 + x**2 - 2*x, 0)"],["wentworth-first-steps-in-algebra-1894/ex-42/5",4,"Wentworth 1894, Exercise 42 (5)"],["form/e894bf81e4",5,"identity: a**3*b**3/(3*c**2)"],["wentworth-first-steps-in-algebra-1894/ex-42/6",4,"Wentworth 1894, Exercise 42 (6)"],["de-morgan-elementary-illustrations-calculus-1899/eq-ad87ee0ddb",16,"De Morgan 1899, p. 31: OF = b"],["shape/10cc008d57",6,"identity: N*a**N/(b*c)"],["shape/052ac14ef5",6,"solve: Eq(N*x - a**N + x**N, 0)"],["hardy-course-of-pure-mathematics-1921/ex-xiv/3b",4,"Hardy 1921, Exercise XIV (3b)"],["shape/a3321864ea",6,"factor: 0"],["form/89e41e1d0b",5,"solve: Eq(a**2 + x**2 - 2*x, 0)"],["shape/ecc5c0b3f3",6,"solve: Eq(N*x + a**N + x**N, 0)"],["hardy-course-of-pure-mathematics-1921/x-1b4fe40821",15,"Hardy 1921, p. 120: Euclid’s proof is as follows. If there are only ..."],["hardy-course-of-pure-mathematics-1921/x-785e65a464",15,"Hardy 1921, p. 125: That a theorem is ‘obvious’ in this sense does ..."],["boyden-first-book-in-algebra-1895/ex-31/5",4,"Boyden 1895, Exercise 31 (5)"],["form/e196e0ab4a",5,"factor: a**3 - a**2"],["hardy-course-of-pure-mathematics-1921/ex-xiv/3c",4,"Hardy 1921, Exercise XIV (3c)"],["whitehead-introduction-to-mathematics-1911/x-004ac1bf71",15,"Whitehead 1911, p. 8: The object of the following Chapters is not to ..."],["form/be21c1f71c",5,"solve: Eq(a**2 + x**4 - 2*x**2, 0)"],["shape/37c13cb60d",6,"solve: Eq(N*x**N + a**N + x**N, 0)"],["hardy-course-of-pure-mathematics-1921/ex-xiv/4a",4,"Hardy 1921, Exercise XIV (4a)"],["cap/other:eliminate_radicals",17,"other:eliminate_radicals"],["hardy-course-of-pure-mathematics-1921/ex-xiv/4b",4,"Hardy 1921, Exercise XIV (4b)"],["hardy-course-of-pure-mathematics-1921/ex-xiv/4c",4,"Hardy 1921, Exercise XIV (4c)"],["form/49a830d69f",5,"identity: 2*a/(3*b)"],["form/a90c671231",5,"identity: 3*b**2*c/(4*a**3)"],["hardy-course-of-pure-mathematics-1921/ex-xiv/4d",4,"Hardy 1921, Exercise XIV (4d)"],["hardy-course-of-pure-mathematics-1921/ex-xiv/5",4,"Hardy 1921, Exercise XIV (5)"],["hardy-course-of-pure-mathematics-1921/ex-xiv/6",4,"Hardy 1921, Exercise XIV (6)"],["whitehead-introduction-to-mathematics-1911/x-f3e2549e93",15,"Whitehead 1911, p. 8: But it is equally an error to confine attention ..."],["whitehead-introduction-to-mathematics-1911/x-501bc7829f",15,"Whitehead 1911, p. 11: In the eye of science, the fall of an ..."],["hardy-course-of-pure-mathematics-1921/ex-xiv/7",4,"Hardy 1921, Exercise XIV (7)"],["whitehead-introduction-to-mathematics-1911/x-a2a22dd6c1",15,"Whitehead 1911, p. 13: Pythagoras had a glimpse of it when he proclaimed ..."],["todhunter-spherical-trigonometry-1886/x-d2baa13134",15,"Todhunter 1886, scan 63: If \\sin b \\sin A be greater than \\sin ..."],["form/8080493388",5,"identity: (-a**2 + x**2 + 2)/(-a + x)"],["hardy-course-of-pure-mathematics-1921/ex-xix",3,"Hardy 1921, Exercise XIX"],["hardy-course-of-pure-mathematics-1921/ex-xix/1",4,"Hardy 1921, Exercise XIX (1)"],["hardy-course-of-pure-mathematics-1921/ex-xix/2",4,"Hardy 1921, Exercise XIX (2)"],["hardy-course-of-pure-mathematics-1921/ex-xix/3",4,"Hardy 1921, Exercise XIX (3)"],["hardy-course-of-pure-mathematics-1921/ex-xix/4",4,"Hardy 1921, Exercise XIX (4)"],["todhunter-spherical-trigonometry-1886/ch-great-and-small-circles",2,"Todhunter 1886, Great and Small Circles","../books/todhunter-spherical-trigonometry-1886/ch/ch-great-and-small-circles/index.html"],["todhunter-spherical-trigonometry-1886/eq-e1147ea536",16,"Todhunter 1886, scan 11: CD=\\surd(OD^2-OC^2)"],["theorem/concurrence-of-altitudes",9,"concurrence of altitudes","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-concurrence-of-altitudes"],["hardy-course-of-pure-mathematics-1921/ex-xci",3,"Hardy 1921, Exercise XCI"],["hardy-course-of-pure-mathematics-1921/ex-xix/5",4,"Hardy 1921, Exercise XIX (5)"],["hardy-course-of-pure-mathematics-1921/ex-xix/6",4,"Hardy 1921, Exercise XIX (6)"],["hardy-course-of-pure-mathematics-1921/ex-xix/7",4,"Hardy 1921, Exercise XIX (7)"],["todhunter-spherical-trigonometry-1886/eq-f86b7866b7",16,"Todhunter 1886, scan 12: PD=\\surd(PC^2+CD^2)"],["wentworth-plane-geometry-1899/eq-ca50a6b923",16,"Wentworth 1899, scan 144: a:b = c:d"],["form/c89be3f99e",5,"identity: 2*a*x/(-a + x) - a + x"],["hardy-course-of-pure-mathematics-1921/ex-xix/8",4,"Hardy 1921, Exercise XIX (8)"],["hardy-course-of-pure-mathematics-1921/ex-xix/9",4,"Hardy 1921, Exercise XIX (9)"],["todhunter-spherical-trigonometry-1886/eq-543ae7f36f",16,"Todhunter 1886, scan 14: AOB = AOM - BOM = BON - BOM = MON"],["method/finding-the-angular-radius-of-the-circumscribed-circle-of-a-spherical-triangle",8,"finding the angular radius of the circumscribed circle of a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-finding-the-angular-radius-of-the-circumscribed-circle-of-a-spherical-triangle"],["form/bf7e61ea5f",5,"identity: -2*x + (x - 3)/(x - 2) + 1"],["hardy-course-of-pure-mathematics-1921/ex-xix/10",4,"Hardy 1921, Exercise XIX (10)"],["todhunter-spherical-trigonometry-1886/eq-46c72377a0",16,"Todhunter 1886, scan 16: \\frac{\\operatorname{arc} ab} {\\operatorname{radius} Ca}=\\frac{\\operatorname{arc} AB} {\\operatorname{radius} OA}"],["wentworth-plane-geometry-1899/eq-4762affdaf",16,"Wentworth 1899, scan 144: a:b = b:c = c:d = d:e"],["form/5d22efaa78",5,"identity: x/(-a + x)"],["hardy-course-of-pure-mathematics-1921/ex-xl/1",4,"Hardy 1921, Exercise XL (1)"],["hardy-course-of-pure-mathematics-1921/ex-xl/2",4,"Hardy 1921, Exercise XL (2)"],["todhunter-spherical-trigonometry-1886/eq-a2eba91f59",16,"Todhunter 1886, scan 16: \\frac{\\operatorname{arc} ab}{\\operatorname{arc} AB}=\\frac{Ca}{OA} =\\frac{Ca}{Oa}=\\sin POa"],["dickson-theory-of-equations-1922/eq-802dd70860",16,"Dickson 1922, p. 158: |f(z_1)|\\leqq |f(z)|"],["hardy-course-of-pure-mathematics-1921/ex-xli/1",4,"Hardy 1921, Exercise XLI (1)"],["hardy-course-of-pure-mathematics-1921/ex-xli/2",4,"Hardy 1921, Exercise XLI (2)"],["whitehead-introduction-to-mathematics-1911/ch-ii",2,"Whitehead 1911, ch. II: Variables","../books/whitehead-introduction-to-mathematics-1911/ch/ch-ii/index.html"],["whitehead-introduction-to-mathematics-1911/eq-47a7093dac",16,"Whitehead 1911, p. 16: x + 2 = 2 + x"],["whitehead-introduction-to-mathematics-1911/eq-4918b8bdf4",16,"Whitehead 1911, p. 15: x + y = y + x"],["whitehead-introduction-to-mathematics-1911/eq-f9ea264b61",16,"Whitehead 1911, p. 16: x + 2 = 3"],["whitehead-introduction-to-mathematics-1911/eq-30eea3d4c4",16,"Whitehead 1911, p. 16: x + 2 > 3"],["whitehead-introduction-to-mathematics-1911/eq-ec53b9efae",16,"Whitehead 1911, p. 15: y > x"],["dickson-theory-of-equations-1922/eq-c13a38941d",16,"Dickson 1922, p. 158: |f(z_1)|\\leqq |f(z')| < P"],["dickson-theory-of-equations-1922/eq-7ae83c12ce",16,"Dickson 1922, p. 158: |f(z)| < |f(z_1)|"],["form/e9e1f138e1",5,"identity: 1/(x**2 - 5*x + 6)"],["shape/30e24a5eb2",6,"identity: 1/(N*x + N + x**N)"],["hardy-course-of-pure-mathematics-1921/ex-xli/3",4,"Hardy 1921, Exercise XLI (3)"],["hardy-course-of-pure-mathematics-1921/ex-xli/4",4,"Hardy 1921, Exercise XLI (4)"],["hardy-course-of-pure-mathematics-1921/ex-xli/5a",4,"Hardy 1921, Exercise XLI (5a)"],["form/901cd6eb98",5,"solve: Eq(x**4 + 3*x**3 - 3*x**2 - 11*x - 6, 0)"],["whitehead-introduction-to-mathematics-1911/eq-0be5ba2e17",16,"Whitehead 1911, p. 19: 6x + 6y = 6"],["form/a4dcb77d94",5,"identity: 4*x/5 + 2/5"],["shape/7d7004a93c",6,"identity: N*x + N"],["form/faaf06f714",5,"identity: 13*x/12 + 1/4"],["hardy-course-of-pure-mathematics-1921/ex-xli/5b",4,"Hardy 1921, Exercise XLI (5b)"],["form/ac7b8f2d0b",5,"solve: Eq(x**6 + 2*x**5 - 8*x**4 - 14*x**3 + 11*x**2 + 28*x + 12, 0)"],["shape/6267cebbfb",6,"solve: Eq(N*x + 4*N*x**N + N + x**N, 0)"],["hardy-course-of-pure-mathematics-1921/ex-xli/6",4,"Hardy 1921, Exercise XLI (6)"],["hardy-course-of-pure-mathematics-1921/ex-xli/7",4,"Hardy 1921, Exercise XLI (7)"],["whitehead-introduction-to-mathematics-1911/eq-bc66d0e9ba",16,"Whitehead 1911, p. 19: y^{2} = x"],["whitehead-introduction-to-mathematics-1911/eq-44c574dd98",16,"Whitehead 1911, p. 19: x + y > 1"],["whitehead-introduction-to-mathematics-1911/eq-f48010edab",16,"Whitehead 1911, p. 22: pv = 1"],["theorem/power-difference-inequality",9,"power difference inequality","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-power-difference-inequality"],["form/4a33b8247d",5,"identity: 11*x/15 - 97/30"],["hardy-course-of-pure-mathematics-1921/ex-xli/8",4,"Hardy 1921, Exercise XLI (8)"],["hardy-course-of-pure-mathematics-1921/ex-xli/9",4,"Hardy 1921, Exercise XLI (9)"],["hardy-course-of-pure-mathematics-1921/ex-xli/10",4,"Hardy 1921, Exercise XLI (10)"],["todhunter-spherical-trigonometry-1886/ch-spherical-triangles",2,"Todhunter 1886, Spherical Triangles","../books/todhunter-spherical-trigonometry-1886/ch/ch-spherical-triangles/index.html"],["todhunter-spherical-trigonometry-1886/eq-c8e5a5b19f",16,"Todhunter 1886, scan 18: \\dfrac{\\operatorname{arc} AB} {\\operatorname{radius} OA}"],["theorem/limit-of-n-nth-root-of-x-1",9,"limit of n(nth root of x - 1)","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-of-n-nth-root-of-x-1"],["hardy-course-of-pure-mathematics-1921/ex-xlii",3,"Hardy 1921, Exercise XLII"],["hardy-course-of-pure-mathematics-1921/ex-xlii/1",4,"Hardy 1921, Exercise XLII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xlii/2",4,"Hardy 1921, Exercise XLII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xlii/3",4,"Hardy 1921, Exercise XLII (3)"],["todhunter-spherical-trigonometry-1886/eq-14d3380c7e",16,"Todhunter 1886, scan 18: C = 90^\\circ"],["todhunter-spherical-trigonometry-1886/eq-b540f47156",16,"Todhunter 1886, scan 18: C = \\dfrac{\\pi}{2}"],["wentworth-plane-geometry-1899/eq-8cb65d4ce0",16,"Wentworth 1899, scan 148: a+c+e+g : b+d+f+h = a:b"],["hardy-course-of-pure-mathematics-1921/ex-xlii/4",4,"Hardy 1921, Exercise XLII (4)"],["theorem/cauchy-s-condensation-test",9,"Cauchy's condensation test","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-cauchy-s-condensation-test"],["wentworth-plane-geometry-1899/eq-f857efa993",16,"Wentworth 1899, scan 152: \\dfrac{EB}{AE} = \\dfrac{FC}{AF}"],["wentworth-plane-geometry-1899/eq-e74f8c6099",16,"Wentworth 1899, scan 159: \\dfrac{AB}{A'B'}"],["wentworth-plane-geometry-1899/eq-aa03064f98",16,"Wentworth 1899, scan 157: AB:A'B' = BC:B'C' = CD:C'D'"],["boyden-first-book-in-algebra-1895/ex-12/7",4,"Boyden 1895, Exercise 12 (7)"],["hardy-course-of-pure-mathematics-1921/ex-xliii",3,"Hardy 1921, Exercise XLIII"],["hardy-course-of-pure-mathematics-1921/ex-xliii/1a",4,"Hardy 1921, Exercise XLIII (1a)"],["form/5c63e7944f",5,"differentiate: sqrt((x + 1)/(-x + 1))"],["shape/8d6ee613cf",6,"differentiate: ((x + 1)/(-x + 1))**N"],["hardy-course-of-pure-mathematics-1921/ex-xliii/1b",4,"Hardy 1921, Exercise XLIII (1b)"],["form/bf42b378a8",5,"differentiate: sqrt((a*x + b)/(c*x + d))"],["shape/aa87451dca",6,"differentiate: ((a*x + b)/(c*x + d))**N"],["hardy-course-of-pure-mathematics-1921/ex-xliii/1c",4,"Hardy 1921, Exercise XLIII (1c)"],["form/aa08f88f17",5,"differentiate: sqrt((d*x**2 + 2*e*x + f)/(a*x**2 + 2*b*x + c))"],["concept/proof",7,"proof","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-proof"],["wentworth-first-steps-in-algebra-1894/ex-48/1",4,"Wentworth 1894, Exercise 48 (1)"],["shape/f81f8092dc",6,"differentiate: ((N*e*x + d*x**N + f)/(N*b*x + a*x**N + c))**N"],["hardy-course-of-pure-mathematics-1921/ex-xliii/1d",4,"Hardy 1921, Exercise XLIII (1d)"],["form/c41b3afc06",5,"differentiate: (a*x + b)**e*(c*x + d)**f"],["shape/c41b3afc06",6,"differentiate: (a*x + b)**e*(c*x + d)**f"],["hardy-course-of-pure-mathematics-1921/ex-xliii/2a",4,"Hardy 1921, Exercise XLIII (2a)"],["concept/theorem",7,"theorem","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-theorem"],["method/solution-of-a-problem",8,"solution of a problem","../books/wentworth-plane-geometry-1899/terms/index.html#t-method-solution-of-a-problem"],["concept/conclusion",7,"conclusion","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-conclusion"],["concept/postulate",7,"postulate","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-postulate"],["boyden-first-book-in-algebra-1895/ex-12/8",4,"Boyden 1895, Exercise 12 (8)"],["wentworth-first-steps-in-algebra-1894/ex-48/5",4,"Wentworth 1894, Exercise 48 (5)"],["hardy-course-of-pure-mathematics-1921/ex-xliii/2b",4,"Hardy 1921, Exercise XLIII (2b)"],["hardy-course-of-pure-mathematics-1921/ex-xliii/3i",4,"Hardy 1921, Exercise XLIII (3i)"],["form/b179afe45b",5,"differentiate: a*x**2 + b*g**2 + c + 2*d*g + 2*e*x + 2*f*g*x"],["shape/93ba989b39",6,"differentiate: N*d*g + N*e*x + N*f*g*x + a*x**N + b*g**N + c"],["hardy-course-of-pure-mathematics-1921/ex-xliii/3ii",4,"Hardy 1921, Exercise XLIII (3ii)"],["form/78483fbd6c",5,"differentiate: -5*a*b**2*x**2 + b**5 + x**5"],["shape/9cce036872",6,"differentiate: N*a*b**N*x**N + b**N + x**N"],["concept/contradictory-of-a-theorem",7,"contradictory of a theorem","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-contradictory-of-a-theorem"],["concept/opposite-of-a-theorem",7,"opposite of a theorem","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-opposite-of-a-theorem"],["wentworth-plane-geometry-1899/eq-3f9e0f71e4",16,"Wentworth 1899, scan 158: AB:A'B' = AC:A'C' = BC:B'C'"],["hardy-course-of-pure-mathematics-1921/ex-xliv/1",4,"Hardy 1921, Exercise XLIV (1)"],["form/3ebc15fe37",5,"differentiate: (sin(x)*cos(x)/sqrt(a**2*cos(x)**2 + b**2*sin(x)**2), -sqrt(x) + (x + 1)*atan(sqrt(x)), sqrt(a**2*cos(x)**2 + b**2*sin(x)**2), x*asin(x) + sqrt(-x**2 + 1), sin(x)**c, sin(x**c), sin(cos(x)), cos(x)**c, cos(x**c), cos(sin(x)))"],["shape/bf859c1875",6,"differentiate: ((a**N*cos(x)**N + b**N*sin(x)**N)**N*sin(x)*cos(x), -x**N + (x + 1)*atan(x**N), (a**N*cos(x)**N + b**N*sin(x)**N)**N, x*asin(x) + (-x**N + 1)**N, sin(x)**c, sin(x**c), sin(cos(x)), cos(x)**c, cos(x**c), cos(sin(x)))"],["hardy-course-of-pure-mathematics-1921/ex-xliv/2",4,"Hardy 1921, Exercise XLIV (2)"],["boyden-first-book-in-algebra-1895/ex-12/9",4,"Boyden 1895, Exercise 12 (9)"],["wentworth-plane-geometry-1899/eq-17fe6592bf",16,"Wentworth 1899, scan 162: \\dfrac {CO}{C'O'}=\\dfrac {AC}{A'C'}=\\dfrac {AB}{A'B'}=\\dfrac {BC}{B'C'}"],["hardy-course-of-pure-mathematics-1921/ex-xliv/3",4,"Hardy 1921, Exercise XLIV (3)"],["form/f8cf1ab982",5,"differentiate: (asin(2*x*sqrt(-x**2 + 1)), asin(sqrt(-x**2 + 1)), atan((a + x)/(-a*x + 1)))"],["shape/0378074d01",6,"differentiate: (asin(N*x*(-x**N + 1)**N), asin((-x**N + 1)**N), atan((a + x)/(-a*x + 1)))"],["hardy-course-of-pure-mathematics-1921/ex-xliv/4",4,"Hardy 1921, Exercise XLIV (4)"],["form/5b8238f09e",5,"differentiate: (-asin((a*x + b)/sqrt(-a*c + b**2))/sqrt(-a), atan((a*x + b)/sqrt(a*c - b**2))/sqrt(a*c - b**2))"],["shape/4ed5581516",6,"differentiate: (-(-a)**N*asin((-a*c + b**N)**N*(a*x + b)), (a*c - b**N)**N*atan((a*c - b**N)**N*(a*x + b)))"],["wentworth-plane-geometry-1899/x-3481c15b88",15,"Wentworth 1899, scan 10: the straight edge in every part will touch the ..."],["wentworth-plane-geometry-1899/x-4e1fd0630d",15,"Wentworth 1899, scan 11: A surface has only two dimensions, length and breadth."],["wentworth-plane-geometry-1899/x-c6692155d6",15,"Wentworth 1899, scan 11: A point has no dimension, but denotes position simply."],["hardy-course-of-pure-mathematics-1921/ex-xliv/5",4,"Hardy 1921, Exercise XLIV (5)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/6",4,"Hardy 1921, Exercise XLIV (6)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/7",4,"Hardy 1921, Exercise XLIV (7)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/8",4,"Hardy 1921, Exercise XLIV (8)"],["wentworth-plane-geometry-1899/x-ae86efbc5d",15,"Wentworth 1899, scan 11: It must be distinctly understood at the outset that ..."],["wentworth-plane-geometry-1899/x-e6d26d1d71",15,"Wentworth 1899, scan 12: If a surface moves in space, it generates, in ..."],["form/8c0e8d9868",5,"identity: (a**2 + 3*a + 2)*(a**2 + 7*a + 12)/((a**2 + 5*a + 6)*(a**2 + 9*a + 20))"],["hardy-course-of-pure-mathematics-1921/ex-xliv/9",4,"Hardy 1921, Exercise XLIV (9)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/10",4,"Hardy 1921, Exercise XLIV (10)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/11",4,"Hardy 1921, Exercise XLIV (11)"],["wentworth-plane-geometry-1899/x-7b8052a143",15,"Wentworth 1899, scan 14: Thus, Every horse is a quadruped is true, but ..."],["theorem/half-angle-formulae-for-a-spherical-triangle",9,"half-angle formulae for a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-half-angle-formulae-for-a-spherical-triangle"],["theorem/delambre-s-analogies",9,"Delambre's analogies","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-delambre-s-analogies"],["de-morgan-elementary-illustrations-calculus-1899/ch-limiting-ratios-of-magnitudes-that-increase-without-limit",2,"De Morgan 1899, Limiting Ratios of Magnitudes that Increase Without Limit","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-limiting-ratios-of-magnitudes-that-increase-without-limit/index.html"],["hardy-course-of-pure-mathematics-1921/ex-xliv/12",4,"Hardy 1921, Exercise XLIV (12)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/13",4,"Hardy 1921, Exercise XLIV (13)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/14",4,"Hardy 1921, Exercise XLIV (14)"],["form/d71933f396",5,"differentiate: (sin(x)/x, x*cos(x))"],["shape/d71933f396",6,"differentiate: (sin(x)/x, x*cos(x))"],["wentworth-plane-geometry-1899/eq-54d88bc220",16,"Wentworth 1899, scan 18: AB = AC+CB"],["wentworth-plane-geometry-1899/eq-0befb31442",16,"Wentworth 1899, scan 18: AC = AB-CB"],["form/00762fe2a0",5,"evaluate: 9*a - 2*b*c at a=5, b=4, c=3"],["boyden-first-book-in-algebra-1895/ex-56",3,"Boyden 1895, Exercise 56"],["boyden-first-book-in-algebra-1895/ex-55",3,"Boyden 1895, Exercise 55"],["hardy-course-of-pure-mathematics-1921/ex-xliv/15",4,"Hardy 1921, Exercise XLIV (15)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/16",4,"Hardy 1921, Exercise XLIV (16)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/17",4,"Hardy 1921, Exercise XLIV (17)"],["wentworth-plane-geometry-1899/eq-d76d0e6458",16,"Wentworth 1899, scan 18: AC = 2AB"],["concept/multiple",7,"multiple"],["wentworth-plane-geometry-1899/eq-abf75d4f4f",16,"Wentworth 1899, scan 18: AD = 3AB"],["boyden-first-book-in-algebra-1895/ex-31/6",4,"Boyden 1895, Exercise 31 (6)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/18",4,"Hardy 1921, Exercise XLIV (18)"],["hardy-course-of-pure-mathematics-1921/ex-xliv/19",4,"Hardy 1921, Exercise XLIV (19)"],["wentworth-plane-geometry-1899/eq-6eca0f5ed9",16,"Wentworth 1899, scan 18: AE = 4AB"],["wentworth-plane-geometry-1899/eq-b1a8924bfa",16,"Wentworth 1899, scan 24: \\angle ACB = \\angle DEF"],["boyden-first-book-in-algebra-1895/ex-12/10",4,"Boyden 1895, Exercise 12 (10)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/1",4,"Hardy 1921, Exercise XLIX (1)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/2a",4,"Hardy 1921, Exercise XLIX (2a)"],["form/a8ef372095",5,"integrate: 1/sqrt(a**2 - x**2)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/2b",4,"Hardy 1921, Exercise XLIX (2b)"],["wentworth-plane-geometry-1899/eq-0953b4e9b1",16,"Wentworth 1899, scan 28: CE = CK"],["wentworth-plane-geometry-1899/eq-e31f9e17b2",16,"Wentworth 1899, scan 28: \\angle FCE = \\angle FCK"],["shape/8f96cac6fe",6,"factor: N*a**N + a**N"],["form/8cfd198775",5,"factor: a**5 + 3*a**2"],["form/74d5812b49",5,"evaluate: 2*a + b + c at a=5, b=4, c=3"],["shape/6a77dd34d7",6,"evaluate: N*a + b + c"],["hardy-course-of-pure-mathematics-1921/ex-xlix/3",4,"Hardy 1921, Exercise XLIX (3)"],["form/b73367b1e8",5,"integrate: x*(a + x)**b"],["shape/b73367b1e8",6,"integrate: x*(a + x)**b"],["hardy-course-of-pure-mathematics-1921/ex-xlix/4a",4,"Hardy 1921, Exercise XLIX (4a)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/4b",4,"Hardy 1921, Exercise XLIX (4b)"],["wentworth-plane-geometry-1899/eq-f688e35c9c",16,"Wentworth 1899, scan 32: OE > OG"],["wentworth-plane-geometry-1899/eq-c2c8b51f73",16,"Wentworth 1899, scan 38: \\angle BHK + \\angle HKD = \\text{a st.\\ }\\angle"],["hardy-course-of-pure-mathematics-1921/x-df6a9893e1",15,"Hardy 1921, p. 154: The reader will find no difficulty in proving such ..."],["form/f628be774c",5,"evaluate: -a + b + 2*c at a=5, b=4, c=3"],["hardy-course-of-pure-mathematics-1921/ex-xlix/5",4,"Hardy 1921, Exercise XLIX (5)"],["form/e60aa2dc73",5,"integrate: 1/sqrt((-a + x)*(b - x))"],["shape/56c7a79aee",6,"integrate: ((-a + x)*(b - x))**N"],["hardy-course-of-pure-mathematics-1921/ex-xlix/6a",4,"Hardy 1921, Exercise XLIX (6a)"],["form/17c40075ee",5,"integrate: sqrt((-a + x)*(b - x))"],["hardy-course-of-pure-mathematics-1921/ex-xlix/6b",4,"Hardy 1921, Exercise XLIX (6b)"],["form/ec86050004",5,"integrate: sqrt((b - x)/(-a + x))"],["shape/594ecc2892",6,"integrate: ((b - x)/(-a + x))**N"],["de-morgan-elementary-illustrations-calculus-1899/eq-1456e3d9a6",16,"De Morgan 1899, p. 6: \\dfrac{2}{x(x + 1)}"],["dickson-theory-of-equations-1922/eq-a0687933dd",16,"Dickson 1922, p. 156: f^{(n)}(a) = n!"],["hardy-course-of-pure-mathematics-1921/ex-xlix/7",4,"Hardy 1921, Exercise XLIX (7)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/8",4,"Hardy 1921, Exercise XLIX (8)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/9",4,"Hardy 1921, Exercise XLIX (9)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/10",4,"Hardy 1921, Exercise XLIX (10)"],["de-morgan-elementary-illustrations-calculus-1899/eq-344f2569cd",16,"De Morgan 1899, p. 6: \\dfrac{1}{x^{2}}"],["de-morgan-elementary-illustrations-calculus-1899/eq-2aef2fdd2d",16,"De Morgan 1899, p. 6: \\dfrac{M}{N} = \\dfrac{2x^{2}}{x(x + 1)}"],["form/b094ef3f2e",5,"identity: (a + x)**2/((-a + x)*(-a**2 + x**2))"],["shape/bcf6dbdcaa",6,"identity: (a + x)**N/((-a + x)*(-a**N + x**N))"],["hardy-course-of-pure-mathematics-1921/ex-xlix/11",4,"Hardy 1921, Exercise XLIX (11)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/12",4,"Hardy 1921, Exercise XLIX (12)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/13",4,"Hardy 1921, Exercise XLIX (13)"],["de-morgan-elementary-illustrations-calculus-1899/eq-7fe6d9b045",16,"De Morgan 1899, p. 6: \\dfrac{2x}{x + 1}"],["de-morgan-elementary-illustrations-calculus-1899/eq-4bb1604b75",16,"De Morgan 1899, p. 7: 1 - \\dfrac{1}{x + 1}"],["de-morgan-elementary-illustrations-calculus-1899/eq-a496cdeed7",16,"De Morgan 1899, p. 6: 1 + 2 + 3 + \\dots + x,\\quad\\text{or}\\quad \\frac{x(x + 1)}{2}"],["de-morgan-elementary-illustrations-calculus-1899/eq-ad8fd0ddd0",16,"De Morgan 1899, p. 36: y + dy = (x + dx)^{2}"],["form/8326529be9",5,"identity: x**2*(1/x - 2/x**2 + x**(-3))/(-x + 1)**2"],["hardy-course-of-pure-mathematics-1921/ex-xlix/14",4,"Hardy 1921, Exercise XLIX (14)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/15a",4,"Hardy 1921, Exercise XLIX (15a)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/15b",4,"Hardy 1921, Exercise XLIX (15b)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/16",4,"Hardy 1921, Exercise XLIX (16)"],["hardy-course-of-pure-mathematics-1921/x-dfed52a158",15,"Hardy 1921, p. 157: If \\phi(n) steadily increases, and \\psi(n) steadily decreases, as ..."],["form/9776f72f3f",5,"solve: Eq(x/2 - 1/2, x/3 + 1/3)"],["form/5550a96a26",5,"solve: Eq(3*x/4 - 1/4, 2*x/3 + 1/3)"],["hardy-course-of-pure-mathematics-1921/ex-xlix/17",4,"Hardy 1921, Exercise XLIX (17)"],["concept/doubly-even-square-board",7,"doubly even square 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Moon","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-r-moon"],["hardy-course-of-pure-mathematics-1921/ex-xlv/5",4,"Hardy 1921, Exercise XLV (5)"],["person/minding",1,"Minding","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-minding"],["theorem/two-great-circles-bisect-each-other",9,"two great circles bisect each other","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-two-great-circles-bisect-each-other"],["theorem/points-of-a-circle-on-a-sphere-are-equidistant-from-its-pole",9,"points of a circle on a sphere are equidistant from its pole","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-points-of-a-circle-on-a-sphere-are-equidistant-from-its-pole"],["theorem/spherical-angle-measured-by-arc-of-great-circle",9,"spherical angle measured by arc of great circle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-spherical-angle-measured-by-arc-of-great-circle"],["hardy-course-of-pure-mathematics-1921/x-c8e47abfa6",15,"Hardy 1921, p. 433: We can represent the values of z and Z ..."],["form/b23c56f651",5,"factor: a**2 - a*b**2"],["hardy-course-of-pure-mathematics-1921/ex-xlv/6",4,"Hardy 1921, Exercise XLV (6)"],["hardy-course-of-pure-mathematics-1921/ex-xlv/7",4,"Hardy 1921, Exercise XLV (7)"],["hardy-course-of-pure-mathematics-1921/ex-xlv/8",4,"Hardy 1921, Exercise XLV (8)"],["hardy-course-of-pure-mathematics-1921/ex-xlv/9",4,"Hardy 1921, Exercise XLV (9)"],["theorem/arc-of-a-small-circle-compared-with-arc-of-a-great-circle",9,"arc of a small circle compared with arc of a great circle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-arc-of-a-small-circle-compared-with-arc-of-a-great-circle"],["dickson-theory-of-equations-1922/eq-188881ec7f",16,"Dickson 1922, p. 155: z = x+iy"],["hardy-course-of-pure-mathematics-1921/ex-xlv/10",4,"Hardy 1921, Exercise XLV (10)"],["hardy-course-of-pure-mathematics-1921/ex-xlv/11",4,"Hardy 1921, Exercise XLV (11)"],["todhunter-spherical-trigonometry-1886/x-15efe6643b",15,"Todhunter 1886, scan 11: A sphere is a solid bounded by a surface ..."],["todhunter-spherical-trigonometry-1886/x-970cc3b75f",15,"Todhunter 1886, scan 12: The section of the surface of a sphere by ..."],["shape/a422859e72",6,"factor: -a*b**N + a**N"],["hardy-course-of-pure-mathematics-1921/ex-xlv/12",4,"Hardy 1921, Exercise XLV (12)"],["hardy-course-of-pure-mathematics-1921/ex-xlv/13",4,"Hardy 1921, Exercise XLV (13)"],["todhunter-spherical-trigonometry-1886/x-8f6fab5596",15,"Todhunter 1886, scan 12: When only one great circle can be drawn through ..."],["todhunter-spherical-trigonometry-1886/x-a793b4f83a",15,"Todhunter 1886, scan 13: Then PO is at right angles to the plane ..."],["todhunter-spherical-trigonometry-1886/x-7ecdd48a80",15,"Todhunter 1886, scan 13: Thus the distance of a pole of a circle ..."],["hardy-course-of-pure-mathematics-1921/x-9334469a23",15,"Hardy 1921, p. 433: Thus \\am Z denotes a one-valued and continuous function ..."],["form/6b196f39f2",5,"solve: Eq(-1 + (8*x + 7)/(5*x + 4), -2*x/(5*x + 1) + 1)"],["hardy-course-of-pure-mathematics-1921/ex-xlv/14",4,"Hardy 1921, Exercise XLV (14)"],["cap/other:determinant",17,"other:determinant"],["hardy-course-of-pure-mathematics-1921/ex-xlv/15",4,"Hardy 1921, Exercise XLV (15)"],["hardy-course-of-pure-mathematics-1921/ex-xlv/16",4,"Hardy 1921, Exercise XLV (16)"],["ball-mathematical-recreations-1905/ch-xiv",2,"Ball 1905, ch. XIV: Matter and Ether Theories","../books/ball-mathematical-recreations-1905/ch/ch-xiv/index.html"],["concept/electron",7,"electron","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-electron"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/1a",4,"Hardy 1921, Exercise XLVI (1a)"],["ball-mathematical-recreations-1905/x-2c63073b01",15,"Ball 1905, scan 212: The annulus may be divided into four closed circuits, ..."],["hardy-course-of-pure-mathematics-1921/x-64f2da1adc",15,"Hardy 1921, p. 133: It should be noticed that the limit may be ..."],["concept/ether",7,"ether","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-ether"],["hardy-course-of-pure-mathematics-1921/x-0a202f649d",15,"Hardy 1921, p. 131: Since -\\phi(n) always increases if \\phi(n) always decreases, it ..."],["concept/vortex-ring",7,"vortex ring","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-vortex-ring"],["wentworth-first-steps-in-algebra-1894/ex-53/6",4,"Wentworth 1894, Exercise 53 (6)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/1b",4,"Hardy 1921, Exercise XLVI (1b)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/2",4,"Hardy 1921, Exercise XLVI (2)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/3a",4,"Hardy 1921, Exercise XLVI (3a)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/3b",4,"Hardy 1921, Exercise XLVI (3b)"],["ball-mathematical-recreations-1905/x-efd4a32989",15,"Ball 1905, scan 213: Thus if the initial cell is on a, we ..."],["ball-mathematical-recreations-1905/x-84714008de",15,"Ball 1905, scan 213: By following these rules we always can connect the ..."],["form/4551b015e3",5,"solve: Eq(a*(-a + x), b*(-b + x))"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/3c",4,"Hardy 1921, Exercise XLVI (3c)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/4",4,"Hardy 1921, Exercise XLVI (4)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/5",4,"Hardy 1921, Exercise XLVI (5)"],["concept/radioactivity",7,"radioactivity","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-radioactivity"],["form/5a2fc9c23a",5,"solve: Eq((-a + x)*(-b + x), x*(c + x))"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/6",4,"Hardy 1921, Exercise XLVI (6)"],["law/law-of-gravity",10,"law of gravity","../books/ball-mathematical-recreations-1905/terms/index.html#t-law-law-of-gravity"],["ball-mathematical-recreations-1905/x-0039e0cab3",15,"Ball 1905, scan 213: It is convenient to take the cells in each ..."],["hardy-course-of-pure-mathematics-1921/x-c68f957677",15,"Hardy 1921, p. 142: The reader may be tempted to think that the ..."],["ball-mathematical-recreations-1905/x-c8800b3640",15,"Ball 1905, scan 214: and on the other hand is greater than 31,054144---since ..."],["hardy-course-of-pure-mathematics-1921/x-5282f70798",15,"Hardy 1921, p. 438: Show that the roots of f'(z) = 0 are ..."],["hardy-course-of-pure-mathematics-1921/ex-xlvi/7",4,"Hardy 1921, Exercise XLVI (7)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/8a",4,"Hardy 1921, Exercise XLVI (8a)"],["form/69638aa306",5,"extremum: (x - 1)**2*(x + 2)"],["shape/867d352941",6,"extremum: (N + x)*(x - 1)**N"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/8b",4,"Hardy 1921, Exercise XLVI (8b)"],["form/0651d2ad01",5,"extremum: x**3 - 3*x"],["ball-mathematical-recreations-1905/x-dd49f1f5a8",15,"Ball 1905, scan 213: It leads to eight forms, similar to that in ..."],["ball-mathematical-recreations-1905/x-dc328b26d5",15,"Ball 1905, scan 213: It is as yet impossible to say how many ..."],["form/a2a1bd316c",5,"solve: Eq(2*x/35, 2)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/8c",4,"Hardy 1921, Exercise XLVI (8c)"],["form/762bded3bc",5,"extremum: 2*x**3 - 3*x**2 - 36*x + 10"],["shape/767f10dd83",6,"extremum: N*x + 2*N*x**N + N"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/8d",4,"Hardy 1921, Exercise XLVI (8d)"],["form/9491dbda1a",5,"extremum: 4*x**3 - 18*x**2 + 27*x - 7"],["hardy-course-of-pure-mathematics-1921/x-fdec29db62",15,"Hardy 1921, p. 141: The reader should be warned that the words ‘divergent’ ..."],["ball-mathematical-recreations-1905/eq-2f7f1c04ff",16,"Ball 1905, scan 282: N=2^p - 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1)**2/(x + 1)**3"],["hardy-course-of-pure-mathematics-1921/x-9d3c843bbf",15,"Hardy 1921, p. 149: This number M is not exceeded by any member ..."],["concept/series-of-complex-terms",7,"series of complex terms","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-series-of-complex-terms"],["planck-treatise-on-thermodynamics-1903/x-e0d8002041",15,"Planck 1903, p. 47: He put the two communicating vessels, one filled with ..."],["ball-mathematical-recreations-1905/eq-181d323397",16,"Ball 1905, scan 292: (2^p-1)(2^p + 1)\\equiv 0"],["de-morgan-elementary-illustrations-calculus-1899/eq-87411b1cad",16,"De Morgan 1899, p. 108: \\phi(x, y) = 0"],["shape/c5c549f47d",6,"extremum: (x - 1)**N*(x + 1)**N"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/21a",4,"Hardy 1921, Exercise XLVI (21a)"],["wentworth-first-steps-in-algebra-1894/ex-5/2",4,"Wentworth 1894, Exercise 5 (2)"],["form/3a8409b602",5,"extremum: x*(x - 1)/(x**2 + 3*x + 3)"],["wentworth-first-steps-in-algebra-1894/ex-5/3",4,"Wentworth 1894, Exercise 5 (3)"],["shape/666cb13d1c",6,"extremum: x*(x - 1)/(N*x + N + x**N)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/21b",4,"Hardy 1921, Exercise XLVI (21b)"],["wentworth-first-steps-in-algebra-1894/ex-5/4",4,"Wentworth 1894, Exercise 5 (4)"],["form/a7f535b335",5,"extremum: x**4/((x - 3)**3*(x - 1))"],["de-morgan-elementary-illustrations-calculus-1899/eq-6f88fad200",16,"De Morgan 1899, p. 110: \\dfrac{du}{dx} = y - 1"],["shape/6a58da63d9",6,"extremum: x**N*(N + x)**N/(x - 1)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/21c",4,"Hardy 1921, Exercise XLVI (21c)"],["form/60d84ff68d",5,"extremum: (x - 1)**2*(3*x**2 - 2*x - 37)/((x + 5)**2*(3*x**2 - 14*x - 1))"],["concept/limit-inferior",7,"limit inferior","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-limit-inferior"],["planck-treatise-on-thermodynamics-1903/x-e2b31ca6d8",15,"Planck 1903, p. 48: In other words, the internal energy of a perfect ..."],["planck-treatise-on-thermodynamics-1903/x-0dadecce8b",15,"Planck 1903, p. 50: Thus, a gas may be compressed very slowly to ..."],["todhunter-spherical-trigonometry-1886/x-ffe7d5c760",15,"Todhunter 1886, scan 29: The sines of the angles of a spherical triangle ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-a9323a3005",16,"De Morgan 1899, p. 110: \\dfrac{du}{dy} = x"],["de-morgan-elementary-illustrations-calculus-1899/eq-dac4df72f3",16,"De Morgan 1899, p. 110: xy - x - 1 = 0"],["ball-mathematical-recreations-1905/eq-5bd61aa02c",16,"Ball 1905, scan 292: 2^p\\equiv1"],["form/194bf19703",5,"solve: Eq(2/15 + 1/x, 1/6)"],["shape/66af19d932",6,"extremum: (N + x)**N*(x - 1)**N*(N*x + N*x**N + N)/(N*x + N*x**N - 1)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/22",4,"Hardy 1921, Exercise XLVI (22)"],["form/fb79e9320c",5,"extremum: a*cos(x) + b*sin(x)"],["shape/fb79e9320c",6,"extremum: a*cos(x) + b*sin(x)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/23a",4,"Hardy 1921, Exercise XLVI (23a)"],["form/89972e8804",5,"extremum: a**2*cos(x)**2 + b**2*sin(x)**2"],["shape/bd404097db",6,"extremum: a**N*cos(x)**N + b**N*sin(x)**N"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/23b",4,"Hardy 1921, Exercise XLVI (23b)"],["whitehead-introduction-to-mathematics-1911/ch-iii",2,"Whitehead 1911, ch. III: Methods of Application","../books/whitehead-introduction-to-mathematics-1911/ch/ch-iii/index.html"],["whitehead-introduction-to-mathematics-1911/eq-558ca1186d",16,"Whitehead 1911, p. 25: 20y = x"],["ball-mathematical-recreations-1905/eq-33ce914910",16,"Ball 1905, scan 290: 2^{p+y} \\equiv z"],["form/63d2a70309",5,"extremum: a*cos(x)**2 + b*sin(x)**2 + 2*c*sin(x)*cos(x)"],["shape/9aab0e4614",6,"extremum: N*c*sin(x)*cos(x) + a*cos(x)**N + b*sin(x)**N"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/24",4,"Hardy 1921, Exercise XLVI (24)"],["form/74165da30b",5,"extremum: sin(a + x)/sin(b + x)"],["shape/74165da30b",6,"extremum: sin(a + x)/sin(b + x)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/25",4,"Hardy 1921, Exercise XLVI (25)"],["form/e6ad65d1ac",5,"extremum: sin(x)**2/(sin(a + x)*sin(b + x))"],["shape/eb7b495981",6,"extremum: sin(x)**N/(sin(a + x)*sin(b + x))"],["whitehead-introduction-to-mathematics-1911/eq-31b65eeaa1",16,"Whitehead 1911, p. 29: F = k\\dfrac{mM}{d^{2}}"],["whitehead-introduction-to-mathematics-1911/eq-1e679a11d7",16,"Whitehead 1911, p. 39: F = W - w"],["concept/buoyancy",7,"buoyancy"],["ball-mathematical-recreations-1905/eq-ab57689dd0",16,"Ball 1905, scan 290: 2^y (2^p - 1) \\equiv 0"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/26",4,"Hardy 1921, Exercise XLVI (26)"],["form/84007cca19",5,"extremum: a**2*sec(x)**2 + b**2*csc(x)**2"],["shape/8eda328f47",6,"extremum: a**N*sec(x)**N + b**N*csc(x)**N"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/27",4,"Hardy 1921, Exercise XLVI (27)"],["wentworth-first-steps-in-algebra-1894/ex-6/7",4,"Wentworth 1894, Exercise 6 (7)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/28",4,"Hardy 1921, Exercise XLVI (28)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/29a",4,"Hardy 1921, Exercise XLVI (29a)"],["whitehead-introduction-to-mathematics-1911/eq-6d143eb486",16,"Whitehead 1911, p. 39: w = W - F"],["de-morgan-elementary-illustrations-calculus-1899/eq-dcf7f0ca40",16,"De Morgan 1899, p. 37: y - dy = (x - dx)^{2}"],["form/4e5e0a44f6",5,"solve: Eq(6*x, 15*x/2 - 15)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/29b",4,"Hardy 1921, Exercise XLVI (29b)"],["whitehead-introduction-to-mathematics-1911/eq-d22e01c187",16,"Whitehead 1911, p. 39: \\frac{W}{w} = \\frac{W}{W - F}"],["todhunter-spherical-trigonometry-1886/x-24ff309928",15,"Todhunter 1886, scan 29: The radical on the right-hand side must be taken ..."],["todhunter-spherical-trigonometry-1886/x-b21c40ba20",15,"Todhunter 1886, scan 37: The last four formul are commonly, but improperly, called ..."],["ball-mathematical-recreations-1905/eq-48b26d669b",16,"Ball 1905, scan 286: 2^p = 1"],["concept/congruent-figures",7,"congruent figures","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-congruent-figures"],["ball-mathematical-recreations-1905/eq-05fce4282e",16,"Ball 1905, scan 291: 2^{u-v} \\equiv 1"],["boyden-first-book-in-algebra-1895/ex-31/8",4,"Boyden 1895, Exercise 31 (8)"],["hardy-course-of-pure-mathematics-1921/ex-lv",3,"Hardy 1921, Exercise LV"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/29c",4,"Hardy 1921, Exercise XLVI (29c)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/30",4,"Hardy 1921, Exercise XLVI (30)"],["concept/indeterminate-form",7,"indeterminate form","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-indeterminate-form"],["form/052e50a4ea",5,"factor: a**2 + a*b"],["form/4b5d07b158",5,"solve: (Eq(9*a/7, b + 60), Eq(5*b, 6*a))"],["hardy-course-of-pure-mathematics-1921/ch-vii",2,"Hardy 1921, ch. VII: ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS","../books/hardy-course-of-pure-mathematics-1921/ch/ch-vii/index.html"],["shape/14600e2ffa",6,"solve: (Eq(N*a, N + b), Eq(N*b, N*a))"],["ball-mathematical-recreations-1905/eq-c5755f26a3",16,"Ball 1905, scan 291: 2^u \\equiv 2^v"],["ball-mathematical-recreations-1905/eq-75e76307c1",16,"Ball 1905, scan 282: 2047 = 23 \\times 89"],["ball-mathematical-recreations-1905/eq-5aee7c8c4b",16,"Ball 1905, scan 282: 137438953471 = 223 \\times 616318177"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/31",4,"Hardy 1921, Exercise XLVI (31)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/32",4,"Hardy 1921, Exercise XLVI (32)"],["form/cd69da1b65",5,"extremum: x**b*(a - x)**c"],["shape/cd69da1b65",6,"extremum: x**b*(a - x)**c"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/33",4,"Hardy 1921, Exercise XLVI (33)"],["form/5767047331",5,"extremum: a*x + b*c"],["shape/5767047331",6,"extremum: a*x + b*c"],["dickson-theory-of-equations-1922/eq-7d302667ca",16,"Dickson 1922, p. 11: ax^2 + bx + c = 0 \\quad (a \\ne 0)"],["dickson-theory-of-equations-1922/eq-82980b7e36",16,"Dickson 1922, p. 11: (2ax + b)^2 = \\Delta"],["ball-mathematical-recreations-1905/eq-c030590cc2",16,"Ball 1905, scan 286: 8i \\pm 1"],["form/f2d51f8bb9",5,"solve: Eq(x, x/12 + 40)"],["hardy-course-of-pure-mathematics-1921/ex-xlvi/34",4,"Hardy 1921, Exercise XLVI (34)"],["form/1fbfac77e1",5,"extremum: Eq(a*sec(x) + b*sec(d), c)"],["shape/1fbfac77e1",6,"extremum: Eq(a*sec(x) + b*sec(d), c)"],["form/3514466e52",5,"solve: (Eq(a, (x - 6)*(x + 7)), Eq((x - 6)*(x + 7), x**2))"],["hardy-course-of-pure-mathematics-1921/eq-f751accd9b",16,"Hardy 1921, p. 154: \\lim\\phi(n + p) = l"],["hardy-course-of-pure-mathematics-1921/eq-5d9e596179",16,"Hardy 1921, p. 154: \\lim\\{\\phi(n) + \\psi(n)\\} = l + m"],["hardy-course-of-pure-mathematics-1921/ex-xlvii/1",4,"Hardy 1921, Exercise XLVII (1)"],["cap/other:geometric_interpretation",17,"other:geometric_interpretation"],["concept/order-of-greatness",7,"order of greatness","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-order-of-greatness"],["concept/one-valued-function",7,"one-valued function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-one-valued-function"],["form/a88079854b",5,"solve: (Eq(a, x + 10), Eq((a + 2)*(x + 2), a*x + 144))"],["hardy-course-of-pure-mathematics-1921/ex-xlvii/2",4,"Hardy 1921, Exercise XLVII (2)"],["shape/4f590a05eb",6,"solve: (Eq(a, N + x), Eq((N + a)*(N + x), N + a*x))"],["hardy-course-of-pure-mathematics-1921/eq-de0f42f7aa",16,"Hardy 1921, p. 154: \\lim k\\phi(n) = kl"],["hardy-course-of-pure-mathematics-1921/eq-f22be5bbc3",16,"Hardy 1921, p. 154: \\lim \\phi(n)\\psi(n) = lm"],["cap/other:verification",17,"other:verification"],["hardy-course-of-pure-mathematics-1921/ex-xlvii/3",4,"Hardy 1921, Exercise XLVII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xlvii/4",4,"Hardy 1921, Exercise XLVII (4)"],["wentworth-first-steps-in-algebra-1894/ex-6/8a",4,"Wentworth 1894, Exercise 6 (8a)"],["wentworth-first-steps-in-algebra-1894/ex-6/8b",4,"Wentworth 1894, Exercise 6 (8b)"],["form/6c57978d05",5,"evaluate: 400/3"],["form/11b422cc9c",5,"solve: Eq(57*x/50, 2280)"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/1",4,"Hardy 1921, Exercise XLVIII (1)"],["wentworth-first-steps-in-algebra-1894/ex-6/9",4,"Wentworth 1894, Exercise 6 (9)"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/2",4,"Hardy 1921, Exercise XLVIII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/3",4,"Hardy 1921, Exercise XLVIII (3)"],["dickson-theory-of-equations-1922/eq-959a6e3ba9",16,"Dickson 1922, p. 11: x_{1} = \\frac{-b + \\sqrt{\\Delta}}{2a}"],["form/011d862e76",5,"solve: Eq(1000*x + 250, 300)"],["de-morgan-elementary-illustrations-calculus-1899/eq-b31bfc3890",16,"De Morgan 1899, p. 37: dy = 2x\\, dx - (dx)^{2}"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/4",4,"Hardy 1921, Exercise XLVIII (4)"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/5",4,"Hardy 1921, Exercise XLVIII (5)"],["form/5273f917dd",5,"integrate: 1/((x - 1)**2*(x**2 + 1)**2)"],["shape/2d815cbc38",6,"integrate: (x - 1)**N*(x**N + 1)**N"],["hardy-course-of-pure-mathematics-1921/eq-34b58259b8",16,"Hardy 1921, p. 154: \\lim u_{n} = 0"],["theorem/limit-of-a-quotient",9,"limit of a quotient","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-of-a-quotient"],["hardy-course-of-pure-mathematics-1921/x-6cefa55de2",15,"Hardy 1921, p. 170: The equation (2) expresses the fact that if we ..."],["dickson-theory-of-equations-1922/eq-42f482698a",16,"Dickson 1922, p. 11: x_{2} = \\frac{-b - \\sqrt{\\Delta}}{2a}"],["form/cc4989be69",5,"solve: Eq(x/200, 100)"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/6a",4,"Hardy 1921, Exercise XLVIII (6a)"],["form/a4162f06de",5,"integrate: x/((-a + x)*(-b + x)*(-c + x))"],["shape/a4162f06de",6,"integrate: x/((-a + x)*(-b + x)*(-c + x))"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/6b",4,"Hardy 1921, Exercise XLVIII (6b)"],["form/6885361a15",5,"integrate: x/((-a + x)**2*(-b + x))"],["shape/0dac758dff",6,"integrate: x*(-a + x)**N/(-b + x)"],["dickson-theory-of-equations-1922/eq-65ac0615df",16,"Dickson 1922, p. 11: x_{1} + x_{2} = \\frac{-b}{a}"],["dickson-theory-of-equations-1922/eq-49606eed28",16,"Dickson 1922, p. 11: x_{1} x_{2} = \\frac{ c}{a}"],["form/facf8e0923",5,"solve: (Eq(-3*a + 2*x, 7), Eq(2*a + 5*x, 27))"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/6c",4,"Hardy 1921, Exercise XLVIII (6c)"],["form/855c84ce1e",5,"integrate: x/((-a + x)**2*(-b + x)**2)"],["shape/f720290fca",6,"integrate: x*(-a + x)**N*(-b + x)**N"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/6d",4,"Hardy 1921, Exercise XLVIII (6d)"],["form/779372f367",5,"integrate: x/(-a + x)**3"],["shape/5f1b8b9583",6,"integrate: x*(-a + x)**N"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/6e",4,"Hardy 1921, Exercise XLVIII (6e)"],["hardy-course-of-pure-mathematics-1921/eq-06338a424d",16,"Hardy 1921, p. 155: |\\phi(n)| = \\sqrtbr{\\{\\rho(n)\\}^{2} + \\{\\sigma(n)\\}^{2}}"],["boyden-first-book-in-algebra-1895/ex-12/13",4,"Boyden 1895, Exercise 12 (13)"],["form/a9df1eb011",5,"integrate: x/((a**2 + x**2)*(b**2 + x**2))"],["wentworth-first-steps-in-algebra-1894/ex-6/10",4,"Wentworth 1894, Exercise 6 (10)"],["shape/d013643c4c",6,"integrate: x/((a**N + x**N)*(b**N + x**N))"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/6f",4,"Hardy 1921, Exercise XLVIII (6f)"],["form/30cb2d9c48",5,"solve: Eq(x - 15, 5)"],["form/fc38f440d8",5,"integrate: x**2/((a**2 + x**2)*(b + x**2)**2)"],["shape/2549533719",6,"integrate: x**N*(b + x**N)**N/(a**N + x**N)"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/6g",4,"Hardy 1921, Exercise XLVIII (6g)"],["shape/b9aa726bc8",6,"solve: Eq(N + x, N)"],["form/acf093128f",5,"integrate: (-a**2 + x**2)/(x**2*(a**2 + x**2))"],["dickson-theory-of-equations-1922/eq-81959ac762",16,"Dickson 1922, p. 11: a(x - x_1)(x - x_2) \\equiv ax^2 - a(x_1 + x_2)x + ax_1 x_2 \\equiv ax^2 + bx + c"],["form/d62805ca38",5,"solve: (Eq(8*a + x, 17), Eq(-3*a + 7*x, 1))"],["shape/27fb4e3ec0",6,"integrate: x**N*(-a**N + x**N)/(a**N + x**N)"],["wentworth-first-steps-in-algebra-1894/ex-70/1",4,"Wentworth 1894, Exercise 70 (1)"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/6h",4,"Hardy 1921, Exercise XLVIII (6h)"],["form/13ca69be11",5,"solve: (Eq(a/2 + x/2, 20), Eq(-5*a + 5*x, 20))"],["form/e3b50aaa30",5,"integrate: (-a**2 + x**2)/(x*(a**2 + x**2)**2)"],["shape/2b0c0feeed",6,"integrate: (-a**N + x**N)*(a**N + x**N)**N/x"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/7a",4,"Hardy 1921, Exercise XLVIII (7a)"],["form/d8785fa681",5,"integrate: 1/(x**4 + 1)"],["dickson-theory-of-equations-1922/eq-d535ca0ab0",16,"Dickson 1922, p. 11: 0 = ax_1^2 + bx_1 + c"],["form/028533d33d",5,"solve: (Eq(31*a/44 - 17*x/44, 3), Eq(-2*a/7 + x/5 + 1, 0))"],["shape/7d1b8bcb6b",6,"solve: (Eq(N*a + N*x, N), Eq(N*a + N*x + 1, 0))"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/7b",4,"Hardy 1921, Exercise XLVIII (7b)"],["form/f217a9b947",5,"integrate: x**2/(x**4 + 1)"],["shape/17f75862f7",6,"integrate: x**N/(x**N + 1)"],["hardy-course-of-pure-mathematics-1921/ex-xlviii/7c",4,"Hardy 1921, Exercise XLVIII (7c)"],["form/c4270c5ff1",5,"integrate: 1/(x**4 + x**2 + 1)"],["shape/8012a0d8e7",6,"integrate: 1/(2*x**N + 1)"],["macfarlane-vector-analysis-quaternions-1906/eq-b011b13c41",16,"Macfarlane 1906: \\beta^\\theta = \\cos\\theta \\cdot \\beta^\\theta + \\sin\\theta \\cdot \\beta^\\frac{\\pi}{2}"],["dickson-theory-of-equations-1922/eq-3d686061ea",16,"Dickson 1922, p. 11: 0 = ax_2^2 + bx_2 + c"],["dickson-theory-of-equations-1922/eq-24c3dbbbf9",16,"Dickson 1922, p. 11: \\Delta = b^2 - 4ac"],["hardy-course-of-pure-mathematics-1921/ex-xv/1a",4,"Hardy 1921, Exercise XV (1a)"],["form/ddf937092e",5,"solve: (Eq(x - 10, 3*a + 30), Eq(x + 5, 4*a - 20))"],["hardy-course-of-pure-mathematics-1921/ex-xv/1b",4,"Hardy 1921, Exercise XV (1b)"],["hardy-course-of-pure-mathematics-1921/ex-xv/1c",4,"Hardy 1921, Exercise XV (1c)"],["wentworth-first-steps-in-algebra-1894/ex-70/6",4,"Wentworth 1894, Exercise 70 (6)"],["hardy-course-of-pure-mathematics-1921/ex-xv/2a",4,"Hardy 1921, Exercise XV (2a)"],["form/d50a15e03a",5,"solve: (Eq(2*a/3 + x/2, 55), Eq(a + x, 100))"],["hardy-course-of-pure-mathematics-1921/ex-xv/2b",4,"Hardy 1921, Exercise XV (2b)"],["shape/d5caa2a537",6,"solve: (Eq(N*a + N*x, N), Eq(a + x, N))"],["dickson-theory-of-equations-1922/eq-3dc51bfdb1",16,"Dickson 1922, p. 12: b^2 = 4ac"],["form/5ca5ca3f87",5,"solve: (Eq(a/2 + x/2, 20), Eq(-3*a + 3*x, 18))"],["hardy-course-of-pure-mathematics-1921/ex-xv/2c",4,"Hardy 1921, Exercise XV (2c)"],["hardy-course-of-pure-mathematics-1921/ex-xv/3",4,"Hardy 1921, Exercise XV (3)"],["hardy-course-of-pure-mathematics-1921/eq-d64b7d87c1",16,"Hardy 1921, p. 155: \\phi(n)\\psi(n) = \\rho\\rho' - \\sigma\\sigma' + i(\\rho\\sigma' + \\rho'\\sigma)"],["boyden-first-book-in-algebra-1895/ex-31/9",4,"Boyden 1895, Exercise 31 (9)"],["macfarlane-vector-analysis-quaternions-1906/eq-d6a46957c1",16,"Macfarlane 1906: R = r\\beta^\\theta A"],["hardy-course-of-pure-mathematics-1921/eq-753468e100",16,"Hardy 1921, p. 155: \\phi(n) = \\rho(n) + i\\sigma(n)"],["hardy-course-of-pure-mathematics-1921/x-465b066400",15,"Hardy 1921, p. 266: This expansion of f(a + h) is known as ..."],["macfarlane-vector-analysis-quaternions-1906/eq-d586ce79f7",16,"Macfarlane 1906: A = \\dfrac{1}{r}\\beta^{-\\theta}R"],["hardy-course-of-pure-mathematics-1921/ex-xv/4",4,"Hardy 1921, Exercise XV (4)"],["hardy-course-of-pure-mathematics-1921/ex-xv/5a",4,"Hardy 1921, Exercise XV (5a)"],["hardy-course-of-pure-mathematics-1921/ex-xv/5b",4,"Hardy 1921, Exercise XV (5b)"],["hardy-course-of-pure-mathematics-1921/ex-xv/5c",4,"Hardy 1921, Exercise XV (5c)"],["hardy-course-of-pure-mathematics-1921/ex-xv/5d",4,"Hardy 1921, Exercise XV (5d)"],["hardy-course-of-pure-mathematics-1921/x-a528198bc2",15,"Hardy 1921, p. 171: When we put x = 0 in \\phi(x) we ..."],["boyden-first-book-in-algebra-1895/ex-29/13",4,"Boyden 1895, Exercise 29 (13)"],["hardy-course-of-pure-mathematics-1921/ex-xv/6",4,"Hardy 1921, Exercise XV (6)"],["hardy-course-of-pure-mathematics-1921/ex-xv/7",4,"Hardy 1921, Exercise XV (7)"],["dickson-theory-of-equations-1922/eq-0828096f6e",16,"Dickson 1922, p. 12: f(x) \\equiv c_0 x^n + c_1 x^{n-1} + \\dotsb + c_{n-1} x + c_n"],["dickson-theory-of-equations-1922/eq-62542de319",16,"Dickson 1922, p. 12: f(x) \\equiv (x-c)q(x) + r"],["dickson-theory-of-equations-1922/eq-d92a6b31ec",16,"Dickson 1922, p. 12: f(c) = r"],["dickson-theory-of-equations-1922/eq-2f0fc55a5e",16,"Dickson 1922, p. 14: b_1 = a_1 + cb_0"],["concept/quotient-by-synthetic-division",7,"quotient by synthetic division"],["dickson-theory-of-equations-1922/eq-c492f00446",16,"Dickson 1922, p. 14: r = a_n + cb_{n-1}"],["hardy-course-of-pure-mathematics-1921/eq-86ab960aca",16,"Hardy 1921, p. 156: z^{n} = r^{n} (\\cos n\\theta + i\\sin n\\theta)"],["hardy-course-of-pure-mathematics-1921/ex-xv/8a",4,"Hardy 1921, Exercise XV (8a)"],["hardy-course-of-pure-mathematics-1921/ex-xv/8b",4,"Hardy 1921, Exercise XV (8b)"],["hardy-course-of-pure-mathematics-1921/ex-xv/8c",4,"Hardy 1921, Exercise XV (8c)"],["hardy-course-of-pure-mathematics-1921/ex-xv/8d",4,"Hardy 1921, Exercise XV (8d)"],["hardy-course-of-pure-mathematics-1921/eq-9c3a2cb72c",16,"Hardy 1921, p. 156: |z^{n}| = r^{n}"],["hardy-course-of-pure-mathematics-1921/eq-27342cdaf9",16,"Hardy 1921, p. 156: \\lim z^{n} = 0"],["hardy-course-of-pure-mathematics-1921/eq-cfa79400ab",16,"Hardy 1921, p. 156: l = zl"],["boyden-first-book-in-algebra-1895/ex-12/14a",4,"Boyden 1895, Exercise 12 (14a)"],["hardy-course-of-pure-mathematics-1921/ex-xv/8e",4,"Hardy 1921, Exercise XV (8e)"],["shape/1de44fb048",6,"solve: Eq(x, N*a + N)"],["hardy-course-of-pure-mathematics-1921/ex-xv/8f",4,"Hardy 1921, Exercise XV (8f)"],["wentworth-first-steps-in-algebra-1894/ex-70/7",4,"Wentworth 1894, Exercise 70 (7)"],["hardy-course-of-pure-mathematics-1921/ex-xv/8g",4,"Hardy 1921, Exercise XV (8g)"],["hardy-course-of-pure-mathematics-1921/ex-xv/9a",4,"Hardy 1921, Exercise XV (9a)"],["form/ddcd7c4f8e",5,"solve: (Eq(8*a + 3*x, 30), Eq(3*a + 6*x, 21))"],["hardy-course-of-pure-mathematics-1921/ex-xv/9b",4,"Hardy 1921, Exercise XV (9b)"],["shape/83e586217a",6,"solve: Eq(x**N, N*a**N)"],["shape/bac6f92f57",6,"factor: 3*N*a**N"],["form/279348c164",5,"solve: Eq(b*x, a)"],["form/9ba844129c",5,"evaluate: 16"],["hardy-course-of-pure-mathematics-1921/ex-xv/9c",4,"Hardy 1921, Exercise XV (9c)"],["hardy-course-of-pure-mathematics-1921/ex-xv/10a",4,"Hardy 1921, Exercise XV (10a)"],["hardy-course-of-pure-mathematics-1921/ex-xv/10b",4,"Hardy 1921, Exercise XV (10b)"],["hardy-course-of-pure-mathematics-1921/ex-xv/11a",4,"Hardy 1921, Exercise XV (11a)"],["hardy-course-of-pure-mathematics-1921/eq-9e00f0ef42",16,"Hardy 1921, p. 156: s_{n} = 1 + z + z^{2} + \\dots + z^{n-1} = (1 - z^{n})/(1 - z)"],["hardy-course-of-pure-mathematics-1921/x-478eeefecb",15,"Hardy 1921, p. 171: Thus y = x/x is a function which differs ..."],["dickson-theory-of-equations-1922/eq-6b6d9c3e1c",16,"Dickson 1922, p. 15: f(x) \\equiv (x - \\alpha_1)Q(x)"],["form/f1a9b1c6d0",5,"solve: Eq(-a + x, 8)"],["shape/90672e213f",6,"solve: Eq(-a + x, N)"],["form/66daa3974e",5,"solve: Eq(x + 20, 30)"],["hardy-course-of-pure-mathematics-1921/ex-xv/11b",4,"Hardy 1921, Exercise XV (11b)"],["hardy-course-of-pure-mathematics-1921/ex-xv/11c",4,"Hardy 1921, Exercise XV (11c)"],["hardy-course-of-pure-mathematics-1921/ex-xv/11d",4,"Hardy 1921, Exercise XV (11d)"],["hardy-course-of-pure-mathematics-1921/ex-xv/11e",4,"Hardy 1921, Exercise XV (11e)"],["hardy-course-of-pure-mathematics-1921/ex-xv/11f",4,"Hardy 1921, Exercise XV (11f)"],["dickson-theory-of-equations-1922/eq-68ed424c59",16,"Dickson 1922, p. 15: f(x) \\equiv c_0(x - \\alpha_1)(x - \\alpha_2) \\dotsm (x - \\alpha_n)"],["boyden-first-book-in-algebra-1895/ex-29/14",4,"Boyden 1895, Exercise 29 (14)"],["ball-mathematical-recreations-1905/ch-xi",2,"Ball 1905, ch. XI: Cryptographs and Ciphers","../books/ball-mathematical-recreations-1905/ch/ch-xi/index.html"],["hardy-course-of-pure-mathematics-1921/ex-xv/11g",4,"Hardy 1921, Exercise XV (11g)"],["hardy-course-of-pure-mathematics-1921/x-8f00730d44",15,"Hardy 1921, p. 172: It must not be imagined that this scale of ..."],["hardy-course-of-pure-mathematics-1921/ex-xv/11h",4,"Hardy 1921, Exercise XV (11h)"],["concept/leading-coefficient",7,"leading coefficient"],["hardy-course-of-pure-mathematics-1921/x-b976622ded",15,"Hardy 1921, p. 164: The function \\phi(x) = x - [x] oscillates between ..."],["wentworth-first-steps-in-algebra-1894/ex-70/8",4,"Wentworth 1894, Exercise 70 (8)"],["form/3f8ee599dd",5,"solve: (Eq(x - 10, 2*a), Eq(x - 10, a + 10))"],["shape/a6e6e362e7",6,"solve: (Eq(N + x, N*a), Eq(N + x, N + a))"],["hardy-course-of-pure-mathematics-1921/ex-xv/12",4,"Hardy 1921, Exercise XV (12)"],["theorem/limit-of-a-sum",9,"limit of a sum","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-of-a-sum"],["de-morgan-elementary-illustrations-calculus-1899/x-d350e40051",15,"De Morgan 1899, p. 2: We are not, therefore, entitled to say that because ..."],["theorem/limit-of-a-product",9,"limit of a product","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-of-a-product"],["concept/polar-form-of-a-complex-number",7,"polar form of a complex number","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-polar-form-of-a-complex-number"],["form/b06c52d951",5,"identity: 27*a**3"],["boyden-first-book-in-algebra-1895/ex-29/15",4,"Boyden 1895, Exercise 29 (15)"],["form/77bb82a095",5,"solve: Eq(x**3, 64*a**3)"],["cap/other:general_branch_formulae",17,"other:general_branch_formulae"],["de-morgan-elementary-illustrations-calculus-1899/x-c5eaa1c118",15,"De Morgan 1899, p. 2: Let the given ratio be that of the numbers ..."],["ball-mathematical-recreations-1905/x-e7715cc976",15,"Ball 1905, scan 215: “No man of science should think it a waste ..."],["concept/chemical-element",7,"chemical element","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-chemical-element"],["dickson-theory-of-equations-1922/eq-52bbe6e89b",16,"Dickson 1922, p. 16: f(x) \\equiv c_0(x-\\alpha_1)^{m_1} (x-\\alpha_2)^{m_2} \\dotsm (x-\\alpha_k)^{m_k}, \\quad m_1 + m_2 + \\dotsb + m_k = n"],["hardy-course-of-pure-mathematics-1921/ex-xv/13a",4,"Hardy 1921, Exercise XV (13a)"],["hardy-course-of-pure-mathematics-1921/ex-xv/13b",4,"Hardy 1921, Exercise XV (13b)"],["ball-mathematical-recreations-1905/x-fe4332394e",15,"Ball 1905, scan 219: I was badgered for two hours with arguments given ..."],["hardy-course-of-pure-mathematics-1921/ex-xv/13c",4,"Hardy 1921, Exercise XV (13c)"],["ball-mathematical-recreations-1905/x-e37dc8df0a",15,"Ball 1905, scan 228: and according to tradition, on one occasion the candidates ..."],["concept/number-of-roots",7,"number of roots"],["hardy-course-of-pure-mathematics-1921/ex-xv/13d",4,"Hardy 1921, Exercise XV (13d)"],["whitehead-introduction-to-mathematics-1911/ch-iv",2,"Whitehead 1911, ch. IV: Dynamics","../books/whitehead-introduction-to-mathematics-1911/ch/ch-iv/index.html"],["dickson-theory-of-equations-1922/eq-f1e0fc6d07",16,"Dickson 1922, p. 17: a_0 = b_0"],["dickson-theory-of-equations-1922/eq-d1f8cc0481",16,"Dickson 1922, p. 17: f(x) \\equiv c_0 x^n + c_1 x^{n-1} + \\dotsb + c_n = 0\\qquad (c_0 \\ne 0)"],["boyden-first-book-in-algebra-1895/ex-12/14b",4,"Boyden 1895, Exercise 12 (14b)"],["form/0b53e869e7",5,"solve: Eq(4*x**2 - 50, x**2 + 25)"],["ball-mathematical-recreations-1905/eq-0e9e3b5a0b",16,"Ball 1905, scan 337: 10^5"],["hardy-course-of-pure-mathematics-1921/ex-xv/14",4,"Hardy 1921, Exercise XV (14)"],["ball-mathematical-recreations-1905/eq-926aad6674",16,"Ball 1905, scan 338: 26^4"],["ball-mathematical-recreations-1905/eq-f7daf7d70b",16,"Ball 1905, scan 338: 36^3"],["hardy-course-of-pure-mathematics-1921/ex-xv/15",4,"Hardy 1921, Exercise XV (15)"],["person/bakhuis-roozeboom",1,"Bakhuis Roozeboom","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-bakhuis-roozeboom"],["ball-mathematical-recreations-1905/eq-6d3380a388",16,"Ball 1905, scan 337: 5n"],["hardy-course-of-pure-mathematics-1921/ex-xv/16",4,"Hardy 1921, Exercise XV (16)"],["form/a9a53fb09f",5,"solve: Eq((x - 6)*(x + 6), 28)"],["shape/668524416e",6,"solve: Eq((N + x)**2, N)"],["wentworth-first-steps-in-algebra-1894/ex-71/5",4,"Wentworth 1894, Exercise 71 (5)"],["hardy-course-of-pure-mathematics-1921/x-7609bf0dc0",15,"Hardy 1921, p. 161: The reader, if he desires to become expert in ..."],["hardy-course-of-pure-mathematics-1921/x-865bd1d385",15,"Hardy 1921, p. 156: If z^{n} \\to l then z^{n+1} \\to l, by ..."],["concept/thermal-equilibrium",7,"thermal equilibrium","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-thermal-equilibrium"],["hardy-course-of-pure-mathematics-1921/ex-xvi/1",4,"Hardy 1921, Exercise XVI (1)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/2",4,"Hardy 1921, Exercise XVI (2)"],["planck-treatise-on-thermodynamics-1903/x-65e22f9071",15,"Planck 1903, p. 180: The composition of all the phases is then completely ..."],["hardy-course-of-pure-mathematics-1921/ex-xvi/3",4,"Hardy 1921, Exercise XVI (3)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/4",4,"Hardy 1921, Exercise XVI (4)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/5a",4,"Hardy 1921, Exercise XVI (5a)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/5b",4,"Hardy 1921, Exercise XVI (5b)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/6a",4,"Hardy 1921, Exercise XVI (6a)"],["form/69358d1b66",5,"solve: Eq(x, 4*a - 8)"],["dickson-theory-of-equations-1922/eq-e1a8a45424",16,"Dickson 1922, p. 18: (x - \\alpha_1)(x - \\alpha_2) &\\equiv x^2 - (\\alpha_1 + \\alpha_2)x + \\alpha_1\\alpha_2"],["dickson-theory-of-equations-1922/eq-841c5f4bce",16,"Dickson 1922, p. 18: (x - \\alpha_1)(x - \\alpha_2) \\dotsm (x - \\alpha_n) \\equiv x^n - (\\alpha_1 + \\dotsb + \\alpha_n)x^{n-1} \\\\ + (\\alpha_1\\alp"],["form/8fd4273657",5,"solve: Eq(5*x**2 + 73, 198)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/6b",4,"Hardy 1921, Exercise XVI (6b)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/6c",4,"Hardy 1921, Exercise XVI (6c)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/6d",4,"Hardy 1921, Exercise XVI (6d)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/6e",4,"Hardy 1921, Exercise XVI (6e)"],["dickson-theory-of-equations-1922/eq-72621e8045",16,"Dickson 1922, p. 18: \\alpha_1 + \\alpha_2 + \\dotsb + \\alpha_n &= -c_1 / c_0"],["concept/sum-of-the-roots",7,"sum of the roots"],["dickson-theory-of-equations-1922/eq-5c36b0cdd4",16,"Dickson 1922, p. 18: \\alpha_1\\alpha_2 \\dotsm \\alpha_{n-1}\\alpha_n &= (-1)^n c_n / c_0"],["boyden-first-book-in-algebra-1895/ex-31/10",4,"Boyden 1895, Exercise 31 (10)"],["form/7ce80fc3ea",5,"factor: 14*a**4 - 7*a**3 + 7*a"],["shape/57c58e843d",6,"factor: N*a + 2*N*a**N"],["hardy-course-of-pure-mathematics-1921/ex-xvi/7",4,"Hardy 1921, Exercise XVI (7)"],["wentworth-first-steps-in-algebra-1894/ex-9/1",4,"Wentworth 1894, Exercise 9 (1)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/8",4,"Hardy 1921, Exercise XVI (8)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/9",4,"Hardy 1921, Exercise XVI (9)"],["shape/1b73f6dce0",6,"solve: Eq(N*x + N + x**N, 0)"],["hardy-course-of-pure-mathematics-1921/x-d9a95da2e7",15,"Hardy 1921, p. 155: If \\rho(n) and \\sigma(n) both converge to zero then ..."],["concept/product-of-roots",7,"product of roots"],["form/b771bbbda2",5,"solve: Eq(x**2 - 12*x + 27, 0)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/10",4,"Hardy 1921, Exercise XVI (10)"],["hardy-course-of-pure-mathematics-1921/ex-xvi/11",4,"Hardy 1921, Exercise XVI (11)"],["shape/17a89e8959",6,"solve: Eq(N*x + N*x**N + N, 0)"],["concept/oscillatory-discontinuity",7,"oscillatory discontinuity","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-oscillatory-discontinuity"],["form/c14b09ab38",5,"solve: Eq(9*x**2 - 24*x + 16, 0)"],["hardy-course-of-pure-mathematics-1921/ex-xvii/1",4,"Hardy 1921, Exercise XVII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xvii/2a",4,"Hardy 1921, Exercise XVII (2a)"],["wentworth-first-steps-in-algebra-1894/ex-7/7",4,"Wentworth 1894, Exercise 7 (7)"],["form/d9a19d8e8e",5,"solve: Eq(x**2 + 2*x - 3, 0)"],["hardy-course-of-pure-mathematics-1921/ex-xvii/2b",4,"Hardy 1921, Exercise XVII (2b)"],["form/fdbb7fd337",5,"solve: Eq(x**2 - 7*x + 4, 0)"],["concept/set-of-intervals",7,"set of intervals","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-set-of-intervals"],["hardy-course-of-pure-mathematics-1921/ex-xvii/2c",4,"Hardy 1921, Exercise XVII (2c)"],["wentworth-first-steps-in-algebra-1894/ex-7/8",4,"Wentworth 1894, Exercise 7 (8)"],["form/f946a8e0a4",5,"solve: Eq(3*x**2 + 2*x - 2, 0)"],["hardy-course-of-pure-mathematics-1921/ex-xvii/3",4,"Hardy 1921, Exercise XVII (3)"],["wentworth-first-steps-in-algebra-1894/ex-7/9",4,"Wentworth 1894, Exercise 7 (9)"],["hardy-course-of-pure-mathematics-1921/ex-xvii/4",4,"Hardy 1921, Exercise XVII (4)"],["form/aed17cb902",5,"evaluate: 40"],["form/57705d95c9",5,"solve: Eq((2*x - 1)**2 + 9, 12*x - 6)"],["shape/831204f15f",6,"solve: Eq(N + (N*x - 1)**N, N*x + N)"],["hardy-course-of-pure-mathematics-1921/ex-xvii/5a",4,"Hardy 1921, Exercise XVII (5a)"],["hardy-course-of-pure-mathematics-1921/ex-xvii/5b",4,"Hardy 1921, Exercise XVII (5b)"],["hardy-course-of-pure-mathematics-1921/ex-xvii/5c",4,"Hardy 1921, Exercise XVII (5c)"],["hardy-course-of-pure-mathematics-1921/ex-xvii/5d",4,"Hardy 1921, Exercise XVII (5d)"],["concept/dedekind-section",7,"Dedekind section","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-dedekind-section"],["boyden-first-book-in-algebra-1895/ex-31/11",4,"Boyden 1895, Exercise 31 (11)"],["hardy-course-of-pure-mathematics-1921/ex-xvii/6",4,"Hardy 1921, Exercise XVII (6)"],["form/2d55538029",5,"factor: 3*a**3 - a**2 + a"],["macfarlane-vector-analysis-quaternions-1906/eq-5096f7aaa5",16,"Macfarlane 1906: \\dfrac{1}{A}R = r\\beta^\\theta"],["macfarlane-vector-analysis-quaternions-1906/eq-eb1f9b55a7",16,"Macfarlane 1906: r = \\sqrt{p^2 + q^2}"],["shape/1480f8209e",6,"factor: N*a**N + a - a**N"],["form/f8d1309eb8",5,"solve: Eq((3*x - 3)/(x + 1) - (2*x + 2)/(x - 1), 5)"],["hardy-course-of-pure-mathematics-1921/ex-xviii/1",4,"Hardy 1921, Exercise XVIII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xviii/2",4,"Hardy 1921, Exercise XVIII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xviii/3",4,"Hardy 1921, Exercise XVIII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xviii/4a",4,"Hardy 1921, Exercise XVIII (4a)"],["planck-treatise-on-thermodynamics-1903/x-ffcdcd56e9",15,"Planck 1903, p. 54: In isothermal changes C is evidently = \\pm\\infty, because ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-fbcd5110c0",16,"De Morgan 1899, p. 37: \\dfrac{dy}{dx} = 2x - dx"],["hardy-course-of-pure-mathematics-1921/ex-xviii/4b",4,"Hardy 1921, Exercise XVIII (4b)"],["wentworth-first-steps-in-algebra-1894/ex-75/11",4,"Wentworth 1894, Exercise 75 (11)"],["planck-treatise-on-thermodynamics-1903/x-f0ed71f2a8",15,"Planck 1903, p. 2: Two bodies of equal temperature are, therefore, in thermal ..."],["planck-treatise-on-thermodynamics-1903/x-8f23357591",15,"Planck 1903, p. 2: If a body, A, be in thermal equilibrium with ..."],["hardy-course-of-pure-mathematics-1921/ex-xx/1a",4,"Hardy 1921, Exercise XX (1a)"],["cap/other:vector_algebra",17,"other:vector_algebra"],["wentworth-first-steps-in-algebra-1894/ex-75/12",4,"Wentworth 1894, Exercise 75 (12)"],["wentworth-first-steps-in-algebra-1894/ex-80/1",4,"Wentworth 1894, Exercise 80 (1)"],["cap/other:polynomial_cube_root",17,"other:polynomial_cube_root"],["form/db1f64eb7a",5,"solve: Eq(3*x, x + 8)"],["dickson-theory-of-equations-1922/eq-61b7702517",16,"Dickson 1922, p. 19: (x-a)^2 + b^2 \\equiv (x - a-bi)(x - a+bi)"],["hardy-course-of-pure-mathematics-1921/ex-xx/1b",4,"Hardy 1921, Exercise XX (1b)"],["planck-treatise-on-thermodynamics-1903/x-bbf798c0fd",15,"Planck 1903, p. 3: The definition of temperature remains arbitrary in cases where ..."],["planck-treatise-on-thermodynamics-1903/x-777a55ea24",15,"Planck 1903, p. 7: Coefficient of elasticity is the ratio of an infinitely ..."],["planck-treatise-on-thermodynamics-1903/x-a62ec68d6b",15,"Planck 1903, p. 13: For lower pressures (i.e. to the left of the ..."],["planck-treatise-on-thermodynamics-1903/x-d70c2f9ee5",15,"Planck 1903, p. 17: Above the critical temperature and critical pressure, condensation does ..."],["form/36cab76fea",5,"solve: (Eq(a, x + 2), Eq(a*x, 120))"],["hardy-course-of-pure-mathematics-1921/ex-xx/1c",4,"Hardy 1921, Exercise XX (1c)"],["wentworth-first-steps-in-algebra-1894/ex-76/1",4,"Wentworth 1894, Exercise 76 (1)"],["hardy-course-of-pure-mathematics-1921/ex-xx/1d",4,"Hardy 1921, Exercise XX (1d)"],["form/280b514003",5,"evaluate: 440"],["wentworth-first-steps-in-algebra-1894/ex-76/2",4,"Wentworth 1894, Exercise 76 (2)"],["form/baa7a52459",5,"solve: Eq(30/(x - 1) - 30/x, 1)"],["shape/d69e45f199",6,"solve: Eq(N/(x - 1) + N/x, 1)"],["hardy-course-of-pure-mathematics-1921/ex-xx/1e",4,"Hardy 1921, Exercise XX (1e)"],["planck-treatise-on-thermodynamics-1903/x-bb8953ede5",15,"Planck 1903, p. 18: The earlier fundamental distinction between liquids, vapours, and gases ..."],["planck-treatise-on-thermodynamics-1903/x-945f66236b",15,"Planck 1903, p. 20: Only for gases and vapours does Dalton’s law hold, ..."],["dickson-theory-of-equations-1922/eq-1a457adaa8",16,"Dickson 1922, p. 19: f(x) \\equiv Q(x)\\bigl\\{(x-a)^2 + b^2\\bigr\\} + rx + s"],["dickson-theory-of-equations-1922/eq-ed0dc2a88e",16,"Dickson 1922, p. 21: x \\geqq 1 + \\sqrt[k]{G / a_0}"],["theorem/upper-limit-to-the-real-roots-theorem-i",9,"upper limit to the real roots (Theorem I)"],["dickson-theory-of-equations-1922/eq-fcd915bb34",16,"Dickson 1922, p. 21: x^{n-k} + \\dotsb + x + 1 \\equiv \\frac{x^{n-k+1} - 1}{x - 1}"],["form/f8e8c8632b",5,"solve: Eq(-sqrt(x) + x, 30)"],["shape/3181125055",6,"solve: Eq(x - x**N, N)"],["hardy-course-of-pure-mathematics-1921/ex-xx/2",4,"Hardy 1921, Exercise XX (2)"],["hardy-course-of-pure-mathematics-1921/ex-xx/3",4,"Hardy 1921, Exercise XX (3)"],["form/5913704173",5,"evaluate: 201"],["hardy-course-of-pure-mathematics-1921/ex-xx/4",4,"Hardy 1921, Exercise XX (4)"],["wentworth-first-steps-in-algebra-1894/ex-76/3",4,"Wentworth 1894, Exercise 76 (3)"],["form/b6d2f9aa11",5,"evaluate: 25/6"],["wentworth-first-steps-in-algebra-1894/ex-79/1",4,"Wentworth 1894, Exercise 79 (1)"],["form/ebdd6e93ae",5,"evaluate: 75"],["form/c19622d1f6",5,"evaluate: 38"],["hardy-course-of-pure-mathematics-1921/ex-xx/5",4,"Hardy 1921, Exercise XX (5)"],["hardy-course-of-pure-mathematics-1921/ex-xx/6",4,"Hardy 1921, Exercise XX (6)"],["form/676859f08f",5,"solve: (Eq(a/2 + x/2, 13), Eq(3*a/2 + x, 67/2))"],["hardy-course-of-pure-mathematics-1921/ex-xx/7",4,"Hardy 1921, Exercise XX (7)"],["shape/eb13b78b71",6,"solve: (Eq(N*a + N*x, N), Eq(N*a + x, N))"],["wentworth-first-steps-in-algebra-1894/ex-76/14",4,"Wentworth 1894, Exercise 76 (14)"],["dickson-theory-of-equations-1922/eq-63afa8a92e",16,"Dickson 1922, p. 22: x^4 \\equiv (x-1) (x^3 + x^2 + x + 1) + 1"],["hardy-course-of-pure-mathematics-1921/ex-xxii",3,"Hardy 1921, Exercise XXII"],["hardy-course-of-pure-mathematics-1921/ex-xxii/1",4,"Hardy 1921, Exercise XXII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxii/2",4,"Hardy 1921, Exercise XXII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxii/3",4,"Hardy 1921, Exercise XXII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxii/4",4,"Hardy 1921, Exercise XXII (4)"],["hardy-course-of-pure-mathematics-1921/x-12a0a7d696",15,"Hardy 1921, p. 183: The proof just given is somewhat subtle and indirect, ..."],["hardy-course-of-pure-mathematics-1921/ex-xxii/5",4,"Hardy 1921, Exercise XXII (5)"],["hardy-course-of-pure-mathematics-1921/ex-xxii/6",4,"Hardy 1921, Exercise XXII (6)"],["hardy-course-of-pure-mathematics-1921/ex-xxii/7",4,"Hardy 1921, Exercise XXII (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxii/8",4,"Hardy 1921, Exercise XXII (8)"],["hardy-course-of-pure-mathematics-1921/x-7132fce577",15,"Hardy 1921, p. 164: It is equal to zero whenever x is an ..."],["form/822c7000b4",5,"evaluate: 300"],["form/a574c68925",5,"evaluate: 2550"],["hardy-course-of-pure-mathematics-1921/ex-xxii/9",4,"Hardy 1921, Exercise XXII (9)"],["form/d9ed6c251d",5,"evaluate: 12"],["form/2bb8377478",5,"solve: Eq(x**6 - 2*x**3 + 2, 0)"],["hardy-course-of-pure-mathematics-1921/ex-xxii/10",4,"Hardy 1921, Exercise XXII (10)"],["dickson-theory-of-equations-1922/eq-c463c3d0b2",16,"Dickson 1922, p. 22: x^2 \\equiv (x-1) (x+1) + 1"],["dickson-theory-of-equations-1922/eq-50dc27ba42",16,"Dickson 1922, p. 23: x \\geqq 1 + \\frac{-a_{k_i}}{\\sum a_m}"],["macfarlane-vector-analysis-quaternions-1906/eq-8e38079903",16,"Macfarlane 1906: \\theta = \\tan^{-1} \\frac{p}{q}"],["boyden-first-book-in-algebra-1895/x-d04db172ac",15,"Boyden 1895: A man travels a miles north, then x miles ..."],["hardy-course-of-pure-mathematics-1921/ex-xxiii/1",4,"Hardy 1921, Exercise XXIII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxiii/2",4,"Hardy 1921, Exercise XXIII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxiii/3",4,"Hardy 1921, Exercise XXIII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxiii/4",4,"Hardy 1921, Exercise XXIII (4)"],["hardy-course-of-pure-mathematics-1921/ex-xxiii/5",4,"Hardy 1921, Exercise XXIII (5)"],["hardy-course-of-pure-mathematics-1921/ex-xxiii/6",4,"Hardy 1921, Exercise XXIII (6)"],["theorem/upper-limit-to-the-roots-theorem-ii",9,"upper limit to the roots (Theorem II)"],["dickson-theory-of-equations-1922/eq-3141860cb9",16,"Dickson 1922, p. 23: x \\geqq 1 + \\frac{-a_0}{\\sum a_m}"],["form/50ae212f95",5,"evaluate: 243"],["boyden-first-book-in-algebra-1895/ch-subtraction",2,"Boyden 1895, SUBTRACTION","../books/boyden-first-book-in-algebra-1895/ch/ch-subtraction/index.html"],["form/8220c54065",5,"evaluate: 192"],["hardy-course-of-pure-mathematics-1921/ex-xxiii/7",4,"Hardy 1921, Exercise XXIII (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxiii/8",4,"Hardy 1921, Exercise XXIII (8)"],["form/d8d8370cb8",5,"evaluate: 18"],["hardy-course-of-pure-mathematics-1921/ex-xxiii/9",4,"Hardy 1921, Exercise XXIII (9)"],["wentworth-first-steps-in-algebra-1894/ex-79/2",4,"Wentworth 1894, Exercise 79 (2)"],["form/20ab5e5862",5,"evaluate: 21"],["form/18877a81e8",5,"solve: Eq(3*x + 10, x + 20)"],["form/d6444d9724",5,"evaluate: 1092"],["form/ec3595965e",5,"evaluate: 765"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/1a",4,"Hardy 1921, Exercise XXIV (1a)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/1b",4,"Hardy 1921, Exercise XXIV (1b)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/1c",4,"Hardy 1921, Exercise XXIV (1c)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/1d",4,"Hardy 1921, Exercise XXIV (1d)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/2a",4,"Hardy 1921, Exercise XXIV (2a)"],["whitehead-introduction-to-mathematics-1911/eq-bf6ed0c745",16,"Whitehead 1911, p. 102: (x, y) × (x', y') = \\{(xx' - yy'), (xy' + x'y)\\}"],["wentworth-first-steps-in-algebra-1894/ex-77/13",4,"Wentworth 1894, Exercise 77 (13)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/2b",4,"Hardy 1921, Exercise XXIV (2b)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/3a",4,"Hardy 1921, Exercise XXIV (3a)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/3b",4,"Hardy 1921, Exercise XXIV (3b)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/4",4,"Hardy 1921, Exercise XXIV (4)"],["hardy-course-of-pure-mathematics-1921/x-be43c7358a",15,"Hardy 1921, p. 172: If it were complete, then every function \\phi(x) which ..."],["whitehead-introduction-to-mathematics-1911/eq-063e8496ee",16,"Whitehead 1911, p. 101: (x, y) × (x', y') = (x', y') × (x, y)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/5",4,"Hardy 1921, Exercise XXIV (5)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/6a",4,"Hardy 1921, Exercise XXIV (6a)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/6b",4,"Hardy 1921, Exercise XXIV (6b)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/6c",4,"Hardy 1921, Exercise XXIV (6c)"],["hardy-course-of-pure-mathematics-1921/x-7878560696",15,"Hardy 1921, p. 172: We shall say that \\phi(x) is of the kth ..."],["form/538f0f4362",5,"evaluate: 23"],["form/828005b546",5,"evaluate: 31"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/7",4,"Hardy 1921, Exercise XXIV (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/8a",4,"Hardy 1921, Exercise XXIV (8a)"],["form/15e2f9a050",5,"evaluate: 478"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/8b",4,"Hardy 1921, Exercise XXIV (8b)"],["wentworth-first-steps-in-algebra-1894/ex-81/8",4,"Wentworth 1894, Exercise 81 (8)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/8c",4,"Hardy 1921, Exercise XXIV (8c)"],["form/26f2f6e55d",5,"evaluate: 638"],["boyden-first-book-in-algebra-1895/ex-12/15",4,"Boyden 1895, Exercise 12 (15)"],["form/28b6bfce6a",5,"solve: Eq(9*a, x)"],["form/d0de3b0ed8",5,"evaluate: 589"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/8d",4,"Hardy 1921, Exercise XXIV (8d)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/9a",4,"Hardy 1921, Exercise XXIV (9a)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/9b",4,"Hardy 1921, Exercise XXIV (9b)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/9c",4,"Hardy 1921, Exercise XXIV (9c)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/10",4,"Hardy 1921, Exercise XXIV (10)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/11",4,"Hardy 1921, Exercise XXIV (11)"],["cap/other:verbal-to-symbol",17,"other:verbal-to-symbol"],["shape/7b0ff3409c",6,"solve: Eq(N*a, x)"],["form/c2d97e3e99",5,"evaluate: sqrt(6)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/12",4,"Hardy 1921, Exercise XXIV (12)"],["ball-mathematical-recreations-1905/eq-ed7d62a9b1",16,"Ball 1905, scan 348: \\phi(x, y, z) = 0"],["whitehead-introduction-to-mathematics-1911/x-f37d92a7a6",15,"Whitehead 1911, p. 15: The ideas of any and of some are introduced ..."],["whitehead-introduction-to-mathematics-1911/x-29aec76035",15,"Whitehead 1911, p. 17: When we have asked the question implied in the ..."],["whitehead-introduction-to-mathematics-1911/x-69b52ebac8",15,"Whitehead 1911, p. 18: One of the causes of the apparent triviality of ..."],["whitehead-introduction-to-mathematics-1911/x-b1afbecec7",15,"Whitehead 1911, p. 19: Thus the “field” of the relation for x is ..."],["ball-mathematical-recreations-1905/eq-cae42653b7",16,"Ball 1905, scan 348: \\phi(x, y, z, \\omega) = 0"],["form/0b00419a55",5,"evaluate: sqrt(10)/4"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/13",4,"Hardy 1921, Exercise XXIV (13)"],["whitehead-introduction-to-mathematics-1911/x-77da967540",15,"Whitehead 1911, p. 22: Then the law, known as Boyle’s law, expressing the ..."],["hardy-course-of-pure-mathematics-1921/ex-xxiv/14",4,"Hardy 1921, Exercise XXIV (14)"],["whitehead-introduction-to-mathematics-1911/x-6b1a67c688",15,"Whitehead 1911, p. 23: In other words the really fundamental idea is that ..."],["hardy-course-of-pure-mathematics-1921/ex-xxiv/15a",4,"Hardy 1921, Exercise XXIV (15a)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/15b",4,"Hardy 1921, Exercise XXIV (15b)"],["form/064e19a6ff",5,"solve: Eq(-b + x, a)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/16a",4,"Hardy 1921, Exercise XXIV (16a)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/16b",4,"Hardy 1921, Exercise XXIV (16b)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/17a",4,"Hardy 1921, Exercise XXIV (17a)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/17b",4,"Hardy 1921, Exercise XXIV (17b)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/17c",4,"Hardy 1921, Exercise XXIV (17c)"],["whitehead-introduction-to-mathematics-1911/x-204b9aa8e9",15,"Whitehead 1911, p. 16: The Romans would have stated the number of the ..."],["whitehead-introduction-to-mathematics-1911/eq-76b2ae3912",16,"Whitehead 1911, p. 101: \\{(x, y) × (x', y')\\} × (u, v) = (x, y) × \\{(x', y') × (u, v)\\}"],["form/4e5df2fd87",5,"solve: Eq(a + b + x, 100)"],["shape/97451230c8",6,"solve: Eq(a + b + x, N)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/18",4,"Hardy 1921, Exercise XXIV (18)"],["cap/other:number_theory",17,"other:number_theory"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/19",4,"Hardy 1921, Exercise XXIV (19)"],["hardy-course-of-pure-mathematics-1921/ex-xxiv/20",4,"Hardy 1921, Exercise XXIV (20)"],["cap/other:calendar",17,"other:calendar"],["form/c5c3d5ad22",5,"solve: Eq(x/3, a)"],["de-morgan-elementary-illustrations-calculus-1899/eq-178b195afa",16,"De Morgan 1899, p. 9: OD ÷ OA = BM ÷ BA"],["de-morgan-elementary-illustrations-calculus-1899/eq-b47e845802",16,"De Morgan 1899, p. 9: \\sin\\theta = .0174524"],["de-morgan-elementary-illustrations-calculus-1899/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities",2,"De Morgan 1899, On the Ratios of Continuously Increasing or Decreasing Quantities","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-on-the-ratios-of-continuously-increasing-or-decreasing-quantities/index.html"],["boyden-first-book-in-algebra-1895/ex-12/16",4,"Boyden 1895, Exercise 12 (16)"],["shape/7d861f980a",6,"solve: Eq(N*x, a)"],["hardy-course-of-pure-mathematics-1921/ex-xxix/1",4,"Hardy 1921, Exercise XXIX (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxix/2",4,"Hardy 1921, Exercise XXIX (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxix/3",4,"Hardy 1921, Exercise XXIX (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxix/4",4,"Hardy 1921, Exercise XXIX (4)"],["de-morgan-elementary-illustrations-calculus-1899/eq-3f581fc90d",16,"De Morgan 1899, p. 9: 2\\sin\\frac{1}{2}\\theta ÷ \\sin\\theta = 1.00003"],["hardy-course-of-pure-mathematics-1921/ex-xxix/5",4,"Hardy 1921, Exercise XXIX (5)"],["wentworth-first-steps-in-algebra-1894/ex-8/1",4,"Wentworth 1894, Exercise 8 (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxix/6",4,"Hardy 1921, Exercise XXIX (6)"],["form/d4618d0a33",5,"evaluate: 503"],["hardy-course-of-pure-mathematics-1921/ex-xxix/7",4,"Hardy 1921, Exercise XXIX (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxix/8",4,"Hardy 1921, Exercise XXIX (8)"],["form/a7344a0e05",5,"evaluate: 1/4"],["hardy-course-of-pure-mathematics-1921/ex-xxix/9",4,"Hardy 1921, Exercise XXIX (9)"],["de-morgan-elementary-illustrations-calculus-1899/eq-c57fc5661a",16,"De Morgan 1899, p. 9: BM ÷ MA = 114.589"],["de-morgan-elementary-illustrations-calculus-1899/eq-9f68caeb53",16,"De Morgan 1899, p. 9: \\angle BOA = \\theta"],["form/d3a57ba8ec",5,"evaluate: 5**(2/3)/5"],["form/f8d916d59f",5,"evaluate: 2**(2/3)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/1",4,"Hardy 1921, Exercise XXV (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/2",4,"Hardy 1921, Exercise XXV (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/3",4,"Hardy 1921, Exercise XXV (3)"],["todhunter-spherical-trigonometry-1886/eq-9da182692e",16,"Todhunter 1886, scan 22: A' &= \\pi - a"],["todhunter-spherical-trigonometry-1886/eq-a3167bef84",16,"Todhunter 1886, scan 22: B' &= \\pi - b"],["todhunter-spherical-trigonometry-1886/eq-a3ec71e761",16,"Todhunter 1886, scan 22: C' &= \\pi - c"],["todhunter-spherical-trigonometry-1886/eq-ca97c139f1",16,"Todhunter 1886, scan 22: a' &= \\pi - A"],["boyden-first-book-in-algebra-1895/ex-12/17",4,"Boyden 1895, Exercise 12 (17)"],["form/56e85a2ab7",5,"solve: Eq(x/5, a)"],["form/d8ee5b137f",5,"evaluate: 3020**(1/3)/10"],["hardy-course-of-pure-mathematics-1921/ex-xxv/4",4,"Hardy 1921, Exercise XXV (4)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/5",4,"Hardy 1921, Exercise XXV (5)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/6",4,"Hardy 1921, Exercise XXV (6)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/7",4,"Hardy 1921, Exercise XXV (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/8",4,"Hardy 1921, Exercise XXV (8)"],["todhunter-spherical-trigonometry-1886/eq-8912284cb1",16,"Todhunter 1886, scan 22: b' &= \\pi - B"],["cap/other:word-to-expression",17,"other:word-to-expression"],["hardy-course-of-pure-mathematics-1921/ex-xxv/9a",4,"Hardy 1921, Exercise XXV (9a)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/9b",4,"Hardy 1921, Exercise XXV (9b)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/10a",4,"Hardy 1921, Exercise XXV (10a)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/10b",4,"Hardy 1921, Exercise XXV (10b)"],["wentworth-first-steps-in-algebra-1894/ex-8/8",4,"Wentworth 1894, Exercise 8 (8)"],["form/05a43ac69f",5,"solve: Eq(a*x, 36)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/11a",4,"Hardy 1921, Exercise XXV (11a)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/11b",4,"Hardy 1921, Exercise XXV (11b)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/12a",4,"Hardy 1921, Exercise XXV (12a)"],["hardy-course-of-pure-mathematics-1921/ex-xxv/12b",4,"Hardy 1921, Exercise XXV (12b)"],["form/36ac96e8a7",5,"solve: Eq(x, 3*a*b/50)"],["shape/3535fd13ff",6,"solve: Eq(x, N*a*b)"],["hardy-course-of-pure-mathematics-1921/ex-xxvi",3,"Hardy 1921, Exercise XXVI"],["todhunter-spherical-trigonometry-1886/eq-537af0cb96",16,"Todhunter 1886, scan 22: c' &= \\pi - C"],["hardy-course-of-pure-mathematics-1921/ex-xxvi/1",4,"Hardy 1921, Exercise XXVI (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxvi/2",4,"Hardy 1921, Exercise XXVI (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxvi/3",4,"Hardy 1921, Exercise XXVI (3)"],["todhunter-spherical-trigonometry-1886/eq-758185b4d1",16,"Todhunter 1886, scan 23: \\dfrac{AB}{OA} + \\dfrac{BC}{OA} + \\dfrac{CA}{OA} \\text{ is less than } 2\\pi"],["todhunter-spherical-trigonometry-1886/eq-5d7dea565b",16,"Todhunter 1886, scan 23: AB+BC+CD \\text{ is less than } 2\\pi\\times OA"],["form/7494a57adf",5,"identity: 36*a"],["shape/d81017e60a",6,"identity: N*a"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/1",4,"Hardy 1921, Exercise XXVII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/2",4,"Hardy 1921, Exercise XXVII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/3",4,"Hardy 1921, Exercise XXVII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/4",4,"Hardy 1921, Exercise XXVII (4)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/5",4,"Hardy 1921, Exercise XXVII (5)"],["todhunter-spherical-trigonometry-1886/eq-fa9d499657",16,"Todhunter 1886, scan 23: AD+BC\\text{ is greater than }AC"],["todhunter-spherical-trigonometry-1886/eq-38db1db2bc",16,"Todhunter 1886, scan 23: AB+BC+CD\\text{ is greater than }AC+CD"],["form/46bad189fe",5,"solve: Eq(3*x, 2*x + 5)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/6",4,"Hardy 1921, Exercise XXVII (6)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/7",4,"Hardy 1921, Exercise XXVII (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/8",4,"Hardy 1921, Exercise XXVII (8)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/9",4,"Hardy 1921, Exercise XXVII (9)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/10",4,"Hardy 1921, Exercise XXVII (10)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/11",4,"Hardy 1921, Exercise XXVII (11)"],["wentworth-first-steps-in-algebra-1894/ex-9/9",4,"Wentworth 1894, Exercise 9 (9)"],["todhunter-spherical-trigonometry-1886/eq-c6e76fa328",16,"Todhunter 1886, scan 23: A+B+C\\text{ is greater than }\\pi"],["form/155bf35e5a",5,"solve: Eq(x, 3*a - 8)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/12",4,"Hardy 1921, Exercise XXVII (12)"],["hardy-course-of-pure-mathematics-1921/ex-xxvii/13",4,"Hardy 1921, Exercise XXVII (13)"],["boyden-first-book-in-algebra-1895/ex-13",3,"Boyden 1895, Exercise 13"],["wentworth-first-steps-in-algebra-1894/ex-9/15",4,"Wentworth 1894, Exercise 9 (15)"],["form/e3e064e3ee",5,"solve: Eq(7*x - 10, 32)"],["form/9439b4557c",5,"solve: Eq(6*x - 50, 10)"],["hardy-course-of-pure-mathematics-1921/ex-xxviii",3,"Hardy 1921, Exercise XXVIII"],["hardy-course-of-pure-mathematics-1921/ex-xxviii/1",4,"Hardy 1921, Exercise XXVIII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxviii/2",4,"Hardy 1921, Exercise XXVIII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxviii/3",4,"Hardy 1921, Exercise XXVIII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxviii/4",4,"Hardy 1921, Exercise XXVIII (4)"],["hardy-course-of-pure-mathematics-1921/ex-xxviii/5",4,"Hardy 1921, Exercise XXVIII (5)"],["wentworth-first-steps-in-algebra-1894/ex-9/21",4,"Wentworth 1894, Exercise 9 (21)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/1",4,"Hardy 1921, Exercise XXX (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/2",4,"Hardy 1921, Exercise XXX (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/3a",4,"Hardy 1921, Exercise XXX (3a)"],["wentworth-first-steps-in-algebra-1894/ex-9/27",4,"Wentworth 1894, Exercise 9 (27)"],["whitehead-introduction-to-mathematics-1911/eq-0b01b5e8b1",16,"Whitehead 1911, p. 60: (x + y) + z = x + (y + z)"],["cap/other:construct",17,"other:construct"],["hardy-course-of-pure-mathematics-1921/ex-xxx/3b",4,"Hardy 1921, Exercise XXX (3b)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/3c",4,"Hardy 1921, Exercise XXX (3c)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/4",4,"Hardy 1921, Exercise XXX (4)"],["law/associative-law-of-addition",10,"associative law of addition"],["whitehead-introduction-to-mathematics-1911/eq-875b6ddbdb",16,"Whitehead 1911, p. 60: x × y = y × x"],["law/commutative-law-of-multiplication",10,"commutative law of multiplication"],["form/27e6f63d7e",5,"solve: Eq(9*x + 5, 2*x + 47)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/5",4,"Hardy 1921, Exercise XXX (5)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/6",4,"Hardy 1921, Exercise XXX (6)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/7",4,"Hardy 1921, Exercise XXX (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/8",4,"Hardy 1921, Exercise XXX (8)"],["whitehead-introduction-to-mathematics-1911/eq-e78d91f228",16,"Whitehead 1911, p. 60: (x × y) × z = x × (y × z)"],["whitehead-introduction-to-mathematics-1911/eq-bd4a33de69",16,"Whitehead 1911, p. 60: x × (y + z) = (x × y) + (x × z)"],["form/1f2f462566",5,"identity: 8*x"],["hardy-course-of-pure-mathematics-1921/ex-xxx/9a",4,"Hardy 1921, Exercise XXX (9a)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/9b",4,"Hardy 1921, Exercise XXX (9b)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/9c",4,"Hardy 1921, Exercise XXX (9c)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/9d",4,"Hardy 1921, Exercise XXX (9d)"],["whitehead-introduction-to-mathematics-1911/eq-2199efc8dd",16,"Whitehead 1911, p. 65: x + y - 1 = 0"],["whitehead-introduction-to-mathematics-1911/eq-6b2ba8fcf5",16,"Whitehead 1911, p. 69: ax + by - c = 0"],["concept/linear",7,"linear"],["hardy-course-of-pure-mathematics-1921/ex-xxx/10a",4,"Hardy 1921, Exercise XXX (10a)"],["concept/linear-form",7,"linear form"],["whitehead-introduction-to-mathematics-1911/eq-b89e2577df",16,"Whitehead 1911, p. 67: 0 × x = 0"],["whitehead-introduction-to-mathematics-1911/eq-f5aa07468e",16,"Whitehead 1911, p. 67: x + 0 = x"],["whitehead-introduction-to-mathematics-1911/eq-df1e46abc5",16,"Whitehead 1911, p. 68: x^{2} + (0 × x) - 4 = 0"],["whitehead-introduction-to-mathematics-1911/eq-6697d12633",16,"Whitehead 1911, p. 101: (x, y) × (a, b) = (c, d)"],["whitehead-introduction-to-mathematics-1911/eq-6713816e59",16,"Whitehead 1911, p. 101: (x, y) = \\frac{(c, d)}{(a, b)}"],["whitehead-introduction-to-mathematics-1911/eq-5745fa9041",16,"Whitehead 1911, p. 102: (x,y) × \\{(a, b) + (c, d)\\} = \\{(x, y) × (a, b)\\} + \\{(x, y) × (c, d)\\}"],["whitehead-introduction-to-mathematics-1911/eq-5db9163105",16,"Whitehead 1911, p. 103: (x, y) + (0, 0) = (x, y)"],["whitehead-introduction-to-mathematics-1911/eq-80b23b3091",16,"Whitehead 1911, p. 103: (x, y) × (0, 0) = (0, 0)"],["form/bf84a89c4e",5,"solve: (Eq(-a + x, 18), Eq(a + x, 74))"],["hardy-course-of-pure-mathematics-1921/ex-xxx/10b",4,"Hardy 1921, Exercise XXX (10b)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/10c",4,"Hardy 1921, Exercise XXX (10c)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/10d",4,"Hardy 1921, Exercise XXX (10d)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/11",4,"Hardy 1921, Exercise XXX (11)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/12",4,"Hardy 1921, Exercise XXX (12)"],["form/835da1d275",5,"identity: x**9"],["hardy-course-of-pure-mathematics-1921/ex-xxx/13",4,"Hardy 1921, Exercise XXX (13)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/14a",4,"Hardy 1921, Exercise XXX (14a)"],["form/f812fce105",5,"identity: -a - 3*b"],["hardy-course-of-pure-mathematics-1921/ex-xxx/14b",4,"Hardy 1921, Exercise XXX (14b)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/15",4,"Hardy 1921, Exercise XXX (15)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/16",4,"Hardy 1921, Exercise XXX (16)"],["whitehead-introduction-to-mathematics-1911/eq-4e291b291c",16,"Whitehead 1911, p. 103: x × 1 = x"],["whitehead-introduction-to-mathematics-1911/eq-0f02213b66",16,"Whitehead 1911, p. 103: (x, y) × (1, 0) = \\{(x - 0), (y + 0)\\} = (x, y)"],["boyden-first-book-in-algebra-1895/ex-12",3,"Boyden 1895, Exercise 12"],["boyden-first-book-in-algebra-1895/ex-10",3,"Boyden 1895, Exercise 10"],["hardy-course-of-pure-mathematics-1921/ex-xxx/17",4,"Hardy 1921, Exercise XXX (17)"],["concept/neighbourhood",7,"neighbourhood","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-neighbourhood"],["todhunter-spherical-trigonometry-1886/x-7031fc74b2",15,"Todhunter 1886, scan 16: Spherical Trigonometry investigates the relations which subsist between the ..."],["todhunter-spherical-trigonometry-1886/x-9ba9bf0470",15,"Todhunter 1886, scan 17: Thus a figure will be formed on the surface ..."],["whitehead-introduction-to-mathematics-1911/eq-e33102fc05",16,"Whitehead 1911, p. 103: \\sqrt{(-1)} × \\sqrt{(-1)} = -1"],["whitehead-introduction-to-mathematics-1911/eq-9ceeac8775",16,"Whitehead 1911, p. 104: (0, 1) × (0, 1) = \\{(0 - 1), (0 + 0)\\} = (-1, 0)"],["whitehead-introduction-to-mathematics-1911/eq-af5d7d8f30",16,"Whitehead 1911, p. 104: (0, -1) × (0, -1) = (-1, 0)"],["form/f71d5e0361",5,"solve: Eq(x, 12*a - 39)"],["hardy-course-of-pure-mathematics-1921/ex-xxx/18",4,"Hardy 1921, Exercise XXX (18)"],["todhunter-spherical-trigonometry-1886/x-5a086e42fa",15,"Todhunter 1886, scan 18: As in the case of plane triangles, A, B, ..."],["hardy-course-of-pure-mathematics-1921/ex-xxxi",3,"Hardy 1921, Exercise XXXI"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/1",4,"Hardy 1921, Exercise XXXI (1)"],["shape/25dd6b26c6",6,"identity: N*b - a"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/2",4,"Hardy 1921, Exercise XXXI (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/3",4,"Hardy 1921, Exercise XXXI (3)"],["todhunter-spherical-trigonometry-1886/x-db04fdbe3d",15,"Todhunter 1886, scan 19: In spherical triangles each side is restricted to be ..."],["whitehead-introduction-to-mathematics-1911/eq-8dd0dfa263",16,"Whitehead 1911, p. 105: (a, 0) × (x, y) = (ax, ay)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/4a",4,"Hardy 1921, Exercise XXXI (4a)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/4b",4,"Hardy 1921, Exercise XXXI (4b)"],["form/6912af4735",5,"solve: (Eq(x, 2*E/3), Eq(x + E, 30))"],["shape/95b064cf62",6,"solve: (Eq(x, E*N), Eq(x + E, N))"],["todhunter-spherical-trigonometry-1886/x-3d97267e4a",15,"Todhunter 1886, scan 19: From the restriction of the preceding Article it will ..."],["whitehead-introduction-to-mathematics-1911/eq-2b904d9b2c",16,"Whitehead 1911, p. 105: (0, b) × (x, y) = (-by, bx)"],["whitehead-introduction-to-mathematics-1911/eq-cffdca1d92",16,"Whitehead 1911, p. 105: (a, 0) × (0, b) = (0, ab)"],["whitehead-introduction-to-mathematics-1911/eq-983c9f3c65",16,"Whitehead 1911, p. 105: (a, 0) × (a', 0) =( aa', 0)"],["whitehead-introduction-to-mathematics-1911/eq-a5aa93f352",16,"Whitehead 1911, p. 105: (0, b) × (0, b') = (-bb', 0)"],["whitehead-introduction-to-mathematics-1911/eq-0451322401",16,"Whitehead 1911, p. 108: \\text{the angle } QOR = \\text{the angle } XOP"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/5",4,"Hardy 1921, Exercise XXXI (5)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/6",4,"Hardy 1921, Exercise XXXI (6)"],["boyden-first-book-in-algebra-1895/ex-16",3,"Boyden 1895, Exercise 16"],["boyden-first-book-in-algebra-1895/ex-23/16",4,"Boyden 1895, Exercise 23 (16)"],["form/967a9081b0",5,"solve: (Eq(x, a + 36), Eq(a + x, 317))"],["shape/66e5a13c29",6,"solve: (Eq(x, N + a), Eq(a + x, N))"],["whitehead-introduction-to-mathematics-1911/eq-59ac801585",16,"Whitehead 1911, p. 110: (u, v) + (3, 0) = (2, 0)"],["cap/other:arrange.polynomial",17,"other:arrange.polynomial"],["boyden-first-book-in-algebra-1895/ex-23/17",4,"Boyden 1895, Exercise 23 (17)"],["boyden-first-book-in-algebra-1895/ex-23/18",4,"Boyden 1895, Exercise 23 (18)"],["form/927a982ca2",5,"identity: 2*x + 4"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/7",4,"Hardy 1921, Exercise XXXI (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/8",4,"Hardy 1921, Exercise XXXI (8)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/9a",4,"Hardy 1921, Exercise XXXI (9a)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/9b",4,"Hardy 1921, Exercise XXXI (9b)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/9c",4,"Hardy 1921, Exercise XXXI (9c)"],["whitehead-introduction-to-mathematics-1911/eq-581898a0dd",16,"Whitehead 1911, p. 110: \\{(u, v) + (3, 0)\\}^{2} = (-2, 0)"],["form/9c55e68fb4",5,"evaluate: 2*a + 3*b + c at a=1, b=2, c=3"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/9d",4,"Hardy 1921, Exercise XXXI (9d)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/9e",4,"Hardy 1921, Exercise XXXI (9e)"],["concept/steadily-increasing-function",7,"steadily increasing function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-steadily-increasing-function"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/10a",4,"Hardy 1921, Exercise XXXI (10a)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/10b",4,"Hardy 1921, Exercise XXXI (10b)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/10c",4,"Hardy 1921, Exercise XXXI (10c)"],["de-morgan-elementary-illustrations-calculus-1899/x-84795adbf1",15,"De Morgan 1899, p. 5: Here both M and N decrease at every step, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-c4fcbb2886",15,"De Morgan 1899, p. 7: This is what we mean by saying that \\dfrac{M}{N} ..."],["boyden-first-book-in-algebra-1895/ex-14/19",4,"Boyden 1895, Exercise 14 (19)"],["form/be5e88a9dd",5,"solve: (Eq(b, 2*a), Eq(3*a + 4*b, 66))"],["shape/616806f266",6,"solve: (Eq(b, N*a), Eq(N*a + N*b, N))"],["form/9b9e0c2429",5,"solve: Eq(x/6, 6)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/11",4,"Hardy 1921, Exercise XXXI (11)"],["boyden-first-book-in-algebra-1895/ex-16/1",4,"Boyden 1895, Exercise 16 (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/12a",4,"Hardy 1921, Exercise XXXI (12a)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/12b",4,"Hardy 1921, Exercise XXXI (12b)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/13",4,"Hardy 1921, Exercise XXXI (13)"],["boyden-first-book-in-algebra-1895/ex-15/1",4,"Boyden 1895, Exercise 15 (1)"],["form/afebea405b",5,"identity: 25*a*x"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/14",4,"Hardy 1921, Exercise XXXI (14)"],["form/06c82fdf55",5,"identity: 2*x**3"],["boyden-first-book-in-algebra-1895/ex-23/19",4,"Boyden 1895, Exercise 23 (19)"],["form/a7dfb1b4f8",5,"identity: -(-2*a + x)**2 + (2*a + x)**2"],["boyden-first-book-in-algebra-1895/ex-23/20",4,"Boyden 1895, Exercise 23 (20)"],["form/96134cde8e",5,"identity: (a/3 - x/2 + 1)*(a/3 + x/2 + 1)"],["shape/84c75705a2",6,"identity: (N*a + N*x + 1)**2"],["boyden-first-book-in-algebra-1895/ex-31/12",4,"Boyden 1895, Exercise 31 (12)"],["theorem/wilson-s-theorem",9,"Wilson's theorem","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-wilson-s-theorem"],["form/915be175c7",5,"identity: -6*a**2 + 6*a*b + 5*x**2"],["boyden-first-book-in-algebra-1895/x-6416c9abd4",15,"Boyden 1895: Any number of terms may be enclosed in a ..."],["shape/18ee2acab6",6,"identity: N*a*b + N*a**N + N*x**N"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/15",4,"Hardy 1921, Exercise XXXI (15)"],["boyden-first-book-in-algebra-1895/ex-24/6",4,"Boyden 1895, Exercise 24 (6)"],["boyden-first-book-in-algebra-1895/ex-19",3,"Boyden 1895, Exercise 19"],["boyden-first-book-in-algebra-1895/ex-20",3,"Boyden 1895, Exercise 20"],["form/89e9a9c670",5,"identity: -11*a**3*b"],["boyden-first-book-in-algebra-1895/ex-23",3,"Boyden 1895, Exercise 23"],["boyden-first-book-in-algebra-1895/ex-24/1",4,"Boyden 1895, Exercise 24 (1)"],["concept/sub-atom",7,"sub-atom","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-sub-atom"],["form/176bb5abac",5,"identity: 5*a*b"],["form/f155cae980",5,"factor: a**3 - a**2*b + a*b**2"],["boyden-first-book-in-algebra-1895/ex-24/4",4,"Boyden 1895, Exercise 24 (4)"],["boyden-first-book-in-algebra-1895/ex-24/5",4,"Boyden 1895, Exercise 24 (5)"],["shape/e9163da284",6,"factor: a*b**N - a**N*b + a**N"],["boyden-first-book-in-algebra-1895/ex-31/13",4,"Boyden 1895, Exercise 31 (13)"],["form/8c614d449a",5,"factor: 3*a*(e + f) + 5*b*d*(e + f) - 9*c**2*e*(e + f)"],["shape/d2b1d37e93",6,"factor: N*a*(e + f) + N*b*d*(e + f) + N*c**N*e*(e + f)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/16",4,"Hardy 1921, Exercise XXXI (16)"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/17",4,"Hardy 1921, Exercise XXXI (17)"],["shape/9d574e0c94",6,"identity: N*x + N*x**N"],["hardy-course-of-pure-mathematics-1921/ex-xxxi/18",4,"Hardy 1921, Exercise XXXI (18)"],["hardy-course-of-pure-mathematics-1921/x-a0fa14271c",15,"Hardy 1921, p. 192: We shall call such a square a neighbourhood of ..."],["ball-mathematical-recreations-1905/x-d7c72baa1f",15,"Ball 1905, scan 230: In the cubic equation x^3 + qx + r ..."],["hardy-course-of-pure-mathematics-1921/ex-xxxii",3,"Hardy 1921, Exercise XXXII"],["hardy-course-of-pure-mathematics-1921/ex-xxxii/1",4,"Hardy 1921, Exercise XXXII (1)"],["cap/other:limit_points_of_sequence",17,"other:limit_points_of_sequence"],["hardy-course-of-pure-mathematics-1921/ex-xxxii/2",4,"Hardy 1921, Exercise XXXII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxii/3",4,"Hardy 1921, Exercise XXXII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxxii/4",4,"Hardy 1921, Exercise XXXII (4)"],["ball-mathematical-recreations-1905/x-ce0ba96fc9",15,"Ball 1905, scan 237: I was not up to the differential calculus, and ..."],["ball-mathematical-recreations-1905/x-11ec03ee3a",15,"Ball 1905, scan 233: they would require every candidate to show a competent ..."],["form/0aa9c943b8",5,"identity: 23*x**3 + 20*x**2 + 27*x + 6"],["hardy-course-of-pure-mathematics-1921/ex-xxxii/5",4,"Hardy 1921, Exercise XXXII (5)"],["hardy-course-of-pure-mathematics-1921/ex-xxxii/6",4,"Hardy 1921, Exercise XXXII (6)"],["hardy-course-of-pure-mathematics-1921/x-b41a634909",15,"Hardy 1921, p. 190: This theorem is of fundamental importance in the theory ..."],["hardy-course-of-pure-mathematics-1921/x-00ed3dc415",15,"Hardy 1921, p. 193: There is nothing to show that the \\EPSILON_{1} of ..."],["todhunter-spherical-trigonometry-1886/eq-5d516fcddc",16,"Todhunter 1886, scan 47: \\sin b = \\sin B \\sin c"],["ball-mathematical-recreations-1905/x-f9d40bcd5f",15,"Ball 1905, scan 239: ‘It was a quadratic,’ said he, ‘made me senior ..."],["hardy-course-of-pure-mathematics-1921/ex-xxxii/7",4,"Hardy 1921, Exercise XXXII (7)"],["form/060a3253c7",5,"identity: 2*a**2*b**3"],["boyden-first-book-in-algebra-1895/ex-17",3,"Boyden 1895, Exercise 17"],["boyden-first-book-in-algebra-1895/ex-24/7",4,"Boyden 1895, Exercise 24 (7)"],["form/fc93ab7652",5,"identity: 4*a*b**2"],["boyden-first-book-in-algebra-1895/ex-24/8",4,"Boyden 1895, Exercise 24 (8)"],["form/66fda8a106",5,"identity: 9*a**2*b**2"],["boyden-first-book-in-algebra-1895/ex-24/9",4,"Boyden 1895, Exercise 24 (9)"],["todhunter-spherical-trigonometry-1886/eq-b2677c517d",16,"Todhunter 1886, scan 47: \\sin a = \\sin A \\sin c"],["ball-mathematical-recreations-1905/eq-5522dd5ee8",16,"Ball 1905, scan 363: a = N - 4 \\{N/4\\}"],["form/463f253d7f",5,"identity: 12*a*x**2"],["ball-mathematical-recreations-1905/ch-xiii",2,"Ball 1905, ch. XIII: Time and its Measurement","../books/ball-mathematical-recreations-1905/ch/ch-xiii/index.html"],["shape/31cd263045",6,"identity: N*a*x**N"],["form/dadd49784f",5,"identity: -9*a**3*x"],["hardy-course-of-pure-mathematics-1921/ex-xxxiii/1",4,"Hardy 1921, Exercise XXXIII (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxxiii/2",4,"Hardy 1921, Exercise XXXIII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxiii/3",4,"Hardy 1921, Exercise XXXIII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxxiii/4",4,"Hardy 1921, Exercise XXXIII (4)"],["ball-mathematical-recreations-1905/eq-3f7c290c16",16,"Ball 1905, scan 363: b = N - 7 \\{N/7\\}"],["ball-mathematical-recreations-1905/eq-8d71b26aac",16,"Ball 1905, scan 363: c = N - 19 \\{N/19\\}"],["concept/golden-number",7,"golden number"],["hardy-course-of-pure-mathematics-1921/ex-xxxiii/5",4,"Hardy 1921, Exercise XXXIII (5)"],["boyden-first-book-in-algebra-1895/ex-17/12",4,"Boyden 1895, Exercise 17 (12)"],["boyden-first-book-in-algebra-1895/ex-12/18",4,"Boyden 1895, Exercise 12 (18)"],["ball-mathematical-recreations-1905/eq-60aefbe501",16,"Ball 1905, scan 363: \\xi = \\{N/100\\}\\allowbreak - \\{N/400\\} - \\{N/300\\}"],["hardy-course-of-pure-mathematics-1921/ex-xxxiv/1",4,"Hardy 1921, Exercise XXXIV (1)"],["hardy-course-of-pure-mathematics-1921/x-3d0413ab71",15,"Hardy 1921, p. 194: This class has an upper bound \\eta, and plainly ..."],["hardy-course-of-pure-mathematics-1921/ex-xxxiv/2",4,"Hardy 1921, Exercise XXXIV (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxiv/3",4,"Hardy 1921, Exercise XXXIV (3)"],["form/10e8a68cf7",5,"identity: 4*x**2/3 - 7*x/2 - 1/2"],["form/ec2270f7bc",5,"solve: Eq(x, 2*a + b - c)"],["shape/b0059ac10c",6,"solve: Eq(x, N*a + b - c)"],["hardy-course-of-pure-mathematics-1921/ex-xxxiv/4",4,"Hardy 1921, Exercise XXXIV (4)"],["hardy-course-of-pure-mathematics-1921/ex-xxxiv/5",4,"Hardy 1921, Exercise XXXIV (5)"],["hardy-course-of-pure-mathematics-1921/x-1f759f67ed",15,"Hardy 1921, p. 189: It is indeed obvious that there are an infinity ..."],["ball-mathematical-recreations-1905/eq-9ae461c7d0",16,"Ball 1905, scan 363: \\eta = \\{N/100\\}\\allowbreak - \\{N/400\\} - 2"],["hardy-course-of-pure-mathematics-1921/ex-xxxix",3,"Hardy 1921, Exercise XXXIX"],["hardy-course-of-pure-mathematics-1921/ex-xxxix/1",4,"Hardy 1921, Exercise XXXIX (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxxix/2",4,"Hardy 1921, Exercise XXXIX (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxix/3",4,"Hardy 1921, Exercise XXXIX (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxxix/4",4,"Hardy 1921, Exercise XXXIX (4)"],["ball-mathematical-recreations-1905/eq-13cd4ecc8b",16,"Ball 1905, scan 363: \\xi = 0"],["ball-mathematical-recreations-1905/eq-983eed9437",16,"Ball 1905, scan 363: \\eta = 0"],["ball-mathematical-recreations-1905/eq-a66d643746",16,"Ball 1905, scan 362: m=15"],["hardy-course-of-pure-mathematics-1921/ex-xxxix/5",4,"Hardy 1921, Exercise XXXIX (5)"],["shape/86f1acf5a5",6,"solve: Eq(a*x, N)"],["ball-mathematical-recreations-1905/eq-167c7be436",16,"Ball 1905, scan 362: n = 6"],["hardy-course-of-pure-mathematics-1921/x-e1d4284e75",15,"Hardy 1921, p. 191: This statement is apparently simpler; but it contains phrases ..."],["ball-mathematical-recreations-1905/eq-42a7585dea",16,"Ball 1905, scan 362: m = 22"],["ball-mathematical-recreations-1905/eq-0d33326592",16,"Ball 1905, scan 362: n = 2"],["ball-mathematical-recreations-1905/eq-a18095a9e4",16,"Ball 1905, scan 362: m = 23"],["ball-mathematical-recreations-1905/eq-773083705f",16,"Ball 1905, scan 362: n = 3"],["boyden-first-book-in-algebra-1895/ex-16/29",4,"Boyden 1895, Exercise 16 (29)"],["form/1837d39ed8",5,"solve: Eq(a*x, 23)"],["hardy-course-of-pure-mathematics-1921/ex-xxxix/6",4,"Hardy 1921, Exercise XXXIX (6)"],["ball-mathematical-recreations-1905/eq-60b73e758c",16,"Ball 1905, scan 362: c > 10"],["boyden-first-book-in-algebra-1895/ex-17/1",4,"Boyden 1895, Exercise 17 (1)"],["form/cfb12bfdf6",5,"identity: a + b + c - d + 2*e"],["hardy-course-of-pure-mathematics-1921/ex-xxxv/1",4,"Hardy 1921, Exercise XXXV (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxxv/2",4,"Hardy 1921, Exercise XXXV (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxv/3",4,"Hardy 1921, Exercise XXXV (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxxv/4",4,"Hardy 1921, Exercise XXXV (4)"],["boyden-first-book-in-algebra-1895/ex-17/6",4,"Boyden 1895, Exercise 17 (6)"],["form/e82278d3c7",5,"identity: -2*a"],["form/d1dbb4d7a6",5,"identity: -5*a + 4*b - 6"],["hardy-course-of-pure-mathematics-1921/ex-xxxv/5",4,"Hardy 1921, Exercise XXXV (5)"],["hardy-course-of-pure-mathematics-1921/ex-xxxv/6",4,"Hardy 1921, Exercise XXXV (6)"],["hardy-course-of-pure-mathematics-1921/ex-xxxv/7",4,"Hardy 1921, Exercise XXXV (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxxv/8",4,"Hardy 1921, Exercise XXXV (8)"],["theorem/quotient-rule-for-derivatives",9,"quotient rule for derivatives","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-quotient-rule-for-derivatives"],["cap/core.eqn",17,"core.eqn"],["todhunter-spherical-trigonometry-1886/eq-41ce285f78",16,"Todhunter 1886, scan 44: \\cos B = \\sin A \\cos b"],["todhunter-spherical-trigonometry-1886/eq-c258282ed1",16,"Todhunter 1886, scan 44: \\cos A = \\sin B \\cos a"],["todhunter-spherical-trigonometry-1886/eq-9c35c7f2e6",16,"Todhunter 1886, scan 51: \\tan b = \\tan c \\cos A"],["todhunter-spherical-trigonometry-1886/eq-2eb4c207e8",16,"Todhunter 1886, scan 51: \\cot B = \\cos c \\tan A"],["todhunter-spherical-trigonometry-1886/eq-082cf7ae05",16,"Todhunter 1886, scan 51: \\sin a = \\sin c \\sin A"],["ball-mathematical-recreations-1905/eq-bae39274f2",16,"Ball 1905, scan 366: \\sin^{-1} (\\cos \\omega \\sec \\tfrac{1}{2} \\omega) - \\cot^{-1} \\{ \\sin \\omega \\cos(l - \\tfrac{1}{2} \\omega) (\\cos^2 l - \\s"],["concept/latitude",7,"latitude"],["concept/obliquity-of-the-ecliptic",7,"obliquity of the ecliptic"],["boyden-first-book-in-algebra-1895/ex-36/2",4,"Boyden 1895, Exercise 36 (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/1",4,"Hardy 1921, Exercise XXXVI (1)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/2",4,"Hardy 1921, Exercise XXXVI (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/3",4,"Hardy 1921, Exercise XXXVI (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/4",4,"Hardy 1921, Exercise XXXVI (4)"],["de-morgan-elementary-illustrations-calculus-1899/eq-9f105d9495",16,"De Morgan 1899, p. 11: (a + h)^{2} = a^{2} + 2ah + h^{2}"],["todhunter-spherical-trigonometry-1886/eq-f69419a136",16,"Todhunter 1886, scan 51: \\tan c=\\dfrac{\\tan b}{\\cos A}"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/5",4,"Hardy 1921, Exercise XXXVI (5)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/6",4,"Hardy 1921, Exercise XXXVI (6)"],["de-morgan-elementary-illustrations-calculus-1899/eq-cc1f10d393",16,"De Morgan 1899, p. 11: 1:h :: h:h^{2}"],["todhunter-spherical-trigonometry-1886/eq-f4b2adc053",16,"Todhunter 1886, scan 51: \\tan a = \\tan A \\sin b"],["hardy-course-of-pure-mathematics-1921/eq-fd818fa0fc",16,"Hardy 1921, p. 163: |\\phi(x) - l| < \\DELTA"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/7",4,"Hardy 1921, Exercise XXXVI (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/8",4,"Hardy 1921, Exercise XXXVI (8)"],["concept/linear-factor",7,"linear factor","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-linear-factor"],["theorem/remainder-theorem",9,"remainder theorem","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-remainder-theorem"],["method/quadratic-formula",8,"quadratic formula","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-quadratic-formula"],["todhunter-spherical-trigonometry-1886/eq-22c30c146b",16,"Todhunter 1886, scan 47: \\cos B = \\cos b \\sin A"],["method/synthetic-division",8,"synthetic division","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-synthetic-division"],["hardy-course-of-pure-mathematics-1921/eq-62e4d754e9",16,"Hardy 1921, p. 163: \\lim_{x \\to \\infty} \\phi(x) = l"],["boyden-first-book-in-algebra-1895/ex-12/19",4,"Boyden 1895, Exercise 12 (19)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/9",4,"Hardy 1921, Exercise XXXVI (9)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/10",4,"Hardy 1921, Exercise XXXVI (10)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/11",4,"Hardy 1921, Exercise XXXVI (11)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/12",4,"Hardy 1921, Exercise XXXVI (12)"],["hardy-course-of-pure-mathematics-1921/eq-6fe7f44971",16,"Hardy 1921, p. 163: \\phi(x) \\to -\\infty"],["theorem/equation-of-degree-n-has-at-most-n-roots",9,"equation of degree n has at most n roots","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-equation-of-degree-n-has-at-most-n-roots"],["hardy-course-of-pure-mathematics-1921/eq-f3d240eed3",16,"Hardy 1921, p. 166: \\lim_{y \\to +0} \\phi(y) = l"],["form/f919083faa",5,"identity: a**2*x**2*(a**2 + a*x + x**2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/13",4,"Hardy 1921, Exercise XXXVI (13)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/14",4,"Hardy 1921, Exercise XXXVI (14)"],["boyden-first-book-in-algebra-1895/ex-20/2",4,"Boyden 1895, Exercise 20 (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/15",4,"Hardy 1921, Exercise XXXVI (15)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/16",4,"Hardy 1921, Exercise XXXVI (16)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/17a",4,"Hardy 1921, Exercise XXXVI (17a)"],["hardy-course-of-pure-mathematics-1921/eq-41491ea29b",16,"Hardy 1921, p. 166: \\lim_{y \\to 0} \\phi(y) = l"],["hardy-course-of-pure-mathematics-1921/eq-e16a95e23c",16,"Hardy 1921, p. 167: \\lim_{x \\to a} \\phi(x) = l"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/17b",4,"Hardy 1921, Exercise XXXVI (17b)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/18",4,"Hardy 1921, Exercise XXXVI (18)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/19",4,"Hardy 1921, Exercise XXXVI (19)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/20",4,"Hardy 1921, Exercise XXXVI (20)"],["theorem/imaginary-roots-occur-in-pairs",9,"imaginary roots occur in pairs","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-imaginary-roots-occur-in-pairs"],["theorem/upper-limit-theorem-by-greatest-negative-coefficient",9,"upper limit theorem by greatest negative coefficient","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-upper-limit-theorem-by-greatest-negative-coefficient"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/21",4,"Hardy 1921, Exercise XXXVI (21)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvi/22",4,"Hardy 1921, Exercise XXXVI (22)"],["theorem/upper-limit-theorem-by-coefficient-ratios",9,"upper limit theorem by coefficient ratios","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-upper-limit-theorem-by-coefficient-ratios"],["dickson-theory-of-equations-1922/x-a05f5e1381",15,"Dickson 1922, p. 11: is called a quadratic equation or equation of the ..."],["form/e631f13168",5,"identity: (x**2 - 2*x - 1)*(x**4 + 3*x**3 + 9*x**2 + 10*x + 12)"],["dickson-theory-of-equations-1922/x-8ad9ce7628",15,"Dickson 1922, p. 12: If a polynomial f(x) be divided by x - ..."],["dickson-theory-of-equations-1922/x-e00a7f71e7",15,"Dickson 1922, p. 12: If f(c) is zero, the polynomial f(x) has the ..."],["dickson-theory-of-equations-1922/x-25f2cde19e",15,"Dickson 1922, p. 14: First we bring down the first coefficient 1. Then ..."],["dickson-theory-of-equations-1922/x-b35ac51792",15,"Dickson 1922, p. 12: We thus obtain the useful result that ax^2 + ..."],["dickson-theory-of-equations-1922/x-4fe43fe1cf",15,"Dickson 1922, p. 16: An equation of degree n cannot have more than ..."],["dickson-theory-of-equations-1922/x-9f8bfb4329",15,"Dickson 1922, p. 21: For example, in x^5 + 4x^4 - 7x^2 - ..."],["todhunter-spherical-trigonometry-1886/eq-3f37ec6aa9",16,"Todhunter 1886, scan 47: \\cos c = \\cos a \\cos b"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/1",4,"Hardy 1921, Exercise XXXVII (1)"],["cap/other:continuity",17,"other:continuity"],["form/af89d2e230",5,"identity: (x + 1)*(x + 6)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/2",4,"Hardy 1921, Exercise XXXVII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/3",4,"Hardy 1921, Exercise XXXVII (3)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/4",4,"Hardy 1921, Exercise XXXVII (4)"],["concept/curved-line",7,"curved line","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-curved-line"],["concept/broken-line",7,"broken line","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-broken-line"],["concept/two-dimensional-figure",7,"two-dimensional figure","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-two-dimensional-figure"],["form/b0c3a8d3cf",5,"identity: 5*a*(b**2 + c)"],["shape/a16a02eb5e",6,"identity: (N + x)*(x + 1)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/5",4,"Hardy 1921, Exercise XXXVII (5)"],["wentworth-first-steps-in-algebra-1894/ex-2/16",4,"Wentworth 1894, Exercise 2 (16)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/6",4,"Hardy 1921, Exercise XXXVII (6)"],["form/b76b90d548",5,"identity: a*(b - c)"],["hardy-course-of-pure-mathematics-1921/x-9613d15e91",15,"Hardy 1921, p. 197: The reader will probably remember that in elementary geometry ..."],["shape/b76b90d548",6,"identity: a*(b - c)"],["wentworth-first-steps-in-algebra-1894/ex-2/18",4,"Wentworth 1894, Exercise 2 (18)"],["concept/similar-solids",7,"similar solids","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-similar-solids"],["shape/6ec65f0cd9",6,"identity: N*a*(b - c)"],["form/39bc2ae01c",5,"solve: (Eq(a, 3*x), Eq(a, x + 12))"],["wentworth-first-steps-in-algebra-1894/ex-2/19",4,"Wentworth 1894, Exercise 2 (19)"],["shape/af7a48c8db",6,"solve: (Eq(a, N*x), Eq(a, N + x))"],["shape/b1be36b67f",6,"identity: N*a*(b**N + c)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/7",4,"Hardy 1921, Exercise XXXVII (7)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/8",4,"Hardy 1921, Exercise XXXVII (8)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/9",4,"Hardy 1921, Exercise XXXVII (9)"],["concept/equivalent-solids",7,"equivalent solids","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-equivalent-solids"],["concept/superposition",7,"superposition","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-superposition"],["form/a2a58d76b4",5,"solve: (Eq(a, 8*x), Eq(a - x, 63/10))"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/10",4,"Hardy 1921, Exercise XXXVII (10)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/11",4,"Hardy 1921, Exercise XXXVII (11)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/12",4,"Hardy 1921, Exercise XXXVII (12)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/13",4,"Hardy 1921, Exercise XXXVII (13)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/14",4,"Hardy 1921, Exercise XXXVII (14)"],["hardy-course-of-pure-mathematics-1921/eq-4b6d4ffc85",16,"Hardy 1921, p. 167: \\lim_{x \\to a+0} \\phi(x) = l"],["hardy-course-of-pure-mathematics-1921/eq-f8126b6910",16,"Hardy 1921, p. 168: \\lim_{x \\to a+0} \\phi(x) = \\phi(a+0)"],["form/5f16b546db",5,"identity: (x - 5)*(x - 2)"],["form/d0c682c9cb",5,"identity: (x - 2)*(x + 5)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/15",4,"Hardy 1921, Exercise XXXVII (15)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/16",4,"Hardy 1921, Exercise XXXVII (16)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/17",4,"Hardy 1921, Exercise XXXVII (17)"],["hardy-course-of-pure-mathematics-1921/eq-c93ee7d619",16,"Hardy 1921, p. 168: \\phi(a-0) \\leq \\phi(a) \\leq \\phi(a+0)"],["todhunter-spherical-trigonometry-1886/eq-36de107c6c",16,"Todhunter 1886, scan 51: \\cot A = \\cot a \\sin b"],["hardy-course-of-pure-mathematics-1921/eq-72a05b191f",16,"Hardy 1921, p. 163: \\phi(x) \\to \\infty"],["form/1ccbd02473",5,"identity: (x - 9)*(x + 7)"],["form/4a888c3a9b",5,"identity: (x + 3)*(x + 17)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/18",4,"Hardy 1921, Exercise XXXVII (18)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/19",4,"Hardy 1921, Exercise XXXVII (19)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/20",4,"Hardy 1921, Exercise XXXVII (20)"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/21",4,"Hardy 1921, Exercise XXXVII (21)"],["hardy-course-of-pure-mathematics-1921/eq-4fe17211a8",16,"Hardy 1921, p. 170: \\lim\\phi(x) = 0"],["hardy-course-of-pure-mathematics-1921/eq-173e0b8f6d",16,"Hardy 1921, p. 170: \\lim\\psi(x) = 0"],["hardy-course-of-pure-mathematics-1921/ex-xxxvii/22",4,"Hardy 1921, Exercise XXXVII (22)"],["cap/other:limits",17,"other:limits"],["hardy-course-of-pure-mathematics-1921/eq-d22cfaa9c6",16,"Hardy 1921, p. 170: \\psi(x) = [1 - x^{2}]"],["hardy-course-of-pure-mathematics-1921/eq-c4b2c5b181",16,"Hardy 1921, p. 171: \\lim(x/x) = 1"],["form/d706c727a0",5,"identity: (-a**4 + x**4)/(a + x)"],["hardy-course-of-pure-mathematics-1921/ex-xxxviii/1",4,"Hardy 1921, Exercise XXXVIII (1)"],["shape/b67a7e9122",6,"identity: (-a**N + x**N)/(a + x)"],["wentworth-first-steps-in-algebra-1894/ex-30/24",4,"Wentworth 1894, Exercise 30 (24)"],["wentworth-first-steps-in-algebra-1894/ex-30/25",4,"Wentworth 1894, Exercise 30 (25)"],["shape/adac001b14",6,"identity: (a**N + x**N)/(a + x)"],["concept/foot-of-a-line",7,"foot of a line","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-foot-of-a-line"],["concept/transversal",7,"transversal","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-transversal"],["todhunter-spherical-trigonometry-1886/eq-f06e05b729",16,"Todhunter 1886, scan 51: \\cot B = \\cot b \\sin a"],["todhunter-spherical-trigonometry-1886/eq-291084b2c9",16,"Todhunter 1886, scan 51: \\cos b=\\dfrac{\\cos c}{\\cos a}"],["form/546210b078",5,"identity: (x - 3)*(x + 3)"],["form/b0659e90c1",5,"identity: (x - 5)*(x + 5)"],["hardy-course-of-pure-mathematics-1921/ex-xxxviii/2",4,"Hardy 1921, Exercise XXXVIII (2)"],["hardy-course-of-pure-mathematics-1921/ex-xxxviii/3",4,"Hardy 1921, Exercise XXXVIII (3)"],["dickson-theory-of-equations-1922/eq-498dcea9b8",16,"Dickson 1922, p. 1: i^2 = -1"],["dickson-theory-of-equations-1922/eq-ae3fd510eb",16,"Dickson 1922, p. 1: (\\sqrt{p})^2 i^2 = -p"],["hardy-course-of-pure-mathematics-1921/eq-d21fe0ccc9",16,"Hardy 1921, p. 171: \\lim\\phi(x) = 2"],["todhunter-spherical-trigonometry-1886/eq-78cf8bd21e",16,"Todhunter 1886, scan 51: \\cos B=\\dfrac{\\tan a}{\\tan c}"],["hardy-course-of-pure-mathematics-1921/ex-xxxviii/4",4,"Hardy 1921, Exercise XXXVIII (4)"],["hardy-course-of-pure-mathematics-1921/ex-xxxviii/5",4,"Hardy 1921, Exercise XXXVIII (5)"],["dickson-theory-of-equations-1922/eq-a091be94e3",16,"Dickson 1922, p. 2: (a+bi) + (c+di) = (a+c) + (b+d)i"],["todhunter-spherical-trigonometry-1886/eq-c87d0e3888",16,"Todhunter 1886, scan 51: \\sin A=\\dfrac{\\sin a}{\\sin c}"],["form/d1ad57b069",5,"identity: -729*a**3*b**15*c**6"],["todhunter-spherical-trigonometry-1886/ex-iv/1",4,"Todhunter 1886, Exercise IV (1)"],["todhunter-spherical-trigonometry-1886/ex-iv/2",4,"Todhunter 1886, Exercise IV (2)"],["todhunter-spherical-trigonometry-1886/ex-iv/3",4,"Todhunter 1886, Exercise IV (3)"],["todhunter-spherical-trigonometry-1886/ex-iv/4",4,"Todhunter 1886, Exercise IV (4)"],["dickson-theory-of-equations-1922/eq-0961b35b5e",16,"Dickson 1922, p. 2: (a+bi) - (c+di) = (a-c) + (b-d)i"],["dickson-theory-of-equations-1922/eq-0756b518ce",16,"Dickson 1922, p. 2: (a+bi)(c+di) = ac-bd+(ad+bc)i"],["form/29e207f026",5,"identity: (-a + b)**4"],["todhunter-spherical-trigonometry-1886/ex-iv/5",4,"Todhunter 1886, Exercise IV (5)"],["todhunter-spherical-trigonometry-1886/ex-iv/6",4,"Todhunter 1886, Exercise IV (6)"],["todhunter-spherical-trigonometry-1886/ex-iv/7",4,"Todhunter 1886, Exercise IV (7)"],["todhunter-spherical-trigonometry-1886/ex-iv/8",4,"Todhunter 1886, Exercise IV (8)"],["dickson-theory-of-equations-1922/eq-7d97384470",16,"Dickson 1922, p. 2: (a+bi)(a-bi) = a^2-b^2i^2 = a^2+b^2"],["hardy-course-of-pure-mathematics-1921/eq-4b6a526511",16,"Hardy 1921, p. 171: \\phi(x) = \\{(x + 1)^{2} - 1\\}/x = x + 2"],["todhunter-spherical-trigonometry-1886/eq-423b109bc0",16,"Todhunter 1886, scan 44: \\cos c = \\cot A \\cot B"],["todhunter-spherical-trigonometry-1886/ex-iv/9",4,"Todhunter 1886, Exercise IV (9)"],["todhunter-spherical-trigonometry-1886/ex-iv/10",4,"Todhunter 1886, Exercise IV (10)"],["todhunter-spherical-trigonometry-1886/ex-iv/11",4,"Todhunter 1886, Exercise IV (11)"],["todhunter-spherical-trigonometry-1886/ex-iv/12",4,"Todhunter 1886, Exercise IV (12)"],["dickson-theory-of-equations-1922/eq-728c6eae67",16,"Dickson 1922, p. 2: \\frac{e+fi}{a+bi} = \\frac{(e+fi)(a-bi)}{a^2+b^2} = \\frac{ae+bf}{a^2+b^2} + \\frac{af-be}{a^2+b^2} i"],["dickson-theory-of-equations-1922/eq-4338bef33c",16,"Dickson 1922, p. 2: a^2+b^2 = 0"],["form/d803f8b407",5,"identity: (a - 1)**2"],["shape/841c531ea3",6,"identity: (a - 1)**N"],["todhunter-spherical-trigonometry-1886/ex-iv/13",4,"Todhunter 1886, Exercise IV (13)"],["todhunter-spherical-trigonometry-1886/ex-iv/14",4,"Todhunter 1886, Exercise IV (14)"],["todhunter-spherical-trigonometry-1886/ex-iv/15",4,"Todhunter 1886, Exercise IV (15)"],["todhunter-spherical-trigonometry-1886/ex-iv/16",4,"Todhunter 1886, Exercise IV (16)"],["dickson-theory-of-equations-1922/eq-c3515f4ce7",16,"Dickson 1922, p. 1: a+bi=0"],["dickson-theory-of-equations-1922/eq-7932989444",16,"Dickson 1922, p. 3: r = \\sqrt{a^2+b^2}"],["todhunter-spherical-trigonometry-1886/eq-a1789ce80c",16,"Todhunter 1886, scan 52: \\cos a = \\frac{\\cos A}{\\sin B}"],["form/a346f9dfe9",5,"identity: (3*a*b - c**2*d)**3"],["form/b8283be22f",5,"solve: (Eq(x, 5*b - d), Eq(a, 4*c), Eq(b, a*c))"],["shape/123683fe25",6,"solve: (Eq(x, N*b - d), Eq(a, N*c), Eq(b, a*c))"],["cap/cas.simplify",17,"cas.simplify"],["form/90e4155ebe",5,"solve: (Eq(a, -c + 25), Eq(x, a*b))"],["shape/d571ec7b62",6,"solve: (Eq(a, N - c), Eq(x, a*b))"],["dickson-theory-of-equations-1922/eq-d42a1beb66",16,"Dickson 1922, p. 3: \\cos \\theta = a/r"],["dickson-theory-of-equations-1922/eq-98a5aaf591",16,"Dickson 1922, p. 3: \\sin \\theta = b/r"],["hardy-course-of-pure-mathematics-1921/eq-32726bbc64",16,"Hardy 1921, p. 173: \\lim\\dfrac{\\sin x}{x} = 1"],["todhunter-spherical-trigonometry-1886/ex-iv/18",4,"Todhunter 1886, Exercise IV (18)"],["todhunter-spherical-trigonometry-1886/ex-ix/1",4,"Todhunter 1886, Exercise IX (1)"],["boyden-first-book-in-algebra-1895/ex-24/2",4,"Boyden 1895, Exercise 24 (2)"],["todhunter-spherical-trigonometry-1886/ex-ix/2",4,"Todhunter 1886, Exercise IX (2)"],["form/2ff3e122b5",5,"identity: 13*a*b"],["boyden-first-book-in-algebra-1895/ex-24/3",4,"Boyden 1895, Exercise 24 (3)"],["form/c1d06b0c78",5,"identity: 3*a**2*b"],["dickson-theory-of-equations-1922/eq-238d308122",16,"Dickson 1922, p. 3: a+bi = r(\\cos\\theta + i\\sin\\theta)"],["form/9bae335478",5,"identity: 7*a**2 + 10*x**2"],["shape/b2271cbdae",6,"identity: N*a**N + N*x**N"],["todhunter-spherical-trigonometry-1886/ex-ix/3",4,"Todhunter 1886, Exercise IX (3)"],["todhunter-spherical-trigonometry-1886/ex-ix/5",4,"Todhunter 1886, Exercise IX (5)"],["todhunter-spherical-trigonometry-1886/ex-ix/9",4,"Todhunter 1886, Exercise IX (9)"],["todhunter-spherical-trigonometry-1886/ex-ix/11",4,"Todhunter 1886, Exercise IX (11)"],["dickson-theory-of-equations-1922/eq-f6de462048",16,"Dickson 1922, p. 3: \\omega = -\\tfrac{1}{2} + \\tfrac{1}{2}\\sqrt{3}i"],["dickson-theory-of-equations-1922/eq-bf5c9ba49e",16,"Dickson 1922, p. 3: x^3-1 = (x-1) (x^2+x+1)"],["hardy-course-of-pure-mathematics-1921/eq-11c25a8710",16,"Hardy 1921, p. 173: \\sin x < x < \\tan x"],["todhunter-spherical-trigonometry-1886/ex-ix/12",4,"Todhunter 1886, Exercise IX (12)"],["boyden-first-book-in-algebra-1895/ex-25/1",4,"Boyden 1895, Exercise 25 (1)"],["form/8756b09d9b",5,"identity: (18*a**3*x**4 - 42*a**2*x**3 + 90*a*b*x**6)/(6*a*x**3)"],["shape/21ee28d74d",6,"identity: N*x**N*(N*a*b*x**N + 2*N*a**N*x**N)/a"],["boyden-first-book-in-algebra-1895/ex-25/2",4,"Boyden 1895, Exercise 25 (2)"],["dickson-theory-of-equations-1922/eq-c0a9d3cd5a",16,"Dickson 1922, p. 3: (x + \\tfrac{1}{2})^2 = -\\tfrac{3}{4}"],["todhunter-spherical-trigonometry-1886/ex-v/3",4,"Todhunter 1886, Exercise V (3)"],["todhunter-spherical-trigonometry-1886/ex-v/5a",4,"Todhunter 1886, Exercise V (5a)"],["todhunter-spherical-trigonometry-1886/ex-v/6",4,"Todhunter 1886, Exercise V (6)"],["dickson-theory-of-equations-1922/eq-5e9b19b016",16,"Dickson 1922, p. 3: \\omega^2 + \\omega+1 = 0"],["dickson-theory-of-equations-1922/eq-dbefbba7cb",16,"Dickson 1922, p. 3: \\omega^3 = 1"],["dickson-theory-of-equations-1922/eq-c5a6b9f2ee",16,"Dickson 1922, p. 3: \\omega \\omega' = 1"],["dickson-theory-of-equations-1922/eq-f7cdff540f",16,"Dickson 1922, p. 3: \\omega' = \\omega^2"],["dickson-theory-of-equations-1922/eq-8140068f41",16,"Dickson 1922, p. 3: \\cos \\theta + i \\sin \\theta"],["hardy-course-of-pure-mathematics-1921/eq-f356980e32",16,"Hardy 1921, p. 173: \\lim \\dfrac{1 - \\cos x}{x^{2}} = \\frac{1}{2}"],["form/79827dd0c7",5,"solve: (Eq(a, 3*x/4), Eq(a + x, 350))"],["todhunter-spherical-trigonometry-1886/ex-v/7",4,"Todhunter 1886, Exercise V (7)"],["form/9bf89ba55b",5,"identity: (-18*a**4*x**4 + 10*a**2*x**5 + 6*a**2*x**3)/(2*a*x**3)"],["todhunter-spherical-trigonometry-1886/ex-v/10",4,"Todhunter 1886, Exercise V (10)"],["shape/66194c9344",6,"identity: 3*N**2*a**N*x**(2*N)/a"],["boyden-first-book-in-algebra-1895/ex-25/3",4,"Boyden 1895, Exercise 25 (3)"],["form/e798277e7b",5,"identity: (72*a**6*x**5 - 36*a**3*x**4 - 18*a**2*x**2)/(9*a*x**2)"],["hardy-course-of-pure-mathematics-1921/eq-bdae8fdd0c",16,"Hardy 1921, p. 173: \\lim \\dfrac{\\arcsin x}{x} = 1"],["form/200a891b00",5,"identity: 5*a**2*b**3"],["form/b3728c1184",5,"identity: -17*a"],["todhunter-spherical-trigonometry-1886/ex-v/15",4,"Todhunter 1886, Exercise V (15)"],["todhunter-spherical-trigonometry-1886/ex-v/16",4,"Todhunter 1886, Exercise V (16)"],["todhunter-spherical-trigonometry-1886/ex-v/17",4,"Todhunter 1886, Exercise V (17)"],["dickson-theory-of-equations-1922/eq-b19df795f5",16,"Dickson 1922, p. 4: (\\cos \\theta + i \\sin \\theta) (\\cos \\alpha + i \\sin \\alpha) = \\cos (\\theta + \\alpha) + i \\sin (\\theta + \\alpha)"],["todhunter-spherical-trigonometry-1886/eq-8812eb9fda",16,"Todhunter 1886, scan 52: \\cos b = \\frac{\\cos B}{\\sin A}"],["form/b7923fd3af",5,"identity: -3*a**2*b"],["todhunter-spherical-trigonometry-1886/ex-vi/2b",4,"Todhunter 1886, Exercise VI (2b)"],["todhunter-spherical-trigonometry-1886/ex-vi/4",4,"Todhunter 1886, Exercise VI (4)"],["todhunter-spherical-trigonometry-1886/ex-vi/5a",4,"Todhunter 1886, Exercise VI (5a)"],["boyden-first-book-in-algebra-1895/ex-26/1",4,"Boyden 1895, Exercise 26 (1)"],["form/a3350667aa",5,"identity: (x**2 + 8*x - 105)/(x + 15)"],["boyden-first-book-in-algebra-1895/ex-26/2",4,"Boyden 1895, Exercise 26 (2)"],["form/c27979c7f4",5,"identity: (x**2 + 8*x - 33)/(x + 11)"],["todhunter-spherical-trigonometry-1886/eq-efcaf3aa22",16,"Todhunter 1886, scan 52: \\sin c = \\dfrac{\\sin a}{\\sin A}"],["todhunter-spherical-trigonometry-1886/ex-vi/5b",4,"Todhunter 1886, Exercise VI (5b)"],["todhunter-spherical-trigonometry-1886/ex-vi/6",4,"Todhunter 1886, Exercise VI (6)"],["todhunter-spherical-trigonometry-1886/ex-vi/8",4,"Todhunter 1886, Exercise VI (8)"],["dickson-theory-of-equations-1922/eq-7887749d4b",16,"Dickson 1922, p. 4: \\frac{\\cos \\beta + i \\sin \\beta} {\\cos \\theta + i \\sin \\theta} = \\cos(\\beta - \\theta) + i \\sin(\\beta - \\theta)"],["dickson-theory-of-equations-1922/eq-2fa905e224",16,"Dickson 1922, p. 5: \\frac{1}{\\cos\\theta + i \\sin\\theta} = \\cos\\theta - i \\sin\\theta"],["dickson-theory-of-equations-1922/eq-4e493c3b1b",16,"Dickson 1922, p. 5: (\\cos\\theta + i \\sin\\theta)^n = \\cos n\\theta + i \\sin n\\theta"],["todhunter-spherical-trigonometry-1886/eq-fe81930014",16,"Todhunter 1886, scan 52: \\sin b = \\tan a\\, \\cot A"],["form/a39eb1981f",5,"identity: (a**3*b**2 - 3*a*b)**3"],["todhunter-spherical-trigonometry-1886/ex-vi/9",4,"Todhunter 1886, Exercise VI (9)"],["form/7809420bbb",5,"identity: (a**3 + x**3)/(a + x)"],["boyden-first-book-in-algebra-1895/ex-26/9",4,"Boyden 1895, Exercise 26 (9)"],["form/818f3b5db5",5,"identity: (-81*a**4 + 16*x**4)/(-3*a + 2*x)"],["boyden-first-book-in-algebra-1895/ex-26/10",4,"Boyden 1895, Exercise 26 (10)"],["dickson-theory-of-equations-1922/eq-9ac58b7b2a",16,"Dickson 1922, p. 5: 4\\sqrt{2} + 4\\sqrt{2} i = 8(\\cos 45° + i \\sin 45°)"],["todhunter-spherical-trigonometry-1886/ex-vii/3",4,"Todhunter 1886, Exercise VII (3)"],["todhunter-spherical-trigonometry-1886/ex-vii/4",4,"Todhunter 1886, Exercise VII (4)"],["form/fab6ae4477",5,"identity: (-a**4 + 81*x**8)/(-a + 3*x**2)"],["todhunter-spherical-trigonometry-1886/ex-vii/6",4,"Todhunter 1886, Exercise VII (6)"],["shape/cf5756df31",6,"identity: (N*x**N - a**N)/(N*x**N - a)"],["boyden-first-book-in-algebra-1895/ex-26/11",4,"Boyden 1895, Exercise 26 (11)"],["form/77851e5192",5,"identity: (-12*a**5 - 17*a**4*x - 5*a**3*x**2 - 2*a**2*x**3 - a*x**4 + x**5)/(-3*a**2 - 2*a*x + x**2)"],["shape/f942bfc1b0",6,"identity: (N*a**N*x + 2*N*a**N*x**N + N*a**N - a*x**N + x**N)/(N*a*x + N*a**N + x**N)"],["dickson-theory-of-equations-1922/eq-92e15a3d1c",16,"Dickson 1922, p. 6: 3 \\theta = 45°+ k·360°"],["form/adf32cb280",5,"identity: (-186*a**7*b**6*x**5 + 120*a**3*b**5*x**4)/(6*a**3*b**4*x**3)"],["todhunter-spherical-trigonometry-1886/ex-vii/7",4,"Todhunter 1886, Exercise VII (7)"],["dickson-theory-of-equations-1922/eq-53ce9296a1",16,"Dickson 1922, p. 6: \\theta = 15°+k·120°"],["dickson-theory-of-equations-1922/eq-8f53e62748",16,"Dickson 1922, p. 7: r^n(\\cos n\\theta + i \\sin n\\theta) = \\cos A + i \\sin A"],["dickson-theory-of-equations-1922/eq-aaea467059",16,"Dickson 1922, p. 7: n\\theta = A + k·360°"],["hardy-course-of-pure-mathematics-1921/eq-32955812a8",16,"Hardy 1921, p. 173: \\lim \\dfrac{\\sin \\alpha x}{x} = \\alpha"],["todhunter-spherical-trigonometry-1886/eq-18b885383b",16,"Todhunter 1886, scan 52: \\sin B = \\dfrac{\\cos A}{\\cos a}"],["todhunter-spherical-trigonometry-1886/ex-viii/2",4,"Todhunter 1886, Exercise VIII (2)"],["todhunter-spherical-trigonometry-1886/ex-viii/3",4,"Todhunter 1886, Exercise VIII (3)"],["todhunter-spherical-trigonometry-1886/ex-viii/5",4,"Todhunter 1886, Exercise VIII (5)"],["dickson-theory-of-equations-1922/eq-b88d06a9bb",16,"Dickson 1922, p. 7: \\cos\\left(\\frac{A + k·360°}{n}\\right) + i \\sin\\left(\\frac{A + k·360°}{n}\\right)"],["dickson-theory-of-equations-1922/eq-09d00b99d9",16,"Dickson 1922, p. 8: \\cos\\frac{2k \\pi}{n} + i \\sin\\frac{2k \\pi}{n}"],["hardy-course-of-pure-mathematics-1921/eq-27965bce4f",16,"Hardy 1921, p. 173: \\lim \\dfrac{\\tan \\alpha x}{x}= \\alpha"],["todhunter-spherical-trigonometry-1886/ex-viii/6",4,"Todhunter 1886, Exercise VIII (6)"],["todhunter-spherical-trigonometry-1886/ex-viii/7",4,"Todhunter 1886, Exercise VIII (7)"],["todhunter-spherical-trigonometry-1886/ex-viii/10",4,"Todhunter 1886, Exercise VIII (10)"],["dickson-theory-of-equations-1922/eq-db6f90e194",16,"Dickson 1922, p. 8: R = \\cos\\frac{2\\pi}{n} + i \\sin\\frac{2\\pi}{n}"],["concept/primitive-root-of-unity",7,"primitive root of unity"],["dickson-theory-of-equations-1922/eq-74d3968fc7",16,"Dickson 1922, p. 8: R,\\ R^2,\\ R^3,\\dotsc,\\ R^{n-1},\\ R^n = 1"],["dickson-theory-of-equations-1922/eq-4c66e7b31b",16,"Dickson 1922, p. 8: R = \\cos\\pi/2 + i \\sin\\pi/2 = i"],["form/a2f5e4b2a1",5,"identity: (15*a**3*x**2 - 4*a**3 - 14*a**2*x**3 + a*x**4 + x**5)/(2*a**2 - 3*a*x + x**2)"],["todhunter-spherical-trigonometry-1886/ex-viii/11",4,"Todhunter 1886, Exercise VIII (11)"],["todhunter-spherical-trigonometry-1886/ex-viii/13",4,"Todhunter 1886, Exercise VIII (13)"],["todhunter-spherical-trigonometry-1886/ex-viii/16",4,"Todhunter 1886, Exercise VIII (16)"],["dickson-theory-of-equations-1922/eq-57c8d2fa81",16,"Dickson 1922, p. 9: \\rho^n=1"],["dickson-theory-of-equations-1922/eq-504c7a4acc",16,"Dickson 1922, p. 9: \\rho^l \\neq 1"],["dickson-theory-of-equations-1922/eq-cf992cb854",16,"Dickson 1922, p. 9: (R^k)^{\\frac{n}{d}} = (R^n)^{\\frac{k}{d}} = 1"],["form/34308d7289",5,"identity: (x**6 - x**4 - 2*x**3 - x**2)/(x**2 + x + 1)"],["todhunter-spherical-trigonometry-1886/ex-xi/2",4,"Todhunter 1886, Exercise XI (2)"],["todhunter-spherical-trigonometry-1886/ex-xi/3",4,"Todhunter 1886, Exercise XI (3)"],["dickson-theory-of-equations-1922/eq-82fe1399d0",16,"Dickson 1922, p. 10: R^{kl} = \\cos\\frac{2kl\\pi}{n} + i \\sin\\frac{2kl\\pi}{n}"],["concept/homologous-angles",7,"homologous angles","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-homologous-angles"],["todhunter-spherical-trigonometry-1886/ex-xi/5",4,"Todhunter 1886, Exercise XI (5)"],["concept/alternate-interior-angles",7,"alternate-interior angles","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-alternate-interior-angles"],["theorem/all-straight-angles-are-equal",9,"all straight angles are equal","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-all-straight-angles-are-equal"],["theorem/vertical-angles-are-equal",9,"vertical angles are equal","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-vertical-angles-are-equal"],["todhunter-spherical-trigonometry-1886/ex-xii/1",4,"Todhunter 1886, Exercise XII (1)"],["todhunter-spherical-trigonometry-1886/ex-xii/2",4,"Todhunter 1886, Exercise XII (2)"],["todhunter-spherical-trigonometry-1886/ex-xii/3",4,"Todhunter 1886, Exercise XII (3)"],["todhunter-spherical-trigonometry-1886/ex-xii/4",4,"Todhunter 1886, Exercise XII (4)"],["theorem/perpendicular-is-the-shortest-distance-from-a-point-to-a-line",9,"perpendicular is the shortest distance from a point to a line","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-perpendicular-is-the-shortest-distance-from-a-point-to-a-line"],["theorem/alternate-interior-angles-of-parallel-lines-are-equal",9,"alternate-interior angles of parallel lines are equal","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-alternate-interior-angles-of-parallel-lines-are-equal"],["hardy-course-of-pure-mathematics-1921/eq-dee76a8f42",16,"Hardy 1921, p. 173: \\lim \\dfrac{\\cosec x - \\cot x}{x} = \\frac{1}{2}"],["todhunter-spherical-trigonometry-1886/ex-xii/6",4,"Todhunter 1886, Exercise XII (6)"],["wentworth-plane-geometry-1899/x-a7d77f19da",15,"Wentworth 1899, scan 16: A straight line is a line such that any ..."],["wentworth-plane-geometry-1899/x-621a62826d",15,"Wentworth 1899, scan 18: The size of an angle depends upon the extent ..."],["wentworth-plane-geometry-1899/x-ccc6172099",15,"Wentworth 1899, scan 21: Suppose the straight line OC (Fig. 15) to move ..."],["wentworth-plane-geometry-1899/x-38137a24a8",15,"Wentworth 1899, scan 22: The natural angular unit is one complete revolution. But ..."],["wentworth-plane-geometry-1899/x-61570d908d",15,"Wentworth 1899, scan 28: Fold over CFA, on CF as an axis, until ..."],["todhunter-spherical-trigonometry-1886/ex-xii/7",4,"Todhunter 1886, Exercise XII (7)"],["todhunter-spherical-trigonometry-1886/ex-xii/9",4,"Todhunter 1886, Exercise XII (9)"],["todhunter-spherical-trigonometry-1886/ex-xii/11",4,"Todhunter 1886, Exercise XII (11)"],["form/eb21d661d9",5,"identity: 2*a**2*(-b**15)**(1/5)"],["wentworth-plane-geometry-1899/x-b90976307c",15,"Wentworth 1899, scan 24: The beginner must not forget that in Plane Geometry ..."],["wentworth-plane-geometry-1899/x-236943ac8f",15,"Wentworth 1899, scan 30: The perpendicular is the shortest line that can be ..."],["todhunter-spherical-trigonometry-1886/ex-xiii/1",4,"Todhunter 1886, Exercise XIII (1)"],["todhunter-spherical-trigonometry-1886/ex-xiii/2",4,"Todhunter 1886, Exercise XIII (2)"],["todhunter-spherical-trigonometry-1886/ex-xiii/3",4,"Todhunter 1886, Exercise XIII (3)"],["form/909d6f57a8",5,"identity: 3*b**4*(-a**9)**(1/3)/4"],["shape/93637be627",6,"identity: N*b**N*(-a**N)**N"],["planck-treatise-on-thermodynamics-1903/eq-9690d8638e",16,"Planck 1903, p. 5: \\frac{V}{M} = v"],["planck-treatise-on-thermodynamics-1903/eq-53ff5ff8ae",16,"Planck 1903, p. 5: p = f(v, t)"],["hardy-course-of-pure-mathematics-1921/eq-5f1c9d1de8",16,"Hardy 1921, p. 173: \\lim\\limits_{x \\to 1} \\dfrac{1 + \\cos \\pi x}{\\tan^{2}\\pi x} = \\frac{1}{2}"],["form/96379c04c8",5,"identity: 2*a**2*Abs(b)"],["shape/46ddc2908e",6,"identity: N*a**N*Abs(b)"],["todhunter-spherical-trigonometry-1886/ex-xiii/5",4,"Todhunter 1886, Exercise XIII (5)"],["form/38562016bb",5,"identity: 2*(-a**5*b**15)**(1/5)/3"],["shape/a4505b591b",6,"identity: N*(-a**N*b**N)**N"],["todhunter-spherical-trigonometry-1886/ex-xiii/6",4,"Todhunter 1886, Exercise XIII (6)"],["todhunter-spherical-trigonometry-1886/ex-xiii/7",4,"Todhunter 1886, Exercise XIII (7)"],["todhunter-spherical-trigonometry-1886/ex-xiii/8",4,"Todhunter 1886, Exercise XIII (8)"],["planck-treatise-on-thermodynamics-1903/eq-eddb0eef33",16,"Planck 1903, p. 5: pv = T"],["planck-treatise-on-thermodynamics-1903/eq-c9fe1f44ed",16,"Planck 1903, p. 5: t = (v - v_{0})P"],["concept/isobaric-change",7,"isobaric change"],["form/344a6f41fd",5,"identity: c**2*Abs(a - b)"],["shape/b8b9e4ff1c",6,"identity: c**N*Abs(a - b)"],["todhunter-spherical-trigonometry-1886/ex-xiii/9",4,"Todhunter 1886, Exercise XIII (9)"],["todhunter-spherical-trigonometry-1886/ex-xiii/11",4,"Todhunter 1886, Exercise XIII (11)"],["todhunter-spherical-trigonometry-1886/ex-xiii/12",4,"Todhunter 1886, Exercise XIII (12)"],["form/ece3068643",5,"identity: -a**2*(-b**3)**(1/3) + a**2*(-b**5)**(1/5)/2 + a**2*(b**3)**(1/3)/2 - a**2*Abs(b)/2"],["shape/49f98a45a4",6,"identity: N*a**N*(-b**N)**N + N*a**N*(b**N)**N + N*a**N*Abs(b) - a**N*(-b**N)**N"],["planck-treatise-on-thermodynamics-1903/eq-f166490237",16,"Planck 1903, p. 5: pv_{0} = T_{0}"],["planck-treatise-on-thermodynamics-1903/eq-ac192091cd",16,"Planck 1903, p. 6: v - v_{0} = \\alpha v_{0}"],["hardy-course-of-pure-mathematics-1921/eq-13dd732abe",16,"Hardy 1921, p. 171: \\lim\\limits_{x \\to a} (x^{2} - a^{2})/(x - a) = 2a"],["form/0e5e346797",5,"identity: -15*a**2*b**2*c**4*(-a**3)**(1/3)*Abs(b)*Abs(c)"],["shape/fd5217a9d7",6,"identity: N*a**N*b**N*c**N*(-a**N)**N*Abs(b)*Abs(c)"],["todhunter-spherical-trigonometry-1886/ex-xiii/15",4,"Todhunter 1886, Exercise XIII (15)"],["planck-treatise-on-thermodynamics-1903/eq-bde3d0cdce",16,"Planck 1903, p. 6: 1 = \\alpha v_{0} P\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-b3510bc128",16,"Planck 1903, p. 6: T = T_{0} (1 + \\alpha t)"],["form/82f19f94f7",5,"identity: 5*a**2*b**6*Abs(a)/(a**15*b**25)**(1/5)"],["hardy-course-of-pure-mathematics-1921/eq-f381ef55bd",16,"Hardy 1921, p. 171: \\lim\\limits_{x \\to a} (x^{m} - a^{m})/(x - a) = ma^{m-1}"],["todhunter-spherical-trigonometry-1886/ex-xv/2",4,"Todhunter 1886, Exercise XV (2)"],["planck-treatise-on-thermodynamics-1903/eq-d341be6921",16,"Planck 1903, p. 6: p = \\frac{T_{0}}{v} (1 + \\alpha t)"],["planck-treatise-on-thermodynamics-1903/eq-2d719e3138",16,"Planck 1903, p. 6: t + \\dfrac{1}{\\alpha} = \\theta"],["planck-treatise-on-thermodynamics-1903/eq-66a0b8bd64",16,"Planck 1903, p. 6: \\alpha T_{0} = C"],["planck-treatise-on-thermodynamics-1903/eq-dc3e77bfb0",16,"Planck 1903, p. 6: p = \\frac{C}{v} \\theta = \\frac{CM}{V} \\theta\\Add{.}"],["boyden-first-book-in-algebra-1895/ex-13/1",4,"Boyden 1895, Exercise 13 (1)"],["planck-treatise-on-thermodynamics-1903/eq-0225f4d7d6",16,"Planck 1903, p. 7: \\frac{1}{273} = \\alpha"],["hardy-course-of-pure-mathematics-1921/eq-15df879609",16,"Hardy 1921, p. 171: \\lim\\limits_{x \\to 1} (x^{7} - 2x^{5} + 1)/(x^{3} - 3x^{2} + 2) = 1"],["hardy-course-of-pure-mathematics-1921/eq-c9750fc98e",16,"Hardy 1921, p. 169: \\lim\\limits_{x \\to a} x^{m} = a^{m}"],["form/13efc14f6d",5,"solve: (Eq(x, a - 6), Eq(b - 6, x/2))"],["todhunter-spherical-trigonometry-1886/ex-xv/3",4,"Todhunter 1886, Exercise XV (3)"],["todhunter-spherical-trigonometry-1886/ex-xv/4",4,"Todhunter 1886, Exercise XV (4)"],["todhunter-spherical-trigonometry-1886/ex-xv/6",4,"Todhunter 1886, Exercise XV (6)"],["planck-treatise-on-thermodynamics-1903/eq-1aebc0e362",16,"Planck 1903, p. 8: dV = \\frac{CM\\theta}{p^{2}}\\, dp = \\frac{V}{p}\\, dp"],["todhunter-spherical-trigonometry-1886/ex-xv/7",4,"Todhunter 1886, Exercise XV (7)"],["planck-treatise-on-thermodynamics-1903/eq-5f34473412",16,"Planck 1903, p. 8: -\\frac{dV}{V} = \\frac{dp}{p}"],["planck-treatise-on-thermodynamics-1903/eq-25549c8500",16,"Planck 1903, p. 8: \\frac{\\;\\;dp\\;\\;}{\\dfrac{dp}{p}} = p"],["theorem/coefficient-of-elasticity-of-a-perfect-gas",9,"coefficient of elasticity of a perfect gas"],["hardy-course-of-pure-mathematics-1921/eq-5953441da9",16,"Hardy 1921, p. 169: \\lim\\limits_{x \\to a} R(x) = R(a)"],["hardy-course-of-pure-mathematics-1921/eq-e1444cb45d",16,"Hardy 1921, p. 169: \\lim\\limits_{x \\to 0} (a + bx + cx^{2} + \\dots + kx^{m}) = a"],["form/63f6c0b694",5,"solve: Eq(a - 50, -x + 40)"],["todhunter-spherical-trigonometry-1886/ex-xv/8",4,"Todhunter 1886, Exercise XV (8)"],["todhunter-spherical-trigonometry-1886/ex-xv/11",4,"Todhunter 1886, Exercise XV (11)"],["todhunter-spherical-trigonometry-1886/ex-xv/14",4,"Todhunter 1886, Exercise XV (14)"],["planck-treatise-on-thermodynamics-1903/eq-1d0ee8e408",16,"Planck 1903, p. 8: dp = \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v} d\\theta + \\left(\\frac{\\dd p}{\\dd v}\\right)_{\\theta} dv"],["todhunter-spherical-trigonometry-1886/ex-xv/15",4,"Todhunter 1886, Exercise XV (15)"],["todhunter-spherical-trigonometry-1886/ex-xv/16",4,"Todhunter 1886, Exercise XV (16)"],["planck-treatise-on-thermodynamics-1903/eq-506a29afad",16,"Planck 1903, p. 8: \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p} = -\\frac{\\left(\\dfrac{\\dd p}{\\dd \\theta}\\right)_{v}}{\\left(\\dfrac{\\dd p}{\\dd v"],["hardy-course-of-pure-mathematics-1921/eq-0e226d43ac",16,"Hardy 1921, p. 169: \\lim\\limits_{x \\to 0} \\left\\{(a + bx + \\dots + kx^{m})/(\\alpha + \\beta x + \\dots + \\kappa x^{\\mu})\\right\\} = a/\\alpha"],["form/5c2c352098",5,"solve: Eq(x**2, 16*a**2)"],["todhunter-spherical-trigonometry-1886/ex-xv/18",4,"Todhunter 1886, Exercise XV (18)"],["form/3f26f324e5",5,"solve: Eq(exp(3), 64*a**3)"],["shape/a31b65044b",6,"solve: Eq(exp(N), N*a**N)"],["todhunter-spherical-trigonometry-1886/ex-xv/20",4,"Todhunter 1886, Exercise XV (20)"],["planck-treatise-on-thermodynamics-1903/eq-7c3f6b8493",16,"Planck 1903, p. 9: \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v} = -\\left(\\frac{\\dd p}{\\dd v}\\right)_{\\theta} · \\left(\\frac{\\dd v}{\\dd \\theta}\\"],["planck-treatise-on-thermodynamics-1903/eq-b8684b27cc",16,"Planck 1903, p. 9: \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p} · \\frac{1}{v_{0}} = 0.00018"],["planck-treatise-on-thermodynamics-1903/eq-9fea9ca451",16,"Planck 1903, p. 9: -\\left(\\frac{\\dd v}{\\dd p}\\right)_{\\theta} · \\frac{1}{v_{0}} = 0.000003"],["planck-treatise-on-thermodynamics-1903/eq-1a780895d3",16,"Planck 1903, p. 10: p = \\frac{C_{1}M_{1} \\theta}{V_{1}}"],["hardy-course-of-pure-mathematics-1921/eq-ab462dc14d",16,"Hardy 1921, p. 168: \\phi(x) + \\psi(x) \\to l + l'"],["todhunter-spherical-trigonometry-1886/ex-xvi",3,"Todhunter 1886, Exercise XVI"],["todhunter-spherical-trigonometry-1886/ex-xvi/1",4,"Todhunter 1886, Exercise XVI (1)"],["todhunter-spherical-trigonometry-1886/ex-xvi/2",4,"Todhunter 1886, Exercise XVI (2)"],["todhunter-spherical-trigonometry-1886/ex-xvi/3",4,"Todhunter 1886, Exercise XVI (3)"],["todhunter-spherical-trigonometry-1886/ex-xvi/4",4,"Todhunter 1886, Exercise XVI (4)"],["planck-treatise-on-thermodynamics-1903/eq-7d83b5837e",16,"Planck 1903, p. 10: V = V_{1} + V_{2} + \\dots"],["planck-treatise-on-thermodynamics-1903/eq-b59dacb538",16,"Planck 1903, p. 10: p_{1} = \\frac{C_{1}M_{1} \\theta}{V} = \\frac{V_{1}}{V} p"],["form/2be0c60d7f",5,"solve: (Eq(a, 5*c), Eq(b, 4*c), Eq(a + b + c, 15/2))"],["todhunter-spherical-trigonometry-1886/ex-xvi/5",4,"Todhunter 1886, Exercise XVI (5)"],["todhunter-spherical-trigonometry-1886/ex-xvi/6",4,"Todhunter 1886, Exercise XVI (6)"],["todhunter-spherical-trigonometry-1886/ex-xvi/7",4,"Todhunter 1886, Exercise XVI (7)"],["todhunter-spherical-trigonometry-1886/ex-xvi/8",4,"Todhunter 1886, Exercise XVI (8)"],["planck-treatise-on-thermodynamics-1903/eq-e2a61a8415",16,"Planck 1903, p. 10: p_{1} + p_{2} + \\dots = \\frac{V_{1} + V_{2} + \\dots}{V} p = p\\Add{.}"],["hardy-course-of-pure-mathematics-1921/eq-5e46353463",16,"Hardy 1921, p. 168: \\phi(x)\\psi(x) \\to ll'"],["planck-treatise-on-thermodynamics-1903/eq-9595a4c42d",16,"Planck 1903, p. 11: p_{1} : p_{2} : \\dots = V_{1} : V_{2} : \\dots = C_{1}M_{1} : C_{2}M_{2} : \\dots\\Add{,}"],["planck-treatise-on-thermodynamics-1903/eq-302e25586a",16,"Planck 1903, p. 11: p = (C_{1}M_{1} + C_{2}M_{2} + \\dots) \\frac{\\theta}{V}"],["form/a122633c22",5,"solve: (Eq(b, 3*a), Eq(c, 2*b), Eq(a + b + c, 45000))"],["theorem/characteristic-equation-of-a-gas-mixture",9,"characteristic equation of a gas mixture"],["planck-treatise-on-thermodynamics-1903/eq-10b71dd3eb",16,"Planck 1903, p. 11: C = \\frac{C_{1}M_{1} + C_{2}M_{2} + \\dots}{M_{1} + M_{2} + \\dots}"],["theorem/characteristic-constant",9,"characteristic constant"],["hardy-course-of-pure-mathematics-1921/eq-4a93597be2",16,"Hardy 1921, p. 168: \\phi(x)/\\psi(x) \\to l/l'"],["planck-treatise-on-thermodynamics-1903/eq-24a672e192",16,"Planck 1903, p. 11: 0.0014291 : 0.0012571 : 0.0012930 = \\frac{1}{C_{1}} : \\frac{1}{C_{2}} : \\frac{1}{C_{3}}"],["hardy-course-of-pure-mathematics-1921/eq-43684d7c82",16,"Hardy 1921, p. 172: \\lim_{x\\to 0} (x^{2}/x) = 0"],["form/27317f2ba6",5,"evaluate: 97"],["planck-treatise-on-thermodynamics-1903/eq-4f736f3404",16,"Planck 1903, p. 12: C = \\frac{C_{1}M_{1} + C_{2}M_{2}}{M_{1} + M_{2}}"],["planck-treatise-on-thermodynamics-1903/eq-defb0bc16e",16,"Planck 1903, p. 12: M_{1} : M_{2} = 0.2998"],["concept/composition-of-gas-mixture",7,"composition of gas mixture"],["concept/air",7,"air"],["form/0c098b75c6",5,"evaluate: 7/2500"],["form/1358c0635a",5,"evaluate: 39/10"],["planck-treatise-on-thermodynamics-1903/eq-c76c2da803",16,"Planck 1903, p. 12: C_{1}M_{1} : C_{2}M_{2} = p_{1} : p_{2} = V_{1} : V_{2} = 0.2637"],["hardy-course-of-pure-mathematics-1921/eq-5f131319e9",16,"Hardy 1921, p. 172: \\phi(x)/x^{-k} = x^{k}\\phi(x)"],["form/5b50350996",5,"evaluate: 101/2"],["planck-treatise-on-thermodynamics-1903/eq-58dd8f0b82",16,"Planck 1903, p. 13: pv = \\const"],["planck-treatise-on-thermodynamics-1903/eq-b1f744f3ab",16,"Planck 1903, p. 13: p = \\frac{R\\theta}{v - b} - \\frac{a}{v^{2}}"],["planck-treatise-on-thermodynamics-1903/eq-df53eb51b4",16,"Planck 1903, p. 14: p = \\frac{R\\theta}{v - a} - \\frac{c}{\\theta(v + b)^{2}}"],["planck-treatise-on-thermodynamics-1903/eq-4c2e4e323f",16,"Planck 1903, p. 18: \\left(\\frac{\\dd p}{\\dd v}\\right)_{\\theta} = 0"],["planck-treatise-on-thermodynamics-1903/eq-a7054d6a70",16,"Planck 1903, p. 18: \\left(\\frac{\\dd^{2} p}{\\dd v^{2}}\\right)_{\\theta} = 0"],["hardy-course-of-pure-mathematics-1921/eq-765c57ba10",16,"Hardy 1921, p. 172: \\lim\\sqrtp{1 + x} = \\lim\\sqrtp{1 - x} = 1"],["form/65b781316f",5,"solve: Eq(x**3, 1155000)"],["form/ea2399b033",5,"solve: Eq(x**2, 8464)"],["planck-treatise-on-thermodynamics-1903/eq-e83cbe875a",16,"Planck 1903, p. 19: \\theta^{2} = \\frac{8c}{27(a + b)R}"],["hardy-course-of-pure-mathematics-1921/eq-a56e34edc4",16,"Hardy 1921, p. 172: \\lim\\{\\sqrtp{1 + x} - \\sqrtp{1 - x}\\}/x = 1"],["de-morgan-elementary-illustrations-calculus-1899/eq-5112129174",16,"De Morgan 1899, p. 38: y = \\log x"],["planck-treatise-on-thermodynamics-1903/eq-9bfc6ab936",16,"Planck 1903, p. 19: p^{2} = \\frac{cR}{216(a + b)^{3}}"],["planck-treatise-on-thermodynamics-1903/eq-f4e79629cc",16,"Planck 1903, p. 19: v = 3a + 2b"],["form/718981f08a",5,"factor: 4*a**5 + 2*a**4 + 6*a**3"],["whitehead-introduction-to-mathematics-1911/eq-658b227017",16,"Whitehead 1911, p. 83: x + a = b"],["whitehead-introduction-to-mathematics-1911/eq-3295306d5b",16,"Whitehead 1911, p. 83: x = b - a"],["de-morgan-elementary-illustrations-calculus-1899/eq-cef0aaf15d",16,"De Morgan 1899, p. 40: MM' = dx"],["de-morgan-elementary-illustrations-calculus-1899/eq-869ac9c8cf",16,"De Morgan 1899, p. 15: \\phi x = x + x^{2}"],["todhunter-spherical-trigonometry-1886/eq-0f31c72245",16,"Todhunter 1886, scan 49: a_1 + p_1 = a_2 + p_2 = a_5 + p_5 = \\dfrac{\\pi}{2}"],["form/94883260ed",5,"factor: 3*a*b**7 - 9*a*b**6 + 9*a*b**5 - 3*a*b**4 - 24*a*b"],["wentworth-plane-geometry-1899/eq-c4cadae58d",16,"Wentworth 1899, scan 41: \\angle A+\\angle B+\\angle BCA = 2"],["wentworth-plane-geometry-1899/eq-934ea52728",16,"Wentworth 1899, scan 42: AB + BC > AC"],["hardy-course-of-pure-mathematics-1921/eq-d299d6057e",16,"Hardy 1921, p. 173: \\lim\\{\\sqrtp{1 + x + x^{2}} - 1\\}/x = \\frac{1}{2}"],["wentworth-plane-geometry-1899/eq-bd03195dd6",16,"Wentworth 1899, scan 42: AC - BC < AB"],["wentworth-plane-geometry-1899/eq-2333a4036b",16,"Wentworth 1899, scan 42: AB > AC"],["hardy-course-of-pure-mathematics-1921/eq-900c16a3a7",16,"Hardy 1921, p. 168: H < \\phi(x) < K"],["form/d4618d09fa",5,"factor: 2*x**4 - x**3 + 4*x - 2"],["wentworth-plane-geometry-1899/eq-93a402e5e4",16,"Wentworth 1899, scan 61: AO = OC"],["wentworth-plane-geometry-1899/eq-2bdaf7feae",16,"Wentworth 1899, scan 61: BO = OE"],["hardy-course-of-pure-mathematics-1921/eq-1597cea1ff",16,"Hardy 1921, p. 168: \\lambda = \\Lambda = l"],["wentworth-plane-geometry-1899/eq-1f81b4de1f",16,"Wentworth 1899, scan 58: BC = AE"],["boyden-first-book-in-algebra-1895/ex-36/3",4,"Boyden 1895, Exercise 36 (3)"],["boyden-first-book-in-algebra-1895/ex-36/4",4,"Boyden 1895, Exercise 36 (4)"],["form/04078b61eb",5,"factor: 4*a**2 + 4*a*x + x**2"],["boyden-first-book-in-algebra-1895/ex-36/5",4,"Boyden 1895, Exercise 36 (5)"],["boyden-first-book-in-algebra-1895/ex-36/6",4,"Boyden 1895, Exercise 36 (6)"],["form/2e35985ad5",5,"factor: a**2 - 8*a*x + 16*x**2"],["wentworth-plane-geometry-1899/eq-1f0cb18888",16,"Wentworth 1899, scan 58: AB = EC"],["form/c6cd1a58a3",5,"factor: a**2 + 4*a*x + 4*x**2"],["wentworth-plane-geometry-1899/eq-2497f58cdc",16,"Wentworth 1899, scan 63: AB = BC = CD"],["wentworth-plane-geometry-1899/eq-989cdce328",16,"Wentworth 1899, scan 64: BF=FC = \\frac{1}{2}BC"],["form/0545849eb8",5,"factor: -a**6*b**4 + b**2*x**4"],["concept/midpoint",7,"midpoint"],["wentworth-plane-geometry-1899/eq-089f849d62",16,"Wentworth 1899, scan 64: DE = BF = \\frac{1}{2}BC"],["todhunter-spherical-trigonometry-1886/eq-bdfe4a7981",16,"Todhunter 1886, scan 49: p_3 = a_3"],["form/ee6404cdb5",5,"factor: -64*a**2 + 121*x**2"],["form/2fcc7f1c0a",5,"factor: -a**4 + x**4"],["wentworth-plane-geometry-1899/eq-7db4069704",16,"Wentworth 1899, scan 64: \\frac{1}{2} (AB + DC)"],["concept/median-of-a-trapezoid",7,"median of a trapezoid"],["todhunter-spherical-trigonometry-1886/eq-990d540bee",16,"Todhunter 1886, scan 49: p_4 = a_4"],["de-morgan-elementary-illustrations-calculus-1899/eq-c52798580e",16,"De Morgan 1899, p. 38: y + dy = \\log x + \\dfrac{1}{x}\\, dx - \\dfrac{1}{2x^{2}}\\, dx^{2}"],["hardy-course-of-pure-mathematics-1921/ex-lx",3,"Hardy 1921, Exercise LX"],["hardy-course-of-pure-mathematics-1921/ex-lxi",3,"Hardy 1921, Exercise LXI"],["hardy-course-of-pure-mathematics-1921/ex-lxii",3,"Hardy 1921, Exercise LXII"],["hardy-course-of-pure-mathematics-1921/x-6ad252a68c",15,"Hardy 1921, p. 200: The notion of a derivative or differential coefficient was ..."],["hardy-course-of-pure-mathematics-1921/x-a3c2603998",15,"Hardy 1921, p. 200: The notion of ‘velocity’ is in fact merely a ..."],["boyden-first-book-in-algebra-1895/ex-13/2",4,"Boyden 1895, Exercise 13 (2)"],["hardy-course-of-pure-mathematics-1921/eq-988f8eec7c",16,"Hardy 1921, p. 168: |\\phi(x_{2}) - \\phi(x_{1})| < \\DELTA"],["form/df743bb26a",5,"factor: -a**2 - a + x**2 - x"],["hardy-course-of-pure-mathematics-1921/eq-c8d50f3c9b",16,"Hardy 1921, p. 165: \\phi(x_{2}) \\geq \\phi(x_{1})"],["hardy-course-of-pure-mathematics-1921/eq-77f7a27b29",16,"Hardy 1921, p. 165: \\phi(x_{2}) > \\phi(x_{1})"],["whitehead-introduction-to-mathematics-1911/x-7e26e20772",15,"Whitehead 1911, p. 88: The equation x^{2} + 1 = 3 becomes x^{2} ..."],["form/08d73aa506",5,"factor: a**3 + x**3"],["theorem/existence-of-an-integral-of-a-continuous-function",9,"existence of an integral of a continuous function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-existence-of-an-integral-of-a-continuous-function"],["planck-treatise-on-thermodynamics-1903/eq-9419f48d5a",16,"Planck 1903, p. 110: U - \\theta\\Phi = F"],["form/460b759f77",5,"factor: a**6 + 64*x**6"],["whitehead-introduction-to-mathematics-1911/x-347838c23c",15,"Whitehead 1911, p. 91: Nothing can be proved by a succession of blots, ..."],["whitehead-introduction-to-mathematics-1911/x-c05ff2faef",15,"Whitehead 1911, p. 97: All these requisites are satisfied by taking (x, y) ..."],["form/37b878ccca",5,"factor: x**3/27 + 1"],["wentworth-plane-geometry-1899/eq-34832eefd9",16,"Wentworth 1899, scan 217: \\dfrac{a}{2} × \\dfrac{a\\sqrt{3}}{2} = \\dfrac{a^2\\sqrt{3}}{4}"],["dickson-theory-of-equations-1922/eq-5fc07deaa2",16,"Dickson 1922, p. 47: (A-B)(A-\\omega B)(A-\\omega^2 B) = A^3 - B^3 = 2 \\sqrt{R}"],["dickson-theory-of-equations-1922/eq-9d97af5edc",16,"Dickson 1922, p. 47: (1-\\omega)(1-\\omega^2) = 3"],["hardy-course-of-pure-mathematics-1921/x-331bef581e",15,"Hardy 1921, p. 217: An immediate deduction from Theorem A is the following ..."],["hardy-course-of-pure-mathematics-1921/x-99a41d97cd",15,"Hardy 1921, p. 220: Thus if y = x^{3} then \\phi'(x) = 3x^{2}, ..."],["hardy-course-of-pure-mathematics-1921/x-6d93bb8b96",15,"Hardy 1921, p. 220: Suppose, e.g., that \\phi''(\\xi) < 0. Then, by Theorem ..."],["dickson-theory-of-equations-1922/eq-77315f5426",16,"Dickson 1922, p. 47: \\omega - \\omega^2 = \\sqrt{3}i"],["form/3c1e7a3f39",5,"solve: Eq(x + 8, a)"],["dickson-theory-of-equations-1922/ch-iv",2,"Dickson 1922, ch. IV: Solution of Cubic and Quartic Equations; Their Discriminants","../books/dickson-theory-of-equations-1922/ch/ch-iv/index.html"],["dickson-theory-of-equations-1922/eq-e7621149f0",16,"Dickson 1922, p. 45: x^3 + bx^2 + cx + d = 0"],["dickson-theory-of-equations-1922/eq-3c8c1052e2",16,"Dickson 1922, p. 45: y^3 + py + q = 0"],["form/40234a2745",5,"factor: -a**3 + x**3"],["form/4805cbc22e",5,"factor: -a**3*b**3 + x**3"],["concept/integration-by-substitution",7,"integration by substitution","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-integration-by-substitution"],["method/integration-by-rationalisation",8,"integration by rationalisation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-integration-by-rationalisation"],["form/9014d9f172",5,"factor: 64*a**3*b**3*x**6 - 125*c**3*d**9*e**6"],["theorem/extreme-value-theorem",9,"extreme value theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-extreme-value-theorem"],["theorem/boundedness-of-a-continuous-function",9,"boundedness of a continuous function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-boundedness-of-a-continuous-function"],["form/40950dc36a",5,"factor: -a**3 + x**6"],["boyden-first-book-in-algebra-1895/ex-35/18",4,"Boyden 1895, Exercise 35 (18)"],["method/integration-of-rational-functions",8,"integration of rational functions","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-integration-of-rational-functions"],["concept/chord",7,"chord","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-chord"],["hardy-course-of-pure-mathematics-1921/x-b3f984015f",15,"Hardy 1921, p. 188: The reader may be tempted to think that this ..."],["form/162c62590c",5,"factor: a*b**3 + a*x**3 - b - x"],["shape/20ee8dab6d",6,"factor: a*b**N + a*x**N - b - x"],["dickson-theory-of-equations-1922/eq-6596361c2f",16,"Dickson 1922, p. 45: p = c - \\frac{b^2}{3}"],["dickson-theory-of-equations-1922/eq-4e83469fb5",16,"Dickson 1922, p. 45: q = d - \\frac{bc}{3} + \\frac{2b^3}{27}"],["form/bc9b7c9e40",5,"factor: a**2 - 2*a*x + x**2"],["hardy-course-of-pure-mathematics-1921/x-a184042fb3",15,"Hardy 1921, p. 227: For \\phi'(\\xi) is the tangent of the angle which ..."],["form/d4adb9824f",5,"factor: x**2 - 10*x + 25"],["hardy-course-of-pure-mathematics-1921/x-6e3c43f393",15,"Hardy 1921, p. 228: In the first place we want to know whether ..."],["concept/senate-house-examination",7,"Senate-House examination","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-senate-house-examination"],["form/79c41867d8",5,"factor: 4*x**4 - 12*x**2 + 9"],["concept/poll-examination",7,"Poll examination","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-poll-examination"],["de-morgan-elementary-illustrations-calculus-1899/eq-f221481e7c",16,"De Morgan 1899, p. 46: (a + da)^{2} + (b - db)^{2} = l^{2}"],["concept/analytical-society",7,"Analytical Society","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-analytical-society"],["de-morgan-elementary-illustrations-calculus-1899/eq-7586f33bb7",16,"De Morgan 1899, p. 46: 2a\\, da + (da)^{2} - 2b\\, db + (db)^{2} = 0\\Add{,}"],["de-morgan-elementary-illustrations-calculus-1899/eq-8b54a8bd74",16,"De Morgan 1899, p. 46: \\frac{db}{da} = \\frac{2a + da}{2b - db}\\Add{.}"],["form/4345fe519d",5,"factor: a**4 - 2*a**2*(b + x) + (b + x)**2"],["shape/470c771a7f",6,"factor: N*a**N*(b + x) + a**N + (b + x)**N"],["de-morgan-elementary-illustrations-calculus-1899/eq-f0f7008416",16,"De Morgan 1899, p. 47: ay + bx = ab\\Add{.}"],["hardy-course-of-pure-mathematics-1921/x-5ce26a510f",15,"Hardy 1921, p. 229: Whether there are continuous functions which never have derivatives, ..."],["person/robert-woodhouse",1,"Robert Woodhouse","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-robert-woodhouse"],["planck-treatise-on-thermodynamics-1903/eq-6fbf277d92",16,"Planck 1903, p. 110: dF = W"],["planck-treatise-on-thermodynamics-1903/eq-d5b1ed871a",16,"Planck 1903, p. 110: F_{2} - F_{1} = \\tsum W"],["person/george-peacock",1,"George Peacock","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-george-peacock"],["person/charles-babbage",1,"Charles Babbage","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-charles-babbage"],["person/john-herschel",1,"John Herschel","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-john-herschel"],["person/william-whewell",1,"William Whewell","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-william-whewell"],["person/george-airy",1,"George Airy","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-george-airy"],["ball-mathematical-recreations-1905/x-a0cb48a10b",15,"Ball 1905, scan 241: They created an Analytical Society which Babbage explained was ..."],["ball-mathematical-recreations-1905/x-91acd04955",15,"Ball 1905, scan 241: In 1817 Peacock, who was moderator, introduced the symbols ..."],["ball-mathematical-recreations-1905/x-25c04b851f",15,"Ball 1905, scan 242: It is by silent perseverance only that we can ..."],["form/795982f51b",5,"solve: Eq(3*x - 27, a - 9)"],["shape/d8ff0b242a",6,"solve: Eq(N*x + N, N + a)"],["hardy-course-of-pure-mathematics-1921/x-8666fffac5",15,"Hardy 1921, p. 230: It is hardly necessary to point out that \\int\\dots ..."],["boyden-first-book-in-algebra-1895/ex-37/9",4,"Boyden 1895, Exercise 37 (9)"],["form/abab68a2e9",5,"factor: x**2 + 8*x - 65"],["ball-mathematical-recreations-1905/x-1e6fc1d27b",15,"Ball 1905, scan 243: We are employed from seven in the morning till ..."],["concept/radius",7,"radius","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-radius"],["method/integration-of-algebraic-functions",8,"integration of algebraic functions","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-integration-of-algebraic-functions"],["method/tangent-half-angle-substitution",8,"tangent half-angle substitution","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-tangent-half-angle-substitution"],["ball-mathematical-recreations-1905/x-e8f924ac14",15,"Ball 1905, scan 250: The assignment of marks to groups of subjects was ..."],["method/integration-of-an-inverse-function",8,"integration of an inverse function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-integration-of-an-inverse-function"],["boyden-first-book-in-algebra-1895/x-ffcfaee34b",15,"Boyden 1895: Negative numbers are usually spoken of as less than ..."],["form/0ec2f1c87c",5,"factor: a*x**2 - 5*a*x + 6*a + b*x**2 - 5*b*x + 6*b"],["hardy-course-of-pure-mathematics-1921/x-16e8961eb8",15,"Hardy 1921, p. 276: The fact is of course that \\dd x/\\dd r ..."],["boyden-first-book-in-algebra-1895/ex-14",3,"Boyden 1895, Exercise 14"],["form/8f3d96da51",5,"identity: (-a + x)*(a + x)*(a**2 + x**2)"],["shape/b553331a04",6,"identity: (-a + x)*(a + x)*(a**N + x**N)"],["theorem/fundamental-theorem-of-calculus",9,"fundamental theorem of calculus","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-fundamental-theorem-of-calculus"],["concept/conic",7,"conic","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-conic"],["boyden-first-book-in-algebra-1895/ex-38/16",4,"Boyden 1895, Exercise 38 (16)"],["form/2db45efaed",5,"identity: -(-a + 2*x)**2 + (2*a + x)**2"],["ball-mathematical-recreations-1905/x-8960e69a4c",15,"Ball 1905, scan 254: In 1895 the proctors and moderators, without consulting the ..."],["concept/resolvent-cubic",7,"resolvent cubic","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-resolvent-cubic"],["theorem/number-of-real-roots-of-a-cubic",9,"number of real roots of a cubic","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-number-of-real-roots-of-a-cubic"],["theorem/cardan-s-formulas",9,"Cardan's formulas","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-cardan-s-formulas"],["theorem/discriminant-formula-for-the-cubic",9,"discriminant formula for the cubic","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-discriminant-formula-for-the-cubic"],["boyden-first-book-in-algebra-1895/ex-39/1",4,"Boyden 1895, Exercise 39 (1)"],["form/a866744e96",5,"hcf: (18*a**3*x + 9*a**2*x**3, 6*a**5*x**2 + 3*a**3*x**4)"],["method/ferrari-s-solution-of-the-quartic-equation",8,"Ferrari's solution of the quartic equation","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-ferrari-s-solution-of-the-quartic-equation"],["boyden-first-book-in-algebra-1895/ex-39/6",4,"Boyden 1895, Exercise 39 (6)"],["hardy-course-of-pure-mathematics-1921/x-46b739e02e",15,"Hardy 1921, p. 247: The integral of any rational function of \\cos x ..."],["method/descartes-solution-of-the-quartic-equation",8,"Descartes' solution of the quartic equation","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-descartes-solution-of-the-quartic-equation"],["method/trigonometric-solution-of-a-cubic",8,"trigonometric solution of a cubic","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-trigonometric-solution-of-a-cubic"],["form/ee959b530a",5,"hcf: (-a**2 + x**2, -a**3 + a**2*x - 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4*e)*(a*e**2 + b*c*e)/((a**2*d - 2*a*e)*(a*c*e + b*c**2))"],["shape/f47a6aab2a",6,"identity: (a*e**N + b*c*e)*(N*a*d + N*e)/((N*a*e + a**N*d)*(a*c*e + b*c**N))"],["boyden-first-book-in-algebra-1895/ex-48/7",4,"Boyden 1895, Exercise 48 (7)"],["form/b96b642f1e",5,"identity: (a**2 - a - 6)*(a**2 + 3*a - 4)/((a**2 - 2*a - 3)*(a**2 + a - 2))"],["theorem/sides-opposite-equal-angles-of-a-spherical-triangle-are-equal",9,"sides opposite equal angles of a spherical triangle are equal","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-sides-opposite-equal-angles-of-a-spherical-triangle-are-equal"],["theorem/greater-angle-lies-opposite-greater-side-in-a-spherical-triangle",9,"greater angle lies opposite greater side in a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-greater-angle-lies-opposite-greater-side-in-a-spherical-triangle"],["theorem/greater-side-lies-opposite-greater-angle-in-a-spherical-triangle",9,"greater side lies opposite greater angle in a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-greater-side-lies-opposite-greater-angle-in-a-spherical-triangle"],["form/76ea36c04c",5,"identity: 3*c**3/(-b**3 + c**3) + 1/(b - c) - 1/(3*a + 3)"],["shape/2f5ea125d0",6,"identity: (N - a + a**N)/(N + a + a**N)"],["boyden-first-book-in-algebra-1895/ex-48/8",4,"Boyden 1895, Exercise 48 (8)"],["form/b3595c55ce",5,"identity: (a**2 + 2*a - 3)*(a**2 + 7*a + 10)/((a + 3)*(a**2 + a - 2))"],["shape/b01455d717",6,"identity: (N*a + N + a**N)**2/((N + a)*(N + a + a**N))"],["boyden-first-book-in-algebra-1895/ex-58/6",4,"Boyden 1895, Exercise 58 (6)"],["todhunter-spherical-trigonometry-1886/x-7266e99e1f",15,"Todhunter 1886, scan 20: Since there are two poles for each side of ..."],["todhunter-spherical-trigonometry-1886/x-213a6a4986",15,"Todhunter 1886, scan 21: The sides and angles of the polar triangle are ..."],["todhunter-spherical-trigonometry-1886/x-a087f483bf",15,"Todhunter 1886, scan 22: Thus any such theorem will remain true when the ..."],["todhunter-spherical-trigonometry-1886/x-3908216c5c",15,"Todhunter 1886, scan 22: Any two sides of a spherical triangle are together ..."],["todhunter-spherical-trigonometry-1886/x-c0dcf14836",15,"Todhunter 1886, scan 23: The three angles of a spherical triangle are together ..."],["todhunter-spherical-trigonometry-1886/x-97d3e6c824",15,"Todhunter 1886, scan 25: This Chapter might be extended; but it is unnecessary ..."],["dickson-theory-of-equations-1922/x-f8e49eaff1",15,"Dickson 1922, p. 48: This is called the irreducible case since it may ..."],["form/d7a2d25bbc",5,"solve: Eq(c, a*x/b)"],["shape/d7a2d25bbc",6,"solve: Eq(c, a*x/b)"],["form/73fb0f2744",5,"solve: (Eq(a, 8*x/9), Eq(-a + x, 28))"],["form/8175025ab6",5,"identity: (a**3 - a**2*x**3 - 2*a**2*x + a*x**4 + x**5 + x**3)/(-a + x**3 + x)"],["shape/dc441418cf",6,"identity: (N*a**N*x + a*x**N - a**N*x**N + a**N + 2*x**N)/(-a + x + x**N)"],["boyden-first-book-in-algebra-1895/ex-58/7",4,"Boyden 1895, Exercise 58 (7)"],["shape/2e47cbf765",6,"solve: (Eq(a, N*x), Eq(-a + x, N))"],["hardy-course-of-pure-mathematics-1921/eq-e690d352de",16,"Hardy 1921, p. 177: R(x) = P(x)/Q(x)"],["hardy-course-of-pure-mathematics-1921/eq-1e6c249c34",16,"Hardy 1921, p. 177: \\sin(x + h) - \\sin x = 2\\sin \\tfrac{1}{2}h \\cos(x + \\tfrac{1}{2}h)"],["concept/trigonometrical-identity",7,"trigonometrical identity","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-trigonometrical-identity"],["form/02e204c95c",5,"identity: b*(-a + 2*x)/(3*b**2 - b*c)"],["shape/c511d2ed38",6,"identity: b*(N*x - a)/(N*b**N - b*c)"],["boyden-first-book-in-algebra-1895/ex-49/1",4,"Boyden 1895, Exercise 49 (1)"],["form/939baddc67",5,"identity: 6*a**2*b*c/(7*d*e**2)"],["shape/8307edb327",6,"identity: N*a**N*b*c*e**N/d"],["boyden-first-book-in-algebra-1895/ex-5/18",4,"Boyden 1895, Exercise 5 (18)"],["concept/principle-of-duality",7,"principle of duality","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-principle-of-duality"],["theorem/finite-subdivision-with-small-oscillation",9,"finite subdivision with small oscillation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-finite-subdivision-with-small-oscillation"],["form/e9f6b562bd",5,"identity: 3*a**2*b/(4*c**2*d)"],["shape/c09e68c92d",6,"identity: N*a**N*b*c**N/d"],["theorem/exterior-angle-of-a-triangle",9,"exterior angle of a triangle","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-exterior-angle-of-a-triangle"],["theorem/angles-opposite-equal-sides-of-a-triangle-are-equal",9,"angles opposite equal sides of a triangle are equal","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-angles-opposite-equal-sides-of-a-triangle-are-equal"],["de-morgan-elementary-illustrations-calculus-1899/eq-c8717ae212",16,"De Morgan 1899, p. 48: BQ = \\dfrac{b^{2}}{l}"],["hardy-course-of-pure-mathematics-1921/x-d2d7d0ce8b",15,"Hardy 1921, p. 249: The reader is of course familiar with the idea ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-d26b964259",16,"De Morgan 1899, p. 48: BP = AQ"],["boyden-first-book-in-algebra-1895/ex-49/7",4,"Boyden 1895, Exercise 49 (7)"],["form/de94a0fa72",5,"identity: (a - 2)*(a**2 - 10*a + 21)*(a**2 - 5*a + 4)/((a - 7)*(a**2 - 9*a + 20)*(a**2 - 4*a + 3))"],["boyden-first-book-in-algebra-1895/ex-4/6",4,"Boyden 1895, Exercise 4 (6)"],["dickson-theory-of-equations-1922/eq-6925d38e51",16,"Dickson 1922, p. 45: R = \\left(\\frac{p}{3}\\right)^3 + \\left(\\frac{q}{2}\\right)^2"],["hardy-course-of-pure-mathematics-1921/x-1fb1aa1c65",15,"Hardy 1921, p. 251: The notion of the length of a curve, other ..."],["dickson-theory-of-equations-1922/eq-99077da007",16,"Dickson 1922, p. 46: \\omega = -\\tfrac{1}{2} + \\tfrac{1}{2} \\sqrt{3}i"],["dickson-theory-of-equations-1922/eq-2de7d35a68",16,"Dickson 1922, p. 46: \\omega^2 = -\\tfrac{1}{2} - \\tfrac{1}{2} \\sqrt{3}i"],["wentworth-plane-geometry-1899/x-9f85565493",15,"Wentworth 1899, scan 39: A triangle is a portion of a plane bounded ..."],["wentworth-plane-geometry-1899/x-67b0818c7e",15,"Wentworth 1899, scan 41: The sum of the three angles of a triangle ..."],["wentworth-plane-geometry-1899/x-1d80a0e940",15,"Wentworth 1899, scan 53: All points in a plane that satisfy a single ..."],["dickson-theory-of-equations-1922/eq-838f091d7a",16,"Dickson 1922, p. 46: A = \\sqrt[3]{-\\frac{q}{2} + \\sqrt{R}}"],["wentworth-plane-geometry-1899/x-b7dea26f46",15,"Wentworth 1899, scan 53: The word locus (pronounced lo kus) is a Latin ..."],["hardy-course-of-pure-mathematics-1921/x-1c2069ca30",15,"Hardy 1921, p. 252: The explanation of this is of course that between ..."],["form/62b42bf614",5,"solve: Eq(3*x - 25, 47)"],["boyden-first-book-in-algebra-1895/ex-4/7",4,"Boyden 1895, Exercise 4 (7)"],["form/d83141556f",5,"solve: Eq(5*x + 14, 69)"],["form/341f903df8",5,"solve: (Eq(-a + x, 48), Eq(a + x, 216))"],["todhunter-spherical-trigonometry-1886/eq-1db9267540",16,"Todhunter 1886, scan 73: \\tan R = \\dfrac{\\sin\\tfrac{1}{2}a } {\\sin A\\cos\\tfrac{1}{2}b\\cos\\tfrac{1}{2}c }"],["wentworth-plane-geometry-1899/x-8442de289e",15,"Wentworth 1899, scan 44: In § 139 we have given two angles and ..."],["boyden-first-book-in-algebra-1895/ex-6/1",4,"Boyden 1895, Exercise 6 (1)"],["form/9b213ee782",5,"solve: Eq(x + 36, 10*x)"],["form/537510bfe3",5,"solve: (Eq(a, b + 2), Eq(15*a, 16*b))"],["wentworth-plane-geometry-1899/x-2e21731e52",15,"Wentworth 1899, scan 42: The sum of two sides of a triangle is ..."],["boyden-first-book-in-algebra-1895/ex-7/1",4,"Boyden 1895, Exercise 7 (1)"],["form/d673d7b9d0",5,"solve: (Eq(a, 4*x), Eq(a + x, 70))"],["whitehead-introduction-to-mathematics-1911/eq-9002e32a61",16,"Whitehead 1911, p. 89: x = ±\\sqrt{(b - a)}"],["form/491fcffb5e",5,"solve: (Eq(x, 5*a + 61/100), Eq(a + x, 1039/100))"],["whitehead-introduction-to-mathematics-1911/eq-2d110c8238",16,"Whitehead 1911, p. 90: \\sqrt{(-1)} \\sqrt{c^{2}} = c\\sqrt{(-1)}"],["whitehead-introduction-to-mathematics-1911/eq-0a9bbcdb1f",16,"Whitehead 1911, p. 97: (x, y) + (x', y') = (x + x', y + y')"],["form/f22a4b67a1",5,"solve: (Eq(x, a + 2000), Eq(b, 2*x), Eq(a + b + x, 18000))"],["whitehead-introduction-to-mathematics-1911/eq-34e9a2a67e",16,"Whitehead 1911, p. 96: (x, y) = (c, d) - (a, b)"],["boyden-first-book-in-algebra-1895/ex-36/11",4,"Boyden 1895, Exercise 36 (11)"],["dickson-theory-of-equations-1922/eq-5e33576228",16,"Dickson 1922, p. 46: B = \\sqrt[3]{-\\frac{q}{2} - \\sqrt{R}}"],["whitehead-introduction-to-mathematics-1911/eq-d8eb32fa8a",16,"Whitehead 1911, p. 97: (x, y) - (u, v) = (x - u, y - v)"],["boyden-first-book-in-algebra-1895/ex-8/1",4,"Boyden 1895, Exercise 8 (1)"],["form/8f7e6d41ef",5,"solve: Eq(7*x/2, 14)"],["whitehead-introduction-to-mathematics-1911/eq-a35a7d4dbe",16,"Whitehead 1911, p. 98: (x, y) - (u, v) = (x, y) + (-u, -v)"],["boyden-first-book-in-algebra-1895/ex-51/10",4,"Boyden 1895, Exercise 51 (10)"],["whitehead-introduction-to-mathematics-1911/eq-21081019cf",16,"Whitehead 1911, p. 98: (x, y) - (x, y) = (0, 0)"],["boyden-first-book-in-algebra-1895/ex-13/3",4,"Boyden 1895, Exercise 13 (3)"],["boyden-first-book-in-algebra-1895/ex-51/3",4,"Boyden 1895, Exercise 51 (3)"],["planck-treatise-on-thermodynamics-1903/eq-304dbefb1c",16,"Planck 1903, p. 26: p = \\frac{C_{0} \\theta}{v_{0}}"],["dickson-theory-of-equations-1922/eq-137814bd93",16,"Dickson 1922, p. 46: AB = -\\frac{p}{3}"],["form/e959836dce",5,"factor: 9*x**2 + 24*x + 16"],["form/6b05971957",5,"identity: 4*d**2*(1/(a**4*b**12))**(1/4)*Abs(c)/3"],["form/66d807e4f9",5,"identity: d/(a + b/c)"],["shape/66d807e4f9",6,"identity: d/(a + b/c)"],["boyden-first-book-in-algebra-1895/ex-51/4",4,"Boyden 1895, Exercise 51 (4)"],["form/681c079c7c",5,"identity: (a - 1/a**2)/(1 - 1/a)"],["shape/0f63c0a0da",6,"identity: (a - a**N)/(1 - 1/a)"],["planck-treatise-on-thermodynamics-1903/eq-00e2a9c407",16,"Planck 1903, p. 26: p = \\frac{C\\theta}{v}"],["form/32a61e2cff",5,"identity: (a**2 + a + 1)/(1 + 1/a + a**(-2))"],["form/35b415833c",5,"identity: 2*(a + b)**2*Abs(a)/(5*b**2)"],["planck-treatise-on-thermodynamics-1903/eq-cd1f212d42",16,"Planck 1903, p. 26: C = \\frac{m_{0}C_{0}}{m}"],["boyden-first-book-in-algebra-1895/ex-37/26",4,"Boyden 1895, Exercise 37 (26)"],["dickson-theory-of-equations-1922/eq-a2d7477e1e",16,"Dickson 1922, p. 46: y_1 = A + B"],["boyden-first-book-in-algebra-1895/ex-51/1",4,"Boyden 1895, Exercise 51 (1)"],["planck-treatise-on-thermodynamics-1903/eq-9293f7771e",16,"Planck 1903, p. 27: C = \\frac{m_{0}C_{0}}{m} = \\frac{m_{0}}{m} · \\frac{pv_{0}}{\\theta} = \\frac{2 · 1013650}{m · 273 · 0.00008988} = \\frac{82"],["form/5a5d3a97f6",5,"factor: x**6 - 12*x**5 + 35*x**4"],["form/388aaee3f6",5,"identity: (a/b + 1)*(a**2/b**2 - a/b + 1)"],["planck-treatise-on-thermodynamics-1903/eq-b75812a8d3",16,"Planck 1903, p. 27: 82600000 = R"],["form/90a6a3dc97",5,"identity: (a/b - 5)*(a/b + 2)"],["form/007cb9b199",5,"identity: (a/b + 1)/(-1 + c/b)"],["shape/007cb9b199",6,"identity: (a/b + 1)/(-1 + c/b)"],["boyden-first-book-in-algebra-1895/ex-51/2",4,"Boyden 1895, Exercise 51 (2)"],["form/9cafeb89ad",5,"identity: (a/e + d)/(-b/e + c)"],["shape/9cafeb89ad",6,"identity: (a/e + d)/(-b/e + c)"],["planck-treatise-on-thermodynamics-1903/eq-c4eb3de3f8",16,"Planck 1903, p. 27: m = \\frac{R}{C}"],["form/5503198188",5,"factor: 8*a**3/d**3 + b**3/c**3"],["planck-treatise-on-thermodynamics-1903/eq-32276a2a3d",16,"Planck 1903, p. 27: v = \\dfrac{V}{M}"],["planck-treatise-on-thermodynamics-1903/eq-8afd7f3492",16,"Planck 1903, p. 27: V = \\frac{R\\theta}{p} · \\frac{M}{m}"],["form/937b08f598",5,"factor: -8 - 2*b/a + b**2/a**2"],["shape/9dd90734e8",6,"factor: N + N*b/a + a**N*b**N"],["planck-treatise-on-thermodynamics-1903/eq-6443a8be92",16,"Planck 1903, p. 27: \\dfrac{M}{m} = n"],["planck-treatise-on-thermodynamics-1903/eq-90a28a87ce",16,"Planck 1903, p. 27: V = \\frac{R\\theta}{p} · n"],["form/a0f26f4190",5,"identity: (a/b - b/a)/(a - b)"],["shape/a0f26f4190",6,"identity: (a/b - b/a)/(a - b)"],["form/07e7ec0292",5,"solve: Eq(26*x/15 + 8, 2*x)"],["boyden-first-book-in-algebra-1895/ex-8/12",4,"Boyden 1895, Exercise 8 (12)"],["form/d8ce310f6f",5,"solve: (Eq(a, x/2), Eq(b, a + 3*x), Eq(a + b + x, 90))"],["shape/3c5c4de34d",6,"solve: (Eq(a, N*x), Eq(b, N*x + a), Eq(a + b + x, N))"],["planck-treatise-on-thermodynamics-1903/eq-513dae5b74",16,"Planck 1903, p. 28: p_{1} : p_{2} : \\dots = C_{1}M_{1} : C_{2}M_{2}"],["planck-treatise-on-thermodynamics-1903/eq-089de8f735",16,"Planck 1903, p. 28: p_{1} : p_{2} : \\dots = \\frac{M_{1}}{m_{1}} : \\frac{M_{2}}{m_{2}} : \\dots = n_{1} : n_{2} : \\dots"],["planck-treatise-on-thermodynamics-1903/eq-42878e3e41",16,"Planck 1903, p. 28: \\frac{M_{1} + M_{2} + \\dots}{m} = \\frac{M_{1}}{m_{1}} + \\frac{M_{2}}{m_{2}} + \\dots"],["form/589a0e3fb2",5,"identity: 1 - 1/(1 + 2/(a - 2))"],["planck-treatise-on-thermodynamics-1903/eq-219b4db4b3",16,"Planck 1903, p. 28: m = \\frac{M_{1} + M_{2} + \\dots}{\\dfrac{M_{1}}{m_{1}} + \\dfrac{M_{2}}{m_{2}} + \\dots}"],["planck-treatise-on-thermodynamics-1903/eq-bc49f96784",16,"Planck 1903, p. 30: \\ce{C5H11Br} = \\ce{C5H10 + HBr}"],["concept/normal",7,"normal","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-normal"],["de-morgan-elementary-illustrations-calculus-1899/x-0ef727e97c",15,"De Morgan 1899, p. 14: Such are x^{2} + a^{2}, \\dfrac{a + x}{a - ..."],["de-morgan-elementary-illustrations-calculus-1899/x-5383d86947",15,"De Morgan 1899, p. 14: Thus if in x^{2} + a^{2} x only is ..."],["wentworth-plane-geometry-1899/eq-40d487f911",16,"Wentworth 1899, scan 67: (n-2)2"],["wentworth-plane-geometry-1899/eq-d6c8e01f32",16,"Wentworth 1899, scan 67: \\displaystyle \\frac{2(n-2)}{n}"],["todhunter-spherical-trigonometry-1886/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle",2,"Todhunter 1886, Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle","../books/todhunter-spherical-trigonometry-1886/ch/ch-relations-between-the-trigonometrical-functions-of-the-sides-and-the-angles-of-a-spherical-triangle/index.html"],["form/f69df07817",5,"factor: 3*x**2 - 3*x - 216"],["form/02c77f7757",5,"solve: (Eq(a, 2*x), Eq(b, 7500), Eq(a + b + x, 6*x))"],["shape/c889473c55",6,"solve: (Eq(a, N*x), Eq(b, N), Eq(a + b + x, N*x))"],["boyden-first-book-in-algebra-1895/ex-8/17",4,"Boyden 1895, Exercise 8 (17)"],["form/e075e88869",5,"solve: (Eq(x, a/9), Eq(x + 72, a))"],["shape/c10e92fa23",6,"solve: (Eq(x, N*a), Eq(N + x, a))"],["dickson-theory-of-equations-1922/eq-e057477cfa",16,"Dickson 1922, p. 46: y_2 = \\omega A + \\omega^2 B"],["todhunter-spherical-trigonometry-1886/eq-9b25003591",16,"Todhunter 1886, scan 28: \\cos b = \\cos c \\cos a + \\sin c \\sin a \\cos B"],["todhunter-spherical-trigonometry-1886/eq-bd9923af51",16,"Todhunter 1886, scan 28: \\cos c = \\cos a \\cos b + \\sin a \\sin b \\cos C"],["todhunter-spherical-trigonometry-1886/eq-64857c7cd8",16,"Todhunter 1886, scan 28: \\cos a = \\sin b \\cos A"],["todhunter-spherical-trigonometry-1886/eq-ce6db00535",16,"Todhunter 1886, scan 29: \\sin A=\\dfrac{\\surd(1-\\cos^2 a-\\cos^2 b-\\cos^2 c+2\\cos a\\cos b\\cos c)}{\\sin b \\sin c}"],["todhunter-spherical-trigonometry-1886/eq-b391a8c265",16,"Todhunter 1886, scan 29: \\dfrac{\\sin A}{\\sin a}=\\dfrac{\\sin B}{\\sin b}=\\dfrac{\\sin C}{\\sin c}"],["law/sine-rule-for-spherical-triangles",10,"sine rule for spherical triangles"],["todhunter-spherical-trigonometry-1886/eq-e97756a3ef",16,"Todhunter 1886, scan 30: \\dfrac{\\sin B}{\\sin C}=\\dfrac{\\sin b}{\\sin c}"],["hardy-course-of-pure-mathematics-1921/x-83c434d7b3",15,"Hardy 1921, p. 191: Another method of stating the definition is this: \\phi(x, ..."],["todhunter-spherical-trigonometry-1886/eq-a6e7f009e2",16,"Todhunter 1886, scan 30: \\cot a \\sin b = \\cot A \\sin C + \\cos b \\cos C"],["boyden-first-book-in-algebra-1895/ex-13/4",4,"Boyden 1895, Exercise 13 (4)"],["todhunter-spherical-trigonometry-1886/eq-10946a306d",16,"Todhunter 1886, scan 31: \\sin^2 \\dfrac{A}{2} = \\dfrac{\\sin \\tfrac{1}{2}(a+b-c)\\sin\\tfrac{1}{2}(a-b+c)}{\\sin b \\sin c}"],["concept/half-angle-formula",7,"half-angle formula"],["form/10e6c68833",5,"solve: (Eq(-5*a + 2*x, -11), Eq(a + 3*x, 9))"],["form/f81eab0db9",5,"solve: (Eq(a + 2*x, 12), Eq(-3*a + 7*x, 41))"],["todhunter-spherical-trigonometry-1886/eq-ce76df0ac1",16,"Todhunter 1886, scan 31: \\sin^2 \\dfrac{A}{2} = \\dfrac{\\sin(s - b)\\sin(s - c)}{\\sin b \\sin c}"],["form/ce971693b1",5,"solve: (Eq(4*a - 6*x, 7/3), Eq(3*a - 2*x, 3))"],["form/84c0d06895",5,"solve: (Eq(2*a + 3*x, 11), Eq(-5*a + 7*x, 190))"],["todhunter-spherical-trigonometry-1886/eq-cd80df2828",16,"Todhunter 1886, scan 32: \\tan\\dfrac{A}{2}=\\Surd{\\left\\{ \\dfrac{\\sin(s-b)\\sin(s-c)}{\\sin s \\sin(s-a)} \\right\\}}"],["todhunter-spherical-trigonometry-1886/eq-0e6a10feee",16,"Todhunter 1886, scan 32: \\sin A = \\dfrac{2}{\\sin b \\sin c}\\{\\sin s \\sin(s-a)\\sin(s-b)\\sin(s-c)\\}^{\\tfrac{1}{2}}"],["form/9bcda617f8",5,"solve: (Eq(7*a + 2*x/3, 189), Eq(-3*a/5 + x, 6))"],["todhunter-spherical-trigonometry-1886/eq-8fc3b3b3f2",16,"Todhunter 1886, scan 33: \\cos A =-\\cos B \\cos C + \\sin B \\sin C \\cos a"],["boyden-first-book-in-algebra-1895/ex-13/5",4,"Boyden 1895, Exercise 13 (5)"],["form/b21312884b",5,"solve: (Eq(-c + x, b), Eq(c + x, a))"],["todhunter-spherical-trigonometry-1886/eq-40ca774a25",16,"Todhunter 1886, scan 33: \\sin^{2}\\frac{a}{2}=-\\frac{\\cos S\\cos(S-A)}{\\sin B\\sin C}"],["dickson-theory-of-equations-1922/eq-2cb1eaaa80",16,"Dickson 1922, p. 46: y_3 = \\omega^2 A + \\omega B"],["form/9f80e4e58a",5,"solve: (Eq((x - 2)/(a - 2), 3/5), Eq((x + 1)/(a + 1), 2/3))"],["boyden-first-book-in-algebra-1895/ex-8/18",4,"Boyden 1895, Exercise 8 (18)"],["form/fc6d1a0cee",5,"solve: Eq(61*x/120, 6100)"],["todhunter-spherical-trigonometry-1886/eq-2ecde172ad",16,"Todhunter 1886, scan 34: \\sin a=\\dfrac{2}{\\sin B\\sin C}\\left\\{-\\cos S\\cos (S-A)\\cos(S-B)\\cos (S-C)\\right\\}^{\\tfrac{1}{2}}"],["todhunter-spherical-trigonometry-1886/eq-8958eae439",16,"Todhunter 1886, scan 35: \\tan\\tfrac{1}{2}(A + B) = \\frac{\\cos\\tfrac{1}{2}(a - b)}{\\cos\\tfrac{1}{2}(a + b)}\\cot\\frac{C}{2}"],["theorem/napier-s-analogy",9,"Napier's analogy"],["todhunter-spherical-trigonometry-1886/eq-919dc6e313",16,"Todhunter 1886, scan 35: \\tan\\frac{1}{2}(A - B) = \\frac{\\sin\\tfrac{1}{2}(a - b)} {\\sin\\tfrac{1}{2}(a + b)} \\cot\\frac{C}{2}"],["planck-treatise-on-thermodynamics-1903/eq-a4c9292168",16,"Planck 1903, p. 110: F_{2} - F_{1} < \\tsum W"],["planck-treatise-on-thermodynamics-1903/eq-953e98471a",16,"Planck 1903, p. 110: dF < W"],["boyden-first-book-in-algebra-1895/x-ef6f872f0a",15,"Boyden 1895: ac-bc-dc = a - b -dc"],["form/9c0355b5dc",5,"solve: (Eq(6*a + 4*x, 466), Eq(9*a + 5*x, 638))"],["boyden-first-book-in-algebra-1895/ex-9/1",4,"Boyden 1895, Exercise 9 (1)"],["boyden-first-book-in-algebra-1895/ex-9/2",4,"Boyden 1895, Exercise 9 (2)"],["shape/50371e2b9f",6,"solve: Eq(N*x**N + N, N)"],["form/e5ae00c48c",5,"solve: Eq(5*x**2 - 12, 33)"],["todhunter-spherical-trigonometry-1886/eq-12f4998788",16,"Todhunter 1886, scan 35: \\tan\\tfrac{1}{2}(a + b) = \\frac{\\cos\\tfrac{1}{2}(A - B)} {\\cos\\tfrac{1}{2}(A + B)} \\tan\\frac{c}{2}"],["macfarlane-vector-analysis-quaternions-1906/eq-f36bb919ba",16,"Macfarlane 1906: R &= \\frac{1}{\\sum F}\\sum \\left(\\mathrm{V}AF \\right) \\tag{1}"],["todhunter-spherical-trigonometry-1886/eq-81b7d10954",16,"Todhunter 1886, scan 35: \\tan\\tfrac{1}{2}(a - b) = \\frac{\\sin\\tfrac{1}{2}(A - B)} {\\sin\\tfrac{1}{2}(A + B)} \\tan\\frac{c}{2}"],["dickson-theory-of-equations-1922/eq-74c69d1aeb",16,"Dickson 1922, p. 47: (y_1-y_2)(y_1-y_3)(y_2-y_3) = 6\\sqrt{3}\\sqrt{R}i"],["todhunter-spherical-trigonometry-1886/eq-600fbc0a24",16,"Todhunter 1886, scan 36: \\cos^2\\tfrac{1}{2}c = \\cos^2\\tfrac{1}{2}(a - b) \\cos^2\\tfrac{1}{2}C + \\cos^2\\tfrac{1}{2}(a + b) \\sin^2\\tfrac{1}{2}C"],["theorem/delambre-s-analogy",9,"Delambre's analogy"],["todhunter-spherical-trigonometry-1886/eq-a7a71fa57d",16,"Todhunter 1886, scan 36: \\cos\\tfrac{1}{2}(A + B) \\cos\\tfrac{1}{2}c = \\cos\\tfrac{1}{2}(a + b) \\sin\\tfrac{1}{2}C"],["todhunter-spherical-trigonometry-1886/eq-5d7b6aedba",16,"Todhunter 1886, scan 36: \\cos\\tfrac{1}{2}(A - B) \\sin\\tfrac{1}{2}c = \\sin\\tfrac{1}{2}(a + b) \\sin\\tfrac{1}{2}C"],["macfarlane-vector-analysis-quaternions-1906/eq-36b868e20d",16,"Macfarlane 1906: \\mathrm{V}\\left\\{\\sum \\mathrm{V}AF - \\mathrm{V}R\\sum F\\right\\} \\sum F = 0"],["dickson-theory-of-equations-1922/eq-6471c37b12",16,"Dickson 1922, p. 47: -108R = -4p^3 - 27q^2"],["dickson-theory-of-equations-1922/eq-f0881680b8",16,"Dickson 1922, p. 47: \\Delta = 18bcd - 4b^3 d + b^2 c^2 - 4c^3 - 27d^2"],["boyden-first-book-in-algebra-1895/ex-9/3",4,"Boyden 1895, Exercise 9 (3)"],["boyden-first-book-in-algebra-1895/ex-9/4",4,"Boyden 1895, Exercise 9 (4)"],["form/41dd32253e",5,"solve: Eq(x**2, 10*x - 21)"],["shape/4fcde7fe81",6,"solve: Eq(x**N, N*x + N)"],["boyden-first-book-in-algebra-1895/ex-9/5",4,"Boyden 1895, Exercise 9 (5)"],["todhunter-spherical-trigonometry-1886/eq-ccb91821f8",16,"Todhunter 1886, scan 36: \\sin\\tfrac{1}{2}(A + B) \\cos\\tfrac{1}{2}c = \\cos\\tfrac{1}{2}(a - b) \\cos\\tfrac{1}{2}C"],["theorem/gauss-s-theorem-delambre-s",9,"Gauss's theorem (Delambre's)"],["dickson-theory-of-equations-1922/eq-2370e9c8e6",16,"Dickson 1922, p. 152: f(x) \\equiv a_0(x - \\alpha_1)(x - \\alpha_2) \\dotsm (x - \\alpha_m)"],["boyden-first-book-in-algebra-1895/ex-58/5",4,"Boyden 1895, Exercise 58 (5)"],["todhunter-spherical-trigonometry-1886/eq-2b782b6695",16,"Todhunter 1886, scan 36: \\sin\\tfrac{1}{2}(A - B) \\sin\\tfrac{1}{2}c = \\sin\\tfrac{1}{2}(a - b) \\cos\\tfrac{1}{2}C"],["boyden-first-book-in-algebra-1895/ex-13/6",4,"Boyden 1895, Exercise 13 (6)"],["todhunter-spherical-trigonometry-1886/eq-f799456ece",16,"Todhunter 1886, scan 40: \\cos a \\cos b + \\sin a \\sin b \\cos C = \\cos c"],["planck-treatise-on-thermodynamics-1903/eq-eb36f42371",16,"Planck 1903, p. 110: U - F = \\theta\\Phi"],["shape/45715cf893",6,"solve: (Eq(a, N*b + N*x), Eq(x, b), Eq(a + b + x, N))"],["boyden-first-book-in-algebra-1895/ex-57/23",4,"Boyden 1895, Exercise 57 (23)"],["form/09c48cd0c4",5,"solve: (Eq(a, 5*b + 5*x), Eq(x, b), Eq(a + b + x, 36))"],["form/5e9754dcb4",5,"solve: Eq(47*x/360, 1)"],["shape/175253c74a",6,"solve: Eq(N*x, 1)"],["de-morgan-elementary-illustrations-calculus-1899/eq-e56c305587",16,"De Morgan 1899, p. 18: a\\left(1 + \\frac{b}{a} + \\frac{c}{b}\\, \\frac{b}{a} + \\frac{d}{c}\\, \\frac{c}{b}\\, \\frac{b}{a} + \\etc.\\right)"],["planck-treatise-on-thermodynamics-1903/eq-e8422977e1",16,"Planck 1903, p. 111: \\tsum W = 0"],["de-morgan-elementary-illustrations-calculus-1899/ch-convergent-and-divergent-series",2,"De Morgan 1899, Convergent and Divergent Series","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-convergent-and-divergent-series/index.html"],["de-morgan-elementary-illustrations-calculus-1899/eq-a148c73f49",16,"De Morgan 1899, p. 18: k + l + m + \\etc. = k\\left(1 + \\frac{l}{k} + \\frac{m}{l}\\, \\frac{l}{k} + \\etc.\\right)"],["de-morgan-elementary-illustrations-calculus-1899/eq-cdd13e9a99",16,"De Morgan 1899, p. 18: \\dfrac{l}{k} > \\dfrac{m}{l} > \\dfrac{n}{m}"],["planck-treatise-on-thermodynamics-1903/eq-e34ae472cf",16,"Planck 1903, p. 111: F_{2} - F_{1} < 0"],["planck-treatise-on-thermodynamics-1903/eq-df6cf39e19",16,"Planck 1903, p. 113: dF = dU - \\theta\\, d\\Phi - \\Phi\\, d\\theta"],["planck-treatise-on-thermodynamics-1903/eq-c6ff0b5f02",16,"Planck 1903, p. 33: \\frac{Q}{\\Delta\\theta} = c_{m}."],["boyden-first-book-in-algebra-1895/ex-13/7",4,"Boyden 1895, Exercise 13 (7)"],["planck-treatise-on-thermodynamics-1903/eq-66bf3ac1cb",16,"Planck 1903, p. 33: \\frac{Q}{d\\theta} = c."],["concept/convex-polygon",7,"convex polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-convex-polygon"],["form/6a3e3ab3e9",5,"solve: (Eq(a, 2*x), Eq(5*a + 2*x, 144))"],["form/891b558d93",5,"solve: (Eq(a, 2*x + 3), Eq(a + 2*x, 131))"],["form/401fdc224c",5,"solve: (Eq(a, x - 1200), Eq(a + 3*x, 16800))"],["planck-treatise-on-thermodynamics-1903/eq-a73da99b73",16,"Planck 1903, p. 196: \\frac{L}{\\theta^{2}}\\, d\\theta - \\frac{s}{\\theta}\\, dp - \\varphi\\, dc = 0"],["todhunter-spherical-trigonometry-1886/eq-8b99ce8883",16,"Todhunter 1886, scan 151: \\mathrm{F + S = E + 1 }"],["boyden-first-book-in-algebra-1895/ex-9/12",4,"Boyden 1895, Exercise 9 (12)"],["concept/salient-angle",7,"salient angle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-salient-angle"],["hardy-course-of-pure-mathematics-1921/x-1cfae5f090",15,"Hardy 1921, p. 190: It is impossible, without the use of this or ..."],["boyden-first-book-in-algebra-1895/ex-24/10",4,"Boyden 1895, Exercise 24 (10)"],["todhunter-spherical-trigonometry-1886/eq-362e343216",16,"Todhunter 1886, scan 75: PA' = PB' = PC' = \\dfrac{\\pi}{2}-r"],["wentworth-plane-geometry-1899/eq-e0d3c9d91a",16,"Wentworth 1899, scan 170: \\dfrac{\\overline{AC}^2}{\\overline{BC}^2} = \\dfrac{AB × AF}{AB × BF} = \\dfrac{AF}{BF}"],["form/88c12451c4",5,"solve: (Eq(b, 5*a), Eq(a + b, 726))"],["form/0ff7260c69",5,"solve: (Eq(b, 7*a), Eq(-a + b, 852))"],["form/945cc0b3da",5,"solve: (Eq(a, b + 10), Eq(3*a, 5*b))"],["form/fc387c2ffb",5,"solve: Eq(4*x + 48, 7*x)"],["hardy-course-of-pure-mathematics-1921/x-9caaa1162e",15,"Hardy 1921, p. 254: The constituents of a determinant are functions of x. ..."],["de-morgan-elementary-illustrations-calculus-1899/x-04971a73cc",15,"De Morgan 1899, p. 12: In this reasoning there is evidently an absolute error; ..."],["hardy-course-of-pure-mathematics-1921/x-3bb9e68d13",15,"Hardy 1921, p. 258: This theorem reduces to the Mean Value Theorem ([§]125) ..."],["concept/heptagon",7,"heptagon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-heptagon"],["dickson-theory-of-equations-1922/eq-8034c0ec51",16,"Dickson 1922, p. 47: ax^3 + bx^2 + cx +d = 0 \\quad (a \\neq 0)"],["form/8b7fa63fc5",5,"solve: (Eq(x, a/5), Eq(a + x, 360))"],["de-morgan-elementary-illustrations-calculus-1899/x-b7b542f990",15,"De Morgan 1899, p. 13: The smaller h is made, the more near does ..."],["wentworth-plane-geometry-1899/eq-1e6f8875cf",16,"Wentworth 1899, scan 170: \\dfrac{\\overline{AB}^2}{\\overline{AC}^2} = \\dfrac{AB × AB}{AB × AF} = \\dfrac{AB}{AF}"],["de-morgan-elementary-illustrations-calculus-1899/x-a999a7b983",15,"De Morgan 1899, p. 13: The proposition, therefore, that h can be taken so ..."],["form/7162727cc4",5,"solve: Eq(11*x/15, 22000)"],["form/e90bb01f40",5,"solve: Eq(17*x/20, 255)"],["hardy-course-of-pure-mathematics-1921/eq-c6b16feeeb",16,"Hardy 1921, p. 190: |\\phi(x, y) - \\phi(\\xi, \\eta) | < \\DELTA"],["concept/re-entrant-angle",7,"re-entrant angle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-re-entrant-angle"],["boyden-first-book-in-algebra-1895/ex-13/8",4,"Boyden 1895, Exercise 13 (8)"],["form/1b35255b6d",5,"solve: Eq(43*x/72, 43)"],["hardy-course-of-pure-mathematics-1921/ex-lxiii",3,"Hardy 1921, Exercise LXIII"],["hardy-course-of-pure-mathematics-1921/x-84e79e0edc",15,"Hardy 1921, p. 257: In an equilateral triangle (the triangle of minimum perimeter ..."],["hardy-course-of-pure-mathematics-1921/eq-29c828c6c8",16,"Hardy 1921, p. 191: \\phi(x, y) = \\frac{2xy}{x^{2} + y^{2}}"],["form/f76cae81be",5,"solve: Eq(32*x/15 + 3600, 10000)"],["hardy-course-of-pure-mathematics-1921/eq-fc5a85d683",16,"Hardy 1921, p. 191: \\lim\\phi(x, y) = \\frac{2a}{1 + a^{2}}"],["form/1c5243ea36",5,"solve: Eq(19*x/12 + 34, 3*x)"],["form/6fc47a099e",5,"solve: Eq(25*x/14 + 31, 4*x)"],["concept/intersection-of-curves",7,"intersection of curves","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-intersection-of-curves"],["form/6cfd91a6db",5,"solve: (Eq(x, 2*a + 2*b), Eq(b, a/3), Eq(a + b + x, 84))"],["concept/touching-of-curves",7,"touching of curves","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-touching-of-curves"],["hardy-course-of-pure-mathematics-1921/eq-ad8a7ad3d0",16,"Hardy 1921, p. 191: y^{5} - xy - y - x = 0"],["form/196392adc2",5,"solve: Eq(55*x/63, 110)"],["boyden-first-book-in-algebra-1895/x-de60a748c3",15,"Boyden 1895: To change fractions to equivalent fractions having a common ..."],["concept/curvature",7,"curvature","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-curvature"],["concept/complex-fraction",7,"complex fraction","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-complex-fraction"],["hardy-course-of-pure-mathematics-1921/eq-3a4d6a61fd",16,"Hardy 1921, p. 193: f(x, y) - f(x, y') = (y - y') (y^{4} + y^{3}y' + y^{2}y'^{2} + yy'^{3} + y'^{4} - x - 1)"],["hardy-course-of-pure-mathematics-1921/eq-437fdd70ef",16,"Hardy 1921, p. 192: f(a, \\lambda) = 0"],["hardy-course-of-pure-mathematics-1921/eq-cb8106688e",16,"Hardy 1921, p. 193: y = \\tfrac{1}{2}\\{1 + x - \\sqrtp{1 + 6 x + x^{2}}\\}"],["hardy-course-of-pure-mathematics-1921/x-d784c60bec",15,"Hardy 1921, p. 439: Another way of expressing this fact is to say ..."],["boyden-first-book-in-algebra-1895/ex-13/9",4,"Boyden 1895, Exercise 13 (9)"],["hardy-course-of-pure-mathematics-1921/x-759b71e8ce",15,"Hardy 1921, p. 441: We can always, by the exercise of a little ..."],["theorem/central-angle-is-measured-by-its-intercepted-arc",9,"central angle is measured by its intercepted arc","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-central-angle-is-measured-by-its-intercepted-arc"],["concept/signed-quantity",7,"signed quantity","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-signed-quantity"],["boyden-first-book-in-algebra-1895/ex-13/10",4,"Boyden 1895, Exercise 13 (10)"],["boyden-first-book-in-algebra-1895/ex-45",3,"Boyden 1895, Exercise 45"],["concept/escribed-circle",7,"escribed circle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-escribed-circle"],["concept/construction",7,"construction","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-construction"],["concept/cubic-equation",7,"cubic equation","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cubic-equation"],["hardy-course-of-pure-mathematics-1921/x-028a4a1f73",15,"Hardy 1921, p. 264: In view of the great importance of this theorem ..."],["hardy-course-of-pure-mathematics-1921/x-12adf02b79",15,"Hardy 1921, p. 265: Apply this process to the equation x^{2} = 2, ..."],["hardy-course-of-pure-mathematics-1921/x-8592e9daea",15,"Hardy 1921, p. 268: In order that there should be a maximum or ..."],["hardy-course-of-pure-mathematics-1921/x-e6f0c66b5a",15,"Hardy 1921, p. 270: Two curves are said to intersect (or cut) at ..."],["hardy-course-of-pure-mathematics-1921/x-8d7d446b69",15,"Hardy 1921, p. 267: But a difficulty arises if -1 < x < ..."],["concept/incommensurable-magnitudes",7,"incommensurable magnitudes","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-incommensurable-magnitudes"],["theorem/right-spherical-triangle-relations",9,"right spherical triangle relations","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-right-spherical-triangle-relations"],["planck-treatise-on-thermodynamics-1903/eq-cf373073f6",16,"Planck 1903, p. 68: Q + W = U_{2} - U_{1}"],["planck-treatise-on-thermodynamics-1903/eq-3392e5ed4a",16,"Planck 1903, p. 69: \\ce{[Pb] + [S] - [PbS]} = 18,400~\\Unit{cal.}"],["boyden-first-book-in-algebra-1895/ex-46",3,"Boyden 1895, Exercise 46"],["person/apollonius-of-perga",1,"Apollonius of Perga","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-apollonius-of-perga"],["person/pappus-of-alexandria",1,"Pappus of Alexandria","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-pappus-of-alexandria"],["concept/minor-arc",7,"minor arc","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-minor-arc"],["concept/major-arc",7,"major arc","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-major-arc"],["person/diocles",1,"Diocles","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-diocles"],["person/johann-heinrich-lambert",1,"Johann Heinrich Lambert","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-johann-heinrich-lambert"],["person/ren-descartes",1,"René Descartes","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-ren-descartes"],["hardy-course-of-pure-mathematics-1921/x-1e65c2c74f",15,"Hardy 1921, p. 273: The circle which has contact of the second order ..."],["method/implicit-differentiation",8,"implicit differentiation","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-implicit-differentiation"],["concept/central-angle",7,"central angle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-central-angle"],["concept/concentric-circles",7,"concentric circles","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-concentric-circles"],["person/ferdinand-von-lindemann",1,"Ferdinand von Lindemann","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-ferdinand-von-lindemann"],["person/alexis-clairaut",1,"Alexis Clairaut","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-alexis-clairaut"],["planck-treatise-on-thermodynamics-1903/eq-04e665298e",16,"Planck 1903, p. 69: \\ce{[PbS]} = -18,400~\\Unit{cal.}"],["planck-treatise-on-thermodynamics-1903/eq-483ffde56b",16,"Planck 1903, p. 69: \\ce{(H2O) - [H2O]} = 80 × 18 = 1440~\\Unit{cal.}"],["planck-treatise-on-thermodynamics-1903/eq-6c66a6cd07",16,"Planck 1903, p. 70: \\ce{(H2SO4) + 5(H2O) - (H2SO4 . 5H2O)} = 13,100~\\Unit{cal.}"],["ball-mathematical-recreations-1905/x-0f652319a3",15,"Ball 1905, scan 255: Among the more interesting geometrical problems of antiquity are ..."],["ball-mathematical-recreations-1905/x-bd1ad02bd8",15,"Ball 1905, scan 255: To duplicate a cube the length of whose side ..."],["method/euclidean-algorithm",8,"Euclidean algorithm","../books/wentworth-plane-geometry-1899/terms/index.html#t-method-euclidean-algorithm"],["ball-mathematical-recreations-1905/x-03b43af4ff",15,"Ball 1905, scan 258: He did not give a geometrical construction, but he ..."],["wentworth-plane-geometry-1899/x-840658d23e",15,"Wentworth 1899, scan 84: By the definition of a circle, all its radii ..."],["wentworth-plane-geometry-1899/x-2bdd72ca57",15,"Wentworth 1899, scan 101: No quantity is great or small except by comparison ..."],["wentworth-plane-geometry-1899/x-224387316b",15,"Wentworth 1899, scan 102: If a variable, by having different successive values, can ..."],["wentworth-plane-geometry-1899/x-7fde54b7df",15,"Wentworth 1899, scan 103: Then it is evident that the moving point may ..."],["concept/polar-coordinates",7,"polar coordinates","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-polar-coordinates"],["concept/lower-sum",7,"lower sum","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-lower-sum"],["ball-mathematical-recreations-1905/x-2b2598d4f4",15,"Ball 1905, scan 257: It is probable that the Greeks were aware that ..."],["wentworth-first-steps-in-algebra-1894/ex-31/1",4,"Wentworth 1894, Exercise 31 (1)"],["planck-treatise-on-thermodynamics-1903/eq-60eec4b213",16,"Planck 1903, p. 70: \\ce{(H2SO4) + 10(H2O) - (H2SO4 . 10H2O)} = 15,100~\\Unit{cal.}"],["ball-mathematical-recreations-1905/x-290c978456",15,"Ball 1905, scan 266: It is however a mere accident that \\pi is ..."],["ball-mathematical-recreations-1905/x-63445412de",15,"Ball 1905, scan 267: In reality the fact that the ratio of the ..."],["hardy-course-of-pure-mathematics-1921/x-37a658040d",15,"Hardy 1921, p. 278: Suppose for example that y = 1 - x ..."],["form/4d69a9e805",5,"factor: 2*x**2 - 4*x"],["planck-treatise-on-thermodynamics-1903/eq-196e28ac0a",16,"Planck 1903, p. 70: \\ce{(H2SO4 . 5H2O) + 5(H2O) - (H2SO4 . 10H2O)} = 2000~\\Unit{cal.}"],["planck-treatise-on-thermodynamics-1903/eq-68575e9c54",16,"Planck 1903, p. 70: \\ce{(H2SO4) + ($\\aq$) - (H2SO4 $\\aq$)} = 17,900~\\Unit{cal.}"],["planck-treatise-on-thermodynamics-1903/eq-dc220dd2d2",16,"Planck 1903, p. 71: U_{2} - U_{1} = Q"],["planck-treatise-on-thermodynamics-1903/eq-c6f498da01",16,"Planck 1903, p. 71: W = -\\int_{1}^{2} p_{0}\\, dV = p_{0} (V_{1} - V_{2})"],["wentworth-first-steps-in-algebra-1894/ex-31/2",4,"Wentworth 1894, Exercise 31 (2)"],["concept/algebraic-form",7,"algebraic form","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-algebraic-form"],["concept/generality",7,"generality","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-generality"],["form/68a7879b46",5,"factor: 3*x**3 - 6*x"],["hardy-course-of-pure-mathematics-1921/x-3f1bec515e",15,"Hardy 1921, p. 280: The symbol dy/dx thus acquires a double meaning; but ..."],["person/michael-stifel",1,"Michael Stifel","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-michael-stifel"],["person/georg-cantor",1,"Georg Cantor","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-georg-cantor"],["whitehead-introduction-to-mathematics-1911/x-189259e71a",15,"Whitehead 1911, p. 72: The Greeks thought of this subject rather in the ..."],["whitehead-introduction-to-mathematics-1911/x-101f0f2b10",15,"Whitehead 1911, p. 72: For example, the diagonal of a square cannot be ..."],["whitehead-introduction-to-mathematics-1911/x-8274a844dd",15,"Whitehead 1911, p. 75: One very simple way of doing this is to ..."],["whitehead-introduction-to-mathematics-1911/x-28e1548dc2",15,"Whitehead 1911, p. 83: But if we now interpret our symbols as “operations,” ..."],["boyden-first-book-in-algebra-1895/x-a549763b75",15,"Boyden 1895: To divide one fraction by another, invert the divisor ..."],["form/417f412818",5,"factor: -a**2*x**2 + 1"],["hardy-course-of-pure-mathematics-1921/x-978278d377",15,"Hardy 1921, p. 280: This is sometimes expressed by saying that dy is ..."],["hardy-course-of-pure-mathematics-1921/x-4a92ceff5d",15,"Hardy 1921, p. 205: the reader must however be careful to remember that ..."],["de-morgan-elementary-illustrations-calculus-1899/x-ed1e74bb52",15,"De Morgan 1899, p. 17: On the other hand, a series is said to ..."],["hardy-course-of-pure-mathematics-1921/x-df7215ab28",15,"Hardy 1921, p. 284: But nothing which we know so far provides us ..."],["hardy-course-of-pure-mathematics-1921/x-89912e2f29",15,"Hardy 1921, p. 286: We define the area of PpqQ as being the ..."],["boyden-first-book-in-algebra-1895/ch-complex-fractions",2,"Boyden 1895, COMPLEX FRACTIONS","../books/boyden-first-book-in-algebra-1895/ch/ch-complex-fractions/index.html"],["shape/b6ef634770",6,"factor: -a**N*x**N + 1"],["wentworth-first-steps-in-algebra-1894/ex-33/10",4,"Wentworth 1894, Exercise 33 (10)"],["hardy-course-of-pure-mathematics-1921/x-1553873da7",15,"Hardy 1921, p. 200: The geometry of curves is merely one of many ..."],["concept/real-root",7,"real root","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-real-root"],["macfarlane-vector-analysis-quaternions-1906",0,"Macfarlane, Vector Analysis and Quaternions (1906)","../books/macfarlane-vector-analysis-quaternions-1906/index.html"],["planck-treatise-on-thermodynamics-1903/x-26d051c382",15,"Planck 1903, p. 262: When chemical interchanges between the different substances in solution ..."],["form/f488eeeecd",5,"factor: -16*c**2 + (a + 3*b)**2"],["shape/893f6dfc22",6,"factor: N*c**N + (N*b + a)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/13",4,"Wentworth 1894, Exercise 34 (13)"],["form/2654c45d2c",5,"factor: -9*c**2 + (a - 5*b)**2"],["concept/properties-of-the-definite-integral",7,"properties of the definite integral","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-properties-of-the-definite-integral"],["theorem/generalised-mean-value-theorem-for-integrals",9,"generalised mean value theorem for integrals","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-generalised-mean-value-theorem-for-integrals"],["theorem/fundamental-theorem-of-the-integral-calculus",9,"fundamental theorem of the integral calculus","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-fundamental-theorem-of-the-integral-calculus"],["concept/ordinary-tangent",7,"ordinary tangent","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-ordinary-tangent"],["whitehead-introduction-to-mathematics-1911/ch-xii",2,"Whitehead 1911, ch. XII: Periodicity in Nature","../books/whitehead-introduction-to-mathematics-1911/ch/ch-xii/index.html"],["wentworth-first-steps-in-algebra-1894/ex-35",3,"Wentworth 1894, Exercise 35"],["wentworth-first-steps-in-algebra-1894/ex-35/1",4,"Wentworth 1894, Exercise 35 (1)"],["method/polygon-perimeter-method",8,"polygon perimeter method","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-polygon-perimeter-method"],["concept/point-of-tangency",7,"point of tangency","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-point-of-tangency"],["boyden-first-book-in-algebra-1895/x-6d7658455f",15,"Boyden 1895: To multiply a monomial by a monomial, multiply the ..."],["concept/factorial",7,"factorial","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-factorial"],["concept/reduced-cubic-equation",7,"reduced cubic equation","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-reduced-cubic-equation"],["theorem/cauchy-s-form-of-the-remainder",9,"Cauchy's form of the remainder","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-cauchy-s-form-of-the-remainder"],["hardy-course-of-pure-mathematics-1921/x-8a14b08931",15,"Hardy 1921, p. 286: As it is not always practicable actually to determine ..."],["hardy-course-of-pure-mathematics-1921/x-6436ccac57",15,"Hardy 1921, p. 287: But when we are considering ‘indefinite integrals’ or ‘integral ..."],["form/0bb8f7b52f",5,"factor: x**2 + 3*x - 18"],["hardy-course-of-pure-mathematics-1921/x-a00866be0e",15,"Hardy 1921, p. 288: The whole difficulty lies in the question, what is ..."],["dickson-theory-of-equations-1922/x-d3c80f6dcb",15,"Dickson 1922, p. 60: In this sense the tangent at O is said ..."],["dickson-theory-of-equations-1922/x-b64fc54d3f",15,"Dickson 1922, p. 65: Hence x^3 - 3lx + q = 0 has ..."],["dickson-theory-of-equations-1922/x-adb0691fc0",15,"Dickson 1922, p. 69: Between two consecutive real roots a and b of ..."],["dickson-theory-of-equations-1922/x-ea7d4d2020",15,"Dickson 1922, p. 61: For example, x^4 + 2x^3 =0 has the triple ..."],["dickson-theory-of-equations-1922/x-3bf441d44c",15,"Dickson 1922, p. 58: The use of the bend points insures greater accuracy ..."],["dickson-theory-of-equations-1922/eq-7764c3a6b3",16,"Dickson 1922, p. 51: y_1 = x_1 x_2 + x_3 x_4"],["form/48fc1c8277",5,"factor: -96*a**2*b**2 - 4*a*b*x + x**2"],["method/solving-a-triangle",8,"solving a triangle","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-method-solving-a-triangle"],["method/solving-a-right-angled-triangle",8,"solving a right-angled triangle","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-method-solving-a-right-angled-triangle"],["method/solving-a-triangle-from-its-three-sides",8,"solving a triangle from its three sides","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-method-solving-a-triangle-from-its-three-sides"],["method/solving-a-triangle-from-two-sides-and-the-included-angle",8,"solving a triangle from two sides and the included angle","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-method-solving-a-triangle-from-two-sides-and-the-included-angle"],["method/solving-a-triangle-from-two-angles-and-the-included-side",8,"solving a triangle from two angles and the included side","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-solving-a-triangle-from-two-angles-and-the-included-side"],["method/solving-a-triangle-from-two-sides-and-the-angle-opposite-one-of-them",8,"solving a triangle from two sides and the angle opposite one of them","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-solving-a-triangle-from-two-sides-and-the-angle-opposite-one-of-them"],["method/solving-a-triangle-from-two-angles-and-the-side-opposite-one-of-them",8,"solving a triangle from two angles and the side opposite one of them","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-solving-a-triangle-from-two-angles-and-the-side-opposite-one-of-them"],["planck-treatise-on-thermodynamics-1903/eq-6c807eb9d2",16,"Planck 1903, p. 71: U_{2} - U_{1} = Q + p_{0} (V_{1} - V_{2})"],["theorem/law-of-sines",9,"law of sines","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-theorem-law-of-sines"],["theorem/cosine-formula-for-an-angle-of-a-spherical-triangle",9,"cosine formula for an angle of a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-cosine-formula-for-an-angle-of-a-spherical-triangle"],["concept/regular-polygon",7,"regular polygon","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-regular-polygon"],["wentworth-plane-geometry-1899/eq-498ab4bcdc",16,"Wentworth 1899, scan 171: AB(AF + BF) = \\overline{AB}^2"],["theorem/napier-s-analogies",9,"Napier's analogies","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-napier-s-analogies"],["hardy-course-of-pure-mathematics-1921/x-f4956cb1aa",15,"Hardy 1921, p. 288: We must therefore found our definition on the notion ..."],["ball-mathematical-recreations-1905/x-f2f7bacebd",15,"Ball 1905, scan 267: The use of a single symbol to denote this ..."],["concept/right-spherical-triangle",7,"right spherical triangle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-right-spherical-triangle"],["todhunter-spherical-trigonometry-1886/x-8fc5a5d324",15,"Todhunter 1886, scan 63: Hence, when a = b, there will be no ..."],["dickson-theory-of-equations-1922/eq-a11aa4e10c",16,"Dickson 1922, p. 51: y_2 = x_1 x_3 + x_2 x_4"],["dickson-theory-of-equations-1922/eq-93454282cb",16,"Dickson 1922, p. 51: y_3 = x_1 x_4 + x_2 x_3"],["dickson-theory-of-equations-1922/eq-1cfa74d7d1",16,"Dickson 1922, p. 51: x_1 x_2 = \\tfrac{1}{2} y_1 - n"],["hardy-course-of-pure-mathematics-1921/x-4c006bdc1e",15,"Hardy 1921, p. 292: This follows from (7). For we can take H ..."],["ball-mathematical-recreations-1905/x-64182d0beb",15,"Ball 1905, scan 269: With a polygon of n sides this process gives ..."],["boyden-first-book-in-algebra-1895/ex-24/11",4,"Boyden 1895, Exercise 24 (11)"],["wentworth-plane-geometry-1899/eq-1c7555f123",16,"Wentworth 1899, scan 171: \\overline{AC}^2 = \\overline{AB}^2 + \\overline{BC}^2 = 2 \\overline{AB}^2"],["wentworth-plane-geometry-1899/eq-4192b4dc1f",16,"Wentworth 1899, scan 171: AC = AB \\sqrt{2}"],["wentworth-plane-geometry-1899/eq-cf8a09db25",16,"Wentworth 1899, scan 172: \\overline{AB}^2 = \\overline{BC}^2 + \\overline{AC}^2 - 2 BC × DC"],["wentworth-plane-geometry-1899/eq-ebf9e51630",16,"Wentworth 1899, scan 173: \\overline{AB}^2 = \\overline{BC}^2 + \\overline{AC}^2 + 2 BC × DC"],["wentworth-plane-geometry-1899/eq-016b3e0d9e",16,"Wentworth 1899, scan 189: a^2+b^2 = 2m^2+2\\left(\\dfrac{c}{2}\\right)^2"],["ball-mathematical-recreations-1905/x-eb68e7624e",15,"Ball 1905, scan 273: The reason is that Archimedes, having calculated the lengths ..."],["ball-mathematical-recreations-1905/x-c9cbe33140",15,"Ball 1905, scan 277: If the experiment is repeated many hundreds of times, ..."],["planck-treatise-on-thermodynamics-1903/eq-58ddde3223",16,"Planck 1903, p. 72: V_{1} - V_{2} = R \\frac{\\theta}{p_{0}} (n_{1} - n_{2})"],["form/5af513e62b",5,"identity: -5*a**3*b**3"],["boyden-first-book-in-algebra-1895/x-71db7ea47e",15,"Boyden 1895: To divide a polynomial by a polynomial, arrange the ..."],["boyden-first-book-in-algebra-1895/ex-24/12",4,"Boyden 1895, Exercise 24 (12)"],["form/aba2ca55f6",5,"identity: -15*a**3*c**4/b"],["shape/b4ff48e190",6,"identity: N*a**N*c**N/b"],["ball-mathematical-recreations-1905/x-4bac922b2a",15,"Ball 1905, scan 277: Inscribe in the given circle a square, and to ..."],["hardy-course-of-pure-mathematics-1921/x-f14c861585",15,"Hardy 1921, p. 295: That the value of a definite integral may sometimes ..."],["planck-treatise-on-thermodynamics-1903/x-f13b977df3",15,"Planck 1903, p. 48: The results of the experiment show that when the ..."],["planck-treatise-on-thermodynamics-1903/x-a22359d3fb",15,"Planck 1903, p. 49: Strictly speaking, this expression is vague, since a process ..."],["boyden-first-book-in-algebra-1895/ex-24/13",4,"Boyden 1895, Exercise 24 (13)"],["form/aea8c9e1a3",5,"identity: 4*a**3*b/5"],["planck-treatise-on-thermodynamics-1903/x-1a8e89ac78",15,"Planck 1903, p. 50: Wherever external pressure enters---as, for instance, in the calculation ..."],["planck-treatise-on-thermodynamics-1903/x-8e460c2cc5",15,"Planck 1903, p. 53: Only for infinitesimal changes, i.e. when 1 and 2 ..."],["hardy-course-of-pure-mathematics-1921/eq-f341da88eb",16,"Hardy 1921, p. 198: \\lim_{h \\to 0} \\frac{\\phi(x + h) - \\phi(x)}{h} = \\tan\\psi"],["hardy-course-of-pure-mathematics-1921/eq-9d83f30ec6",16,"Hardy 1921, p. 200: \\phi'(x) = \\lim_{h \\to 0} \\frac{\\phi(x + h) - \\phi(x)}{h}"],["hardy-course-of-pure-mathematics-1921/eq-02dc9f4a51",16,"Hardy 1921, p. 207: \\phi(x) = a_{0}x^{n} + a_{1}x^{n-1} + \\dots + a_{n}"],["theorem/euler-s-theorem-on-homogeneous-functions",9,"Euler's theorem on homogeneous functions","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-euler-s-theorem-on-homogeneous-functions"],["boyden-first-book-in-algebra-1895/ex-26",3,"Boyden 1895, Exercise 26"],["hardy-course-of-pure-mathematics-1921/ch-vi",2,"Hardy 1921, ch. VI: DERIVATIVES AND INTEGRALS","../books/hardy-course-of-pure-mathematics-1921/ch/ch-vi/index.html"],["boyden-first-book-in-algebra-1895/ex-27",3,"Boyden 1895, Exercise 27"],["wentworth-plane-geometry-1899/eq-da3be39f6e",16,"Wentworth 1899, scan 163: \\dfrac{AB}{A'B'}= \\dfrac{BC}{B'C'}= \\dfrac{CD}{C'D'}= \\dfrac{DE}{D'E'}"],["concept/dependent",7,"dependent","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-dependent"],["hardy-course-of-pure-mathematics-1921/eq-a6b0135800",16,"Hardy 1921, p. 207: \\phi'(x) = n \\left\\{ a_{0}x^{n-1} + \\binom{n - 1}{1} a_{1}x^{n-2} + \\binom{n - 1}{2} a_{2}x^{n-3} + \\dots + a_{n-1} \\rig"],["theorem/taylor-s-theorem",9,"Taylor's theorem","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-taylor-s-theorem"],["hardy-course-of-pure-mathematics-1921/eq-c009902e90",16,"Hardy 1921, p. 207: \\phi(x) = a_{0}(x - \\alpha_{1})(x - \\alpha_{2}) \\dots (x - \\alpha_{n})"],["hardy-course-of-pure-mathematics-1921/eq-b72250a6a6",16,"Hardy 1921, p. 207: \\phi'(x) = na_{0}x^{n-1} + (n - 1)a_{1}x^{n-2} + \\dots + a_{n-1}"],["hardy-course-of-pure-mathematics-1921/eq-3621f7612e",16,"Hardy 1921, p. 207: \\phi'(x) = a_{0}\\tsum (x - \\alpha_{2})(x - \\alpha_{3}) \\dots (x - \\alpha_{n})"],["theorem/lagrange-s-form-of-the-remainder",9,"Lagrange's form of the remainder","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-lagrange-s-form-of-the-remainder"],["de-morgan-elementary-illustrations-calculus-1899/x-ef442b718b",15,"De Morgan 1899, p. 21: Thus, when \\phi x = x^{n}, \\phi' x = ..."],["de-morgan-elementary-illustrations-calculus-1899/x-09a9cd96b8",15,"De Morgan 1899, p. 22: Here the coefficient of h is -\\dfrac{1}{x^{2}}, which is ..."],["de-morgan-elementary-illustrations-calculus-1899/x-2c23f169af",15,"De Morgan 1899, p. 21: The proof of this is equivalent to Taylor’s Theorem ..."],["theorem/schwarz-s-inequality-for-integrals",9,"Schwarz's inequality for integrals","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-schwarz-s-inequality-for-integrals"],["hardy-course-of-pure-mathematics-1921/eq-97734f487a",16,"Hardy 1921, p. 207: \\phi'(x) = a_{0} \\tsum m_{1}(x - \\alpha_{1})^{m_{1}-1} (x - \\alpha_{2})^{m_{2}}\\dots (x - \\alpha_{\\nu})^{m_{\\nu}}"],["hardy-course-of-pure-mathematics-1921/eq-7bf632eb97",16,"Hardy 1921, p. 210: -\\frac{pA(x -\\alpha)^{p-1}}{(x - \\alpha)^{2p}} = -\\frac{pA}{(x - \\alpha)^{p+1}}"],["quantity/radius-of-curvature",11,"radius of curvature","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-quantity-radius-of-curvature"],["concept/circle-of-curvature",7,"circle of curvature","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-circle-of-curvature"],["planck-treatise-on-thermodynamics-1903/eq-f391bcebd0",16,"Planck 1903, p. 72: \\frac{W}{J} = \\frac{p_{0} (V_{1} - V_{2})}{J}"],["concept/heat-equivalent",7,"heat equivalent"],["hardy-course-of-pure-mathematics-1921/eq-58ae381ee4",16,"Hardy 1921, p. 203: \\phi'(x) = f'(x) + F'(x)"],["hardy-course-of-pure-mathematics-1921/eq-5def5cb8e6",16,"Hardy 1921, p. 209: R'(x) = \\frac{P'(x)Q(x) - P(x)Q'(x)}{\\{Q(x)\\}^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-2068343824",16,"Hardy 1921, p. 203: \\phi'(x) = kf'(x)"],["hardy-course-of-pure-mathematics-1921/eq-ad5e8e6771",16,"Hardy 1921, p. 203: \\phi'(x) = f(x)F'(x) + f'(x)F(x)"],["boyden-first-book-in-algebra-1895/ex-34",3,"Boyden 1895, Exercise 34"],["boyden-first-book-in-algebra-1895/ex-35",3,"Boyden 1895, Exercise 35"],["boyden-first-book-in-algebra-1895/ch-factoring",2,"Boyden 1895, FACTORING","../books/boyden-first-book-in-algebra-1895/ch/ch-factoring/index.html"],["hardy-course-of-pure-mathematics-1921/eq-6c875f4e2a",16,"Hardy 1921, p. 204: \\phi'(x) = -\\frac{f'(x)}{\\{f(x)\\}^{2}}"],["concept/contact-of-the-nth-order",7,"contact of the nth order","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-contact-of-the-nth-order"],["whitehead-introduction-to-mathematics-1911/x-5f4ac51206",15,"Whitehead 1911, p. 89: Hence, if our symbols are to mean the ordinary ..."],["hardy-course-of-pure-mathematics-1921/eq-819ca38553",16,"Hardy 1921, p. 204: \\phi'(x) = \\frac{f'(x)F(x) - f(x)F'(x)}{\\{F(x)\\}^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-a4377b8037",16,"Hardy 1921, p. 204: \\phi'(x) = F'\\{f(x)\\} f'(x)"],["concept/composite-function",7,"composite function"],["boyden-first-book-in-algebra-1895/ex-36",3,"Boyden 1895, Exercise 36"],["boyden-first-book-in-algebra-1895/ex-37",3,"Boyden 1895, Exercise 37"],["theorem/difference-of-two-cubes",9,"difference of two cubes","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-theorem-difference-of-two-cubes"],["wentworth-plane-geometry-1899/x-6d8bd6d617",15,"Wentworth 1899, scan 111: A circumference is divided into 360 equal parts, called ..."],["wentworth-plane-geometry-1899/x-b8cf97efac",15,"Wentworth 1899, scan 110: We cannot make DB' equal to zero, since, by ..."],["wentworth-plane-geometry-1899/x-082dad3dcc",15,"Wentworth 1899, scan 115: Thus, if OA is considered positive, then OC may ..."],["wentworth-plane-geometry-1899/x-d996b3f53e",15,"Wentworth 1899, scan 114: An angle included by a tangent and a chord ..."],["concept/ambiguous-case",7,"ambiguous case","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-ambiguous-case"],["hardy-course-of-pure-mathematics-1921/eq-b9269f29fc",16,"Hardy 1921, p. 204: \\phi'(x) = \\frac{1}{\\psi'(y)}"],["hardy-course-of-pure-mathematics-1921/eq-809d734789",16,"Hardy 1921, p. 206: \\frac{dy}{dx} = \\frac{dy_{1}}{dx} + \\frac{dy_{2}}{dx}"],["hardy-course-of-pure-mathematics-1921/eq-a59990528a",16,"Hardy 1921, p. 206: \\frac{dy}{dx} = k\\frac{dy_{1}}{dx}"],["concept/definite-integral",7,"definite integral","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-definite-integral"],["hardy-course-of-pure-mathematics-1921/x-93927056b7",15,"Hardy 1921, p. 303: Two functions u and v are said to be ..."],["hardy-course-of-pure-mathematics-1921/eq-2aedb80959",16,"Hardy 1921, p. 206: \\frac{dy}{dx} = y_{1}\\frac{dy_{2}}{dx} + y_{2}\\frac{dy_{1}}{dx}"],["hardy-course-of-pure-mathematics-1921/eq-c27fac1e4a",16,"Hardy 1921, p. 206: \\frac{dy}{dx} = -\\frac{1}{y_{1}^{2}}\\, \\frac{dy_{1}}{dx}"],["hardy-course-of-pure-mathematics-1921/eq-c7226dd428",16,"Hardy 1921, p. 206: \\frac{dy}{dx} = \\biggl(y_{2}\\frac{dy_{1}}{dx} - y_{1}\\frac{dy_{2}}{dx}\\biggr) \\bigg/ y_{2}^{2}"],["hardy-course-of-pure-mathematics-1921/eq-4a30ab2dda",16,"Hardy 1921, p. 206: \\frac{dz}{dx} = \\frac{dz}{dy}\\, \\frac{dy}{dx}"],["hardy-course-of-pure-mathematics-1921/x-84d51e9b6f",15,"Hardy 1921, p. 303: This condition is therefore necessary for the existence of ..."],["hardy-course-of-pure-mathematics-1921/x-246cd666ef",15,"Hardy 1921, p. 307: This rule, which gives a very good approximation, is ..."],["hardy-course-of-pure-mathematics-1921/eq-259bf41fcb",16,"Hardy 1921, p. 206: \\dfrac{dy}{dx} = 1 \\bigg/ \\biggl(\\dfrac{dx}{dy}\\biggr)"],["theorem/equation-of-the-tangent",9,"equation of the tangent","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-equation-of-the-tangent"],["concept/quartic-equation",7,"quartic equation","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-quartic-equation"],["concept/reduced-quartic-equation",7,"reduced quartic equation","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-reduced-quartic-equation"],["hardy-course-of-pure-mathematics-1921/eq-26903dc53f",16,"Hardy 1921, p. 201: y - y_{0} = (x - x_{0}) \\phi'(x_{0})"],["dickson-theory-of-equations-1922/x-fb1bd97f16",15,"Dickson 1922, p. 71: Thus, in x^5 - 2x^3 - 4x^2 + 3 ..."],["hardy-course-of-pure-mathematics-1921/x-94c8253472",15,"Hardy 1921, p. 307: It should be observed that if \\phi(x) is any ..."],["hardy-course-of-pure-mathematics-1921/eq-3d86c0f0f8",16,"Hardy 1921, p. 201: (y - y_{0}) \\phi'(x_{0}) + x - x_{0} = 0"],["theorem/equation-of-the-normal",9,"equation of the normal"],["dickson-theory-of-equations-1922/x-12c4c34000",15,"Dickson 1922, p. 72: For example, x^6 - 3x^2 + x + 1 ..."],["dickson-theory-of-equations-1922/x-c9944bcbfb",15,"Dickson 1922, p. 76: Experience shows that most students make some error in ..."],["dickson-theory-of-equations-1922/x-58569773b3",15,"Dickson 1922, p. 71: Unfortunately it rarely tells us the exact number of ..."],["dickson-theory-of-equations-1922/x-364ec5bdb9",15,"Dickson 1922, p. 78: A violation of this Corollary usually indicates an error ..."],["theorem/comparison-theorem",9,"comparison theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-comparison-theorem"],["de-morgan-elementary-illustrations-calculus-1899/eq-8dd22968a7",16,"De Morgan 1899, p. 41: P'Q = \\phi' x\\, dx + \\phi'' x\\, \\frac{(dx)^{2}}{2} + \\phi''' x\\, \\frac{(dx)^{3}}{2·3} + \\etc."],["quantity/semi-perimeter",11,"semi-perimeter","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-quantity-semi-perimeter"],["method/finding-the-angular-radius-of-the-inscribed-circle-of-a-spherical-triangle",8,"finding the angular radius of the inscribed circle of a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-finding-the-angular-radius-of-the-inscribed-circle-of-a-spherical-triangle"],["concept/maximum",7,"maximum","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-maximum"],["concept/angular-radius",7,"angular radius","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-angular-radius"],["concept/spherical-angle",7,"spherical angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-spherical-angle"],["todhunter-spherical-trigonometry-1886/x-9944470e3f",15,"Todhunter 1886, scan 69: Let ABC be the triangle; bisect the angles A ..."],["todhunter-spherical-trigonometry-1886/x-695650b82d",15,"Todhunter 1886, scan 73: Then P will be the pole of the small ..."],["todhunter-spherical-trigonometry-1886/x-68f65f62d8",15,"Todhunter 1886, scan 74: Many examples may be proposed involving properties of the ..."],["todhunter-spherical-trigonometry-1886/x-2b7dd60977",15,"Todhunter 1886, scan 75: Thus P is the pole of the small circle ..."],["todhunter-spherical-trigonometry-1886/x-7905c724cb",15,"Todhunter 1886, scan 76: Shew that in an equilateral triangle \\tan R = ..."],["concept/harmonic-series",7,"harmonic series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-harmonic-series"],["de-morgan-elementary-illustrations-calculus-1899/x-d955f0ee8c",15,"De Morgan 1899, p. 24: Therefore to find the coefficient of h in the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-426a08cf16",15,"De Morgan 1899, p. 24: Hence the ratio of the increments of \\phi x ..."],["dickson-theory-of-equations-1922/eq-4e6014cb4b",16,"Dickson 1922, p. 51: x_1 x_2 + x_3 x_4 = y_1"],["dickson-theory-of-equations-1922/eq-44bd77719d",16,"Dickson 1922, p. 51: x_3 x_4 = \\tfrac{1}{2} y_1 + n"],["todhunter-spherical-trigonometry-1886/eq-38d65d06bc",16,"Todhunter 1886, scan 152: F-1+S = E + 1"],["todhunter-spherical-trigonometry-1886/eq-652302df59",16,"Todhunter 1886, scan 152: F + S = E + 2"],["de-morgan-elementary-illustrations-calculus-1899/x-b160c237b0",15,"De Morgan 1899, p. 24: It follows, therefore, that if, instead of the full ..."],["concept/sufficiently-large-values",7,"sufficiently large values","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-sufficiently-large-values"],["hardy-course-of-pure-mathematics-1921/x-b577dfc56f",15,"Hardy 1921, p. 309: Moreover, in inferring the convergence or divergence of \\sum ..."],["whitehead-introduction-to-mathematics-1911/x-085a6f17a4",15,"Whitehead 1911, p. 103: Hence both for addition and for multiplication the couple ..."],["dickson-theory-of-equations-1922/eq-2e8f3c53ee",16,"Dickson 1922, p. 155: f(z) \\equiv z^n + a_1 z^{n-1} + \\dotsb + a_n = 0"],["dickson-theory-of-equations-1922/eq-bcdc18a1be",16,"Dickson 1922, p. 155: f(z) = \\phi(x,y) + i\\psi(x,y)"],["dickson-theory-of-equations-1922/eq-cde9467aee",16,"Dickson 1922, p. 156: f(z) = z^n(1+D)"],["whitehead-introduction-to-mathematics-1911/x-9458afe428",15,"Whitehead 1911, p. 111: It was receiving its final form about the same ..."],["hardy-course-of-pure-mathematics-1921/x-bb69e23e33",15,"Hardy 1921, p. 217: If \\phi(a) = 0 and \\phi(b) = 0, then ..."],["hardy-course-of-pure-mathematics-1921/x-617aa8f056",15,"Hardy 1921, p. 311: None the less d’Alembert’s test is very useful in ..."],["hardy-course-of-pure-mathematics-1921/x-6ebc9f7340",15,"Hardy 1921, p. 211: But there is no practical difficulty in the actual ..."],["hardy-course-of-pure-mathematics-1921/x-77644e4f5f",15,"Hardy 1921, p. 214: This function is called the second derivative or second ..."],["hardy-course-of-pure-mathematics-1921/x-6f424e68d5",15,"Hardy 1921, p. 219: A condition for a maximum or minimum value of ..."],["hardy-course-of-pure-mathematics-1921/x-6854c5fbfa",15,"Hardy 1921, p. 313: Let s be the sum of the series of ..."],["hardy-course-of-pure-mathematics-1921/x-251ae42b73",15,"Hardy 1921, p. 315: The fact is that the geometric series, by comparison ..."],["hardy-course-of-pure-mathematics-1921/x-b6ddd3b2da",15,"Hardy 1921, p. 317: The converse of Abel’s theorem is not true, i.e. ..."],["hardy-course-of-pure-mathematics-1921/eq-c0a72b3ee8",16,"Hardy 1921, p. 211: \\frac{dy}{dx} = \\biggl(\\frac{dy}{dz}\\biggr) \\bigg/ \\biggl(\\frac{dx}{dz}\\biggr) = \\frac{p}{q} z^{p-q} = mx^{m-1}"],["hardy-course-of-pure-mathematics-1921/eq-39633c3aa2",16,"Hardy 1921, p. 211: \\phi'(x) = \\lim_{h \\to 0} \\frac{(x + h)^{m} - x^{m}}{h}"],["hardy-course-of-pure-mathematics-1921/eq-2d3bba3762",16,"Hardy 1921, p. 211: \\lim_{\\xi \\to x} \\frac{\\xi^{m} - x^{m}}{\\xi - x} = mx^{m-1}"],["hardy-course-of-pure-mathematics-1921/eq-9f403d3b67",16,"Hardy 1921, p. 211: \\frac{d}{dx} (ax + b)^{m} = ma(ax + b)^{m-1}"],["hardy-course-of-pure-mathematics-1921/eq-9337aa93d4",16,"Hardy 1921, p. 211: x^{3} + y^{3} - 3axy = 0"],["hardy-course-of-pure-mathematics-1921/eq-1060cda363",16,"Hardy 1921, p. 211: x^{2} + y^{2} \\frac{dy}{dx} - a\\left(y + x \\frac{dy}{dx}\\right) = 0"],["hardy-course-of-pure-mathematics-1921/eq-19dc168d58",16,"Hardy 1921, p. 211: \\frac{dy}{dx} = -\\frac{x^{2} - ay}{y^{2} - ax}"],["hardy-course-of-pure-mathematics-1921/eq-e872fea371",16,"Hardy 1921, p. 212: D_{x} \\sin x = \\cos x"],["hardy-course-of-pure-mathematics-1921/eq-2353069910",16,"Hardy 1921, p. 212: D_{x} \\cos x = -\\sin x"],["hardy-course-of-pure-mathematics-1921/eq-75b0665818",16,"Hardy 1921, p. 212: D_{x} \\tan x = \\sec^{2} x"],["hardy-course-of-pure-mathematics-1921/eq-222aa0473b",16,"Hardy 1921, p. 212: D_{x} \\cot x = -\\cosec^{2} x"],["hardy-course-of-pure-mathematics-1921/eq-1ba0d29d78",16,"Hardy 1921, p. 212: D_{x} \\sec x = \\tan x \\sec x"],["hardy-course-of-pure-mathematics-1921/eq-20bd599b40",16,"Hardy 1921, p. 212: D_{x} \\cosec x = -\\cot x\\cosec x"],["hardy-course-of-pure-mathematics-1921/x-f136d8c28e",15,"Hardy 1921, p. 315: But this is far from being the case; if ..."],["concept/isopiestic-curve",7,"isopiestic curve","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-isopiestic-curve"],["dickson-theory-of-equations-1922/eq-736c0060d8",16,"Dickson 1922, p. 51: x_1 + x_2 + x_3 + x_4 = -b"],["hardy-course-of-pure-mathematics-1921/eq-fe9494ba04",16,"Hardy 1921, p. 212: D_{x} \\arcsin x = ±1/\\sqrtp{1 - x^{2}}"],["boyden-first-book-in-algebra-1895/ex-13/11a",4,"Boyden 1895, Exercise 13 (11a)"],["dickson-theory-of-equations-1922/eq-3ce9cec79a",16,"Dickson 1922, p. 51: x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4 = -d"],["dickson-theory-of-equations-1922/eq-ee84fe1e31",16,"Dickson 1922, p. 51: x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4 = c"],["dickson-theory-of-equations-1922/eq-7b4fcb73f0",16,"Dickson 1922, p. 51: x_1 x_2 x_3 x_4 = e"],["concept/ambiguous-sign",7,"ambiguous sign"],["boyden-first-book-in-algebra-1895/ex-36/12",4,"Boyden 1895, Exercise 36 (12)"],["hardy-course-of-pure-mathematics-1921/eq-886ad49ca3",16,"Hardy 1921, p. 212: D_{x} \\arccos x = \\mp 1/\\sqrtp{1 - x^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-34f067fa70",16,"Hardy 1921, p. 212: D_{x} \\arctan x = 1/(1 + x^{2})"],["form/365aa31d3e",5,"factor: 81*x**4 - 18*x**2 + 1"],["hardy-course-of-pure-mathematics-1921/eq-87d9d63b3c",16,"Hardy 1921, p. 212: D_{x} \\arccot x = -1/(1 + x^{2})"],["hardy-course-of-pure-mathematics-1921/eq-ffc07ff567",16,"Hardy 1921, p. 212: D_{x} \\arcsec x = ± 1/\\{x\\sqrtp{x^{2} - 1}\\}"],["hardy-course-of-pure-mathematics-1921/eq-c8c56a8e3d",16,"Hardy 1921, p. 212: D_{x} \\arccosec x = \\mp 1/\\{x\\sqrtp{x^{2} - 1}\\}"],["hardy-course-of-pure-mathematics-1921/x-733fa055db",15,"Hardy 1921, p. 337: if \\sum u_{n} is absolutely convergent then it is ..."],["boyden-first-book-in-algebra-1895/ex-36/13",4,"Boyden 1895, Exercise 36 (13)"],["form/f5ba5fa970",5,"factor: 121*a**2 + 66*a*x + 9*x**2"],["boyden-first-book-in-algebra-1895/ex-36/14",4,"Boyden 1895, Exercise 36 (14)"],["shape/d078b1ec87",6,"factor: 2*N*x**N + 1"],["hardy-course-of-pure-mathematics-1921/eq-9e77107220",16,"Hardy 1921, p. 212: D_{x} \\arcsin(x/a) = ±1/\\sqrtp{a^{2} - x^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-30bb09d5df",16,"Hardy 1921, p. 212: D_{x} \\arctan(x/a) = a/(x^{2} + a^{2})"],["hardy-course-of-pure-mathematics-1921/eq-6dea22caa2",16,"Hardy 1921, p. 212: a\\sqrtb{1 - (x^{2}/a^{2})} = ±\\sqrtp{a^{2} - x^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-04f82a280a",16,"Hardy 1921, p. 217: \\phi'(x) = 0"],["hardy-course-of-pure-mathematics-1921/eq-edfae2f6c5",16,"Hardy 1921, p. 217: \\phi'(x) > 0"],["hardy-course-of-pure-mathematics-1921/eq-fdf4695acd",16,"Hardy 1921, p. 219: \\phi'(\\xi) = 0"],["concept/necessary-condition",7,"necessary condition"],["concept/repeated-limit",7,"repeated limit","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-repeated-limit"],["form/c7ae4a4761",5,"factor: 4*x**2 - 36*x + 81"],["boyden-first-book-in-algebra-1895/ex-36/15",4,"Boyden 1895, Exercise 36 (15)"],["form/1a99a2a493",5,"factor: 9*a**2 - 6*a*x**2 + x**4"],["shape/12e7c52bc4",6,"factor: N*a*x**N + N*a**N + x**N"],["planck-treatise-on-thermodynamics-1903/x-a3a269a53e",15,"Planck 1903, p. 69: This means that the internal energy of lead and ..."],["planck-treatise-on-thermodynamics-1903/x-8955f873bc",15,"Planck 1903, p. 71: It, therefore, depends on the initial and final states ..."],["planck-treatise-on-thermodynamics-1903/x-524fc04014",15,"Planck 1903, p. 69: The pressure has, however, very little influence on the ..."],["planck-treatise-on-thermodynamics-1903/x-cc77ecc37f",15,"Planck 1903, p. 68: In our equations we shall therefore use Q (the ..."],["todhunter-spherical-trigonometry-1886/eq-10f8472e19",16,"Todhunter 1886, scan 74: \\tan R_1 = \\frac{2\\sin\\frac{1}{2}a\\cos\\frac{1}{2}b\\cos\\frac{1}{2}c} {\\surd{\\{\\sin s\\sin(s-a)\\sin(s-b)\\sin(s-c)\\}}}"],["hardy-course-of-pure-mathematics-1921/eq-88eb0962de",16,"Hardy 1921, p. 220: \\phi'(x) = 3x^{2}"],["concept/inflection",7,"inflection"],["hardy-course-of-pure-mathematics-1921/eq-25807df4db",16,"Hardy 1921, p. 220: y = 1 - \\sqrtp{x^{2}}"],["concept/angle-bisector",7,"angle bisector","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-angle-bisector"],["hardy-course-of-pure-mathematics-1921/eq-d4652908f9",16,"Hardy 1921, p. 214: \\phi(x) = x^{2}\\sin(1/x)"],["hardy-course-of-pure-mathematics-1921/eq-62671e421f",16,"Hardy 1921, p. 214: \\phi'(x) = 2x \\sin(1/x) - \\cos(1/x)"],["hardy-course-of-pure-mathematics-1921/eq-87ea73ff88",16,"Hardy 1921, p. 221: \\phi'(0) = \\lim_{h \\to 0} \\frac{h^{2}\\sin(1/h)}{h} = 0"],["hardy-course-of-pure-mathematics-1921/eq-79e89e77fc",16,"Hardy 1921, p. 221: \\phi(x) = x^{2}\\sin(1/x) + ax"],["hardy-course-of-pure-mathematics-1921/x-5497f88e1a",15,"Hardy 1921, p. 320: The second of the two tests mentioned in [§]172 ..."],["concept/complementary-vector",7,"complementary vector","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-complementary-vector"],["hardy-course-of-pure-mathematics-1921/eq-ff42e19701",16,"Hardy 1921, p. 221: \\phi'(x) = 2x\\sin(1/x) - \\cos(1/x) + a"],["concept/isosceles-triangle",7,"isosceles triangle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-isosceles-triangle"],["concept/circumscribed-circle",7,"circumscribed circle","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-circumscribed-circle"],["concept/median-of-a-triangle",7,"median of a triangle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-median-of-a-triangle"],["hardy-course-of-pure-mathematics-1921/x-99581912c1",15,"Hardy 1921, p. 321: For our present purposes the field of application of ..."],["wentworth-plane-geometry-1899/eq-d844f1603d",16,"Wentworth 1899, scan 175: \\dfrac{OM}{OQ} = \\dfrac{OP}{ON}"],["hardy-course-of-pure-mathematics-1921/x-703020f67a",15,"Hardy 1921, p. 322: The integral \\ds\\int_{a}^{x} \\phi(t)\\, dt was defined in [§§]156 ..."],["quantity/distance",11,"distance","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-quantity-distance"],["wentworth-plane-geometry-1899/x-fe9b4629af",15,"Wentworth 1899, scan 143: Let r and r' denote the radii of the ..."],["wentworth-plane-geometry-1899/x-dbef2fe938",15,"Wentworth 1899, scan 140: Let ABC be the \\triangle required, EF the given ..."],["wentworth-plane-geometry-1899/x-726fa69ba5",15,"Wentworth 1899, scan 143: To bisect the angle formed by two lines, without ..."],["wentworth-plane-geometry-1899/x-ba4bdf36d0",15,"Wentworth 1899, scan 143: To draw the internal tangents use an auxiliary \\odot ..."],["concept/complete-divisor",7,"complete divisor","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-complete-divisor"],["todhunter-spherical-trigonometry-1886/eq-04fc2dc9b4",16,"Todhunter 1886, scan 70: \\tan r = \\Surd{\\left\\{ \\dfrac{\\sin (s - a) \\sin (s - b) \\sin (s - c)}{\\sin s} \\right\\}} = \\dfrac{n}{\\sin s}"],["todhunter-spherical-trigonometry-1886/eq-62601d0b83",16,"Todhunter 1886, scan 70: \\tan r = \\dfrac{\\sin\\tfrac{1}{2}B \\sin \\tfrac{1}{2}C}{\\cos \\tfrac{1}{2}A} \\sin a"],["todhunter-spherical-trigonometry-1886/eq-768fc7d450",16,"Todhunter 1886, scan 70: \\tan r = \\dfrac{\\surd\\{-\\cos S \\cos (S - A) \\cos (S - B) \\cos (S - C)\\}}{2 \\cos \\tfrac{1}{2}A \\cos \\tfrac{1}{2}B \\cos \\t"],["todhunter-spherical-trigonometry-1886/eq-f0f1cf0ad3",16,"Todhunter 1886, scan 70: 4 \\cos\\tfrac{1}{2}A \\cos\\tfrac{1}{2}B \\cos\\tfrac{1}{2}C = \\cos S + \\cos (S - A) + \\cos (S - B) + \\cos (S - C)"],["todhunter-spherical-trigonometry-1886/eq-73e871dd4c",16,"Todhunter 1886, scan 70: \\cot r = \\frac{1}{2N} \\bigl\\{\\cos S + \\cos (S - A) + \\cos (S - B) + \\cos (S - C)\\bigr\\}"],["wentworth-plane-geometry-1899/eq-990f5b2b71",16,"Wentworth 1899, scan 176: AC : AD = AD : AB"],["wentworth-plane-geometry-1899/eq-d0aa994786",16,"Wentworth 1899, scan 176: AC × AB = \\overline{AD}^2"],["todhunter-spherical-trigonometry-1886/eq-da7876b4b3",16,"Todhunter 1886, scan 71: \\tan r_1 = \\Surd{\\left\\{\\dfrac{\\sin s \\sin(s-b)\\sin(s-c)} {\\sin(s-a)}\\right\\}} = \\dfrac{n}{\\sin(s-a)}"],["todhunter-spherical-trigonometry-1886/eq-916e3ae166",16,"Todhunter 1886, scan 71: \\tan r_1 = \\dfrac{\\cos\\tfrac{1}{2}B \\cos\\tfrac{1}{2}C} {\\cos\\tfrac{1}{2}A} \\sin a"],["todhunter-spherical-trigonometry-1886/eq-9153800a52",16,"Todhunter 1886, scan 71: \\cot r_1 = \\dfrac{1}{2N} \\{-c\\cos S - \\cos(S-A) + \\cos(S-B) + \\cos(S-C) \\}"],["todhunter-spherical-trigonometry-1886/eq-d91254e1dc",16,"Todhunter 1886, scan 73: \\tan R = \\dfrac{\\tan\\tfrac{1}{2}a}{\\cos(S-A)}"],["todhunter-spherical-trigonometry-1886/eq-74a8f7d01c",16,"Todhunter 1886, scan 73: \\tan R = \\Surd{\\left\\{\\dfrac{-\\cos S}{\\cos(S-A)\\cos(S-B)\\cos(S-C)}\\right\\}} = \\dfrac{\\cos S}{N}"],["hardy-course-of-pure-mathematics-1921/x-a983cd4f5d",15,"Hardy 1921, p. 324: There is one fundamental property of a convergent infinite ..."],["todhunter-spherical-trigonometry-1886/eq-032e2247dc",16,"Todhunter 1886, scan 73: \\tan R = \\dfrac{2\\sin\\tfrac{1}{2}a \\sin\\tfrac{1}{2}b \\sin\\tfrac{1}{2}c} {\\surd{\\left\\{\\sin s\\sin(s-a) \\sin(s-b) \\sin(s-c"],["todhunter-spherical-trigonometry-1886/eq-1575f54e6c",16,"Todhunter 1886, scan 74: 4\\sin\\tfrac{1}{2}a\\sin\\tfrac{1}{2}b\\sin\\tfrac{1}{2}c = \\sin(s-a) + \\sin(s-b) + \\sin(s-c)-\\sin s"],["todhunter-spherical-trigonometry-1886/eq-3a8158aafa",16,"Todhunter 1886, scan 74: \\tan R=\\dfrac{1}{2n}\\{ \\sin(s-a)+\\sin(s-b)+\\sin(s-c)-\\sin s \\}"],["todhunter-spherical-trigonometry-1886/eq-a3c3848cce",16,"Todhunter 1886, scan 74: \\tan R_1 = \\frac{\\tan\\frac{1}{2}a}{-\\cos S}"],["todhunter-spherical-trigonometry-1886/eq-69a917517f",16,"Todhunter 1886, scan 74: \\tan R_1 = \\Surd{\\left\\{ \\frac{\\cos(S-A)}{-\\cos S\\cos(S-B)\\cos(S-C)}\\right\\}} = \\frac{\\cos(S-A)}{N}"],["todhunter-spherical-trigonometry-1886/eq-d9ce4adeb9",16,"Todhunter 1886, scan 74: \\tan R_1 = \\frac{\\sin\\frac{1}{2}a} {\\sin A\\sin\\frac{1}{2}b\\sin\\frac{1}{2}c}"],["todhunter-spherical-trigonometry-1886/eq-f893970be1",16,"Todhunter 1886, scan 74: \\tan R_1 =\\frac{1}{2n} \\{\\sin s - \\sin(s-a) + \\sin(s-b) + \\sin(s-c) \\}"],["todhunter-spherical-trigonometry-1886/eq-1586a4479e",16,"Todhunter 1886, scan 75: (\\cot r + \\tan R)^2=\\dfrac{1}{4n^2}(\\sin a+\\sin b+\\sin c)^2 -1"],["todhunter-spherical-trigonometry-1886/eq-8e0b5f1392",16,"Todhunter 1886, scan 75: (\\cot r_1-\\tan R)^2=\\dfrac{1}{4n^2}(\\sin b+\\sin c-\\sin a)^2 -1"],["todhunter-spherical-trigonometry-1886/eq-beff7f0256",16,"Todhunter 1886, scan 75: 4n^2=1-\\cos^2 a-\\cos^2 b-\\cos^2 c+2\\cos a\\cos b\\cos c"],["todhunter-spherical-trigonometry-1886/eq-1b9000ea4f",16,"Todhunter 1886, scan 75: \\cot r + \\tan R = \\dfrac{1}{2n}\\Bigl\\{\\sin s + \\sin(s-a)+\\sin(s-b)+\\sin (s-c)\\Bigr\\}"],["dickson-theory-of-equations-1922/eq-2174b10c59",16,"Dickson 1922, p. 51: y_1 + y_2 + y_3 = c"],["de-morgan-elementary-illustrations-calculus-1899/x-fe5b8decb8",15,"De Morgan 1899, p. 25: Thus, if x becomes x + dx, x^{2} becomes ..."],["de-morgan-elementary-illustrations-calculus-1899/x-23d4ec6aea",15,"De Morgan 1899, p. 25: Care must be taken not to confound d.x^{2}, the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-9ed1161a79",15,"De Morgan 1899, p. 26: It must not be imagined that because x occurs ..."],["de-morgan-elementary-illustrations-calculus-1899/x-47c9095102",15,"De Morgan 1899, p. 28: If x = 4 and dx = .01, then ..."],["de-morgan-elementary-illustrations-calculus-1899/x-ae58aafec0",15,"De Morgan 1899, p. 29: We must adopt the first of these explanations when ..."],["de-morgan-elementary-illustrations-calculus-1899/x-c04c26be53",15,"De Morgan 1899, p. 29: The equations which we thus use are not absolutely ..."],["concept/quadratic-form",7,"quadratic form","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-quadratic-form"],["de-morgan-elementary-illustrations-calculus-1899/x-64d8e94343",15,"De Morgan 1899, p. 26: In the Differential Calculus, the limit of the ratio ..."],["concept/lune",7,"lune","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-lune"],["concept/table-of-power-residues",7,"table of power residues","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-table-of-power-residues"],["theorem/area-of-a-spherical-polygon",9,"area of a spherical polygon","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-area-of-a-spherical-polygon"],["theorem/area-of-a-lune",9,"area of a lune","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-area-of-a-lune"],["theorem/cagnoli-s-theorem",9,"Cagnoli's theorem","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-cagnoli-s-theorem"],["wentworth-plane-geometry-1899/eq-ccf7423350",16,"Wentworth 1899, scan 177: \\overline{NO}^2 = NM × NP - OM × OP"],["method/trial-division",8,"trial division","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-trial-division"],["theorem/area-of-a-sphere",9,"area of a sphere","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-area-of-a-sphere"],["method/modular-exponentiation",8,"modular exponentiation","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-modular-exponentiation"],["concept/spherical-polygon",7,"spherical polygon","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-spherical-polygon"],["concept/symmetrical-spherical-triangles",7,"symmetrical spherical triangles","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-symmetrical-spherical-triangles"],["dickson-theory-of-equations-1922/eq-dab3b0efc5",16,"Dickson 1922, p. 52: 64k^6 + 32qk^4 + 4(q^2 - 4s)k^2 - r^2 = 0"],["dickson-theory-of-equations-1922/eq-bf5a941106",16,"Dickson 1922, p. 53: k_1^2 + k_2^2 + k_3^2 = -\\tfrac{1}{2}q"],["dickson-theory-of-equations-1922/eq-c008449efe",16,"Dickson 1922, p. 53: k_1^2 k_2^2 k_3^2 = \\frac{r^2}{64}"],["todhunter-spherical-trigonometry-1886/x-e1388788ed",15,"Todhunter 1886, scan 79: This expression is true even when the polygon has ..."],["person/carl-friedrich-gauss",1,"Carl Friedrich Gauss","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-carl-friedrich-gauss"],["person/marin-mersenne",1,"Marin Mersenne","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-marin-mersenne"],["person/adrien-marie-legendre",1,"Adrien Marie Legendre","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-adrien-marie-legendre"],["person/f-n-cole",1,"F.N. Cole","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-f-n-cole"],["person/c-e-bickmore",1,"C.E. Bickmore","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-c-e-bickmore"],["law/parallelogram-law-of-vector-addition",10,"parallelogram law of vector addition","../books/ball-mathematical-recreations-1905/terms/index.html#t-law-parallelogram-law-of-vector-addition"],["ball-mathematical-recreations-1905/x-3394fa6d4d",15,"Ball 1905, scan 283: that, if 4n + 3 and 8n + 7 ..."],["ball-mathematical-recreations-1905/x-96f0c80f17",15,"Ball 1905, scan 283: Comme de 25, qui est un quarré, ôtez 2; ..."],["whitehead-introduction-to-mathematics-1911/x-826fef33c1",15,"Whitehead 1911, p. 112: This conception, simple as it looks, is the main ..."],["whitehead-introduction-to-mathematics-1911/x-9f106f14e0",15,"Whitehead 1911, p. 121: A locus is the curve (or surface, if we ..."],["whitehead-introduction-to-mathematics-1911/x-363fba015b",15,"Whitehead 1911, p. 124: Consider y - x = 1: the corresponding locus ..."],["whitehead-introduction-to-mathematics-1911/x-6ec6d53d26",15,"Whitehead 1911, p. 125: We each of us refer our sensible perceptions of ..."],["whitehead-introduction-to-mathematics-1911/x-3d917500dd",15,"Whitehead 1911, p. 119: Euclid always contemplates a straight line as drawn between ..."],["whitehead-introduction-to-mathematics-1911/x-cbf48a7731",15,"Whitehead 1911, p. 117: Variables, like a, b, and c above, which are ..."],["hardy-course-of-pure-mathematics-1921/x-eeffed6e32",15,"Hardy 1921, p. 333: It often happens that the subject of integration has ..."],["de-morgan-elementary-illustrations-calculus-1899/x-4c22d3bc92",15,"De Morgan 1899, p. 29: If two straight lines be drawn at right angles ..."],["planck-treatise-on-thermodynamics-1903/eq-742e0cfea6",16,"Planck 1903, p. 89: \\Phi = M\\phi = M \\left(c_{v} \\log \\theta + \\frac{R}{m} \\log v + \\const\\right)"],["de-morgan-elementary-illustrations-calculus-1899/x-2d4c94dc26",15,"De Morgan 1899, p. 29: for, though there is an infinite number of points ..."],["de-morgan-elementary-illustrations-calculus-1899/x-4cc06a163f",15,"De Morgan 1899, p. 29: The line OA is called the axis of x, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-e3e4d89b00",15,"De Morgan 1899, p. 30: It is moreover usual to call the co-ordinate OM, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-c2d5d96512",15,"De Morgan 1899, p. 30: As O moves towards A, the point P will, ..."],["hardy-course-of-pure-mathematics-1921/x-8c92041bd2",15,"Hardy 1921, p. 228: It is natural to consider the converse question, that ..."],["hardy-course-of-pure-mathematics-1921/x-df67d80e11",15,"Hardy 1921, p. 231: There is however one case of exception to the ..."],["hardy-course-of-pure-mathematics-1921/x-f6b275e91c",15,"Hardy 1921, p. 230: the \\int and the dx no more mean anything ..."],["planck-treatise-on-thermodynamics-1903/eq-d42a342646",16,"Planck 1903, p. 89: d\\Phi = M \\left(c_{v}\\, \\frac{d\\theta}{\\theta} + \\frac{R}{m}\\, \\frac{dv}{v}\\right) = \\frac{M · q}{\\theta} = \\frac{Q}{\\th"],["planck-treatise-on-thermodynamics-1903/eq-a36e82dd9d",16,"Planck 1903, p. 89: d\\Phi = M \\left(c_{v}\\, \\frac{d\\theta}{\\theta} + \\frac{R}{m}\\, \\frac{dv}{v}\\right) = \\frac{dU + p\\, dV}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-fd0de5c196",16,"Planck 1903, p. 89: Q + W = dU"],["planck-treatise-on-thermodynamics-1903/eq-15461fd21a",16,"Planck 1903, p. 90: \\Phi_{1} + \\Phi_{2} = \\const"],["planck-treatise-on-thermodynamics-1903/eq-2a94b0480b",16,"Planck 1903, p. 91: \\Phi_{1} + \\Phi_{2} = \\Phi_{1}' + \\Phi_{2}'\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-4a40bfb19e",16,"Planck 1903, p. 91: \\Phi_{1} + \\Phi_{2} + \\dots + \\Phi_{n} = \\Phi_{1}' + \\Phi_{2}' + \\dots + \\Phi_{n}'"],["hardy-course-of-pure-mathematics-1921/x-588d2ac54b",15,"Hardy 1921, p. 236: If the equation Q(x) = 0 cannot be solved ..."],["boyden-first-book-in-algebra-1895/ex-36/16",4,"Boyden 1895, Exercise 36 (16)"],["planck-treatise-on-thermodynamics-1903/eq-856c171dcb",16,"Planck 1903, p. 97: \\int_{1\\; (\\alpha)}^{2} \\frac{dU + p\\, dV}{\\theta} + \\int_{2\\; (\\beta)}^{1} \\frac{dU + p\\, dV}{\\theta} = 0"],["planck-treatise-on-thermodynamics-1903/eq-c7b1f71a97",16,"Planck 1903, p. 97: \\int_{1\\; (\\alpha)}^{2} \\frac{dU + p\\, dV}{\\theta} = \\int_{1\\; (\\beta)}^{2} \\frac{dU + p\\, dV}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-71ad805282",16,"Planck 1903, p. 98: \\Phi = \\int \\frac{dU + p\\, dV}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-75d40f792e",16,"Planck 1903, p. 98: d\\Phi = \\frac{dU + p\\, dV}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-7d92ce9621",16,"Planck 1903, p. 98: d\\phi = \\frac{du + p\\, dv}{\\theta}\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-08681809d4",16,"Planck 1903, p. 99: d\\Phi = \\frac{Q}{\\theta}\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-0d8f61ef4a",16,"Planck 1903, p. 98: U = Mu"],["planck-treatise-on-thermodynamics-1903/eq-a311cc7376",16,"Planck 1903, p. 98: V = Mv"],["concept/horoscope",7,"horoscope","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-horoscope"],["concept/astrological-house",7,"astrological house","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-astrological-house"],["shape/243e4f1cb3",6,"factor: 2*N*x**N + N"],["whitehead-introduction-to-mathematics-1911/eq-7aec5d73b3",16,"Whitehead 1911, p. 238: \\frac{\\sin A}{a} = \\frac{\\sin B}{b} = \\frac{\\sin C}{c}"],["whitehead-introduction-to-mathematics-1911/eq-f4fb62a4cd",16,"Whitehead 1911, p. 238: a^{2} = b^{2} + c^{2} - 2bc \\cos A"],["theorem/law-of-cosines",9,"law of cosines"],["whitehead-introduction-to-mathematics-1911/eq-8f3934de24",16,"Whitehead 1911, p. 239: A + B + C = 180°"],["whitehead-introduction-to-mathematics-1911/eq-b006ad1e88",16,"Whitehead 1911, p. 239: a < b + c"],["concept/zodiacal-sign",7,"zodiacal sign","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-zodiacal-sign"],["person/john-flamsteed",1,"John Flamsteed","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-john-flamsteed"],["concept/great-circle",7,"great circle","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-great-circle"],["person/girolamo-cardano",1,"Girolamo Cardano","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-girolamo-cardano"],["hardy-course-of-pure-mathematics-1921/x-b0b56cb7d2",15,"Hardy 1921, p. 330: An integral in which the subject of integration tends ..."],["method/isolation-of-a-root",8,"isolation of a root","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-isolation-of-a-root"],["ball-mathematical-recreations-1905/x-d773587e74",15,"Ball 1905, scan 306: It is easier to give instances of success in ..."],["ball-mathematical-recreations-1905/x-837f078ce6",15,"Ball 1905, scan 305: The old lady dug in the spot thus indicated, ..."],["concept/bend-point",7,"bend point","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-bend-point"],["shape/2838202be0",6,"factor: 2*N*a**N*x**N + x**N"],["dickson-theory-of-equations-1922/x-ef66ec0ef1",15,"Dickson 1922, p. 91: Newton used the close approximation 0.1 to p, in ..."],["dickson-theory-of-equations-1922/x-5ded088744",15,"Dickson 1922, p. 91: Given an approximate value a of a real root, ..."],["dickson-theory-of-equations-1922/x-92b26befd5",15,"Dickson 1922, p. 86: W. G. Horner, London Philosophical Transactions, 1819. Earlier (1804) ..."],["dickson-theory-of-equations-1922/x-b113785214",15,"Dickson 1922, p. 92: Failure is certain if we use a point P_2 ..."],["dickson-theory-of-equations-1922/x-09a9a335ac",15,"Dickson 1922, p. 93: The advantage of having c at each step is ..."],["dickson-theory-of-equations-1922/x-b88978a295",15,"Dickson 1922, p. 98: To find the imaginary roots x+yi of an equation ..."],["hardy-course-of-pure-mathematics-1921/eq-46f193eeeb",16,"Hardy 1921, p. 226: \\phi(b) - \\phi(a) = (b - a)\\phi'(\\xi)"],["hardy-course-of-pure-mathematics-1921/eq-38f962ce53",16,"Hardy 1921, p. 227: \\phi(b) = \\phi(a) + (b - a) \\phi'\\{a + \\theta(b - a)\\}"],["concept/alternating-series",7,"alternating series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-alternating-series"],["concept/series-of-positive-and-negative-terms",7,"series of positive and negative terms","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-series-of-positive-and-negative-terms"],["hardy-course-of-pure-mathematics-1921/eq-1dcca4b25f",16,"Hardy 1921, p. 227: \\phi(a + h) = \\phi(a) + h\\phi'(a + \\theta h)"],["hardy-course-of-pure-mathematics-1921/eq-6b46c3ac6c",16,"Hardy 1921, p. 230: \\phi(x) = \\int \\psi(x)\\, dx"],["concept/circle-of-convergence",7,"circle of convergence","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-circle-of-convergence"],["hardy-course-of-pure-mathematics-1921/eq-f8e4650609",16,"Hardy 1921, p. 230: \\int x^{m}\\, dx = \\frac{x^{m+1}}{m + 1}"],["hardy-course-of-pure-mathematics-1921/eq-4394e2ee53",16,"Hardy 1921, p. 230: \\int \\cos x\\, dx = \\sin x"],["hardy-course-of-pure-mathematics-1921/eq-f0ef56cd30",16,"Hardy 1921, p. 230: \\int \\sin x\\, dx = -\\cos x"],["concept/rearrangement-of-a-series",7,"rearrangement of a series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-rearrangement-of-a-series"],["theorem/abel-s-test",9,"Abel's test","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-abel-s-test"],["hardy-course-of-pure-mathematics-1921/eq-b5d004d098",16,"Hardy 1921, p. 231: \\int \\frac{dx}{x} = \\log(±x) = \\log|x|"],["hardy-course-of-pure-mathematics-1921/eq-f0d3f2539b",16,"Hardy 1921, p. 231: \\int \\frac{dx}{x} = \\log x"],["hardy-course-of-pure-mathematics-1921/eq-e03bdeef09",16,"Hardy 1921, p. 231: \\int \\frac{dx}{x} = \\log(-x)"],["hardy-course-of-pure-mathematics-1921/eq-408281831a",16,"Hardy 1921, p. 232: \\int \\frac{dx}{x} = \\tfrac{1}{2}\\log x^{2}"],["hardy-course-of-pure-mathematics-1921/eq-d4eff72c43",16,"Hardy 1921, p. 232: \\int \\frac{dx}{1 + x^{2}} = \\arctan x"],["theorem/cauchy-schwarz-inequality",9,"Cauchy–Schwarz inequality","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-cauchy-schwarz-inequality"],["hardy-course-of-pure-mathematics-1921/eq-3d50b49f33",16,"Hardy 1921, p. 232: \\int \\{f(x) + F(x)\\}\\, dx = \\int f(x) dx + \\int F(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-0d5a338121",16,"Hardy 1921, p. 232: \\int \\frac{x}{\\sqrtp{1 - x^{2}}} = ±\\arcsin x"],["hardy-course-of-pure-mathematics-1921/eq-401be09736",16,"Hardy 1921, p. 232: \\log 1 = 0"],["hardy-course-of-pure-mathematics-1921/eq-a293242816",16,"Hardy 1921, p. 232: \\log (1/x) = -\\log x"],["hardy-course-of-pure-mathematics-1921/eq-b91c95345f",16,"Hardy 1921, p. 232: \\log xy = \\log x + \\log y"],["hardy-course-of-pure-mathematics-1921/eq-6c4bff4a7e",16,"Hardy 1921, p. 232: \\int kf(x)\\, dx = k\\int f(x)\\, dx"],["concept/complex-variable",7,"complex variable","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-complex-variable"],["hardy-course-of-pure-mathematics-1921/x-fe469c998e",15,"Hardy 1921, p. 338: In the first place we note that, if \\sum ..."],["person/peter-gustav-lejeune-dirichlet",1,"Peter Gustav Lejeune Dirichlet","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-peter-gustav-lejeune-dirichlet"],["hardy-course-of-pure-mathematics-1921/x-476da3035d",15,"Hardy 1921, p. 337: The reader should carefully guard himself against supposing that ..."],["hardy-course-of-pure-mathematics-1921/x-33b493c1ae",15,"Hardy 1921, p. 339: In the first instance there are no comparison tests ..."],["hardy-course-of-pure-mathematics-1921/eq-7a175027ca",16,"Hardy 1921, p. 232: \\int (a_{0}x^{n} + a_{1}x^{n-1} + \\dots + a_{n})\\, dx = \\frac{a_{0}x^{n+1}}{n + 1} + \\frac{a_{1}x^{n}}{n} + \\dots + a_{n"],["de-morgan-elementary-illustrations-calculus-1899/eq-db9055b953",16,"De Morgan 1899, p. 47: (a + da)y + (b - db)x = (a + da)(b - db)\\Add{;}"],["de-morgan-elementary-illustrations-calculus-1899/eq-f669eb9351",16,"De Morgan 1899, p. 47: y\\, da - x\\, db = b\\, da - a\\, db - da\\, db\\Add{.}"],["de-morgan-elementary-illustrations-calculus-1899/eq-c538f03ded",16,"De Morgan 1899, p. 47: y - x\\, \\frac{2a + da}{2b - db} = b - a\\, \\frac{2a + da}{2b - db} - db\\Add{.}"],["de-morgan-elementary-illustrations-calculus-1899/eq-10bbd0fdea",16,"De Morgan 1899, p. 47: y - \\frac{a}{b}\\, x = b - \\frac{a^{2}}{b}"],["de-morgan-elementary-illustrations-calculus-1899/eq-9f03fe101d",16,"De Morgan 1899, p. 47: by - ax = b^{2} - a^{2}\\Add{.}"],["de-morgan-elementary-illustrations-calculus-1899/eq-8ebcb7b59b",16,"De Morgan 1899, p. 47: x = OM = \\frac{a^{3}}{a^{2} + b^{2}} = \\frac{a^{3}}{l^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-a9a6b159e7",16,"Hardy 1921, p. 233: \\int \\frac{A}{(x - \\alpha)^{p}}\\, dx = -\\frac{A}{p - 1}\\, \\frac{1}{(x - \\alpha)^{p-1}}"],["hardy-course-of-pure-mathematics-1921/eq-b7d4f19a67",16,"Hardy 1921, p. 233: \\int F'\\{f(x)\\}\\, f'(x)\\, dx = F\\{f(x)\\}"],["theorem/product-of-extremes-and-means",9,"product of extremes and means","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-product-of-extremes-and-means"],["theorem/division-of-proportions",9,"division of proportions","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-division-of-proportions"],["concept/equimultiples",7,"equimultiples","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-equimultiples"],["de-morgan-elementary-illustrations-calculus-1899/x-58078c3273",15,"De Morgan 1899, p. 33: The latter method is preferable, inasmuch as it enables ..."],["method/composition-of-proportions",8,"composition of proportions","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-composition-of-proportions"],["concept/quantities-of-the-same-kind",7,"quantities of the same kind","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-quantities-of-the-same-kind"],["hardy-course-of-pure-mathematics-1921/eq-cbe550f7c7",16,"Hardy 1921, p. 233: \\int \\psi(ax + b)\\, dx = \\frac{1}{a}\\phi(ax + b)"],["concept/linear-substitution",7,"linear substitution"],["boyden-first-book-in-algebra-1895/ex-13/11b",4,"Boyden 1895, Exercise 13 (11b)"],["hardy-course-of-pure-mathematics-1921/eq-8f1ce82386",16,"Hardy 1921, p. 233: \\int \\frac{dx}{ax + b} = \\frac{1}{a} \\log|ax + b|"],["concept/harmonic-division",7,"harmonic division","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-harmonic-division"],["hardy-course-of-pure-mathematics-1921/x-c0312ffff2",15,"Hardy 1921, p. 340: We shall see shortly that the series 1 - ..."],["todhunter-spherical-trigonometry-1886/eq-826316e317",16,"Todhunter 1886, scan 82: \\sin^2\\tfrac{1}{4}E = \\frac{\\sin\\frac{1}{2}s \\sin\\frac{1}{2}(s-a) \\sin\\frac{1}{2}(s-b) \\sin\\frac{1}{2}(s-c) } {\\cos\\frac"],["concept/mutually-equiangular-polygons",7,"mutually equiangular polygons","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-mutually-equiangular-polygons"],["hardy-course-of-pure-mathematics-1921/eq-03c855c260",16,"Hardy 1921, p. 233: \\int \\frac{dx}{x - \\alpha} = \\log|x - \\alpha|"],["hardy-course-of-pure-mathematics-1921/eq-6df063ce58",16,"Hardy 1921, p. 234: \\lambda = A/2a"],["hardy-course-of-pure-mathematics-1921/eq-2b7fc05ecf",16,"Hardy 1921, p. 234: \\mu = -D/(2a\\sqrt{\\Delta})"],["hardy-course-of-pure-mathematics-1921/x-e3298f2207",15,"Hardy 1921, p. 342: It can indeed be proved that a conditionally convergent ..."],["wentworth-plane-geometry-1899/x-4d554c39ed",15,"Wentworth 1899, scan 145: The mean proportional between two quantities is equal to ..."],["wentworth-plane-geometry-1899/x-f7c6bd1691",15,"Wentworth 1899, scan 150: In the treatment of proportion, it is assumed that ..."],["wentworth-plane-geometry-1899/x-54b8ecb75b",15,"Wentworth 1899, scan 152: By increasing the number of equal parts into which ..."],["wentworth-plane-geometry-1899/x-74a576bd46",15,"Wentworth 1899, scan 157: Similar polygons are polygons that have their homologous angles ..."],["wentworth-plane-geometry-1899/x-e95b2632ac",15,"Wentworth 1899, scan 157: The primary idea of similarity is likeness of form."],["hardy-course-of-pure-mathematics-1921/eq-0973eeccc2",16,"Hardy 1921, p. 234: \\gamma = -b/a"],["hardy-course-of-pure-mathematics-1921/eq-334da62431",16,"Hardy 1921, p. 234: \\delta = \\sqrt{\\Delta}/a"],["hardy-course-of-pure-mathematics-1921/eq-da195dd392",16,"Hardy 1921, p. 234: \\Delta = ac - b^{2}"],["hardy-course-of-pure-mathematics-1921/eq-bee92aa2cf",16,"Hardy 1921, p. 234: D = aB - bA"],["boyden-first-book-in-algebra-1895/ex-24/14",4,"Boyden 1895, Exercise 24 (14)"],["hardy-course-of-pure-mathematics-1921/eq-29b95145cf",16,"Hardy 1921, p. 234: \\int \\frac{f'(x)}{f(x)}\\, dx = \\log |f(x)|"],["hardy-course-of-pure-mathematics-1921/eq-0d0350eb0d",16,"Hardy 1921, p. 234: \\int \\frac{2(x - \\lambda)}{(x - \\lambda)^{2} + \\mu^{2}}\\, dx = \\log\\{(x - \\lambda)^{2} + \\mu^{2}\\}"],["hardy-course-of-pure-mathematics-1921/eq-a829ff1619",16,"Hardy 1921, p. 234: \\int \\frac{-2\\delta\\mu}{(x - \\lambda)^{2} + \\mu^{2}}\\, dx = -2\\delta \\arctan \\left(\\frac{x - \\lambda}{\\mu}\\right)"],["hardy-course-of-pure-mathematics-1921/eq-b07cecfdaa",16,"Hardy 1921, p. 237: ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0"],["hardy-course-of-pure-mathematics-1921/eq-31ff66cc6e",16,"Hardy 1921, p. 238: aX^{2} + 2hXY + bY^{2} + 2GX + 2FY = 0"],["hardy-course-of-pure-mathematics-1921/eq-4ea7627168",16,"Hardy 1921, p. 238: F = h\\xi + b\\eta + f"],["form/fef6b2ab87",5,"identity: -3*a**3*b**3"],["concept/inverse-tangent",7,"inverse tangent","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-inverse-tangent"],["boyden-first-book-in-algebra-1895/ex-24/15",4,"Boyden 1895, Exercise 24 (15)"],["form/7f25257f46",5,"identity: 8*a**3*b**4"],["hardy-course-of-pure-mathematics-1921/eq-4b33d3173e",16,"Hardy 1921, p. 238: G = a\\xi + h\\eta + g"],["hardy-course-of-pure-mathematics-1921/x-5a2b92905c",15,"Hardy 1921, p. 343: This theorem may be stated as follows: Resulta convergent ..."],["hardy-course-of-pure-mathematics-1921/eq-46aa0121c2",16,"Hardy 1921, p. 238: x - \\xi = -\\frac{2 (G + Ft)}{a + 2ht + bt^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-69d4504380",16,"Hardy 1921, p. 238: y - \\eta = -\\frac{2t(G + Ft)}{a + 2ht + bt^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-5a08bfbf62",16,"Hardy 1921, p. 238: hx + by + f = -\\tfrac{1}{2}(a + 2ht + bt^{2}) \\frac{dx}{dt}"],["planck-treatise-on-thermodynamics-1903/x-f4015effd1",15,"Planck 1903, p. 171: If this point lie within one of the regions ..."],["planck-treatise-on-thermodynamics-1903/x-52549a3ef9",15,"Planck 1903, p. 80: If a perfect gas be allowed to expand, doing ..."],["hardy-course-of-pure-mathematics-1921/eq-d3c80cf3ec",16,"Hardy 1921, p. 238: 2y = \\frac{(t^{2} + c)\\sqrt{a} + 2bt}{t\\sqrt{a} + b}"],["method/approximate-solution-of-a-spherical-triangle",8,"approximate solution of a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-approximate-solution-of-a-spherical-triangle"],["theorem/d-alembert-s-ratio-test",9,"d'Alembert's ratio test","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-d-alembert-s-ratio-test"],["hardy-course-of-pure-mathematics-1921/eq-7d4b582b40",16,"Hardy 1921, p. 238: \\int \\frac{dx}{hx + by + f}= -2\\int \\frac{dt}{a + 2ht + bt^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-29b0476ca8",16,"Hardy 1921, p. 238: y^{2} = ax^{2} + 2bx + c"],["hardy-course-of-pure-mathematics-1921/eq-cbfb755ae8",16,"Hardy 1921, p. 238: 2\\frac{dx}{dt} = \\frac{(t^{2} + c)\\sqrt{a} + 2bt}{(t\\sqrt{a} + b)^{2}}"],["boyden-first-book-in-algebra-1895/ex-24/16",4,"Boyden 1895, Exercise 24 (16)"],["theorem/dirichlet-s-rearrangement-theorem",9,"Dirichlet's rearrangement theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-dirichlet-s-rearrangement-theorem"],["hardy-course-of-pure-mathematics-1921/eq-a66c184613",16,"Hardy 1921, p. 238: \\int \\frac{dx}{y} = \\int \\frac{dt}{t\\sqrt{a} + b} = \\frac{1}{\\sqrt{a}} \\log \\left|x\\sqrt{a} + y + \\frac{b}{\\sqrt{a}}\\rig"],["de-morgan-elementary-illustrations-calculus-1899/eq-c09da155b8",16,"De Morgan 1899, p. 50: Aa : A'a :: OA : OB :: a : b"],["concept/binomial-series",7,"binomial series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-binomial-series"],["planck-treatise-on-thermodynamics-1903/x-ab901604c4",15,"Planck 1903, p. 171: The conditions of stable equilibrium of any substance can ..."],["concept/diameter",7,"diameter","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-diameter"],["todhunter-spherical-trigonometry-1886/x-a3dd0ae03f",15,"Todhunter 1886, scan 86: If the sides of a spherical triangle be small ..."],["todhunter-spherical-trigonometry-1886/x-9670b9ee3a",15,"Todhunter 1886, scan 86: Thus when the sides of the spherical triangle and ..."],["todhunter-spherical-trigonometry-1886/x-16fe05fe8a",15,"Todhunter 1886, scan 88: therefore \\dfrac{S}{r^2} is approximately equal to the spherical excess ..."],["hardy-course-of-pure-mathematics-1921/eq-6c8e85acee",16,"Hardy 1921, p. 239: \\int \\frac{dx}{\\sqrtp{x^{2} + a^{2}}} = \\log \\{x + \\sqrtp{x^{2} + a^{2}}\\}"],["hardy-course-of-pure-mathematics-1921/eq-0da67353ab",16,"Hardy 1921, p. 239: \\int \\frac{dx}{\\sqrtp{x^{2} - a^{2}}} = \\log |x + \\sqrtp{x^{2} - a^{2}}|"],["form/0ebf9188c7",5,"identity: -6*a**2*b**2*c**3"],["todhunter-spherical-trigonometry-1886/eq-2207369818",16,"Todhunter 1886, scan 80: \\sin\\tfrac{1}{2}E = \\dfrac{\\surd\\{\\sin s \\sin(s-a) \\sin(s-b) \\sin(s-c) \\} x} {2\\cos\\tfrac{1}{2}a \\cos\\tfrac{1}{2}b \\cos\\"],["todhunter-spherical-trigonometry-1886/eq-ba76121da9",16,"Todhunter 1886, scan 80: \\tan\\tfrac{1}{4}E = \\surd\\{\\tan\\tfrac{1}{2}s \\tan\\tfrac{1}{2}(s-a) \\tan\\tfrac{1}{2}(s-b) \\tan\\tfrac{1}{2}(s-c) \\}"],["todhunter-spherical-trigonometry-1886/eq-298ba1333f",16,"Todhunter 1886, scan 81: \\sin\\tfrac{1}{2}E = \\sin C \\sin\\tfrac{1}{2}a \\sin\\tfrac{1}{2}b \\sec\\tfrac{1}{2}c;"],["todhunter-spherical-trigonometry-1886/eq-e24709451a",16,"Todhunter 1886, scan 81: \\tan\\tfrac{1}{2}E = \\frac{\\sin\\tfrac{1}{2}a \\sin\\tfrac{1}{2}b \\sin C } {\\cos\\tfrac{1}{2}a \\cos\\tfrac{1}{2}b + \\sin\\tfrac"],["todhunter-spherical-trigonometry-1886/eq-4d203e9088",16,"Todhunter 1886, scan 81: \\frac{\\cos^2\\tfrac{1}{2}a + \\cos^2\\tfrac{1}{2}b + \\cos^2\\tfrac{1}{2}c-1 } {2\\cos\\tfrac{1}{2}a \\cos\\tfrac{1}{2}b \\cos\\tfr"],["hardy-course-of-pure-mathematics-1921/eq-6d67995f75",16,"Hardy 1921, p. 239: \\int \\frac{dx}{\\sqrtp{a^{2} - x^{2}}} = \\arcsin(x/a)"],["hardy-course-of-pure-mathematics-1921/eq-6baf3ff312",16,"Hardy 1921, p. 239: \\lambda x + \\mu = (\\lambda/a) (ax + b) + \\mu - (\\lambda b/a)"],["concept/algebraic-identity",7,"algebraic identity"],["todhunter-spherical-trigonometry-1886/eq-abaf048ff8",16,"Todhunter 1886, scan 82: \\cos^2\\tfrac{1}{4}E = \\frac{\\cos\\frac{1}{2}s \\cos\\frac{1}{2}(s-a) \\cos\\frac{1}{2}(s-b) \\cos\\frac{1}{2}(s-c) } {\\cos\\frac"],["todhunter-spherical-trigonometry-1886/eq-6da0507ea5",16,"Todhunter 1886, scan 82: \\sin(C-\\tfrac12E) = \\frac{\\surd\\{\\sin s \\sin(s-a) \\sin(s-b) \\sin(s-c) \\} } {2\\sin\\frac12a \\sin\\frac12b \\cos\\frac12c }"],["concept/negative-direction",7,"negative direction","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-negative-direction"],["concept/sign",7,"sign","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-sign"],["hardy-course-of-pure-mathematics-1921/eq-980b70a11b",16,"Hardy 1921, p. 239: \\int \\frac{ax + b}{\\sqrtp{ax^{2} + 2bx + c}}\\, dx = \\sqrtp{ax^{2} + 2bx + c}"],["hardy-course-of-pure-mathematics-1921/eq-f786c92a8c",16,"Hardy 1921, p. 239: \\int \\frac{(\\lambda x + \\mu)\\, dx}{\\sqrtp{ax^{2} + 2bx + c}} = \\frac{\\lambda}{a} \\sqrtp{ax^{2} + 2bx + c} + \\left(\\mu - "],["hardy-course-of-pure-mathematics-1921/eq-16fa454b03",16,"Hardy 1921, p. 239: \\kappa = (ac - b^{2})/a"],["de-morgan-elementary-illustrations-calculus-1899/x-4d829ef47c",15,"De Morgan 1899, p. 31: And conversely it may be proved by any number ..."],["de-morgan-elementary-illustrations-calculus-1899/x-84547d81c6",15,"De Morgan 1899, p. 32: Hence the equation ay + bx = ab belongs ..."],["de-morgan-elementary-illustrations-calculus-1899/x-f99c8a41e0",15,"De Morgan 1899, p. 35: In this manner, if four points be taken similarly ..."],["hardy-course-of-pure-mathematics-1921/eq-00d4e3f617",16,"Hardy 1921, p. 240: \\int(\\lambda x + \\mu) \\sqrtp{ax^{2} + 2bx + c}\\, dx \\\\ = \\left(\\frac{\\lambda}{3a}\\right) (ax^{2} + 2bx + c)^{3/2} + \\lef"],["de-morgan-elementary-illustrations-calculus-1899/x-c35fd79311",15,"De Morgan 1899, p. 33: Thus, if OE = 4, and OF = 5, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-5f0f367225",15,"De Morgan 1899, p. 36: And thus, if y be any function of x, ..."],["wentworth-plane-geometry-1899/x-6618b16436",15,"Wentworth 1899, scan 176: If from a fixed point without a circle a ..."],["wentworth-plane-geometry-1899/x-f26d3ad3a2",15,"Wentworth 1899, scan 173: The last three theorems enable us to compute the ..."],["wentworth-plane-geometry-1899/x-c211b0108d",15,"Wentworth 1899, scan 171: The projection of any line upon a second line ..."],["wentworth-plane-geometry-1899/x-cf843dae46",15,"Wentworth 1899, scan 175: that is, the ratio of two corresponding segments is ..."],["boyden-first-book-in-algebra-1895/ex-24/17",4,"Boyden 1895, Exercise 24 (17)"],["de-morgan-elementary-illustrations-calculus-1899/eq-63be0f0666",16,"De Morgan 1899, p. 52: q\\left(1 + \\frac{\\mu}{dp}\\right) : (p - dp)\\left(1 + \\frac{\\nu}{dp}\\right) :: a(a - da) : b(b + db)"],["de-morgan-elementary-illustrations-calculus-1899/eq-9244f5e9ef",16,"De Morgan 1899, p. 52: q &: p &&:: a^{2} &&: b^{2}"],["de-morgan-elementary-illustrations-calculus-1899/eq-1f7848bcf0",16,"De Morgan 1899, p. 52: q + p = l &: p &&:: a^{2} + b^{2} = l^{2} &&: b^{2}"],["concept/vertex-of-a-conic-section",7,"vertex of a conic section","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-vertex-of-a-conic-section"],["theorem/focus-directrix-property-of-conic-sections",9,"focus-directrix property of conic sections","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-theorem-focus-directrix-property-of-conic-sections"],["dickson-theory-of-equations-1922/eq-129f5917fd",16,"Dickson 1922, p. 47: (y_1 - y_2)^2 (y_1 - y_3)^2 (y_2 - y_3)^2 = -4p^3 - 27q^2"],["person/johannes-kepler",1,"Johannes Kepler","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-johannes-kepler"],["whitehead-introduction-to-mathematics-1911/x-c2b5f59c6c",15,"Whitehead 1911, p. 136: No more impressive warning can be given to those ..."],["whitehead-introduction-to-mathematics-1911/x-8f49b29eb2",15,"Whitehead 1911, p. 131: There are accordingly three types of conic sections, namely, ..."],["whitehead-introduction-to-mathematics-1911/x-cbf28ac952",15,"Whitehead 1911, p. 136: Here we have finally found the desired property of ..."],["whitehead-introduction-to-mathematics-1911/x-d469d2784b",15,"Whitehead 1911, p. 138: (1) The orbits of the planets are ellipses, the ..."],["concept/abstractness",7,"abstractness","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-abstractness"],["whitehead-introduction-to-mathematics-1911/x-d357bb1c05",15,"Whitehead 1911, p. 139: This sweeping general law, coupled with the three laws ..."],["quantity/length-of-a-curve",11,"length of a curve","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-quantity-length-of-a-curve"],["theorem/abel-s-theorem",9,"Abel's theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-abel-s-theorem"],["person/niels-henrik-abel",1,"Niels Henrik Abel","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-person-niels-henrik-abel"],["concept/linear-difference-equation",7,"linear difference-equation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-linear-difference-equation"],["form/32d9ed2072",5,"identity: 2*c**2*(a + b)**2"],["shape/ec27407abf",6,"identity: N*c**N*(a + b)**N"],["hardy-course-of-pure-mathematics-1921/x-8e0d53b77c",15,"Hardy 1921, p. 240: Suppose, for example, that \\phi(x) = x\\psi(x), where \\psi(x) ..."],["hardy-course-of-pure-mathematics-1921/x-f779a30245",15,"Hardy 1921, p. 249: It is indeed one which needs and has received ..."],["hardy-course-of-pure-mathematics-1921/x-c212a39b29",15,"Hardy 1921, p. 250: Calculate \\Phi(x), the integral of \\phi(x). This involves an ..."],["boyden-first-book-in-algebra-1895/ex-24/18",4,"Boyden 1895, Exercise 24 (18)"],["hardy-course-of-pure-mathematics-1921/x-f3c6556fad",15,"Hardy 1921, p. 251: It is however easy to see what the formula ..."],["hardy-course-of-pure-mathematics-1921/x-f22ae4b4f4",15,"Hardy 1921, p. 253: This integral cannot however be evaluated in terms of ..."],["concept/infinite-sequence",7,"infinite sequence","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-infinite-sequence"],["method/taylor-s-theorem",8,"Taylor's theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-taylor-s-theorem"],["method/partial-fractions",8,"partial fractions","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-partial-fractions"],["concept/infinite-integral",7,"infinite integral","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-infinite-integral"],["hardy-course-of-pure-mathematics-1921/x-e6cb28bb4c",15,"Hardy 1921, p. 443: The cosine and sine are continuous for all values ..."],["hardy-course-of-pure-mathematics-1921/x-8d9a321f9d",15,"Hardy 1921, p. 346: For \\lim a_{n}z_{1}^{n} = 0, since \\sum a_{n}z_{1}^{n} is ..."],["theorem/limit-representation-of-the-exponential-function",9,"limit representation of the exponential function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-representation-of-the-exponential-function"],["form/e615737e74",5,"identity: 15*c*(a - b)**3/(3*a - 3*b)"],["hardy-course-of-pure-mathematics-1921/x-531ace9542",15,"Hardy 1921, p. 361: The logarithm of x tends to infinity with x, ..."],["wentworth-plane-geometry-1899/eq-f873cfefbf",16,"Wentworth 1899, scan 202: S:S'=\\overline{AB}^2:\\overline{A'B'^2}"],["wentworth-plane-geometry-1899/eq-028c911437",16,"Wentworth 1899, scan 203: BE \\Bumpeq CH + AF"],["law/pythagorean-theorem-area-form",10,"Pythagorean theorem (area form)"],["wentworth-plane-geometry-1899/eq-66a8987c79",16,"Wentworth 1899, scan 210: \\overline{NP}^2 = MN × NO = a × b"],["wentworth-plane-geometry-1899/eq-868646ac61",16,"Wentworth 1899, scan 205: \\overline{BD}^2 + \\overline{AC}^2 = \\overline{AB}^2 + \\overline{DC}^2"],["wentworth-plane-geometry-1899/eq-a15819d0c1",16,"Wentworth 1899, scan 217: = \\sqrt{s(s - a)(s - b)(s - c)}"],["wentworth-plane-geometry-1899/eq-deae2d0caa",16,"Wentworth 1899, scan 217: = \\dfrac{abc}{4R}"],["hardy-course-of-pure-mathematics-1921/x-fff2e494a7",15,"Hardy 1921, p. 346: In other words, if the series converges at P ..."],["boyden-first-book-in-algebra-1895/ex-13/11c",4,"Boyden 1895, Exercise 13 (11c)"],["dickson-theory-of-equations-1922/eq-db2a962045",16,"Dickson 1922, p. 47: a^4 P = 18 abcd - 4b^3 d + b^2 c^2 - 4ac^3 - 27a^2 d^2"],["dickson-theory-of-equations-1922/eq-6a497c2f9d",16,"Dickson 1922, p. 50: x^4 +bx^3 +cx^2 +dx+e=0"],["dickson-theory-of-equations-1922/eq-91be38c8e2",16,"Dickson 1922, p. 50: (x^2 + \\tfrac{1}{2}bx + \\tfrac{1}{2}y)^2 = (\\tfrac{1}{4}b^2 - c + y)x^2 + (\\tfrac{1}{2}by - d)x + \\tfrac{1}{4}y^2 - e"],["dickson-theory-of-equations-1922/eq-265ee3ae11",16,"Dickson 1922, p. 50: (\\tfrac{1}{2}by - d)^2 - 4(\\tfrac{1}{4}b^2 - c + y)(\\tfrac{1}{4}y^2 - e) = 0"],["dickson-theory-of-equations-1922/eq-5aaac54a36",16,"Dickson 1922, p. 50: y^3 - cy^2 + (bd - 4e)y - b^{2}e + 4ce - d^2 = 0"],["dickson-theory-of-equations-1922/eq-00e50a8546",16,"Dickson 1922, p. 50: x^2 + \\tfrac{1}{2}bx + \\tfrac{1}{2}y = mx+n"],["shape/4aa712f59e",6,"identity: N*c*(a - b)**N/(N*a + N*b)"],["macfarlane-vector-analysis-quaternions-1906/eq-aa5672c189",16,"Macfarlane 1906: \\beta^b \\times \\gamma^c = (\\beta^\\frac{b}{2}\\gamma^\\frac{c}{2})^2"],["macfarlane-vector-analysis-quaternions-1906/eq-a2fd828b9a",16,"Macfarlane 1906: \\cos\\frac{b}{2}\\,\\cos\\frac{c}{2}-\\sin\\frac{b}{2}\\,\\sin\\frac{c}{2}"],["dickson-theory-of-equations-1922/eq-ad4801ebf6",16,"Dickson 1922, p. 51: \\Delta = ( x_1 - x_2 )^2 ( x_1 - x_3 )^2 ( x_1 - x_4 )^2 ( x_2 - x_3 )^2 ( x_2 - x_4 )^2 ( x_3 - x_4 )^2"],["dickson-theory-of-equations-1922/eq-1f55d0cb8b",16,"Dickson 1922, p. 52: y_1 - y_2 = (x_1-x_4)(x_2-x_3)"],["dickson-theory-of-equations-1922/eq-21c1b15cff",16,"Dickson 1922, p. 52: (y_1-y_2)^2 (y_1-y_3)^2 (y_2- y_3)^2 = \\Delta"],["dickson-theory-of-equations-1922/eq-c7221081c9",16,"Dickson 1922, p. 52: p = bd - 4e - \\tfrac{1}{3} c^2"],["dickson-theory-of-equations-1922/eq-cac9022cf1",16,"Dickson 1922, p. 52: q = -b^2 e + \\tfrac{1}{3} bcd + \\tfrac{8}{3} ce - d^2 - \\tfrac{2}{27} c^3"],["dickson-theory-of-equations-1922/eq-439a41945c",16,"Dickson 1922, p. 52: z^4 + qz^2 + rz + s = 0"],["dickson-theory-of-equations-1922/eq-91019bd7ed",16,"Dickson 1922, p. 52: (z^2 + 2kz + l)(z^2 - 2kz + m) = z^4 + (l + m - 4k^2)z^2 + 2k(m - l)z + lm"],["hardy-course-of-pure-mathematics-1921/x-eeacb5a5f0",15,"Hardy 1921, p. 347: It should be observed that this general result gives ..."],["dickson-theory-of-equations-1922/eq-42f45803e8",16,"Dickson 1922, p. 53: 64y^3 + 32qy^2 + 4(q^2 - 4s)y - r^2 = 0"],["dickson-theory-of-equations-1922/eq-61a93dd16a",16,"Dickson 1922, p. 53: z = \\sqrt{y_1} + \\sqrt{y_2} + \\sqrt{y_3}"],["dickson-theory-of-equations-1922/eq-9d740d680a",16,"Dickson 1922, p. 54: \\sqrt{y_1}·\\sqrt{y_2}·\\sqrt{y_3} = -\\frac{r}{8}"],["dickson-theory-of-equations-1922/eq-786adabbfb",16,"Dickson 1922, p. 49: z^3 - \\tfrac{3}{4}z - \\tfrac{1}{4}\\cos 3A = 0 \\qquad (z = \\cos A)"],["dickson-theory-of-equations-1922/eq-cd6c867298",16,"Dickson 1922, p. 49: \\cos 3A = 4\\cos^3 A - 3\\cos A"],["dickson-theory-of-equations-1922/eq-312e8df0ad",16,"Dickson 1922, p. 49: n = \\sqrt{-\\tfrac{4}{3}p}"],["dickson-theory-of-equations-1922/eq-5b68683450",16,"Dickson 1922, p. 49: \\cos{3A} = -\\tfrac{1}{2}q ÷ \\sqrt{-p^{3}/27}"],["dickson-theory-of-equations-1922/eq-40e6fa3a96",16,"Dickson 1922, p. 49: z^3 + \\frac{p}{n^2}z + \\frac{q}{n^3} = 0"],["instrument/grille",13,"grille","../books/ball-mathematical-recreations-1905/terms/index.html#t-instrument-grille"],["instrument/scytale",13,"scytale","../books/ball-mathematical-recreations-1905/terms/index.html#t-instrument-scytale"],["ball-mathematical-recreations-1905/x-06997623fc",15,"Ball 1905, scan 311: The art of constructing a cryptograph lies in the ..."],["ball-mathematical-recreations-1905/x-80a4555193",15,"Ball 1905, scan 311: A simple example is when each letter is replaced ..."],["concept/heat-engine",7,"heat engine","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-heat-engine"],["de-morgan-elementary-illustrations-calculus-1899/eq-becba1758a",16,"De Morgan 1899, p. 110: \\frac{dy}{dx} = -\\frac{y - 1}{x}"],["de-morgan-elementary-illustrations-calculus-1899/eq-dca2f35d6c",16,"De Morgan 1899, p. 110: y = 1 + \\dfrac{1}{x}"],["de-morgan-elementary-illustrations-calculus-1899/eq-2beccc8947",16,"De Morgan 1899, p. 109: \\phi x = \\psi y"],["de-morgan-elementary-illustrations-calculus-1899/eq-37df0c573a",16,"De Morgan 1899, p. 110: -\\dfrac{1}{x^{2}}"],["concept/limit-of-the-ratio-of-the-increment",7,"limit of the ratio of the increment"],["de-morgan-elementary-illustrations-calculus-1899/eq-f200ed188d",16,"De Morgan 1899, p. 109: u = \\Chg{\\phi(x)}{\\phi x}"],["de-morgan-elementary-illustrations-calculus-1899/eq-8ca84612db",16,"De Morgan 1899, p. 109: u = \\psi y"],["ball-mathematical-recreations-1905/x-c644cb21bb",15,"Ball 1905, scan 312: The majority of stories dealing with secret communications are ..."],["ball-mathematical-recreations-1905/x-b63b92df0a",15,"Ball 1905, scan 320: It is said that during the Indian Mutiny messages ..."],["ball-mathematical-recreations-1905/x-313814a71f",15,"Ball 1905, scan 322: In English the letter which occurs most frequently is ..."],["hardy-course-of-pure-mathematics-1921/x-01a7ce3cf5",15,"Hardy 1921, p. 444: The formulae for differentiation of the circular functions may ..."],["hardy-course-of-pure-mathematics-1921/x-c1a22ef130",15,"Hardy 1921, p. 353: Such equations may be solved by a method which ..."],["boyden-first-book-in-algebra-1895/ex-13/11d",4,"Boyden 1895, Exercise 13 (11d)"],["concept/e",7,"e","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-e"],["concept/scale-of-infinity",7,"scale of infinity","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-scale-of-infinity"],["concept/common-logarithm",7,"common logarithm","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-common-logarithm"],["de-morgan-elementary-illustrations-calculus-1899/eq-f4987d528b",16,"De Morgan 1899, p. 109: du = \\psi(y + dy) - \\psi y"],["de-morgan-elementary-illustrations-calculus-1899/eq-2c0b9770b1",16,"De Morgan 1899, p. 109: \\phi' x\\, dx + \\etc. = \\psi' y\\, dy + \\etc."],["de-morgan-elementary-illustrations-calculus-1899/eq-70bf118f8e",16,"De Morgan 1899, p. 109: \\frac{dy}{dx} = \\frac{\\phi' x}{\\psi' y} = \\frac{\\;\\dfrac{du}{dx}\\;}{\\dfrac{du}{dy}}"],["de-morgan-elementary-illustrations-calculus-1899/x-fae104f011",15,"De Morgan 1899, p. 37: The line TPV indicates the direction in which the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-d6648d2d48",15,"De Morgan 1899, p. 37: If, therefore, a line PV be drawn through P, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-9d809bb568",15,"De Morgan 1899, p. 36: Since the relation y = x^{2} is true for ..."],["de-morgan-elementary-illustrations-calculus-1899/x-ecca7e3ba8",15,"De Morgan 1899, p. 38: There is some confusion between these different uses of ..."],["de-morgan-elementary-illustrations-calculus-1899/x-020e3c0b22",15,"De Morgan 1899, p. 37: If the curve were the interior of a small ..."],["de-morgan-elementary-illustrations-calculus-1899/x-a8b3bf9814",15,"De Morgan 1899, p. 38: This problem, of drawing a tangent to any curve, ..."],["theorem/logarithm-grows-more-slowly-than-any-positive-power",9,"logarithm grows more slowly than any positive power","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-logarithm-grows-more-slowly-than-any-positive-power"],["planck-treatise-on-thermodynamics-1903/x-fc6f56c954",15,"Planck 1903, p. 86: Such an engine could be used simultaneously as a ..."],["planck-treatise-on-thermodynamics-1903/x-0e16568dd4",15,"Planck 1903, p. 87: From the impossibility of perpetual motion of the second ..."],["planck-treatise-on-thermodynamics-1903/eq-acabf38579",16,"Planck 1903, p. 105: \\frac{Q_{1}}{\\theta_{1}} + \\frac{Q_{2}}{\\theta_{2}} = 0"],["planck-treatise-on-thermodynamics-1903/eq-f003fe1084",16,"Planck 1903, p. 105: Q_{1} : Q_{2} : W = (-\\theta_{1}) : \\theta_{2} : (\\theta_{1} - \\theta_{2})"],["concept/line",7,"line","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-line"],["wentworth-first-steps-in-algebra-1894/ex-21/2",4,"Wentworth 1894, Exercise 21 (2)"],["form/d3e06df3e3",5,"identity: 4*a*(2*a - 3*b)"],["shape/acabb3a269",6,"identity: N*a*(N*a + N*b)"],["planck-treatise-on-thermodynamics-1903/x-52909a0434",15,"Planck 1903, p. 97: The entropy of a body in a given state, ..."],["concept/geodesy",7,"geodesy","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-geodesy"],["theorem/derivative-of-the-exponential-function",9,"derivative of the exponential function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-derivative-of-the-exponential-function"],["concept/figure-of-the-earth",7,"figure of the Earth","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-figure-of-the-earth"],["concept/station",7,"station","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-station"],["theorem/exponential-as-a-limit",9,"exponential as a limit","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-exponential-as-a-limit"],["method/triangulation",8,"triangulation","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-triangulation"],["wentworth-first-steps-in-algebra-1894/ex-33/2",4,"Wentworth 1894, Exercise 33 (2)"],["method/general-roy-s-rule",8,"General Roy's rule","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-general-roy-s-rule"],["theorem/legendre-s-theorem",9,"Legendre's theorem","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-legendre-s-theorem"],["concept/homogeneous-linear-equations",7,"homogeneous linear equations","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-homogeneous-linear-equations"],["quantity/spherical-excess",11,"spherical excess","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-quantity-spherical-excess"],["wentworth-first-steps-in-algebra-1894/ex-33/4",4,"Wentworth 1894, Exercise 33 (4)"],["theorem/area-of-a-spherical-triangle",9,"area of a spherical triangle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-area-of-a-spherical-triangle"],["person/delambre",1,"Delambre","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-person-delambre"],["person/george-everest",1,"George Everest","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-person-george-everest"],["concept/spheroid",7,"spheroid","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-spheroid"],["hardy-course-of-pure-mathematics-1921/x-1a8d2b1e8c",15,"Hardy 1921, p. 357: These new functions have generally been introduced because it ..."],["concept/homophonic-cipher",7,"homophonic cipher","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-homophonic-cipher"],["hardy-course-of-pure-mathematics-1921/x-ef8cf3960a",15,"Hardy 1921, p. 358: We define \\log x, the logarithm of x, by ..."],["todhunter-spherical-trigonometry-1886/x-290670c8fc",15,"Todhunter 1886, scan 98: Now the area is not known exactly unless the ..."],["todhunter-spherical-trigonometry-1886/x-8c55cfa267",15,"Todhunter 1886, scan 103: This process, by which we find the angle COD ..."],["concept/polyalphabetic-cipher",7,"polyalphabetic cipher","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-polyalphabetic-cipher"],["concept/known-term",7,"known term","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-known-term"],["concept/many-to-one-cipher",7,"many-to-one cipher","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-many-to-one-cipher"],["hardy-course-of-pure-mathematics-1921/x-f04fce79c2",15,"Hardy 1921, p. 361: Perhaps the most interesting feature of the function \\log ..."],["de-morgan-elementary-illustrations-calculus-1899/ch-fluxions-and-the-idea-of-time",2,"De Morgan 1899, Fluxions, and the Idea of Time","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-fluxions-and-the-idea-of-time/index.html"],["hardy-course-of-pure-mathematics-1921/x-42fdf3d995",15,"Hardy 1921, p. 364: We now define the exponential function e^{y} for all ..."],["theorem/cramer-s-rule",9,"Cramer's rule","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-cramer-s-rule"],["theorem/consistency-of-a-linear-system",9,"consistency of a linear system","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-consistency-of-a-linear-system"],["concept/rank-of-a-determinant",7,"rank of a determinant","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-rank-of-a-determinant"],["concept/minor",7,"minor","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-minor"],["concept/complementary-minor",7,"complementary minor","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-complementary-minor"],["concept/matrix",7,"matrix","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-matrix"],["planck-treatise-on-thermodynamics-1903/eq-c8503d68e9",16,"Planck 1903, p. 105: Q_{2} = W' + Q_{1}'"],["dickson-theory-of-equations-1922/x-68c56e1822",15,"Dickson 1922, p. 115: If D denotes the determinant of the coefficients of ..."],["dickson-theory-of-equations-1922/x-6360bed985",15,"Dickson 1922, p. 121: A system of m linear equations in n unknowns ..."],["dickson-theory-of-equations-1922/x-eb95b57f20",15,"Dickson 1922, p. 116: For example, a determinant D of order 3 is ..."],["dickson-theory-of-equations-1922/x-8a150dc650",15,"Dickson 1922, p. 119: A necessary and sufficient condition that n linear homogeneous ..."],["dickson-theory-of-equations-1922/x-ee75d546a5",15,"Dickson 1922, p. 122: For r = 1, this development becomes the known ..."],["dickson-theory-of-equations-1922/x-f07466ded6",15,"Dickson 1922, p. 115: The theorem was discovered by induction in 1750 by ..."],["concept/cipher-requisites",7,"cipher requisites","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cipher-requisites"],["planck-treatise-on-thermodynamics-1903/eq-fc0a546426",16,"Planck 1903, p. 106: Q_{1}' = \\frac{\\theta_{1}}{\\theta_{2} - \\theta_{1}} W'"],["planck-treatise-on-thermodynamics-1903/eq-2f7542b1c4",16,"Planck 1903, p. 106: W' = \\frac{\\theta_{2} - \\theta_{1}}{\\theta_{1}} Q_{1}'"],["planck-treatise-on-thermodynamics-1903/eq-c67635035f",16,"Planck 1903, p. 106: - \\frac{Q_{1}}{\\theta_{1}} - \\frac{Q_{2}}{\\theta_{2}} > 0"],["planck-treatise-on-thermodynamics-1903/eq-3ae8f63545",16,"Planck 1903, p. 106: \\frac{Q_{1}}{\\theta_{1}} + \\frac{Q_{2}}{\\theta_{2}} < 0"],["planck-treatise-on-thermodynamics-1903/eq-54ff2f2157",16,"Planck 1903, p. 106: W' < \\frac{\\theta_{2} - \\theta_{1}}{\\theta_{1}} Q_{1}'"],["planck-treatise-on-thermodynamics-1903/eq-f4cb016daf",16,"Planck 1903, p. 107: Q_{2} \\left(\\frac{1}{\\theta_{2}} - \\frac{1}{\\theta_{1}}\\right) < 0"],["planck-treatise-on-thermodynamics-1903/eq-e30dfb0d5e",16,"Planck 1903, p. 107: W + Q = 0"],["instrument/cipher-machine",13,"cipher machine","../books/ball-mathematical-recreations-1905/terms/index.html#t-instrument-cipher-machine"],["concept/shorthand",7,"shorthand","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-shorthand"],["planck-treatise-on-thermodynamics-1903/eq-d9e7a6e00e",16,"Planck 1903, p. 114: \\Phi = M\\phi = M(c_{v} \\log \\theta + \\frac{R}{m} \\log v + \\const)"],["planck-treatise-on-thermodynamics-1903/eq-6edd246f27",16,"Planck 1903, p. 114: F = M\\{c_{v} \\theta (\\const - \\log \\theta) - \\frac{R\\theta}{m} \\log v + \\const\\}"],["planck-treatise-on-thermodynamics-1903/eq-0a1d30c910",16,"Planck 1903, p. 114: dF = -\\frac{M\\theta R}{m} · \\frac{dv}{v} = -p\\, dV \\leq W"],["planck-treatise-on-thermodynamics-1903/eq-10633e48da",16,"Planck 1903, p. 114: d\\left(\\Phi - \\frac{U + pV}{\\theta}\\right) \\geq 0"],["planck-treatise-on-thermodynamics-1903/eq-98fe49b50b",16,"Planck 1903, p. 114: \\Phi - \\frac{U + pV}{\\theta} = \\Psi"],["person/charles-i",1,"Charles I","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-charles-i"],["planck-treatise-on-thermodynamics-1903/eq-35a5246563",16,"Planck 1903, p. 117: \\delta\\left(\\Phi - \\frac{U}{\\theta}\\right) + \\frac{W}{\\theta} = 0"],["planck-treatise-on-thermodynamics-1903/eq-fb8fa599dd",16,"Planck 1903, p. 117: - \\delta F = -W"],["planck-treatise-on-thermodynamics-1903/eq-50057181f1",16,"Planck 1903, p. 117: \\delta F = 0"],["planck-treatise-on-thermodynamics-1903/eq-a8b2b09b7c",16,"Planck 1903, p. 118: W = -p\\, \\delta V"],["planck-treatise-on-thermodynamics-1903/eq-423a81a84c",16,"Planck 1903, p. 118: \\delta\\Psi = 0"],["planck-treatise-on-thermodynamics-1903/eq-fe81ffe3a2",16,"Planck 1903, p. 115: \\delta\\Phi - \\frac{\\delta U - W}{\\theta} \\leq 0"],["planck-treatise-on-thermodynamics-1903/eq-30d368b06e",16,"Planck 1903, p. 116: \\delta\\Phi - \\frac{\\delta U - W}{\\theta} = 0"],["planck-treatise-on-thermodynamics-1903/eq-d50bdf4ffa",16,"Planck 1903, p. 117: \\delta U = W"],["planck-treatise-on-thermodynamics-1903/eq-2f39329449",16,"Planck 1903, p. 117: \\delta \\Phi = 0"],["person/charles-wheatstone",1,"Charles Wheatstone","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-charles-wheatstone"],["ball-mathematical-recreations-1905/x-21f10aa65b",15,"Ball 1905, scan 324: A cipher of the second type is one in ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-444ce5df8b",16,"De Morgan 1899, p. 58: dx = r \\sin\\theta\\, d\\theta"],["de-morgan-elementary-illustrations-calculus-1899/eq-c06d810e30",16,"De Morgan 1899, p. 58: dy = r \\cos\\theta\\, d\\theta"],["de-morgan-elementary-illustrations-calculus-1899/eq-634dbc45b5",16,"De Morgan 1899, p. 59: \\theta = \\phi t"],["de-morgan-elementary-illustrations-calculus-1899/eq-4c765366e2",16,"De Morgan 1899, p. 59: \\frac{dx}{dt} = r \\sin\\theta\\, \\frac{d\\theta}{dt}"],["de-morgan-elementary-illustrations-calculus-1899/eq-96d490aed6",16,"De Morgan 1899, p. 59: \\frac{dy}{dt} = r \\cos\\theta\\, \\frac{d\\theta}{dt}"],["de-morgan-elementary-illustrations-calculus-1899/eq-a64391c055",16,"De Morgan 1899, p. 59: a \\sin\\theta"],["de-morgan-elementary-illustrations-calculus-1899/eq-064bb6686a",16,"De Morgan 1899, p. 59: a \\cos\\theta"],["ball-mathematical-recreations-1905/x-f180ed1910",15,"Ball 1905, scan 324: A disadvantage of this cipher is that since each ..."],["hardy-course-of-pure-mathematics-1921/x-ef2fad94d7",15,"Hardy 1921, p. 365: Thus the derivative of the exponential function is equal ..."],["todhunter-spherical-trigonometry-1886/eq-3061b14a0f",16,"Todhunter 1886, scan 85: \\cos EF=\\cos DE \\cos DF + \\sin DE \\sin DF \\cos A"],["todhunter-spherical-trigonometry-1886/eq-6330ffbec6",16,"Todhunter 1886, scan 86: \\theta = \\tan\\tfrac12A \\sin^2\\tfrac14(b + c) - \\cot\\tfrac12A \\sin^2\\tfrac14(b - c)"],["concept/angle-between-the-chords",7,"angle between the chords"],["todhunter-spherical-trigonometry-1886/eq-2fe41806f3",16,"Todhunter 1886, scan 86: 2r\\sin\\dfrac{\\alpha}{2r}"],["todhunter-spherical-trigonometry-1886/eq-aae4e1aee6",16,"Todhunter 1886, scan 87: \\cos A = \\frac{\\cos a - \\cos b\\cos c}{\\sin b \\sin c}"],["todhunter-spherical-trigonometry-1886/eq-cfb1c6ec64",16,"Todhunter 1886, scan 87: 1 - \\frac{\\alpha^2}{2r^2} + \\frac{\\alpha^4}{24r^4} - \\ldots"],["concept/series-expansion",7,"series expansion"],["concept/radius-of-the-sphere",7,"radius of the sphere"],["todhunter-spherical-trigonometry-1886/eq-d4ab6f42aa",16,"Todhunter 1886, scan 87: \\frac{\\alpha}{r} - \\frac{\\alpha^3}{6r^3} +\\ldots"],["planck-treatise-on-thermodynamics-1903/eq-e80ab95979",16,"Planck 1903, p. 207: \\Psi = \\Phi - \\frac{U + pV}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-b1eb743090",16,"Planck 1903, p. 208: V = \\frac{R\\theta}{p} (n_{1} + n_{2} + \\dots) = \\frac{R\\theta}{p} \\tsum n_{1}"],["todhunter-spherical-trigonometry-1886/eq-9f08f7c802",16,"Todhunter 1886, scan 87: \\theta = \\frac{\\beta \\gamma \\sin A'}{6r^2}=\\frac{S}{3r^2}"],["todhunter-spherical-trigonometry-1886/eq-8fae053762",16,"Todhunter 1886, scan 88: B = B' + \\frac{S}{3r^2}"],["todhunter-spherical-trigonometry-1886/eq-25e34dbffd",16,"Todhunter 1886, scan 88: A+B+C = A'+B'+C'+\\frac{S}{r^2} = \\pi + \\frac{S}{r^2}"],["todhunter-spherical-trigonometry-1886/eq-f8dde2dd0a",16,"Todhunter 1886, scan 88: S = \\tfrac{1}{2}\\beta\\gamma\\sin A' = \\tfrac{1}{2}\\beta\\gamma\\sin A"],["todhunter-spherical-trigonometry-1886/eq-bbe33dc6e2",16,"Todhunter 1886, scan 89: \\sin B' = \\frac{\\beta}{\\alpha}\\sin A' = \\frac{\\beta}{\\alpha}\\sin A"],["todhunter-spherical-trigonometry-1886/eq-611426c391",16,"Todhunter 1886, scan 89: S=\\frac{\\gamma^2 \\sin A' \\sin B'}{2\\sin(A'+B')}"],["planck-treatise-on-thermodynamics-1903/eq-ec089e71f7",16,"Planck 1903, p. 208: U_{1} = \\tsum n_{1} (c_{v_{1}}\\theta + h_{1})"],["todhunter-spherical-trigonometry-1886/eq-047d92b0ae",16,"Todhunter 1886, scan 89: S=\\frac{\\alpha^2\\sin B' \\sin C'}{2\\sin(B'+C')}"],["todhunter-spherical-trigonometry-1886/eq-ff1920fea4",16,"Todhunter 1886, scan 88: A = A' + \\frac{S}{3r^2}"],["todhunter-spherical-trigonometry-1886/eq-7b05326c6c",16,"Todhunter 1886, scan 90: \\sin\\frac{1}{2}E = \\frac{\\sin\\tfrac{1}{2}a \\sin\\tfrac{1}{2}b \\sin C} {\\cos\\tfrac{1}{2}c}"],["todhunter-spherical-trigonometry-1886/eq-8c133698d2",16,"Todhunter 1886, scan 90: \\sin C' \\frac{\\alpha\\beta}{2r^2} \\left( 1 + \\frac{\\alpha^2+\\beta^2+\\gamma^2}{24r^2} \\right)"],["todhunter-spherical-trigonometry-1886/eq-b07d086024",16,"Todhunter 1886, scan 90: \\frac{\\operatorname{Sin} A}{\\operatorname{Sin} B} = \\frac{\\sin a}{\\sin b}"],["todhunter-spherical-trigonometry-1886/eq-c9d4bebbcf",16,"Todhunter 1886, scan 91: \\frac{\\alpha}{\\beta} \\left\\{1 + \\frac{\\beta^2 - \\alpha^2}{6r^2} \\left(1 + \\frac{7\\beta^2-3\\alpha^2}{60r^2}\\right)\\right\\"],["ball-mathematical-recreations-1905/x-25464e8aa9",15,"Ball 1905, scan 328: He took a key-word, such as prudentia, and constructed ..."],["todhunter-spherical-trigonometry-1886/eq-ce2a1367b1",16,"Todhunter 1886, scan 91: = \\frac{\\alpha^2-\\beta^2}{\\alpha\\gamma\\sin B} \\left( 1 - \\frac{\\beta^2+\\gamma^2-\\alpha^2}{12r^2} \\right)"],["todhunter-spherical-trigonometry-1886/eq-70dc21c690",16,"Todhunter 1886, scan 93: x = \\alpha - \\dfrac{\\beta\\sin A}{\\sin B} - \\dfrac{\\mu (\\alpha^2-\\beta^2) }{\\gamma\\sin B}"],["todhunter-spherical-trigonometry-1886/eq-f26e7db759",16,"Todhunter 1886, scan 93: \\mu=\\dfrac{\\alpha\\gamma\\sin B}{6r^2}"],["todhunter-spherical-trigonometry-1886/eq-fcf71dbcfb",16,"Todhunter 1886, scan 93: x=\\frac{\\alpha(\\beta^2-\\alpha^2)(3\\alpha^2-7\\beta^2)}{360r^4}"],["todhunter-spherical-trigonometry-1886/eq-a9b000eac4",16,"Todhunter 1886, scan 93: = \\frac{\\alpha (\\beta^2-\\alpha^2) (\\alpha^2 + \\beta^2 - 5\\gamma^2)} {720r^4}"],["wentworth-plane-geometry-1899/x-bb179e78d5",15,"Wentworth 1899, scan 193: In propositions relating to areas, the words “rectangle,” “triangle,” ..."],["ball-mathematical-recreations-1905/x-43aa59e6f1",15,"Ball 1905, scan 325: cells, and each cell is determined uniquely by the ..."],["concept/equivalent-plane-figures",7,"equivalent plane figures","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-equivalent-plane-figures"],["theorem/area-of-rectangles-with-equal-altitudes",9,"area of rectangles with equal altitudes","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-area-of-rectangles-with-equal-altitudes"],["theorem/area-of-a-parallelogram",9,"area of a parallelogram","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-area-of-a-parallelogram"],["theorem/areas-of-similar-polygons",9,"areas of similar polygons","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-areas-of-similar-polygons"],["concept/logarithmic-series",7,"logarithmic series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-logarithmic-series"],["wentworth-plane-geometry-1899/x-95a089455f",15,"Wentworth 1899, scan 196: The area of a rectangle is equal to the ..."],["ball-mathematical-recreations-1905/x-4def7fbc08",15,"Ball 1905, scan 331: Lastly, no ambiguity should be possible in deciphering the ..."],["ball-mathematical-recreations-1905/x-41616d7ca8",15,"Ball 1905, scan 335: For twenty-four hours de Rohan pored over the message, ..."],["concept/altitude-of-a-triangle",7,"altitude of a triangle","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-altitude-of-a-triangle"],["ball-mathematical-recreations-1905/x-07018d380d",15,"Ball 1905, scan 333: The Queen seems to have found writing in cipher ..."],["concept/arc-of-a-curve",7,"arc of a curve","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-arc-of-a-curve"],["concept/hyperbolic-function",7,"hyperbolic function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-hyperbolic-function"],["boyden-first-book-in-algebra-1895/ex-13/12",4,"Boyden 1895, Exercise 13 (12)"],["hardy-course-of-pure-mathematics-1921/x-6580ef02c8",15,"Hardy 1921, p. 374: We saw however in [§]200 that with the aid ..."],["concept/scholium",7,"scholium","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-scholium"],["hardy-course-of-pure-mathematics-1921/x-f2c3026afd",15,"Hardy 1921, p. 376: Conversely, the rate of increase of the higher logarithmic ..."],["wentworth-plane-geometry-1899/x-f933653a17",15,"Wentworth 1899, scan 193: The unit of surface is a square whose side ..."],["wentworth-plane-geometry-1899/x-19205e1914",15,"Wentworth 1899, scan 193: Plane figures that have equal areas but cannot be ..."],["wentworth-first-steps-in-algebra-1894/ex-21/3",4,"Wentworth 1894, Exercise 21 (3)"],["hardy-course-of-pure-mathematics-1921/x-95ca970ed4",15,"Hardy 1921, p. 379: The reader will observe that the exponential series has ..."],["form/1c370b54b9",5,"identity: 7*b*(2*a - 3*b)"],["shape/f15aca3adb",6,"identity: N*b*(N*a + N*b)"],["wentworth-plane-geometry-1899/x-aa9ed859ce",15,"Wentworth 1899, scan 196: When the base and altitude each contain the linear ..."],["wentworth-plane-geometry-1899/x-d4e051bbd7",15,"Wentworth 1899, scan 199: The area of an irregular polygon may be found ..."],["wentworth-plane-geometry-1899/x-dd4082ade1",15,"Wentworth 1899, scan 203: The square on the hypotenuse of a right triangle ..."],["concept/continuity",7,"continuity","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-continuity"],["hardy-course-of-pure-mathematics-1921/x-17a97e9a2c",15,"Hardy 1921, p. 374: The approximations are of course very rough, but suffice ..."],["hardy-course-of-pure-mathematics-1921/x-5d28bfd0c6",15,"Hardy 1921, p. 372: Verify that these formulae may be deduced from the ..."],["boyden-first-book-in-algebra-1895/ex-13/13",4,"Boyden 1895, Exercise 13 (13)"],["whitehead-introduction-to-mathematics-1911/x-652d54770d",15,"Whitehead 1911, p. 145: If a train has been travelling at the rate ..."],["whitehead-introduction-to-mathematics-1911/x-b1bd03f61f",15,"Whitehead 1911, p. 147: With these explanations and cautions, we write y = ..."],["whitehead-introduction-to-mathematics-1911/x-b3b31327d4",15,"Whitehead 1911, p. 147: Thus in y = f(x), we may determine, if ..."],["todhunter-spherical-trigonometry-1886/x-ad2197bc85",15,"Todhunter 1886, scan 106: If A and C are constant, and b be ..."],["boyden-first-book-in-algebra-1895/eq-11fe4d7c70",16,"Boyden 1895: m - y"],["boyden-first-book-in-algebra-1895/eq-7924acabcd",16,"Boyden 1895: x - b + m + y - z"],["whitehead-introduction-to-mathematics-1911/x-948d87850b",15,"Whitehead 1911, p. 152: The train certainly cannot be running at forty miles ..."],["concept/perimeter",7,"perimeter","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-perimeter"],["concept/curvilinear-integral",7,"curvilinear integral","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-curvilinear-integral"],["boyden-first-book-in-algebra-1895/eq-8e65cfdb09",16,"Boyden 1895: x \\cdot x = xx = x^2"],["boyden-first-book-in-algebra-1895/eq-1b147006ff",16,"Boyden 1895: x \\cdot x \\cdot x = xxx = x^3"],["de-morgan-elementary-illustrations-calculus-1899/x-5ab95e6a55",15,"De Morgan 1899, p. 39: We shall proceed to a strict proof of this; ..."],["de-morgan-elementary-illustrations-calculus-1899/x-2fef6a54a3",15,"De Morgan 1899, p. 39: However the original arc may be diminished, let the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-4ebfca4919",15,"De Morgan 1899, p. 41: Hence PQ can be taken so small that VQ ..."],["de-morgan-elementary-illustrations-calculus-1899/x-f217b8fbac",15,"De Morgan 1899, p. 40: Let PP' (4) be a part of a curve, ..."],["hardy-course-of-pure-mathematics-1921/ex-lxxxviii",3,"Hardy 1921, Exercise LXXXVIII"],["hardy-course-of-pure-mathematics-1921/ex-lxxxix",3,"Hardy 1921, Exercise LXXXIX"],["boyden-first-book-in-algebra-1895/eq-eac7109220",16,"Boyden 1895: a + (b - c - x) = a + b - c - x"],["hardy-course-of-pure-mathematics-1921/ex-lxxxvii",3,"Hardy 1921, Exercise LXXXVII"],["concept/observational-error",7,"observational error","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-observational-error"],["hardy-course-of-pure-mathematics-1921/x-54247cc4a6",15,"Hardy 1921, p. 257: This equation has three real roots if s^{4} > ..."],["theorem/cosine-formula-for-a-side-of-a-spherical-triangle",9,"cosine formula for a side of a spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-cosine-formula-for-a-side-of-a-spherical-triangle"],["todhunter-spherical-trigonometry-1886/x-35033ae7a2",15,"Todhunter 1886, scan 103: A side and the opposite angle of a spherical ..."],["todhunter-spherical-trigonometry-1886/x-a63246acaf",15,"Todhunter 1886, scan 103: then we require the ratio of \\delta a to ..."],["hardy-course-of-pure-mathematics-1921/x-13a749bc1f",15,"Hardy 1921, p. 258: If \\phi(x) \\to a as x \\to \\infty, then ..."],["wentworth-first-steps-in-algebra-1894/ex-39/1",4,"Wentworth 1894, Exercise 39 (1)"],["form/f5d515dcbd",5,"factor: x**3 - 7*x"],["shape/a62e25ae82",6,"factor: N*x + x**N"],["concept/book-cipher",7,"book cipher","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-book-cipher"],["concept/key",7,"key","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-key"],["concept/cipher",7,"cipher","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cipher"],["concept/cryptography",7,"cryptography","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cryptography"],["shape/c34c084cee",6,"factor: N - x + x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/10",4,"Wentworth 1894, Exercise 39 (10)"],["concept/cryptograph",7,"cryptograph","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cryptograph"],["hardy-course-of-pure-mathematics-1921/x-3c07ad2ec8",15,"Hardy 1921, p. 395: But it will be found, on closer examination, that ..."],["boyden-first-book-in-algebra-1895/eq-17f167238d",16,"Boyden 1895: a + c - d + e = a + (c - d + e)"],["wentworth-plane-geometry-1899/eq-893cd7fb48",16,"Wentworth 1899, scan 225: \\dfrac{(n-2) 2 \\text{ rt.\\ } \\angle_s}{n}"],["boyden-first-book-in-algebra-1895/eq-d23e379903",16,"Boyden 1895: x - (y + z - c) = x - y - z + c"],["method/dividing-a-monomial-by-a-monomial",8,"dividing a monomial by a monomial","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-method-dividing-a-monomial-by-a-monomial"],["hardy-course-of-pure-mathematics-1921/x-69505ed6e5",15,"Hardy 1921, p. 401: Since |\\zeta | = \\rho, and the different angles ..."],["hardy-course-of-pure-mathematics-1921/x-1d43932253",15,"Hardy 1921, p. 402: An equation such as (1), in which every value ..."],["hardy-course-of-pure-mathematics-1921/x-c2e0e106f5",15,"Hardy 1921, p. 403: It would not be unnatural to suppose that, conversely, ..."],["hardy-course-of-pure-mathematics-1921/x-c0d76eff2f",15,"Hardy 1921, p. 404: It might seem natural, as \\exp \\zeta = e^{\\zeta} ..."],["form/93ab4dc44d",5,"factor: 6*a**2 + 5*a*x + x**2"],["de-morgan-elementary-illustrations-calculus-1899/eq-cfaabb0e12",16,"De Morgan 1899, p. 60: \\dfrac{dy}{dt} = 2x\\, \\dfrac{dx}{dt} + \\dfrac{dx}{dt}\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/eq-f080928982",16,"De Morgan 1899, p. 60: \\dot{y} = 2x\\, \\dot{x}"],["de-morgan-elementary-illustrations-calculus-1899/eq-6897ffb1b4",16,"De Morgan 1899, p. 60: dy = 2x\\, dx"],["concept/digit",7,"digit","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-digit"],["concept/alphabet",7,"alphabet","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-alphabet"],["ball-mathematical-recreations-1905/x-c8bbf37465",15,"Ball 1905, scan 337: A code dictionary is prepared in which every word ..."],["ball-mathematical-recreations-1905/x-6557bd1602",15,"Ball 1905, scan 337: This is a cipher with 10^5 symbols, and as ..."],["wentworth-plane-geometry-1899/eq-4f814b5170",16,"Wentworth 1899, scan 227: \\overline{OA}^2 - \\overline{OP}^2 = \\overline{AP}^2"],["wentworth-plane-geometry-1899/eq-fca07b2e3a",16,"Wentworth 1899, scan 241: \\overline{AD}^2 = DH × DC"],["wentworth-plane-geometry-1899/eq-292f074c4a",16,"Wentworth 1899, scan 241: AD = \\sqrt{2-\\sqrt{4-a^2}}"],["wentworth-plane-geometry-1899/eq-fe10d94775",16,"Wentworth 1899, scan 241: \\sqrt{R(2R - \\sqrt{4R^2 - a^2})}"],["wentworth-plane-geometry-1899/eq-29776951fe",16,"Wentworth 1899, scan 232: S = \\frac{1}{2}R × P"],["wentworth-plane-geometry-1899/eq-2ac9524257",16,"Wentworth 1899, scan 233: S' = \\frac{1}{2} R × P"],["wentworth-plane-geometry-1899/eq-c03156d208",16,"Wentworth 1899, scan 233: S = \\frac{1}{2}R× C"],["wentworth-plane-geometry-1899/eq-56a60be0c2",16,"Wentworth 1899, scan 233: \\odot = \\frac{1}{2} R × C = \\frac{1}{2} R × 2\\pi R = \\pi R^2"],["ball-mathematical-recreations-1905/x-66ec4f7a86",15,"Ball 1905, scan 338: If the clue number is the same all through ..."],["boyden-first-book-in-algebra-1895/ex-13/14",4,"Boyden 1895, Exercise 13 (14)"],["wentworth-plane-geometry-1899/eq-90091e8382",16,"Wentworth 1899, scan 226: P:P' = OA:O'A' = OM:O'M'"],["wentworth-plane-geometry-1899/eq-e39fd6b4df",16,"Wentworth 1899, scan 225: AB:A'B' = BC:B'C'"],["concept/similar",7,"similar","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-similar"],["concept/resonance",7,"resonance","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-resonance"],["de-morgan-elementary-illustrations-calculus-1899/eq-6b1007824d",16,"De Morgan 1899, p. 63: at + \\frac{1}{2}gt^{2}"],["concept/measure-of-time",7,"measure of time","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-measure-of-time"],["quantity/velocity",11,"velocity","../books/ball-mathematical-recreations-1905/terms/index.html#t-quantity-velocity"],["boyden-first-book-in-algebra-1895/ex-13/15",4,"Boyden 1895, Exercise 13 (15)"],["boyden-first-book-in-algebra-1895/eq-5c412b5885",16,"Boyden 1895: a - b - c + d = a - (b + c - d)"],["quantity/free-period-of-vibration",11,"free period of vibration","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-quantity-free-period-of-vibration"],["whitehead-introduction-to-mathematics-1911/x-f74b8f7495",15,"Whitehead 1911, p. 164: the existence of successive events so analogous to each ..."],["whitehead-introduction-to-mathematics-1911/x-522c3dfb50",15,"Whitehead 1911, p. 168: Hence the velocity of the train has not been ..."],["whitehead-introduction-to-mathematics-1911/x-169e98a85b",15,"Whitehead 1911, p. 168: Thus the question as to whether the train has ..."],["ball-mathematical-recreations-1905/x-fce91791af",15,"Ball 1905, scan 337: When a code message is published by the Government ..."],["whitehead-introduction-to-mathematics-1911/x-dcd7f88142",15,"Whitehead 1911, p. 166: It has been one of the first tasks of ..."],["whitehead-introduction-to-mathematics-1911/x-31fd18280c",15,"Whitehead 1911, p. 170: Any one wanting to upset a rocking stone will ..."],["form/22dd6862c1",5,"evaluate: 9*a at a=7, b=5, c=3"],["ball-mathematical-recreations-1905/x-2c1f83c06e",15,"Ball 1905, scan 338: For if we make a similar code with the ..."],["ball-mathematical-recreations-1905/x-49ec722d57",15,"Ball 1905, scan 339: Poe wrote an essay on cryptography in which he ..."],["ball-mathematical-recreations-1905/x-928984a2f8",15,"Ball 1905, scan 339: More than one of his correspondents did not play ..."],["dickson-theory-of-equations-1922/eq-cacc5c228e",16,"Dickson 1922, p. 69: \\frac{(x-a)(x-b)f'(x)}{f(x)} \\equiv r(x-b) + s(x-a) + (x-a)(x-b) \\frac{Q'(x)}{Q(x)}"],["form/f4470117ae",5,"evaluate: a*b*c**2 at a=7, b=5, c=3"],["shape/2f7ccbbd4d",6,"evaluate: a*b*c**N"],["wentworth-first-steps-in-algebra-1894/ex-3/10",4,"Wentworth 1894, Exercise 3 (10)"],["whitehead-introduction-to-mathematics-1911/x-ea0fb9a934",15,"Whitehead 1911, p. 170: Thus a pendulum has but one period of vibration, ..."],["hardy-course-of-pure-mathematics-1921/eq-575153a90c",16,"Hardy 1921, p. 240: \\int f'(x)F(x)\\, dx = f(x)F(x) - \\int f(x)F'(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-2e7cfb1d4d",16,"Hardy 1921, p. 240: \\int\\phi(x)\\, dx = \\int x\\chi''(x)\\, dx = x\\chi'(x) - \\int \\chi'(x)\\, dx = x\\chi'(x) - \\chi(x)"],["hardy-course-of-pure-mathematics-1921/eq-6b20dc6dbe",16,"Hardy 1921, p. 240: F(x) = \\sqrtp{ax^{2} + 2bx + c} = y"],["hardy-course-of-pure-mathematics-1921/eq-3adf3950b8",16,"Hardy 1921, p. 240: \\int y\\, dx = \\frac{(ax + b)y}{2a} + \\frac{ac - b^{2}}{2a} \\int \\frac{dx}{y}"],["hardy-course-of-pure-mathematics-1921/eq-fadb59cc27",16,"Hardy 1921, p. 242: \\int R(x, \\sqrt{X})\\, dx"],["form/9269e74d06",5,"hcf: (42*a*b**2, 60*a**2*b)"],["shape/70343737ac",6,"hcf: (N*a*b**N, N*a**N*b)"],["wentworth-first-steps-in-algebra-1894/ex-40/4",4,"Wentworth 1894, Exercise 40 (4)"],["hardy-course-of-pure-mathematics-1921/eq-1f6b5848c7",16,"Hardy 1921, p. 242: \\frac{A + B\\sqrt{X}}{C + D\\sqrt{X}} = \\frac{(A + B\\sqrt{X})(C - D\\sqrt{X})}{C^{2} - D^{2}X} = E + F\\sqrt{X}"],["hardy-course-of-pure-mathematics-1921/eq-5012b5ce2b",16,"Hardy 1921, p. 242: \\int \\frac{G}{\\sqrt{X}}\\, dx"],["method/approximation-of-surds-by-the-binomial-series",8,"approximation of surds by the binomial series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-approximation-of-surds-by-the-binomial-series"],["planck-treatise-on-thermodynamics-1903/x-9dea7c9d12",15,"Planck 1903, p. 109: It cannot in general be integrated, since the left-hand ..."],["theorem/equal-order-from-a-finite-limiting-ratio",9,"equal order from a finite limiting ratio","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-equal-order-from-a-finite-limiting-ratio"],["hardy-course-of-pure-mathematics-1921/eq-eb34c122ac",16,"Hardy 1921, p. 242: \\int \\frac{x^{m}}{\\sqrt{X}}\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/x-5993730d2d",15,"De Morgan 1899, p. 43: In the language of Leibnitz if h be an ..."],["todhunter-spherical-trigonometry-1886/eq-3fe6422f17",16,"Todhunter 1886, scan 100: \\sin (A - \\delta B - \\delta C) = \\sin A - (\\delta B + \\delta C) \\cos A"],["hardy-course-of-pure-mathematics-1921/eq-c21ae385c4",16,"Hardy 1921, p. 243: \\frac{d}{dx}(x^{m-1}\\sqrt{X}) = (m - 1)x^{m-2} \\sqrt{X} + \\frac{(ax + b) x^{m-1}}{\\sqrt{X}} = \\frac{\\alpha x^{m} + \\beta"],["theorem/addition-formula-for-the-exponential-function",9,"addition formula for the exponential function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-addition-formula-for-the-exponential-function"],["de-morgan-elementary-illustrations-calculus-1899/x-768d8cff7c",15,"De Morgan 1899, p. 43: Hence (1) is said to be comparable to the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-895da47c64",15,"De Morgan 1899, p. 42: As h is diminished, all these expressions decrease without ..."],["de-morgan-elementary-illustrations-calculus-1899/x-a537f429e6",15,"De Morgan 1899, p. 43: Nevertheless this decrease increases the ratio of the first ..."],["de-morgan-elementary-illustrations-calculus-1899/x-f16acbbf3a",15,"De Morgan 1899, p. 44: The converse proposition is readily shown, that if the ..."],["concept/apothem",7,"apothem","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-apothem"],["concept/radius-of-a-regular-polygon",7,"radius of a regular polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-radius-of-a-regular-polygon"],["concept/centre-of-a-regular-polygon",7,"centre of a regular polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-centre-of-a-regular-polygon"],["theorem/area-of-a-regular-polygon",9,"area of a regular polygon","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-area-of-a-regular-polygon"],["concept/decagon",7,"decagon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-decagon"],["concept/hexagon",7,"hexagon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-hexagon"],["concept/octagon",7,"octagon","../books/wentworth-plane-geometry-1899/terms/index.html#t-concept-octagon"],["concept/euler-s-number",7,"Euler's number","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-euler-s-number"],["hardy-course-of-pure-mathematics-1921/eq-8fc3be7fa3",16,"Hardy 1921, p. 243: \\int \\frac{dx}{(x - p)^{m}\\sqrt{X}}"],["hardy-course-of-pure-mathematics-1921/eq-7b3d5bbf61",16,"Hardy 1921, p. 243: \\int \\frac{Lx + M}{(Ax^{2} + 2Bx + C) \\sqrt{ax^{2} + 2bx + c}}\\, dx"],["person/john-napier",1,"John Napier","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-person-john-napier"],["wentworth-plane-geometry-1899/x-9b3ddfe50a",15,"Wentworth 1899, scan 220: A regular polygon is a polygon which is both ..."],["wentworth-plane-geometry-1899/x-7c7dd49317",15,"Wentworth 1899, scan 231: The constant ratio of the circumference of a circle ..."],["wentworth-plane-geometry-1899/x-d7bb5d452f",15,"Wentworth 1899, scan 238: Therefore, to inscribe a regular decagon, divide the radius ..."],["hardy-course-of-pure-mathematics-1921/eq-591c679177",16,"Hardy 1921, p. 243: x = \\frac{\\mu t + \\nu}{t + 1}"],["hardy-course-of-pure-mathematics-1921/eq-4cf186d8de",16,"Hardy 1921, p. 243: a\\mu\\nu + b(\\mu + \\nu) + c = 0"],["hardy-course-of-pure-mathematics-1921/eq-e8f5a4d533",16,"Hardy 1921, p. 243: A\\mu\\nu + B(\\mu + \\nu) + C = 0"],["boyden-first-book-in-algebra-1895/eq-8a3bb4d944",16,"Boyden 1895: a^3 = a \\cdot a \\cdot a"],["form/67040ebb53",5,"hcf: (49*a*b**3, 35*a**2*b**2)"],["todhunter-spherical-trigonometry-1886/eq-57583349f0",16,"Todhunter 1886, scan 100: \\cot C + \\cot A = \\dfrac{\\sin (A + C)}{\\sin A \\sin C} = \\dfrac{\\sin B}{\\sin A \\sin C}"],["todhunter-spherical-trigonometry-1886/eq-0a019d6e23",16,"Todhunter 1886, scan 100: \\frac{a \\sin B}{\\sin^2 A} \\delta C + \\frac{a \\sin C \\cos A}{\\sin^2 A)} \\delta B"],["todhunter-spherical-trigonometry-1886/eq-bba8f1585a",16,"Todhunter 1886, scan 100: \\frac{a \\sin C}{\\sin^2 A} \\delta B + \\frac{a \\sin B \\cos A}{\\sin^2 A} \\delta C"],["todhunter-spherical-trigonometry-1886/eq-0b4b0082ab",16,"Todhunter 1886, scan 102: \\cos (\\theta + x)=\\frac{\\cos \\theta-\\sin h \\,\\sin k}{\\cos h \\,\\cos k}"],["todhunter-spherical-trigonometry-1886/eq-cd94f9a08e",16,"Todhunter 1886, scan 102: \\cos \\theta-x \\sin \\theta=\\frac{\\cos \\theta-hk} {1-\\frac{1}{2}(h^2+k^2)}"],["hardy-course-of-pure-mathematics-1921/eq-d16a30c26d",16,"Hardy 1921, p. 243: (aB - bA)\\xi^{2} - (cA - aC)\\xi + (bC - cB) = 0"],["boyden-first-book-in-algebra-1895/eq-f00a950ee4",16,"Boyden 1895: (x-y)^4=x^4-4x^3y+6x^2y^2-4xy^3+y^4"],["boyden-first-book-in-algebra-1895/eq-a27e4bc9ae",16,"Boyden 1895: (x-1)^3=x^3-3x^2+3x-1"],["shape/27e8275b6f",6,"hcf: (N*a*b**N, N*a**N*b**N)"],["person/augustin-louis-cauchy",1,"Augustin-Louis Cauchy","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-augustin-louis-cauchy"],["method/euler-s-method-of-elimination",8,"Euler's method of elimination","../books/dickson-theory-of-equations-1922/terms/index.html#t-method-euler-s-method-of-elimination"],["hardy-course-of-pure-mathematics-1921/eq-7199c2f73d",16,"Hardy 1921, p. 243: H\\int \\frac{t\\, dt}{(\\alpha t^{2} + \\beta)\\sqrtp{\\gamma t^{2} + \\delta}} + K\\int \\frac{dt}{(\\alpha t^{2} + \\beta)\\sqrtp{"],["concept/elementary-symmetric-function",7,"elementary symmetric function","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-elementary-symmetric-function"],["de-morgan-elementary-illustrations-calculus-1899/eq-94325beae8",16,"De Morgan 1899, p. 63: at - \\frac{1}{2}gt^{2}"],["hardy-course-of-pure-mathematics-1921/eq-1fd04971f7",16,"Hardy 1921, p. 243: \\frac{t}{\\sqrtp{\\gamma t^{2} + \\delta}} = u"],["hardy-course-of-pure-mathematics-1921/eq-b48dc807c7",16,"Hardy 1921, p. 243: \\int \\frac{dt}{(\\alpha t^{2} + \\beta) \\sqrtp{\\gamma t^{2} + \\delta}} = \\int \\frac{du}{\\beta + (\\alpha\\delta - \\beta\\gamm"],["person/james-joseph-sylvester",1,"James Joseph Sylvester","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-james-joseph-sylvester"],["hardy-course-of-pure-mathematics-1921/x-7978249b91",15,"Hardy 1921, p. 382: The only difference is that the proof is a ..."],["dickson-theory-of-equations-1922/x-e155ab5f83",15,"Dickson 1922, p. 151: The student should employ only methods of elimination (such ..."],["hardy-course-of-pure-mathematics-1921/eq-97d11bbc04",16,"Hardy 1921, p. 247: \\cos x = \\frac{1 - t^{2}}{1 + t^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-b69b24503c",16,"Hardy 1921, p. 248: \\int \\phi(y)\\, dy = \\int xf'(x)\\, dx = xf(x) - \\int f(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/x-9a9aa74939",15,"Hardy 1921, p. 385: Let us consider the error committed in taking 8\\frac{3}{16} ..."],["hardy-course-of-pure-mathematics-1921/eq-de58b281bf",16,"Hardy 1921, p. 247: \\sin x = \\frac{2t}{1 + t^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-fc0d10a20e",16,"Hardy 1921, p. 247: \\frac{dx}{dt} = \\frac{2}{1 + t^{2}}"],["boyden-first-book-in-algebra-1895/eq-16cf008035",16,"Boyden 1895: (a + b)^{2} = a^{2} + 2ab + b^{2} = a^{2} + (2a + b)b"],["shape/6b78d80a32",6,"hcf: (-c**N + (a - b)**N, a*b - b*c - b**N)"],["de-morgan-elementary-illustrations-calculus-1899/eq-8ad6e02cb8",16,"De Morgan 1899, p. 65: \\phi' t\\, dt + \\phi'' t\\, \\frac{(dt)^{2}}{2} + \\phi''' t\\, \\frac{(dt)^{3}}{2·3} + \\etc."],["de-morgan-elementary-illustrations-calculus-1899/eq-4eb619ecc8",16,"De Morgan 1899, p. 65: \\phi' t\\, dt + \\frac{1}{2}\\phi'' t (dt)^{2}"],["hardy-course-of-pure-mathematics-1921/eq-ce11d5e7b0",16,"Hardy 1921, p. 249: \\int x^{m}(\\log x)^{n}\\, dx = \\frac{x^{m+1} (\\log x)^{n}}{m + 1} - \\frac{n}{m + 1} \\int x^{m}(\\log x)^{n-1}\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-c762f7884b",16,"Hardy 1921, p. 250: (PRP') + (NN'RP) = (NN'P'P)"],["hardy-course-of-pure-mathematics-1921/eq-0ac3ea7cfc",16,"Hardy 1921, p. 250: \\Phi(x + h) - \\Phi(x) = h\\{\\phi(x) + \\mu(h)\\}"],["hardy-course-of-pure-mathematics-1921/eq-78add9b1fb",16,"Hardy 1921, p. 250: |\\mu(h)| < \\lambda(h)"],["hardy-course-of-pure-mathematics-1921/eq-cbf9e72a4c",16,"Hardy 1921, p. 250: \\Phi'(x) = \\lim_{h \\to 0} \\frac{\\Phi(x + h) - \\Phi(x)}{h} = \\lim_{h \\to 0} \\{\\phi(x) + \\mu(h)\\} = \\phi(x)"],["hardy-course-of-pure-mathematics-1921/x-4567fddcc1",15,"Hardy 1921, p. 392: This formula is interesting historically as having been employed ..."],["form/0901978f7b",5,"lcm: (21*a*b**3, 27*a**3*b**5)"],["macfarlane-vector-analysis-quaternions-1906/ch-addition-of-vectors-in-space",2,"Macfarlane 1906, Addition of Vectors in Space","../books/macfarlane-vector-analysis-quaternions-1906/ch/ch-addition-of-vectors-in-space/index.html"],["planck-treatise-on-thermodynamics-1903/eq-768f20f649",16,"Planck 1903, p. 119: d\\phi = \\frac{du + p\\, dv}{\\theta} = \\frac{1}{\\theta} \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{v} d\\theta + \\frac{\\left(\\d"],["planck-treatise-on-thermodynamics-1903/eq-1fa697f9f9",16,"Planck 1903, p. 120: \\left(\\frac{\\dd u}{\\dd v}\\right)_{\\theta} = \\theta \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v} - p"],["planck-treatise-on-thermodynamics-1903/eq-5893c35fbe",16,"Planck 1903, p. 120: \\left(\\frac{\\dd \\phi}{\\dd \\theta}\\right)_{v} = \\frac{c_{v}}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-2b8108f944",16,"Planck 1903, p. 120: \\left(\\frac{\\dd \\phi}{\\dd v}\\right)_{\\theta} = \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v}"],["planck-treatise-on-thermodynamics-1903/eq-20db95c796",16,"Planck 1903, p. 120: c_{p} - c_{v} = \\theta \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v} · \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}"],["hardy-course-of-pure-mathematics-1921/eq-8c8728dc78",16,"Hardy 1921, p. 251: \\{S(x + h) - S(x)\\}/h = \\{PP'\\}/h = (PP'/h) × (\\{PP'\\}/PP')"],["planck-treatise-on-thermodynamics-1903/eq-a4afe4cffb",16,"Planck 1903, p. 123: \\left(\\frac{\\dd \\phi}{\\dd p}\\right)_{\\theta} = -\\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}"],["hardy-course-of-pure-mathematics-1921/x-5857bcebc6",15,"Hardy 1921, p. 387: If n is not divisible by 10, and \\log_{10}n ..."],["shape/b1730dcdd8",6,"lcm: (N*a*b**N, N*a**N*b**N)"],["planck-treatise-on-thermodynamics-1903/eq-f826ba780d",16,"Planck 1903, p. 120: c_{p} - c_{v} = -\\theta \\left(\\frac{\\dd p}{\\dd v}\\right)_{\\theta} · \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}^{2}"],["planck-treatise-on-thermodynamics-1903/eq-1b5e9694e6",16,"Planck 1903, p. 123: \\left(\\frac{\\dd c_{p}}{\\dd p}\\right)_{\\theta} = -\\theta \\left(\\frac{\\dd^{2} v}{\\dd \\theta^{2}}\\right)_{p}"],["planck-treatise-on-thermodynamics-1903/eq-9994c48efa",16,"Planck 1903, p. 121: c_{p} - c_{v} = \\left\\{\\left(\\frac{\\dd u}{\\dd v}\\right)_{\\theta} + p\\right\\} \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p}"],["planck-treatise-on-thermodynamics-1903/eq-9fc151304c",16,"Planck 1903, p. 121: \\frac{\\theta}{p} · \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{v} - 1"],["planck-treatise-on-thermodynamics-1903/eq-943dcd1692",16,"Planck 1903, p. 122: \\dfrac{c_{p}}{c_{v}} = \\gamma"],["planck-treatise-on-thermodynamics-1903/eq-caf2fa48bd",16,"Planck 1903, p. 123: \\left(\\frac{\\dd u}{\\dd p}\\right)_{\\theta} = -\\theta \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p} - p\\left(\\frac{\\dd v}{\\dd "],["hardy-course-of-pure-mathematics-1921/eq-dc89131cfe",16,"Hardy 1921, p. 251: PP' + \\sqrtp{PR^{2} + RP'^{2}} = h\\bigsqrtp{1 + \\frac{k^{2}}{h^{2}}}"],["hardy-course-of-pure-mathematics-1921/eq-1b16a0d109",16,"Hardy 1921, p. 251: k = \\phi(x + h) - \\phi(x) = h\\phi'(\\xi)"],["shape/5a9ee3a5bd",6,"lcm: (a**N - 1, a + a**N)"],["planck-treatise-on-thermodynamics-1903/eq-f6ca83f8c0",16,"Planck 1903, p. 125: \\Delta \\theta = \\frac{\\alpha}{\\theta^{2}}\\, \\Delta p"],["planck-treatise-on-thermodynamics-1903/eq-be48d98703",16,"Planck 1903, p. 125: \\theta \\left(\\frac{\\dd v}{\\dd \\theta}\\right)_{p} - v = c_{p} \\frac{\\alpha}{\\theta^{2}}"],["planck-treatise-on-thermodynamics-1903/eq-f4beb4ab67",16,"Planck 1903, p. 126: c_{p} = \\frac{c_{p}^{(0)}}{\\left(1 - \\dfrac{3\\alpha p}{\\theta^{3}}\\right)^{\\efrac{2}{3}}}"],["planck-treatise-on-thermodynamics-1903/eq-41b6ba2ef4",16,"Planck 1903, p. 126: c_{p} = \\theta^{2} · f(\\theta^{3} - 3\\alpha p)"],["planck-treatise-on-thermodynamics-1903/eq-0a9298afa0",16,"Planck 1903, p. 126: v = \\frac{c_{p}^{(0)} \\theta}{3p} \\left(\\sqrt[3]{1 - \\frac{3\\alpha p}{\\theta^{3}}} + \\beta\\right)"],["planck-treatise-on-thermodynamics-1903/eq-715f54862f",16,"Planck 1903, p. 130: \\theta = t + \\frac{1}{\\alpha'}"],["hardy-course-of-pure-mathematics-1921/eq-0506545013",16,"Hardy 1921, p. 251: \\lim (PP'/h) = \\lim \\sqrtb{1 + [\\phi'(\\xi)]^{2}} = \\sqrtb{1 + [\\phi'(x)]^{2}}"],["boyden-first-book-in-algebra-1895/ex-31/14",4,"Boyden 1895, Exercise 31 (14)"],["planck-treatise-on-thermodynamics-1903/eq-9f22c6d0dd",16,"Planck 1903, p. 124: p_{1}v_{1} - p_{2}v_{2} = W"],["planck-treatise-on-thermodynamics-1903/eq-930cf79e4f",16,"Planck 1903, p. 124: W = -\\Delta(pv)"],["planck-treatise-on-thermodynamics-1903/eq-8e95b578e1",16,"Planck 1903, p. 124: \\Delta u = W + Q = -\\Delta(pv)"],["planck-treatise-on-thermodynamics-1903/eq-bf2b77f35c",16,"Planck 1903, p. 128: \\left(\\frac{q}{dv}\\right)_{t} = \\left(\\frac{\\dd u}{dv}\\right)_{t} + p"],["planck-treatise-on-thermodynamics-1903/eq-a22e92b01f",16,"Planck 1903, p. 129: \\theta = \\frac{100 e^{J}}{e^{J_{1}} - 1}"],["planck-treatise-on-thermodynamics-1903/eq-62d10c7412",16,"Planck 1903, p. 129: \\alpha = \\frac{1}{\\theta_{0}} = \\frac{e^{J_{1}} - 1}{100}"],["hardy-course-of-pure-mathematics-1921/eq-737007c76b",16,"Hardy 1921, p. 251: \\lim \\{PP'\\}/PP' = 1"],["hardy-course-of-pure-mathematics-1921/eq-2044544237",16,"Hardy 1921, p. 251: S'(x) = \\lim \\{S(x + h) - S(x)\\}/h = \\sqrtb{1 + [\\phi'(x)]^{2}}"],["form/e34d84579f",5,"factor: -5*a**2*b*(a - b) + 5*a - 5*b - 15*c*d*(a - b)"],["planck-treatise-on-thermodynamics-1903/eq-e08af2a16f",16,"Planck 1903, p. 130: \\left(\\frac{\\dd v}{\\dd t}\\right)_{p} = \\alpha' v_{0}"],["hardy-course-of-pure-mathematics-1921/eq-541872891d",16,"Hardy 1921, p. 251: S(x) = \\int \\sqrtb{1 + [\\phi'(x)]^{2}}\\, dx"],["concept/limit-of-intersections",7,"limit of intersections","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-limit-of-intersections"],["concept/intersection-of-two-curves",7,"intersection of two curves","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-intersection-of-two-curves"],["de-morgan-elementary-illustrations-calculus-1899/x-cae4f31855",15,"De Morgan 1899, p. 46: As A'B' moves towards AB, da and db are ..."],["concept/angular-coordinates-on-a-sphere",7,"angular coordinates on a sphere","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-angular-coordinates-on-a-sphere"],["shape/5b143a9df7",6,"factor: N*a + N*a**N*b*(a - b) + N*b + N*c*d*(a - b)"],["de-morgan-elementary-illustrations-calculus-1899/x-c12ca703d6",15,"De Morgan 1899, p. 45: But here it is necessary to remark that AB ..."],["de-morgan-elementary-illustrations-calculus-1899/x-6639f7b054",15,"De Morgan 1899, p. 46: Let P be the point of separation; then every ..."],["de-morgan-elementary-illustrations-calculus-1899/x-5281fe27c4",15,"De Morgan 1899, p. 48: This limit of the intersections is different for every ..."],["de-morgan-elementary-illustrations-calculus-1899/x-6b23f9dfda",15,"De Morgan 1899, p. 48: Hence BP = AQ and AP = BQ, or ..."],["concept/non-euclidean-geometry",7,"non-Euclidean geometry","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-non-euclidean-geometry"],["concept/hyperbolic-geometry",7,"hyperbolic geometry","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-hyperbolic-geometry"],["concept/parabolic-geometry",7,"parabolic geometry","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-parabolic-geometry"],["concept/small-circle",7,"small circle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-small-circle"],["method/expansion-in-powers",8,"expansion in powers","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-expansion-in-powers"],["concept/pole-of-a-circle",7,"pole of a circle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-pole-of-a-circle"],["concept/plane-triangle",7,"plane triangle","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-plane-triangle"],["boyden-first-book-in-algebra-1895/ex-13/16",4,"Boyden 1895, Exercise 13 (16)"],["concept/nine-points-circle",7,"nine points circle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-nine-points-circle"],["concept/excircle",7,"excircle","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-excircle"],["theorem/parallel-postulate",9,"parallel postulate","../books/ball-mathematical-recreations-1905/terms/index.html#t-theorem-parallel-postulate"],["concept/side",7,"side","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-side"],["method/limiting-case",8,"limiting case","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-limiting-case"],["form/a279cebf2d",5,"solve: (Eq(x, 2*a/3), Eq(a + x, 30))"],["shape/c9e660a631",6,"solve: (Eq(x, N*a), Eq(a + x, N))"],["todhunter-spherical-trigonometry-1886/x-ee53284ef9",15,"Todhunter 1886, scan 106: The student must have perceived that many of the ..."],["todhunter-spherical-trigonometry-1886/x-a372706cfa",15,"Todhunter 1886, scan 106: if we suppose r to become indefinitely great, the ..."],["todhunter-spherical-trigonometry-1886/x-55d8335dcd",15,"Todhunter 1886, scan 107: Let O be the pole of a small circle, ..."],["todhunter-spherical-trigonometry-1886/x-a3088fdec3",15,"Todhunter 1886, scan 108: this gives a relation between the angular co-ordinates of ..."],["todhunter-spherical-trigonometry-1886/x-77339bfbed",15,"Todhunter 1886, scan 108: this result corresponds to the well-known property of a ..."],["todhunter-spherical-trigonometry-1886/x-52fdf6c1fe",15,"Todhunter 1886, scan 110: The student should convince himself by examination that the ..."],["todhunter-spherical-trigonometry-1886/x-c5ae896b0c",15,"Todhunter 1886, scan 121: The results which have been demonstrated with respect to ..."],["concept/even-function",7,"even function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-even-function"],["person/giovanni-girolamo-saccheri",1,"Giovanni Girolamo Saccheri","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-giovanni-girolamo-saccheri"],["person/henri-poincar",1,"Henri Poincaré","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-henri-poincar"],["person/charles-howard-hinton",1,"Charles Howard Hinton","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-charles-howard-hinton"],["ball-mathematical-recreations-1905/x-f105beeec5",15,"Ball 1905, scan 350: In the hyperbolic system there are no similar figures ..."],["ball-mathematical-recreations-1905/x-265817c877",15,"Ball 1905, scan 341: Space, time, and matter cannot be defined; but the ..."],["ball-mathematical-recreations-1905/x-9e8c4c8356",15,"Ball 1905, scan 342: If an inhabitant of flatland was able to move ..."],["ball-mathematical-recreations-1905/x-943ec7c623",15,"Ball 1905, scan 348: I have assumed that the thickness of the supporting ..."],["concept/odd-function",7,"odd function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-odd-function"],["concept/modulus-of-a-complex-number",7,"modulus of a complex number","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-modulus-of-a-complex-number"],["ball-mathematical-recreations-1905/x-b4c9fdd4cd",15,"Ball 1905, scan 349: which is usually stated in the form that if ..."],["quantity/specific-energy",11,"specific energy","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-specific-energy"],["concept/homogeneous-geometry",7,"homogeneous geometry","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-homogeneous-geometry"],["concept/principal-value-of-a-logarithm",7,"principal value of a logarithm","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-principal-value-of-a-logarithm"],["concept/functional-equation",7,"functional equation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-functional-equation"],["theorem/logarithmic-series",9,"logarithmic series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-logarithmic-series"],["planck-treatise-on-thermodynamics-1903/x-4ee133e4be",15,"Planck 1903, p. 128: The numerator of this expression may be found directly ..."],["planck-treatise-on-thermodynamics-1903/x-6584d7ae2c",15,"Planck 1903, p. 122: For gases, \\gamma is large; and, in fact, the ..."],["planck-treatise-on-thermodynamics-1903/x-c3daaad29c",15,"Planck 1903, p. 103: It would be absurd to assume that the validity ..."],["planck-treatise-on-thermodynamics-1903/x-0cdfc8bb69",15,"Planck 1903, p. 124: This expression vanishes in the case of perfect gases, ..."],["concept/periodic-function",7,"periodic function","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-periodic-function"],["concept/maclaurin-s-series",7,"Maclaurin's series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-maclaurin-s-series"],["hardy-course-of-pure-mathematics-1921/x-a5e9d2789d",15,"Hardy 1921, p. 445: The point of importance is this. The infinite of ..."],["concept/line-at-infinity",7,"line at infinity","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-line-at-infinity"],["boyden-first-book-in-algebra-1895/ex-14/1",4,"Boyden 1895, Exercise 14 (1)"],["de-morgan-elementary-illustrations-calculus-1899/eq-dba7d0871b",16,"De Morgan 1899, p. 67: \\dfrac{(x + 1)^{m}}{x^{m}} = 1 + \\dfrac{mx^{m-1} + \\etc.}{x^{m}}"],["de-morgan-elementary-illustrations-calculus-1899/eq-cf520c6df8",16,"De Morgan 1899, p. 67: (x + 1)^{m+1} = x^{m+1} + (m + 1)x^{m} + \\frac{1}{2}(m + 1)m x^{m-1} + \\etc."],["de-morgan-elementary-illustrations-calculus-1899/eq-c89962e0ff",16,"De Morgan 1899, p. 71: \\frac{a}{pa + b} = \\frac{1}{p + \\dfrac{b}{a}}"],["de-morgan-elementary-illustrations-calculus-1899/eq-49d8c2ef0c",16,"De Morgan 1899, p. 71: \\frac{A + (a + a' + a'' + \\etc.)}{B + p(a + a' + a'' + \\etc.) + b + b' + b'' + \\etc.}"],["de-morgan-elementary-illustrations-calculus-1899/eq-2b10c9de0e",16,"De Morgan 1899, p. 72: \\frac{(x + 1)^{3} + (x + 2)^{3} + \\dots + (x + n)^{3}}{(x + 1)^{4} - x^{4}}"],["de-morgan-elementary-illustrations-calculus-1899/eq-41581a5c44",16,"De Morgan 1899, p. 73: \\frac{1^{3} + 2^{3} + 3^{3} + \\dots + x^{3} + (x + 1)^{3} + \\dots + (x + n)^{3}}{(x + n)^{4}}"],["de-morgan-elementary-illustrations-calculus-1899/eq-8cb6210e2c",16,"De Morgan 1899, p. 73: x\\, \\dfrac{x - 1}{2} ÷ x^{2} = \\dfrac{x - 1}{2x}"],["hardy-course-of-pure-mathematics-1921/x-489e9d5000",15,"Hardy 1921, p. 409: The left-hand sides of these equations are defined, by ..."],["wentworth-plane-geometry-1899/x-bdd2c0f44e",15,"Wentworth 1899, scan 243: Isoperimetric polygons are polygons which have equal perimeters."],["hardy-course-of-pure-mathematics-1921/x-2ff921f815",15,"Hardy 1921, p. 267: The reader should be careful to guard himself against ..."],["hardy-course-of-pure-mathematics-1921/x-578c66120e",15,"Hardy 1921, p. 410: All the ordinary formulae of elementary Trigonometry are algebraical ..."],["method/harmonic-analysis",8,"harmonic analysis","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-method-harmonic-analysis"],["concept/periodicity",7,"periodicity","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-periodicity"],["law/henry-s-law",10,"Henry's law","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-law-henry-s-law"],["theorem/fourier-s-theorem",9,"Fourier's theorem","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-theorem-fourier-s-theorem"],["hardy-course-of-pure-mathematics-1921/x-deb8f5be41",15,"Hardy 1921, p. 410: The same process of transformation may be applied to ..."],["whitehead-introduction-to-mathematics-1911/x-162220dcaa",15,"Whitehead 1911, p. 179: This peculiar property of the triangle, which is not ..."],["instrument/transit-instrument",13,"transit instrument","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-instrument-transit-instrument"],["hardy-course-of-pure-mathematics-1921/x-c83c7d390b",15,"Hardy 1921, p. 272: A point at which the tangent to a curve ..."],["hardy-course-of-pure-mathematics-1921/x-6918bbf023",15,"Hardy 1921, p. 268: if there is to be a maximum or minimum ..."],["hardy-course-of-pure-mathematics-1921/x-d92f0f582c",15,"Hardy 1921, p. 264: The formula f(x + h) = f(x) + hf'(x ..."],["quantity/time",11,"time","../books/ball-mathematical-recreations-1905/terms/index.html#t-quantity-time"],["hardy-course-of-pure-mathematics-1921/x-7d942f780e",15,"Hardy 1921, p. 413: These facts suggest the existence of some functional connection ..."],["hardy-course-of-pure-mathematics-1921/x-cecab47661",15,"Hardy 1921, p. 414: Moreover we saw in [§]191 that the series on ..."],["whitehead-introduction-to-mathematics-1911/x-9b1c34f8d4",15,"Whitehead 1911, p. 174: The origin of trigonometry was practical; it was invented ..."],["whitehead-introduction-to-mathematics-1911/x-dfa1ceb749",15,"Whitehead 1911, p. 177: Thus if the scale of a plan be an ..."],["hardy-course-of-pure-mathematics-1921/x-05f8ce615e",15,"Hardy 1921, p. 420: The sums of the series, for other values of ..."],["whitehead-introduction-to-mathematics-1911/x-65c9c73fec",15,"Whitehead 1911, p. 183: We have called v the sine of u, and ..."],["whitehead-introduction-to-mathematics-1911/x-fc6f4eac7d",15,"Whitehead 1911, p. 186: It can be proved that \\pi is an incommensurable ..."],["whitehead-introduction-to-mathematics-1911/x-b511d1f679",15,"Whitehead 1911, p. 193: We are here in the presence of one of ..."],["whitehead-introduction-to-mathematics-1911/x-fb6279194e",15,"Whitehead 1911, p. 174: Characteristically enough conic sections were invented about 150 years ..."],["concept/sidereal-day",7,"sidereal day","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-sidereal-day"],["concept/true-solar-day",7,"true solar day","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-true-solar-day"],["person/jahn",1,"Jahn","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-jahn"],["theorem/binomial-theorem",9,"binomial theorem","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-theorem-binomial-theorem"],["concept/julian-calendar",7,"Julian calendar","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-julian-calendar"],["theorem/taylor-s-theorem-with-integral-remainder",9,"Taylor's theorem with integral remainder","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-taylor-s-theorem-with-integral-remainder"],["dickson-theory-of-equations-1922/x-e82f9974dc",15,"Dickson 1922, p. 155: This simplified proof consists in showing that the two ..."],["concept/equation-of-time",7,"equation of time","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-equation-of-time"],["concept/leap-year",7,"leap year","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-leap-year"],["concept/gregorian-calendar",7,"Gregorian calendar","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-gregorian-calendar"],["method/date-of-easter",8,"date of Easter","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-date-of-easter"],["dickson-theory-of-equations-1922/x-da3afa5ffb",15,"Dickson 1922, p. 155: The proof differs from that of the auxiliary theorem ..."],["instrument/sun-dial",13,"sun-dial","../books/ball-mathematical-recreations-1905/terms/index.html#t-instrument-sun-dial"],["instrument/sun-ring",13,"sun-ring","../books/ball-mathematical-recreations-1905/terms/index.html#t-instrument-sun-ring"],["concept/principal-value-of-a-power",7,"principal value of a power","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-principal-value-of-a-power"],["law/inertia",10,"inertia","../books/ball-mathematical-recreations-1905/terms/index.html#t-law-inertia"],["ball-mathematical-recreations-1905/x-7dfe7c1887",15,"Ball 1905, scan 353: Thus to measure a length we may take a ..."],["ball-mathematical-recreations-1905/x-6541445974",15,"Ball 1905, scan 356: The true solar day is not however always of ..."],["boyden-first-book-in-algebra-1895/eq-cefd196911",16,"Boyden 1895: (a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3} = a^{3} + (3 a^{2} + 3 a b + b^{2}) b"],["unit/minute",12,"minute","../books/ball-mathematical-recreations-1905/terms/index.html#t-unit-minute"],["todhunter-spherical-trigonometry-1886/eq-6a7b9d112f",16,"Todhunter 1886, scan 103: \\cos c = \\cos a \\,\\cos b + \\sin a \\,\\sin b \\,\\cos C"],["law/spherical-law-of-cosines-for-sides",10,"spherical law of cosines for sides"],["todhunter-spherical-trigonometry-1886/eq-31ee1ad3c2",16,"Todhunter 1886, scan 104: \\delta a \\,\\cos B + \\delta b \\,\\cos A = 0"],["todhunter-spherical-trigonometry-1886/eq-768a831f1c",16,"Todhunter 1886, scan 104: \\delta A \\,\\cos b + \\delta B \\,\\cos a = 0"],["todhunter-spherical-trigonometry-1886/eq-16b4d8c2b3",16,"Todhunter 1886, scan 103: \\cos (a + \\delta a) = \\cos a - \\sin a \\,\\delta a"],["todhunter-spherical-trigonometry-1886/eq-61e2903576",16,"Todhunter 1886, scan 103: \\sin (a + \\delta a) = \\sin a + \\cos a \\,\\delta a"],["todhunter-spherical-trigonometry-1886/eq-0b08793202",16,"Todhunter 1886, scan 104: \\sin A \\sin c = \\sin C \\,\\sin a"],["ball-mathematical-recreations-1905/x-9df2cff115",15,"Ball 1905, scan 365: I believe it is not generally known that a ..."],["ball-mathematical-recreations-1905/x-996a122b30",15,"Ball 1905, scan 359: The Julian calendar made the year, on an average, ..."],["todhunter-spherical-trigonometry-1886/eq-bc495b38a7",16,"Todhunter 1886, scan 104: \\delta A \\cot A = \\delta a \\cot a"],["todhunter-spherical-trigonometry-1886/eq-ee63d785e7",16,"Todhunter 1886, scan 105: \\cot C \\sin B = \\cot c \\sin a - \\cos B \\cos a"],["todhunter-spherical-trigonometry-1886/eq-260df06816",16,"Todhunter 1886, scan 105: \\delta B \\cos A = - \\delta a \\cot b \\sin B"],["todhunter-spherical-trigonometry-1886/eq-df708001e4",16,"Todhunter 1886, scan 105: -\\frac{\\cos A}{\\sin C} \\delta B = \\frac{\\cos b}{\\sin c} \\delta a"],["todhunter-spherical-trigonometry-1886/eq-33c75a520f",16,"Todhunter 1886, scan 105: \\frac{\\delta a}{\\surd{(1 - n^2 \\sin^2 a)}} + \\frac{\\delta b}{\\surd{(1 - n^2 \\sin^2 b)}} = 0"],["todhunter-spherical-trigonometry-1886/eq-22071d4562",16,"Todhunter 1886, scan 105: n = \\frac{\\sin C}{\\sin c}\\,"],["ball-mathematical-recreations-1905/x-c1685ae55f",15,"Ball 1905, scan 357: The French were the last civilized nation to abandon ..."],["ball-mathematical-recreations-1905/x-76a56a21c6",15,"Ball 1905, scan 355: Our experiences are consistent with this statement, and that ..."],["todhunter-spherical-trigonometry-1886/eq-66bd77ebfc",16,"Todhunter 1886, scan 105: \\sin C \\delta b = \\sin a \\delta B"],["todhunter-spherical-trigonometry-1886/eq-2e09b66a59",16,"Todhunter 1886, scan 105: \\delta b \\sin C = -\\delta C \\tan a"],["todhunter-spherical-trigonometry-1886/eq-c60b71bab7",16,"Todhunter 1886, scan 105: \\delta a \\tan C = \\delta B \\sin a"],["todhunter-spherical-trigonometry-1886/eq-f54a77482f",16,"Todhunter 1886, scan 105: \\delta a \\tan C = -\\delta C \\tan a"],["todhunter-spherical-trigonometry-1886/eq-6e45612880",16,"Todhunter 1886, scan 105: \\delta b \\cos C = \\delta a"],["todhunter-spherical-trigonometry-1886/eq-2697c6b93d",16,"Todhunter 1886, scan 105: \\delta B \\cos a = -\\delta C"],["todhunter-spherical-trigonometry-1886/eq-c8ef95b0b4",16,"Todhunter 1886, scan 105: \\delta B \\tan C = \\delta C \\tan B"],["concept/equiangular-spiral",7,"equiangular spiral","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-equiangular-spiral"],["todhunter-spherical-trigonometry-1886/eq-6e3a6a4ea2",16,"Todhunter 1886, scan 105: \\delta a \\cot C = -\\delta B \\sin a"],["todhunter-spherical-trigonometry-1886/eq-5c59a9ea1e",16,"Todhunter 1886, scan 105: \\delta a = \\delta A \\sin c \\sin B"],["todhunter-spherical-trigonometry-1886/eq-b620e7d2fe",16,"Todhunter 1886, scan 105: \\delta A \\sin B \\cos C = -\\delta B \\sin A"],["todhunter-spherical-trigonometry-1886/eq-dd3b1e5e9a",16,"Todhunter 1886, scan 106: \\delta b \\tan c = \\delta c \\tan b"],["todhunter-spherical-trigonometry-1886/eq-3bab42ac05",16,"Todhunter 1886, scan 106: \\delta A \\cot c = \\delta b \\sin A"],["todhunter-spherical-trigonometry-1886/eq-8772bfd74d",16,"Todhunter 1886, scan 106: \\delta A = \\delta a \\sin b \\sin C"],["todhunter-spherical-trigonometry-1886/eq-97c62aeed0",16,"Todhunter 1886, scan 106: \\delta a \\sin B \\cos c = \\delta b \\sin A"],["concept/perpendicular",7,"perpendicular","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-perpendicular"],["hardy-course-of-pure-mathematics-1921/x-4ddb5faaff",15,"Hardy 1921, p. 428: This projection gives a map of the sphere on ..."],["concept/function-of-two-variables",7,"function of two variables","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-function-of-two-variables"],["hardy-course-of-pure-mathematics-1921/x-7b75252960",15,"Hardy 1921, p. 421: This is a generalisation of the result proved in ..."],["concept/similarity",7,"similarity","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-similarity"],["hardy-course-of-pure-mathematics-1921/x-3b380435a6",15,"Hardy 1921, p. 423: A more complete discussion of the binomial series, taking ..."],["hardy-course-of-pure-mathematics-1921/x-808132e548",15,"Hardy 1921, p. 426: To one value of Z corresponds one of z, ..."],["concept/right-triangle",7,"right triangle","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-right-triangle"],["hardy-course-of-pure-mathematics-1921/x-ee4a968cfc",15,"Hardy 1921, p. 429: The reader will probably find but little difficulty in ..."],["de-morgan-elementary-illustrations-calculus-1899/x-8d8a7b2402",15,"De Morgan 1899, p. 49: The following cannot be regarded as a demonstration, except ..."],["de-morgan-elementary-illustrations-calculus-1899/x-026ab20d0f",15,"De Morgan 1899, p. 49: The beginner will be struck with the extraordinary assertions ..."],["de-morgan-elementary-illustrations-calculus-1899/x-3274a6bc8f",15,"De Morgan 1899, p. 50: But at the same time we observe that every ..."],["de-morgan-elementary-illustrations-calculus-1899/x-6e17544c12",15,"De Morgan 1899, p. 51: Hence the ratio A\\alpha to \\beta B' or dp ..."],["de-morgan-elementary-illustrations-calculus-1899/x-beb97445d0",15,"De Morgan 1899, p. 49: An infinitely small arc of a circle is a ..."],["concept/amplitude-of-a-complex-number",7,"amplitude of a complex number","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-amplitude-of-a-complex-number"],["concept/triangle",7,"triangle","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-triangle"],["concept/rectangle",7,"rectangle","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-rectangle"],["concept/positive-direction",7,"positive direction","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-positive-direction"],["dickson-theory-of-equations-1922/eq-bff8156735",16,"Dickson 1922, p. 72: f(x) \\equiv a_0 x^n + a_1 x^{n-1} + \\dotsb + a_l x^{n-l}"],["dickson-theory-of-equations-1922/eq-a2fcb3e3ba",16,"Dickson 1922, p. 72: F(x) \\equiv (x-r)f(x) \\equiv A_0 x^{n+1} + A_1 x^n + \\dotsb + A_{l+1}x^{n-l}"],["dickson-theory-of-equations-1922/eq-b3b77d9f0f",16,"Dickson 1922, p. 72: A_1 = a_1 - ra_0"],["dickson-theory-of-equations-1922/eq-1eccab8397",16,"Dickson 1922, p. 72: A_{l+1} = -r a_l"],["dickson-theory-of-equations-1922/eq-ec76e989a8",16,"Dickson 1922, p. 73: f(x) \\equiv (x - r_1)\\dotsm (x - r_k)\\phi(x)"],["dickson-theory-of-equations-1922/eq-02a3690bff",16,"Dickson 1922, p. 76: f = q_1 f_1 - f_2"],["dickson-theory-of-equations-1922/eq-3d4b8d70fb",16,"Dickson 1922, p. 77: f_{i-1}(x) = q_i f_i(x) - f_{i+1}(x)"],["hardy-course-of-pure-mathematics-1921/x-b0d070702c",15,"Hardy 1921, p. 276: Thus if u = x + y + z, ..."],["hardy-course-of-pure-mathematics-1921/x-00348b36a2",15,"Hardy 1921, p. 280: We have up to the present attributed no meaning ..."],["dickson-theory-of-equations-1922/eq-4ece3b817e",16,"Dickson 1922, p. 77: f_{i-1}(\\rho) = -f_{i+1}(\\rho) \\ne 0"],["concept/function-value",7,"function value"],["dickson-theory-of-equations-1922/eq-dce102a460",16,"Dickson 1922, p. 78: V_{r-p} - V_{r+p} = 1"],["dickson-theory-of-equations-1922/eq-688575c9b0",16,"Dickson 1922, p. 78: V_a\\geqq V_{b}"],["dickson-theory-of-equations-1922/eq-22c1a91c9f",16,"Dickson 1922, p. 79: c_{i+1} F_{i-1}(\\rho) = -k_{i+1}(\\rho) F_{i+1}(\\rho)"],["dickson-theory-of-equations-1922/eq-777063e486",16,"Dickson 1922, p. 80: f = z^4 + qz^2 + rz + s"],["dickson-theory-of-equations-1922/eq-f0f118b2a3",16,"Dickson 1922, p. 80: f_1 = 4z^3 + 2qz + r"],["dickson-theory-of-equations-1922/eq-1752c87f49",16,"Dickson 1922, p. 80: f_2 = -2qz^2 - 3rz - 4s"],["hardy-course-of-pure-mathematics-1921/x-0e3bf586a2",15,"Hardy 1921, p. 280: The advantages of the ‘differential’ notation are in reality ..."],["dickson-theory-of-equations-1922/eq-d66f92d186",16,"Dickson 1922, p. 80: f_3 = Lz - 12rs - rq^2"],["dickson-theory-of-equations-1922/eq-4e68fe907a",16,"Dickson 1922, p. 80: L = 8qs - 2q^3 - 9r^2"],["dickson-theory-of-equations-1922/eq-1eb93a9c1c",16,"Dickson 1922, p. 80: \\Delta = -4P^3 - 27Q^2"],["dickson-theory-of-equations-1922/eq-b8b56945d9",16,"Dickson 1922, p. 80: P = -4s - \\frac{q^2}{3}"],["dickson-theory-of-equations-1922/eq-bc6db44bd5",16,"Dickson 1922, p. 80: Q = \\tfrac{8}{3}qs - r^2 - \\tfrac{2}{27}q^3"],["dickson-theory-of-equations-1922/eq-34c02cbed8",16,"Dickson 1922, p. 80: 4s = -P - \\frac{q^2}{3}"],["dickson-theory-of-equations-1922/eq-a16b8fc2d7",16,"Dickson 1922, p. 80: r^2 = -Q - \\tfrac{2}{3}qP - \\tfrac{8}{27}q^3"],["dickson-theory-of-equations-1922/eq-7091be904d",16,"Dickson 1922, p. 80: f_3 = Lz + 3rP"],["hardy-course-of-pure-mathematics-1921/eq-89e230522a",16,"Hardy 1921, p. 253: u_{0}^{2} u_{3} - 3u_{0} u_{1} u_{2} + 2u_{1}^{3}"],["hardy-course-of-pure-mathematics-1921/eq-2148dba5f1",16,"Hardy 1921, p. 253: u_{0} u_{4} - 4u_{1} u_{3} + 3u_{2}^{2}"],["dickson-theory-of-equations-1922/eq-185a40dbe0",16,"Dickson 1922, p. 80: L = 9Q + 4qP"],["dickson-theory-of-equations-1922/eq-03da7a5a48",16,"Dickson 1922, p. 80: 18r^2 qP^2 - 9r^2 LP + 4sL^2 = q^2 \\Delta"],["dickson-theory-of-equations-1922/eq-e3d047fc4d",16,"Dickson 1922, p. 80: f_4 = \\Delta"],["dickson-theory-of-equations-1922/eq-50e7b4244d",16,"Dickson 1922, p. 85: V_0 - V_{\\infty} = V"],["dickson-theory-of-equations-1922/eq-39cd545676",16,"Dickson 1922, p. 85: f(x) \\equiv a_0 x^n + a_1 x^{n-1} + \\dotsb + a_{n-1}x + a_n = 0"],["hardy-course-of-pure-mathematics-1921/eq-1f372e71dc",16,"Hardy 1921, p. 254: U_{0}U_{2n} - 2nU_{1}U_{2n-1} + \\frac{2n(2n - 1)}{1·2} U_{2}U_{2n-2} - \\dots + U_{2n}U_{0}"],["hardy-course-of-pure-mathematics-1921/eq-a16f6f6ff3",16,"Hardy 1921, p. 254: U_{r}' = rU_{r-1}"],["hardy-course-of-pure-mathematics-1921/eq-9bf983ccb7",16,"Hardy 1921, p. 254: y^{3} + 3yx + 2x^{3} = 0"],["hardy-course-of-pure-mathematics-1921/eq-2553f6b481",16,"Hardy 1921, p. 254: x^{2}(1 + x^{3})y'' - \\frac{3}{2}xy' + y = 0"],["de-morgan-elementary-illustrations-calculus-1899/eq-9151e9ad10",16,"De Morgan 1899, p. 74: \\phi' x\\, dx + \\phi'' x\\, \\frac{(dx)^{2}}{2} + \\phi''' x\\, \\frac{(dx)^{3}}{2·3} + \\etc."],["hardy-course-of-pure-mathematics-1921/eq-02e7c6bc5c",16,"Hardy 1921, p. 254: y = \\phi\\{\\psi(y_{1})\\} + \\phi\\{x - \\psi(y_{1})\\}"],["hardy-course-of-pure-mathematics-1921/eq-27b62ea6eb",16,"Hardy 1921, p. 254: y = \\phi(c) + \\phi(x - c)"],["hardy-course-of-pure-mathematics-1921/eq-4e72c182d1",16,"Hardy 1921, p. 254: y = 2\\phi(\\frac{1}{2}x)"],["hardy-course-of-pure-mathematics-1921/eq-f16f7c3bed",16,"Hardy 1921, p. 254: y = \\{x/\\psi(y_{1})\\} \\phi\\{\\psi(y_{1})\\}"],["hardy-course-of-pure-mathematics-1921/eq-d2e2421fd9",16,"Hardy 1921, p. 254: y = c\\phi(x/c)"],["hardy-course-of-pure-mathematics-1921/eq-05ed1fd955",16,"Hardy 1921, p. 254: y = \\beta x"],["hardy-course-of-pure-mathematics-1921/eq-3ff03767ed",16,"Hardy 1921, p. 254: \\beta = \\phi(\\alpha)/\\alpha"],["hardy-course-of-pure-mathematics-1921/eq-b7ee109918",16,"Hardy 1921, p. 254: \\phi(\\alpha) - \\alpha\\phi'(\\alpha) = 0"],["concept/absolute-value",7,"absolute value","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-absolute-value"],["de-morgan-elementary-illustrations-calculus-1899/eq-839bad55f8",16,"De Morgan 1899, p. 75: \\phi x + \\phi' x\\, h"],["de-morgan-elementary-illustrations-calculus-1899/eq-fe29db1e47",16,"De Morgan 1899, p. 75: \\phi x + \\phi' x\\, k"],["de-morgan-elementary-illustrations-calculus-1899/eq-d8c1d9e29f",16,"De Morgan 1899, p. 75: \\phi' x\\, h"],["de-morgan-elementary-illustrations-calculus-1899/eq-fed5548ff2",16,"De Morgan 1899, p. 75: \\sqrt{3^{2} + 4^{2}}"],["theorem/pythagorean-theorem-hypotenuse-of-the-right-triangle-with-base-3-and-other-side-4",9,"Pythagorean theorem (hypotenuse of the right triangle with base 3 and other side 4)"],["hardy-course-of-pure-mathematics-1921/eq-b49ea70b07",16,"Hardy 1921, p. 254: y_{2} = 0"],["planck-treatise-on-thermodynamics-1903/x-18026c3026",15,"Planck 1903, p. 146: This was first verified by the measurements of W. ..."],["concept/root-of-an-equation",7,"root of an equation","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-root-of-an-equation"],["hardy-course-of-pure-mathematics-1921/eq-a176985c56",16,"Hardy 1921, p. 254: 1 + y_{1}^{2} + yy_{2} = 0"],["hardy-course-of-pure-mathematics-1921/eq-e55d32c919",16,"Hardy 1921, p. 254: y_{1}^{2} + yy_{2} = 0"],["hardy-course-of-pure-mathematics-1921/eq-6e820aa4c1",16,"Hardy 1921, p. 254: y_{3} = 0"],["hardy-course-of-pure-mathematics-1921/eq-723cb15edc",16,"Hardy 1921, p. 254: (1 + y_{1}^{2}) y_{3} = 3y_{1} y_{2}^{2}"],["concept/plane-angle",7,"plane angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-plane-angle"],["concept/multiple-root",7,"multiple root","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-multiple-root"],["concept/octahedron",7,"octahedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-octahedron"],["concept/icosahedron",7,"icosahedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-icosahedron"],["hardy-course-of-pure-mathematics-1921/eq-68422c1d2e",16,"Hardy 1921, p. 254: 5y_{3}^{2} = 3y_{2} y_{4}"],["boyden-first-book-in-algebra-1895/ex-14/2",4,"Boyden 1895, Exercise 14 (2)"],["hardy-course-of-pure-mathematics-1921/eq-e4bbcb7afd",16,"Hardy 1921, p. 254: 9y_{2}^{2} y_{5} - 45y_{2} y_{3} y_{4} + 40y_{3}^{3} = 0"],["hardy-course-of-pure-mathematics-1921/ch-x",2,"Hardy 1921, ch. X: THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS","../books/hardy-course-of-pure-mathematics-1921/ch/ch-x/index.html"],["concept/focus",7,"focus","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-focus"],["hardy-course-of-pure-mathematics-1921/x-63c7d8f2ab",15,"Hardy 1921, p. 434: Thus if its path is like (b) in [fig:B]Fig. ..."],["concept/sphere-inscribed-in-a-polyhedron",7,"sphere inscribed in a polyhedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-sphere-inscribed-in-a-polyhedron"],["concept/surface-of-a-polyhedron",7,"surface of a polyhedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-surface-of-a-polyhedron"],["boyden-first-book-in-algebra-1895/ex-14/3",4,"Boyden 1895, Exercise 14 (3)"],["hardy-course-of-pure-mathematics-1921/eq-1b308413ac",16,"Hardy 1921, p. 254: D_{x}^{2} (y_{2}^{-2/3}) = 0"],["hardy-course-of-pure-mathematics-1921/eq-86888ad5a4",16,"Hardy 1921, p. 255: y_{2} = ±(pr - q^{2})/(px^{2} + 2qx + r)^{3/2}"],["hardy-course-of-pure-mathematics-1921/eq-d0917f8c63",16,"Hardy 1921, p. 254: D_{x}^{3} (y_{2}^{-2/3}) = 0"],["hardy-course-of-pure-mathematics-1921/eq-59b8baa10a",16,"Hardy 1921, p. 255: 4ac - 5b^{2} = (4\\alpha\\gamma - 5\\beta^{2})/\\tau^{8}"],["hardy-course-of-pure-mathematics-1921/eq-df901e7a8f",16,"Hardy 1921, p. 255: bt - a^{2} = - (\\beta\\tau - \\alpha^{2})/\\tau^{6}"],["theorem/polyhedral-formula",9,"polyhedral formula","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-polyhedral-formula"],["theorem/five-regular-polyhedra",9,"five regular polyhedra","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-five-regular-polyhedra"],["theorem/sum-of-plane-angles-of-a-polyhedron",9,"sum of plane angles of a polyhedron","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-sum-of-plane-angles-of-a-polyhedron"],["todhunter-spherical-trigonometry-1886/x-322bf3d4de",15,"Todhunter 1886, scan 124: A polyhedron is a solid bounded by any number ..."],["todhunter-spherical-trigonometry-1886/x-38cd10975e",15,"Todhunter 1886, scan 125: It will be seen that the demonstration establishes something ..."],["todhunter-spherical-trigonometry-1886/x-c0c5bad25c",15,"Todhunter 1886, scan 124: If \\mathrm{S} be the number of solid angles in ..."],["hardy-course-of-pure-mathematics-1921/eq-dc058a11b8",16,"Hardy 1921, p. 255: (1 - x^{2})y_{k+2} - (2k + 1)xy_{k+1} + (n^{2} - k^{2})y_{k} = 0"],["todhunter-spherical-trigonometry-1886/x-32f9d5e38f",15,"Todhunter 1886, scan 124: Take any point within the polyhedron as centre, and ..."],["todhunter-spherical-trigonometry-1886/x-def1a53745",15,"Todhunter 1886, scan 125: but n cannot be less than 3, so that ..."],["todhunter-spherical-trigonometry-1886/x-fd27761983",15,"Todhunter 1886, scan 134: A regular octahedron is inscribed in a cube so ..."],["todhunter-spherical-trigonometry-1886/x-9f65b68cb8",15,"Todhunter 1886, scan 130: Or we may adopt Carnot’s method, in which this ..."],["hardy-course-of-pure-mathematics-1921/eq-fc9a75e31b",16,"Hardy 1921, p. 255: vD_{x}^{n}u = D_{x}^{n}(uv) - nD_{x}^{n-1}(uD_{x}v)"],["de-morgan-elementary-illustrations-calculus-1899/x-dd41dee829",15,"De Morgan 1899, p. 52: The number of units of length described in a ..."],["hardy-course-of-pure-mathematics-1921/ex-xciii",3,"Hardy 1921, Exercise XCIII"],["boyden-first-book-in-algebra-1895/ex-14/4",4,"Boyden 1895, Exercise 14 (4)"],["de-morgan-elementary-illustrations-calculus-1899/x-b27aa8fd63",15,"De Morgan 1899, p. 52: Suppose a point moving along a straight line uniformly; ..."],["de-morgan-elementary-illustrations-calculus-1899/x-227c7d7705",15,"De Morgan 1899, p. 57: And since, when x is the space described, \\phi' ..."],["hardy-course-of-pure-mathematics-1921/eq-b5b713363c",16,"Hardy 1921, p. 255: x = a(2\\cos t + \\cos 2t)"],["hardy-course-of-pure-mathematics-1921/eq-2b280c2b4a",16,"Hardy 1921, p. 255: y = a(2\\sin t - \\sin 2t)"],["hardy-course-of-pure-mathematics-1921/eq-579f64ece5",16,"Hardy 1921, p. 255: x\\sin \\tfrac{1}{2} t + y\\cos \\tfrac{1}{2} t = a\\sin \\tfrac{3}{2} t"],["quantity/psi-function",11,"Psi function","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-psi-function"],["hardy-course-of-pure-mathematics-1921/x-0e7fdeaf7b",15,"Hardy 1921, p. 434: Thus PQ will have been described twice, once from ..."],["hardy-course-of-pure-mathematics-1921/eq-1873cdb6a2",16,"Hardy 1921, p. 255: x\\cos \\tfrac{1}{2} t - y\\sin \\tfrac{1}{2} t = 3a\\cos \\tfrac{3}{2} t"],["hardy-course-of-pure-mathematics-1921/x-b717e344b8",15,"Hardy 1921, p. 435: But the latter contour evidently lies inside the circle ..."],["hardy-course-of-pure-mathematics-1921/eq-24271f3efa",16,"Hardy 1921, p. 255: QR = 4a"],["hardy-course-of-pure-mathematics-1921/eq-9cd3d4f531",16,"Hardy 1921, p. 255: x^{2} + y^{2} = 9a^{2}"],["hardy-course-of-pure-mathematics-1921/x-f97d852043",15,"Hardy 1921, p. 436: We can then show, by an argument similar to ..."],["hardy-course-of-pure-mathematics-1921/eq-39f550ba9a",16,"Hardy 1921, p. 255: (x^{2} + y^{2} + 12ax + 9a^{2})^{2} = 4a(2x + 3a)^{3}"],["planck-treatise-on-thermodynamics-1903/x-93a41a8154",15,"Planck 1903, p. 133: The entropy may in general, however, as we shall ..."],["hardy-course-of-pure-mathematics-1921/eq-a62ba4fe79",16,"Hardy 1921, p. 255: u^{2}\\xi - u\\eta = a(u^{3} - 1)"],["hardy-course-of-pure-mathematics-1921/eq-0344bba0fa",16,"Hardy 1921, p. 255: u^{2}\\xi + u\\eta = 3a(u^{3} + 1)"],["hardy-course-of-pure-mathematics-1921/x-8bcd76fb5c",15,"Hardy 1921, p. 436: This assumption is obviously legitimate, for to suppose the ..."],["wentworth-first-steps-in-algebra-1894/ex-9/4",4,"Wentworth 1894, Exercise 9 (4)"],["form/34f44d1b4f",5,"solve: Eq(4*x + 6, x + 9)"],["planck-treatise-on-thermodynamics-1903/x-6269520229",15,"Planck 1903, p. 137: Experience immediately shows, however, that in any state of ..."],["planck-treatise-on-thermodynamics-1903/x-6c79695e6d",15,"Planck 1903, p. 136: The equations % [eqn:(99)](99)% might therefore be called the ..."],["hardy-course-of-pure-mathematics-1921/eq-ca13778c8f",16,"Hardy 1921, p. 255: (p + q)^{2/3} - (p - q)^{2/3} = 1"],["hardy-course-of-pure-mathematics-1921/eq-2784e7b875",16,"Hardy 1921, p. 257: \\begin{vmatrix} f(a) & \\phi(a) & \\psi(a)\\\\ f(b) & \\phi(b) & \\psi(b)\\\\ f'(\\xi) & \\phi'(\\xi) & \\psi'(\\xi) \\end{vmatrix} =0"],["hardy-course-of-pure-mathematics-1921/eq-317adfc478",16,"Hardy 1921, p. 258: \\frac{f(b) - f(a)}{\\phi(b) - \\phi(a)} = \\frac{f'(\\xi)}{\\phi'(\\xi)}\\Add{.}"],["hardy-course-of-pure-mathematics-1921/eq-0afd7d6097",16,"Hardy 1921, p. 258: \\phi(x) - \\phi(x_{0}) = (x - x_{0})\\phi'(\\xi)"],["hardy-course-of-pure-mathematics-1921/eq-484e018461",16,"Hardy 1921, p. 257: \\phi(x) = 1/(1 + x^{2})"],["boyden-first-book-in-algebra-1895/ex-14/5",4,"Boyden 1895, Exercise 14 (5)"],["hardy-course-of-pure-mathematics-1921/eq-a0f44d1801",16,"Hardy 1921, p. 256: \\pi < \\frac{\\sin \\pi x}{x(1 - x)} \\leq 4"],["boyden-first-book-in-algebra-1895/ex-14/6",4,"Boyden 1895, Exercise 14 (6)"],["concept/function",7,"function","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-function"],["hardy-course-of-pure-mathematics-1921/eq-080d8676a0",16,"Hardy 1921, p. 256: \\frac{dy}{dx} = \\frac{(6x^{2} + x - 1) (x - 1)^{2} (x + 1)^{3}}{x^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-a3b0227f5a",16,"Hardy 1921, p. 257: \\phi^{n} (x) = Q_{n}(x)/(1 + x^{2})^{n+1}"],["hardy-course-of-pure-mathematics-1921/eq-4e77b7e06a",16,"Hardy 1921, p. 257: Q_{n} = (-1)^{n} n!\\left\\{(n + 1)x^{n} - \\dfrac{(n + 1)n(n - 1)}{3!} x^{n-2} + \\dots\\right\\}"],["hardy-course-of-pure-mathematics-1921/eq-b2ff7ce302",16,"Hardy 1921, p. 256: \\lambda(ax^{2} + bx + c) + \\mu(a'x^{2} + b'x + c') = 0"],["hardy-course-of-pure-mathematics-1921/eq-5c8f8af742",16,"Hardy 1921, p. 256: \\arctan\\{(a^{2} - b^{2})/2ab\\}"],["hardy-course-of-pure-mathematics-1921/eq-e446461b37",16,"Hardy 1921, p. 256: s(x - s) x^{2} + 4\\Delta^{2} = 0"],["wentworth-first-steps-in-algebra-1894/ex-21/4",4,"Wentworth 1894, Exercise 21 (4)"],["hardy-course-of-pure-mathematics-1921/x-ca7d559d41",15,"Hardy 1921, p. 440: The operations of proceeding to the limit zero with ..."],["hardy-course-of-pure-mathematics-1921/x-30bc3e54a1",15,"Hardy 1921, p. 440: It is natural to suppose so: but that is ..."],["boyden-first-book-in-algebra-1895/ex-14/7",4,"Boyden 1895, Exercise 14 (7)"],["hardy-course-of-pure-mathematics-1921/eq-e916be595d",16,"Hardy 1921, p. 257: 2\\Delta + \\frac{a^{2} + b^{2} + c^{2}}{2\\sqrt{3}}"],["hardy-course-of-pure-mathematics-1921/eq-36370b0289",16,"Hardy 1921, p. 256: s(s - a)(s - b)(s - c) = \\Delta^{2}"],["hardy-course-of-pure-mathematics-1921/eq-abc3e60521",16,"Hardy 1921, p. 256: a + b + c = 2s"],["hardy-course-of-pure-mathematics-1921/eq-dc076b8428",16,"Hardy 1921, p. 257: 256\\Delta\\Delta' = 25a^{4}\\sqrt{5}"],["concept/argand-diagram",7,"argand diagram","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-argand-diagram"],["form/658d83a800",5,"identity: 2*a*(-2*a + b)"],["hardy-course-of-pure-mathematics-1921/x-69da6b1285",15,"Hardy 1921, p. 442: In practice, a result obtained by assuming that two ..."],["concept/complete-equation",7,"complete equation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-complete-equation"],["hardy-course-of-pure-mathematics-1921/eq-7f5b9564fa",16,"Hardy 1921, p. 257: x^{2}y - 4x^{2} - 4xy + y^{2} + 16x - 2y - 7 = 0"],["concept/base-of-a-logarithm-system",7,"base of a logarithm system","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-base-of-a-logarithm-system"],["hardy-course-of-pure-mathematics-1921/eq-cb382596c1",16,"Hardy 1921, p. 257: (x^{2} - 4y + 8)/(y^{2} - 6x + 3)"],["hardy-course-of-pure-mathematics-1921/eq-bc56253f7a",16,"Hardy 1921, p. 257: \\frac{d}{da}\\{\\lim_{x \\to a} f(x)\\} - \\lim_{x \\to a}f'(x) = \\tfrac{3}{4} \\sec^{3} a - \\tfrac{5}{12} \\sec a"],["hardy-course-of-pure-mathematics-1921/eq-99d00f5968",16,"Hardy 1921, p. 258: \\int \\frac{dx}{(1 + x^{2})^{3}}"],["form/2d6c423ab5",5,"identity: -a**2*(-a**3 - a**2*b**2)"],["shape/7753edcb60",6,"identity: -a**N*(-a**N*b**N - a**N)"],["concept/circular-function",7,"circular function","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-circular-function"],["hardy-course-of-pure-mathematics-1921/eq-4c0bb7150a",16,"Hardy 1921, p. 259: 2(n - 1)(q - \\tfrac{1}{4}p^{2}) \\int \\frac{dx}{(x^{2} + px + q)^{n}} \\\\ = \\frac{x + \\frac{1}{2}p}{(x^{2} + px + q)^{n-1}"],["concept/addition-formulae",7,"addition formulae","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-addition-formulae"],["hardy-course-of-pure-mathematics-1921/eq-bb37a6a8a5",16,"Hardy 1921, p. 259: (p + 1) I_{p, q} = x^{p+1}(1 + x)^{q} - qI_{p+1, q-1}"],["planck-treatise-on-thermodynamics-1903/eq-42f682eb81",16,"Planck 1903, p. 133: M_{1} + M_{2} + M_{3} = M"],["planck-treatise-on-thermodynamics-1903/eq-a0564c7b66",16,"Planck 1903, p. 133: M_{1} v_{1} + M_{2} v_{2} + M_{3} v_{3} = V"],["planck-treatise-on-thermodynamics-1903/eq-bbbff32189",16,"Planck 1903, p. 133: M_{1} u_{1} + M_{2} u_{2} + M_{3} u_{3} = U"],["hardy-course-of-pure-mathematics-1921/x-d7e839ceb5",15,"Hardy 1921, p. 408: Explain the fallacy in the following argument: since e^{2m\\pi ..."],["concept/domain-of-definition",7,"domain of definition","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-domain-of-definition"],["hardy-course-of-pure-mathematics-1921/eq-8cf336b4da",16,"Hardy 1921, p. 259: I_{p, q} = (-1)^{p+1} \\int y^{p} (1 + y)^{-p-q-2}\\, dy"],["hardy-course-of-pure-mathematics-1921/eq-c0a5643391",16,"Hardy 1921, p. 259: \\int xX^{-1/3}\\, dx = -3(3a - 2bx) X^{2/3}/10b^{2}"],["hardy-course-of-pure-mathematics-1921/eq-1ad2aa12d2",16,"Hardy 1921, p. 259: \\int x^{2}X^{-1/3}\\, dx = 3(9a^{2} - 6abx + 5b^{2}x^{2}) X^{2/3}/40b^{3}"],["hardy-course-of-pure-mathematics-1921/eq-fe30e3d75a",16,"Hardy 1921, p. 259: 2(n - 1)I_{m, n} = -x^{m-1} (1 + x^{2})^{-(n-1)} + (m - 1)I_{m-2, n-1}"],["hardy-course-of-pure-mathematics-1921/eq-3e3d97e8eb",16,"Hardy 1921, p. 259: \\beta I_{n} = x^{n} \\sin\\beta x - nJ_{n-1}"],["hardy-course-of-pure-mathematics-1921/eq-800f2708ac",16,"Hardy 1921, p. 259: \\beta J_{n} = -x^{n} \\cos\\beta x + nI_{n-1}"],["planck-treatise-on-thermodynamics-1903/eq-8aa7e87c09",16,"Planck 1903, p. 134: \\Phi = M_{1} \\phi_{1} + M_{2} \\phi_{2} + M_{3} \\phi_{3}"],["planck-treatise-on-thermodynamics-1903/eq-144dcc2370",16,"Planck 1903, p. 134: \\delta \\phi = \\frac{\\delta u + p\\, \\delta v}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-da7931c349",16,"Planck 1903, p. 135: \\theta_{1} = \\theta_{2} = \\theta_{3} (= \\theta)"],["planck-treatise-on-thermodynamics-1903/eq-33db367916",16,"Planck 1903, p. 135: p_{1} = p_{2} = p_{3}"],["planck-treatise-on-thermodynamics-1903/eq-c239bcbfbc",16,"Planck 1903, p. 135: \\phi_{1} - \\phi_{2} = \\frac{(u_{1} - u_{2}) + p_{1}(v_{1} - v_{2})}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-9de0df905f",16,"Planck 1903, p. 135: \\phi_{2} - \\phi_{3} = \\frac{(u_{2} - u_{3}) + p_{2}(v_{2} - v_{3})}{\\theta}"],["hardy-course-of-pure-mathematics-1921/eq-d24f052b3e",16,"Hardy 1921, p. 260: nI_{n} = \\sin x\\cos^{n-1} x + (n - 1) I_{n-2}"],["de-morgan-elementary-illustrations-calculus-1899/eq-3dfaefda20",16,"De Morgan 1899, p. 132: \\frac{1}{3}bwx^{3}"],["de-morgan-elementary-illustrations-calculus-1899/eq-7352dd5663",16,"De Morgan 1899, p. 132: \\frac{1}{3}bwa^{3}"],["planck-treatise-on-thermodynamics-1903/eq-46502a4bf4",16,"Planck 1903, p. 137: \\theta\\, \\delta^{2} \\Phi = -\\tsum M_{1} \\left(\\frac{(c_{v})_{1}}{\\theta}\\, \\delta \\theta_{1}^{2} - \\left(\\frac{\\dd p_{1}"],["planck-treatise-on-thermodynamics-1903/eq-3ea07f64af",16,"Planck 1903, p. 136: \\int_{v_{2}}^{v_{1}} p\\, dv = p_{1} (v_{1} - v_{2})"],["planck-treatise-on-thermodynamics-1903/eq-903d77ccfb",16,"Planck 1903, p. 136: \\int_{v_{3}}^{v_{2}} p\\, dv = p_{2}(v_{2} - v_{3})"],["planck-treatise-on-thermodynamics-1903/eq-a670e7d455",16,"Planck 1903, p. 135: \\phi_{1} - \\phi_{2} = \\frac{u_{1} - u_{2}}{\\theta} + \\frac{1}{\\theta} \\int_{v_{2}}^{v_{1}} p\\, dv"],["planck-treatise-on-thermodynamics-1903/eq-b88a6d8656",16,"Planck 1903, p. 140: \\frac{R\\theta}{v_{1} - a} - \\frac{c}{\\theta (v_{1} + b)^{2}} = \\frac{R\\theta}{v_{2} - a} - \\frac{c}{\\theta (v_{2} + b)^{"],["hardy-course-of-pure-mathematics-1921/eq-a56ef41322",16,"Hardy 1921, p. 260: nJ_{n} = -\\cos x\\sin^{n-1} x + (n - 1) J_{n-2}"],["planck-treatise-on-thermodynamics-1903/eq-b424247de9",16,"Planck 1903, p. 241: -\\log c_{1} = \\log K"],["planck-treatise-on-thermodynamics-1903/eq-653455d2d2",16,"Planck 1903, p. 240: c_{1} = \\frac{n_{1}}{n_{0} + n_{1}}"],["planck-treatise-on-thermodynamics-1903/eq-0f0b2f3608",16,"Planck 1903, p. 140: R\\theta \\log \\frac{v_{1} - a}{v_{2} - a} - \\frac{c}{\\theta} \\left(\\frac{1}{v_{2} + b} - \\frac{1}{v_{1} + b}\\right) = (v_"],["planck-treatise-on-thermodynamics-1903/eq-1d2aa1d4ab",16,"Planck 1903, p. 140: u - \\theta \\phi = f"],["planck-treatise-on-thermodynamics-1903/eq-21276ad018",16,"Planck 1903, p. 141: f_{2} - f_{1} = p_{1} (v_{1} - v_{2})"],["planck-treatise-on-thermodynamics-1903/eq-92ebe5a75d",16,"Planck 1903, p. 141: \\left(\\frac{\\dd f}{\\dd \\theta}\\right)_{v} = -\\phi"],["planck-treatise-on-thermodynamics-1903/eq-b731185761",16,"Planck 1903, p. 141: \\left(\\frac{\\dd f}{\\dd v}\\right)_{\\theta} = -p"],["planck-treatise-on-thermodynamics-1903/eq-b4323c460a",16,"Planck 1903, p. 142: (u_{1} - u_{2}) + p_{1} (v_{1} - v_{2}) = \\theta (v_{1} - v_{2})\\, \\frac{dp_{1}}{d\\theta}"],["hardy-course-of-pure-mathematics-1921/eq-99c21aa1f2",16,"Hardy 1921, p. 260: (n - 1)(I_{n} + I_{n-2}) = \\tan^{n-1}x"],["hardy-course-of-pure-mathematics-1921/eq-962ff1cc4a",16,"Hardy 1921, p. 260: (m+n)I_{m, n} = -\\cos^{m+1}x \\sin^{n-1}x + (n - 1) I_{m, n-2}"],["planck-treatise-on-thermodynamics-1903/eq-b600679148",16,"Planck 1903, p. 142: \\phi_{1} - \\phi_{2} = (v_{1} - v_{2})\\, \\frac{dp_{1}}{d\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-a80d3df803",16,"Planck 1903, p. 142: L = u_{1} - u_{2} + p_{1}(v_{1} - v_{2})"],["planck-treatise-on-thermodynamics-1903/eq-1e8aa2a618",16,"Planck 1903, p. 142: W = -p_{1}(v_{1} - v_{2})"],["planck-treatise-on-thermodynamics-1903/eq-61f8624ddc",16,"Planck 1903, p. 142: L = \\theta (v_{1} - v_{2})"],["planck-treatise-on-thermodynamics-1903/eq-cc1a2bbf03",16,"Planck 1903, p. 144: v_{1} = \\frac{R\\theta}{mp_{1}}"],["planck-treatise-on-thermodynamics-1903/eq-5e15281954",16,"Planck 1903, p. 145: u_{1} = c_{v} \\theta + \\const"],["planck-treatise-on-thermodynamics-1903/eq-d5d509585f",16,"Planck 1903, p. 146: \\frac{d\\theta}{dp_{1}} = \\frac{\\theta (v_{1} - v_{2})}{L}"],["hardy-course-of-pure-mathematics-1921/eq-d52f5c5230",16,"Hardy 1921, p. 260: (n - 1)(n - 2)I_{m, n} = (n - 2)^{2}I_{m, n-2} + m(m - 1)I_{m-2, n-2} \\\\ -x^{m-1} \\cosec^{n-1}x \\{m\\sin x + (n - 2) x\\co"],["boyden-first-book-in-algebra-1895/ex-14/8",4,"Boyden 1895, Exercise 14 (8)"],["planck-treatise-on-thermodynamics-1903/eq-088141d2af",16,"Planck 1903, p. 147: \\frac{L}{\\theta} = \\phi_{1} - \\phi_{2}"],["concept/coordinate-geometry",7,"coordinate geometry","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-coordinate-geometry"],["concept/special-element",7,"special element","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-special-element"],["concept/limiting-infinity",7,"limiting infinity","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-limiting-infinity"],["concept/actual-infinity",7,"actual infinity","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-actual-infinity"],["concept/uniform-convergence",7,"uniform convergence","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-uniform-convergence"],["hardy-course-of-pure-mathematics-1921/eq-bd21af0ad6",16,"Hardy 1921, p. 260: (n - 1)(a^{2} - b^{2}) I_{n} = -b\\sin x (a + b\\cos x)^{-(n-1)} + (2n - 3)aI_{n-1} - (n - 2)I_{n-2}"],["hardy-course-of-pure-mathematics-1921/eq-5c45e2c263",16,"Hardy 1921, p. 260: 4n(n + 1)(ab - h^{2})I_{n+2} - 2n(2n + 1)(a + b)I_{n+1} + 4n^{2}I_{n} = -\\frac{d^{2} I_{n}}{dx^{2}}"],["concept/incomplete-symbol",7,"incomplete symbol","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-incomplete-symbol"],["concept/non-uniform-convergence",7,"non-uniform convergence","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-non-uniform-convergence"],["theorem/addition-theorem",9,"addition-theorem","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-theorem-addition-theorem"],["concept/curve-of-normal-error",7,"curve of normal error","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-curve-of-normal-error"],["concept/damped-vibration",7,"damped vibration","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-concept-damped-vibration"],["person/seidel",1,"Seidel","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-person-seidel"],["whitehead-introduction-to-mathematics-1911/x-68b09adc7a",15,"Whitehead 1911, p. 200: The summation of a series approximates to a limit ..."],["whitehead-introduction-to-mathematics-1911/x-2177f55248",15,"Whitehead 1911, p. 200: But this description of the meaning of approximating to ..."],["whitehead-introduction-to-mathematics-1911/x-462fbb605a",15,"Whitehead 1911, p. 203: Mathematics would be a much easier science than it ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-4554e2d1be",16,"De Morgan 1899, p. 77: \\phi(x + h) = (x + h)^{2} + x + h - 4 = x^{2} + x - 4 + (2x + 1)h + h^{2}"],["whitehead-introduction-to-mathematics-1911/x-d66abb8814",15,"Whitehead 1911, p. 198: The statesman in framing his speech puts the dominating ..."],["hardy-course-of-pure-mathematics-1921/eq-dc9b228533",16,"Hardy 1921, p. 260: (m + 1)I_{m, n} = x^{m+1}(\\log x)^{n} - nI_{m, n-1}"],["hardy-course-of-pure-mathematics-1921/eq-8e6c2b61c3",16,"Hardy 1921, p. 260: x^{m+1} \\left\\{\\frac{(\\log x)^{n}}{m + 1} - \\frac{n(\\log x)^{n-1}}{(m + 1)^{2}} + \\frac{n(n - 1)(\\log x)^{n-2}}{(m + 1)^"],["hardy-course-of-pure-mathematics-1921/eq-d9c9f27216",16,"Hardy 1921, p. 260: \\phi'' + a^{2}\\phi = 0"],["hardy-course-of-pure-mathematics-1921/eq-52ce7faf3c",16,"Hardy 1921, p. 261: \\phi'^{2} + a^{2}\\phi^{2} = a^{2}b^{2}"],["hardy-course-of-pure-mathematics-1921/eq-043b847356",16,"Hardy 1921, p. 261: y' + \\omega z = 0"],["hardy-course-of-pure-mathematics-1921/eq-83361a063b",16,"Hardy 1921, p. 261: z' - \\omega y = 0"],["hardy-course-of-pure-mathematics-1921/eq-fc3da95a8c",16,"Hardy 1921, p. 261: x = \\cos\\phi + \\frac{\\sin\\alpha \\sin\\phi}{1 - \\cos^{2}\\alpha \\sin^{2}\\phi}"],["hardy-course-of-pure-mathematics-1921/eq-acf4b78adf",16,"Hardy 1921, p. 261: \\frac{1}{2}\\pi(1 + \\sin\\alpha)^{2}/\\sin\\alpha"],["concept/limit",7,"limit","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-limit"],["dickson-theory-of-equations-1922/eq-34a5c32978",16,"Dickson 1922, p. 91: f(a+h) = f(a) + f'(a)h + f''(a) \\frac{h^2}{2} + \\dotsb"],["dickson-theory-of-equations-1922/eq-31c3fd2191",16,"Dickson 1922, p. 91: f(a) + f'(a)h = 0"],["dickson-theory-of-equations-1922/eq-64b87a678e",16,"Dickson 1922, p. 91: h = \\frac{-f(a)}{f'(a)}"],["theorem/newton-s-correction",9,"Newton's correction"],["dickson-theory-of-equations-1922/eq-78e020cb16",16,"Dickson 1922, p. 92: f'(a) = \\tan XTP"],["dickson-theory-of-equations-1922/eq-4f85f5694d",16,"Dickson 1922, p. 93: c = \\frac{\\alpha f(\\beta) - \\beta f(\\alpha)}{f(\\beta) - f(\\alpha)}"],["theorem/regula-falsi-formula",9,"regula falsi formula"],["dickson-theory-of-equations-1922/eq-b0ab76ea7d",16,"Dickson 1922, p. 93: -f(\\alpha) : c - \\alpha = f(\\beta) : \\beta - c"],["dickson-theory-of-equations-1922/eq-2bc1e14efe",16,"Dickson 1922, p. 94: f'(x) = 4x^3 + 3x^2 - 6x - 1"],["hardy-course-of-pure-mathematics-1921/eq-7ff7275368",16,"Hardy 1921, p. 261: a^{2}(\\beta - \\cos\\beta\\sin\\beta)"],["hardy-course-of-pure-mathematics-1921/eq-1a1bce2514",16,"Hardy 1921, p. 261: \\pi(a^{2} + \\frac{1}{2}b^{2})"],["concept/quadrantal-triangle",7,"quadrantal triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-quadrantal-triangle"],["theorem/sum-of-squared-cosines-to-the-vertices-of-a-quadrantal-triangle-is-one",9,"sum of squared cosines to the vertices of a quadrantal triangle is one","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-sum-of-squared-cosines-to-the-vertices-of-a-quadrantal-triangle-is-one"],["theorem/sum-of-cosines-to-fixed-points-varies-as-the-cosine-of-one-arc",9,"sum of cosines to fixed points varies as the cosine of one arc","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-sum-of-cosines-to-fixed-points-varies-as-the-cosine-of-one-arc"],["theorem/sum-of-cosines-to-the-solid-angles-of-a-regular-polyhedron-is-zero",9,"sum of cosines to the solid angles of a regular polyhedron is zero","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-sum-of-cosines-to-the-solid-angles-of-a-regular-polyhedron-is-zero"],["theorem/sum-of-squared-cosines-to-the-solid-angles-of-a-regular-polyhedron-is-one-third-of-their-number",9,"sum of squared cosines to the solid angles of a regular polyhedron is one third of their number","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-sum-of-squared-cosines-to-the-solid-angles-of-a-regular-polyhedron-is-one-third-of-their-number"],["todhunter-spherical-trigonometry-1886/x-0ce78283f5",15,"Todhunter 1886, scan 138: Thus, whatever may be the position of T, the ..."],["todhunter-spherical-trigonometry-1886/x-773969ff87",15,"Todhunter 1886, scan 143: Thus the sum of the squares of the cosines ..."],["concept/correlation-between-homogeneous-and-common-geometry",7,"correlation between homogeneous and common geometry","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-correlation-between-homogeneous-and-common-geometry"],["concept/uniform-motion",7,"uniform motion","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-uniform-motion"],["theorem/cosine-addition-formula",9,"cosine addition formula","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-cosine-addition-formula"],["hardy-course-of-pure-mathematics-1921/eq-21f616cdf1",16,"Hardy 1921, p. 261: \\int \\frac{dx}{(lx + my + n)(hx + by + f)} = \\alpha\\log \\frac{PT}{PT'} + \\beta"],["hardy-course-of-pure-mathematics-1921/eq-a85f5c7cc3",16,"Hardy 1921, p. 261: \\alpha e + \\gamma = 0"],["hardy-course-of-pure-mathematics-1921/x-6ec8e6afa1",15,"Hardy 1921, p. 445: In a system of real homogeneous geometry a point ..."],["boyden-first-book-in-algebra-1895/ex-31/15",4,"Boyden 1895, Exercise 31 (15)"],["form/2a2e9dd4f7",5,"factor: -12*a*b**2 + 4*b**3*c - 8*b*c**3"],["hardy-course-of-pure-mathematics-1921/x-f237829a1c",15,"Hardy 1921, p. 445: Thus, in what may be called ‘real homogeneous Cartesian ..."],["de-morgan-elementary-illustrations-calculus-1899/x-192413e739",15,"De Morgan 1899, p. 58: we have equations, which, though never exactly true, are ..."],["de-morgan-elementary-illustrations-calculus-1899/x-f226a23356",15,"De Morgan 1899, p. 59: [The motion of the point M or the point ..."],["de-morgan-elementary-illustrations-calculus-1899/x-99a02d809f",15,"De Morgan 1899, p. 59: The same considerations of velocity which have been applied ..."],["instrument/hourglass",13,"hourglass","../books/ball-mathematical-recreations-1905/terms/index.html#t-instrument-hourglass"],["instrument/clock",13,"clock","../books/ball-mathematical-recreations-1905/terms/index.html#t-instrument-clock"],["instrument/water-clock",13,"water clock","../books/ball-mathematical-recreations-1905/terms/index.html#t-instrument-water-clock"],["concept/isochronism",7,"isochronism","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-isochronism"],["shape/d7e60353fd",6,"factor: N*a*b**N + N*b*c**N + N*b**N*c"],["todhunter-spherical-trigonometry-1886/eq-90526d7351",16,"Todhunter 1886, scan 107: \\cos A = \\frac{\\cos a - \\cos b \\cos c }{\\sin b \\sin c}"],["todhunter-spherical-trigonometry-1886/eq-85a23ecd78",16,"Todhunter 1886, scan 107: \\cos A = \\frac{\\beta^2 + \\gamma^2 - \\alpha^2}{2 \\beta \\gamma}\\,"],["todhunter-spherical-trigonometry-1886/eq-69c081ae40",16,"Todhunter 1886, scan 107: \\frac{\\sin A}{\\sin B} = \\frac{\\sin a}{\\sin b}"],["law/law-of-sines-spherical",10,"law of sines (spherical)"],["todhunter-spherical-trigonometry-1886/eq-76fb85a5cc",16,"Todhunter 1886, scan 107: \\frac{\\sin A}{\\sin B} = \\frac{\\alpha}{\\beta}"],["law/law-of-sines-plane",10,"law of sines (plane)"],["todhunter-spherical-trigonometry-1886/eq-be7f1476db",16,"Todhunter 1886, scan 108: \\cos r = \\cos \\alpha \\cos \\theta + \\sin \\alpha \\sin \\theta \\cos (\\phi - \\beta)"],["concept/angular-co-ordinates-on-a-sphere",7,"angular co-ordinates on a sphere"],["person/huygens",1,"Huygens","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-huygens"],["person/hooke",1,"Hooke","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-hooke"],["todhunter-spherical-trigonometry-1886/eq-8d706c5694",16,"Todhunter 1886, scan 108: 0 = \\cos \\alpha \\cos \\theta + \\sin \\alpha \\sin \\theta \\cos (\\phi - \\beta)"],["todhunter-spherical-trigonometry-1886/eq-e593165e50",16,"Todhunter 1886, scan 108: \\tan \\frac{\\theta_1}{2} \\tan \\frac{\\theta_2}{2}= \\frac{\\cos r - \\cos \\alpha}{\\cos r + \\cos \\alpha}"],["todhunter-spherical-trigonometry-1886/eq-ae226de395",16,"Todhunter 1886, scan 108: \\frac{\\cos r - \\cos \\alpha}{\\cos r + \\cos \\alpha} = \\tan \\frac{\\alpha + r}{2} \\tan \\frac{\\alpha - r}{2}"],["todhunter-spherical-trigonometry-1886/eq-cfe0db7585",16,"Todhunter 1886, scan 109: \\sin PM = \\sin OP \\sin AOB"],["todhunter-spherical-trigonometry-1886/eq-349dd6820f",16,"Todhunter 1886, scan 109: \\frac{\\sin PM}{\\sin PN} = \\frac{\\sin AOB}{\\sin COB}"],["todhunter-spherical-trigonometry-1886/eq-4b0def777d",16,"Todhunter 1886, scan 109: \\sin x = \\frac{\\sin \\theta_2}{\\sin(\\theta_1 + \\theta_2)} \\sin x_1 + \\frac{\\sin \\theta_1}{\\sin(\\theta_1 + \\theta_2)} \\sin"],["method/finding-south-from-a-watch",8,"finding south from a watch","../books/ball-mathematical-recreations-1905/terms/index.html#t-method-finding-south-from-a-watch"],["boyden-first-book-in-algebra-1895/eq-15eb597cb6",16,"Boyden 1895: 2 a b + 6 a c + 4 a d = 2 a \\left(b + 3 c + 2 d \\right)"],["todhunter-spherical-trigonometry-1886/eq-fb1c5f1a1d",16,"Todhunter 1886, scan 110: \\cos B = \\cos CF \\sin FCB"],["todhunter-spherical-trigonometry-1886/eq-661c96c233",16,"Todhunter 1886, scan 111: \\frac{\\sin x}{\\cos B \\cos C} = \\frac{\\sin y}{\\cos C \\cos A} = \\frac{\\sin z}{\\cos A \\cos B}"],["todhunter-spherical-trigonometry-1886/eq-ae89c15e6c",16,"Todhunter 1886, scan 111: \\frac{\\sin x}{\\sin B \\sin C} = \\frac{\\sin y}{\\sin C \\sin A} = \\frac{\\sin z}{\\sin A \\sin B}"],["concept/median",7,"median"],["todhunter-spherical-trigonometry-1886/eq-6da907218a",16,"Todhunter 1886, scan 112: \\cos(\\rho-r) = \\cos\\alpha\\cos\\beta + \\sin\\alpha\\sin\\beta\\cos\\gamma"],["todhunter-spherical-trigonometry-1886/eq-8f42d22be2",16,"Todhunter 1886, scan 112: \\cos(\\rho+r_1) = \\cos\\alpha_1\\cos\\beta + \\sin\\alpha_1\\sin\\beta\\cos\\gamma"],["ball-mathematical-recreations-1905/x-811e3172b3",15,"Ball 1905, scan 368: The time occupied by a given amount of some ..."],["boyden-first-book-in-algebra-1895/ex-36/24",4,"Boyden 1895, Exercise 36 (24)"],["todhunter-spherical-trigonometry-1886/eq-5216e40cfd",16,"Todhunter 1886, scan 112: \\cos(\\rho+r_2) = \\cos\\alpha_2\\cos\\beta + \\sin\\alpha_2\\sin\\beta \\cos\\left(\\frac{\\pi}{2}-\\gamma\\right)"],["todhunter-spherical-trigonometry-1886/eq-ca9cd2c245",16,"Todhunter 1886, scan 112: \\cos(\\rho+r_3) = \\cos\\alpha_3\\cos\\beta + \\sin\\alpha_3\\sin\\beta \\cos\\left(\\frac{\\pi}{2}+\\gamma\\right)"],["todhunter-spherical-trigonometry-1886/eq-4a37097765",16,"Todhunter 1886, scan 113: 2\\cos\\rho \\sin\\frac{a}{2}\\cos\\frac{b+c}{2} + 2n\\sin\\rho=\\cos\\beta\\sin a"],["todhunter-spherical-trigonometry-1886/eq-39ddfa0aba",16,"Todhunter 1886, scan 113: 2\\cos\\rho\\sin\\frac{a}{2}\\cos\\frac{b-c}{2}-2n\\sin\\rho=\\cos\\beta\\sin a"],["todhunter-spherical-trigonometry-1886/eq-ec40b21ace",16,"Todhunter 1886, scan 113: \\tan\\rho= \\frac{\\sin\\dfrac{a}{2}\\sin\\dfrac{b}{2}\\sin\\dfrac{c}{2}}{n} = \\frac{1}{2}\\tan R"],["ball-mathematical-recreations-1905/x-c283ac394a",15,"Ball 1905, scan 369: Most of these early clocks were regulated by horizontal ..."],["ball-mathematical-recreations-1905/x-dc76ba32f7",15,"Ball 1905, scan 371: so arranged that the cylinder descended with uniform velocity ..."],["todhunter-spherical-trigonometry-1886/eq-f514b47875",16,"Todhunter 1886, scan 113: \\cos\\beta= \\dfrac{\\cos\\dfrac{b}{2}\\cos\\dfrac{c}{2}\\cos\\rho}{\\cos\\dfrac{a}{2}}"],["todhunter-spherical-trigonometry-1886/eq-c78ecbf863",16,"Todhunter 1886, scan 117: \\tan\\frac{\\lambda}{2} \\tan\\frac{\\mu}{2} = \\frac{\\cos\\rho - \\cos\\beta}{\\cos\\rho + \\cos\\beta}"],["todhunter-spherical-trigonometry-1886/eq-6bf6267c57",16,"Todhunter 1886, scan 118: \\tan\\frac{\\lambda}{2} = \\frac{\\cos \\dfrac{a}{2} - \\cos \\dfrac{b}{2} \\cos \\dfrac{c}{2} } {\\cos \\dfrac{b}{2} \\sin \\dfrac{c"],["todhunter-spherical-trigonometry-1886/eq-963e982eec",16,"Todhunter 1886, scan 118: \\tan\\frac{\\mu}{2} = \\frac{\\cos \\dfrac{b}{2} \\sin \\dfrac{c}{2} } {\\cos \\dfrac{a}{2} + \\cos \\dfrac{b}{2} \\cos \\dfrac{c}{2}"],["todhunter-spherical-trigonometry-1886/eq-1476ffaf7f",16,"Todhunter 1886, scan 120: \\sin z = \\frac{\\cos \\rho}{n} \\sin\\frac{a}{2} \\sin\\frac{b}{2} \\sin\\frac{c}{2} \\cos(A - B)"],["boyden-first-book-in-algebra-1895/eq-9ff5c9ec8a",16,"Boyden 1895: 2am+2ax+bm+bx=(m+x)(2a+b)."],["de-morgan-elementary-illustrations-calculus-1899/eq-e95faec009",16,"De Morgan 1899, p. 78: du = (2xy + 2y^{3})\\, dx + \\etc."],["de-morgan-elementary-illustrations-calculus-1899/eq-f31ad677eb",16,"De Morgan 1899, p. 79: du = (x^{2} + 6xy^{2})\\, dy + \\etc."],["todhunter-spherical-trigonometry-1886/eq-2df09c5e30",16,"Todhunter 1886, scan 120: \\frac{\\sin x}{\\cos(B - C)} = \\frac{\\sin y}{\\cos(C - A)} = \\frac{\\sin z}{\\cos(A - B)}"],["todhunter-spherical-trigonometry-1886/eq-e95ecb044c",16,"Todhunter 1886, scan 121: z = \\dfrac12 R \\cos(A - B)"],["todhunter-spherical-trigonometry-1886/eq-770ab45613",16,"Todhunter 1886, scan 121: \\lambda = \\dfrac{b^2 + c^2 - a^2}{2c}"],["todhunter-spherical-trigonometry-1886/eq-09d63ce8ae",16,"Todhunter 1886, scan 121: \\mu = \\dfrac{c}{2}"],["todhunter-spherical-trigonometry-1886/eq-e72d80c6db",16,"Todhunter 1886, scan 118: \\sin z = \\sin \\beta \\sin \\left(\\frac{A}{2} + \\gamma\\right)"],["ball-mathematical-recreations-1905/x-25f2bb7297",15,"Ball 1905, scan 372: Both move round in the same direction, but the ..."],["ball-mathematical-recreations-1905/x-7baa90162f",15,"Ball 1905, scan 372: To do this it is sufficient to point the ..."],["boyden-first-book-in-algebra-1895/eq-31aeee1cab",16,"Boyden 1895: a^8+a^6-a^5-a^3+a^2+1=(a^2+1)(a^6-a^3+1)."],["form/5f2d1ab79e",5,"factor: 8*a**5 + 24*a**3*x + 18*a*x**2"],["de-morgan-elementary-illustrations-calculus-1899/eq-a350f39a3f",16,"De Morgan 1899, p. 79: du = (2xy + 2y^{3})\\, dx + (x^{2} + 6xy^{2})\\, dy + \\etc."],["de-morgan-elementary-illustrations-calculus-1899/eq-6d4ebb0a79",16,"De Morgan 1899, p. 81: \\dfrac{du}{dx}\\, dx = (2xy + 2y^{3})\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/eq-290a27a568",16,"De Morgan 1899, p. 81: \\frac{du}{dx} = 2xy + 2y^{3}"],["de-morgan-elementary-illustrations-calculus-1899/eq-ca2f22cd08",16,"De Morgan 1899, p. 81: \\dfrac{du}{dy}\\, dy = (x^{2} + 6xy^{2})\\, dy"],["de-morgan-elementary-illustrations-calculus-1899/eq-df8b8e80f3",16,"De Morgan 1899, p. 81: \\frac{du}{dy} = x^{2} + 6xy^{2}"],["de-morgan-elementary-illustrations-calculus-1899/eq-ebcb732414",16,"De Morgan 1899, p. 81: d.u = \\frac{du}{dx}\\, dx + \\frac{du}{dy}\\,dy"],["de-morgan-elementary-illustrations-calculus-1899/eq-66a9ed8fc6",16,"De Morgan 1899, p. 82: \\ux\\, dx + \\uy\\, dy + \\etc"],["hardy-course-of-pure-mathematics-1921/x-7b6baaf42b",15,"Hardy 1921, p. 445: The infinite of analysis is a ‘limiting’ and not ..."],["shape/c7b83c41fb",6,"factor: N*a*x**N + N*a**N*x + N*a**N"],["de-morgan-elementary-illustrations-calculus-1899/eq-375a3051d8",16,"De Morgan 1899, p. 83: dz = \\frac{dz}{dp}\\, dp + \\frac{dz}{dq}\\, dq + \\frac{dz}{dr}\\, dr + \\frac{dz}{ds}\\, ds + \\etc."],["hardy-course-of-pure-mathematics-1921/x-31127faa34",15,"Hardy 1921, p. 445: When (x, y, z) varies on the first line, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-005f3361f0",15,"De Morgan 1899, p. 60: If we suppose y to be any function of ..."],["de-morgan-elementary-illustrations-calculus-1899/x-3e6b912648",15,"De Morgan 1899, p. 60: If we diminish dt, the term \\dfrac{dx}{dt}\\, dx will ..."],["de-morgan-elementary-illustrations-calculus-1899/x-814afcca55",15,"De Morgan 1899, p. 60: The processes are the same in both methods, since ..."],["hardy-course-of-pure-mathematics-1921/x-229fd39a33",15,"Hardy 1921, p. 287: The number _a^b f(x) dx is called a integral; ..."],["hardy-course-of-pure-mathematics-1921/x-39c3a4dd26",15,"Hardy 1921, p. 288: It has however, for our present purpose, a fatal ..."],["hardy-course-of-pure-mathematics-1921/x-70b2aee08d",15,"Hardy 1921, p. 445: It is however no more than an illustration, and ..."],["hardy-course-of-pure-mathematics-1921/x-009ce096e9",15,"Hardy 1921, p. 287: And when we are considering a ‘definite integral’ we ..."],["hardy-course-of-pure-mathematics-1921/x-1930f35ad1",15,"Hardy 1921, p. 295: which is the formula for the transformation of a ..."],["hardy-course-of-pure-mathematics-1921/x-c4f80120ea",15,"Hardy 1921, p. 445: This correlation is historically important, for it is from ..."],["whitehead-introduction-to-mathematics-1911/x-e8718671da",15,"Whitehead 1911, p. 225: When x increases to x + h, the function ..."],["whitehead-introduction-to-mathematics-1911/x-f56181d402",15,"Whitehead 1911, p. 227: In reading over the Newtonian method of statement, it ..."],["whitehead-introduction-to-mathematics-1911/x-40ca4ae07b",15,"Whitehead 1911, p. 229: A function f(x) has the limit l at a ..."],["whitehead-introduction-to-mathematics-1911/x-cc425bc87f",15,"Whitehead 1911, p. 223: It is a well-founded historical generalization, that the last ..."],["hardy-course-of-pure-mathematics-1921/eq-bb6f6a7c21",16,"Hardy 1921, p. 262: f(b) - f(a) = (b - a) f'(\\xi)"],["hardy-course-of-pure-mathematics-1921/eq-1d94bdebf3",16,"Hardy 1921, p. 262: f(a + h) - f(a) = hf'(a + \\theta_{1} h)"],["de-morgan-elementary-illustrations-calculus-1899/eq-09c4a3b473",16,"De Morgan 1899, p. 84: \\dfrac{dz}{dp}\\, dp + \\dfrac{dz}{dq}\\, dq + \\dfrac{dz}{dr}\\, dr + \\dfrac{dz}{ds}\\, ds"],["concept/theorem-total-differential",7,"theorem: total differential"],["hardy-course-of-pure-mathematics-1921/eq-cad52e3821",16,"Hardy 1921, p. 263: f(a + h) = f(a) + hf'(a) + \\tfrac{1}{2}h^{2} f''(a + \\theta_{2}h)"],["hardy-course-of-pure-mathematics-1921/eq-449ff3d8ca",16,"Hardy 1921, p. 266: f(a + h) - S_{n} = R_{n}"],["hardy-course-of-pure-mathematics-1921/eq-8a0a2b6010",16,"Hardy 1921, p. 266: f(a + h) = \\lim_{n\\to\\infty} S_{n}"],["hardy-course-of-pure-mathematics-1921/eq-60ff671089",16,"Hardy 1921, p. 266: R_{n} = \\frac{h^{n}}{n!} f^{(n)}(a + \\theta_{n} h)"],["hardy-course-of-pure-mathematics-1921/eq-98db441875",16,"Hardy 1921, p. 266: f(h) = f(0) + hf'(0) + \\frac{h^{2}}{2!} f''(0) + \\dots"],["method/comparing-charges",8,"comparing charges","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-comparing-charges"],["hardy-course-of-pure-mathematics-1921/eq-a7b003e918",16,"Hardy 1921, p. 267: \\sin h = h - \\frac{h^{3}}{3!} + \\frac{h^{5}}{5!} - \\dots"],["hardy-course-of-pure-mathematics-1921/eq-9b36974cc6",16,"Hardy 1921, p. 265: x = \\xi - \\frac{f(\\xi)}{f'(\\xi)}"],["law/equal-and-opposite-inner-electrification",10,"equal and opposite inner electrification","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-law-equal-and-opposite-inner-electrification"],["hardy-course-of-pure-mathematics-1921/eq-34e3509ef0",16,"Hardy 1921, p. 272: y - f(\\xi) = (x - \\xi)f'(\\xi)"],["hardy-course-of-pure-mathematics-1921/eq-5051cb77e1",16,"Hardy 1921, p. 268: \\psi(x) = f(x)/\\phi(x)"],["hardy-course-of-pure-mathematics-1921/eq-b55d730a26",16,"Hardy 1921, p. 269: f(x)/\\phi(x) \\to f'(\\xi)/\\phi'(\\xi)"],["person/watt",1,"Watt","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-watt"],["person/hirn",1,"Hirn","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-hirn"],["planck-treatise-on-thermodynamics-1903/x-b430526c8f",15,"Planck 1903, p. 153: In the critical state the compressibility is infinite; so ..."],["planck-treatise-on-thermodynamics-1903/x-4bf41f0ef0",15,"Planck 1903, p. 155: these curves will meet in one point, the fundamental ..."],["planck-treatise-on-thermodynamics-1903/x-d6714cf392",15,"Planck 1903, p. 157: The existence of a sharp bend in the curve, ..."],["hardy-course-of-pure-mathematics-1921/eq-4c431ddcde",16,"Hardy 1921, p. 269: f(x)/\\phi(x) \\to f^{(p)}(\\xi)/\\phi^{(p)}(\\xi)"],["hardy-course-of-pure-mathematics-1921/eq-5416fc63cf",16,"Hardy 1921, p. 268: \\phi(\\xi + h) - \\phi(\\xi) = \\frac{h^{n}}{n!} \\phi^{(n)} (\\xi + \\theta_{n} h)"],["hardy-course-of-pure-mathematics-1921/eq-28916bf52e",16,"Hardy 1921, p. 271: \\lim \\frac{QR}{h^{2}} = \\tfrac{1}{2}\\{\\phi''(\\xi) - f''(\\xi)\\}"],["boyden-first-book-in-algebra-1895/eq-efe3242df1",16,"Boyden 1895: a^2-b^2=(a+b)(a-b)"],["boyden-first-book-in-algebra-1895/eq-f8c037cebf",16,"Boyden 1895: x^2 + 14x + 45 = (x + 9)(x + 5)"],["dickson-theory-of-equations-1922/eq-df8a4e28cd",16,"Dickson 1922, p. 101: a_1 b_2 - a_2 b_1"],["concept/method-solving-simultaneous-equations-by-determinants",7,"method: solving simultaneous equations by determinants"],["dickson-theory-of-equations-1922/eq-e9aa4680d4",16,"Dickson 1922, p. 102: x = \\frac{k_1 b_2 - k_2 b_1}{D}"],["concept/theorem-cramer-s-rule",7,"theorem: Cramer's rule"],["dickson-theory-of-equations-1922/eq-b971437ccd",16,"Dickson 1922, p. 102: y = \\frac{a_1 k_2 - a_2 k_1}{D}"],["dickson-theory-of-equations-1922/eq-ce9c5ced09",16,"Dickson 1922, p. 102: a_1 b_2 c_3 - a_1 b_3 c_2 + a_2 b_3 c_1 - a_2 b_1 c_3 + a_3 b_1 c_2 - a_3 b_2 c_1"],["dickson-theory-of-equations-1922/eq-daab0d8ccf",16,"Dickson 1922, p. 105: \\sum_{(24)} ± a_q b_r c_s d_t"],["hardy-course-of-pure-mathematics-1921/eq-9ee587c987",16,"Hardy 1921, p. 271: \\lim \\frac{QR}{h^{n}} = \\frac{1}{n!}\\{\\phi^{(n)}(\\xi) - f^{(n)}(\\xi)\\}"],["boyden-first-book-in-algebra-1895/eq-f0e694f3da",16,"Boyden 1895: x^2 - 6x + 5 = (x - 5)(x - 1)"],["dickson-theory-of-equations-1922/eq-497c76589d",16,"Dickson 1922, p. 106: (-1)^i e_{{i_1}1} e_{{i_2}2} \\dotsm e_{{i_n}n}"],["dickson-theory-of-equations-1922/eq-dbf1e7bd6c",16,"Dickson 1922, p. 110: D = e_{11}E_{11} - e_{21}E_{21} + e_{31}E_{31} - \\dotsb + (-1)^{n-1} e_{n1}E_{n1}"],["concept/expansion",7,"expansion","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-expansion"],["dickson-theory-of-equations-1922/eq-a354ca37fe",16,"Dickson 1922, p. 111: D = \\sum_{j=1}^n (-1)^{j+k} e_{jk} E_{jk}"],["dickson-theory-of-equations-1922/eq-5e61aafa7e",16,"Dickson 1922, p. 109: D = -a_2A_2 + b_2B_2 - c_2C_2"],["dickson-theory-of-equations-1922/eq-f135ad4b00",16,"Dickson 1922, p. 109: D = -b_1B_1 + b_2B_2 - b_3B_3"],["dickson-theory-of-equations-1922/eq-abfb6d9f72",16,"Dickson 1922, p. 105: (-1)^m P \\equiv (-1)^t P"],["concept/interchange",7,"interchange"],["concept/spherical-distance",7,"spherical distance","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-spherical-distance"],["concept/symmetry",7,"symmetry","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-symmetry"],["concept/polyhedron",7,"polyhedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-polyhedron"],["concept/regular-polyhedron",7,"regular polyhedron","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-regular-polyhedron"],["method/charging-a-vessel-with-multiples-of-a-given-charge",8,"charging a vessel with multiples of a given charge","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-charging-a-vessel-with-multiples-of-a-given-charge"],["concept/polyhedral-angle",7,"polyhedral angle","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-polyhedral-angle"],["theorem/volume-of-a-tetrahedron",9,"volume of a tetrahedron","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-volume-of-a-tetrahedron"],["todhunter-spherical-trigonometry-1886/x-b1ec984451",15,"Todhunter 1886, scan 146: Let P denote the pole of the inscribed circle, ..."],["todhunter-spherical-trigonometry-1886/x-675c2a3867",15,"Todhunter 1886, scan 150: then the spherical triangle which corresponds to the three ..."],["todhunter-spherical-trigonometry-1886/x-d3abf4f9d0",15,"Todhunter 1886, scan 150: Again, we know in mechanics that if three forces ..."],["todhunter-spherical-trigonometry-1886/x-8b0b466615",15,"Todhunter 1886, scan 151: The result, however, is generally true, even in cases ..."],["todhunter-spherical-trigonometry-1886/x-b595b4f71e",15,"Todhunter 1886, scan 151: We begin with a theorem which is due to ..."],["todhunter-spherical-trigonometry-1886/x-65746fd40b",15,"Todhunter 1886, scan 150: Hence it may be inferred that any change of ..."],["hardy-course-of-pure-mathematics-1921/eq-728dc254aa",16,"Hardy 1921, p. 273: r = \\frac{(1 + \\eta_{1}^{2})^{3/2}}{\\eta_{2}}"],["hardy-course-of-pure-mathematics-1921/eq-ba49ff925c",16,"Hardy 1921, p. 273: (x - a)^{2} + (y - b)^{2} = r^{2}"],["hardy-course-of-pure-mathematics-1921/eq-1fbd60b077",16,"Hardy 1921, p. 266: f(x) = a_{0} + a_{1}x + a_{2}x^{2} + \\dots + (a_{n} + \\epsilon_{x}) x^{n}"],["hardy-course-of-pure-mathematics-1921/eq-e75faa969a",16,"Hardy 1921, p. 266: a_{r} = f^{(r)}(0)/r!"],["law/equal-production-of-positive-and-negative-electricity",10,"equal production of positive and negative electricity","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-law-equal-production-of-positive-and-negative-electricity"],["hardy-course-of-pure-mathematics-1921/x-2124d88208",15,"Hardy 1921, p. 445: The confusion about these matters so prevalent among students ..."],["hardy-course-of-pure-mathematics-1921/eq-57b2a49afd",16,"Hardy 1921, p. 270: (\\sin x\\arcsin x - x^{2})/x^{6} \\to \\frac{1}{18}"],["hardy-course-of-pure-mathematics-1921/eq-e50e76b6f2",16,"Hardy 1921, p. 270: \\lim_{x \\to n} (x - n)\\cosec x\\pi = \\frac{(-1)^{n}}{\\pi}"],["law/charge-on-a-closed-conducting-surface",10,"charge on a closed conducting surface","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-law-charge-on-a-closed-conducting-surface"],["concept/closed-conducting-surface",7,"closed conducting surface","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-closed-conducting-surface"],["boyden-first-book-in-algebra-1895/ex-37/27",4,"Boyden 1895, Exercise 37 (27)"],["boyden-first-book-in-algebra-1895/ex-32/18",4,"Boyden 1895, Exercise 32 (18)"],["concept/comparison",7,"comparison","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-comparison"],["concept/hypothesis",7,"hypothesis","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-hypothesis"],["concept/negative-electrification",7,"negative electrification","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-negative-electrification"],["hardy-course-of-pure-mathematics-1921/eq-6fc81f793d",16,"Hardy 1921, p. 346: \\lim a_{n}z_{1}^{n} = 0"],["hardy-course-of-pure-mathematics-1921/eq-233bd69fb7",16,"Hardy 1921, p. 346: |a_{n}z^{n}| = |a_{n}z_{1}^{n}| \\left(\\frac{r}{r_{1}}\\right)^{n} < K \\left(\\frac{r}{r_{1}}\\right)^{n}"],["hardy-course-of-pure-mathematics-1921/eq-12b1ed631e",16,"Hardy 1921, p. 348: z = \\cos\\theta + i\\sin\\theta"],["form/35fd00b011",5,"factor: 3*a*x**4 - 39*a*x**2 + 108*a"],["hardy-course-of-pure-mathematics-1921/eq-c6213349b7",16,"Hardy 1921, p. 349: 1/(1 - z)^{2} = 1 + 2z + 3z^{2} + \\dots"],["theorem/binomial-theorem-for-a-negative-integral-exponent",9,"binomial theorem for a negative integral exponent","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-binomial-theorem-for-a-negative-integral-exponent"],["hardy-course-of-pure-mathematics-1921/eq-aa5bff62cf",16,"Hardy 1921, p. 348: \\lim |a_{n+1}z^{n+1}|/|a_{n}z^{n}| = \\lambda |z|"],["hardy-course-of-pure-mathematics-1921/x-9c84b8c1e7",15,"Hardy 1921, p. 420: The function is discontinuous for \\theta = (2k + ..."],["hardy-course-of-pure-mathematics-1921/eq-5b25e7fbcd",16,"Hardy 1921, p. 348: \\frac{|a_{n+1}|}{|a_{n}|} = \\frac{|m - n|}{n + 1} \\to 1"],["hardy-course-of-pure-mathematics-1921/eq-b4aa3963e4",16,"Hardy 1921, p. 349: \\frac{1}{(1 - z)^{m}} = 1 + mz + \\frac{m(m + 1)}{1·2} z^{2} + \\dots"],["hardy-course-of-pure-mathematics-1921/eq-2cb05e003f",16,"Hardy 1921, p. 349: f(m, z) = 1 + \\binom{m}{1} z + \\binom{m}{2} z^{2} + \\dots"],["boyden-first-book-in-algebra-1895/ex-37/29",4,"Boyden 1895, Exercise 37 (29)"],["instrument/electrophorus",13,"electrophorus","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-electrophorus"],["concept/corollary",7,"corollary","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-corollary"],["shape/b3137b9794",6,"factor: 2*N*a*x**N + N*a"],["boyden-first-book-in-algebra-1895/ex-37/28",4,"Boyden 1895, Exercise 37 (28)"],["form/892ea744ea",5,"factor: 12*a**2*x - 12*a*x**2 + 3*x**3"],["shape/d75f0b4d0a",6,"factor: N*a*x**N + N*a**N*x + N*x**N"],["boyden-first-book-in-algebra-1895/ex-37/30",4,"Boyden 1895, Exercise 37 (30)"],["shape/06c22d37fc",6,"factor: N*a*x + N*a + N*b*x + N*b + a*x**N + b*x**N"],["hardy-course-of-pure-mathematics-1921/eq-5948eabcd0",16,"Hardy 1921, p. 349: f(m, z)f(m', z) = f(m + m', z)"],["hardy-course-of-pure-mathematics-1921/eq-21408658da",16,"Hardy 1921, p. 350: f(z)f(z') = f(z + z')"],["person/george-stokes",1,"George Stokes","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-george-stokes"],["boyden-first-book-in-algebra-1895/ex-14/9",4,"Boyden 1895, Exercise 14 (9)"],["hardy-course-of-pure-mathematics-1921/eq-0dda8fc2a1",16,"Hardy 1921, p. 350: C(z) = 1 - \\frac{z^{2}}{2!} + \\frac{z^{4}}{4!} - \\dots"],["hardy-course-of-pure-mathematics-1921/eq-24514bc0e3",16,"Hardy 1921, p. 350: S(z) = z - \\frac{z^{3}}{3!} + \\frac{z^{5}}{5!} - \\dots"],["boyden-first-book-in-algebra-1895/ex-14/10",4,"Boyden 1895, Exercise 14 (10)"],["boyden-first-book-in-algebra-1895/ex-14/11",4,"Boyden 1895, Exercise 14 (11)"],["person/peter-guthrie-tait",1,"Peter Guthrie Tait","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-peter-guthrie-tait"],["form/94b508c4f8",5,"evaluate: 3*a + 5*b - 2*c + 6*d at a=1, b=2, c=3, x=0"],["person/joseph-john-thomson",1,"Joseph John Thomson","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-joseph-john-thomson"],["person/georges-louis-le-sage",1,"Georges-Louis Le Sage","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-georges-louis-le-sage"],["person/osborne-reynolds",1,"Osborne Reynolds","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-osborne-reynolds"],["person/karl-pearson",1,"Karl Pearson","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-karl-pearson"],["person/henri-becquerel",1,"Henri Becquerel","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-henri-becquerel"],["person/wilhelm-conrad-r-ntgen",1,"Wilhelm Conrad Röntgen","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-wilhelm-conrad-r-ntgen"],["boyden-first-book-in-algebra-1895/ex-14/12",4,"Boyden 1895, Exercise 14 (12)"],["shape/e5563842d5",6,"evaluate: N*a + N*b + N*c + N*d"],["form/18c2dfd636",5,"evaluate: -5*a*c - 3*a*d + 6*b*c + 2*b*d + 2*c*d at a=1, b=2, c=3, x=0"],["shape/3a1cead0fd",6,"evaluate: N*a*c + N*a*d + N*b*c + N*b*d + N*c*d"],["de-morgan-elementary-illustrations-calculus-1899/eq-a7cf54886f",16,"De Morgan 1899, p. 86: dy = mx^{m-1}\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/eq-a2fc102871",16,"De Morgan 1899, p. 86: dy = -\\dfrac{m\\, dx}{x^{m+1}}"],["de-morgan-elementary-illustrations-calculus-1899/eq-2c70f078fa",16,"De Morgan 1899, p. 86: dy = -mx^{-m-1}\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/eq-6f949df9af",16,"De Morgan 1899, p. 86: dy = a^{x}\\log a\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/eq-221f780fb1",16,"De Morgan 1899, p. 86: dy = e^{x}\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/eq-b2c37e18de",16,"De Morgan 1899, p. 86: a = 2.7182818 = e"],["de-morgan-elementary-illustrations-calculus-1899/eq-ae66fbe86f",16,"De Morgan 1899, p. 86: dy = \\dfrac{dx}{x}"],["planck-treatise-on-thermodynamics-1903/eq-e80287e960",16,"Planck 1903, p. 153: p_{1} = p_{2} = p_{3}\\Add{,}"],["planck-treatise-on-thermodynamics-1903/eq-1553950291",16,"Planck 1903, p. 153: \\phi_{1} - \\phi_{2} = \\frac{u_{1} - u_{2} + p_{1}(v_{1} - v_{2})}{\\theta}\\Add{,}"],["de-morgan-elementary-illustrations-calculus-1899/eq-0c55da9012",16,"De Morgan 1899, p. 86: dy = -.4342944\\, \\dfrac{dx}{x}"],["de-morgan-elementary-illustrations-calculus-1899/eq-8b36f5e495",16,"De Morgan 1899, p. 86: dy = \\cos x\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/eq-54a8a3b655",16,"De Morgan 1899, p. 86: dy = -\\sin x\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/eq-1322ef33cf",16,"De Morgan 1899, p. 86: dy = \\dfrac{dx}{\\cos^{2} x}"],["ball-mathematical-recreations-1905/x-de0c78fa19",15,"Ball 1905, scan 380: If an observer lived in two dimensional space filled ..."],["theorem/greater-angle-lies-opposite-greater-side-in-a-triangle",9,"greater angle lies opposite greater side in a triangle","../books/whitehead-introduction-to-mathematics-1911/terms/index.html#t-theorem-greater-angle-lies-opposite-greater-side-in-a-triangle"],["theorem/sum-of-the-angles-of-a-triangle",9,"sum of the angles of a triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-theorem-sum-of-the-angles-of-a-triangle"],["theorem/sum-of-two-sides-of-a-triangle-exceeds-the-third",9,"sum of two sides of a triangle exceeds the third","../books/wentworth-plane-geometry-1899/terms/index.html#t-theorem-sum-of-two-sides-of-a-triangle-exceeds-the-third"],["hardy-course-of-pure-mathematics-1921/eq-20a8bed8b7",16,"Hardy 1921, p. 350: C(z + z') = C(z)C(z') - S(z)S(z')"],["ball-mathematical-recreations-1905/x-590fd0d345",15,"Ball 1905, scan 378: The tendency of the particles forming a ring to ..."],["whitehead-introduction-to-mathematics-1911/x-9ba46eba54",15,"Whitehead 1911, p. 236: But we do not need to consider who is ..."],["whitehead-introduction-to-mathematics-1911/x-5d223a0d9e",15,"Whitehead 1911, p. 238: Also there is the still simpler correlation between the ..."],["whitehead-introduction-to-mathematics-1911/x-ebbee8947f",15,"Whitehead 1911, p. 240: It is as though the great science of Anthropology ..."],["whitehead-introduction-to-mathematics-1911/x-ad678ce0a3",15,"Whitehead 1911, p. 242: The peculiarity of geometry is the fixity and overwhelming ..."],["whitehead-introduction-to-mathematics-1911/x-2bd7053c9d",15,"Whitehead 1911, p. 243: The abstract logical form of the propositions when fully ..."],["whitehead-introduction-to-mathematics-1911/x-515371b2fa",15,"Whitehead 1911, p. 241: The number of the archangels can be counted just ..."],["de-morgan-elementary-illustrations-calculus-1899/x-1d77e70501",15,"De Morgan 1899, p. 65: And it must be observed that 6t is the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-4fabc915b5",15,"De Morgan 1899, p. 64: But as the terms involving (dt)^{2} in the velocities, ..."],["ball-mathematical-recreations-1905/x-5f7ad95733",15,"Ball 1905, scan 387: A body by itself in space would receive on ..."],["ball-mathematical-recreations-1905/x-78c815093c",15,"Ball 1905, scan 376: It is essential to the theory that the atom ..."],["hardy-course-of-pure-mathematics-1921/eq-4de0c56ec2",16,"Hardy 1921, p. 350: S(z + z') = S(z)C(z') + C(z)S(z')"],["maxwell-elementary-treatise-electricity-1888/x-8ae2582822",15,"Maxwell 1888, scan 32: This statement, which is approximately true for any deep ..."],["de-morgan-elementary-illustrations-calculus-1899/eq-4b519361f6",16,"De Morgan 1899, p. 87: .4342944 \\left(\\frac{dx}{x} - \\tfrac{1}{2}\\, \\frac{(dx)^{2}}{x^{2}} + \\tfrac{1}{3}\\, \\frac{(dx)^{3}}{x^{3}} - \\etc.\\righ"],["de-morgan-elementary-illustrations-calculus-1899/eq-85808bb1f1",16,"De Morgan 1899, p. 87: y = \\sin x"],["de-morgan-elementary-illustrations-calculus-1899/eq-9ede6b63ec",16,"De Morgan 1899, p. 87: \\cos x\\, dx - \\frac{1}{2}\\sin x\\, (dx)^{2} - \\etc."],["ball-mathematical-recreations-1905/x-1c3e7a5cc1",15,"Ball 1905, scan 378: This is true as far as it goes, but ..."],["ball-mathematical-recreations-1905/x-de1ce17d62",15,"Ball 1905, scan 379: Sir William Thomson gave the number of vibrations per ..."],["concept/limiting-case",7,"limiting case","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-limiting-case"],["maxwell-elementary-treatise-electricity-1888/x-e808f1945d",15,"Maxwell 1888, scan 33: Their electrification however will be found to be of ..."],["boyden-first-book-in-algebra-1895/ex-14/13",4,"Boyden 1895, Exercise 14 (13)"],["hardy-course-of-pure-mathematics-1921/eq-3a4f152e7a",16,"Hardy 1921, p. 350: \\{C(z)\\}^{2} + \\{S(z)\\}^{2} = 1"],["hardy-course-of-pure-mathematics-1921/eq-57332ed8e5",16,"Hardy 1921, p. 350: u_{n} = v_{n} = \\frac{(-1)^{n}}{\\sqrtp{n + 1}}"],["hardy-course-of-pure-mathematics-1921/eq-85d8dfda6c",16,"Hardy 1921, p. 350: w_{n} = (-1)^{n} \\sum_{r=0}^{n} \\frac{1}{\\sqrtb{(r + 1)(n + 1 - r)}}"],["boyden-first-book-in-algebra-1895/eq-0b11f67dc0",16,"Boyden 1895: x^2 + 2x - 3 = (x + 3)(x - 1)"],["boyden-first-book-in-algebra-1895/eq-aaa1990315",16,"Boyden 1895: x^2 - 8x - 20 = (x - 10)(x + 2)"],["form/49c60bb2b7",5,"evaluate: a*b*c - 7*a*b*e + 5*a*c*e + 3*b*c*d at a=1, b=2, c=3, d=4, x=0"],["planck-treatise-on-thermodynamics-1903/x-fa8b56442e",15,"Planck 1903, p. 172: A question which may also be answered directly is ..."],["planck-treatise-on-thermodynamics-1903/x-2cf607f4a1",15,"Planck 1903, p. 171: It will be seen that their ratio is that ..."],["planck-treatise-on-thermodynamics-1903/x-5b615b0880",15,"Planck 1903, p. 170: This quantity is essentially positive, since M_{12}, M_{21}, as ..."],["method/solving-an-oblique-spherical-triangle",8,"solving an oblique spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-solving-an-oblique-spherical-triangle"],["method/solving-a-right-spherical-triangle",8,"solving a right spherical triangle","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-method-solving-a-right-spherical-triangle"],["todhunter-spherical-trigonometry-1886/x-ecb7519ed7",15,"Todhunter 1886, scan 166: Thus by taking only the nearest number of seconds ..."],["todhunter-spherical-trigonometry-1886/x-d8317ddd78",15,"Todhunter 1886, scan 168: The student can obtain more examples, which can be ..."],["hardy-course-of-pure-mathematics-1921/eq-722c219ea4",16,"Hardy 1921, p. 349: \\sum u_{n} × \\sum v_{n} = \\sum w_{n}"],["concept/dominant-term",7,"dominant term","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-dominant-term"],["de-morgan-elementary-illustrations-calculus-1899/x-6ad0cd8e6f",15,"De Morgan 1899, p. 66: The limit of any ratio may be found by ..."],["de-morgan-elementary-illustrations-calculus-1899/x-bc7a92ef4c",15,"De Morgan 1899, p. 66: Divide both numerator and denominator by x^{2}, which gives ..."],["de-morgan-elementary-illustrations-calculus-1899/x-05de4723b7",15,"De Morgan 1899, p. 66: It is easy to show that the increase of ..."],["shape/781b3ce112",6,"evaluate: N*a*b*e + N*a*c*e + N*b*c*d + a*b*c"],["de-morgan-elementary-illustrations-calculus-1899/x-d259aa955c",15,"De Morgan 1899, p. 66: We will now prove the following: That in any ..."],["de-morgan-elementary-illustrations-calculus-1899/x-fc62278ae9",15,"De Morgan 1899, p. 73: This result will be of use when we come ..."],["ball-mathematical-recreations-1905/x-37e84257ba",15,"Ball 1905, scan 389: I should sum up the effect of this discussion ..."],["hardy-course-of-pure-mathematics-1921/eq-510ee9f380",16,"Hardy 1921, p. 349: c_{n} = a_{0}b_{n} + a_{1}b_{n-1} + \\dots + a_{n}b_{0}"],["hardy-course-of-pure-mathematics-1921/eq-8e3f01e948",16,"Hardy 1921, p. 349: f(z)/(1 - z) = \\sum s_{n}z^{n}"],["hardy-course-of-pure-mathematics-1921/eq-7edee13785",16,"Hardy 1921, p. 350: \\sum_{1}^{\\infty} \\frac{n^{2} + 9n + 5}{(n + 1)(2n + 3)(2n + 5)(n + 4)} = \\frac{5}{36}"],["hardy-course-of-pure-mathematics-1921/eq-d6bd18ee9e",16,"Hardy 1921, p. 352: u_{n} = \\frac{x^{n} - x^{-n-1}}{(x^{n} + x^{-n})(x^{n+1} + x^{-n-1}) }"],["boyden-first-book-in-algebra-1895/eq-6b7c844c53",16,"Boyden 1895: 3ac+3bc & = & 3c(a+b)"],["planck-treatise-on-thermodynamics-1903/eq-4c9492ab70",16,"Planck 1903, p. 149: \\left(\\frac{\\dd v_{1}}{\\dd \\theta}\\right)_{p} = \\frac{R}{mp_{1}}"],["planck-treatise-on-thermodynamics-1903/eq-f69d70f052",16,"Planck 1903, p. 149: (c_{p})_{1} - (c_{p})_{2} = \\frac{dL}{d\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-862a1aa452",16,"Planck 1903, p. 149: \\frac{dL}{d\\theta} = (c_{p})_{1} - (c_{p})_{2} + \\frac{L}{\\theta} - \\frac{L}{v_{1} - v_{2}} \\left[\\left(\\frac{\\dd v_{1}}"],["planck-treatise-on-thermodynamics-1903/eq-16f627a1ce",16,"Planck 1903, p. 150: c = \\frac{du}{d\\theta} + p\\, \\frac{dv}{d\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-be4970fddc",16,"Planck 1903, p. 150: h_{1} = \\frac{du_{1}}{d\\theta} + p_{1}\\, \\frac{dv_{1}}{d\\theta}"],["wentworth-first-steps-in-algebra-1894/ex-22/1",4,"Wentworth 1894, Exercise 22 (1)"],["todhunter-spherical-trigonometry-1886/eq-79e7ded3c6",16,"Todhunter 1886, scan 124: \\mathrm{S+F=E+2}"],["theorem/euler-s-polyhedral-formula",9,"Euler's polyhedral formula"],["todhunter-spherical-trigonometry-1886/eq-0fd343d679",16,"Todhunter 1886, scan 124: r^2\\{s-(m-2)\\pi\\}"],["planck-treatise-on-thermodynamics-1903/eq-96bf2f9e25",16,"Planck 1903, p. 151: h_{2} = \\frac{du_{2}}{d\\theta} + p_{2}\\, \\frac{dv_{2}}{d\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-1b759eb264",16,"Planck 1903, p. 151: h_{2} = (c_{p})_{2}"],["planck-treatise-on-thermodynamics-1903/eq-63c03e0514",16,"Planck 1903, p. 151: h_{1} = (c_{p})_{2} + \\frac{dL}{d\\theta} - \\frac{L}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-7e48c39cf6",16,"Planck 1903, p. 153: \\left(\\frac{\\dd p}{\\dd v}\\right)_{2} = 0"],["planck-treatise-on-thermodynamics-1903/eq-51eabeb049",16,"Planck 1903, p. 153: \\left(\\frac{\\dd^{2} p}{\\dd v^{2}}\\right)_{2} = 0"],["planck-treatise-on-thermodynamics-1903/eq-58bcda4bf2",16,"Planck 1903, p. 152: p = p_{2} + \\left(\\frac{\\dd p}{\\dd v}\\right)_{2} (v - v_{2}) + \\tfrac{1}{2} \\left(\\frac{\\dd^{2} p}{\\dd v^{2}}\\right)_{2}"],["theorem/taylor-expansion-of-pressure-about-the-state-2",9,"Taylor expansion of pressure about the state 2"],["maxwell-elementary-treatise-electricity-1888/x-3789e32da9",15,"Maxwell 1888, scan 34: When an electrified body is placed within a closed ..."],["form/d5f1ab61c3",5,"identity: (x + 6)*(x + 7)"],["shape/e7b00e2b4c",6,"identity: (N + x)**2"],["planck-treatise-on-thermodynamics-1903/eq-05b493a4fc",16,"Planck 1903, p. 153: \\phi_{2} - \\phi_{3} = \\frac{u_{2} - u_{3} + p_{1}(v_{2} - v_{3})}{\\theta}\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-c0e9d20b7c",16,"Planck 1903, p. 154: M_{1} + (M_{2} + M_{3}) = M\\Add{,}"],["planck-treatise-on-thermodynamics-1903/eq-e797c03eda",16,"Planck 1903, p. 154: M_{1} v_{1} + M_{2} v_{2} + M_{3} v_{3} = V\\Add{,}"],["planck-treatise-on-thermodynamics-1903/eq-7537f40422",16,"Planck 1903, p. 154: M_{1} u_{1} + M_{2} u_{2} + M_{3} u_{3} = U\\Add{,}"],["planck-treatise-on-thermodynamics-1903/eq-9d5123db3f",16,"Planck 1903, p. 155: p_{12} = p_{21}"],["planck-treatise-on-thermodynamics-1903/eq-410dc688b1",16,"Planck 1903, p. 156: \\frac{dp_{12}}{d\\theta} = \\frac{L_{12}}{\\theta (v_{1} - v_{2})}"],["law/clapeyron-equation-vaporization",10,"Clapeyron equation (vaporization)"],["hardy-course-of-pure-mathematics-1921/eq-820aca6aa2",16,"Hardy 1921, p. 352: \\frac{x^{n} - x^{-n-1}}{(x^{n} + x^{-n})(x^{n+1} + x^{-n-1}) } = \\frac{1}{x - 1} \\left(\\frac{1}{x^{n} + x^{-n}} - \\frac{"],["hardy-course-of-pure-mathematics-1921/eq-108ffc8534",16,"Hardy 1921, p. 352: a_{n} + p_{1}a_{n-1} + p_{2}a_{n-2} + \\dots + p_{k}a_{n-k} = 0"],["planck-treatise-on-thermodynamics-1903/eq-7e7c68c575",16,"Planck 1903, p. 156: \\frac{dp_{23}}{d\\theta} = \\frac{L_{23}}{\\theta (v_{2} - v_{3})}"],["law/clapeyron-equation-fusion",10,"Clapeyron equation (fusion)"],["planck-treatise-on-thermodynamics-1903/eq-af02b771c3",16,"Planck 1903, p. 156: \\frac{dp_{31}}{d\\theta} = \\frac{L_{31}}{\\theta (v_{3} - v_{1})}"],["law/clapeyron-equation-sublimation",10,"Clapeyron equation (sublimation)"],["planck-treatise-on-thermodynamics-1903/eq-37da518888",16,"Planck 1903, p. 157: u = \\dfrac{U}{M}"],["planck-treatise-on-thermodynamics-1903/eq-207e4ca161",16,"Planck 1903, p. 162: \\Phi = M\\phi"],["planck-treatise-on-thermodynamics-1903/eq-9a264085fa",16,"Planck 1903, p. 162: \\Phi' = M\\phi' = M_{12} \\phi_{12} + M_{21} \\phi_{21}"],["concept/matter",7,"matter","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-matter"],["planck-treatise-on-thermodynamics-1903/eq-38120d1693",16,"Planck 1903, p. 162: \\Phi'' = M\\phi'' = M_{1} \\phi_{1} + M_{2} \\phi_{2} + M_{3} \\phi_{3}\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-31c9571ea4",16,"Planck 1903, p. 163: \\phi'' > \\phi' > \\phi"],["planck-treatise-on-thermodynamics-1903/eq-cc53aa4623",16,"Planck 1903, p. 159: \\phi_{12} - \\phi_{21} = \\frac{u_{12} - u_{21} + p_{12} (v_{12} - v_{21})}{\\theta_{12}}"],["planck-treatise-on-thermodynamics-1903/eq-44bd0fd06a",16,"Planck 1903, p. 160: \\frac{du_{12}}{dv_{12}} = \\left(\\frac{\\dd u}{\\dd v}\\right)_{12} + \\left(\\frac{\\dd u}{\\dd \\theta}\\right)_{12} \\frac{d\\the"],["planck-treatise-on-thermodynamics-1903/eq-e82a962627",16,"Planck 1903, p. 160: \\frac{du_{12}}{dv_{12}} = \\theta_{12} \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{12} - p_{12} + (c_{v})_{12}\\, \\frac{d\\theta"],["todhunter-spherical-trigonometry-1886/eq-52f362833c",16,"Todhunter 1886, scan 124: 4\\pi r^2"],["todhunter-spherical-trigonometry-1886/eq-5eac6c4aae",16,"Todhunter 1886, scan 126: 2(S-2)\\pi"],["planck-treatise-on-thermodynamics-1903/eq-5482c8baa1",16,"Planck 1903, p. 175: \\Psi = \\Psi' + \\Psi'' + \\dots +\\Psi^{\\beta}"],["planck-treatise-on-thermodynamics-1903/eq-f9d0f5989c",16,"Planck 1903, p. 160: \\frac{du_{21}}{dv_{21}} = \\theta_{21} \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{21} - p_{12} + (c_{v})_{21}\\, \\frac{d\\theta"],["planck-treatise-on-thermodynamics-1903/eq-f5f66c9c80",16,"Planck 1903, p. 167: \\frac{u_{12} - u_{21}}{v_{12} - v_{21}} = \\theta_{12}\\, \\frac{dp_{12}}{d\\theta_{12}} - p_{12}"],["planck-treatise-on-thermodynamics-1903/eq-d2969e0824",16,"Planck 1903, p. 167: \\frac{\\dd u}{\\dd v} = \\theta\\, \\frac{dp}{d\\theta} - p"],["planck-treatise-on-thermodynamics-1903/eq-cbf43437fe",16,"Planck 1903, p. 167: \\frac{dp_{12}}{d\\theta_{12}} = \\frac{\\dd p}{\\dd \\theta} + \\frac{\\dd p}{\\dd v} · \\frac{dv_{12}}{d\\theta_{12}}"],["planck-treatise-on-thermodynamics-1903/eq-cc4f77f7f5",16,"Planck 1903, p. 164: p_{12} (v - v_{12}) + (u - u_{12}) - \\theta_{12} (\\phi - \\phi_{12}) = 0"],["todhunter-spherical-trigonometry-1886/eq-fd85cd3a6f",16,"Todhunter 1886, scan 127: r = R\\cos{aOc} = R\\cot{eca}\\cot{eac} = R\\cot{\\frac{\\pi}{m}} \\cot{\\frac{\\pi}{n}}"],["todhunter-spherical-trigonometry-1886/eq-308ea681d3",16,"Todhunter 1886, scan 144: P = Q = R = \\dfrac{S}{3}"],["planck-treatise-on-thermodynamics-1903/eq-8eab142a41",16,"Planck 1903, p. 164: v = \\frac{\\lambda v_{12} + \\mu v_{21}}{\\lambda + \\mu}"],["planck-treatise-on-thermodynamics-1903/eq-020abfefa5",16,"Planck 1903, p. 165: \\delta \\phi' = \\frac{\\delta u + p_{12}\\, \\delta v}{\\theta_{12}}"],["planck-treatise-on-thermodynamics-1903/eq-bb15d0ce16",16,"Planck 1903, p. 165: \\delta (\\phi' - \\phi) = \\left(\\frac{1}{\\theta_{12}} - \\frac{1}{\\theta}\\right) \\delta u + \\left(\\frac{p_{12}}{\\theta_{12}"],["planck-treatise-on-thermodynamics-1903/eq-e840de8f24",16,"Planck 1903, p. 166: \\theta\\, \\delta^{2} (\\phi' - \\phi) = (\\delta \\theta - \\delta \\theta_{12})\\, \\delta \\phi + (\\delta p_{12} - \\delta p)\\, \\"],["planck-treatise-on-thermodynamics-1903/eq-eb59fef41f",16,"Planck 1903, p. 166: \\delta \\phi = \\frac{c_{v}}{\\theta}\\, \\delta \\theta + \\frac{\\dd p}{\\dd \\theta}\\, \\delta v"],["boyden-first-book-in-algebra-1895/eq-4852306701",16,"Boyden 1895: 6a^2x+12abx+6b^2x & = & 6x(a+b)(a+b)"],["de-morgan-elementary-illustrations-calculus-1899/ch-total-and-partial-differential-coefficients-implicit-differentiation",2,"De Morgan 1899, Total and Partial Differential Coefficients. Implicit Differentiation","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-total-and-partial-differential-coefficients-implicit-differentiation/index.html"],["de-morgan-elementary-illustrations-calculus-1899/eq-ed44f1d834",16,"De Morgan 1899, p. 95: d.z = \\frac{dz}{dx}\\, dx + \\frac{dz}{dy}\\, p\\, dx"],["planck-treatise-on-thermodynamics-1903/eq-aa13c73477",16,"Planck 1903, p. 166: \\delta p = \\frac{\\dd p}{\\dd \\theta}\\, \\delta \\theta + \\frac{\\dd p}{\\dd v}\\, \\delta v"],["planck-treatise-on-thermodynamics-1903/eq-7216db55c2",16,"Planck 1903, p. 167: \\delta \\theta_{12} = \\frac{c_{v}\\, \\delta \\theta - \\theta\\, \\dfrac{\\dd p}{\\dd v} · \\dfrac{dv_{12}}{d\\theta_{12}}\\, \\delt"],["planck-treatise-on-thermodynamics-1903/eq-c35c28f7dc",16,"Planck 1903, p. 167: \\delta^{2} (\\phi' - \\phi) = -\\frac{\\dd p}{\\dd v} · \\frac{c_{v}}{\\theta} · \\frac{\\left(\\dfrac{dv_{12}}{d\\theta_{12}}\\, \\d"],["concept/molecule",7,"molecule","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-molecule"],["dickson-theory-of-equations-1922/eq-b55c9de054",16,"Dickson 1922, p. 114: a_{11} x_1 + a_{12} x_2 + \\dotsb + a_{1n} x_n = k_1"],["concept/system-of-linear-equations",7,"system of linear equations"],["dickson-theory-of-equations-1922/eq-050cc17f0e",16,"Dickson 1922, p. 114: D = \\begin{vmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ \\Dots{4} \\\\ a_{n1} & a_{n2} & \\cdots & a_{nn} \\end{vmatrix}"],["dickson-theory-of-equations-1922/eq-c14e78caa6",16,"Dickson 1922, p. 114: Dx_1 = K_1,\\qquad Dx_2 = K_2,\\qquad \\dotsc,\\qquad Dx_n = K_n"],["dickson-theory-of-equations-1922/eq-91408a1efd",16,"Dickson 1922, p. 114: K_1 = \\begin{vmatrix} k_1 & a_{12} & \\cdots & a_{1n} \\\\ \\Dots{4} \\\\ k_n & a_{n2} & \\cdots & a_{nn} \\end{vmatrix}"],["dickson-theory-of-equations-1922/eq-c62045401c",16,"Dickson 1922, p. 118: L_i \\equiv a_{i1} x_1 + a_{i2} x_2 + \\dotsb + a_{in} x_n - k_i"],["dickson-theory-of-equations-1922/eq-cb38a63ed9",16,"Dickson 1922, p. 118: K = \\begin{vmatrix} a_{11} & \\cdots & a_{1r} & k_1 \\\\ \\Dots{4} \\\\ a_{r+11} & \\cdots & a_{r+1r} & k_{r+1} \\end{vmatrix}"],["dickson-theory-of-equations-1922/eq-2a32685556",16,"Dickson 1922, p. 118: 0 = ±K"],["concept/linear-combination",7,"linear combination"],["hardy-course-of-pure-mathematics-1921/eq-a723c7ae56",16,"Hardy 1921, p. 352: (1 + p_{1}z + p_{2}z^{2} + \\dots + p_{k}z^{k})f(z) = P_{0} + P_{1}z + \\dots + P_{k-1}z^{k-1}"],["dickson-theory-of-equations-1922/eq-1e4d086572",16,"Dickson 1922, p. 118: d_{r+1} = \\begin{vmatrix} a_{11} & \\cdots & a_{1r} \\\\ \\Dots{3} \\\\ a_{r1} & \\cdots & a_{rr} \\end{vmatrix}"],["dickson-theory-of-equations-1922/eq-ce8cfb1608",16,"Dickson 1922, p. 118: d_1L_1 - d_2L_2 + \\dotsb + (-1)^rd_{r+1}L_{r+1} = \\mp K = 0"],["dickson-theory-of-equations-1922/eq-e1eb73394c",16,"Dickson 1922, p. 120: A = \\begin{pmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ \\Dots{4} \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\end{pmatrix}"],["dickson-theory-of-equations-1922/eq-db2ca6317f",16,"Dickson 1922, p. 121: B = \\begin{pmatrix} a_{11} & a_{12} & \\cdots & a_{1n} & k_1\\\\ \\Dots{5}\\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} & k_m \\end{pm"],["dickson-theory-of-equations-1922/eq-ffb0d502b1",16,"Dickson 1922, p. 122: D = \\begin{vmatrix} a_1 & b_1 & c_1 & d_1 \\\\ a_2 & b_2 & c_2 & d_2 \\\\ a_3 & b_3 & c_3 & d_3 \\\\ a_4 & b_4 & c_4 & d_4 \\en"],["dickson-theory-of-equations-1922/eq-62f1e40ac4",16,"Dickson 1922, p. 122: M = \\begin{vmatrix} a_1 & b_1 \\\\ a_3 & b_3 \\end{vmatrix}"],["dickson-theory-of-equations-1922/eq-a512c0f2f4",16,"Dickson 1922, p. 122: M' = \\begin{vmatrix} c_2 & d_2 \\\\ c_4 & d_4 \\end{vmatrix}"],["dickson-theory-of-equations-1922/eq-f7988def99",16,"Dickson 1922, p. 124: \\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} · \\begin{vmatrix} e & f \\\\ g & h \\end{vmatrix} = \\begin{vmatrix} ae + bg & "],["dickson-theory-of-equations-1922/eq-15b4864de1",16,"Dickson 1922, p. 124: \\begin{vmatrix} \\Neg a_1 & \\Neg b_1 & \\Neg c_1 & 0 & 0 & 0 \\\\ \\Neg a_2 & \\Neg b_2 & \\Neg c_2 & 0 & 0 & 0 \\\\ \\Neg a_3 & \\"],["dickson-theory-of-equations-1922/eq-78c942b9d6",16,"Dickson 1922, p. 119: x_1 = 0, \\dotsc, x_n = 0"],["todhunter-spherical-trigonometry-1886/eq-a8bb94e492",16,"Todhunter 1886, scan 130: 144\\hspace{3pt}V^2=2a^6"],["quantity/area",11,"area","../books/ball-mathematical-recreations-1905/terms/index.html#t-quantity-area"],["maxwell-elementary-treatise-electricity-1888/x-13102932a5",15,"Maxwell 1888, scan 35: We have thus a method of comparing the electric ..."],["boyden-first-book-in-algebra-1895/ex-38/6",4,"Boyden 1895, Exercise 38 (6)"],["form/dd17a5ce8d",5,"factor: x**2 - 11*x + 10"],["concept/number",7,"number","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-number"],["whitehead-introduction-to-mathematics-1911/x-672dcfcbf7",15,"Whitehead 1911, p. 245: Lengths are measured by the foot-rule. By transporting the ..."],["whitehead-introduction-to-mathematics-1911/x-19fbb8d81d",15,"Whitehead 1911, p. 246: These preconceived conditions when accurately formulated may be called ..."],["whitehead-introduction-to-mathematics-1911/x-9c66e36fc4",15,"Whitehead 1911, p. 248: A rule which made days of violently different lengths, ..."],["hardy-course-of-pure-mathematics-1921/eq-3e3028563d",16,"Hardy 1921, p. 353: a_{n} - a_{n-1} - 8a_{n-2} + 12a_{n-3} = 0"],["form/d8a6b7cebf",5,"factor: -a**9 + 8*x**6"],["boyden-first-book-in-algebra-1895/ex-38/11",4,"Boyden 1895, Exercise 38 (11)"],["form/10b6af969c",5,"factor: 64*x**3 + 1"],["whitehead-introduction-to-mathematics-1911/x-946352486e",15,"Whitehead 1911, p. 248: For example, astronomers tell us that the earth’s rotation ..."],["whitehead-introduction-to-mathematics-1911/x-d2e66bcecd",15,"Whitehead 1911, p. 250: The mathematical sciences associated with them do not form ..."],["concept/dimension",7,"dimension","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-dimension"],["boyden-first-book-in-algebra-1895/ex-36/25",4,"Boyden 1895, Exercise 36 (25)"],["planck-treatise-on-thermodynamics-1903/x-6f509e7d29",15,"Planck 1903, p. 122: This means that in solids and liquids the energy ..."],["form/f43e4307cd",5,"factor: 75*a**2*b**3*x - 30*a*b**4*x**2 + 3*b**5*x**3"],["shape/2e9572729b",6,"factor: N*a*b**N*x**N + N*a**N*b**N*x + N*b**N*x**N"],["shape/cf4ef32c70",6,"hcf: (N*a**N*x + N*a**N*x**N, 2*N*a**N*x**N)"],["todhunter-spherical-trigonometry-1886/eq-189d427919",16,"Todhunter 1886, scan 125: mF=nS=2E"],["todhunter-spherical-trigonometry-1886/eq-20359ce215",16,"Todhunter 1886, scan 125: S = \\frac{4 m}{2(m+n)-mn}"],["todhunter-spherical-trigonometry-1886/eq-b9a3b34cea",16,"Todhunter 1886, scan 125: E = \\frac{2mn}{2(m+n)-mn}"],["todhunter-spherical-trigonometry-1886/eq-fabebb2c1f",16,"Todhunter 1886, scan 125: F = \\frac{4 n}{2(m+n)-mn}"],["todhunter-spherical-trigonometry-1886/eq-125ecfb5bb",16,"Todhunter 1886, scan 125: \\frac{1}{m}+\\frac{1}{n} \\text{ must be greater than } \\frac{1}{2}"],["todhunter-spherical-trigonometry-1886/eq-048dde3aef",16,"Todhunter 1886, scan 127: \\sin{\\frac{I}{2}}=\\frac{\\cos \\dfrac{\\pi}{n}} {\\sin{\\dfrac{\\pi}{m}}}"],["todhunter-spherical-trigonometry-1886/eq-1b8bbfe0ac",16,"Todhunter 1886, scan 127: CE = AE\\cot{ACE} = \\frac{a}{2}\\cot{\\frac{\\pi}{m}}"],["todhunter-spherical-trigonometry-1886/eq-ac9b3748cf",16,"Todhunter 1886, scan 127: r = CE\\tan{CEO} = CE\\tan{\\frac{I}{2}} = \\frac{a}{2}\\cot{\\frac{\\pi}{m}}\\tan{\\frac{I}{2}}"],["todhunter-spherical-trigonometry-1886/eq-121ebd5f42",16,"Todhunter 1886, scan 127: R = r\\tan{\\frac{\\pi}{m}} \\tan{\\frac{\\pi}{n}} = \\frac{a}{2} \\tan{\\frac{I}{2}} \\tan{\\frac{\\pi}{n}}"],["todhunter-spherical-trigonometry-1886/eq-732d278f02",16,"Todhunter 1886, scan 127: \\dfrac{ma^2}{4}\\cot{\\dfrac{\\pi}{m}}"],["todhunter-spherical-trigonometry-1886/eq-4f8a7f8e26",16,"Todhunter 1886, scan 127: \\dfrac{mFa^2}{4}\\cot{\\dfrac{\\pi}{m}}"],["todhunter-spherical-trigonometry-1886/eq-e9ce095f9f",16,"Todhunter 1886, scan 127: \\dfrac{mFra^2}{12}\\cot{\\dfrac{\\pi}{m}}"],["todhunter-spherical-trigonometry-1886/eq-68cce3b6e6",16,"Todhunter 1886, scan 128: = abc\\surd{( 1 - \\cos^2{\\alpha} - \\cos^2{\\beta} - \\cos^2{\\gamma} + 2\\cos{\\alpha} \\cos{\\beta} \\cos{\\gamma} )}"],["todhunter-spherical-trigonometry-1886/eq-64f572916f",16,"Todhunter 1886, scan 129: OD^2 = a^2+b^2+c^2 + 2ab\\cos{\\gamma} + 2bc\\cos{\\alpha} + 2ca\\cos{\\beta}"],["person/dmitri-mendeleev",1,"Dmitri Mendeleev","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-dmitri-mendeleev"],["todhunter-spherical-trigonometry-1886/eq-b05425a14d",16,"Todhunter 1886, scan 130: 144\\hspace{3pt}V^2 = -a'^2 b'^2 c'^2 + a^2a'^2 (b'^2+c'^2-a'^2) + b^2b'^2 (c'^2+a'^2-b'^2) + c^2c'^2 (a'^2+b'^2-c'^2) - "],["todhunter-spherical-trigonometry-1886/eq-89ffab1455",16,"Todhunter 1886, scan 130: \\cos{ADB}=\\dfrac{a'^2+b'^2-c^2}{2a'b'}"],["todhunter-spherical-trigonometry-1886/eq-df1d298afc",16,"Todhunter 1886, scan 130: 1=\\cos^2{ADB}+\\cos^2{BDC}+\\cos^2{CDA}-2\\cos{ADB}\\cos{BDC}\\cos{CDA}"],["todhunter-spherical-trigonometry-1886/eq-778dfb0254",16,"Todhunter 1886, scan 131: 0=-a^2b^2c^2 + a'^2a^2(b^2+c^2-a^2) + b'^2b^2(c^2+a^2-b^2) + c'^2c^2(a^2+b^2-c^2) - a^2(a'^2-b'^2)(a'^2-c'^2) - b^2(b'^2"],["todhunter-spherical-trigonometry-1886/eq-22c8b9899a",16,"Todhunter 1886, scan 131: p^2(2a^2b^2+2b^2c^2+2c^2a^2 -a^4-b^4-c^4) = -a^2b^2c^2 + a'^2a^2(b^2+c^2-a^2) + b'^2b^2(c^2+a^2-b^2) + c'^2c^2(a^2+b^2-c"],["ball-mathematical-recreations-1905/x-4fc721b312",15,"Ball 1905, scan 391: Thus, if the disturbance is represented by the swinging ..."],["todhunter-spherical-trigonometry-1886/eq-bccf076a5d",16,"Todhunter 1886, scan 132: \\cos{ADB}=\\dfrac{\\cos{\\gamma}-\\cos{\\alpha'}\\cos{\\beta'}} {\\sin{\\alpha'}\\sin{\\beta}'}"],["theorem/spherical-law-of-cosines-sides",9,"spherical law of cosines (sides)"],["todhunter-spherical-trigonometry-1886/eq-bf65169ac5",16,"Todhunter 1886, scan 132: \\cos{\\alpha}=1-2\\sin^2{\\frac{\\alpha}{2}}"],["todhunter-spherical-trigonometry-1886/eq-be81949a9f",16,"Todhunter 1886, scan 132: aa'+bb'+cc'=2\\sigma"],["todhunter-spherical-trigonometry-1886/eq-1e1de4d61b",16,"Todhunter 1886, scan 132: 36\\hspace{3pt}V^2r^2=\\sigma(\\sigma-aa')(\\sigma-bb')(\\sigma-cc')"],["ball-mathematical-recreations-1905/x-9dc52ebaa7",15,"Ball 1905, scan 392: I should say rather that we have obtained an ..."],["ball-mathematical-recreations-1905/x-3f563735a2",15,"Ball 1905, scan 390: just as two clocks whose rates are nearly the ..."],["ball-mathematical-recreations-1905/x-877a98ab61",15,"Ball 1905, scan 390: The objection to this is that no explanation is ..."],["hardy-course-of-pure-mathematics-1921/eq-b882dc77ab",16,"Hardy 1921, p. 353: a_{n} = 2^{n}\\{A_{1} + (n + 1) A_{2}\\} + (-3)^{n} B"],["maxwell-elementary-treatise-electricity-1888/x-a507850337",15,"Maxwell 1888, scan 37: By means of Thomson’s Quadrant Electrometer it is easy ..."],["hardy-course-of-pure-mathematics-1921/eq-5d4af092cf",16,"Hardy 1921, p. 353: f(n) = (\\omega_{3}^{n} - \\omega_{3}^{2n})/(\\omega_{3} - \\omega_{3}^{2})"],["hardy-course-of-pure-mathematics-1921/eq-ae71d4c304",16,"Hardy 1921, p. 353: u_{n} - 2\\cos\\theta u_{n-1} + u_{n-2} = 0"],["hardy-course-of-pure-mathematics-1921/eq-ffb45f6b9d",16,"Hardy 1921, p. 353: u_{n} = A\\cos n\\theta + B\\sin n\\theta"],["hardy-course-of-pure-mathematics-1921/eq-84fcf7c305",16,"Hardy 1921, p. 353: f(n) + f(n - 1) + f(n - 2) = 0"],["hardy-course-of-pure-mathematics-1921/eq-c23c1d994f",16,"Hardy 1921, p. 353: z/(1 + z + z^{2}) = z(1 - z)/(1 - z^{3})"],["hardy-course-of-pure-mathematics-1921/eq-505d1c71d3",16,"Hardy 1921, p. 353: p_{n} = \\frac{1}{2} (p_{n-1} + p_{n-2})"],["hardy-course-of-pure-mathematics-1921/eq-483582be78",16,"Hardy 1921, p. 353: \\frac{1}{3}\\{2 + (-\\frac{1}{2})^{n}\\}"],["ball-mathematical-recreations-1905/x-8f885cac54",15,"Ball 1905, scan 394: Thus it would seem that a cubic inch of ..."],["ball-mathematical-recreations-1905/x-4fadfb6d2b",15,"Ball 1905, scan 394: and probably the size of a molecule would be ..."],["maxwell-elementary-treatise-electricity-1888/ch-i",2,"Maxwell 1888, ch. I: Electrification by Friction","../books/maxwell-elementary-treatise-electricity-1888/ch/ch-i/index.html"],["hardy-course-of-pure-mathematics-1921/eq-8c8213b98a",16,"Hardy 1921, p. 354: \\frac{1}{a + 1} + \\frac{1}{a + 2} + \\dots + \\frac{1}{a + n} = \\binom{n}{1}\\frac{1}{a + 1} - \\binom{n}{2}\\frac{1!}{(a + 1"],["hardy-course-of-pure-mathematics-1921/eq-b105edb288",16,"Hardy 1921, p. 354: \\int_{0}^{1} x^{a}\\frac{1 - x^{n}}{1 - x}\\, dx = \\int_{0}^{1} (1 - x)^{a}\\{1 - (1 - x)^{n}\\}\\frac{dx}{x}"],["hardy-course-of-pure-mathematics-1921/eq-a30ed911b3",16,"Hardy 1921, p. 354: \\sum_{0}^{\\infty} \\frac{z^{n}}{n!} \\sum_{1}^{\\infty} \\frac{(-1)^{n-1}z^{n}}{n·n!} = \\sum_{1}^{\\infty} \\left(1 + \\frac{1}"],["hardy-course-of-pure-mathematics-1921/eq-b78588f364",16,"Hardy 1921, p. 354: (A_{1}B_{n} + A_{2}B_{n-1} + \\dots + A_{n}B_{1})/n \\to AB"],["boyden-first-book-in-algebra-1895/ex-14/14",4,"Boyden 1895, Exercise 14 (14)"],["hardy-course-of-pure-mathematics-1921/x-532ef63f0f",15,"Hardy 1921, p. 436: There is another proof, proceeding on different lines, which ..."],["hardy-course-of-pure-mathematics-1921/x-788e292a5c",15,"Hardy 1921, p. 436: when z describes any contour \\gamma in the positive ..."],["hardy-course-of-pure-mathematics-1921/eq-585cf34373",16,"Hardy 1921, p. 354: c_{n} = a_{1}b_{n} + a_{2}b_{n-1} + \\dots + a_{n}b_{1}"],["hardy-course-of-pure-mathematics-1921/eq-df75c33db6",16,"Hardy 1921, p. 354: C_{n} = a_{1}B_{n} + a_{2}B_{n-1} + \\dots + a_{n}B_{1}"],["concept/partial-sums",7,"partial sums"],["planck-treatise-on-thermodynamics-1903/x-1417ae5822",15,"Planck 1903, p. 126: Equations % [eqn:(88)](88)% and % [eqn:(89)](89)%, like Thomson and ..."],["form/df56da60da",5,"evaluate: 4*a**4 + 3*b**3 + 2*c**2 at a=1, b=2, c=3"],["de-morgan-elementary-illustrations-calculus-1899/eq-349b6d3752",16,"De Morgan 1899, p. 91: \\Delta^{2} y_{1} - \\Delta^{2} y = \\Delta^{3} y"],["de-morgan-elementary-illustrations-calculus-1899/eq-eb85e3d917",16,"De Morgan 1899, p. 92: y_{2} = y_{1} + \\Delta y_{1}"],["de-morgan-elementary-illustrations-calculus-1899/eq-a90b8b3363",16,"De Morgan 1899, p. 92: y_{1} = y + \\Delta y"],["de-morgan-elementary-illustrations-calculus-1899/eq-6e3fc40df8",16,"De Morgan 1899, p. 92: \\Delta y_{2} = \\Delta y + 2\\Delta^{2} y + \\Delta^{3} y"],["de-morgan-elementary-illustrations-calculus-1899/eq-747d61a173",16,"De Morgan 1899, p. 93: n\\Delta x = h"],["de-morgan-elementary-illustrations-calculus-1899/eq-6c7d2d00b0",16,"De Morgan 1899, p. 93: \\phi(x + h) = y + \\frac{dy}{dx}\\, h + \\frac{d^{2} y}{dx^{2}}\\, \\frac{h^{2}}{2} + \\frac{d^{3} y}{dx^{3}}\\, \\frac{h^{3}}{2"],["boyden-first-book-in-algebra-1895/eq-70f6a00386",16,"Boyden 1895: 3(a+b)=3a+3b"],["de-morgan-elementary-illustrations-calculus-1899/eq-da90e188c0",16,"De Morgan 1899, p. 95: \\frac{d.z}{dx} = \\frac{dz}{dx} + \\frac{dz}{dx}\\, p"],["de-morgan-elementary-illustrations-calculus-1899/x-6673b8db03",15,"De Morgan 1899, p. 74: That \\phi'' x is derived in the same manner ..."],["boyden-first-book-in-algebra-1895/ex-39/5",4,"Boyden 1895, Exercise 39 (5)"],["form/fabf88c0fd",5,"hcf: (81*x**8 - 16, 81*x**8 - 72*x**4 + 16)"],["shape/6c390058ab",6,"hcf: (N*x**N + N, 2*N*x**N + N)"],["concept/method-reducing-a-fraction-to-an-integral-or-mixed-expression",7,"method: reducing a fraction to an integral or mixed expression"],["boyden-first-book-in-algebra-1895/eq-641bd3efa5",16,"Boyden 1895: \\frac{5a^2b \\div 5ab}{10ab^2 \\div 5ab} = \\frac{a}{2b}"],["boyden-first-book-in-algebra-1895/eq-03fca7add8",16,"Boyden 1895: \\frac{a^2bx-b^3x}{a^2bx-ab^2x} = \\frac{bx(a^2-b^2) \\div bx(a-b)}{abx(a-b) \\div bx(a-b)} = \\frac{a+b}{a}"],["boyden-first-book-in-algebra-1895/eq-ef6f872f0a",16,"Boyden 1895: \\frac{ac-bc-d}{c} = a - b -\\frac{d}{c}"],["concept/entire-number",7,"entire number"],["boyden-first-book-in-algebra-1895/eq-3a9b5f53a1",16,"Boyden 1895: b + \\frac{a}{c} = \\frac{bc + a}{c}"],["planck-treatise-on-thermodynamics-1903/eq-096537c457",16,"Planck 1903, p. 168: u &= \\frac{\\lambda u_{1} + \\mu u_{2} + \\nu u_{3}}{\\lambda + \\mu + \\nu}"],["planck-treatise-on-thermodynamics-1903/eq-216ee73835",16,"Planck 1903, p. 169: M\\, \\delta\\phi'' = \\phi_{1}\\, \\delta M_{1} + \\phi_{2}\\, \\delta M_{2} + \\phi_{3}\\, \\delta M_{3}"],["planck-treatise-on-thermodynamics-1903/eq-d51c825f14",16,"Planck 1903, p. 169: \\delta M_{1} + \\delta M_{2} + \\delta M_{3} &= 0\\Add{,}"],["planck-treatise-on-thermodynamics-1903/eq-e1c8b41c66",16,"Planck 1903, p. 169: v_{1}\\, \\delta M_{1} + v_{2}\\, \\delta M_{2} + v_{3}\\, \\delta M_{3} &= M\\, \\delta v\\Add{,}"],["planck-treatise-on-thermodynamics-1903/eq-3e68783718",16,"Planck 1903, p. 169: u_{1}\\, \\delta M_{1} + u_{2}\\, \\delta M_{2} + u_{3}\\, \\delta M_{3} &= M\\, \\delta u\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-29a3789d2a",16,"Planck 1903, p. 169: \\delta \\phi'' = \\frac{\\delta u + p_{1}\\, \\delta v}{\\theta_{1}}"],["hardy-course-of-pure-mathematics-1921/eq-8368cf46ea",16,"Hardy 1921, p. 354: C_{1} + C_{2} + \\dots + C_{n} = A_{1}B_{n} + A_{2}B_{n-1} + \\dots + A_{n}B_{1}"],["hardy-course-of-pure-mathematics-1921/eq-c261ad2262",16,"Hardy 1921, p. 354: (C_{1} + C_{2} + \\dots + C_{n})/n \\to AB"],["form/188b090ffc",5,"identity: (a**2*x**2 - 16*a**2)/(a*x**2 + 9*a*x + 20*a)"],["concept/method-reducing-a-mixed-expression-to-a-fraction",7,"method: reducing a mixed expression to a fraction"],["shape/6d98602aad",6,"identity: (N*a**N + a**N*x**N)/(N*a*x + N*a + a*x**N)"],["planck-treatise-on-thermodynamics-1903/eq-3e97d4f3ce",16,"Planck 1903, p. 170: \\theta_{1}^{2}\\, \\delta^{2} (\\phi'' - \\phi') = \\left[\\delta u - \\left(\\theta_{1}\\, \\frac{dp_{12}}{d\\theta_{12}} - p_{1}\\"],["planck-treatise-on-thermodynamics-1903/eq-d912d81fbb",16,"Planck 1903, p. 170: \\frac{M_{12}\\, \\delta v_{12} + M_{21}\\, \\delta v_{21} - M\\, \\delta v}{v_{12} - v_{21}} = \\frac{M_{12}\\, \\delta u_{12} + "],["hardy-course-of-pure-mathematics-1921/eq-3838bad0f2",16,"Hardy 1921, p. 355: \\int_{-1}^{1} \\frac{dx}{(a - x) \\sqrtp{1 - x^{2}}} = \\frac{\\pi}{\\sqrtp{a^{2} - 1}}"],["planck-treatise-on-thermodynamics-1903/x-2acf688f4a",15,"Planck 1903, p. 131: As soon as accurate measurement of even a single ..."],["boyden-first-book-in-algebra-1895/ex-14/15",4,"Boyden 1895, Exercise 14 (15)"],["form/054b97c279",5,"evaluate: a**3*c/2 - 3*a*b*c**3/4 - b**3 - c**3 at a=1, b=2, c=3"],["shape/a7e173100e",6,"evaluate: N*a*b*c**N + N*a**N*c - b**N - c**N"],["whitehead-introduction-to-mathematics-1911/x-6b64ff05e2",15,"Whitehead 1911, p. 250: We perform first the operations in brackets and obtain ..."],["whitehead-introduction-to-mathematics-1911/x-5681a760cf",15,"Whitehead 1911, p. 250: This fundamental ratio \\dfrac{SP}{PN} is called the eccentricity of ..."],["whitehead-introduction-to-mathematics-1911/x-5c560cfcb8",15,"Whitehead 1911, p. 250: An ellipse with small eccentricity is very nearly a ..."],["whitehead-introduction-to-mathematics-1911/x-a5a3942699",15,"Whitehead 1911, p. 251: But it is possible for a series with terms ..."],["whitehead-introduction-to-mathematics-1911/x-035cad72ec",15,"Whitehead 1911, p. 251: Such convergent series, which are not absolutely convergent, are ..."],["concept/carnot-cycle",7,"Carnot cycle","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-carnot-cycle"],["concept/sum-to-infinity",7,"sum to infinity","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-sum-to-infinity"],["concept/divariant-system",7,"divariant system","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-divariant-system"],["planck-treatise-on-thermodynamics-1903/x-53ec4848ce",15,"Planck 1903, p. 174: An aqueous solution of sulphuric acid forms a system ..."],["hardy-course-of-pure-mathematics-1921/eq-37051b676b",16,"Hardy 1921, p. 355: \\int_{0}^{\\infty} \\frac{dx}{\\{\\sqrtp{x^{2} + 1} + x\\}^{n}} = \\int_{0}^{\\infty} \\{\\sqrtp{x^{2} + 1} - x\\}^{n}\\, dx = \\fra"],["hardy-course-of-pure-mathematics-1921/eq-fde36f8348",16,"Hardy 1921, p. 355: 2y = ax - (b/x)"],["form/40abf5a897",5,"identity: (x - 6)*(x + 7)"],["concept/reversible-process",7,"reversible process","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-reversible-process"],["wentworth-first-steps-in-algebra-1894/ex-22/2",4,"Wentworth 1894, Exercise 22 (2)"],["wentworth-first-steps-in-algebra-1894/ex-22/4",4,"Wentworth 1894, Exercise 22 (4)"],["wentworth-first-steps-in-algebra-1894/ex-22/3",4,"Wentworth 1894, Exercise 22 (3)"],["form/9617e6e6aa",5,"identity: (x - 7)*(x - 6)"],["planck-treatise-on-thermodynamics-1903/x-02d92ed8a6",15,"Planck 1903, p. 174: The question as to the number of the independent ..."],["hardy-course-of-pure-mathematics-1921/x-b7e96953b3",15,"Hardy 1921, p. 342: There is another test, due to Abel, which, though ..."],["hardy-course-of-pure-mathematics-1921/eq-a0913a27fe",16,"Hardy 1921, p. 355: 2y = ax + (b/x)"],["hardy-course-of-pure-mathematics-1921/eq-be6cfd16f8",16,"Hardy 1921, p. 356: \\int_{0}^{\\pi} f(\\sec\\tfrac{1}{2}x + \\tan\\tfrac{1}{2}x)\\frac{dx}{\\sqrtp{\\sin x}} = \\int_{0}^{\\pi} f(\\cosec x)\\frac{dx}{\\"],["hardy-course-of-pure-mathematics-1921/eq-526c9f4a28",16,"Hardy 1921, p. 356: \\int_{0}^{\\infty} \\frac{dx}{(x^{2} + a^{2})(x^{2} + b^{2})} = \\frac{\\pi}{2ab(a + b)}"],["hardy-course-of-pure-mathematics-1921/eq-63dbe8d1d2",16,"Hardy 1921, p. 356: \\int_{0}^{\\infty} \\frac{x^{2}\\, dx}{(x^{2} + a^{2})(x^{2} + b^{2})} = \\frac{\\pi}{2(a + b)}"],["hardy-course-of-pure-mathematics-1921/eq-ae8b9f040e",16,"Hardy 1921, p. 356: A = \\beta + \\sqrtp{\\alpha\\gamma}"],["boyden-first-book-in-algebra-1895/ex-14/16",4,"Boyden 1895, Exercise 14 (16)"],["hardy-course-of-pure-mathematics-1921/eq-16465b396c",16,"Hardy 1921, p. 275: \\frac{\\dd x}{\\dd \\theta} = -r\\sin\\theta"],["hardy-course-of-pure-mathematics-1921/eq-a54aba2168",16,"Hardy 1921, p. 274: \\lim_{h\\to 0}\\frac{f(x + h, y) - f(x, y)}{h}"],["de-morgan-elementary-illustrations-calculus-1899/eq-4dbfd578e9",16,"De Morgan 1899, p. 101: \\frac{dy}{dx} = \\cos x"],["planck-treatise-on-thermodynamics-1903/eq-60973405e7",16,"Planck 1903, p. 179: \\beta \\leq \\alpha + 2"],["planck-treatise-on-thermodynamics-1903/eq-7f591321af",16,"Planck 1903, p. 184: \\delta\\, d\\Psi = 0"],["hardy-course-of-pure-mathematics-1921/eq-ba21be7b42",16,"Hardy 1921, p. 275: \\frac{\\dd r}{\\dd x} = \\frac{x}{\\sqrtp{x^{2} + y^{2}}}"],["hardy-course-of-pure-mathematics-1921/eq-7016aea767",16,"Hardy 1921, p. 275: \\frac{\\dd \\theta}{\\dd x} = -\\frac{y}{x^{2} + y^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-83a9184f5d",16,"Hardy 1921, p. 275: \\frac{\\dd x}{\\dd r} = \\cos\\theta"],["hardy-course-of-pure-mathematics-1921/eq-3d08d29cd8",16,"Hardy 1921, p. 276: \\lim (\\Delta r/\\Delta x) = \\lim (PP_{2}/PP_{1}) = \\sec\\theta"],["form/1f563b303c",5,"evaluate: -2*a*b/(a + b) + 2*a - b at a=1, b=2"],["hardy-course-of-pure-mathematics-1921/eq-1d1f796a14",16,"Hardy 1921, p. 276: \\lim (\\delta r/\\Delta r) = \\cos^{2}\\theta"],["concept/quantity-psi-function",7,"quantity: Psi function"],["planck-treatise-on-thermodynamics-1903/eq-e4e744706e",16,"Planck 1903, p. 175: \\delta \\Psi = 0"],["hardy-course-of-pure-mathematics-1921/eq-84c79d0bd2",16,"Hardy 1921, p. 276: b(\\dd z/\\dd x) = a(\\dd z/\\dd y)"],["hardy-course-of-pure-mathematics-1921/eq-e0ba9a91dc",16,"Hardy 1921, p. 276: \\dd u/\\dd x = 1"],["shape/1c20fe7441",6,"evaluate: N*a*b/(a + b) + N*a - b"],["boyden-first-book-in-algebra-1895/ex-14/17",4,"Boyden 1895, Exercise 14 (17)"],["form/3aefca3990",5,"evaluate: 3*a*b - 2*a + 2*b*c + 4*b*d/15 - 3*c**3/4 - d at a=1, b=2, c=3, x=0"],["shape/0c7161c249",6,"evaluate: N*a*b + N*a + N*b*c + N*b*d + N*c**N - d"],["wentworth-first-steps-in-algebra-1894/ex-22/5",4,"Wentworth 1894, Exercise 22 (5)"],["form/41334ce0eb",5,"identity: (x - 5)*(x + 8)"],["shape/4ebf9e463e",6,"identity: (N*a*x - a**N + x**N)*(N*a*x + a**N - x**N)"],["concept/positive-electrification",7,"positive electrification","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-positive-electrification"],["wentworth-first-steps-in-algebra-1894/ex-22/32",4,"Wentworth 1894, Exercise 22 (32)"],["todhunter-spherical-trigonometry-1886/eq-2366db96d7",16,"Todhunter 1886, scan 136: \\cos TU = \\cos TA \\cos UA + \\cos TB \\cos UB + \\cos TC \\cos UC"],["hardy-course-of-pure-mathematics-1921/eq-480a2fa135",16,"Hardy 1921, p. 277: \\frac{df}{dt} = \\frac{\\dd f}{\\dd x}\\, \\frac{dx}{dt} + \\frac{\\dd f}{\\dd y}\\, \\frac{dy}{dt}"],["hardy-course-of-pure-mathematics-1921/eq-68ae94ca98",16,"Hardy 1921, p. 278: \\frac{df}{dx} = \\frac{\\dd f}{\\dd x} + \\frac{\\dd f}{\\dd y}\\, \\frac{dy}{dx}"],["form/2cfbf8cf63",5,"identity: (-a**2 - b**2 + x**2)*(a**2 + b**2 - b*x + x**2)"],["shape/1d5b75e7e4",6,"identity: (-a**N - b**N + x**N)*(a**N - b*x + b**N + x**N)"],["concept/plane-trigonometry",7,"plane trigonometry","../books/todhunter-spherical-trigonometry-1886/terms/index.html#t-concept-plane-trigonometry"],["concept/solid-geometry",7,"solid geometry","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-solid-geometry"],["concept/geometry",7,"geometry","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-geometry"],["hardy-course-of-pure-mathematics-1921/eq-6a225d3187",16,"Hardy 1921, p. 278: \\frac{\\dd f}{\\dd x}\\, \\frac{dx}{dt} + \\frac{\\dd f}{\\dd y}\\, \\frac{dy}{dt} = 0"],["hardy-course-of-pure-mathematics-1921/eq-8455a58188",16,"Hardy 1921, p. 278: r' = (xx' + yy')/r"],["whitehead-introduction-to-mathematics-1911/x-71ba536915",15,"Whitehead 1911, p. 251: the algebra should be studied graphically, so that in ..."],["whitehead-introduction-to-mathematics-1911/x-81c9e14c58",15,"Whitehead 1911, p. 251: But in all these courses great care should be ..."],["boyden-first-book-in-algebra-1895/ex-32/19",4,"Boyden 1895, Exercise 32 (19)"],["hardy-course-of-pure-mathematics-1921/eq-491e7e0f0e",16,"Hardy 1921, p. 278: \\theta' = (xy' - yx')/r^{2}"],["hardy-course-of-pure-mathematics-1921/eq-2cf83046c4",16,"Hardy 1921, p. 278: \\phi(x + h) - \\phi(x) = hf'(x + \\theta h)"],["hardy-course-of-pure-mathematics-1921/eq-a9a964bc9b",16,"Hardy 1921, p. 278: \\delta z = f(x + h, y + k) - f(x, y)"],["hardy-course-of-pure-mathematics-1921/eq-b028ca9177",16,"Hardy 1921, p. 279: \\delta z = (f_{x}' + \\epsilon)\\, \\delta x + (f_{y}' + \\eta)\\, \\delta y"],["boyden-first-book-in-algebra-1895/ex-33/11",4,"Boyden 1895, Exercise 33 (11)"],["form/ec4cc1bfd3",5,"factor: -25*a**2*b**2 + 81*c**4*x**2"],["shape/c17473308e",6,"factor: N*a**N*b**N + N*c**N*x**N"],["todhunter-spherical-trigonometry-1886/eq-20392bb642",16,"Todhunter 1886, scan 137: \\Sigma = \\cos TH_1 + \\cos TH_2 + \\cos TH_3 + \\ldots"],["todhunter-spherical-trigonometry-1886/eq-6e52058936",16,"Todhunter 1886, scan 138: G=\\surd{(P^2+Q^2+R^2)}"],["todhunter-spherical-trigonometry-1886/eq-d26aa9aeb5",16,"Todhunter 1886, scan 138: \\cos \\alpha = \\frac{P}{G}"],["todhunter-spherical-trigonometry-1886/eq-4cc7277b9c",16,"Todhunter 1886, scan 138: \\cos^2\\alpha +\\cos^2\\beta+\\cos^2\\gamma = 1"],["todhunter-spherical-trigonometry-1886/eq-cf7fe690be",16,"Todhunter 1886, scan 138: \\Sigma=G \\cos TU"],["todhunter-spherical-trigonometry-1886/eq-049a870f1f",16,"Todhunter 1886, scan 138: \\cos TU= \\lambda \\cos \\alpha + \\mu \\cos \\beta + \\nu \\cos \\gamma"],["todhunter-spherical-trigonometry-1886/eq-c2dc6f3362",16,"Todhunter 1886, scan 139: G = 0"],["hardy-course-of-pure-mathematics-1921/eq-5f75daaae5",16,"Hardy 1921, p. 279: \\delta z = f_{x}'\\, \\delta x + f_{y}'\\, \\delta y"],["todhunter-spherical-trigonometry-1886/eq-7596a277f8",16,"Todhunter 1886, scan 139: \\Sigma = \\cos^2 TH_1 + \\cos^2 TH_2 + \\cos^2 TH_3 + \\dots"],["todhunter-spherical-trigonometry-1886/eq-0feaf39c63",16,"Todhunter 1886, scan 139: \\Sigma = P\\lambda^2+Q\\mu^2+R\\nu^2+2p\\mu\\nu+2q\\nu\\lambda+2r\\lambda \\mu"],["todhunter-spherical-trigonometry-1886/eq-30146f55e2",16,"Todhunter 1886, scan 139: \\Sigma=P\\lambda^2+Q\\mu^2+R\\nu^2"],["todhunter-spherical-trigonometry-1886/eq-5f37530b5a",16,"Todhunter 1886, scan 143: 0 = 2p\\mu\\nu + 2q\\nu\\lambda + 2r\\lambda\\mu"],["todhunter-spherical-trigonometry-1886/eq-ecec46ec0a",16,"Todhunter 1886, scan 144: \\lambda\\lambda'P + \\mu\\mu'Q + \\nu\\nu'R"],["hardy-course-of-pure-mathematics-1921/eq-f5dd641ea0",16,"Hardy 1921, p. 280: \\delta y = f'(x)\\, \\delta x"],["todhunter-spherical-trigonometry-1886/eq-1d85f9938a",16,"Todhunter 1886, scan 144: \\dfrac{S}{3} (\\lambda\\lambda' + \\mu\\mu' + \\nu\\nu') = \\dfrac{S}{3} \\cos TU"],["theorem/propagation-of-small-errors",9,"propagation of small errors","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-propagation-of-small-errors"],["concept/hypotenuse",7,"hypotenuse","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-hypotenuse"],["concept/astronomical-ephemeris",7,"astronomical ephemeris","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-astronomical-ephemeris"],["hardy-course-of-pure-mathematics-1921/eq-b681da5a98",16,"Hardy 1921, p. 280: dy = f'(x)\\, \\delta x"],["de-morgan-elementary-illustrations-calculus-1899/x-6f0cc4ce85",15,"De Morgan 1899, p. 75: These last are in the proportion of h to ..."],["dickson-theory-of-equations-1922/ch-ix",2,"Dickson 1922, ch. IX: Symmetric Functions","../books/dickson-theory-of-equations-1922/ch/ch-ix/index.html"],["de-morgan-elementary-illustrations-calculus-1899/x-9c7ed61625",15,"De Morgan 1899, p. 75: The errors of the hypothenuse are then .0008 and ..."],["de-morgan-elementary-illustrations-calculus-1899/x-c7974cf3ad",15,"De Morgan 1899, p. 76: It also follows, that if x increase by successive ..."],["de-morgan-elementary-illustrations-calculus-1899/x-96304af24a",15,"De Morgan 1899, p. 76: And even for this interval, though it can hardly ..."],["hardy-course-of-pure-mathematics-1921/eq-26c646e8b5",16,"Hardy 1921, p. 280: dx = \\delta x"],["hardy-course-of-pure-mathematics-1921/eq-388103fd5d",16,"Hardy 1921, p. 280: dy = f'(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-e7c0e41e3a",16,"Hardy 1921, p. 280: \\frac{dy}{dx} = f'(x)"],["hardy-course-of-pure-mathematics-1921/eq-c9133bfb30",16,"Hardy 1921, p. 280: \\lim \\frac{dy}{\\delta y} = 1"],["boyden-first-book-in-algebra-1895/ex-14/18",4,"Boyden 1895, Exercise 14 (18)"],["de-morgan-elementary-illustrations-calculus-1899/eq-ceb507d891",16,"De Morgan 1899, p. 98: d.z = \\frac{dz}{dx}\\, dx + \\frac{dz}{dy}\\, dy + \\frac{dz}{da}\\, da + \\etc."],["de-morgan-elementary-illustrations-calculus-1899/eq-48c19557c0",16,"De Morgan 1899, p. 98: da = \\frac{da}{dx}\\, dx + \\frac{da}{dy}\\, dy"],["de-morgan-elementary-illustrations-calculus-1899/eq-2097842d11",16,"De Morgan 1899, p. 96: \\frac{d.z}{dx} = \\frac{dz}{dx} + \\frac{dz}{dy}\\, \\frac{dy}{dx} + \\frac{dz}{da}\\, \\frac{da}{dy}\\, \\frac{dy}{dx} + \\frac{d"],["de-morgan-elementary-illustrations-calculus-1899/eq-bbddd9a260",16,"De Morgan 1899, p. 100: \\frac{dz}{dx} = \\frac{dz}{da}\\, \\frac{da}{dy}\\, \\frac{dy}{dx}"],["de-morgan-elementary-illustrations-calculus-1899/eq-878613750f",16,"De Morgan 1899, p. 100: \\dfrac{dz}{dy} = \\dfrac{1}{y}"],["de-morgan-elementary-illustrations-calculus-1899/eq-a9b758ffee",16,"De Morgan 1899, p. 100: \\frac{dz}{da} = \\frac{1}{a}"],["de-morgan-elementary-illustrations-calculus-1899/eq-cdbf9e2f31",16,"De Morgan 1899, p. 101: \\frac{da}{dy} = \\frac{1}{y}"],["hardy-course-of-pure-mathematics-1921/eq-c50f73a27b",16,"Hardy 1921, p. 281: dz = f_{x}'\\, \\delta x + f_{y}'\\, \\delta y"],["dickson-theory-of-equations-1922/eq-e2f3df54b3",16,"Dickson 1922, p. 128: \\Sigma \\alpha = \\alpha + \\beta + \\gamma"],["dickson-theory-of-equations-1922/eq-081f76d03a",16,"Dickson 1922, p. 129: E_1 = \\Sigma x_1"],["planck-treatise-on-thermodynamics-1903/eq-f1e6a049fd",16,"Planck 1903, p. 175: \\Psi' = \\Phi' - \\frac{U' + pV'}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-89590a0259",16,"Planck 1903, p. 176: \\Psi' = \\frac{\\dd \\Psi'}{\\dd M_{1}'}\\, M_{1}' + \\frac{\\dd \\Psi'}{\\dd M_{2}'}\\, M_{2}' + \\dots + \\frac{\\dd \\Psi'}{\\dd M_{"],["concept/theorem-euler-s-theorem-for-homogeneous-functions",7,"theorem: Euler's theorem for homogeneous functions"],["planck-treatise-on-thermodynamics-1903/eq-3d7f12c290",16,"Planck 1903, p. 177: \\frac{\\dd \\Psi'}{\\dd M_{1}'} = \\frac{\\dd \\Psi''}{\\dd M_{1}''} = \\dots = \\frac{\\dd \\Psi^{\\beta}}{\\dd M_{1}^{\\beta}}"],["planck-treatise-on-thermodynamics-1903/eq-607065685d",16,"Planck 1903, p. 177: M_{\\alpha} = M_{\\alpha}' + M_{\\alpha}'' + \\dots + M_{\\alpha}^{\\beta}"],["planck-treatise-on-thermodynamics-1903/eq-9fa770a29d",16,"Planck 1903, p. 178: \\alpha\\beta + 2"],["planck-treatise-on-thermodynamics-1903/eq-5f3e4dda36",16,"Planck 1903, p. 179: \\bigl[(\\alpha - 1)\\beta + 2\\bigr] - \\bigl[\\alpha (\\beta - 1)\\bigr] = \\alpha - \\beta + 2"],["hardy-course-of-pure-mathematics-1921/eq-c302768a00",16,"Hardy 1921, p. 281: dz = f_{x}'\\, dx + f_{y}'\\, dy"],["hardy-course-of-pure-mathematics-1921/eq-edbb136345",16,"Hardy 1921, p. 281: A = \\pi ab"],["form/90d66c5f17",5,"evaluate: (a**2*b*d + a*b**2*c + a*b*c**2 + a*c**2*d)/(a*b*c) at a=1, b=2, c=3, x=0"],["shape/b179fd3f95",6,"evaluate: (a*b*c**N + a*b**N*c + a*c**N*d + a**N*b*d)/(a*b*c)"],["planck-treatise-on-thermodynamics-1903/eq-58c67cad38",16,"Planck 1903, p. 184: \\frac{\\dd \\Psi}{\\dd \\theta} = \\frac{U + pV}{\\theta^{2}}"],["planck-treatise-on-thermodynamics-1903/eq-b4360c8e14",16,"Planck 1903, p. 184: \\frac{\\dd \\Psi}{\\dd p} = -\\frac{V}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-4a988168aa",16,"Planck 1903, p. 183: d\\Phi' = \\frac{dU' + p\\, dV'}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-56c9e829d2",16,"Planck 1903, p. 176: \\Delta \\Psi = \\eps\\Psi'"],["planck-treatise-on-thermodynamics-1903/eq-b5a63bc1dc",16,"Planck 1903, p. 187: \\frac{dp}{d\\theta} = \\frac{Q}{\\theta\\, \\delta V}"],["planck-treatise-on-thermodynamics-1903/eq-dbf91c6b02",16,"Planck 1903, p. 187: L = \\theta\\, \\frac{dp}{d\\theta} (v'' - v')"],["planck-treatise-on-thermodynamics-1903/eq-5bb5d6a375",16,"Planck 1903, p. 188: Q = \\theta · \\frac{dp}{d\\theta} · \\delta V"],["hardy-course-of-pure-mathematics-1921/eq-4f588383f4",16,"Hardy 1921, p. 281: \\frac{dA}{A} = \\frac{da}{a} + \\frac{db}{b}"],["hardy-course-of-pure-mathematics-1921/eq-3c935922eb",16,"Hardy 1921, p. 282: \\frac{d\\Delta}{\\Delta} = \\cot A\\, dA + \\frac{db}{b} + \\frac{dc}{c}"],["planck-treatise-on-thermodynamics-1903/eq-7461956fd9",16,"Planck 1903, p. 189: \\delta V = \\bigl[(v'' + cv''') - (1 + c) v'\\bigr]\\, \\delta M_{1}''"],["planck-treatise-on-thermodynamics-1903/eq-c80a64fa97",16,"Planck 1903, p. 189: L = \\theta\\, \\frac{dp}{d\\theta} \\bigl(v'' + cv''' - (1 + c)v'\\bigr)"],["planck-treatise-on-thermodynamics-1903/eq-012b14d2bf",16,"Planck 1903, p. 189: v'' = \\frac{R}{m} · \\frac{\\theta}{p}"],["law/perfect-gas-equation-for-specific-volume",10,"perfect gas equation for specific volume"],["planck-treatise-on-thermodynamics-1903/eq-9f2feaa4bb",16,"Planck 1903, p. 189: L = \\frac{R}{m} \\theta^{2} · \\frac{d \\log p}{d\\theta}"],["dickson-theory-of-equations-1922/eq-6265949a3f",16,"Dickson 1922, p. 129: E_2 = \\Sigma x_1x_2"],["hardy-course-of-pure-mathematics-1921/eq-7e3e0f2929",16,"Hardy 1921, p. 282: \\frac{d\\Delta}{\\Delta} = 2\\frac{da}{a} + \\frac{c\\, dB}{a\\sin B} + \\frac{b\\, dC}{a\\sin C}"],["hardy-course-of-pure-mathematics-1921/eq-eb080003f2",16,"Hardy 1921, p. 282: d\\Delta = R(\\cos A\\, da + \\cos B\\, db + \\cos C\\, dc)"],["boyden-first-book-in-algebra-1895/ex-31/16",4,"Boyden 1895, Exercise 31 (16)"],["planck-treatise-on-thermodynamics-1903/x-d38baf88b5",15,"Planck 1903, p. 190: If, finally, we dissolve salt sufficient for saturation in ..."],["planck-treatise-on-thermodynamics-1903/x-38f99da6b1",15,"Planck 1903, p. 199: This means that the relative decrease of the vapour ..."],["planck-treatise-on-thermodynamics-1903/x-e58310ee9c",15,"Planck 1903, p. 203: The error committed in putting the rate of diffusion ..."],["planck-treatise-on-thermodynamics-1903/x-534eff6066",15,"Planck 1903, p. 199: This proposition furnishes a means of distinguishing between a ..."],["dickson-theory-of-equations-1922/eq-e1b2f25eee",16,"Dickson 1922, p. 129: E_n = x_1x_2 \\dotsm x_n"],["boyden-first-book-in-algebra-1895/ex-14/20",4,"Boyden 1895, Exercise 14 (20)"],["form/8af402fb94",5,"factor: -2*a**2*b*c**9 + 6*a*b**3*c**5 - 4*a*b**2*c**6 + 2*a*b*c**7"],["shape/88ef715af8",6,"factor: N*a*b*c**N + 2*N*a*b**N*c**N + N*a**N*b*c**N"],["planck-treatise-on-thermodynamics-1903/x-81f958bbb8",15,"Planck 1903, p. 198: Since \\Delta is small for small values of c ..."],["hardy-course-of-pure-mathematics-1921/eq-4d8f438300",16,"Hardy 1921, p. 282: \\frac{\\dd a}{\\dd b} = -\\frac{\\cos B}{\\cos A}"],["hardy-course-of-pure-mathematics-1921/eq-705207139e",16,"Hardy 1921, p. 282: \\frac{\\dd a}{\\dd c} = -\\frac{\\cos C}{\\cos A}"],["hardy-course-of-pure-mathematics-1921/eq-98f4de2e4e",16,"Hardy 1921, p. 282: \\frac{da}{\\cos A} + \\frac{db}{\\cos B} + \\frac{dc}{\\cos C} = 0"],["hardy-course-of-pure-mathematics-1921/eq-f9b130688c",16,"Hardy 1921, p. 282: \\frac{\\dd a}{\\dd b} = -\\frac{\\cos A}{\\cos B}"],["hardy-course-of-pure-mathematics-1921/eq-36195831b8",16,"Hardy 1921, p. 282: \\frac{\\dd z}{\\dd x} = \\frac{\\dd z}{\\dd u}\\, \\frac{\\dd u}{\\dd x} + \\frac{\\dd z}{\\dd v}\\, \\frac{\\dd v}{\\dd x}"],["hardy-course-of-pure-mathematics-1921/eq-5554e7def1",16,"Hardy 1921, p. 356: \\int_{0}^{\\infty} \\frac{x^{2}\\, dx}{(x^{2} - a^{2})^{2} + b^{2}x^{2}} = \\frac{\\pi}{2b}"],["hardy-course-of-pure-mathematics-1921/eq-bd2cdd6e4f",16,"Hardy 1921, p. 356: \\int_{0}^{\\infty} \\phi(x)\\, dx = \\sum_{0}^{\\infty} \\frac{1}{(n + 1)^{2}}"],["dickson-theory-of-equations-1922/eq-755ef18d99",16,"Dickson 1922, p. 139: s_k = k\\sum (-1)^{i+j} \\frac{(i+j-1)!}{i!j!} p^iq^j"],["dickson-theory-of-equations-1922/eq-bcc9d28aef",16,"Dickson 1922, p. 139: s_k = k\\sum_{j=0}^K (-1)^j \\frac{(k-j-1)!}{(k-2j)!j!} p^{k-2j} q^j"],["theorem/waring-s-formula-for-the-quadratic",9,"Waring's formula for the quadratic"],["dickson-theory-of-equations-1922/eq-ca54a59572",16,"Dickson 1922, p. 140: s_k = k\\sum (-1)^{r_1 + \\dotsb + r_n} \\frac{(r_1 + \\dotsb + r_n-1)!}{r_1! \\dotsm r_n!} c_1^{r_1} \\dotsm c_n^{r_n}"],["theorem/waring-s-formula",9,"Waring's formula"],["dickson-theory-of-equations-1922/eq-9aee83ffb8",16,"Dickson 1922, p. 139: x^k + \\left(\\frac{q}{x}\\right)^k = c"],["dickson-theory-of-equations-1922/eq-d0d7f21748",16,"Dickson 1922, p. 140: \\Sigma \\alpha_1^a \\alpha_2^b = \\frac{1}{m} (s_a s_b - s_{a+b})"],["dickson-theory-of-equations-1922/eq-8362dad116",16,"Dickson 1922, p. 141: \\Sigma \\alpha_1^4 \\alpha_2^3 \\alpha_3^2 = s_2s_3s_4 - s_2s_7 - s_3s_6 - s_4s_5 + 2s_9"],["hardy-course-of-pure-mathematics-1921/eq-b1608e240a",16,"Hardy 1921, p. 283: P = -\\frac{a_{1}p + a_{2}q - a_{3}}{c_{1}p + c_{2}q - c_{3}}"],["hardy-course-of-pure-mathematics-1921/eq-64c18584e8",16,"Hardy 1921, p. 283: \\frac{dy}{dx} = -\\frac{f_{a}'}{f_{b}'}"],["boyden-first-book-in-algebra-1895/ex-15/2",4,"Boyden 1895, Exercise 15 (2)"],["concept/cosine",7,"cosine","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-cosine"],["hardy-course-of-pure-mathematics-1921/eq-41ce9a8bf5",16,"Hardy 1921, p. 283: (x - x_{0}) f_{x_{0}}'(x_{0}, y_{0}) + (y - y_{0}) f_{y_{0}}'(x_{0}, y_{0}) = 0"],["hardy-course-of-pure-mathematics-1921/eq-15e7a1924c",16,"Hardy 1921, p. 284: F'(x) = f(x)"],["hardy-course-of-pure-mathematics-1921/eq-f2767503d1",16,"Hardy 1921, p. 284: s = m_{0}\\delta_{0} + m_{1}\\delta_{1} + \\dots + m_{n}\\delta_{n}"],["hardy-course-of-pure-mathematics-1921/eq-7e21b04086",16,"Hardy 1921, p. 284: S \\geq m(b - a)"],["hardy-course-of-pure-mathematics-1921/eq-2bb53da838",16,"Hardy 1921, p. 284: s \\leq M(b - a)"],["hardy-course-of-pure-mathematics-1921/eq-e318568491",16,"Hardy 1921, p. 286: 0 \\leq J - s < \\epsilon"],["concept/continuous-function",7,"continuous function","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-continuous-function"],["de-morgan-elementary-illustrations-calculus-1899/eq-e210e4f558",16,"De Morgan 1899, p. 102: (a + da)^{b} = a^{b} + ba^{b-1}\\, da + \\etc."],["de-morgan-elementary-illustrations-calculus-1899/eq-65ab71632b",16,"De Morgan 1899, p. 102: \\dfrac{dz}{da} = ba^{b-1}"],["de-morgan-elementary-illustrations-calculus-1899/eq-bd356a8f96",16,"De Morgan 1899, p. 102: a^{b+db} = a^{b}\\, a^{db} = a^{b}(1 + \\log a\\, db + \\etc.)"],["de-morgan-elementary-illustrations-calculus-1899/eq-d48dc49a7a",16,"De Morgan 1899, p. 102: \\dfrac{dz}{db} = a^{b} \\log a"],["hardy-course-of-pure-mathematics-1921/eq-62b2eed216",16,"Hardy 1921, p. 286: S - s = \\tsum (M_{\\nu} - m_{\\nu})\\, \\delta_{\\nu} < \\epsilon"],["hardy-course-of-pure-mathematics-1921/eq-47e45f8b14",16,"Hardy 1921, p. 356: \\int_{1}^{\\infty} dx \\left(\\int_{1}^{\\infty} \\frac{x - y}{(x + y)^{3}}\\, dy\\right) = -1"],["boyden-first-book-in-algebra-1895/ex-15/3",4,"Boyden 1895, Exercise 15 (3)"],["concept/order-of-integration",7,"order of integration"],["form/bdc3ed07bf",5,"identity: -18*a**3*x"],["hardy-course-of-pure-mathematics-1921/eq-cf5734d245",16,"Hardy 1921, p. 356: \\int_{1}^{\\infty} dy \\left(\\int_{1}^{\\infty} \\frac{x - y}{(x + y)^{3}}\\, dx\\right) = 1"],["hardy-course-of-pure-mathematics-1921/eq-6cde32c20d",16,"Hardy 1921, p. 286: \\sigma = \\tsum f_{\\nu}\\delta_{\\nu}"],["macfarlane-vector-analysis-quaternions-1906/x-ba6e59d0f7",15,"Macfarlane 1906: The additional angle \\overline{\\phi/} is introduced to specify the ..."],["hardy-course-of-pure-mathematics-1921/eq-38ff9778a9",16,"Hardy 1921, p. 356: \\int_{1}^{\\infty} dx \\left(\\int_{1}^{\\infty} \\frac{x^{2} - y^{2}}{(x^{2} + y^{2})^{2}}\\, dy\\right) = -\\tfrac{1}{4}\\pi"],["hardy-course-of-pure-mathematics-1921/eq-975d6e2226",16,"Hardy 1921, p. 356: \\int_{1}^{\\infty} dy \\left(\\int_{1}^{\\infty} \\frac{x^{2} - y^{2}}{(x^{2} + y^{2})^{2}}\\, dx\\right) = \\tfrac{1}{4}\\pi"],["boyden-first-book-in-algebra-1895/eq-4ba2a5499f",16,"Boyden 1895: b - \\frac{a - x}{c} = \\frac{bc - a + x}{c}"],["de-morgan-elementary-illustrations-calculus-1899/x-6434bd59fe",15,"De Morgan 1899, p. 77: For example, let x^{2} + x - 4 = ..."],["de-morgan-elementary-illustrations-calculus-1899/x-e4f1657e50",15,"De Morgan 1899, p. 77: A near value of x is 1.57; let this ..."],["de-morgan-elementary-illustrations-calculus-1899/x-6e57f96a72",15,"De Morgan 1899, p. 77: If we proceed in the same way with 1.5616, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-517d1d9413",15,"De Morgan 1899, p. 78: We have here chosen an equation of the second ..."],["hardy-course-of-pure-mathematics-1921/eq-67071498ac",16,"Hardy 1921, p. 358: \\log x = \\int \\frac{dx}{x}"],["boyden-first-book-in-algebra-1895/ex-15/4",4,"Boyden 1895, Exercise 15 (4)"],["form/34b21f2a6a",5,"identity: -42*x"],["de-morgan-elementary-illustrations-calculus-1899/eq-a45ffc88c1",16,"De Morgan 1899, p. 101: \\frac{dz}{dx} = \\frac{dz}{da}\\, \\frac{da}{dx} + \\frac{dz}{db}\\, \\frac{db}{dx}"],["de-morgan-elementary-illustrations-calculus-1899/eq-614cda0c07",16,"De Morgan 1899, p. 101: \\dfrac{dz}{db} = b"],["de-morgan-elementary-illustrations-calculus-1899/eq-f8c2ff84cc",16,"De Morgan 1899, p. 101: \\dfrac{dz}{db} = a"],["de-morgan-elementary-illustrations-calculus-1899/eq-67ef441bc6",16,"De Morgan 1899, p. 101: \\frac{dz}{dx} = b\\, \\frac{da}{dx} + a\\, \\frac{db}{dx}"],["de-morgan-elementary-illustrations-calculus-1899/eq-379482b40b",16,"De Morgan 1899, p. 102: z = \\dfrac{a}{b}"],["de-morgan-elementary-illustrations-calculus-1899/eq-e29efdf6ef",16,"De Morgan 1899, p. 102: \\frac{dz}{dx} = \\frac{1}{b}\\, \\frac{da}{dx} - \\frac{a}{b^{2}}\\, \\frac{db}{dx}"],["de-morgan-elementary-illustrations-calculus-1899/eq-fab0ba8d60",16,"De Morgan 1899, p. 102: z = a^{b}"],["dickson-theory-of-equations-1922/eq-99dd9da92e",16,"Dickson 1922, p. 143: x = -\\frac{b}{a} = -\\frac{d}{c}"],["dickson-theory-of-equations-1922/eq-a3f632d227",16,"Dickson 1922, p. 143: R \\equiv ad - bc = 0"],["de-morgan-elementary-illustrations-calculus-1899/eq-bb9f8e5759",16,"De Morgan 1899, p. 102: \\frac{dz}{dx} = ba^{b-1}\\, \\frac{da}{dx} + a^{b} \\log a\\, \\frac{db}{dx}"],["boyden-first-book-in-algebra-1895/ex-15/5",4,"Boyden 1895, Exercise 15 (5)"],["concept/absolute-convergence",7,"absolute convergence","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-absolute-convergence"],["form/34133810a2",5,"identity: 10*x**2"],["boyden-first-book-in-algebra-1895/ex-15/6",4,"Boyden 1895, Exercise 15 (6)"],["form/353f3e2cf4",5,"identity: -10*a*b**2*x"],["shape/3ecf0fc6b7",6,"identity: N*a*b**N*x"],["boyden-first-book-in-algebra-1895/ex-33/1",4,"Boyden 1895, Exercise 33 (1)"],["dickson-theory-of-equations-1922/eq-a7cdbc3f9b",16,"Dickson 1922, p. 152: D = a_0^{2m-2}(\\alpha_1 - \\alpha_2)^2(\\alpha_1 - \\alpha_3)^2 \\dotsm (\\alpha_1 - \\alpha_m)^2(\\alpha_2 - \\alpha_3)^2 \\dots"],["dickson-theory-of-equations-1922/eq-11dd105fd5",16,"Dickson 1922, p. 152: f'(\\alpha_1) = a_0(\\alpha_1 - \\alpha_2)(\\alpha_1 - \\alpha_3) \\dotsm (\\alpha_1 - \\alpha_m)"],["dickson-theory-of-equations-1922/eq-23d142c83a",16,"Dickson 1922, p. 152: D = (-1)^{\\frac{m(m-1)}{2}} \\frac{1}{a_0} R(f, f')"],["dickson-theory-of-equations-1922/eq-12e3237baa",16,"Dickson 1922, p. 147: f \\equiv (x-c)\\alpha"],["concept/uniqueness",7,"uniqueness","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-uniqueness"],["hardy-course-of-pure-mathematics-1921/eq-c51a295a07",16,"Hardy 1921, p. 359: D_{x} \\log x = 1/x"],["hardy-course-of-pure-mathematics-1921/eq-4654bde320",16,"Hardy 1921, p. 359: \\log x = \\int_{1}^{x} \\frac{dt}{t} = -\\int_{x}^{1} \\frac{dt}{t} < 0"],["form/972a05c4d7",5,"factor: -a**2 + x**2"],["shape/19bf1684df",6,"factor: -a**N + x**N"],["hardy-course-of-pure-mathematics-1921/eq-05db6ca01f",16,"Hardy 1921, p. 359: \\log x = \\int_{1}^{x} \\frac{dt}{t} = -\\int_{1}^{1/x} \\frac{du}{u} = -\\log(1/x)"],["todhunter-spherical-trigonometry-1886/eq-cd0a423266",16,"Todhunter 1886, scan 146: \\tan R = \\frac{2 \\sin \\frac{1}{2} a \\sin \\tfrac{1}{2} b \\sin \\tfrac{1}{2} c}{n}"],["hardy-course-of-pure-mathematics-1921/x-58671b2085",15,"Hardy 1921, p. 445: But the infinite of geometry is an actual and ..."],["hardy-course-of-pure-mathematics-1921/eq-9112fa9327",16,"Hardy 1921, p. 360: f(xy) = f(x) + f(y)"],["hardy-course-of-pure-mathematics-1921/eq-c67d6cb998",16,"Hardy 1921, p. 363: \\log x^{n} = n\\log x"],["hardy-course-of-pure-mathematics-1921/eq-e10f0b5935",16,"Hardy 1921, p. 363: \\log e^{n} = n\\log e = n"],["de-morgan-elementary-illustrations-calculus-1899/x-86684b0135",15,"De Morgan 1899, p. 80: Therefore the numerator of each of the fractions \\dfrac{p}{a}, ..."],["de-morgan-elementary-illustrations-calculus-1899/x-5a01cc731b",15,"De Morgan 1899, p. 81: The last equation gives a striking illustration of the ..."],["de-morgan-elementary-illustrations-calculus-1899/x-4cbe7b1bfb",15,"De Morgan 1899, p. 82: The symbol \\phi(x, y) must not be confounded with ..."],["de-morgan-elementary-illustrations-calculus-1899/x-4cf9d381fc",15,"De Morgan 1899, p. 80: Neither are we allowed to say that \\dfrac{p}{a} divided ..."],["de-morgan-elementary-illustrations-calculus-1899/x-380152ec4a",15,"De Morgan 1899, p. 83: The etc. is the representative of an infinite series ..."],["hardy-course-of-pure-mathematics-1921/eq-ad054a0478",16,"Hardy 1921, p. 364: \\log e^{y} = y"],["de-morgan-elementary-illustrations-calculus-1899/x-722801497c",15,"De Morgan 1899, p. 79: in which, however, it must be remembered, that du ..."],["concept/cube-root-of-unity",7,"cube root of unity","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-cube-root-of-unity"],["boyden-first-book-in-algebra-1895/ex-15/8",4,"Boyden 1895, Exercise 15 (8)"],["concept/electric-fluid",7,"electric fluid","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electric-fluid"],["boyden-first-book-in-algebra-1895/ex-15/7",4,"Boyden 1895, Exercise 15 (7)"],["form/a7e0e784a9",5,"identity: -4*a*b + 10*c*x"],["shape/641a86e70e",6,"identity: N*a*b + N*c*x"],["planck-treatise-on-thermodynamics-1903/eq-cd704b4aa5",16,"Planck 1903, p. 191: \\lambda = \\frac{R}{m} \\theta^{2} · \\frac{d \\log \\dfrac{p}{p_{0}}}{d\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-6dcded4acb",16,"Planck 1903, p. 192: c' = \\frac{M_{2}'}{M_{1}'}"],["planck-treatise-on-thermodynamics-1903/eq-c31d64130f",16,"Planck 1903, p. 192: M_{1}'\\, \\frac{\\dd^{2} \\Psi'}{\\dd M_{1}'\\, \\dd M_{2}'} = \\varphi'"],["planck-treatise-on-thermodynamics-1903/eq-b11f7960a4",16,"Planck 1903, p. 193: \\frac{\\dd^{2} \\Psi'}{\\dd M_{1}'^{2}} = -\\frac{M_{2}'}{M_{1}'^{2}} · \\varphi'"],["planck-treatise-on-thermodynamics-1903/eq-6922eedbc9",16,"Planck 1903, p. 193: \\delta^{2} \\Psi < 0"],["planck-treatise-on-thermodynamics-1903/eq-4598c83ef4",16,"Planck 1903, p. 197: \\left(\\frac{\\dd p}{\\dd \\theta}\\right)_{c} = \\frac{L}{\\theta · s}"],["hardy-course-of-pure-mathematics-1921/eq-0e119c5541",16,"Hardy 1921, p. 364: y = \\log x,\\quad x = e^{y}"],["planck-treatise-on-thermodynamics-1903/eq-a508b10cf9",16,"Planck 1903, p. 194: \\frac{L_{1}}{\\theta^{2}}\\, d\\theta - \\frac{s_{1}}{\\theta}\\, dp - \\varphi'\\, dc' + \\varphi''\\, dc'' = 0"],["planck-treatise-on-thermodynamics-1903/eq-9d17b9e0d6",16,"Planck 1903, p. 195: \\frac{L_{2}}{\\theta^{2}}\\, d\\theta - \\frac{s_{2}}{\\theta}\\, dp - \\varphi'\\, \\frac{dc'}{c'} + \\varphi''\\, \\frac{dc''}{c''"],["planck-treatise-on-thermodynamics-1903/eq-70db3afef3",16,"Planck 1903, p. 195: dp = \\frac{\\left(\\dfrac{c''}{c'} - 1\\right) \\theta \\varphi'\\, dc'}{s_{1} + c'' s_{2}}"],["planck-treatise-on-thermodynamics-1903/eq-80d55a3b17",16,"Planck 1903, p. 195: dc'' = \\frac{\\left(\\dfrac{1}{s_{1}} + \\dfrac{1}{c's_{2}}\\right)}{\\left(\\dfrac{1}{s_{1}} + \\dfrac{1}{c'' s_{2}}\\right)} ·"],["planck-treatise-on-thermodynamics-1903/eq-a419dde12a",16,"Planck 1903, p. 194: c'' = 0"],["planck-treatise-on-thermodynamics-1903/eq-d94e45e2ac",16,"Planck 1903, p. 197: s = v = \\frac{R}{m} · \\frac{\\theta}{p}"],["planck-treatise-on-thermodynamics-1903/eq-c74fd04a22",16,"Planck 1903, p. 198: \\Delta = \\frac{R}{m} \\theta^{2} \\left(\\frac{\\dd \\log \\dfrac{p}{p_{0}}}{\\dd \\theta}\\right)_{c}"],["planck-treatise-on-thermodynamics-1903/eq-f64240e7c6",16,"Planck 1903, p. 198: \\left(\\frac{\\dd p}{\\dd c}\\right)_{\\theta} = -\\frac{\\theta\\varphi}{s}"],["planck-treatise-on-thermodynamics-1903/eq-aac5b85949",16,"Planck 1903, p. 199: \\frac{p - p_{0}}{p} = \\frac{cm\\varphi}{R}"],["planck-treatise-on-thermodynamics-1903/eq-2d6bda1396",16,"Planck 1903, p. 200: \\left(\\frac{\\dd \\theta}{\\dd c}\\right)_{p} = \\frac{\\theta^{2} \\varphi}{L}"],["planck-treatise-on-thermodynamics-1903/eq-96aea18a0c",16,"Planck 1903, p. 200: \\theta - \\theta_{0} = \\frac{c\\theta^{2} \\varphi}{L}"],["hardy-course-of-pure-mathematics-1921/x-37c0e31abc",15,"Hardy 1921, p. 349: We can now extend this result to all cases ..."],["planck-treatise-on-thermodynamics-1903/eq-e1820622d2",16,"Planck 1903, p. 201: \\left(\\frac{\\dd \\theta'}{\\dd c}\\right)_{p} = -\\frac{\\theta^{2} \\varphi}{L'}"],["concept/heat-of-solidification",7,"heat of solidification","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-heat-of-solidification"],["concept/heat-of-precipitation",7,"heat of precipitation"],["planck-treatise-on-thermodynamics-1903/eq-803d5eaa28",16,"Planck 1903, p. 202: \\theta_{0}' - \\theta' = \\frac{c\\theta^{2} \\varphi}{L'}"],["planck-treatise-on-thermodynamics-1903/eq-1fb2ee7621",16,"Planck 1903, p. 204: \\frac{L}{\\theta^{2}}\\, d\\theta - \\frac{s'}{\\theta}\\, dp' - \\frac{s''}{\\theta}\\, dp'' - \\varphi\\, dc = 0"],["planck-treatise-on-thermodynamics-1903/eq-596f9caf74",16,"Planck 1903, p. 204: \\left(\\frac{\\dd P}{\\dd c}\\right)_{\\theta} = -\\frac{\\theta\\varphi}{s'}"],["planck-treatise-on-thermodynamics-1903/eq-15b391c81d",16,"Planck 1903, p. 205: P = \\frac{c\\theta\\varphi}{v}"],["hardy-course-of-pure-mathematics-1921/x-773ec13ca6",15,"Hardy 1921, p. 354: Deduce that if \\sum c_{n} is convergent then its ..."],["todhunter-spherical-trigonometry-1886/eq-cb9be38ba7",16,"Todhunter 1886, scan 145: 0 = \\cos \\theta \\cos \\left(\\frac{\\pi}{2} - \\frac{c}{2}\\right) + \\sin \\theta \\sin \\left(\\frac{\\pi}{2} - \\frac{c}{2}\\right"],["todhunter-spherical-trigonometry-1886/eq-df24f3de26",16,"Todhunter 1886, scan 146: \\cos PAQ = cos\\tfrac{1}{2}(B-C)"],["todhunter-spherical-trigonometry-1886/eq-fff9b4163b",16,"Todhunter 1886, scan 146: \\cos PQ = \\cos PA \\cos QA + \\sin PA \\sin QA \\cos \\tfrac{1}{2}(B-C)"],["todhunter-spherical-trigonometry-1886/eq-fcbe7e2519",16,"Todhunter 1886, scan 146: \\sin PA = \\frac{\\sin PE}{\\sin PAE} = \\frac{\\sin r}{\\sin\\tfrac{1}{2}A}"],["todhunter-spherical-trigonometry-1886/eq-57fd2af42d",16,"Todhunter 1886, scan 146: \\cos PQ = \\cos R \\cos r \\cos(s-a) + \\sin R \\sin r \\sin \\tfrac{1}{2}(b+c) \\operatorname{cosec} \\tfrac{1}{2}a"],["todhunter-spherical-trigonometry-1886/eq-f4a8b7299f",16,"Todhunter 1886, scan 146: \\cot r = \\frac{\\sin s}{n}"],["hardy-course-of-pure-mathematics-1921/x-1f16498c12",15,"Hardy 1921, p. 347: In case (3) the circle is called the of ..."],["hardy-course-of-pure-mathematics-1921/eq-31586c938c",16,"Hardy 1921, p. 363: 1 = \\int_{1}^{e} \\frac{dt}{t}"],["hardy-course-of-pure-mathematics-1921/eq-b0f6beedc7",16,"Hardy 1921, p. 365: dy/dx = 1/x"],["todhunter-spherical-trigonometry-1886/eq-3ff02703cc",16,"Todhunter 1886, scan 146: \\frac{\\cos PQ}{\\cos R \\sin r} = \\cot r \\cos(s-a) + \\tan R \\sin \\frac{1}{2}(b+c) \\operatorname{cosec} \\frac{1}{2}a"],["todhunter-spherical-trigonometry-1886/eq-88a959a7cc",16,"Todhunter 1886, scan 146: \\cos^2 PQ = \\cos^2 R \\sin^2 r + \\cos^2 (R-r)"],["todhunter-spherical-trigonometry-1886/eq-8d9436e32a",16,"Todhunter 1886, scan 146: \\sin^2 PQ = \\sin^2 (R-r) - \\cos^2 R \\sin^2 r"],["todhunter-spherical-trigonometry-1886/eq-d6651eee52",16,"Todhunter 1886, scan 147: \\cos QQ_1 = \\cos R \\cos r_1 \\cos (s-c) - \\sin R \\sin r_1 \\sin \\tfrac{1}{2}(C-A) \\sec \\tfrac{1}{2}B"],["todhunter-spherical-trigonometry-1886/eq-dfc4828fcb",16,"Todhunter 1886, scan 147: \\cos^2 QQ_1 = \\cos^2 R \\sin^2 r_1 + \\cos^2 (R + r_1)"],["todhunter-spherical-trigonometry-1886/eq-bab61a2e0c",16,"Todhunter 1886, scan 147: \\sin^2 QQ_1 = \\sin^2 (R + r_1) - \\cos^2 R \\sin^2 r_1"],["todhunter-spherical-trigonometry-1886/eq-37de993ee3",16,"Todhunter 1886, scan 148: \\sin BQ = \\sin CQ"],["hardy-course-of-pure-mathematics-1921/eq-fb6b9c4cac",16,"Hardy 1921, p. 286: F(x) = \\int f(x)\\, dx"],["todhunter-spherical-trigonometry-1886/eq-a791c31890",16,"Todhunter 1886, scan 148: \\sin BD \\sin CE \\sin AF= \\sin CD \\sin AE \\sin BF"],["concept/concurrence-of-arcs",7,"concurrence of arcs"],["todhunter-spherical-trigonometry-1886/eq-83d4ebb4d8",16,"Todhunter 1886, scan 148: \\dfrac{\\sin BD}{\\sin CD}\\, \\dfrac{\\sin CE}{\\sin AE}\\, \\dfrac{\\sin AF}{\\sin BF} = 1"],["todhunter-spherical-trigonometry-1886/eq-160d7d9c36",16,"Todhunter 1886, scan 150: \\surd(1-\\cos^2 \\alpha-\\cos^2 \\beta-\\cos^2 \\gamma+2\\cos \\alpha \\cos \\beta \\cos \\gamma)"],["todhunter-spherical-trigonometry-1886/eq-26519385f6",16,"Todhunter 1886, scan 152: 2E=3a+4b+5c+6d+ \\ldots\\ldots"],["todhunter-spherical-trigonometry-1886/eq-982541c041",16,"Todhunter 1886, scan 152: 2E= 3 \\alpha + 4 \\beta + 5 \\gamma + 6 \\delta + \\ldots\\ldots"],["todhunter-spherical-trigonometry-1886/eq-ab005abd91",16,"Todhunter 1886, scan 152: F=a+b+c+d+ \\ldots\\ldots"],["todhunter-spherical-trigonometry-1886/eq-615696e1eb",16,"Todhunter 1886, scan 152: S= \\alpha + \\beta + \\gamma + \\delta + \\ldots\\ldots"],["hardy-course-of-pure-mathematics-1921/eq-ab1600dd79",16,"Hardy 1921, p. 286: (PpqQ) = \\int_{a}^{b} f(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-a901ed35fb",16,"Hardy 1921, p. 287: \\ds\\int_{a}^{b} f(x)\\, dx = F(b) - F(a)"],["hardy-course-of-pure-mathematics-1921/eq-c3147220ec",16,"Hardy 1921, p. 365: \\frac{dx}{dy} = x = e^{y}"],["todhunter-spherical-trigonometry-1886/eq-f3337c6b61",16,"Todhunter 1886, scan 152: 2E-3F = b + 2c + 3d + \\ldots\\ldots"],["todhunter-spherical-trigonometry-1886/eq-42242cebb6",16,"Todhunter 1886, scan 152: 2E-3S = \\beta + 2 \\gamma + 3 \\delta + \\ldots\\ldots"],["todhunter-spherical-trigonometry-1886/eq-9753ddcc87",16,"Todhunter 1886, scan 153: 2F + 2S=4 + 2E"],["todhunter-spherical-trigonometry-1886/eq-74caa3a88b",16,"Todhunter 1886, scan 153: 2 (\\alpha + \\beta + \\gamma + \\delta + \\ldots) - (a + 2b + 3c + 4d + \\ldots) = 4"],["todhunter-spherical-trigonometry-1886/eq-c0969412fd",16,"Todhunter 1886, scan 153: 2 (a + b + c + d + \\ldots) - (\\alpha + 2 \\beta + 3 \\gamma + 4 \\delta + \\ldots) = 4"],["todhunter-spherical-trigonometry-1886/eq-bfde24d6fd",16,"Todhunter 1886, scan 153: a + \\alpha - (c + \\gamma) - 2 (d + \\delta) - 3 (e + \\epsilon) - \\ldots\\ldots = 8"],["todhunter-spherical-trigonometry-1886/eq-697a3bf215",16,"Todhunter 1886, scan 153: 3a + 2b + c - e - 2f - \\ldots\\ldots -2\\beta - 4\\gamma - \\ldots\\ldots = 12"],["hardy-course-of-pure-mathematics-1921/eq-a3b02b7342",16,"Hardy 1921, p. 287: F(x) = F(a) + \\int_{a}^{x} f(t)\\, dt"],["hardy-course-of-pure-mathematics-1921/eq-8de745a33e",16,"Hardy 1921, p. 289: \\arctan m = \\int_{0}^{m} \\frac{dt}{1 + t^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-388281d235",16,"Hardy 1921, p. 365: dx/dy = ae^{ay}"],["todhunter-spherical-trigonometry-1886/eq-6dc53ac76f",16,"Todhunter 1886, scan 154: 1 + e + e' = s + s' + F"],["todhunter-spherical-trigonometry-1886/eq-5bde5974a3",16,"Todhunter 1886, scan 154: 1 + e' = s' + F"],["todhunter-spherical-trigonometry-1886/eq-7db4fb0f5a",16,"Todhunter 1886, scan 154: \\mathrm{S + F = E + P + 1 }"],["theorem/cauchy-s-extension-of-euler-s-theorem",9,"Cauchy's extension of Euler's theorem"],["todhunter-spherical-trigonometry-1886/eq-01380bf85c",16,"Todhunter 1886, scan 156: \\tan c = \\frac{\\cot A \\cot a + \\cot B \\cot b} {\\cot a \\cot b - \\cos A \\cos B}"],["todhunter-spherical-trigonometry-1886/eq-e0d5206458",16,"Todhunter 1886, scan 156: \\cos \\theta \\sin(b - c) + \\cos \\phi \\sin(c - a) + \\cos \\psi \\sin(a - b) = 0"],["todhunter-spherical-trigonometry-1886/eq-e0f5c63ad2",16,"Todhunter 1886, scan 156: \\cos PA \\cos BC = \\cos PB \\cos CA = \\cos PC \\cos AB"],["dickson-theory-of-equations-1922/eq-4094994b57",16,"Dickson 1922, p. 156: D \\equiv a_1\\left(\\frac{1}{z}\\right) + \\dotsb + a_n\\left(\\frac{1}{z}\\right)^n"],["dickson-theory-of-equations-1922/eq-57f0b061cd",16,"Dickson 1922, p. 156: |f(z)| \\geqq |z|^n \\bigl[1 - |D|\\bigr]"],["dickson-theory-of-equations-1922/eq-763906457f",16,"Dickson 1922, p. 156: |f(z)| > \\rho^n(1-p) \\geqq P"],["todhunter-spherical-trigonometry-1886/eq-c5ede1faa2",16,"Todhunter 1886, scan 156: \\tan \\alpha \\tan \\alpha' = \\tan \\beta \\tan \\beta' = \\tan \\gamma \\tan \\gamma'"],["todhunter-spherical-trigonometry-1886/eq-1eb32181fb",16,"Todhunter 1886, scan 156: \\frac{\\cos p}{\\cos \\alpha \\cos \\alpha'} = \\frac{\\cos q}{\\cos \\beta \\cos \\beta' } = \\frac{\\cos r}{\\cos \\gamma \\cos \\gamma"],["todhunter-spherical-trigonometry-1886/eq-f585cb1c96",16,"Todhunter 1886, scan 157: \\frac{\\sin \\alpha}{\\sin \\alpha'} = 2 \\cos \\frac{a}{2}"],["todhunter-spherical-trigonometry-1886/eq-3fabae2229",16,"Todhunter 1886, scan 157: \\cos AQ \\sin \\frac{a}{2} = \\sin \\frac{c - b}{2} \\sin \\frac{c + b}{2}"],["todhunter-spherical-trigonometry-1886/eq-5fed82367d",16,"Todhunter 1886, scan 157: \\sin AB \\sin CD \\cos P = \\sin AD \\sin BC \\cos Q = \\sin AC \\sin BD \\cos R"],["todhunter-spherical-trigonometry-1886/eq-52db6629ab",16,"Todhunter 1886, scan 157: \\cos A' = \\sin (S - A) \\cos \\frac{a}{2}"],["hardy-course-of-pure-mathematics-1921/eq-5151f6c424",16,"Hardy 1921, p. 365: f(y + z) = f(y)f(z)"],["dickson-theory-of-equations-1922/eq-63021b5876",16,"Dickson 1922, p. 156: \\rho \\geqq \\sqrt[n]{\\frac{P}{1-p}} \\equiv R"],["dickson-theory-of-equations-1922/eq-121cb70165",16,"Dickson 1922, p. 156: f(a+h) = f(a) + f'(a)h + \\dotsb + f^{(r)}(a)·\\frac{h^r}{r!} + \\dotsb + f^{(n)}(a)·\\frac{h^n}{n!}"],["todhunter-spherical-trigonometry-1886/eq-5c0a940a72",16,"Todhunter 1886, scan 158: \\dfrac{\\sin Pa \\cos PA}{\\sin Aa} + \\dfrac{\\sin Pb \\cos PB}{\\sin Bb} + \\dfrac{\\sin Pc \\cos PC}{\\sin Cc} = 1"],["todhunter-spherical-trigonometry-1886/eq-874e33a367",16,"Todhunter 1886, scan 158: a \\cos AP + b \\cos BP = s \\cos SP"],["todhunter-spherical-trigonometry-1886/eq-0a1b0e48a5",16,"Todhunter 1886, scan 158: a \\cos AP + b \\cos BP + c \\cos CP + \\ldots = \\text{constant}"],["de-morgan-elementary-illustrations-calculus-1899/x-1b5cf29bd6",15,"De Morgan 1899, p. 84: For if any result be obtained from a set ..."],["dickson-theory-of-equations-1922/eq-55fdbd4dce",16,"Dickson 1922, p. 156: g(h) \\equiv 1 + bh^r + ch^{r+1} + \\dotsb + lh^n"],["hardy-course-of-pure-mathematics-1921/eq-a518e2622d",16,"Hardy 1921, p. 289: \\phi(m) = \\tfrac{1}{2} m\\mu^{2} + \\int_{\\mu}^{1} \\sqrtp{1 - x^{2}}\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-bc98991963",16,"Hardy 1921, p. 289: \\phi'(m) = \\frac{1}{2(1 + m^{2})}"],["de-morgan-elementary-illustrations-calculus-1899/x-cfa0964183",15,"De Morgan 1899, p. 84: When they are small, the error in the results ..."],["de-morgan-elementary-illustrations-calculus-1899/x-b81e8b4c80",15,"De Morgan 1899, p. 85: Next suppose only the second error, and then only ..."],["de-morgan-elementary-illustrations-calculus-1899/x-d6ff53c393",15,"De Morgan 1899, p. 85: The formulæ employed, like the equations in 28, are ..."],["hardy-course-of-pure-mathematics-1921/eq-86fbfc1e0b",16,"Hardy 1921, p. 289: \\phi(m) = \\tfrac{1}{2} \\int_{0}^{m} \\frac{dt}{1 + t^{2}}"],["hardy-course-of-pure-mathematics-1921/eq-ca4c2a3e11",16,"Hardy 1921, p. 291: \\ds\\int_{a}^{b} f(x)\\, dx = -\\int_{b}^{a} f(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-cc561c4ead",16,"Hardy 1921, p. 292: \\ds\\int_{a}^{a} f(x)\\, dx = 0"],["boyden-first-book-in-algebra-1895/ex-15/9",4,"Boyden 1895, Exercise 15 (9)"],["form/16e39baf0e",5,"identity: -16*x**2"],["planck-treatise-on-thermodynamics-1903/x-fd5296f13a",15,"Planck 1903, p. 243: Van’t Hoff was the first to calculate L by ..."],["hardy-course-of-pure-mathematics-1921/eq-1eec198e49",16,"Hardy 1921, p. 292: \\ds\\int_{a}^{b}f(x)\\, dx + \\int_{b}^{c}f(x)\\, dx = \\int_{a}^{c}f(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-b81f10274f",16,"Hardy 1921, p. 292: \\ds\\int_{a}^{b}kf(x)\\, dx = k \\int_{a}^{b}f(x)\\, dx"],["dickson-theory-of-equations-1922/eq-aae1ab37e0",16,"Dickson 1922, p. 156: h = \\rho(\\cos \\theta + i \\sin \\theta)"],["dickson-theory-of-equations-1922/eq-ab0d37c77b",16,"Dickson 1922, p. 156: b = |b|(\\cos \\beta + i \\sin \\beta)"],["dickson-theory-of-equations-1922/eq-d198694967",16,"Dickson 1922, p. 157: bh^r = |b| \\rho^r \\bigl\\{\\cos(\\beta+r\\theta) + i\\sin (\\beta+r\\theta)\\bigr\\}"],["dickson-theory-of-equations-1922/eq-7b5eabbee8",16,"Dickson 1922, p. 157: g(h) = (1 - |b|\\rho^r) + h^r(ch + \\dotsb + lh^{n-r})"],["dickson-theory-of-equations-1922/eq-ea00ba0510",16,"Dickson 1922, p. 157: |ch + \\dotsb + lh^{n-r}| < |b|"],["dickson-theory-of-equations-1922/eq-3cc9f23397",16,"Dickson 1922, p. 157: |b| \\rho^r < 1"],["dickson-theory-of-equations-1922/eq-b80fa895a6",16,"Dickson 1922, p. 157: |g(h)| < (1 - |b|\\rho^r) + \\rho^r|b|"],["dickson-theory-of-equations-1922/eq-448e97d4c4",16,"Dickson 1922, p. 157: G(x,y) = \\phi^2(x, y) + \\psi^2(x, y)"],["hardy-course-of-pure-mathematics-1921/eq-1d93993c19",16,"Hardy 1921, p. 292: \\ds\\int_{a}^{b}\\{f(x) + \\phi(x)\\}\\, dx = \\int_{a}^{b}f(x)\\, dx + \\int_{a}^{b}\\phi(x)\\, dx"],["hardy-course-of-pure-mathematics-1921/eq-b2e7066e77",16,"Hardy 1921, p. 365: e^{-y} = 1/e^{y}"],["de-morgan-elementary-illustrations-calculus-1899/eq-2603d53bda",16,"De Morgan 1899, p. 107: x^{2} - xy + y^{2} = a"],["de-morgan-elementary-illustrations-calculus-1899/eq-7c136ad3d8",16,"De Morgan 1899, p. 110: xy - x = 1"],["de-morgan-elementary-illustrations-calculus-1899/eq-15d6c67d72",16,"De Morgan 1899, p. 108: u = \\phi(x, y)"],["de-morgan-elementary-illustrations-calculus-1899/eq-fab4d0a130",16,"De Morgan 1899, p. 108: du = \\ux\\, dx + \\uy\\, dy + \\etc."],["de-morgan-elementary-illustrations-calculus-1899/eq-80904a201c",16,"De Morgan 1899, p. 108: \\ux\\, dx + \\uy\\, dy = 0"],["de-morgan-elementary-illustrations-calculus-1899/eq-18aee15c26",16,"De Morgan 1899, p. 108: \\frac{dy}{dx} = -\\frac{\\ux}{\\uy}"],["de-morgan-elementary-illustrations-calculus-1899/eq-249fbb79fa",16,"De Morgan 1899, p. 108: \\frac{dy}{dx} = -\\frac{\\;\\dfrac{du}{dx}\\;}{\\dfrac{du}{dy}}"],["hardy-course-of-pure-mathematics-1921/eq-03e1506c52",16,"Hardy 1921, p. 292: \\ds\\int_{a}^{b}f(x)\\, dx \\geq 0"],["concept/non-negative",7,"non-negative"],["hardy-course-of-pure-mathematics-1921/eq-fcbea156a4",16,"Hardy 1921, p. 292: H(b - a) \\leq \\int_{a}^{b}f(x)\\, dx \\leq K(b - a)"],["boyden-first-book-in-algebra-1895/ex-33/2",4,"Boyden 1895, Exercise 33 (2)"],["todhunter-spherical-trigonometry-1886/eq-7b665ead8b",16,"Todhunter 1886, scan 161: \\sin c=\\dfrac{\\sin a}{\\sin A}"],["todhunter-spherical-trigonometry-1886/eq-1965fa6c3f",16,"Todhunter 1886, scan 161: \\sin b = \\tan a \\cot A"],["hardy-course-of-pure-mathematics-1921/eq-8befb36c3f",16,"Hardy 1921, p. 292: \\ds\\int_{a}^{b}f(x)\\, dx = (b-a)f(\\xi)"],["hardy-course-of-pure-mathematics-1921/eq-15d99a5f83",16,"Hardy 1921, p. 292: F(b) - F(a) = (b - a)F'(\\xi)"],["todhunter-spherical-trigonometry-1886/eq-affc03b8b6",16,"Todhunter 1886, scan 162: \\tan\\tfrac12 A = \\Surd{\\left\\{\\frac{\\sin(s - b)\\sin(s - c)}{\\sin s \\sin(s - a)}\\right\\}}"],["todhunter-spherical-trigonometry-1886/eq-0439ff4fe9",16,"Todhunter 1886, scan 164: \\tan\\dfrac12 (A - B) = \\dfrac{\\sin\\tfrac12 (a - b)}{\\sin\\tfrac12 (a + b)}\\cot\\tfrac12 C"],["todhunter-spherical-trigonometry-1886/eq-92630403c2",16,"Todhunter 1886, scan 164: \\tan\\tfrac12 (A + B) = \\dfrac{\\cos\\tfrac12 (a - b)}{\\cos\\tfrac12 (a + b)}\\cot\\tfrac12 C"],["todhunter-spherical-trigonometry-1886/eq-a744f1059a",16,"Todhunter 1886, scan 164: \\sin c = \\dfrac{\\sin a \\sin C}{\\sin A}"],["hardy-course-of-pure-mathematics-1921/eq-b1def553e1",16,"Hardy 1921, p. 293: H\\int_{a}^{b} \\phi(x)\\, dx \\leq \\int_{a}^{b} f(x)\\phi(x)\\, dx \\leq K\\int_{a}^{b} \\phi(x)\\, dx"],["theorem/generalised-mean-value-theorem-for-integrals-bounds",9,"Generalised Mean Value Theorem for integrals (bounds)"],["hardy-course-of-pure-mathematics-1921/eq-697e376b03",16,"Hardy 1921, p. 293: \\int_{a}^{b} f(x)\\phi(x)\\, dx = f(\\xi) \\int_{a}^{b} \\phi(x)\\, dx"],["todhunter-spherical-trigonometry-1886/eq-6804b95e2b",16,"Todhunter 1886, scan 165: \\cos \\tfrac{1}{2} c = \\dfrac{\\cos \\tfrac{1}{2} (a + b) \\sin \\tfrac{1}{2}C}{\\cos\\tfrac{1}{2}(A+B)}"],["todhunter-spherical-trigonometry-1886/eq-fe2bfe5ccd",16,"Todhunter 1886, scan 165: \\tan \\theta = \\tan b \\cos C"],["todhunter-spherical-trigonometry-1886/eq-ec148c206b",16,"Todhunter 1886, scan 165: \\cos c = \\dfrac{\\cos b \\cos (a - \\theta)}{\\cos \\theta}"],["todhunter-spherical-trigonometry-1886/eq-89542f733c",16,"Todhunter 1886, scan 166: \\sin B = \\dfrac{\\sin b}{\\sin a}\\sin A"],["todhunter-spherical-trigonometry-1886/eq-432e2e689d",16,"Todhunter 1886, scan 166: \\tan \\tfrac12 C = \\dfrac{\\cos \\tfrac12 (b - a)}{\\cos \\tfrac12 (b + a)}\\cot \\tfrac12 (B + A)"],["planck-treatise-on-thermodynamics-1903/eq-694651c1cd",16,"Planck 1903, p. 208: U = \\tsum n_{1} (c_{v_{1}}\\theta + h_{1})"],["hardy-course-of-pure-mathematics-1921/eq-52f8a41000",16,"Hardy 1921, p. 293: F(x) = \\int_{a}^{x} f(t)\\, dt"],["todhunter-spherical-trigonometry-1886/eq-69c3063014",16,"Todhunter 1886, scan 166: \\tan \\tfrac12 c = \\dfrac{\\cos \\tfrac12 (B + A)}{\\cos \\tfrac12 (B - A)}\\tan \\tfrac12 (b + a)"],["boyden-first-book-in-algebra-1895/ex-15/10",4,"Boyden 1895, Exercise 15 (10)"],["form/0dad50badb",5,"identity: 3*x/2"],["planck-treatise-on-thermodynamics-1903/eq-165068cebd",16,"Planck 1903, p. 213: p_{1} = c_{1} p"],["hardy-course-of-pure-mathematics-1921/eq-35e3ffbda3",16,"Hardy 1921, p. 295: \\int_{a}^{b} f(x)\\phi'(x)\\, dx = f(b)\\phi(b) - f(a)\\phi(a) - \\int_{a}^{b} f'(x)\\phi(x)\\, dx"],["form/2987f9d612",5,"identity: 8*a**4*b + 3*a*b - x**5"],["shape/c8d2bad7a7",6,"identity: N*a*b + N*a**N*b - x**N"],["boyden-first-book-in-algebra-1895/ex-15/11",4,"Boyden 1895, Exercise 15 (11)"],["boyden-first-book-in-algebra-1895/ex-15/12",4,"Boyden 1895, Exercise 15 (12)"],["form/0da602567b",5,"identity: -7*x/12"],["de-morgan-elementary-illustrations-calculus-1899/x-a2f813b0aa",15,"De Morgan 1899, p. 86: This term, in theory, is the only one on ..."],["de-morgan-elementary-illustrations-calculus-1899/x-8764efbbaa",15,"De Morgan 1899, p. 86: When the exponent is negative, or when y = ..."],["de-morgan-elementary-illustrations-calculus-1899/x-3d0b68f365",15,"De Morgan 1899, p. 86: The negative sign indicates that an increase in x ..."],["boyden-first-book-in-algebra-1895/ex-15/13",4,"Boyden 1895, Exercise 15 (13)"],["boyden-first-book-in-algebra-1895/ex-15/14",4,"Boyden 1895, Exercise 15 (14)"],["planck-treatise-on-thermodynamics-1903/x-c6d01847e2",15,"Planck 1903, p. 261: Since the addition of silver nitrate increases the number ..."],["theorem/integration-by-parts-for-a-definite-integral",9,"Integration by parts for a definite integral"],["planck-treatise-on-thermodynamics-1903/eq-f5250d52e3",16,"Planck 1903, p. 209: \\Phi = \\tsum n_{1} \\left(c_{v_{1}} \\log \\theta + R \\log \\frac{\\theta}{p}\\right) + C"],["planck-treatise-on-thermodynamics-1903/eq-6af7933b49",16,"Planck 1903, p. 212: n (c_{v} \\log \\theta + R \\log \\frac{\\theta}{p} + k)"],["planck-treatise-on-thermodynamics-1903/eq-9ec19c7703",16,"Planck 1903, p. 213: n = \\dfrac{M}{m}"],["planck-treatise-on-thermodynamics-1903/eq-f9704b4c3c",16,"Planck 1903, p. 213: \\Phi = \\tsum n_{1} (c_{v_{1}} \\log \\theta + R \\log \\frac{\\theta}{p_{1}} + k_{1})"],["planck-treatise-on-thermodynamics-1903/eq-659a2ae639",16,"Planck 1903, p. 213: \\tsum p_{1} = p"],["concept/dalton",7,"Dalton"],["planck-treatise-on-thermodynamics-1903/eq-4b9c9201a5",16,"Planck 1903, p. 213: c_{1} = \\frac{n_{1}}{n_{1} + n_{2} + \\dots}"],["hardy-course-of-pure-mathematics-1921/eq-494df6e75d",16,"Hardy 1921, p. 295: \\int f\\{\\phi(x)\\}\\phi'(x)\\, dx = F\\{\\phi(x)\\}"],["hardy-course-of-pure-mathematics-1921/eq-9fbd544332",16,"Hardy 1921, p. 366: \\lim y^{\\alpha}/e^{y} = \\lim e^{-y}y^{\\alpha} = 0"],["hardy-course-of-pure-mathematics-1921/eq-3857592cf1",16,"Hardy 1921, p. 361: \\frac{\\log x}{x^{\\alpha}} \\to 0"],["planck-treatise-on-thermodynamics-1903/eq-42966d88eb",16,"Planck 1903, p. 213: \\Phi = \\tsum n_{1} (c_{v_{1}} \\log \\theta + R \\log \\frac{\\theta}{pc_{1}} + k_{1})"],["planck-treatise-on-thermodynamics-1903/eq-838ae84f1d",16,"Planck 1903, p. 214: C = \\tsum n_{1} (k_{1} - R \\log c_{1})"],["planck-treatise-on-thermodynamics-1903/eq-3d9f7e917f",16,"Planck 1903, p. 214: -n_{1} R \\log c_{1} - n_{2} R \\log c_{2}"],["concept/increase-of-entropy",7,"increase of entropy"],["planck-treatise-on-thermodynamics-1903/eq-b87721437a",16,"Planck 1903, p. 215: \\Psi = \\tsum n_{1} (\\varphi_{1} - R \\log c_{1})"],["concept/characteristic-function",7,"characteristic function"],["planck-treatise-on-thermodynamics-1903/eq-60cb399748",16,"Planck 1903, p. 215: c_{v_{1}} \\log \\theta - \\frac{h_{1}}{\\theta} + R \\log \\frac{\\theta}{p} + k_{1} - c_{v_{1}} - R = \\varphi_{1}"],["concept/equilibrium-condition",7,"equilibrium condition"],["hardy-course-of-pure-mathematics-1921/eq-bfb7a4880b",16,"Hardy 1921, p. 295: \\int_{c}^{d} f(t)\\, dt = F(d) - F(c) = F\\{\\phi(b)\\} - F\\{\\phi(a)\\} = \\int_{a}^{b} f\\{\\phi(x)\\}\\phi'(x)\\, dx"],["theorem/transformation-of-a-definite-integral-by-substitution",9,"Transformation of a definite integral by substitution"],["planck-treatise-on-thermodynamics-1903/eq-8c9fa502a4",16,"Planck 1903, p. 215: \\tsum (\\varphi_{1} - R \\log c_{1})\\, \\delta n_{1} + \\tsum n_{1}\\, \\delta(\\varphi_{1} - R \\log c_{1}) = 0"],["planck-treatise-on-thermodynamics-1903/eq-6ce771b120",16,"Planck 1903, p. 216: \\delta n_{1} : \\delta n_{2} : \\dots = \\nu_{1} : \\nu_{2} : \\dots"],["planck-treatise-on-thermodynamics-1903/eq-d434911f3d",16,"Planck 1903, p. 216: \\tsum (\\varphi_{1} - R \\log c_{1}) \\nu_{1} = 0"],["planck-treatise-on-thermodynamics-1903/eq-89dea9bc90",16,"Planck 1903, p. 216: \\nu_{1} \\log c_{1} + \\nu_{2} \\log c_{2} + \\dots = \\frac{\\nu_{1} \\varphi_{1} + \\nu_{2} \\varphi_{2} + \\dots}{R}"],["planck-treatise-on-thermodynamics-1903/eq-cff68b7335",16,"Planck 1903, p. 215: c_{1} + c_{2} + \\dots = 1"],["planck-treatise-on-thermodynamics-1903/eq-7077c547d3",16,"Planck 1903, p. 216: \\frac{\\tsum \\nu_{1} (k_{1} - c_{v_{1}} - R)}{R} = \\log a"],["planck-treatise-on-thermodynamics-1903/eq-e869c35663",16,"Planck 1903, p. 216: \\frac{\\tsum \\nu_{1} h_{1}}{R} = b"],["hardy-course-of-pure-mathematics-1921/eq-1399ff3efb",16,"Hardy 1921, p. 298: f(a + h) = f(a) + hf'(a) + \\dots + \\frac{h^{n-1}}{(n - 1)!} f^{(n-1)}(a) + R_{n}"],["hardy-course-of-pure-mathematics-1921/eq-63af1337f7",16,"Hardy 1921, p. 298: R_{n} = \\frac{h^{n}}{(n - 1)!} \\int_{0}^{1} (1 - t)^{n-1} f^{(n)}(a + th)\\, dt"],["planck-treatise-on-thermodynamics-1903/eq-110d8ba3fd",16,"Planck 1903, p. 216: \\frac{\\tsum \\nu_{1} c_{v_{1}}}{R} = c"],["planck-treatise-on-thermodynamics-1903/eq-8d705882b5",16,"Planck 1903, p. 217: \\nu_{1} \\log c_{1} + \\nu_{2} \\log c_{2} + \\dots = \\log a + (\\nu_{1} + \\nu_{2} + \\dots) \\log \\frac{\\theta}{p} - \\frac{b}{"],["planck-treatise-on-thermodynamics-1903/eq-a9c58ed43f",16,"Planck 1903, p. 217: \\prod c_{1}^{\\nu_{1}} = a\\left(\\frac{\\theta}{p}\\right)^{\\tsum \\nu_{1}} e^{-\\efrac{b}{\\theta}} \\theta^{c}"],["planck-treatise-on-thermodynamics-1903/eq-b10a5af58f",16,"Planck 1903, p. 217: \\prod c_{1}^{\\nu_{1}} = a e^{-\\efrac{b}{\\theta}} \\left(\\frac{\\theta}{p}\\right)^{\\tsum \\nu_{1}}"],["planck-treatise-on-thermodynamics-1903/eq-5430b27408",16,"Planck 1903, p. 217: \\prod c_{1}^{\\nu_{1}} = ae^{-\\efrac{b}{\\theta}} \\left(\\frac{\\theta}{p}\\right)^{\\tsum \\nu_{1}}"],["concept/law-of-mass-action",7,"law of mass action"],["hardy-course-of-pure-mathematics-1921/eq-342b9d1453",16,"Hardy 1921, p. 298: R_{n} = \\frac{(1 - \\theta)^{n-p} f^{(n)}(a + \\theta h)h^{n}}{p(n - 1)!}"],["theorem/form-of-the-remainder-with-parameter-p",9,"Form of the remainder with parameter p"],["hardy-course-of-pure-mathematics-1921/eq-67dd247b1c",16,"Hardy 1921, p. 361: \\lim_{y\\to +0} y^{\\alpha} \\log y = -\\lim_{x\\to +\\infty} (\\log x)/x^{\\alpha} = 0"],["planck-treatise-on-thermodynamics-1903/eq-de0938ee8b",16,"Planck 1903, p. 217: Q = \\delta U + p\\, \\delta V"],["planck-treatise-on-thermodynamics-1903/eq-0bfd1e9695",16,"Planck 1903, p. 217: Q = \\tsum (c_{v_{1}} \\theta + h_{1} + R\\theta)\\, \\delta n_{1}"],["planck-treatise-on-thermodynamics-1903/eq-7619cd1f67",16,"Planck 1903, p. 218: L = \\tsum (c_{v_{1}} \\theta + h_{1} + R\\theta) \\nu_{1}"],["planck-treatise-on-thermodynamics-1903/eq-803c6efb62",16,"Planck 1903, p. 218: L = Rb + R\\theta \\tsum \\nu_{1}"],["planck-treatise-on-thermodynamics-1903/eq-92428de629",16,"Planck 1903, p. 218: L = 1.97 (b + \\theta \\tsum \\nu_{1})"],["planck-treatise-on-thermodynamics-1903/eq-3b05497970",16,"Planck 1903, p. 219: L = 1.97 \\{b + (\\nu_{1} + \\nu_{2} + \\dots) \\theta\\}"],["planck-treatise-on-thermodynamics-1903/eq-2b7b2536cd",16,"Planck 1903, p. 219: c_{1}^{-2} c_{2}^{1} c_{3}^{1} = ae^{-\\efrac{b}{\\theta}}"],["concept/hydriodic-acid",7,"hydriodic acid"],["hardy-course-of-pure-mathematics-1921/eq-7cd0649a2f",16,"Hardy 1921, p. 298: R_{n} = \\frac{(1 - \\theta)^{n-1} f^{(n)}(a + \\theta h) h^{n}}{(n - 1)!}"],["planck-treatise-on-thermodynamics-1903/eq-fc8f50bb1a",16,"Planck 1903, p. 220: a = 0.120"],["planck-treatise-on-thermodynamics-1903/eq-dfec285d1e",16,"Planck 1903, p. 220: b = 1300"],["planck-treatise-on-thermodynamics-1903/eq-0fb43ebc55",16,"Planck 1903, p. 221: L = 1.97 (14690 + \\theta) = 28900 + 1.97\\theta"],["hardy-course-of-pure-mathematics-1921/eq-455b947f75",16,"Hardy 1921, p. 299: R_{n} = \\frac{m(m - 1)\\dots (m - n + 1)}{1·2\\dots (n - 1)}\\, \\frac{(1 - \\theta )^{n-1} x^{n}}{(1 + \\theta x)^{n-m}}"],["planck-treatise-on-thermodynamics-1903/eq-90b3d8b451",16,"Planck 1903, p. 219: \\frac{c_{2}c_{3}}{c_{1}^{2}} = \\frac{n_{2}n_{3}}{n_{1}^{2}} = ae^{-\\efrac{b}{\\theta}}"],["hardy-course-of-pure-mathematics-1921/eq-ec8d98bc9c",16,"Hardy 1921, p. 299: |R_{n}| < K |m| \\left|\\binom{m - 1}{n - 1}\\right| |x^{n}| = \\rho_{n}"],["hardy-course-of-pure-mathematics-1921/eq-abe5024ef3",16,"Hardy 1921, p. 361: \\log x < (x^{\\beta} - 1)/\\beta < x^{\\beta}/\\beta"],["hardy-course-of-pure-mathematics-1921/eq-0e9b0c3fe7",16,"Hardy 1921, p. 362: (\\log\\log y)/(\\log y)^{\\alpha} = (\\log x)/x^{\\alpha} \\to 0"],["hardy-course-of-pure-mathematics-1921/eq-85ccd5216b",16,"Hardy 1921, p. 361: (\\log x)/x^{\\alpha} = -y^{\\alpha} \\log y"],["de-morgan-elementary-illustrations-calculus-1899/eq-bdea9e9ab8",16,"De Morgan 1899, p. 118: v = \\dfrac{h}{m + 1}"],["de-morgan-elementary-illustrations-calculus-1899/eq-dd442119ca",16,"De Morgan 1899, p. 118: \\frac{(m + 2)A}{(m + 2)a} = \\frac{A}{a}"],["de-morgan-elementary-illustrations-calculus-1899/eq-dc2d171b07",16,"De Morgan 1899, p. 118: 1 + 2 + \\dots + (m + 1) = \\frac{1}{2}(m + 1)(m + 2)"],["de-morgan-elementary-illustrations-calculus-1899/eq-6f599eb610",16,"De Morgan 1899, p. 119: \\frac{m + 2}{m + 1}\\, ha^{2} + \\frac{m + 2}{m + 1}\\, ha^{2} + (1 + \\alpha)\\, \\frac{h^{3}}{3}"],["de-morgan-elementary-illustrations-calculus-1899/eq-d5bd7f56e9",16,"De Morgan 1899, p. 119: ha^{2} + ha^{2} + \\frac{h^{3}}{3} \\quad\\text{or}\\quad \\frac{(a + h)^{3} - a^{3}}{3}"],["de-morgan-elementary-illustrations-calculus-1899/eq-d533667612",16,"De Morgan 1899, p. 119: \\int_{a}^{a+h} x^{2}\\, dx"],["de-morgan-elementary-illustrations-calculus-1899/x-7c23a73046",15,"De Morgan 1899, p. 87: Let x = 16°, in which case \\sin x ..."],["de-morgan-elementary-illustrations-calculus-1899/x-fd43c4f2f8",15,"De Morgan 1899, p. 87: These examples may serve to show how nearly the ..."],["hardy-course-of-pure-mathematics-1921/eq-56690ae681",16,"Hardy 1921, p. 369: x = a^{y},\\quad y = \\log_{a} x"],["unit/unit",12,"unit","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-unit-unit"],["quantity/radius",11,"radius","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-quantity-radius"],["de-morgan-elementary-illustrations-calculus-1899/x-b036f20ea7",15,"De Morgan 1899, p. 87: The tables give 3.0004341, differing from the former only ..."],["de-morgan-elementary-illustrations-calculus-1899/x-452dd44f98",15,"De Morgan 1899, p. 88: has been found, the result, being a function of ..."],["de-morgan-elementary-illustrations-calculus-1899/x-531305048b",15,"De Morgan 1899, p. 88: Similarly the differential coefficient of the second differential coefficient ..."],["de-morgan-elementary-illustrations-calculus-1899/x-22ba1963fd",15,"De Morgan 1899, p. 88: In order to avoid so cumbrous a system of ..."],["boyden-first-book-in-algebra-1895/ex-15/16",4,"Boyden 1895, Exercise 15 (16)"],["boyden-first-book-in-algebra-1895/ex-15/15",4,"Boyden 1895, Exercise 15 (15)"],["form/57c4806490",5,"identity: 4*a + 5*b + 5*x"],["shape/cb561731d5",6,"identity: N*a + N*b + N*x"],["shape/3b169f8114",6,"identity: a - b + x"],["de-morgan-elementary-illustrations-calculus-1899/eq-16aa2d8765",16,"De Morgan 1899, p. 123: C = -\\psi a"],["de-morgan-elementary-illustrations-calculus-1899/eq-774debd950",16,"De Morgan 1899, p. 123: \\psi x - \\psi a = 0"],["de-morgan-elementary-illustrations-calculus-1899/eq-b18ffa83d9",16,"De Morgan 1899, p. 123: \\psi x - \\psi a"],["boyden-first-book-in-algebra-1895/ex-15/17",4,"Boyden 1895, Exercise 15 (17)"],["hardy-course-of-pure-mathematics-1921/x-717bc8769d",15,"Hardy 1921, p. 357: The process may fairly be compared with that by ..."],["hardy-course-of-pure-mathematics-1921/eq-c45d1d0bf4",16,"Hardy 1921, p. 370: \\log_{10} x = (\\log_{e} x)/(\\log_{e} 10)"],["hardy-course-of-pure-mathematics-1921/eq-e90e79dd44",16,"Hardy 1921, p. 367: (a^{x})^{y} = a^{xy}"],["hardy-course-of-pure-mathematics-1921/eq-00692a1075",16,"Hardy 1921, p. 367: y = e^{x\\log a}"],["planck-treatise-on-thermodynamics-1903/eq-88bce78dd2",16,"Planck 1903, p. 225: U &= n_{0} u_{0} + n_{1} u_{1} + n_{2} u_{2} + \\dots\\Add{,}"],["planck-treatise-on-thermodynamics-1903/eq-aa1712a81e",16,"Planck 1903, p. 225: V &= n_{0} v_{0} + n_{1} v_{1} + n_{2} v_{2} + \\dots\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-704f940bd9",16,"Planck 1903, p. 224: \\frac{U}{n_{0}} = u_{0} + u_{1}\\, \\frac{n_{1}}{n_{0}} + u_{2}\\, \\frac{n_{2}}{n_{0}} + \\dots\\Add{,}"],["planck-treatise-on-thermodynamics-1903/eq-3eb8a73715",16,"Planck 1903, p. 225: \\frac{U}{n_{0}} = u_{0} + u_{1}\\, \\frac{n_{1}}{n_{0}} + \\dots + u_{11} \\left(\\frac{n_{1}}{n_{0}}\\right)^{2} + 2u_{12}\\, "],["planck-treatise-on-thermodynamics-1903/eq-07d0833d6a",16,"Planck 1903, p. 225: V' = (n_{0} + 1) v_{0} + n_{1} v_{1} + n_{2} v_{2} + \\dots"],["planck-treatise-on-thermodynamics-1903/eq-aa055d1009",16,"Planck 1903, p. 225: U' = (n_{0} + 1) u_{0} + n_{1} u_{1} + n_{2} u_{2} + \\dots\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-66b766e18d",16,"Planck 1903, p. 225: U' - (U + u_{0}) + p \\bigl\\{V' - (V + v_{0})\\bigr\\}"],["planck-treatise-on-thermodynamics-1903/eq-555d0a7cf1",16,"Planck 1903, p. 226: d\\phi_{0} = \\frac{du_{0} + p\\, dv_{0}}{\\theta}"],["planck-treatise-on-thermodynamics-1903/eq-8515481e69",16,"Planck 1903, p. 227: \\Phi = n_{0} \\phi_{0} + n_{1} \\phi_{1} + n_{2} \\phi_{2} + \\dots + C,"],["planck-treatise-on-thermodynamics-1903/eq-bec2ec2bc9",16,"Planck 1903, p. 228: C = n_{0} (k_{0} - R \\log c_{0}) + n_{1} (k_{1} - R \\log c_{1}) + \\dots\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-d0d4153600",16,"Planck 1903, p. 228: c_{0} = \\frac{n_{0}}{n_{0} + n_{1} + n_{2} + \\dots}"],["planck-treatise-on-thermodynamics-1903/eq-2dde7430cf",16,"Planck 1903, p. 228: \\Phi = n_{0} (\\phi_{0} + k_{0} - R \\log c_{0}) + n_{1} (\\phi_{1} + k_{1} - R \\log c_{1}) + \\dots\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-42e84bd6ed",16,"Planck 1903, p. 228: \\phi_{0} + k_{0} - \\frac{u_{0} + pv_{0}}{\\theta} &= \\varphi_{0}\\Add{,}"],["boyden-first-book-in-algebra-1895/ex-33/3",4,"Boyden 1895, Exercise 33 (3)"],["form/6411755514",5,"factor: a**4*x**2 - b**2*c**2"],["shape/c17917298c",6,"factor: a**N*x**N - b**N*c**N"],["planck-treatise-on-thermodynamics-1903/eq-152e316d3d",16,"Planck 1903, p. 229: \\Psi = n_{0} (\\varphi_{0} - R \\log c_{0}) &+ n_{1} (\\varphi_{1} - R \\log c_{1}) \\\\ &+ n_{2} (\\varphi_{2} - R \\log c_{2})"],["planck-treatise-on-thermodynamics-1903/eq-1fb2418747",16,"Planck 1903, p. 230: \\tsum \\nu_{0} \\log c_{0} + \\nu_{1} \\log c_{1} + \\nu_{2} \\log c_{2} + \\dots &= \\frac{1}{R} \\tsum \\nu_{0} \\varphi_{0} + \\n"],["planck-treatise-on-thermodynamics-1903/eq-6e430e92a8",16,"Planck 1903, p. 231: \\frac{\\dd \\log K}{\\dd \\theta} = \\frac{L}{R\\theta^{2}}"],["planck-treatise-on-thermodynamics-1903/eq-d58bc2b0d5",16,"Planck 1903, p. 232: \\frac{\\dd \\log K}{\\dd p} = -\\frac{s}{R\\theta}\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-fbeff07c23",16,"Planck 1903, p. 231: s = \\tsum \\nu_{0} v_{0} + \\nu_{1} v_{1} + \\nu_{2} v_{2} + \\dots"],["planck-treatise-on-thermodynamics-1903/eq-f3ffcd9fdd",16,"Planck 1903, p. 231: L = \\tsum (\\nu_{0} u_{0} + \\nu_{1} u_{1} + \\dots) + p(\\nu_{0} v_{0} + \\nu_{1} v_{1} + \\dots);"],["hardy-course-of-pure-mathematics-1921/eq-7099a1643a",16,"Hardy 1921, p. 367: \\log a^{x} = x\\log a"],["hardy-course-of-pure-mathematics-1921/eq-fb3be39d7c",16,"Hardy 1921, p. 367: a^{x} = e^{x\\log a} = e^{\\alpha x}"],["hardy-course-of-pure-mathematics-1921/eq-43c6bd7957",16,"Hardy 1921, p. 367: a^{x} = e^{x\\log a} = e^{-\\beta x}"],["planck-treatise-on-thermodynamics-1903/eq-c3b9a17d33",16,"Planck 1903, p. 236: c_{1} = C e^{-\\efrac{513000}{\\theta^{2}}}"],["planck-treatise-on-thermodynamics-1903/eq-81f0f338f9",16,"Planck 1903, p. 236: c_{1} = 6.1 e^{-\\efrac{513000}{\\theta^{2}}} × 10^{-7}"],["planck-treatise-on-thermodynamics-1903/eq-8dd62d49ba",16,"Planck 1903, p. 237: \\frac{c_{2}^{2}}{c_{1}} = K\\Add{.}"],["planck-treatise-on-thermodynamics-1903/eq-a22c88c2ff",16,"Planck 1903, p. 238: K = \\dfrac{\\lambda_{v}}{\\lambda_{\\infty} (\\lambda_{\\infty} - \\lambda_{v})^{v}}"],["concept/dilution-law",7,"dilution law"],["planck-treatise-on-thermodynamics-1903/eq-a8e3eccac9",16,"Planck 1903, p. 237: c_{1} + c_{2} = c"],["planck-treatise-on-thermodynamics-1903/eq-227c2d381f",16,"Planck 1903, p. 239: \\frac{c_{2} c_{3}}{c_{1}} = K"],["planck-treatise-on-thermodynamics-1903/eq-bb32c0c6a6",16,"Planck 1903, p. 239: \\frac{c_{2} c_{4}}{c_{3}} = K'"],["planck-treatise-on-thermodynamics-1903/eq-471f45b43e",16,"Planck 1903, p. 239: 2c_{4} + c_{3} = c_{2}"],["theorem/logarithmic-test-of-convergence",9,"logarithmic test of convergence","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-logarithmic-test-of-convergence"],["de-morgan-elementary-illustrations-calculus-1899/x-1a32f30e91",15,"De Morgan 1899, p. 90: And the student must recollect, that in like manner ..."],["de-morgan-elementary-illustrations-calculus-1899/x-cea321d8f1",15,"De Morgan 1899, p. 89: If y be a function which decreases when x ..."],["de-morgan-elementary-illustrations-calculus-1899/x-8c38b46153",15,"De Morgan 1899, p. 90: And as we have denoted the operation which deduces ..."],["de-morgan-elementary-illustrations-calculus-1899/x-948378a4a7",15,"De Morgan 1899, p. 92: Hence we have a succession of ratios \\dfrac{dy}{dx}, \\dfrac{d^{2} ..."],["de-morgan-elementary-illustrations-calculus-1899/x-dd489a7eaa",15,"De Morgan 1899, p. 93: Write dx for \\Delta x, etc., and recollect that ..."],["de-morgan-elementary-illustrations-calculus-1899/x-f6b0394639",15,"De Morgan 1899, p. 100: let z = \\log(x^{2} + a^{2}). If we make ..."],["de-morgan-elementary-illustrations-calculus-1899/x-6c0fe33a3d",15,"De Morgan 1899, p. 100: If z = \\log\\log\\sin x, or the logarithm of ..."],["de-morgan-elementary-illustrations-calculus-1899/ch-method-of-indivisibles",2,"De Morgan 1899, Method of Indivisibles","../books/de-morgan-elementary-illustrations-calculus-1899/ch/ch-method-of-indivisibles/index.html"],["de-morgan-elementary-illustrations-calculus-1899/x-746a6d3021",15,"De Morgan 1899, p. 97: When x is contained in y, and y is ..."],["de-morgan-elementary-illustrations-calculus-1899/x-d633470383",15,"De Morgan 1899, p. 101: We must leave \\dfrac{da}{dx} and \\dfrac{db}{dx} as we find ..."],["planck-treatise-on-thermodynamics-1903/eq-529ec179ac",16,"Planck 1903, p. 241: \\frac{\\dd \\log c_{1}}{\\dd p} = \\frac{1}{R} · \\frac{s}{\\theta}"],["concept/cycle-of-operations",7,"cycle of operations","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-cycle-of-operations"],["de-morgan-elementary-illustrations-calculus-1899/x-e8c5bb9df0",15,"De Morgan 1899, p. 102: Let z = \\dfrac{a}{b}. If a become a + ..."],["de-morgan-elementary-illustrations-calculus-1899/x-2dd78d8657",15,"De Morgan 1899, p. 102: Again, a^{b+db} = a^{b}\\, a^{db} = a^{b}(1 + \\log ..."],["de-morgan-elementary-illustrations-calculus-1899/x-189ffb2d36",15,"De Morgan 1899, p. 101: In this case, and part of the following, the ..."],["hardy-course-of-pure-mathematics-1921/x-4e24d5b7f0",15,"Hardy 1921, p. 378: The series on the right-hand side of this equation ..."],["hardy-course-of-pure-mathematics-1921/x-251dda7189",15,"Hardy 1921, p. 381: Another very important expansion in powers of x is ..."],["planck-treatise-on-thermodynamics-1903/eq-d1172cfcb8",16,"Planck 1903, p. 241: \\frac{\\dd \\log c_{1}}{\\dd \\theta} = -\\frac{1}{R} · \\frac{L}{\\Erratum{\\theta_{2}}{\\theta^{2}}}"],["planck-treatise-on-thermodynamics-1903/eq-0ea8992f66",16,"Planck 1903, p. 242: c_{1} = Cp"],["planck-treatise-on-thermodynamics-1903/eq-a29ccb20eb",16,"Planck 1903, p. 242: L = -\\frac{R \\theta^{2}}{C} · \\frac{\\dd C}{\\dd \\theta}"],["planck-treatise-on-thermodynamics-1903/eq-a40093dac2",16,"Planck 1903, p. 241: s = \\frac{R\\theta}{p}"],["planck-treatise-on-thermodynamics-1903/eq-b696c284b2",16,"Planck 1903, p. 241: \\frac{\\dd \\log c_{1}}{\\dd p} = \\frac{1}{p}"],["planck-treatise-on-thermodynamics-1903/eq-b3d9237aee",16,"Planck 1903, p. 243: \\log c_{1} = \\frac{L}{R\\theta} + \\const"],["planck-treatise-on-thermodynamics-1903/eq-2b36ca3272",16,"Planck 1903, p. 243: L = -R\\theta^{2} \\frac{\\dd \\log c_{1}}{\\dd \\theta}"],["hardy-course-of-pure-mathematics-1921/x-3262d9cde1",15,"Hardy 1921, p. 377: The value of \\gamma is in fact .577\\dots, and ..."],["hardy-course-of-pure-mathematics-1921/eq-0704c2859d",16,"Hardy 1921, p. 367: D_{x} e^{x\\log a} = e^{x\\log a} \\log a = a^{x} \\log a"],["planck-treatise-on-thermodynamics-1903/eq-b1aecace6d",16,"Planck 1903, p. 244: \\frac{c_{2}^{2}}{c_{1}} = K'"],["planck-treatise-on-thermodynamics-1903/eq-8487af050f",16,"Planck 1903, p. 246: \\frac{n_{1} + n_{2} + n_{3} + \\dots}{n_{0}} = \\log K"],["planck-treatise-on-thermodynamics-1903/eq-27421b0824",16,"Planck 1903, p. 246: \\frac{n_{1} + n_{2} + n_{3} + \\dots}{n_{0}} = \\frac{1}{R} \\left(\\frac{m_{0}}{m_{0}'}\\, \\varphi_{0}' - \\varphi_{0}\\right)"],["planck-treatise-on-thermodynamics-1903/eq-663aed8c52",16,"Planck 1903, p. 247: \\theta - \\theta_{0} = \\frac{R\\theta^{2}}{n_{0} L} (n_{1} + n_{2} + n_{3} + \\dots)"],["planck-treatise-on-thermodynamics-1903/eq-e65bb543ed",16,"Planck 1903, p. 248: \\theta - \\theta_{0} = \\frac{c \\theta^{2} \\varphi}{L}"],["planck-treatise-on-thermodynamics-1903/eq-6187a23fea",16,"Planck 1903, p. 249: \\varphi = \\frac{R(n_{1} + n_{2} + \\dots)}{n_{1} m_{1} + n_{2} m_{2} + \\dots}"],["hardy-course-of-pure-mathematics-1921/x-77464815f3",15,"Hardy 1921, p. 379: The power series for e^{x} is so important that ..."],["planck-treatise-on-thermodynamics-1903/eq-9abd1c5b2f",16,"Planck 1903, p. 249: c\\varphi = \\frac{R(n_{1} + n_{2} + n_{3} + \\dots)}{n_{0} m_{0}}"],["planck-treatise-on-thermodynamics-1903/eq-3ad5198e96",16,"Planck 1903, p. 249: p_{0} - p = \\frac{R\\theta}{n_{0}s} (n_{1} + n_{2} + n_{3} + \\dots)"],["planck-treatise-on-thermodynamics-1903/eq-60c9a9c2e8",16,"Planck 1903, p. 250: p_{0} - p = \\frac{m_{0}'p (n_{1} + n_{2} + \\dots)}{n_{0} m_{0}}"],["planck-treatise-on-thermodynamics-1903/eq-1358fa44b2",16,"Planck 1903, p. 250: \\frac{p_{0} - p}{p} = (n_{1} + n_{2} + n_{3} + \\dots)\\, \\frac{m_{0}'}{n_{0} m_{0}}"],["planck-treatise-on-thermodynamics-1903/eq-88383fd275",16,"Planck 1903, p. 250: \\theta_{0}' - \\theta' = \\frac{R \\theta^{2}}{n_{0} L'} (n_{1} + n_{2} + n_{3} + \\dots)"],["planck-treatise-on-thermodynamics-1903/eq-ac490ba14c",16,"Planck 1903, p. 251: P = \\frac{R\\theta}{n_{0} m_{0} v} (n_{1} + n_{2} + n_{3} + \\dots)"],["hardy-course-of-pure-mathematics-1921/x-5f1f1622da",15,"Hardy 1921, p. 382: If x lies outside these limits the series is ..."],["hardy-course-of-pure-mathematics-1921/eq-9efb2cc510",16,"Hardy 1921, p. 367: D_{a} e^{x\\log a} = e^{x\\log a} (x/a) = xa^{x-1}"],["planck-treatise-on-thermodynamics-1903/eq-7fa2ed48eb",16,"Planck 1903, p. 251: P = \\frac{R\\theta}{V} (n_{1} + n_{2} + n_{3} + \\dots)"],["planck-treatise-on-thermodynamics-1903/eq-8f2990e83e",16,"Planck 1903, p. 248: c = \\frac{n_{1} m_{1} + n_{2} m_{2} + \\dots}{n_{0} m_{0}}"],["planck-treatise-on-thermodynamics-1903/eq-b1930515a0",16,"Planck 1903, p. 254: c_{1} + c_{2} + \\dots + \\log c_{0}' = \\log K"],["planck-treatise-on-thermodynamics-1903/eq-f6c74d34b7",16,"Planck 1903, p. 241: \\log c_{0}' = \\log K"],["planck-treatise-on-thermodynamics-1903/eq-5b71f79266",16,"Planck 1903, p. 255: (c_{1} + c_{2} + \\dots) - (c_{1}' + c_{2}' + \\dots) = \\log K"],["planck-treatise-on-thermodynamics-1903/eq-58a511a6f5",16,"Planck 1903, p. 253: \\frac{c_{1}'}{c_{1}} = K"],["hardy-course-of-pure-mathematics-1921/eq-991d059070",16,"Hardy 1921, p. 299: \\int_{a}^{b} f(x)\\, dx = \\int_{a}^{b} \\{\\phi(x) + i\\psi(x)\\}\\, dx = \\int_{a}^{b} \\phi(x)\\, dx + i \\int_{a}^{b} \\psi(x)\\,"],["hardy-course-of-pure-mathematics-1921/eq-26b7f21b71",16,"Hardy 1921, p. 299: \\left|\\int_{a}^{b} f(x)\\, dx\\right| \\leq \\int_{a}^{b} |f(x)|\\, dx"],["planck-treatise-on-thermodynamics-1903/eq-8c7a5015f8",16,"Planck 1903, p. 256: \\frac{n_{2}^{2}}{n_{1} n_{0}} = K"],["planck-treatise-on-thermodynamics-1903/eq-c67c59701e",16,"Planck 1903, p. 256: \\frac{n_{2}'^{2}}{n_{1}' n_{0}'} = K'"],["planck-treatise-on-thermodynamics-1903/eq-4275fd758f",16,"Planck 1903, p. 256: \\bar{n}_{0} = n_{0} + n_{0}'"],["concept/conservation-of-molecules",7,"conservation of molecules"],["planck-treatise-on-thermodynamics-1903/eq-d1e0de4f58",16,"Planck 1903, p. 256: \\bar{n}_{2} + \\bar{n}_{4} = n_{1}' + n_{2}'"],["planck-treatise-on-thermodynamics-1903/eq-c8d1708e97",16,"Planck 1903, p. 256: \\bar{n}_{1} + \\bar{n}_{3} = n_{1} + n_{2}"],["planck-treatise-on-thermodynamics-1903/eq-1f7d92a63d",16,"Planck 1903, p. 256: \\bar{n}_{3} + \\bar{n}_{4} = \\bar{n}_{5}"],["planck-treatise-on-thermodynamics-1903/eq-67359c98a1",16,"Planck 1903, p. 257: \\frac{\\bar{c}_{3} \\bar{c}_{5}}{\\bar{c}_{1}} = K"],["hardy-course-of-pure-mathematics-1921/eq-9e9f1cbff1",16,"Hardy 1921, p. 300: |\\tsum f_{\\nu}\\, \\delta_{\\nu}| \\leq \\tsum |f_{\\nu}|\\, \\delta_{\\nu}"],["hardy-course-of-pure-mathematics-1921/eq-9a8d7025df",16,"Hardy 1921, p. 300: \\tan x &= x + \\tfrac{1}{3} x^{3} + \\tfrac{2}{15} x^{5} + \\dots"],["boyden-first-book-in-algebra-1895/ex-31/17",4,"Boyden 1895, Exercise 31 (17)"],["planck-treatise-on-thermodynamics-1903/eq-b59bb3a5bd",16,"Planck 1903, p. 257: \\frac{\\bar{c}_{4} \\bar{c}_{5}}{\\bar{c}_{2}} = K'"],["hardy-course-of-pure-mathematics-1921/eq-d4d082d272",16,"Hardy 1921, p. 300: \\sec x &= 1 + \\tfrac{1}{2} x^{2} + \\tfrac{5}{24} x^{4} + \\dots"],["hardy-course-of-pure-mathematics-1921/eq-0243a2624c",16,"Hardy 1921, p. 300: x\\cosec x &= 1 + \\tfrac{1}{6} x^{2} + \\tfrac{7}{360} x^{4} + \\dots"],["hardy-course-of-pure-mathematics-1921/eq-729e6f7e9a",16,"Hardy 1921, p. 300: x\\cot x &= 1 - \\tfrac{1}{3} x^{2} - \\tfrac{1}{45} x^{4} - \\dots"],["hardy-course-of-pure-mathematics-1921/eq-e969d9ad20",16,"Hardy 1921, p. 300: \\theta_{n} = \\frac{1}{n + 1} + \\frac{n}{2(n + 1)^{2}(n + 2)} \\left\\{\\frac{f^{(n+2)}(0)}{f^{(n+1)}(0)} + \\epsilon_{x}\\rig"],["hardy-course-of-pure-mathematics-1921/eq-7c5fd29dbb",16,"Hardy 1921, p. 300: f(b) = f(a) + \\tfrac{1}{2}(b - a) \\{f'(a) + f'(b)\\} - \\tfrac{1}{12}(b - a)^{3} f'''(\\alpha)"],["hardy-course-of-pure-mathematics-1921/eq-b6814eb243",16,"Hardy 1921, p. 300: f(b) = f(a) + (b - a) f'\\{\\tfrac{1}{2}(a + b)\\} + \\tfrac{1}{24}(b - a)^{3}f'''(\\alpha)"],["hardy-course-of-pure-mathematics-1921/eq-d8fd5b9f8b",16,"Hardy 1921, p. 367: (a^{x} - 1)/x \\to \\log a"],["hardy-course-of-pure-mathematics-1921/eq-c40a30ec41",16,"Hardy 1921, p. 368: \\lim_{n\\to\\infty} \\left(1 + \\frac{x}{n}\\right)^{n} = \\lim_{n\\to\\infty} \\left(1 - \\frac{x}{n}\\right)^{-n} = e^{x}"],["boyden-first-book-in-algebra-1895/eq-abfdb69264",16,"Boyden 1895: \\frac{a}{b}=\\frac{-a}{-b}=-\\frac{-a}{b}=-\\frac{a}{-b}"],["boyden-first-book-in-algebra-1895/ex-31/18",4,"Boyden 1895, Exercise 31 (18)"],["concept/conservative-system",7,"conservative system","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-conservative-system"],["form/6879da2f70",5,"factor: 51*a**5*b - 34*a**4*b**2 + 17*a**2*b**4"],["boyden-first-book-in-algebra-1895/eq-4db100aa7c",16,"Boyden 1895: \\frac{a-b}{x-y}=\\frac{b-a}{y-x}=-\\frac{b-a}{x-y}=-\\frac{a-b}{y-x}"],["boyden-first-book-in-algebra-1895/eq-c78fd82121",16,"Boyden 1895: \\frac{a-b+c}{x+y+z}=\\frac{b-c-a}{-x-y-z}=-\\frac{b-c-a}{x+y+z}=-\\frac{a-b+c}{-x-y-z}"],["shape/98c5e94417",6,"factor: N*a**N*b + 2*N*a**N*b**N"],["boyden-first-book-in-algebra-1895/eq-02f247f875",16,"Boyden 1895: \\frac{b}{c}\\div \\frac{x}{y}=\\frac{b}{c}\\times\\frac{y}{x}=\\frac{by}{cx}"],["concept/cube",7,"cube","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cube"],["boyden-first-book-in-algebra-1895/eq-a2d7b7babd",16,"Boyden 1895: \\left(\\frac{a}{b}\\right)^2 = \\frac{a}{b} \\times \\frac{a}{b} = \\frac{a^2}{b^2}"],["concept/method-raising-a-fraction-to-a-power",7,"method: raising a fraction to a power"],["boyden-first-book-in-algebra-1895/eq-22083ef9c6",16,"Boyden 1895: \\frac{b^4}{a^4} - \\frac{y^4}{x^4} = \\left(\\frac{b^2}{a^2} + \\frac{y^2}{x^2}\\right) \\left(\\frac{b}{a} + \\frac{y}{x}\\right"],["concept/theorem-difference-of-two-squares",7,"theorem: difference of two squares"],["concept/method-factoring",7,"method: factoring"],["boyden-first-book-in-algebra-1895/eq-7e92e07c3b",16,"Boyden 1895: x^4 + x^2 + \\frac{1}{4} = \\left(x^2 + \\frac{1}{2}\\right)^2"],["concept/method-factoring-a-perfect-square-trinomial",7,"method: factoring a perfect square trinomial"],["boyden-first-book-in-algebra-1895/x-e3a409b57e",15,"Boyden 1895: To find the value of x, substitute the value ..."],["boyden-first-book-in-algebra-1895/x-4bc99a8834",15,"Boyden 1895: To solve a pure quadratic equation, reduce to the ..."],["concept/like-terms",7,"like terms","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-like-terms"],["boyden-first-book-in-algebra-1895/eq-235658c1c5",16,"Boyden 1895: \\begin{array}{ccccc} a &+& \\frac{ax}{b} & \\times & abd \\\\ \\cline{1-3} \\frac{c}{d} &+& \\frac{x}{ab} & \\times & abd \\end{a"],["boyden-first-book-in-algebra-1895/eq-0085d942b5",16,"Boyden 1895: \\begin{array}{ccccc} \\frac{1}{1-x} &-& \\frac{1}{1+x} & \\times & (1-x)(1+x) \\\\ \\cline{1-3} \\frac{1}{1-x} &+& \\frac{1}{1+x"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37",3,"Macfarlane 1906, Exercise Probs-28-37"],["method/scalar-product",8,"scalar product","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-method-scalar-product"],["shape/957d2b456c",6,"evaluate: Integral(x**N - log(x), (x, 0, 1))"],["macfarlane-vector-analysis-quaternions-1906/ch-product-of-two-vectors",2,"Macfarlane 1906, Product of Two Vectors","../books/macfarlane-vector-analysis-quaternions-1906/ch/ch-product-of-two-vectors/index.html"],["method/product-of-two-vectors",8,"product of two vectors","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-method-product-of-two-vectors"],["concept/trinomial",7,"trinomial","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-trinomial"],["boyden-first-book-in-algebra-1895/eq-f9a831ae41",16,"Boyden 1895: x = a - b"],["boyden-first-book-in-algebra-1895/eq-455e6e9296",16,"Boyden 1895: x = c + b"],["theorem/energy-of-a-system-of-conductors",9,"energy of a system of conductors","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-energy-of-a-system-of-conductors"],["boyden-first-book-in-algebra-1895/eq-680896f600",16,"Boyden 1895: x^2 = a"],["concept/method-extracting-the-square-root",7,"method: extracting the square root"],["boyden-first-book-in-algebra-1895/eq-a7d5314152",16,"Boyden 1895: x^2 + bx + c = O"],["concept/homogeneous-polynomial",7,"homogeneous polynomial","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-homogeneous-polynomial"],["theorem/volume-of-a-parallelepiped",9,"volume of a parallelepiped","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-theorem-volume-of-a-parallelepiped"],["theorem/increment-of-energy-of-a-fixed-system-of-conductors",9,"increment of energy of a fixed system of conductors","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-increment-of-energy-of-a-fixed-system-of-conductors"],["law/right-handed-screw-rule",10,"right-handed screw rule","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-law-right-handed-screw-rule"],["law/dynamo-rule",10,"dynamo rule","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-law-dynamo-rule"],["law/electric-motor-rule",10,"electric motor rule","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-law-electric-motor-rule"],["boyden-first-book-in-algebra-1895/x-f7acf6d369",15,"Boyden 1895: An algebraic expression is any representation of a number ..."],["theorem/rate-of-change-of-electric-energy-with-potential",9,"rate of change of electric energy with potential","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-rate-of-change-of-electric-energy-with-potential"],["form/d50c923d9c",5,"evaluate: sqrt(Integral(sin(x)**2, (x, 0, pi)))/sqrt(pi)"],["macfarlane-vector-analysis-quaternions-1906/eq-413ac44b43",16,"Macfarlane 1906: f^2 = f_1^2 + f_2^2 + 2f_1f_2 \\cos \\theta_2"],["shape/ab9348e916",6,"evaluate: pi**N*Integral(sin(x)**N, (x, 0, pi))**N"],["macfarlane-vector-analysis-quaternions-1906/eq-2d2ba89262",16,"Macfarlane 1906: \\tan \\theta =\\frac{f_2\\sin\\theta_2}{f_1 + f_2\\cos\\theta_2}"],["macfarlane-vector-analysis-quaternions-1906/x-a1adc39ccb",15,"Macfarlane 1906: Frequently all that is demanded is, given two of ..."],["hardy-course-of-pure-mathematics-1921/x-3066622796",15,"Hardy 1921, p. 6: Thus m = p^{2}, n = q^{2}, as was ..."],["concept/product-of-three-vectors",7,"product of three vectors","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-product-of-three-vectors"],["concept/algebraic-sum",7,"algebraic sum","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-algebraic-sum"],["concept/positive-number",7,"positive number","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-positive-number"],["thompson-calculus-made-easy-1914/ch-vii",2,"Thompson 1914, ch. VII: Successive Differentiation","../books/thompson-calculus-made-easy-1914/ch/ch-vii/index.html"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-38-43",3,"Macfarlane 1906, Exercise Probs-38-43"],["macfarlane-vector-analysis-quaternions-1906/ch-product-of-three-vectors",2,"Macfarlane 1906, Product of Three Vectors","../books/macfarlane-vector-analysis-quaternions-1906/ch/ch-product-of-three-vectors/index.html"],["thompson-calculus-made-easy-1914/ex-iv",3,"Thompson 1914, Exercise IV"],["concept/algebraic-expression",7,"algebraic expression","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-algebraic-expression"],["boyden-first-book-in-algebra-1895/x-b4eafd9be4",15,"Boyden 1895: In combining numbers in algebra it must always be ..."],["boyden-first-book-in-algebra-1895/x-9f88067f32",15,"Boyden 1895: To add similar terms with like signs, add the ..."],["theorem/work-done-in-displacing-conductors-at-constant-potential",9,"work done in displacing conductors at constant potential","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-work-done-in-displacing-conductors-at-constant-potential"],["method/summation",8,"summation","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-summation"],["hardy-course-of-pure-mathematics-1921/x-6170dfabb8",15,"Hardy 1921, p. 14: A section of the rational numbers, in which both ..."],["hardy-course-of-pure-mathematics-1921/x-ecaf3dff35",15,"Hardy 1921, p. 14: What is essential in mathematics is that its symbols ..."],["boyden-first-book-in-algebra-1895/ex-15/18",4,"Boyden 1895, Exercise 15 (18)"],["concept/minuend",7,"minuend","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-minuend"],["thompson-calculus-made-easy-1914/ch-vi",2,"Thompson 1914, ch. VI: Sums, Differences, Products and Quotients","../books/thompson-calculus-made-easy-1914/ch/ch-vi/index.html"],["form/8505630e09",5,"identity: 2*x**3 + 4*x**2 - 2*x + 17"],["concept/subtrahend",7,"subtrahend","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-subtrahend"],["shape/aacace0f45",6,"identity: N*x + 2*N*x**N + N"],["macfarlane-vector-analysis-quaternions-1906/eq-684a8ebf33",16,"Macfarlane 1906: I = \\left\\{\\frac{r}{r^2+(2\\pi nl)^2} - \\frac{2\\pi nl}{r^2+(2\\pi nl)^2}\\cdot \\beta^\\frac{\\pi}{2}\\right\\}E"],["theorem/difference-rule-for-differentiation",9,"difference rule for differentiation","../books/thompson-calculus-made-easy-1914/terms/index.html#t-theorem-difference-rule-for-differentiation"],["maxwell-elementary-treatise-electricity-1888/x-83c4c72316",15,"Maxwell 1888, scan 39: Thus when a fish has swallowed the angler’s hook ..."],["concept/difference",7,"difference","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-difference"],["concept/number-line",7,"number line","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-number-line"],["concept/negative-number",7,"negative number","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-negative-number"],["hardy-course-of-pure-mathematics-1921/ex-ix",3,"Hardy 1921, Exercise IX"],["hardy-course-of-pure-mathematics-1921/ch-i",2,"Hardy 1921, ch. 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X: PHENOMENA OF AN ELECTRIC CURRENT WHICH FLOWS THROUGH HETEROGENEOUS MEDIA","../books/maxwell-elementary-treatise-electricity-1888/ch/ch-x/index.html"],["quantity/electrochemical-equivalent",11,"electrochemical equivalent","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-quantity-electrochemical-equivalent"],["concept/exponential-theorem",7,"exponential theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-exponential-theorem"],["theorem/multinomial-theorem",9,"multinomial theorem","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-theorem-multinomial-theorem"],["concept/parallelogram-rule",7,"parallelogram rule","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-parallelogram-rule"],["method/indirect-determination-of-heat-effect",8,"indirect determination of heat effect","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-method-indirect-determination-of-heat-effect"],["method/calorimetry",8,"calorimetry","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-method-calorimetry"],["boyden-first-book-in-algebra-1895/ex-16/12",4,"Boyden 1895, Exercise 16 (12)"],["boyden-first-book-in-algebra-1895/ex-16/11",4,"Boyden 1895, Exercise 16 (11)"],["form/dec7484c8b",5,"identity: 8*x**4 - 2*x**3 + x**2 - 15*x + 14"],["concept/spherical-triangle",7,"spherical triangle","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-spherical-triangle"],["person/arthur-cayley",1,"Arthur Cayley","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-arthur-cayley"],["macfarlane-vector-analysis-quaternions-1906/x-488bfc4e1d",15,"Macfarlane 1906: A version refers to the change of direction of ..."],["macfarlane-vector-analysis-quaternions-1906/x-04fe220c70",15,"Macfarlane 1906: Suppose that a rigid body rotates \\theta radians round ..."],["shape/19980d533e",6,"identity: N*x + 2*N*x**N + N + x**N"],["form/78bb65005d",5,"identity: 20*a**2*x**2 + 16*a*x**2"],["shape/d86f44108c",6,"identity: N*a*x**N + N*a**N*x**N"],["method/neglecting-higher-order-small-quantities",8,"neglecting higher-order small quantities","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-neglecting-higher-order-small-quantities"],["concept/order-of-smallness",7,"order of smallness","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-order-of-smallness"],["thompson-calculus-made-easy-1914/x-06a9da7d58",15,"Thompson 1914, p. 4: The mathematicians talk about the second order of “magnitude” ..."],["thompson-calculus-made-easy-1914/x-fe80269af9",15,"Thompson 1914, p. 5: Then we see that the smaller a small quantity ..."],["thompson-calculus-made-easy-1914/x-341dc8a359",15,"Thompson 1914, p. 5: But, it must be remembered, that small quantities if ..."],["instrument/grove-s-cell",13,"Grove's cell","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-grove-s-cell"],["instrument/bunsen-s-cell",13,"Bunsen's cell","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-bunsen-s-cell"],["boyden-first-book-in-algebra-1895/ex-26/7",4,"Boyden 1895, Exercise 26 (7)"],["thompson-calculus-made-easy-1914/x-e224d14e50",15,"Thompson 1914, p. 5: Now in the calculus we write dx for a ..."],["thompson-calculus-made-easy-1914/x-ef3b5f8fe7",15,"Thompson 1914, p. 6: Let us think of x as a quantity that ..."],["thompson-calculus-made-easy-1914/x-09414216f6",15,"Thompson 1914, p. 7: Clearly (dx)^2 is negligible if only we consider the ..."],["concept/constant-voltaic-element",7,"constant voltaic element","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-constant-voltaic-element"],["boyden-first-book-in-algebra-1895/ex-26/6",4,"Boyden 1895, Exercise 26 (6)"],["form/aec13939a0",5,"identity: (x**4 - 12*x**2 + 16)/(x**2 - 2*x - 4)"],["form/b4d85a4474",5,"identity: (-a**3 + x**3)/(-a + x)"],["concept/anode",7,"anode","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-anode"],["boyden-first-book-in-algebra-1895/ex-26/8",4,"Boyden 1895, Exercise 26 (8)"],["boyden-first-book-in-algebra-1895/ex-26/12",4,"Boyden 1895, Exercise 26 (12)"],["shape/1453dd21a7",6,"identity: (2*N*a**N*x**N + N*a**N + a*x**N + x**N)/(N*a*x + N*a**N + x**N)"],["maxwell-elementary-treatise-electricity-1888/x-0935dcb7f5",15,"Maxwell 1888, scan 166: Let E be the electromotive force of the circuit; ..."],["maxwell-elementary-treatise-electricity-1888/x-6d3f001e0a",15,"Maxwell 1888, scan 166: ‘The electromotive force of an electrochemical apparatus is in ..."],["boyden-first-book-in-algebra-1895/ex-32/12",4,"Boyden 1895, Exercise 32 (12)"],["form/dc52765f04",5,"factor: a**2*x + a*b*d + a*b*x + a*c + b**2*d + b*c"],["shape/8e340f1da9",6,"factor: a*b*d + a*b*x + a*c + a**N*x + b*c + b**N*d"],["maxwell-elementary-treatise-electricity-1888/x-54cff3e837",15,"Maxwell 1888, scan 165: Thus in a battery the electrodes of which are ..."],["thompson-calculus-made-easy-1914/x-aafdc7c3f5",15,"Thompson 1914, p. 19: Then (dx)^2 will mean a little bit of a ..."],["thompson-calculus-made-easy-1914/x-45feec2834",15,"Thompson 1914, p. 20: But, you will say, we neglected a whole unit."],["thompson-calculus-made-easy-1914/x-236f2ba0fb",15,"Thompson 1914, p. 25: To differentiate x^n, multiply by the power and reduce ..."],["concept/ratio",7,"ratio","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-ratio"],["concept/relation-between-variables",7,"relation between variables","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-relation-between-variables"],["maxwell-elementary-treatise-electricity-1888/x-fbfa7ca40d",15,"Maxwell 1888, scan 167: It is only the strictly reversible processes that must ..."],["maxwell-elementary-treatise-electricity-1888/x-17ba27d8f3",15,"Maxwell 1888, scan 170: The only difference is that diffusion is always going ..."],["maxwell-elementary-treatise-electricity-1888/x-6a7bd92141",15,"Maxwell 1888, scan 172: These defects, however, are more than counterbalanced in all ..."],["maxwell-elementary-treatise-electricity-1888/x-54a75a0488",15,"Maxwell 1888, scan 167: In a voltaic circuit the sum of the electromotive ..."],["unit/calorie",12,"calorie","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-unit-calorie"],["boyden-first-book-in-algebra-1895/ex-32/13",4,"Boyden 1895, Exercise 32 (13)"],["method/measuring-large-potential-differences-in-absolute-measure",8,"measuring large potential differences in absolute measure","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-measuring-large-potential-differences-in-absolute-measure"],["thompson-calculus-made-easy-1914/x-f00a19abb0",15,"Thompson 1914, p. 9: Those which we regard as of fixed value, and ..."],["boyden-first-book-in-algebra-1895/ex-32/14",4,"Boyden 1895, Exercise 32 (14)"],["instrument/quadrant-electrometer",13,"quadrant electrometer","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-quadrant-electrometer"],["form/1431af56e7",5,"factor: 3*a*c + 3*a*x - 2*b*c - 2*b*x"],["shape/96fe15d5ad",6,"factor: N*a*c + N*a*x + N*b*c + N*b*x"],["thompson-calculus-made-easy-1914/x-846c195b12",15,"Thompson 1914, p. 17: It will never do to fall into the schoolboy ..."],["theorem/volume-of-a-cylinder",9,"volume of a cylinder","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-theorem-volume-of-a-cylinder"],["method/measuring-the-potential-of-a-charged-conductor-of-finite-size",8,"measuring the potential of a charged conductor of finite size","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-measuring-the-potential-of-a-charged-conductor-of-finite-size"],["theorem/derivative-of-an-added-constant",9,"derivative of an added constant","../books/thompson-calculus-made-easy-1914/terms/index.html#t-theorem-derivative-of-an-added-constant"],["method/measuring-air-potential-with-an-earthed-insulated-sphere",8,"measuring air potential with an earthed insulated sphere","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-measuring-air-potential-with-an-earthed-insulated-sphere"],["theorem/derivative-of-a-constant-multiple",9,"derivative of a constant multiple","../books/thompson-calculus-made-easy-1914/terms/index.html#t-theorem-derivative-of-a-constant-multiple"],["instrument/pyrometer",13,"pyrometer","../books/thompson-calculus-made-easy-1914/terms/index.html#t-instrument-pyrometer"],["method/measuring-potential-without-touching-the-conductor",8,"measuring potential without touching the conductor","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-measuring-potential-without-touching-the-conductor"],["thompson-calculus-made-easy-1914/x-9b15e9e874",15,"Thompson 1914, p. 26: We usually think of x as a quantity that ..."],["thompson-calculus-made-easy-1914/x-ef0056bc24",15,"Thompson 1914, p. 27: So the 5 has quite disappeared. It added nothing ..."],["thompson-calculus-made-easy-1914/x-b34480ce00",15,"Thompson 1914, p. 31: If r = 5.5 in. and h=20 in. this ..."],["thompson-calculus-made-easy-1914/x-dfaac2754d",15,"Thompson 1914, p. 32: The sensitiveness is approximately doubled from 800° to 1000°, ..."],["maxwell-elementary-treatise-electricity-1888/x-1c734f3b86",15,"Maxwell 1888, scan 191: To ascertain the potential of a charged conductor of ..."],["macfarlane-vector-analysis-quaternions-1906/eq-0fc9493788",16,"Macfarlane 1906: OC = \\sqrt{f_1^2 + f_2^2 + 2f_1f_2\\cos\\theta_2} \\underline{\\left/\\tan^{-1} \\frac{f_2\\sin \\theta_2}{f_1 + f_2\\cos\\theta_2"],["macfarlane-vector-analysis-quaternions-1906/eq-e0713c2875",16,"Macfarlane 1906: OC = 2f_1\\cos\\frac{\\theta_2}{2} \\underline{\\left/\\frac{\\theta_2}{2}\\right.}"],["macfarlane-vector-analysis-quaternions-1906/eq-276ebfa92a",16,"Macfarlane 1906: f_2\\underline{/\\theta_2} = \\sqrt{f^2 + f_1^2 - 2ff_1\\cos\\theta} \\underline{\\left/\\tan^{-1} \\frac{f\\sin\\theta}{-f_1 + f\\c"],["macfarlane-vector-analysis-quaternions-1906/eq-635657bf0f",16,"Macfarlane 1906: f_1 + f_2\\cos(\\theta_2 - \\theta_1) = f\\cos(\\theta - \\theta_1)"],["macfarlane-vector-analysis-quaternions-1906/eq-e6fbde41ea",16,"Macfarlane 1906: f_1\\cos(\\theta_2 - \\theta_1) + f_2 = f\\cos(\\theta_2 - \\theta)"],["wentworth-first-steps-in-algebra-1894/ex-24/2",4,"Wentworth 1894, Exercise 24 (2)"],["form/82a3c8f630",5,"identity: (x**2 - 15*x + 56)/(x - 7)"],["macfarlane-vector-analysis-quaternions-1906/eq-7d4b1c60e4",16,"Macfarlane 1906: f_1 = f\\frac{ \\{\\cos(\\theta - \\theta_1) - \\cos(\\theta_2 - \\theta)\\cos(\\theta_2 - \\theta_1)\\} } {1 - \\cos^2(\\theta_2 - \\t"],["macfarlane-vector-analysis-quaternions-1906/eq-507c036a66",16,"Macfarlane 1906: \\sqrt{\\left( \\sum f\\cos\\theta \\right)^2 + \\left( \\sum f\\sin\\theta \\right )^2} \\cdot \\tan^{-1}\\frac{\\sum f\\sin\\theta}{\\su"],["macfarlane-vector-analysis-quaternions-1906/eq-90ecd2c6bb",16,"Macfarlane 1906: A + B = -C"],["maxwell-elementary-treatise-electricity-1888/x-a12903328e",15,"Maxwell 1888, scan 192: Now let the sphere thus discharged be carried to ..."],["maxwell-elementary-treatise-electricity-1888/x-67146d2af8",15,"Maxwell 1888, scan 193: In this way it has been ascertained by Sir ..."],["thompson-calculus-made-easy-1914/x-383421d8e3",15,"Thompson 1914, p. 49: This is to employ the general symbol f(x) for ..."],["thompson-calculus-made-easy-1914/x-83820ff740",15,"Thompson 1914, p. 49: The corresponding symbol for the differential coefficient is f'(x), ..."],["thompson-calculus-made-easy-1914/x-fd057f44dd",15,"Thompson 1914, p. 50: Suppose we differentiate over again, we shall get the ..."],["maxwell-elementary-treatise-electricity-1888/x-64fc2c6286",15,"Maxwell 1888, scan 193: If by any means we can cause a succession ..."],["theorem/product-rule-for-differentiation",9,"product rule for differentiation","../books/thompson-calculus-made-easy-1914/terms/index.html#t-theorem-product-rule-for-differentiation"],["thompson-calculus-made-easy-1914/x-89c1dff4ad",15,"Thompson 1914, p. 37: The result will certainly not be 2x × 4ax^3; ..."],["law/zeroth-law-of-thermodynamics",10,"zeroth law of thermodynamics","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-law-zeroth-law-of-thermodynamics"],["quantity/specific-volume",11,"specific volume","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-specific-volume"],["hardy-course-of-pure-mathematics-1921",0,"Hardy, A Course of Pure Mathematics (1921)","../books/hardy-course-of-pure-mathematics-1921/index.html"],["hardy-course-of-pure-mathematics-1921/x-564251fa04",15,"Hardy 1921, p. 66: All these constructions were what may be called Euclidean ..."],["concept/sum",7,"sum","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-sum"],["hardy-course-of-pure-mathematics-1921/ex-ii",3,"Hardy 1921, Exercise II"],["instrument/frictional-electric-machine",13,"frictional electric machine","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-frictional-electric-machine"],["hardy-course-of-pure-mathematics-1921/x-61f7c4919f",15,"Hardy 1921, p. 67: This expression contains a fourth root, but this is ..."],["hardy-course-of-pure-mathematics-1921/x-b4241a419d",15,"Hardy 1921, p. 67: Conversely, only irrationals of this kind can be constructed ..."],["instrument/voltaic-battery",13,"voltaic battery","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-voltaic-battery"],["concept/frustum-of-a-pyramid",7,"frustum of a pyramid","../books/slaught-lennes-solid-geometry-1919/terms/index.html#t-concept-frustum-of-a-pyramid"],["hardy-course-of-pure-mathematics-1921/x-f506c64d37",15,"Hardy 1921, p. 67: Hence Euclidean methods will construct any surd expression involving ..."],["boyden-first-book-in-algebra-1895/ex-32/15",4,"Boyden 1895, Exercise 32 (15)"],["instrument/magneto-electric-machine",13,"magneto-electric machine","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-magneto-electric-machine"],["thompson-calculus-made-easy-1914/x-42695b5aab",15,"Thompson 1914, p. 38: To differentiate the product of two functions, multiply each ..."],["thompson-calculus-made-easy-1914/x-49893f9384",15,"Thompson 1914, p. 38: You should note that this process amounts to the ..."],["thompson-calculus-made-easy-1914/x-bad55a502a",15,"Thompson 1914, p. 37: Now du · dv is a small quantity of ..."],["form/77823e2470",5,"factor: -2*a*d + 2*a*x - 3*b*d + 3*b*x + c*d - c*x"],["shape/2c1372bef7",6,"factor: N*a*d + N*a*x + N*b*d + N*b*x + c*d - c*x"],["thompson-calculus-made-easy-1914/x-83e471f307",15,"Thompson 1914, p. 40: This gives us our instructions as to how to ..."],["thompson-calculus-made-easy-1914/x-6feb96a5af",15,"Thompson 1914, p. 41: The working out of quotients is often tedious, but ..."],["concept/electric-glow",7,"electric glow","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electric-glow"],["boyden-first-book-in-algebra-1895/ex-32/16",4,"Boyden 1895, Exercise 32 (16)"],["concept/rational-number",7,"rational number","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-rational-number"],["theorem/density-of-rational-points",9,"density of rational points","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-density-of-rational-points"],["theorem/archimedean-property",9,"Archimedean property","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-archimedean-property"],["concept/infinity",7,"infinity","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-infinity"],["form/58466ba7a0",5,"factor: 3*a*b*d + 6*a*b*x - 3*a*c*d - 6*a*c*x"],["concept/laws-of-algebra",7,"laws of algebra","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-laws-of-algebra"],["theorem/irrationality-of-square-root-of-2",9,"irrationality of square root of 2","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-irrationality-of-square-root-of-2"],["theorem/rational-root-theorem",9,"rational root theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-rational-root-theorem"],["concept/potential-gradient",7,"potential gradient","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-potential-gradient"],["concept/silk-flap",7,"silk flap","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-silk-flap"],["shape/092ac8d925",6,"factor: N*a*b*d + N*a*b*x + N*a*c*d + N*a*c*x"],["boyden-first-book-in-algebra-1895/ex-32/17",4,"Boyden 1895, Exercise 32 (17)"],["concept/irrational-number",7,"irrational number","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-irrational-number"],["instrument/carrier",13,"carrier","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-carrier"],["concept/inductor",7,"inductor","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-inductor"],["boyden-first-book-in-algebra-1895/ex-16/13",4,"Boyden 1895, Exercise 16 (13)"],["hardy-course-of-pure-mathematics-1921/x-08ee59a4bf",15,"Hardy 1921, p. 14: Mr Bertrand Russell has said that ‘mathematics is the ..."],["hardy-course-of-pure-mathematics-1921/x-2e2a047e53",15,"Hardy 1921, p. 15: Moreover, for a beginner, the chief difficulty in the ..."],["form/f24a053658",5,"identity: 4*x**3 - 2"],["shape/c4456b6d2d",6,"identity: N*x**N + N"],["boyden-first-book-in-algebra-1895/ex-16/14",4,"Boyden 1895, Exercise 16 (14)"],["form/860fcf4f6c",5,"identity: -x**(3*a) - x**(2*a) + 2*x**a"],["shape/3fe863f433",6,"identity: N*x**a - 2*x**(N*a)"],["theorem/inequality-of-arithmetic-and-geometric-means",9,"inequality of arithmetic and geometric means","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-inequality-of-arithmetic-and-geometric-means"],["macfarlane-vector-analysis-quaternions-1906/eq-fe92afc7df",16,"Macfarlane 1906: R = r\\rho = xi + yj + zk"],["instrument/condenser",13,"condenser","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-condenser"],["concept/continued-fraction",7,"continued fraction","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-continued-fraction"],["concept/golden-section",7,"golden section","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-golden-section"],["hardy-course-of-pure-mathematics-1921/x-7d985d805e",15,"Hardy 1921, p. 29: A system of real numbers, or of the points ..."],["instrument/water-dropping-accumulator",13,"water dropping accumulator","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-water-dropping-accumulator"],["planck-treatise-on-thermodynamics-1903/x-fa99663a57",15,"Planck 1903, p. 3: The definition of temperature is therefore somewhat arbitrary. This ..."],["hardy-course-of-pure-mathematics-1921/x-de0f061426",15,"Hardy 1921, p. 29: Suppose, for example, that S consists of the points ..."],["hardy-course-of-pure-mathematics-1921/x-c5206f415a",15,"Hardy 1921, p. 31: This point may of course coincide with \\alpha or ..."],["instrument/coulomb-s-torsion-balance",13,"Coulomb's torsion balance","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-coulomb-s-torsion-balance"],["concept/moment-of-inertia",7,"moment of inertia","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-moment-of-inertia"],["planck-treatise-on-thermodynamics-1903/x-0f68ff6468",15,"Planck 1903, p. 2: From this follows the important proposition: If a body, ..."],["boyden-first-book-in-algebra-1895/ex-26/13",4,"Boyden 1895, Exercise 26 (13)"],["instrument/attracted-disk-electrometer",13,"attracted disk electrometer","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-attracted-disk-electrometer"],["hardy-course-of-pure-mathematics-1921/x-ac3b21e0ca",15,"Hardy 1921, p. 32: When p = q = 1 in (1), or ..."],["hardy-course-of-pure-mathematics-1921/x-0cf667c01f",15,"Hardy 1921, p. 36: Such irrational numbers are called algebraical numbers: all other ..."],["boyden-first-book-in-algebra-1895/ex-26/14",4,"Boyden 1895, Exercise 26 (14)"],["form/d122b6a80e",5,"identity: (x**6 - x**5 + 5*x**4 - 5*x**3 + 10*x**2 - 10*x + 3)/(x**2 - x + 3)"],["shape/4b8c11fb02",6,"identity: (N*x + 3*N*x**N + N)/(N - x + x**N)"],["form/2a295098bb",5,"identity: (x**6 - 3*x**5 + 2*x**4 - 2*x**3 - 5*x**2 + x - 2)/(x**3 + x + 2)"],["shape/4d20badcd5",6,"identity: (4*N*x**N + N + x + x**N)/(N + x + x**N)"],["macfarlane-vector-analysis-quaternions-1906/eq-e2f8987fb1",16,"Macfarlane 1906: OPQ = a_1 b_2 - \\frac{1}{2} a_2 a_2 - \\frac{1}{2} b_1 b_2 - \\frac{1}{2} (a_1 - b_1)(b_2 - a_2) = \\frac{1}{2}(a_1 b_2 - a"],["macfarlane-vector-analysis-quaternions-1906/eq-409ff69629",16,"Macfarlane 1906: A^2 = a_1^2 + a_2^2 = a^2"],["person/william-snow-harris",1,"William Snow Harris","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-william-snow-harris"],["macfarlane-vector-analysis-quaternions-1906/eq-d29b46a649",16,"Macfarlane 1906: A^{-1} = \\frac{1}{a}\\alpha = \\frac{a\\alpha}{a^2} = \\frac{a_1i + a_2j}{a_1^2 + a_2^2}"],["macfarlane-vector-analysis-quaternions-1906/eq-0d6238127e",16,"Macfarlane 1906: \\mathrm{S}BA = \\mathrm{S}AB"],["maxwell-elementary-treatise-electricity-1888/x-089a98bfe5",15,"Maxwell 1888, scan 175: In order that the machine may work to the ..."],["maxwell-elementary-treatise-electricity-1888/x-bd48ac009a",15,"Maxwell 1888, scan 176: It was by means of the revolving doubler that ..."],["law/van-der-waals-law",10,"van der Waals' law","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-law-van-der-waals-law"],["maxwell-elementary-treatise-electricity-1888/x-58b8417eea",15,"Maxwell 1888, scan 181: This conductor C’, by which the carrier is enabled ..."],["quantity/coefficient-of-compressibility",11,"coefficient of compressibility","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-coefficient-of-compressibility"],["quantity/mechanical-equivalent-of-heat",11,"mechanical equivalent of heat","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-quantity-mechanical-equivalent-of-heat"],["planck-treatise-on-thermodynamics-1903/x-4247e40124",15,"Planck 1903, p. 122: It follows that, for solids and liquids, the difference ..."],["concept/locus",7,"locus","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-locus"],["maxwell-elementary-treatise-electricity-1888/x-a36a008a14",15,"Maxwell 1888, scan 189: When the distance is too small a small change ..."],["planck-treatise-on-thermodynamics-1903/x-dbdcc94393",15,"Planck 1903, p. 123: This equation contains only quantities that can be directly ..."],["concept/ellipse",7,"ellipse","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-ellipse"],["concept/relative-rate-of-growth",7,"relative rate of growth","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-relative-rate-of-growth"],["boyden-first-book-in-algebra-1895/ex-26/15",4,"Boyden 1895, Exercise 26 (15)"],["form/61616080fb",5,"identity: (x**5 - x**3 - 2*x**2 - x)/(x**3 + x**2 + x)"],["shape/0f6fed1122",6,"identity: (N*x**N - x)/(x + 2*x**N)"],["boyden-first-book-in-algebra-1895/ex-26/16",4,"Boyden 1895, Exercise 26 (16)"],["shape/6f4efec28d",6,"identity: (N*x**N - x**N)/(x + x**N + 1)"],["concept/hyperbola",7,"hyperbola","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-hyperbola"],["concept/polynomial",7,"polynomial","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-polynomial"],["method/comparing-equal-resistances-with-wheatstone-s-bridge",8,"comparing equal resistances with Wheatstone's bridge","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-comparing-equal-resistances-with-wheatstone-s-bridge"],["instrument/slide-wire",13,"slide wire","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-slide-wire"],["theorem/heaviside-s-best-bridge-resistances",9,"Heaviside's best bridge resistances","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-heaviside-s-best-bridge-resistances"],["method/thomson-s-method-of-determining-galvanometer-resistance",8,"Thomson's method of determining galvanometer resistance","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-thomson-s-method-of-determining-galvanometer-resistance"],["form/f8e8064dc1",5,"identity: (x**11 - x**2)/(x**3 - 1)"],["hardy-course-of-pure-mathematics-1921/x-289229b1d2",15,"Hardy 1921, p. 38: In these circumstances y is said to be a ..."],["hardy-course-of-pure-mathematics-1921/x-b57ea25c89",15,"Hardy 1921, p. 39: All that is essential is that there should be ..."],["method/determining-battery-resistance-by-mance-s-method",8,"determining battery resistance by Mance's method","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-determining-battery-resistance-by-mance-s-method"],["boyden-first-book-in-algebra-1895/ex-26/17",4,"Boyden 1895, Exercise 26 (17)"],["boyden-first-book-in-algebra-1895/ex-26/18",4,"Boyden 1895, Exercise 26 (18)"],["form/a46fd54cf4",5,"identity: (x**12 - x**4)/(x**2 + 1)"],["hardy-course-of-pure-mathematics-1921/x-b8bbe12f85",15,"Hardy 1921, p. 47: Consider for example the function x/x, which is a ..."],["hardy-course-of-pure-mathematics-1921/x-b968012cc2",15,"Hardy 1921, p. 48: It is in no way presupposed in the definition ..."],["macfarlane-vector-analysis-quaternions-1906/eq-d986ef8562",16,"Macfarlane 1906: E = \\left(r + 2\\pi n l \\cdot \\beta^\\frac{\\pi}{2} \\right) I"],["macfarlane-vector-analysis-quaternions-1906/eq-505115110d",16,"Macfarlane 1906: I^{-1} E = r + 2\\pi n l \\cdot \\beta^\\frac{\\pi}{2}"],["macfarlane-vector-analysis-quaternions-1906/eq-045c3063e7",16,"Macfarlane 1906: R = \\left(p + q \\cdot \\beta^\\frac{\\pi}{2} \\right) A"],["concept/independent-variable",7,"independent variable","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-independent-variable"],["boyden-first-book-in-algebra-1895/ex-53/17",4,"Boyden 1895, Exercise 53 (17)"],["thompson-calculus-made-easy-1914/x-d87e13e5ff",15,"Thompson 1914, p. 12: Suppose the ladder was so long that when the ..."],["thompson-calculus-made-easy-1914/x-686644f489",15,"Thompson 1914, p. 12: Now right through the differential calculus we are hunting, ..."],["thompson-calculus-made-easy-1914/x-60240dfa8b",15,"Thompson 1914, p. 74: The process can be extended to three or more ..."],["thompson-calculus-made-easy-1914/x-0fcb0bc29a",15,"Thompson 1914, p. 68: By and bye, when you have learned how to ..."],["macfarlane-vector-analysis-quaternions-1906/eq-7c22eb1899",16,"Macfarlane 1906: E = \\frac{ \\sum\\left(\\frac{r}{r^2 + (2\\pi n)^2 l^2}\\right) + 2\\pi n\\sum\\left(\\frac{l}{r^2 + (2\\pi n)^2 l^2}\\right) \\cdot"],["macfarlane-vector-analysis-quaternions-1906/eq-2a78c138d2",16,"Macfarlane 1906: R' = rr'\\beta^{\\theta+\\theta'}A"],["form/cf9c36f018",5,"solve: Eq(-(x - 1)**2 + (x + 2)**2, 10*x + 15)"],["instrument/absolute-galvanometer",13,"absolute galvanometer","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-absolute-galvanometer"],["form/7669c5758e",5,"solve: Eq((x - 2)*(x - 1)*(x + 4), x*(x - 2)*(x + 2))"],["shape/6426b26996",6,"solve: Eq((N + x)**2*(x - 1), x*(N + x)**2)"],["boyden-first-book-in-algebra-1895/ex-53/18",4,"Boyden 1895, Exercise 53 (18)"],["form/63c15bab45",5,"solve: Eq(-2*x - (x - 7)*(x - 2) + (x - 5)*(x + 3) + 2, -12)"],["boyden-first-book-in-algebra-1895/ex-53/19",4,"Boyden 1895, Exercise 53 (19)"],["shape/69d19f363a",6,"solve: Eq((N + x)**N - (x - 1)**N, N*x + N)"],["boyden-first-book-in-algebra-1895/ex-53/20",4,"Boyden 1895, Exercise 53 (20)"],["concept/mercury",7,"mercury","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-mercury"],["concept/fluxional-notation",7,"fluxional notation","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-fluxional-notation"],["quantity/acceleration",11,"acceleration","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-quantity-acceleration"],["quantity/average-velocity",11,"average velocity","../books/thompson-calculus-made-easy-1914/terms/index.html#t-quantity-average-velocity"],["theorem/limit-of-a-constant-multiple",9,"limit of a constant multiple","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-of-a-constant-multiple"],["person/gottfried-wilhelm-leibniz",1,"Gottfried Wilhelm Leibniz","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-person-gottfried-wilhelm-leibniz"],["boyden-first-book-in-algebra-1895/ex-34/1",4,"Boyden 1895, Exercise 34 (1)"],["boyden-first-book-in-algebra-1895/ex-34/2",4,"Boyden 1895, Exercise 34 (2)"],["form/909d66d436",5,"solve: Eq((x + 3)*(2*x + 3) - 14, (x + 1)*(2*x + 1))"],["thompson-calculus-made-easy-1914/x-ba1b4087ab",15,"Thompson 1914, p. 53: It is said that Sandy had not been in ..."],["shape/fa81cdad22",6,"solve: Eq(N + (N + x)*(N*x + N), (x + 1)*(N*x + 1))"],["boyden-first-book-in-algebra-1895/ex-53/21",4,"Boyden 1895, Exercise 53 (21)"],["form/d0167e9c91",5,"solve: Eq((x - 5)**2 + (x + 1)**2, 2*(x + 5)**2)"],["shape/e7d7e86965",6,"solve: Eq((N + x)**N + (x + 1)**N, N*(N + x)**N)"],["unit/volt",12,"volt","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-unit-volt"],["thompson-calculus-made-easy-1914/x-ea2803d89b",15,"Thompson 1914, p. 56: When a railway train has just begun to move, ..."],["thompson-calculus-made-easy-1914/x-3cafa18767",15,"Thompson 1914, p. 57: That is to say, force may be expressed either ..."],["thompson-calculus-made-easy-1914/x-57c709427f",15,"Thompson 1914, p. 58: In this last sentence the word rate is clearly ..."],["person/latimer-clark",1,"Latimer Clark","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-latimer-clark"],["thompson-calculus-made-easy-1914/x-6677eb02bb",15,"Thompson 1914, p. 60: (It is the same velocity as the velocity at ..."],["person/oliver-lodge",1,"Oliver Lodge","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-oliver-lodge"],["maxwell-elementary-treatise-electricity-1888/x-c6819dbe56",15,"Maxwell 1888, scan 208: The comparison which can be effected with the greatest ..."],["maxwell-elementary-treatise-electricity-1888/x-89fbdc0215",15,"Maxwell 1888, scan 209: The coils and are then made to change places, ..."],["concept/abscissa",7,"abscissa","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-abscissa"],["concept/surface",7,"surface","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-surface"],["maxwell-elementary-treatise-electricity-1888/x-69b3d6ba0c",15,"Maxwell 1888, scan 211: It will be observed that though this is not ..."],["concept/ordinate",7,"ordinate","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-ordinate"],["maxwell-elementary-treatise-electricity-1888/x-1890dd46d6",15,"Maxwell 1888, scan 211: By the method now described the galvanometer itself is ..."],["thompson-calculus-made-easy-1914/ex-x",3,"Thompson 1914, Exercise X"],["concept/simultaneous-equations",7,"simultaneous equations","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-simultaneous-equations"],["thompson-calculus-made-easy-1914/x-940b06766f",15,"Thompson 1914, p. 116: The expense C of handling the products of a ..."],["thompson-calculus-made-easy-1914/x-ecc010264a",15,"Thompson 1914, p. 121: If we could split the fraction into two or ..."],["maxwell-elementary-treatise-electricity-1888/x-fafd00188e",15,"Maxwell 1888, scan 213: This method, as has been pointed out by Professor ..."],["thompson-calculus-made-easy-1914/x-ba68deb04f",15,"Thompson 1914, p. 78: If a curve is sloping up at 45° at ..."],["method/polynomial-interpolation",8,"polynomial interpolation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-polynomial-interpolation"],["macfarlane-vector-analysis-quaternions-1906/eq-6b3c4e899c",16,"Macfarlane 1906: \\mathrm{V}(\\mathrm{V}AB)C = -\\mathrm{S}BC \\cdot A + \\mathrm{S}CA \\cdot B"],["macfarlane-vector-analysis-quaternions-1906/eq-2a6adb6713",16,"Macfarlane 1906: \\mathrm{V}(ABC) = \\mathrm{S}AB \\cdot C - \\mathrm{S}BC \\cdot A + \\mathrm{S}CA \\cdot B"],["macfarlane-vector-analysis-quaternions-1906/eq-b498e365cd",16,"Macfarlane 1906: \\mathrm{V}(\\alpha\\beta\\gamma) &= \\cos \\alpha\\beta \\cdot \\gamma - \\cos \\beta\\gamma \\cdot \\alpha + \\cos \\gamma\\alpha \\cdot"],["macfarlane-vector-analysis-quaternions-1906/eq-f5007ca1b0",16,"Macfarlane 1906: \\mathrm{V}(\\beta\\gamma\\alpha) &= \\cos \\beta\\gamma \\cdot \\alpha - \\cos \\gamma \\alpha \\cdot \\beta + \\cos \\alpha \\beta \\cdo"],["planck-treatise-on-thermodynamics-1903/x-4c6b96e5ba",15,"Planck 1903, p. 68: Furthermore, most chemical processes are accompanied by a rise ..."],["boyden-first-book-in-algebra-1895/ex-26/19",4,"Boyden 1895, Exercise 26 (19)"],["form/d1c9028e82",5,"identity: (4*a**4 + x**4)/(a**2 - 2*a*x + x**2)"],["planck-treatise-on-thermodynamics-1903/x-01219dea82",15,"Planck 1903, p. 69: These symbols may be treated like algebraic quantities, whereby ..."],["method/euclidean-construction",8,"Euclidean construction","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-euclidean-construction"],["boyden-first-book-in-algebra-1895/ex-26/20",4,"Boyden 1895, Exercise 26 (20)"],["planck-treatise-on-thermodynamics-1903/x-f42e7b5c9c",15,"Planck 1903, p. 71: But since its change of energy U_{2} - U_{1} ..."],["shape/c76f84c802",6,"identity: (N*a**N + x**N)/(N*a*x + a**N + x**N)"],["form/7ee63575b9",5,"identity: (81*a**4 + 4*x**4)/(9*a**2 + 6*a*x + 2*x**2)"],["shape/8adf734986",6,"identity: (N*a**N + N*x**N)/(N*a*x + N*a**N + N*x**N)"],["theorem/limit-of-a-rational-function-of-sequences",9,"limit of a rational function of sequences","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-of-a-rational-function-of-sequences"],["theorem/limit-of-a-rational-function-of-n",9,"limit of a rational function of n","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-of-a-rational-function-of-n"],["unit/siemens-unit",12,"Siemens unit","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-unit-siemens-unit"],["concept/duplication-of-the-cube",7,"duplication of the cube","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-duplication-of-the-cube"],["hardy-course-of-pure-mathematics-1921/x-4133500dab",15,"Hardy 1921, p. 67: One of the famous problems of antiquity was that ..."],["hardy-course-of-pure-mathematics-1921/x-1c631332c1",15,"Hardy 1921, p. 68: If R is the earth’s radius, the error in ..."],["method/equating-the-derivative-to-zero",8,"equating the derivative to zero","../books/thompson-calculus-made-easy-1914/terms/index.html#t-method-equating-the-derivative-to-zero"],["unit/jacobi-s-etalon",12,"Jacobi's Etalon","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-unit-jacobi-s-etalon"],["concept/electromagnetic-system-of-units",7,"electromagnetic system of units","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electromagnetic-system-of-units"],["concept/explicit-function",7,"explicit function","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-explicit-function"],["planck-treatise-on-thermodynamics-1903/x-733c94fae5",15,"Planck 1903, p. 72: The heat effect, however, is not equal to the ..."],["macfarlane-vector-analysis-quaternions-1906/eq-598b7ab00f",16,"Macfarlane 1906: ij &= k, & jk &= i, & ki &= j"],["thompson-calculus-made-easy-1914/x-8a83287832",15,"Thompson 1914, p. 95: Now it may sound like juggling to be assured ..."],["boyden-first-book-in-algebra-1895/ex-27/1",4,"Boyden 1895, Exercise 27 (1)"],["instrument/rheostat",13,"rheostat","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-rheostat"],["concept/binary-scale",7,"binary scale","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-binary-scale"],["macfarlane-vector-analysis-quaternions-1906/eq-e638710de5",16,"Macfarlane 1906: \\text{velocity flux} = \\text{electromotive-force}"],["macfarlane-vector-analysis-quaternions-1906/eq-af62e707d3",16,"Macfarlane 1906: \\text{current flux} = \\text{mechanical-force}"],["macfarlane-vector-analysis-quaternions-1906/eq-e350f477f1",16,"Macfarlane 1906: \\text{flux force} = \\text{current}"],["method/method-of-multiple-arcs",8,"method of multiple arcs","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-method-of-multiple-arcs"],["form/2e2b0738be",5,"identity: 4*b**2*Abs(a)*Abs(b)"],["shape/fb23245998",6,"identity: N*b**N*Abs(a)*Abs(b)"],["thompson-calculus-made-easy-1914/x-7367fca914",15,"Thompson 1914, p. 107: On plotting the graph it will be found that ..."],["concept/comparison-of-resistances",7,"comparison of resistances","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-comparison-of-resistances"],["theorem/ohm-s-formula",9,"Ohm's formula","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-ohm-s-formula"],["concept/conjugate-conductors",7,"conjugate conductors","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-conjugate-conductors"],["planck-treatise-on-thermodynamics-1903/x-24ab25f3b2",15,"Planck 1903, p. 125: If, under constant pressure, v were proportional to \\theta, ..."],["planck-treatise-on-thermodynamics-1903/x-0236c67f8a",15,"Planck 1903, p. 127: In 4 we defined temperature by means of the ..."],["boyden-first-book-in-algebra-1895/ex-16/15",4,"Boyden 1895, Exercise 16 (15)"],["form/f88c3171f9",5,"identity: -9*b**(2*a) - 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(Eq(b, 5*a), Eq(a + b, 48))"],["concept/bilinear-transformation",7,"bilinear transformation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-bilinear-transformation"],["concept/limiting-point",7,"limiting point","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-limiting-point"],["concept/harmonic-points",7,"harmonic points","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-harmonic-points"],["boyden-first-book-in-algebra-1895/ex-16/17",4,"Boyden 1895, Exercise 16 (17)"],["method/cycle-rule-for-linear-circuits",8,"cycle rule for linear circuits","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-cycle-rule-for-linear-circuits"],["person/augustus-matthiessen",1,"Augustus Matthiessen","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-augustus-matthiessen"],["form/de37ccc110",5,"identity: -a**5 + 5*a**4*x - 3*a**2*x**3 - 2*a*x**4"],["shape/fa9a1708ff",6,"identity: N*a*x**N + N*a**N*x + N*a**N*x**N - a**N"],["form/24c61fe267",5,"solve: (Eq(b, 3*a), Eq(c, a + b), Eq(a + b + c, 120))"],["shape/14de739e67",6,"solve: (Eq(b, N*a), Eq(c, a + b), Eq(a + b + c, N))"],["hardy-course-of-pure-mathematics-1921/x-8fc7c4d150",15,"Hardy 1921, p. 86: It should be observed that it is not always ..."],["boyden-first-book-in-algebra-1895/ex-16/18",4,"Boyden 1895, Exercise 16 (18)"],["maxwell-elementary-treatise-electricity-1888/x-a101d83262",15,"Maxwell 1888, scan 215: There are three classes in which we may place ..."],["form/3b169f8114",5,"identity: a - b + x"],["boyden-first-book-in-algebra-1895/ex-16/19",4,"Boyden 1895, Exercise 16 (19)"],["hardy-course-of-pure-mathematics-1921/x-042da5543a",15,"Hardy 1921, p. 108: This is a case in which attempts to solve ..."],["hardy-course-of-pure-mathematics-1921/x-154843bdaa",15,"Hardy 1921, p. 89: This theorem is sometimes stated as follows: in an ..."],["hardy-course-of-pure-mathematics-1921/x-6443f5add0",15,"Hardy 1921, p. 95: Thus the general linear transformation is equivalent to the ..."],["hardy-course-of-pure-mathematics-1921/x-7024a2841f",15,"Hardy 1921, p. 95: The general bilinear transformation is the most general type ..."],["maxwell-elementary-treatise-electricity-1888/x-39bffe1e9e",15,"Maxwell 1888, scan 218: Finally, by making two different experiments, in one of ..."],["form/396f9c7b77",5,"identity: -3*x**2"],["concept/infinite-set",7,"infinite set","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-infinite-set"],["macfarlane-vector-analysis-quaternions-1906/eq-a37e877cc1",16,"Macfarlane 1906: \\cos a = \\cos b \\cos c + \\sin b \\sin c \\cos A"],["concept/external-angle",7,"external angle"],["maxwell-elementary-treatise-electricity-1888/x-58c84fe670",15,"Maxwell 1888, scan 221: Thus, with Hooper’s insulating material the apparent resistance at ..."],["boyden-first-book-in-algebra-1895/ex-30/1",4,"Boyden 1895, Exercise 30 (1)"],["macfarlane-vector-analysis-quaternions-1906/eq-c119ce8445",16,"Macfarlane 1906: \\beta^\\frac{\\pi}{2}\\gamma^\\frac{\\pi}{2} = -\\cos \\beta\\gamma -\\sin \\beta\\gamma \\cdot \\overline{\\beta\\gamma}^\\frac{\\pi}{2}"],["macfarlane-vector-analysis-quaternions-1906/eq-94eea5b784",16,"Macfarlane 1906: \\beta^b = \\cos b + \\sin b \\cdot \\beta^\\frac{\\pi}{2}"],["boyden-first-book-in-algebra-1895/ex-30/2",4,"Boyden 1895, Exercise 30 (2)"],["boyden-first-book-in-algebra-1895/ex-30/3",4,"Boyden 1895, Exercise 30 (3)"],["form/b1c2a7f172",5,"evaluate: 143"],["concept/function-of-several-variables",7,"function of several variables","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-function-of-several-variables"],["hardy-course-of-pure-mathematics-1921/x-3779019c30",15,"Hardy 1921, p. 116: The function \\phi(n) is said to tend to the ..."],["maxwell-elementary-treatise-electricity-1888/x-53180185a2",15,"Maxwell 1888, scan 222: The whole theory of what has been called residual ..."],["boyden-first-book-in-algebra-1895/ex-16/20",4,"Boyden 1895, Exercise 16 (20)"],["form/a58e4648e3",5,"identity: 8*x**3 - 2*x"],["macfarlane-vector-analysis-quaternions-1906/eq-8324a34c16",16,"Macfarlane 1906: \\cos\\beta^b\\gamma^c = \\cos b\\cos c - \\sin b\\sin c\\cos \\beta\\gamma"],["macfarlane-vector-analysis-quaternions-1906/eq-45237b7bdc",16,"Macfarlane 1906: \\cos b = 1 - \\frac{b^2}{2!} + \\frac{b^4}{4!} - \\frac{b^6}{6!} + \\text{ etc.}"],["macfarlane-vector-analysis-quaternions-1906/eq-4c817396f0",16,"Macfarlane 1906: \\sin b = b - \\frac{b^3}{3!} + \\frac{b^5}{5!} - \\text{ etc.}"],["macfarlane-vector-analysis-quaternions-1906/eq-42924897c7",16,"Macfarlane 1906: e^{b\\beta^\\frac{\\pi}{2}} e^{c\\gamma^\\frac{\\pi}{2}} = e^{b\\beta^\\frac{\\pi}{2} + c\\gamma^\\frac{\\pi}{2}}"],["macfarlane-vector-analysis-quaternions-1906/eq-7fdef1456c",16,"Macfarlane 1906: \\left\\{ b \\cdot \\beta^\\frac{\\pi}{2} + c \\cdot \\gamma^\\frac{\\pi}{2} \\right\\}^n = b^n \\cdot \\beta^{n^\\frac{\\pi}{2}} + nb^{"],["macfarlane-vector-analysis-quaternions-1906/eq-7bdda5c567",16,"Macfarlane 1906: \\sin\\alpha\\beta \\sin\\overline{\\alpha\\beta}\\gamma \\cdot \\overline{\\overline{\\alpha\\beta}\\gamma} = \\cos\\alpha\\gamma \\cdot "],["experiment/faraday-s-ice-pail-experiment",14,"Faraday's ice-pail experiment","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-experiment-faraday-s-ice-pail-experiment"],["boyden-first-book-in-algebra-1895/ex-16/21",4,"Boyden 1895, Exercise 16 (21)"],["law/conservation-of-electric-charge",10,"conservation of electric charge","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-law-conservation-of-electric-charge"],["concept/partial-differential",7,"partial differential","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-partial-differential"],["maxwell-elementary-treatise-electricity-1888/x-50d930adc1",15,"Maxwell 1888, scan 33: Since whatever be the position of the electrified bodies ..."],["thompson-calculus-made-easy-1914/x-6bff6eb201",15,"Thompson 1914, p. 175: Another way of indicating that the differentiation has been ..."],["maxwell-elementary-treatise-electricity-1888/x-41d0d23337",15,"Maxwell 1888, scan 36: In all electrical experiments the electrification of bodies is ..."],["form/ef4fb0a3a9",5,"solve: Eq(-2*a*b**2 - 5*a*c**2 + 15*b**3 + 4*b*c**2 + c**3 - x, 6*a**3 - 2*a*b**2 + 6*a*c**2 - 12*b**3 + 4*c**3)"],["macfarlane-vector-analysis-quaternions-1906/eq-36c6044e96",16,"Macfarlane 1906: -(ll' + mm' + nn') -(mn' - m'n)i^\\frac{\\pi}{2} - (nl' - n'l)j^\\frac{\\pi}{2} -(lm' - l'm)k^\\frac{\\pi}{2}"],["macfarlane-vector-analysis-quaternions-1906/eq-f9f0e2e6fc",16,"Macfarlane 1906: j^\\frac{\\pi}{2}i^\\frac{\\pi}{2} = k^\\frac{\\pi}{2}"],["macfarlane-vector-analysis-quaternions-1906/eq-6619ded5ec",16,"Macfarlane 1906: i^\\frac{\\pi}{2}i^\\frac{\\pi}{2} = -"],["macfarlane-vector-analysis-quaternions-1906/eq-8c0140a590",16,"Macfarlane 1906: y^{-c} = \\cos c - \\sin c \\cdot \\gamma^\\frac{\\pi}{2}"],["concept/electromagnetism",7,"electromagnetism","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electromagnetism"],["form/726f6bee24",5,"solve: Eq(a**3 - 4*a**2 + 16*a + x, a**3 + 64)"],["shape/2ac28b5925",6,"solve: Eq(N*a*b**N + N*a*c**N + N*b*c**N + N*b**N + c**N - x, N*a*b**N + N*a*c**N + N*a**N + N*b**N + N*c**N)"],["boyden-first-book-in-algebra-1895/ex-16/22",4,"Boyden 1895, Exercise 16 (22)"],["shape/df4df93cca",6,"solve: Eq(N*a + N*a**N + a**N + x, N + a**N)"],["concept/total-differential",7,"total differential","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-total-differential"],["concept/pressure",7,"pressure","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-pressure"],["boyden-first-book-in-algebra-1895/ex-16/23",4,"Boyden 1895, Exercise 16 (23)"],["concept/mathematical-physics",7,"mathematical physics","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-mathematical-physics"],["concept/electrical-earth",7,"electrical earth","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electrical-earth"],["form/fe358156a6",5,"solve: Eq(4*a**2 - 6*a*b - 6*b**2 + 8*b*c + x, 0)"],["concept/temperature",7,"temperature","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-temperature"],["instrument/gold-leaf-electroscope",13,"gold-leaf electroscope","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-gold-leaf-electroscope"],["maxwell-elementary-treatise-electricity-1888/x-8eef3cc182",15,"Maxwell 1888, scan 17: Take a stick of sealing-wax, rub it on woollen ..."],["shape/f4eebbce87",6,"solve: Eq(N*a*b + N*a**N + N*b*c + N*b**N + x, 0)"],["thompson-calculus-made-easy-1914/x-b8c4c68085",15,"Thompson 1914, p. 176: But, if you think of it, you will observe ..."],["thompson-calculus-made-easy-1914/x-5df1d1f888",15,"Thompson 1914, p. 177: In the following example F and f denote two ..."],["thompson-calculus-made-easy-1914/x-521e0a2ce1",15,"Thompson 1914, p. 178: This differential equation is of immense importance in mathematical ..."],["thompson-calculus-made-easy-1914/x-810222b0cf",15,"Thompson 1914, p. 180: The truck is a rectangular box open at the ..."],["maxwell-elementary-treatise-electricity-1888/x-7e1328646a",15,"Maxwell 1888, scan 18: The fact that certain bodies after being rubbed appear ..."],["maxwell-elementary-treatise-electricity-1888/x-9e7224e572",15,"Maxwell 1888, scan 19: The discharge, therefore, takes place through metals and through ..."],["maxwell-elementary-treatise-electricity-1888/x-b952d6d72c",15,"Maxwell 1888, scan 20: This is a matter of mere convention, but the ..."],["maxwell-elementary-treatise-electricity-1888/x-cc21851e0d",15,"Maxwell 1888, scan 23: If two vessels containing the same or different fluids ..."],["concept/compound-interest",7,"compound interest","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-compound-interest"],["concept/logarithmic-rate-of-growth",7,"logarithmic rate of growth","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-logarithmic-rate-of-growth"],["maxwell-elementary-treatise-electricity-1888/x-ff262b6808",15,"Maxwell 1888, scan 24: To raise a body to a high temperature may ..."],["maxwell-elementary-treatise-electricity-1888/x-b389bf993b",15,"Maxwell 1888, scan 24: And here we may introduce once for all the ..."],["boyden-first-book-in-algebra-1895/ex-30/4",4,"Boyden 1895, Exercise 30 (4)"],["form/3c6d94aa41",5,"evaluate: 951"],["boyden-first-book-in-algebra-1895/ex-30/5",4,"Boyden 1895, Exercise 30 (5)"],["concept/exponential-function",7,"exponential function","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-exponential-function"],["theorem/work-done-in-charging-a-conductor",9,"work done in charging a conductor","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-work-done-in-charging-a-conductor"],["theorem/exponential-series",9,"exponential series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-exponential-series"],["form/04be8a0a47",5,"evaluate: 42/5"],["boyden-first-book-in-algebra-1895/ex-30/6",4,"Boyden 1895, Exercise 30 (6)"],["form/86453d036b",5,"evaluate: 19/20"],["boyden-first-book-in-algebra-1895/ex-30/7",4,"Boyden 1895, Exercise 30 (7)"],["form/dae61d2eaa",5,"evaluate: 308"],["concept/die-away-factor",7,"die-away factor","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-die-away-factor"],["theorem/reciprocity-of-potentials-and-charges",9,"reciprocity of potentials and charges","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-reciprocity-of-potentials-and-charges"],["theorem/induced-charge-and-potential-ratio-of-two-conductors",9,"induced charge and potential ratio of two conductors","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-induced-charge-and-potential-ratio-of-two-conductors"],["theorem/die-away-curve",9,"die-away curve","../books/thompson-calculus-made-easy-1914/terms/index.html#t-theorem-die-away-curve"],["theorem/increment-of-electrical-energy-when-charges-change",9,"increment of electrical energy when charges change","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-increment-of-electrical-energy-when-charges-change"],["theorem/green-s-theorem",9,"Green's theorem","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-green-s-theorem"],["theorem/rate-of-change-of-electric-energy-with-charge",9,"rate of change of electric energy with charge","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-rate-of-change-of-electric-energy-with-charge"],["theorem/work-done-in-displacing-insulated-conductors",9,"work done in displacing insulated conductors","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-work-done-in-displacing-insulated-conductors"],["boyden-first-book-in-algebra-1895/ex-30/8",4,"Boyden 1895, Exercise 30 (8)"],["concept/pi",7,"pi","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-pi"],["thompson-calculus-made-easy-1914/x-47d869dc69",15,"Thompson 1914, p. 139: Suppose we were to let 1 grow at simple ..."],["boyden-first-book-in-algebra-1895/ex-30/9",4,"Boyden 1895, Exercise 30 (9)"],["boyden-first-book-in-algebra-1895/ex-30/10",4,"Boyden 1895, Exercise 30 (10)"],["hardy-course-of-pure-mathematics-1921/x-7195191e9b",15,"Hardy 1921, p. 121: When \\phi(n) does not tend to a limit, nor ..."],["boyden-first-book-in-algebra-1895/ex-16/24",4,"Boyden 1895, Exercise 16 (24)"],["thompson-calculus-made-easy-1914/x-a19ac3ed18",15,"Thompson 1914, p. 148: Note that x^{-1} is a result that we could ..."],["form/4d0545da34",5,"solve: Eq(-2*a**4 + 3*a**2 - 2*a + x + 5, 1)"],["shape/5000bc2553",6,"solve: Eq(N*a + 2*N*a**N + N + x, 1)"],["boyden-first-book-in-algebra-1895/ex-16/25",4,"Boyden 1895, Exercise 16 (25)"],["form/e9ac840b3d",5,"solve: Eq(3*a**3 + 2*a**2 + 2*a - x - 2, -2*a**3 + 2*a**2 - 4)"],["shape/418c84a587",6,"solve: Eq(N*a + 2*N*a**N + N - x, 2*N*a**N + N)"],["boyden-first-book-in-algebra-1895/ex-16/26",4,"Boyden 1895, Exercise 16 (26)"],["macfarlane-vector-analysis-quaternions-1906/eq-af5c1224ec",16,"Macfarlane 1906: \\beta^\\theta R = \\mathrm{S}\\beta R \\cdot \\beta + \\cos \\theta(\\mathrm{V}\\beta R)\\beta + \\sin \\theta \\mathrm{V}\\beta R"],["concept/radius-vector",7,"radius vector"],["macfarlane-vector-analysis-quaternions-1906/eq-33b9d22621",16,"Macfarlane 1906: \\beta^\\theta R = \\cos \\theta R + \\sin \\theta \\mathrm{V}(\\beta R)"],["macfarlane-vector-analysis-quaternions-1906/eq-1dbb225c03",16,"Macfarlane 1906: \\mathrm{S}\\beta R = lx + my + nz"],["macfarlane-vector-analysis-quaternions-1906/eq-7109661cf2",16,"Macfarlane 1906: l^2 + m^2 + n^2 = 1"],["macfarlane-vector-analysis-quaternions-1906/eq-1218b7829f",16,"Macfarlane 1906: \\beta^b\\rho = \\beta^\\frac{-b}{2}\\rho^\\frac{\\pi}{2}\\beta^\\frac{b}{2}"],["macfarlane-vector-analysis-quaternions-1906/eq-f27f6cfc55",16,"Macfarlane 1906: e^{-\\frac{1}{2}b\\beta^\\frac{\\pi}{2} + \\frac{1}{2}\\pi\\rho^\\frac{\\pi}{2} + \\frac{1}{2}b\\beta^\\frac{\\pi}{2}}"],["maxwell-elementary-treatise-electricity-1888/x-6aa340f4db",15,"Maxwell 1888, scan 38: In fact, this doctrine is the one generalised statement ..."],["form/86fccdcdd8",5,"identity: 6*a**3*x - a**3 - 12*a**2*x**2 - 2*a**2*x - 11*a*x**2"],["macfarlane-vector-analysis-quaternions-1906/eq-c70f70d6b2",16,"Macfarlane 1906: \\beta^b \\times \\gamma^c = m^2 - n^2 + 2mn \\cdot \\nu"],["macfarlane-vector-analysis-quaternions-1906/eq-58ef4d3621",16,"Macfarlane 1906: \\cos\\beta^b \\times \\gamma^c = 1"],["macfarlane-vector-analysis-quaternions-1906/eq-a624560fc0",16,"Macfarlane 1906: \\Sin \\beta^b \\times \\gamma^c = b \\cdot \\beta + c \\cdot \\gamma"],["maxwell-elementary-treatise-electricity-1888/x-d7a6135267",15,"Maxwell 1888, scan 40: The electric potential at a given point of the ..."],["maxwell-elementary-treatise-electricity-1888/x-84f53cfdb2",15,"Maxwell 1888, scan 40: Hence the magnitude of the electric force may be ..."],["maxwell-elementary-treatise-electricity-1888/x-3c9d9afd10",15,"Maxwell 1888, scan 42: By increasing without limit the number of equal parts ..."],["shape/dd205d9f8f",6,"identity: N*a*x**N + 2*N*a**N*x + N*a**N*x**N - a**N"],["form/01cca934ff",5,"evaluate: 73"],["concept/slope-of-a-curve",7,"slope of a curve","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-slope-of-a-curve"],["maxwell-elementary-treatise-electricity-1888/x-306183a6ed",15,"Maxwell 1888, scan 48: This is the first instance we have met with ..."],["boyden-first-book-in-algebra-1895/ex-16/27",4,"Boyden 1895, Exercise 16 (27)"],["form/b82dba3b73",5,"solve: Eq(a + x, b)"],["concept/approximation",7,"approximation","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-approximation"],["shape/b82dba3b73",6,"solve: Eq(a + x, b)"],["boyden-first-book-in-algebra-1895/ex-16/28",4,"Boyden 1895, Exercise 16 (28)"],["form/d2044da90b",5,"identity: x - 3"],["shape/7193b66bbf",6,"identity: N + x"],["form/7190f69ea7",5,"solve: Eq(a + x, 40)"],["shape/cb1316f0cf",6,"solve: Eq(a + x, N)"],["experiment/exploring-the-field-with-a-small-electrified-body",14,"exploring the field with a small electrified body","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-experiment-exploring-the-field-with-a-small-electrified-body"],["theorem/reciprocity-of-potentials",9,"reciprocity of potentials","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-reciprocity-of-potentials"],["maxwell-elementary-treatise-electricity-1888/x-d75087b88c",15,"Maxwell 1888, scan 54: The measurement of small forces is always a difficult ..."],["thompson-calculus-made-easy-1914/x-7eb634817c",15,"Thompson 1914, p. 182: Any one can understand how the whole of anything ..."],["thompson-calculus-made-easy-1914/x-519dd5cae3",15,"Thompson 1914, p. 183: If at any point of the operation we stop, ..."],["thompson-calculus-made-easy-1914/x-7ba65c603d",15,"Thompson 1914, p. 187: As the only information we have is as to ..."],["thompson-calculus-made-easy-1914/x-fd919bb31c",15,"Thompson 1914, p. 189: But x began by being 0, and increases to ..."],["maxwell-elementary-treatise-electricity-1888/x-40fcd7b44a",15,"Maxwell 1888, scan 55: The electrification of each disk is proportional to the ..."],["boyden-first-book-in-algebra-1895/ex-16/30",4,"Boyden 1895, Exercise 16 (30)"],["boyden-first-book-in-algebra-1895/ex-30/11",4,"Boyden 1895, Exercise 30 (11)"],["boyden-first-book-in-algebra-1895/ex-17/2",4,"Boyden 1895, Exercise 17 (2)"],["form/cf21ba09bd",5,"identity: a + b"],["concept/function-of-a-positive-integer-variable",7,"function of a positive integer variable","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-function-of-a-positive-integer-variable"],["shape/ba93585131",6,"identity: N*e + a + b + c - d"],["shape/cf21ba09bd",6,"identity: a + b"],["boyden-first-book-in-algebra-1895/ex-17/3",4,"Boyden 1895, Exercise 17 (3)"],["form/bf10c52aab",5,"identity: -a**3 + 2*a**2*b - a*b**2 - 2*b**3"],["shape/30bb9cda94",6,"identity: N*a**N*b + N*b**N - a*b**N - a**N"],["form/3ddcb94398",5,"evaluate: 623/10"],["boyden-first-book-in-algebra-1895/ex-30/12",4,"Boyden 1895, Exercise 30 (12)"],["form/589c7758b9",5,"evaluate: 839/10"],["boyden-first-book-in-algebra-1895/ex-30/13",4,"Boyden 1895, Exercise 30 (13)"],["form/8532ae3bea",5,"evaluate: 82/25"],["maxwell-elementary-treatise-electricity-1888/x-5529ee5480",15,"Maxwell 1888, scan 57: A convenient way of determining the direction of the ..."],["maxwell-elementary-treatise-electricity-1888/x-db1620b383",15,"Maxwell 1888, scan 57: Place one of the spheres at a fixed point, ..."],["theorem/infinitude-of-primes",9,"infinitude of primes","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-infinitude-of-primes"],["concept/integer-part",7,"integer 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..."],["concept/pitfall",7,"pitfall","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-pitfall"],["person/george-boole",1,"George Boole","../books/thompson-calculus-made-easy-1914/terms/index.html#t-person-george-boole"],["theorem/electric-force-at-a-conductor-s-surface",9,"electric force at a conductor's surface","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-electric-force-at-a-conductor-s-surface"],["theorem/potential-due-to-a-point-charge",9,"potential due to a point charge","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-theorem-potential-due-to-a-point-charge"],["thompson-calculus-made-easy-1914/x-7077d235a4",15,"Thompson 1914, p. 226: A great part of the labour of integrating things ..."],["thompson-calculus-made-easy-1914/x-28d110b7cb",15,"Thompson 1914, p. 226: It is useful in some cases that you can’t ..."],["theorem/attraction-between-two-parallel-charged-planes",9,"attraction between two 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..."],["maxwell-elementary-treatise-electricity-1888/x-e356d0cfd5",15,"Maxwell 1888, scan 98: By this method he has solved problems in electricity ..."],["maxwell-elementary-treatise-electricity-1888/x-4ffe54a311",15,"Maxwell 1888, scan 106: The surface-density is negative on the side next to ..."],["macfarlane-vector-analysis-quaternions-1906/x-b10acc1be9",15,"Macfarlane 1906: By a “quaternion” is meant the operator which changes ..."],["macfarlane-vector-analysis-quaternions-1906/x-0f8f3dd050",15,"Macfarlane 1906: Let A and R be two coinitial vectors; the ..."],["macfarlane-vector-analysis-quaternions-1906/x-b9beda113e",15,"Macfarlane 1906: The resistance is the scalar part of the quaternion, ..."],["macfarlane-vector-analysis-quaternions-1906/x-9f616ffd95",15,"Macfarlane 1906: Etymologically “quaternion” means defined by four elements; which is ..."],["macfarlane-vector-analysis-quaternions-1906/x-9443896fc3",15,"Macfarlane 1906: Note that the product is formed by taking the ..."],["macfarlane-vector-analysis-quaternions-1906/x-2e04b1a022",15,"Macfarlane 1906: The angles are summed because they are indices of ..."],["macfarlane-vector-analysis-quaternions-1906/x-2bf619aed2",15,"Macfarlane 1906: This is the fundamental error in the Argand method."],["boyden-first-book-in-algebra-1895/ex-17/20",4,"Boyden 1895, Exercise 17 (20)"],["boyden-first-book-in-algebra-1895/ex-28/4",4,"Boyden 1895, Exercise 28 (4)"],["concept/discontinuous-function",7,"discontinuous function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-discontinuous-function"],["concept/simple-discontinuity",7,"simple discontinuity","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-simple-discontinuity"],["quantity/electric-charge",11,"electric charge","../books/ball-mathematical-recreations-1905/terms/index.html#t-quantity-electric-charge"],["method/discharging-a-conductor",8,"discharging a 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p. 176: Thus our definition asserts that if we draw two ..."],["shape/e11c8ff5ef",6,"identity: N*a*b*c**N*d**N"],["boyden-first-book-in-algebra-1895/ex-18/6",4,"Boyden 1895, Exercise 18 (6)"],["form/02a8d61d15",5,"identity: 30*a**3*b**2*c**4"],["concept/conductivity-of-gases",7,"conductivity of gases","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-conductivity-of-gases"],["concept/electric-polarity",7,"electric polarity","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electric-polarity"],["hardy-course-of-pure-mathematics-1921/x-ec616d9890",15,"Hardy 1921, p. 179: In other words as x varies from x_{0} to ..."],["hardy-course-of-pure-mathematics-1921/x-6d7afc912f",15,"Hardy 1921, p. 181: Indeed it is not even true that \\phi(x) must ..."],["hardy-course-of-pure-mathematics-1921/x-7d226c53a1",15,"Hardy 1921, p. 181: The net result of this and the last section ..."],["macfarlane-vector-analysis-quaternions-1906/x-0ba1db2b82",15,"Macfarlane 1906: The product is positive when the vector and the ..."],["theorem/product-rule",9,"product rule","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-product-rule"],["macfarlane-vector-analysis-quaternions-1906/x-57e01f3009",15,"Macfarlane 1906: the whole kinetic energy is obtained, not by vector, ..."],["macfarlane-vector-analysis-quaternions-1906/x-e5c8420db3",15,"Macfarlane 1906: We assume that their product is obtained by applying ..."],["macfarlane-vector-analysis-quaternions-1906/x-d6a092501f",15,"Macfarlane 1906: In a sum of vectors, the vectors are necessarily ..."],["theorem/quotient-rule",9,"quotient rule","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-quotient-rule"],["concept/electric-spark",7,"electric 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whole theory of the electric properties of gases ..."],["maxwell-elementary-treatise-electricity-1888/x-2b6bdabfc4",15,"Maxwell 1888, scan 140: It follows from this that neither the electric fluid, ..."],["theorem/small-oscillation-subdivision-theorem",9,"small-oscillation subdivision theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-small-oscillation-subdivision-theorem"],["concept/electric-displacement",7,"electric displacement","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electric-displacement"],["thompson-calculus-made-easy-1914/x-eb8f544e3b",15,"Thompson 1914, p. 252: \\epsilon^x & \\epsilon^x & \\epsilon^x + C"],["macfarlane-vector-analysis-quaternions-1906/x-b5d66d455a",15,"Macfarlane 1906: By the reciprocal of a vector is meant the ..."],["thompson-calculus-made-easy-1914/x-7805b97462",15,"Thompson 1914, p. 252: [-12pt]0pt32ptu\\, \\dfrac{dv}{dx} + v\\, \\dfrac{du}{dx} & uv & No ..."],["boyden-first-book-in-algebra-1895/ex-36/26",4,"Boyden 1895, Exercise 36 (26)"],["theorem/uniform-continuity",9,"uniform continuity","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-uniform-continuity"],["theorem/inverse-function-theorem",9,"inverse function theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-inverse-function-theorem"],["boyden-first-book-in-algebra-1895/ex-18/7",4,"Boyden 1895, Exercise 18 (7)"],["boyden-first-book-in-algebra-1895/ex-18/8",4,"Boyden 1895, Exercise 18 (8)"],["form/c615443e67",5,"identity: -a**4*b**5*c**2"],["concept/continuous-function-of-two-variables",7,"continuous function of two variables","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-continuous-function-of-two-variables"],["concept/separate-continuity",7,"separate 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..."],["theorem/implicit-function-theorem",9,"implicit function theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-implicit-function-theorem"],["concept/implicit-function",7,"implicit function","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-implicit-function"],["method/testing-for-an-exact-differential",8,"testing for an exact differential","../books/thompson-calculus-made-easy-1914/terms/index.html#t-method-testing-for-an-exact-differential"],["concept/integrating-factor",7,"integrating factor","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-integrating-factor"],["concept/electrode",7,"electrode","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electrode"],["quantity/specific-heat-at-constant-volume",11,"specific heat at constant 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..."],["thompson-calculus-made-easy-1914/x-39232f14e6",15,"Thompson 1914, p. 241: This is indeed none other than the equation of ..."],["thompson-calculus-made-easy-1914/x-59e443ef1e",15,"Thompson 1914, p. 243: It is possible in such cases to discover, however, ..."],["thompson-calculus-made-easy-1914/x-10b4ac3681",15,"Thompson 1914, p. 248: You have now been personally conducted over the frontiers ..."],["concept/isothermal-process",7,"isothermal process","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-isothermal-process"],["concept/one-sided-limit",7,"one-sided limit","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-one-sided-limit"],["concept/oscillation",7,"oscillation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-oscillation"],["law/law-of-magnus",10,"law of 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In a circuit formed of any number of metals ..."],["maxwell-elementary-treatise-electricity-1888/x-5de365d398",15,"Maxwell 1888, scan 148: In fact the experiments of Le Roux and others ..."],["theorem/limit-of-sin-x-over-x",9,"limit of sin x over x","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-limit-of-sin-x-over-x"],["hardy-course-of-pure-mathematics-1921/x-c67245227e",15,"Hardy 1921, p. 162: The only difference between the ‘tending of n to ..."],["maxwell-elementary-treatise-electricity-1888/x-1400d132ea",15,"Maxwell 1888, scan 151: The latter, which we may call with Thomson the ..."],["maxwell-elementary-treatise-electricity-1888/x-0a92ea887b",15,"Maxwell 1888, scan 155: We may, however, without any such assumption, make use ..."],["hardy-course-of-pure-mathematics-1921/x-c38906b44c",15,"Hardy 1921, p. 170: It is not a statement about the value of ..."],["law/boyle-s-law",10,"Boyle's 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calculus-tricks are quite easy. Some are enormously difficult. ..."],["method/vector-product",8,"vector product","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-method-vector-product"],["macfarlane-vector-analysis-quaternions-1906/x-7d1c9eef0d",15,"Macfarlane 1906: The formula \\text{velocity flux} = \\text{electromotive-force} is much handier ..."],["macfarlane-vector-analysis-quaternions-1906/x-e15bb25901",15,"Macfarlane 1906: The square combinations give results which are independent of ..."],["macfarlane-vector-analysis-quaternions-1906/x-ef27cfbebe",15,"Macfarlane 1906: The vector product as before is denoted by \\mathrm{V}AB. ..."],["thompson-calculus-made-easy-1914/x-8a1600ab58",15,"Thompson 1914, p. 2: Now any fool can see that if x is ..."],["thompson-calculus-made-easy-1914/x-556b6caf34",15,"Thompson 1914, p. 2: If you think of the duration of time for ..."],["boyden-first-book-in-algebra-1895/ex-18/13",4,"Boyden 1895, Exercise 18 (13)"],["theorem/general-leibniz-rule",9,"general Leibniz rule","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-general-leibniz-rule"],["method/integration",8,"integration","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-integration"],["concept/electric-wind",7,"electric wind","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-electric-wind"],["instrument/revolving-doubler",13,"revolving doubler","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-revolving-doubler"],["concept/receiver",7,"receiver","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-receiver"],["instrument/regenerator",13,"regenerator","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-regenerator"],["form/909d1212ff",5,"solve: Eq(x, 100*a + 10*b + c)"],["theorem/generalised-mean-value-theorem",9,"generalised mean value theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-generalised-mean-value-theorem"],["concept/differential-equation",7,"differential equation","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-differential-equation"],["shape/0aa48e4d65",6,"solve: Eq(x, N*a + N*b + c)"],["boyden-first-book-in-algebra-1895/ex-18/14",4,"Boyden 1895, Exercise 18 (14)"],["form/eede46313d",5,"solve: Eq(Abs(a - x), 7)"],["shape/bb9ecfb811",6,"solve: Eq(Abs(a - x), N)"],["concept/scalar-product",7,"scalar product","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-scalar-product"],["macfarlane-vector-analysis-quaternions-1906/x-64bb2a0831",15,"Macfarlane 1906: The principle here proved is of great use in ..."],["macfarlane-vector-analysis-quaternions-1906/x-7ab39a655e",15,"Macfarlane 1906: The term (2) means the projection of R upon ..."],["hardy-course-of-pure-mathematics-1921/x-35e5ec529c",15,"Hardy 1921, p. 254: In each case we have only to write down ..."],["concept/idiostatic-instrument",7,"idiostatic instrument","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-idiostatic-instrument"],["concept/parallelepiped",7,"parallelepiped","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-parallelepiped"],["concept/projection",7,"projection","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-projection"],["quantity/length",11,"length","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-quantity-length"],["hardy-course-of-pure-mathematics-1921/ex-xlix",3,"Hardy 1921, Exercise XLIX"],["concept/square",7,"square","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-square"],["person/jonathan-swift",1,"Jonathan Swift","../books/ball-mathematical-recreations-1905/terms/index.html#t-person-jonathan-swift"],["thompson-calculus-made-easy-1914/x-71a2d2ee71",15,"Thompson 1914, p. 8: An ox might worry about a flea of ordinary ..."],["method/measuring-a-small-electromotive-force",8,"measuring a small electromotive force","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-measuring-a-small-electromotive-force"],["person/william-nicholson",1,"William Nicholson","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-william-nicholson"],["person/charles-augustin-de-coulomb",1,"Charles-Augustin de Coulomb","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-charles-augustin-de-coulomb"],["method/integration-by-parts",8,"integration by parts","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-integration-by-parts"],["maxwell-elementary-treatise-electricity-1888/x-c5712abe54",15,"Maxwell 1888, scan 174: Thus the rotation of the machine carries the positive ..."],["maxwell-elementary-treatise-electricity-1888/x-62d7728e7b",15,"Maxwell 1888, scan 178: On the other hand, the quantity pU - qV ..."],["maxwell-elementary-treatise-electricity-1888/x-690a9082ee",15,"Maxwell 1888, scan 184: In all electrometers it is of the greatest importance ..."],["maxwell-elementary-treatise-electricity-1888/x-293ae86c2c",15,"Maxwell 1888, scan 190: This method of using an auxiliary electrification besides the ..."],["concept/inverse-function",7,"inverse function","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-inverse-function"],["instrument/replenisher",13,"replenisher","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-replenisher"],["method/measuring-potential-by-a-stream-of-falling-drops",8,"measuring potential by a stream of falling drops","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-measuring-potential-by-a-stream-of-falling-drops"],["maxwell-elementary-treatise-electricity-1888/x-a453ad3e17",15,"Maxwell 1888, scan 191: But if we can make the potential of the ..."],["concept/sphere",7,"sphere","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-sphere"],["concept/rational-function",7,"rational function","../books/dickson-theory-of-equations-1922/terms/index.html#t-concept-rational-function"],["concept/transcendental-function",7,"transcendental function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-transcendental-function"],["planck-treatise-on-thermodynamics-1903/x-ad4aaa4fd3",15,"Planck 1903, p. 205: A better insight into the nature of these quantities ..."],["concept/parameter",7,"parameter","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-parameter"],["hardy-course-of-pure-mathematics-1921/x-eac03d2e12",15,"Hardy 1921, p. 240: The theorem of integration by parts is merely another ..."],["planck-treatise-on-thermodynamics-1903/x-620df8855f",15,"Planck 1903, p. 187: This means that the heat effect in a variation ..."],["person/noyes",1,"Noyes","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-noyes"],["concept/vector-product",7,"vector product","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-vector-product"],["theorem/zero-derivative-implies-constant",9,"zero derivative implies constant","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-zero-derivative-implies-constant"],["theorem/mean-value-theorem",9,"mean value theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-mean-value-theorem"],["maxwell-elementary-treatise-electricity-1888/x-5dc967468a",15,"Maxwell 1888, scan 193: To measure the potential of a conductor without touching ..."],["shape/ef840e04f3",6,"identity: a**N*x**N*(a*x + a**N + x**N)"],["planck-treatise-on-thermodynamics-1903/x-c51ff4a18e",15,"Planck 1903, p. 258: or, the two solutions are isohydric if the concentration ..."],["boyden-first-book-in-algebra-1895/ex-19/1",4,"Boyden 1895, Exercise 19 (1)"],["concept/standard-forms-of-integration",7,"standard forms of integration","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-standard-forms-of-integration"],["concept/arbitrary-constant-of-integration",7,"arbitrary constant of integration","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-arbitrary-constant-of-integration"],["person/wilhelm-weber",1,"Wilhelm Weber","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-wilhelm-weber"],["concept/standard-of-resistance",7,"standard of resistance","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-standard-of-resistance"],["instrument/differential-galvanometer",13,"differential galvanometer","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-differential-galvanometer"],["concept/conic-section",7,"conic section","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-conic-section"],["concept/algebraic-function",7,"algebraic function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-algebraic-function"],["boyden-first-book-in-algebra-1895/ex-19/2",4,"Boyden 1895, Exercise 19 (2)"],["form/8308cf5917",5,"identity: a*x**2*(a**2 - a*x + x**2)"],["shape/9a7f58346d",6,"identity: a*x**N*(-a*x + a**N + x**N)"],["maxwell-elementary-treatise-electricity-1888/x-19bf434cbf",15,"Maxwell 1888, scan 195: To recollect its value in absolute measure it is ..."],["maxwell-elementary-treatise-electricity-1888/x-3553f8c73f",15,"Maxwell 1888, scan 195: In the same way the metre is professedly one ..."],["maxwell-elementary-treatise-electricity-1888/x-0104d48317",15,"Maxwell 1888, scan 200: But this is rather to be taken as an ..."],["hardy-course-of-pure-mathematics-1921/x-821128939e",15,"Hardy 1921, p. 226: Before we give a strict proof of this theorem, ..."],["hardy-course-of-pure-mathematics-1921/x-a52fe76597",15,"Hardy 1921, p. 230: These formulae must be understood as meaning that the ..."],["hardy-course-of-pure-mathematics-1921/x-f03aca2f54",15,"Hardy 1921, p. 237: Thus the integral of R(\\sqrt{x}), where R denotes a ..."],["boyden-first-book-in-algebra-1895/ex-19/3",4,"Boyden 1895, Exercise 19 (3)"],["method/differentiating-from-first-principles",8,"differentiating from first principles","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-differentiating-from-first-principles"],["thompson-calculus-made-easy-1914/x-777cd550fb",15,"Thompson 1914, p. 22: Just look at these results: the operation of differentiating ..."],["thompson-calculus-made-easy-1914/x-739715ec41",15,"Thompson 1914, p. 25: You have now learned how to differentiate powers of ..."],["method/null-method",8,"null method","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-null-method"],["method/comparing-electromotive-forces-by-poggendorff-s-compensation-method",8,"comparing electromotive forces by Poggendorff's compensation method","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-comparing-electromotive-forces-by-poggendorff-s-compensation-method"],["instrument/electrometer",13,"electrometer","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-electrometer"],["form/785530d089",5,"identity: -2*a*x*(a**3 - 3*a*x**2 + x**3)"],["shape/29ac867db4",6,"identity: N*a*x*(N*a*x**N + a**N + x**N)"],["thompson-calculus-made-easy-1914/x-ed451fba05",15,"Thompson 1914, p. 13: If, while x is, as before, the distance of ..."],["thompson-calculus-made-easy-1914/x-7ccec4a881",15,"Thompson 1914, p. 15: It is a solemn scientific name for this very ..."],["thompson-calculus-made-easy-1914/x-1a4eb0dd0b",15,"Thompson 1914, p. 16: You have now to learn to go hunting in ..."],["instrument/resistance-box",13,"resistance box","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-resistance-box"],["thompson-calculus-made-easy-1914/x-c322b99b25",15,"Thompson 1914, p. 29: So that any mere multiplication by a constant reappears ..."],["thompson-calculus-made-easy-1914/x-ad5de5c50a",15,"Thompson 1914, p. 30: As a rule an expression of this kind will ..."],["thompson-calculus-made-easy-1914/x-be021a4257",15,"Thompson 1914, p. 50: Similarly, we may write as the result of thrice ..."],["theorem/power-rule",9,"power rule","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-power-rule"],["maxwell-elementary-treatise-electricity-1888/x-13f15d9736",15,"Maxwell 1888, scan 211: The measurement of the resistance of a battery when ..."],["concept/derivative",7,"derivative","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-derivative"],["concept/versor",7,"versor","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-concept-versor"],["unit/radian",12,"radian","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-unit-radian"],["macfarlane-vector-analysis-quaternions-1906/x-7cf40e92fa",15,"Macfarlane 1906: The expression (\\beta^\\frac{b}{2}\\gamma^\\frac{c}{2})^2 is not, as might be supposed, ..."],["macfarlane-vector-analysis-quaternions-1906/x-46be5ef0b3",15,"Macfarlane 1906: When b and c are infinitesimals, \\cos\\beta^b \\times \\gamma^c ..."],["hardy-course-of-pure-mathematics-1921/ex-lvi",3,"Hardy 1921, Exercise LVI"],["theorem/sum-rule-for-differentiation",9,"sum rule for differentiation","../books/dickson-theory-of-equations-1922/terms/index.html#t-theorem-sum-rule-for-differentiation"],["theorem/quotient-rule-for-differentiation",9,"quotient rule for differentiation","../books/thompson-calculus-made-easy-1914/terms/index.html#t-theorem-quotient-rule-for-differentiation"],["thompson-calculus-made-easy-1914/x-ca635bc95f",15,"Thompson 1914, p. 36: This justifies the procedure. You differentiate each function separately ..."],["thompson-calculus-made-easy-1914/x-07d0fff3d7",15,"Thompson 1914, p. 39: In such a case it is no use to ..."],["theorem/total-differential",9,"total differential","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-theorem-total-differential"],["maxwell-elementary-treatise-electricity-1888/x-957cb978cf",15,"Maxwell 1888, scan 213: Let the electromotive force E of the battery be ..."],["theorem/mean-value-theorem-for-functions-of-two-variables",9,"Mean Value Theorem for functions of two variables","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-mean-value-theorem-for-functions-of-two-variables"],["concept/principal-part-of-an-increment",7,"principal part of an increment","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-principal-part-of-an-increment"],["person/robert-meyer",1,"Robert Meyer","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-person-robert-meyer"],["hardy-course-of-pure-mathematics-1921/ex-lix",3,"Hardy 1921, Exercise LIX"],["hardy-course-of-pure-mathematics-1921/ex-lvii",3,"Hardy 1921, Exercise LVII"],["theorem/area-of-an-ellipse",9,"area of an ellipse","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-area-of-an-ellipse"],["theorem/area-of-a-triangle",9,"area of a triangle","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-area-of-a-triangle"],["concept/upper-sum",7,"upper sum","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-upper-sum"],["hardy-course-of-pure-mathematics-1921/ex-lviii",3,"Hardy 1921, Exercise LVIII"],["planck-treatise-on-thermodynamics-1903/x-39dd5ad811",15,"Planck 1903, p. 46: The term homogeneous is used here in the sense ..."],["concept/polarization",7,"polarization","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-polarization"],["quantity/resistance",11,"resistance","../books/macfarlane-vector-analysis-quaternions-1906/terms/index.html#t-quantity-resistance"],["method/paalzow-s-siphon-method",8,"Paalzow's siphon method","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-method-paalzow-s-siphon-method"],["hardy-course-of-pure-mathematics-1921/x-4e45df982a",15,"Hardy 1921, p. 275: The reader must not suppose, however, that these new ..."],["hardy-course-of-pure-mathematics-1921/x-5594b7ead2",15,"Hardy 1921, p. 276: But the reader must be careful to impress on ..."],["concept/residual-discharge",7,"residual discharge","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-residual-discharge"],["concept/transient-current",7,"transient current","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-transient-current"],["theorem/chain-rule-for-differentiation",9,"chain rule for differentiation","../books/thompson-calculus-made-easy-1914/terms/index.html#t-theorem-chain-rule-for-differentiation"],["thompson-calculus-made-easy-1914/x-f843e50b12",15,"Thompson 1914, p. 69: (We may also write y = (1-x)^{\\efrac{1}{2}} (1+x)^{-\\efrac{1}{2}} and ..."],["thompson-calculus-made-easy-1914/x-30ab00664e",15,"Thompson 1914, p. 68: (1) Differentiate y = \\sqrt{a+x}. Let a+x = u."],["hardy-course-of-pure-mathematics-1921/x-85e79a45e1",15,"Hardy 1921, p. 281: Thus the formula which expresses dz in terms of ..."],["concept/disruptive-discharge",7,"disruptive discharge","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-concept-disruptive-discharge"],["instrument/galvanometer",13,"galvanometer","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-galvanometer"],["boyden-first-book-in-algebra-1895/ex-36/43",4,"Boyden 1895, Exercise 36 (43)"],["method/newton-s-method",8,"Newton's method","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-method-newton-s-method"],["method/higher-derivative-test-for-maxima-and-minima",8,"higher-derivative test for maxima and minima","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-higher-derivative-test-for-maxima-and-minima"],["method/evaluating-limits-by-derivatives",8,"evaluating limits by derivatives","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-evaluating-limits-by-derivatives"],["planck-treatise-on-thermodynamics-1903/x-19e84b6e56",15,"Planck 1903, p. 53: In fact, it assumes an entirely different value along ..."],["theorem/first-mean-value-theorem-for-integrals",9,"first mean value theorem for integrals","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-first-mean-value-theorem-for-integrals"],["instrument/resistance-coil",13,"resistance coil","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-resistance-coil"],["instrument/leyden-jar",13,"Leyden jar","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-instrument-leyden-jar"],["law/ohm-s-law",10,"Ohm's law","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-law-ohm-s-law"],["hardy-course-of-pure-mathematics-1921/ex-lxiv",3,"Hardy 1921, Exercise LXIV"],["person/c-w-siemens",1,"C. W. Siemens","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-c-w-siemens"],["hardy-course-of-pure-mathematics-1921/ex-lxv",3,"Hardy 1921, Exercise LXV"],["person/c-f-varley",1,"C. F. Varley","../books/maxwell-elementary-treatise-electricity-1888/terms/index.html#t-person-c-f-varley"],["hardy-course-of-pure-mathematics-1921/ex-lxvi",3,"Hardy 1921, Exercise LXVI"],["maxwell-elementary-treatise-electricity-1888/x-4696da793a",15,"Maxwell 1888, scan 216: It is of the utmost importance in the electric ..."],["hardy-course-of-pure-mathematics-1921/x-cac89f7c66",15,"Hardy 1921, p. 271: It is evident that the degree of smallness of ..."],["maxwell-elementary-treatise-electricity-1888/x-cb854a1911",15,"Maxwell 1888, scan 220: But even after this current has been allowed to ..."],["concept/rate-of-change",7,"rate of change","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-rate-of-change"],["thompson-calculus-made-easy-1914/x-c95c798100",15,"Thompson 1914, p. 53: What we mean by saying that the rate is ..."],["thompson-calculus-made-easy-1914/x-303d843bf5",15,"Thompson 1914, p. 55: Now the speed was not actually constant all the ..."],["thompson-calculus-made-easy-1914/x-f147508200",15,"Thompson 1914, p. 59: But this notation does not tell us what is ..."],["maxwell-elementary-treatise-electricity-1888/x-31d0c395dd",15,"Maxwell 1888, scan 223: When the maximum polarization is established, the excess of ..."],["maxwell-elementary-treatise-electricity-1888/x-6465b38914",15,"Maxwell 1888, scan 226: ‘Multiply each cycle sign (i.e. current) by the sum ..."],["concept/indefinite-integral",7,"indefinite integral","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-indefinite-integral"],["concept/integrand",7,"integrand","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-integrand"],["theorem/modulus-inequality-for-integrals",9,"modulus inequality for integrals","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-modulus-inequality-for-integrals"],["method/evaluation-of-a-definite-integral-as-the-limit-of-a-sum",8,"evaluation of a definite integral as the limit of a sum","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-evaluation-of-a-definite-integral-as-the-limit-of-a-sum"],["concept/integral-of-a-complex-function",7,"integral of a complex function","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-integral-of-a-complex-function"],["hardy-course-of-pure-mathematics-1921/x-de87f38f5b",15,"Hardy 1921, p. 299: This inequality may be deduced without difficulty from the ..."],["hardy-course-of-pure-mathematics-1921/x-b12adb6c92",15,"Hardy 1921, p. 302: In these circumstances u is called a homogeneous function ..."],["theorem/second-mean-value-theorem-for-integrals",9,"second mean value theorem for integrals","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-second-mean-value-theorem-for-integrals"],["theorem/bonnet-s-form-of-the-second-mean-value-theorem",9,"Bonnet's form of the second mean value theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-bonnet-s-form-of-the-second-mean-value-theorem"],["hardy-course-of-pure-mathematics-1921/x-e7d82bd84d",15,"Hardy 1921, p. 302: This result is known as ’s Theorem on homogeneous ..."],["boyden-first-book-in-algebra-1895/ex-19/4",4,"Boyden 1895, Exercise 19 (4)"],["concept/circular-measure",7,"circular measure","../books/blackburn-elements-plane-trigonometry-1863/terms/index.html#t-concept-circular-measure"],["boyden-first-book-in-algebra-1895/ex-36/44",4,"Boyden 1895, Exercise 36 (44)"],["form/d37ea073c7",5,"identity: 3*a**2*x*(27*a**3 + 36*a*x**2 + 8*x**3)"],["shape/f171ed3ff9",6,"identity: N*a**N*x*(N*a*x**N + N*a**N + N*x**N)"],["boyden-first-book-in-algebra-1895/ex-19/5",4,"Boyden 1895, Exercise 19 (5)"],["boyden-first-book-in-algebra-1895/ex-36/45a",4,"Boyden 1895, Exercise 36 (45a)"],["planck-treatise-on-thermodynamics-1903/x-24e1b2c8b0",15,"Planck 1903, p. 66: By reversing Carnot’s cycle, we have, then, a means ..."],["hardy-course-of-pure-mathematics-1921/x-031e400b96",15,"Hardy 1921, p. 299: It will be remembered that the difficulty in using ..."],["concept/angle-between-two-curves",7,"angle between two curves","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-angle-between-two-curves"],["thompson-calculus-made-easy-1914/x-3c9a0468ca",15,"Thompson 1914, p. 77: We have seen that the short expression “the slope ..."],["thompson-calculus-made-easy-1914/x-033f323dbd",15,"Thompson 1914, p. 77: “The slope of a curve at a point” is, ..."],["boyden-first-book-in-algebra-1895/ex-19/7",4,"Boyden 1895, Exercise 19 (7)"],["form/dd6f4aa793",5,"identity: (x - 1)*(x**4 - 3*x**3 + 2*x**2 - x + 1)"],["shape/8cf105da5c",6,"identity: (x - 1)*(2*N*x**N - x + x**N + 1)"],["thompson-calculus-made-easy-1914/x-5e63489376",15,"Thompson 1914, p. 78: Observe that dx is a short step to the ..."],["thompson-calculus-made-easy-1914/x-701bb55c55",15,"Thompson 1914, p. 79: If a curve slopes downward, as in [fig:11]Fig. 11, ..."],["thompson-calculus-made-easy-1914/x-955ae548cb",15,"Thompson 1914, p. 88: The slope of the tangent is the slope of ..."],["thompson-calculus-made-easy-1914/x-463b0b6006",15,"Thompson 1914, p. 90: In all exercises dealing with curves, students will find ..."],["quantity/atomic-heat",11,"atomic heat","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-atomic-heat"],["concept/solvent",7,"solvent","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-solvent"],["concept/jacobian",7,"Jacobian","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-jacobian"],["method/simpson-s-rule",8,"Simpson's rule","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-simpson-s-rule"],["hardy-course-of-pure-mathematics-1921/x-812159a07d",15,"Hardy 1921, p. 299: We define the integral of a complex function f(x) ..."],["concept/imaginary-root",7,"imaginary root","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-imaginary-root"],["concept/cusp",7,"cusp","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-cusp"],["thompson-calculus-made-easy-1914/x-b7e98d0a47",15,"Thompson 1914, p. 93: One of the principal uses of the process of ..."],["thompson-calculus-made-easy-1914/x-d1dd9cce98",15,"Thompson 1914, p. 95: When there is put before you an equation, and ..."],["thompson-calculus-made-easy-1914/x-73230e63b4",15,"Thompson 1914, p. 97: Ordinarily you are dealing with equations that are true ..."],["thompson-calculus-made-easy-1914/x-fd573b035c",15,"Thompson 1914, p. 98: Quite so. It does not of itself discriminate; it ..."],["thompson-calculus-made-easy-1914/x-90c6627e47",15,"Thompson 1914, p. 99: So now we know that whatever number n may ..."],["thompson-calculus-made-easy-1914/x-b24488d912",15,"Thompson 1914, p. 114: Clearly the change of slope as the curve passes ..."],["thompson-calculus-made-easy-1914/x-174085d72b",15,"Thompson 1914, p. 114: In this case, as the curve passes through M ..."],["method/second-derivative-test",8,"second derivative test","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-method-second-derivative-test"],["theorem/cauchy-s-root-test",9,"Cauchy's root test","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-cauchy-s-root-test"],["law/dulong-and-petit-s-law",10,"Dulong and Petit's law","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-law-dulong-and-petit-s-law"],["concept/heat-of-removal",7,"heat of removal","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-heat-of-removal"],["concept/convergent-series",7,"convergent series","../books/de-morgan-elementary-illustrations-calculus-1899/terms/index.html#t-concept-convergent-series"],["law/neumann-s-law",10,"Neumann's law","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-law-neumann-s-law"],["theorem/integral-test",9,"integral test","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-theorem-integral-test"],["concept/product-of-series",7,"product of series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-product-of-series"],["concept/converse-of-a-theorem",7,"converse of a theorem","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-converse-of-a-theorem"],["concept/carnot-s-theory-of-heat",7,"Carnot's theory of heat","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-carnot-s-theory-of-heat"],["concept/algebraic-fraction",7,"algebraic fraction","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-algebraic-fraction"],["concept/unknown",7,"unknown","../books/boyden-first-book-in-algebra-1895/terms/index.html#t-concept-unknown"],["concept/numerator",7,"numerator","../books/thompson-calculus-made-easy-1914/terms/index.html#t-concept-numerator"],["method/differentiating-an-inverse-function",8,"differentiating an inverse function","../books/thompson-calculus-made-easy-1914/terms/index.html#t-method-differentiating-an-inverse-function"],["thompson-calculus-made-easy-1914/x-7fde6cf143",15,"Thompson 1914, p. 128: We see that it is sufficient to allow for ..."],["thompson-calculus-made-easy-1914/x-33a89eb995",15,"Thompson 1914, p. 129: It is useful to check the results obtained. The ..."],["thompson-calculus-made-easy-1914/x-5312d8d97b",15,"Thompson 1914, p. 132: It follows that, being given a function, if it ..."],["concept/p-series",7,"p-series","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-p-series"],["concept/infinite-integral-of-the-second-kind",7,"infinite integral of the second kind","../books/hardy-course-of-pure-mathematics-1921/terms/index.html#t-concept-infinite-integral-of-the-second-kind"],["concept/indestructibility-of-heat",7,"indestructibility of heat","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-indestructibility-of-heat"],["concept/heat-contained-in-a-body",7,"heat contained in a body","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-heat-contained-in-a-body"],["concept/singular-value",7,"singular 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2"],["boyden-first-book-in-algebra-1895/ex-37/3",4,"Boyden 1895, Exercise 37 (3)"],["boyden-first-book-in-algebra-1895/ex-37/4",4,"Boyden 1895, Exercise 37 (4)"],["form/e662e27fb1",5,"factor: x**2 - 7*x + 10"],["form/c87ff4aec9",5,"factor: x**2 - 5*x + 6"],["shape/744b0f9aff",6,"factor: N*x + N + x**N"],["planck-treatise-on-thermodynamics-1903/x-a01a488b6c",15,"Planck 1903, p. 77: If, for instance, an exchange of heat by conduction ..."],["planck-treatise-on-thermodynamics-1903/x-72d86b068f",15,"Planck 1903, p. 78: If a heavy liquid be initially at rest at ..."],["concept/perpetual-motion",7,"perpetual motion","../books/ball-mathematical-recreations-1905/terms/index.html#t-concept-perpetual-motion"],["concept/free-expansion-of-a-gas",7,"free expansion of a gas","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-free-expansion-of-a-gas"],["concept/heat-reservoir",7,"heat 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equilibrium","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-internal-conditions-of-equilibrium"],["boyden-first-book-in-algebra-1895/ex-19/8",4,"Boyden 1895, Exercise 19 (8)"],["form/4b7b1e83e0",5,"identity: (x**2 + 3*x + 1)*(x**3 - 2*x**2 + x)"],["shape/90b5669460",6,"identity: (N*x + x**N + 1)*(N*x**N + x + x**N)"],["boyden-first-book-in-algebra-1895/ex-19/9",4,"Boyden 1895, Exercise 19 (9)"],["boyden-first-book-in-algebra-1895/ex-37/8",4,"Boyden 1895, Exercise 37 (8)"],["cap/cas.collect",17,"cas.collect"],["boyden-first-book-in-algebra-1895/ex-37/10",4,"Boyden 1895, Exercise 37 (10)"],["form/711ee69185",5,"factor: x**2 - 4*x - 77"],["law/first-law-of-thermodynamics",10,"first law of thermodynamics","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-law-first-law-of-thermodynamics"],["form/e90542b9b1",5,"identity: (-a*b + a*x - b*c + c*x)*(a*b - a*x - b*c + c*x)"],["shape/e90542b9b1",6,"identity: (-a*b + a*x - b*c + c*x)*(a*b - a*x - b*c + c*x)"],["boyden-first-book-in-algebra-1895/ex-19/10",4,"Boyden 1895, Exercise 19 (10)"],["form/25ab53d287",5,"identity: (x**3 + 2*x**2 + 3*x + 2)*(x**4 - x**3 + x**2 - x + 1)"],["shape/1c329fe8cf",6,"identity: (-x + x**N + 1)*(N*x + N*x**N + N + x**N)"],["boyden-first-book-in-algebra-1895/ex-37/11",4,"Boyden 1895, Exercise 37 (11)"],["form/1638113858",5,"factor: x**2 - 2*x - 63"],["quantity/latent-heat",11,"latent heat","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-quantity-latent-heat"],["concept/fundamental-triangle",7,"fundamental triangle","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-fundamental-triangle"],["concept/region-of-validity",7,"region of validity","../books/planck-treatise-on-thermodynamics-1903/terms/index.html#t-concept-region-of-validity"],["concept/developable-surface",7,"developable 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a, N), Eq(N*a + N*x, N))"],["boyden-first-book-in-algebra-1895/ex-55/8",4,"Boyden 1895, Exercise 55 (8)"],["form/ff50d6bcd5",5,"solve: (Eq(9*a + 2*x, -5), Eq(15*a + 11*x, 7))"],["boyden-first-book-in-algebra-1895/ex-55/9",4,"Boyden 1895, Exercise 55 (9)"],["form/1013499aa9",5,"solve: (Eq(4*a - 2*x, 4), Eq(10*a + 3*x, -8))"],["boyden-first-book-in-algebra-1895/ex-56/1",4,"Boyden 1895, Exercise 56 (1)"],["boyden-first-book-in-algebra-1895/ex-56/2",4,"Boyden 1895, Exercise 56 (2)"],["form/77ef8f3d83",5,"solve: Eq(3*x**2 + 4, 16)"],["boyden-first-book-in-algebra-1895/ex-55/10",4,"Boyden 1895, Exercise 55 (10)"],["boyden-first-book-in-algebra-1895/ex-55/11",4,"Boyden 1895, Exercise 55 (11)"],["boyden-first-book-in-algebra-1895/ex-55/12",4,"Boyden 1895, Exercise 55 (12)"],["boyden-first-book-in-algebra-1895/ex-55/13",4,"Boyden 1895, Exercise 55 (13)"],["form/279639bfa4",5,"solve: (Eq(a/3 + x/2, 11), Eq(2*a/5 + 8*x, 102))"],["boyden-first-book-in-algebra-1895/ex-55/14",4,"Boyden 1895, 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109/10))"],["boyden-first-book-in-algebra-1895/ex-55/20",4,"Boyden 1895, Exercise 55 (20)"],["shape/b21312884b",6,"solve: (Eq(-c + x, b), Eq(c + x, a))"],["boyden-first-book-in-algebra-1895/ex-55/21",4,"Boyden 1895, Exercise 55 (21)"],["form/9758c3ae4b",5,"solve: (Eq(13*a/16 + 5*x/4, 3*a/4 + 9*x/8 + 11/8), Eq(3*x/2 - 11/2, a + 17*x/6 - 3/2))"],["shape/89f5e130f8",6,"solve: (Eq(N*a + N*x, N*a + N*x + N), Eq(N*x + N, N*x + N + a))"],["boyden-first-book-in-algebra-1895/ex-55/22",4,"Boyden 1895, Exercise 55 (22)"],["boyden-first-book-in-algebra-1895/ex-55/23",4,"Boyden 1895, Exercise 55 (23)"],["boyden-first-book-in-algebra-1895/ex-55/24",4,"Boyden 1895, Exercise 55 (24)"],["shape/959c1c17bf",6,"solve: (Eq((N + x)/(N + a), N), Eq((x + 1)/(a + 1), N))"],["boyden-first-book-in-algebra-1895/ex-55/25",4,"Boyden 1895, Exercise 55 (25)"],["form/fb869e4112",5,"solve: (Eq((x - 3)/(a - 3), 1/2), Eq((x + 2)/(a + 2), 2/3))"],["shape/b2807acb6b",6,"solve: (Eq((N + x)/(N + a), N), Eq((N + x)/(N + a), N))"],["boyden-first-book-in-algebra-1895/ex-56/3",4,"Boyden 1895, Exercise 56 (3)"],["boyden-first-book-in-algebra-1895/ex-55/26",4,"Boyden 1895, Exercise 55 (26)"],["form/ef3a6fd51a",5,"solve: (Eq((x - 3)/(a + 3), 1/2), Eq((x + 5)/(a - 5), 2))"],["boyden-first-book-in-algebra-1895/ex-55/27",4,"Boyden 1895, Exercise 55 (27)"],["form/1ea9e7c61c",5,"solve: (Eq(-a/2 + x/2, 19), Eq(a/2 + x/2, 43))"],["boyden-first-book-in-algebra-1895/ex-55/28",4,"Boyden 1895, Exercise 55 (28)"],["form/d9ee49ad1e",5,"solve: Eq(a + 4*x**2, -x**2 + 136)"],["shape/e45823778b",6,"solve: Eq(N*x**N + a, N - x**N)"],["boyden-first-book-in-algebra-1895/ex-56/4",4,"Boyden 1895, Exercise 56 (4)"],["form/9ad9c4950e",5,"solve: Eq(15*x**2 - 5, 11*x**2 + 11)"],["boyden-first-book-in-algebra-1895/ex-57/8",4,"Boyden 1895, Exercise 57 (8)"],["boyden-first-book-in-algebra-1895/ex-55/29",4,"Boyden 1895, Exercise 55 (29)"],["form/af0a3aba28",5,"solve: (Eq(x - 5, 4*a - 20), Eq(x + 5, 7*a/3 + 35/3))"],["boyden-first-book-in-algebra-1895/ex-55/30",4,"Boyden 1895, Exercise 55 (30)"],["form/77bea28923",5,"solve: (Eq(x - 7, a/2 - 7/2), Eq(x + 5, 3*a/4 + 15/4))"],["boyden-first-book-in-algebra-1895/ex-55/31",4,"Boyden 1895, Exercise 55 (31)"],["form/af22d82fe5",5,"solve: (Eq(x - 6, a + 6), Eq(2*a - 8, x + 4))"],["shape/ccec5fad2f",6,"solve: (Eq(N + x, N*a + N), Eq(N + x, N*a + N))"],["boyden-first-book-in-algebra-1895/ex-55/32",4,"Boyden 1895, Exercise 55 (32)"],["form/feeb861cb1",5,"solve: (Eq(6*a + 3*x, 354), Eq(7*a + 5*x, 494))"],["boyden-first-book-in-algebra-1895/ex-55/33",4,"Boyden 1895, Exercise 55 (33)"],["boyden-first-book-in-algebra-1895/ex-55/34",4,"Boyden 1895, Exercise 55 (34)"],["form/e2726a26c9",5,"solve: (Eq(55*a + 55*x, 2090), Eq(29*a + 87*x, 1798))"],["boyden-first-book-in-algebra-1895/ex-56/5",4,"Boyden 1895, Exercise 56 (5)"],["form/2fd5ee6388",5,"solve: Eq(1/(15*x**2), 4/15)"],["boyden-first-book-in-algebra-1895/ex-56/6",4,"Boyden 1895, Exercise 56 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Exercise 56 (11)"],["form/680e407b45",5,"solve: Eq(1/((x - 2)*(x - 1)) + 1/((-x + 2)*(-x + 3)) - 2/((-x + 1)*(x - 3)), 1/(-x + 2) + 1/((-x + 2)*(x - 3)*(x - 1)))"],["shape/f800a32102",6,"solve: Eq(N/((-x + 1)*(N + x)) + 1/((N + x)*(x - 1)) + (N - x)**(-2), 1/(N - x) + 1/((N - x)*(N + x)*(x - 1)))"],["boyden-first-book-in-algebra-1895/ex-56/12",4,"Boyden 1895, Exercise 56 (12)"],["form/cbcac964ab",5,"solve: Eq(1/(6*x + 6) - 1/(2*x + 2) + 10/(-3*x**2 + 3), x/(-3*x + 3))"],["shape/7ba6d34b55",6,"solve: Eq(N/(N*x**N + N), x/(N*x + N))"],["boyden-first-book-in-algebra-1895/ex-56/13",4,"Boyden 1895, Exercise 56 (13)"],["form/777f93d396",5,"solve: Eq(x + 30, 3*x + 6)"],["boyden-first-book-in-algebra-1895/ex-56/14",4,"Boyden 1895, Exercise 56 (14)"],["form/8565aa282b",5,"solve: (Eq(x, 3*a - 3*x), Eq(a + x, 112))"],["shape/2099e8c050",6,"solve: (Eq(x, N*a + N*x), Eq(a + x, N))"],["form/c4468471d4",5,"solve: Eq(x**2, -3*a**2 + 4*a*x)"],["shape/afd1081ac4",6,"solve: Eq(x**N, N*a*x + N*a**N)"],["boyden-first-book-in-algebra-1895/ex-57/9",4,"Boyden 1895, Exercise 57 (9)"],["form/6dba2f55fc",5,"solve: Eq(x**2 + x*(a - 1), a)"],["shape/0f5adfeab3",6,"solve: Eq(x*(a - 1) + x**N, a)"],["boyden-first-book-in-algebra-1895/ex-56/15",4,"Boyden 1895, Exercise 56 (15)"],["form/0d2ea9e3af",5,"solve: (Eq(x, a + 4), Eq(a/(x + 30), (a - 10)/x))"],["shape/05dbca9c77",6,"solve: (Eq(x, N + a), Eq(a/(N + x), (N + a)/x))"],["boyden-first-book-in-algebra-1895/ex-57/10",4,"Boyden 1895, Exercise 57 (10)"],["form/876120054a",5,"solve: Eq(-a*c*x**2 + a*d*x, b*c*x - b*d)"],["shape/095b75b02b",6,"solve: Eq(-a*c*x**N + a*d*x, b*c*x - b*d)"],["boyden-first-book-in-algebra-1895/ex-57/11",4,"Boyden 1895, Exercise 57 (11)"],["form/ae54ad6b3d",5,"solve: Eq((x - 3)*(x + 3), 8*x + 24)"],["shape/e33fdb9c71",6,"solve: Eq((N + x)**2, N*x + N)"],["boyden-first-book-in-algebra-1895/ex-57/12",4,"Boyden 1895, Exercise 57 (12)"],["form/9ae6216b3d",5,"solve: Eq((x - 5)*(x + 2), 4*x - 16)"],["boyden-first-book-in-algebra-1895/ex-57/13",4,"Boyden 1895, Exercise 57 (13)"],["form/e839c956cc",5,"solve: Eq(x/5 + 2/x, 7/5)"],["shape/13be325c01",6,"solve: Eq(N*x + N/x, N)"],["shape/0c500d227b",6,"solve: Eq(N + x, N*x + N/(x - 1))"],["boyden-first-book-in-algebra-1895/ex-57/14",4,"Boyden 1895, Exercise 57 (14)"],["form/e913439921",5,"solve: Eq(x/3 - 2, x**2/12 - x/2)"],["shape/9dc3872c7b",6,"solve: Eq(N*x + N, N*x + N*x**N)"],["boyden-first-book-in-algebra-1895/ex-57/15",4,"Boyden 1895, Exercise 57 (15)"],["form/0db5d647f7",5,"solve: Eq(x/7 - 20/7 + 8/(x - 2), 0)"],["shape/383aca668d",6,"solve: Eq(N*x + N + N/(N + x), 0)"],["boyden-first-book-in-algebra-1895/ex-57/16",4,"Boyden 1895, Exercise 57 (16)"],["form/b119808afa",5,"solve: Eq(x/(x + 1) - 13/6 + (x + 1)/x, 0)"],["shape/8bf6eae7ef",6,"solve: Eq(N + x/(x + 1) + (x + 1)/x, 0)"],["boyden-first-book-in-algebra-1895/ex-57/17",4,"Boyden 1895, Exercise 57 (17)"],["form/23b7242a87",5,"solve: Eq(x + 4, 3*x - 24/(x - 1))"],["boyden-first-book-in-algebra-1895/ex-57/18",4,"Boyden 1895, Exercise 57 (18)"],["form/6c6fca9660",5,"solve: Eq((x - 1)/(x + 1) + (x + 3)/(x - 3), (2*x + 4)/(x - 2))"],["boyden-first-book-in-algebra-1895/ex-57/19",4,"Boyden 1895, Exercise 57 (19)"],["form/58da479ebf",5,"solve: Eq((x - 1)/(x + 2) - (3*x**2 + 2)/(x**2 - 4), 3*x/(-x + 2))"],["shape/e0a994ad23",6,"solve: Eq(-(N*x**N + N)/(N + x**N) + (x - 1)/(N + x), N*x/(N - x))"],["boyden-first-book-in-algebra-1895/ex-57/20",4,"Boyden 1895, Exercise 57 (20)"],["form/3428439453",5,"solve: Eq(2*x*(x - 3)/(x**2 - 9) + 2*x/(-x + 3), (x - 3)/(x + 3))"],["shape/38d04b4ca3",6,"solve: Eq(N*x*(N + x)/(N + x**N) + N*x/(N - x), 1)"],["boyden-first-book-in-algebra-1895/ex-57/21",4,"Boyden 1895, Exercise 57 (21)"],["form/e3d88403d0",5,"solve: Eq(6*x, x/2 + 300)"],["boyden-first-book-in-algebra-1895/ex-57/22",4,"Boyden 1895, Exercise 57 (22)"],["form/ae18c3406f",5,"solve: (Eq(a, 3*x), Eq(a - 2, 2*x + 4))"],["boyden-first-book-in-algebra-1895/ex-57/24",4,"Boyden 1895, Exercise 57 (24)"],["boyden-first-book-in-algebra-1895/ex-58/1",4,"Boyden 1895, Exercise 58 (1)"],["form/a0d3f0f187",5,"evaluate: -(a**2*c**2 + 1)/(a**2 + c**2 + d**2) + (a**2 + b**2 + d**2)/(a**2*b**2 + b*d + 1) - (a**2 + 2*a*b + b**2)/(b**2 - 2*b*c + c**2) + (a*d + 4*a + b**2*c**2 + b**2)/(b**2 + c**2 + d**2) at a=1, b=3, c=5, d=0"],["shape/88d7b6aae7",6,"evaluate: -(a**N*c**N + 1)/(a**N + c**N + d**N) + (a**N + b**N + d**N)/(a**N*b**N + b*d + 1) - (N*a*b + a**N + b**N)/(N*b*c + b**N + c**N) + (N*a + a*d + b**N*c**N + b**N)/(b**N + c**N + d**N)"],["boyden-first-book-in-algebra-1895/ex-58/2",4,"Boyden 1895, Exercise 58 (2)"],["boyden-first-book-in-algebra-1895/ex-58/3",4,"Boyden 1895, Exercise 58 (3)"],["form/05d8d6b4c4",5,"solve: Eq(-16*x - (x + 1)**2 + (x + 5)**2, -(x - 5)**2 + (x - 1)**2)"],["shape/76a849ecfa",6,"solve: Eq(N*x + (N + x)**N - (x + 1)**N, -(N + x)**N + (x - 1)**N)"],["boyden-first-book-in-algebra-1895/ex-58/4",4,"Boyden 1895, Exercise 58 (4)"],["form/ccb835b6dc",5,"solve: Eq(1/20 + 1/x, 1/12)"],["form/5af5b7628e",5,"hcf: (x**4 - 1, x**2 - 4*x + 3, x**3 - x**2 - x + 1, x**3 - x**2 + x - 1)"],["boyden-first-book-in-algebra-1895/ex-58/8",4,"Boyden 1895, Exercise 58 (8)"],["form/14883d02ec",5,"identity: (x - 9)*(x - 4) + (x - 5)**2 + (x + 1)*(x + 2) - (2*x + 1)*(2*x + 3)"],["shape/439f99e2ca",6,"identity: (N + x)**2 + (N + x)*(x + 1) + (N + x)**N - (N*x + 1)*(N*x + N)"],["boyden-first-book-in-algebra-1895/ex-58/9",4,"Boyden 1895, Exercise 58 (9)"],["form/df261abe6f",5,"solve: Eq((x - 2)/(x + 2) + (x + 2)/(x - 2), 5/2)"],["boyden-first-book-in-algebra-1895/ex-58/10",4,"Boyden 1895, Exercise 58 (10)"],["form/6212eda761",5,"identity: (1/(x - 1) - 3/((x - 1)*(x + 3)))/(1/(x + 3) + 1/((x - 1)*(x + 3)))"],["shape/e89d6eb176",6,"identity: (N/((N + x)*(x - 1)) + 1/(x - 1))/(1/(N + x) + 1/((N + x)*(x - 1)))"],["boyden-first-book-in-algebra-1895/ex-5/1",4,"Boyden 1895, Exercise 5 (1)"],["form/a39cbfe9eb",5,"solve: (Eq(a, 3*x), Eq(5*a + 7*x, 110))"],["boyden-first-book-in-algebra-1895/ex-5/2",4,"Boyden 1895, Exercise 5 (2)"],["form/83f2790c6b",5,"solve: Eq(12*x, 36)"],["boyden-first-book-in-algebra-1895/ex-5/3",4,"Boyden 1895, Exercise 5 (3)"],["form/cfb77b65ad",5,"solve: (Eq(a, 4*x), Eq(10*a + 4*x, 88))"],["boyden-first-book-in-algebra-1895/ex-5/4",4,"Boyden 1895, Exercise 5 (4)"],["form/4a633e9745",5,"solve: (Eq(a, 2*x), Eq(2*a + 3*x, 14))"],["boyden-first-book-in-algebra-1895/ex-6/7",4,"Boyden 1895, Exercise 6 (7)"],["boyden-first-book-in-algebra-1895/ex-6/8",4,"Boyden 1895, Exercise 6 (8)"],["shape/1fed4d8696",6,"solve: (Eq(b, N*a), Eq(-a + b, N))"],["boyden-first-book-in-algebra-1895/ex-5/5",4,"Boyden 1895, Exercise 5 (5)"],["form/3d17ef6de4",5,"solve: Eq(9*x, 108)"],["boyden-first-book-in-algebra-1895/ex-5/6",4,"Boyden 1895, Exercise 5 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Exercise 5 (17)"],["form/a364899283",5,"solve: (Eq(a, 5*x), Eq(b, 2*a), Eq(a + b + x, 288))"],["shape/a07d473fe1",6,"solve: (Eq(a, N*x), Eq(b, N*a), Eq(a + b + x, N))"],["boyden-first-book-in-algebra-1895/ex-6/2",4,"Boyden 1895, Exercise 6 (2)"],["form/3de2d0aa0b",5,"solve: Eq(2*x + 28, 6*x)"],["boyden-first-book-in-algebra-1895/ex-6/3",4,"Boyden 1895, Exercise 6 (3)"],["form/9b487bd773",5,"solve: Eq(5*x + 24, 7*x)"],["boyden-first-book-in-algebra-1895/ex-6/12",4,"Boyden 1895, Exercise 6 (12)"],["boyden-first-book-in-algebra-1895/ex-6/4",4,"Boyden 1895, Exercise 6 (4)"],["form/b104a95ff8",5,"solve: Eq(5*x + 25, 9*x)"],["boyden-first-book-in-algebra-1895/ex-6/5",4,"Boyden 1895, Exercise 6 (5)"],["form/15fe9f490c",5,"solve: Eq(4*x + 32, 8*x)"],["boyden-first-book-in-algebra-1895/ex-6/6",4,"Boyden 1895, Exercise 6 (6)"],["form/b243a936b3",5,"solve: Eq(7*x + 24, 10*x)"],["boyden-first-book-in-algebra-1895/ex-6/9",4,"Boyden 1895, Exercise 6 (9)"],["form/0e4464e716",5,"solve: (Eq(b, 3*a - 50), Eq(c, 2*b), Eq(a + b + c, 4850))"],["shape/148e9cc6aa",6,"solve: (Eq(b, N*a + N), Eq(c, N*b), Eq(a + b + c, N))"],["boyden-first-book-in-algebra-1895/ex-6/10",4,"Boyden 1895, Exercise 6 (10)"],["form/f5548ac83a",5,"solve: (Eq(a, b + 3), Eq(2*a, 3*b))"],["shape/73263fe7fc",6,"solve: (Eq(a, N + b), Eq(N*a, N*b))"],["boyden-first-book-in-algebra-1895/ex-6/11",4,"Boyden 1895, Exercise 6 (11)"],["boyden-first-book-in-algebra-1895/ex-6/13",4,"Boyden 1895, Exercise 6 (13)"],["form/1177545d2e",5,"solve: (Eq(b, a + 15), Eq(4*b, 7*a))"],["shape/f72c8bf488",6,"solve: (Eq(b, N + a), Eq(N*b, N*a))"],["boyden-first-book-in-algebra-1895/ex-6/14",4,"Boyden 1895, Exercise 6 (14)"],["form/b01a9a0932",5,"solve: (Eq(3*a, 5*b), Eq(a - b, 6))"],["shape/cb2f09620b",6,"solve: (Eq(N*a, N*b), Eq(a - b, N))"],["boyden-first-book-in-algebra-1895/ex-6/15",4,"Boyden 1895, Exercise 6 (15)"],["boyden-first-book-in-algebra-1895/ex-6/16",4,"Boyden 1895, Exercise 6 (16)"],["form/cbab074af3",5,"solve: (Eq(a, b - 15), Eq(4*b, 6*a))"],["shape/6d66e17cee",6,"solve: (Eq(a, N + b), Eq(N*b, N*a))"],["boyden-first-book-in-algebra-1895/ex-6/17",4,"Boyden 1895, Exercise 6 (17)"],["form/cf9c543334",5,"solve: Eq(x, 4*x - 15)"],["boyden-first-book-in-algebra-1895/ex-6/18",4,"Boyden 1895, Exercise 6 (18)"],["form/c496d1da79",5,"solve: (Eq(b, 4*a - 2), Eq(12*a + 6*b, 168))"],["shape/3a60bf407e",6,"solve: (Eq(b, N*a + N), Eq(N*a + N*b, N))"],["boyden-first-book-in-algebra-1895/ex-7/2",4,"Boyden 1895, Exercise 7 (2)"],["boyden-first-book-in-algebra-1895/ex-7/3",4,"Boyden 1895, Exercise 7 (3)"],["form/b37c49820a",5,"solve: (Eq(x, a/3), Eq(a + x, 12000))"],["boyden-first-book-in-algebra-1895/ex-7/4",4,"Boyden 1895, Exercise 7 (4)"],["form/9b7f58657f",5,"solve: (Eq(a, x/2), Eq(a + x, 105))"],["boyden-first-book-in-algebra-1895/ex-7/5",4,"Boyden 1895, Exercise 7 (5)"],["form/f3b6815dd2",5,"solve: (Eq(a, x/16), Eq(-a + x, 675))"],["boyden-first-book-in-algebra-1895/ex-7/7",4,"Boyden 1895, Exercise 7 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x)"],["wentworth-first-steps-in-algebra-1894/ex-29/12",4,"Wentworth 1894, Exercise 29 (12)"],["form/896fd004f2",5,"identity: (x**9 - 27)/(x**3 - 3)"],["wentworth-first-steps-in-algebra-1894/ex-29/13",4,"Wentworth 1894, Exercise 29 (13)"],["wentworth-first-steps-in-algebra-1894/ex-29/14",4,"Wentworth 1894, Exercise 29 (14)"],["form/cf17cfdb0d",5,"identity: (-a**9*b**9 + x**15)/(-a**3*b**3 + x**5)"],["wentworth-first-steps-in-algebra-1894/ex-29/15",4,"Wentworth 1894, Exercise 29 (15)"],["wentworth-first-steps-in-algebra-1894/ex-29/17",4,"Wentworth 1894, Exercise 29 (17)"],["form/7f9874215b",5,"identity: (8*a**3*b**3*x**3 - 27)/(2*a*b*x - 3)"],["wentworth-first-steps-in-algebra-1894/ex-29/18",4,"Wentworth 1894, Exercise 29 (18)"],["form/ece87854af",5,"identity: (-64*a**3*b**3*x**3 + 1)/(-4*a*b*x + 1)"],["shape/1687c7254d",6,"identity: (N*a**N*b**N*x**N + N)/(N*a*b*x + N)"],["wentworth-first-steps-in-algebra-1894/ex-2/20",4,"Wentworth 1894, Exercise 2 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2*N*x**N"],["wentworth-first-steps-in-algebra-1894/ex-31/10",4,"Wentworth 1894, Exercise 31 (10)"],["form/9d09b2dbbf",5,"factor: -6*a**4*x**4 + 3*a**3*x**3 - 9*a**2*x**2"],["shape/44d515379c",6,"factor: 3*N*a**N*x**N"],["wentworth-first-steps-in-algebra-1894/ex-32/1",4,"Wentworth 1894, Exercise 32 (1)"],["wentworth-first-steps-in-algebra-1894/ex-32/2",4,"Wentworth 1894, Exercise 32 (2)"],["wentworth-first-steps-in-algebra-1894/ex-32/3",4,"Wentworth 1894, Exercise 32 (3)"],["wentworth-first-steps-in-algebra-1894/ex-32/4",4,"Wentworth 1894, Exercise 32 (4)"],["wentworth-first-steps-in-algebra-1894/ex-32/5",4,"Wentworth 1894, Exercise 32 (5)"],["form/6d34b0e1a4",5,"factor: -a*b + a*x - b*x + x**2"],["shape/785e89d1e4",6,"factor: -a*b + a*x - b*x + x**N"],["wentworth-first-steps-in-algebra-1894/ex-32/6",4,"Wentworth 1894, Exercise 32 (6)"],["form/bd43b33926",5,"factor: -a*x - 3*a + x**2 + 3*x"],["wentworth-first-steps-in-algebra-1894/ex-32/7",4,"Wentworth 1894, Exercise 32 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x)**N"],["wentworth-first-steps-in-algebra-1894/ex-33/5",4,"Wentworth 1894, Exercise 33 (5)"],["wentworth-first-steps-in-algebra-1894/ex-33/6",4,"Wentworth 1894, Exercise 33 (6)"],["form/453127f111",5,"factor: 16*x**4 - 121"],["wentworth-first-steps-in-algebra-1894/ex-33/7",4,"Wentworth 1894, Exercise 33 (7)"],["form/bd2a6d9534",5,"factor: 121*x**4 - 16"],["wentworth-first-steps-in-algebra-1894/ex-33/8",4,"Wentworth 1894, Exercise 33 (8)"],["form/4bef2b05be",5,"factor: 4*a**2*x**2 - b**2*c**2"],["shape/3caf734e8b",6,"factor: N*a**N*x**N - b**N*c**N"],["wentworth-first-steps-in-algebra-1894/ex-33/9",4,"Wentworth 1894, Exercise 33 (9)"],["wentworth-first-steps-in-algebra-1894/ex-33/11",4,"Wentworth 1894, Exercise 33 (11)"],["shape/e7b321825b",6,"factor: N*a**N*x**N + N"],["wentworth-first-steps-in-algebra-1894/ex-33/12",4,"Wentworth 1894, Exercise 33 (12)"],["form/cc5f50c81d",5,"factor: 25*a**4*x**4 - 9"],["wentworth-first-steps-in-algebra-1894/ex-33/13",4,"Wentworth 1894, Exercise 33 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5*b)**2 + 1"],["shape/83fc2932fc",6,"factor: -(N*a + N*b)**N + 1"],["wentworth-first-steps-in-algebra-1894/ex-34/12",4,"Wentworth 1894, Exercise 34 (12)"],["wentworth-first-steps-in-algebra-1894/ex-34/14",4,"Wentworth 1894, Exercise 34 (14)"],["form/92e606a774",5,"factor: 16*c**2 - (a - 5*b)**2"],["shape/511af0b651",6,"factor: N*c**N - (N*b + a)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/15",4,"Wentworth 1894, Exercise 34 (15)"],["form/85b25fa8ce",5,"factor: 4*a**2 - (b + c)**2"],["shape/2ec0daf410",6,"factor: N*a**N - (b + c)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/16",4,"Wentworth 1894, Exercise 34 (16)"],["form/2aa8fa5943",5,"factor: b**2 - (a - 2*c)**2"],["shape/9c38073b8b",6,"factor: b**N - (N*c + a)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/17",4,"Wentworth 1894, Exercise 34 (17)"],["form/92084f6364",5,"factor: 4*c**2 - (a + 3*b)**2"],["wentworth-first-steps-in-algebra-1894/ex-34/18",4,"Wentworth 1894, Exercise 34 (18)"],["form/b7fbd3b451",5,"factor: -(3*a - 7*b)**2 + 9"],["shape/bb550e9df6",6,"factor: N - (N*a + N*b)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/19",4,"Wentworth 1894, Exercise 34 (19)"],["form/9a2321b7fc",5,"factor: 16*a**2 - (2*b + 5*c)**2"],["shape/954cf52f4f",6,"factor: N*a**N - (N*b + N*c)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/20",4,"Wentworth 1894, Exercise 34 (20)"],["form/e87fbe1d41",5,"factor: 25*b**2 - (3*a - 2*c)**2"],["shape/b8be8994a7",6,"factor: N*b**N - (N*a + N*c)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/21",4,"Wentworth 1894, Exercise 34 (21)"],["form/cc7ad11547",5,"factor: 9*a**2 - (3*b - 5*c)**2"],["wentworth-first-steps-in-algebra-1894/ex-34/22",4,"Wentworth 1894, Exercise 34 (22)"],["form/bdfb2305e6",5,"factor: 16*c**2 - (a - 3*b)**2"],["wentworth-first-steps-in-algebra-1894/ex-34/23",4,"Wentworth 1894, Exercise 34 (23)"],["form/fbd1ec5e00",5,"factor: 49*a**2 - (b + 2*c)**2"],["shape/468e1397d4",6,"factor: N*a**N - (N*c + b)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/24",4,"Wentworth 1894, Exercise 34 (24)"],["form/a1b365e3ba",5,"factor: 36*c**2 - (-2*a + b)**2"],["shape/61773fb09c",6,"factor: N*c**N - (N*a + b)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/25",4,"Wentworth 1894, Exercise 34 (25)"],["form/e94cd6d67b",5,"factor: -(a + b)**2 + (c + d)**2"],["shape/6f534e9ee7",6,"factor: -(a + b)**N + (c + d)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/26",4,"Wentworth 1894, Exercise 34 (26)"],["form/191862e813",5,"factor: -(a - b)**2 + (c - d)**2"],["shape/3e7f01d8f0",6,"factor: -(a - b)**N + (c - d)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/27",4,"Wentworth 1894, Exercise 34 (27)"],["form/a34afe3ae3",5,"factor: -(2*a + b)**2 + (2*c + 3)**2"],["shape/6831b28c24",6,"factor: -(N*a + b)**N + (N*c + N)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/28",4,"Wentworth 1894, Exercise 34 (28)"],["form/92e45cb503",5,"factor: -(a - 2*d)**2 + (b - c)**2"],["shape/d1f4de7136",6,"factor: (b - c)**N - (N*d + a)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/29",4,"Wentworth 1894, Exercise 34 (29)"],["form/523ec19ba2",5,"factor: -(2*a - b)**2 + (3*c - d)**2"],["shape/69fece21c6",6,"factor: -(N*a - b)**N + (N*c - d)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/30",4,"Wentworth 1894, Exercise 34 (30)"],["form/eaa5f32e18",5,"factor: -(a + 2*b)**2 + (c - 3*d)**2"],["shape/27f5e8b6e5",6,"factor: -(N*b + a)**N + (N*d + c)**N"],["wentworth-first-steps-in-algebra-1894/ex-34/31",4,"Wentworth 1894, Exercise 34 (31)"],["form/6f0cae2125",5,"factor: -(a + 3*b)**2 + (c + 2*d)**2"],["wentworth-first-steps-in-algebra-1894/ex-34/32",4,"Wentworth 1894, Exercise 34 (32)"],["form/fa519f3999",5,"factor: -(a - d)**2 + (b + c)**2"],["shape/c8b75f7924",6,"factor: -(a - d)**N + (b + c)**N"],["wentworth-first-steps-in-algebra-1894/ex-35/2",4,"Wentworth 1894, Exercise 35 (2)"],["form/9ec6e8d651",5,"factor: x**3 - 1"],["wentworth-first-steps-in-algebra-1894/ex-35/3",4,"Wentworth 1894, Exercise 35 (3)"],["form/d29e4f25c5",5,"factor: a**3*x**3 - b**3"],["shape/cc56d38d63",6,"factor: a**N*x**N - b**N"],["wentworth-first-steps-in-algebra-1894/ex-35/4",4,"Wentworth 1894, Exercise 35 (4)"],["form/e0c4a26db4",5,"factor: x**3 - 64"],["wentworth-first-steps-in-algebra-1894/ex-35/5",4,"Wentworth 1894, Exercise 35 (5)"],["wentworth-first-steps-in-algebra-1894/ex-35/6",4,"Wentworth 1894, Exercise 35 (6)"],["wentworth-first-steps-in-algebra-1894/ex-35/7",4,"Wentworth 1894, Exercise 35 (7)"],["form/9a07b719db",5,"factor: a**3*x**3 - 27*b**3"],["wentworth-first-steps-in-algebra-1894/ex-35/8",4,"Wentworth 1894, Exercise 35 (8)"],["form/be24e3ed15",5,"factor: a**3*b**3*x**3 - 8"],["wentworth-first-steps-in-algebra-1894/ex-35/9",4,"Wentworth 1894, Exercise 35 (9)"],["form/e32d877bbf",5,"factor: 8*a**3*x**3 - 27*b**6"],["shape/cf143c48c8",6,"factor: N*a**N*x**N + N*b**N"],["wentworth-first-steps-in-algebra-1894/ex-35/10",4,"Wentworth 1894, Exercise 35 (10)"],["form/0d30e88937",5,"factor: -a**9 + 64*x**3"],["wentworth-first-steps-in-algebra-1894/ex-35/11",4,"Wentworth 1894, Exercise 35 (11)"],["wentworth-first-steps-in-algebra-1894/ex-35/12",4,"Wentworth 1894, Exercise 35 (12)"],["form/572ed7338f",5,"factor: a**3*x**3 - 216*b**3"],["wentworth-first-steps-in-algebra-1894/ex-35/13",4,"Wentworth 1894, Exercise 35 (13)"],["form/8457a319b1",5,"factor: -729*a**3 + 64*x**3"],["wentworth-first-steps-in-algebra-1894/ex-35/14",4,"Wentworth 1894, Exercise 35 (14)"],["form/fc9e2526a5",5,"factor: -512*a**3 + 27*x**3"],["wentworth-first-steps-in-algebra-1894/ex-35/15",4,"Wentworth 1894, Exercise 35 (15)"],["form/38cbe7ed62",5,"factor: -125*a**3 + 8*x**6"],["wentworth-first-steps-in-algebra-1894/ex-35/16",4,"Wentworth 1894, Exercise 35 (16)"],["form/7f83c09e26",5,"factor: -27*a**15 + 64*x**12"],["form/f4cee65cb0",5,"factor: -8*x**3 + 216"],["wentworth-first-steps-in-algebra-1894/ex-35/18",4,"Wentworth 1894, Exercise 35 (18)"],["form/e794e9fd77",5,"factor: -27*x**3 + 343"],["shape/f2c98774e2",6,"factor: N*x**N + N"],["wentworth-first-steps-in-algebra-1894/ex-36/1",4,"Wentworth 1894, Exercise 36 (1)"],["form/24f433e226",5,"factor: x**3 + 1"],["wentworth-first-steps-in-algebra-1894/ex-36/2",4,"Wentworth 1894, Exercise 36 (2)"],["shape/3a904843b6",6,"factor: N*x**N + a**N"],["wentworth-first-steps-in-algebra-1894/ex-36/3",4,"Wentworth 1894, Exercise 36 (3)"],["form/13eb4e865d",5,"factor: x**3 + 125"],["wentworth-first-steps-in-algebra-1894/ex-36/4",4,"Wentworth 1894, Exercise 36 (4)"],["form/032e6e31e0",5,"factor: 64*x**3 + 27"],["wentworth-first-steps-in-algebra-1894/ex-36/5",4,"Wentworth 1894, Exercise 36 (5)"],["form/5b2d9882dc",5,"factor: a**3*x**3 + b**3"],["shape/d9c0b431b7",6,"factor: a**N*x**N + b**N"],["wentworth-first-steps-in-algebra-1894/ex-36/6",4,"Wentworth 1894, Exercise 36 (6)"],["form/2fa176426d",5,"factor: x**3 + 64"],["wentworth-first-steps-in-algebra-1894/ex-36/7",4,"Wentworth 1894, Exercise 36 (7)"],["form/d72eb4b33b",5,"factor: a**3 + 8*x**6"],["wentworth-first-steps-in-algebra-1894/ex-36/8",4,"Wentworth 1894, Exercise 36 (8)"],["wentworth-first-steps-in-algebra-1894/ex-36/9",4,"Wentworth 1894, Exercise 36 (9)"],["form/ca1100fca8",5,"factor: a**3*b**3*x**3 + 8"],["shape/754bcf6659",6,"factor: N + a**N*b**N*x**N"],["wentworth-first-steps-in-algebra-1894/ex-36/10",4,"Wentworth 1894, Exercise 36 (10)"],["form/e0e9d2f4cb",5,"factor: 64*a**3 + x**9"],["wentworth-first-steps-in-algebra-1894/ex-36/11",4,"Wentworth 1894, Exercise 36 (11)"],["form/6094344984",5,"factor: a**3*x**3 + 27*b**3"],["shape/bd45ffa747",6,"factor: N*b**N + a**N*x**N"],["wentworth-first-steps-in-algebra-1894/ex-36/12",4,"Wentworth 1894, Exercise 36 (12)"],["form/9545c9cbd3",5,"factor: a**6 + 8*b**3*x**3"],["shape/1d6d0e15ee",6,"factor: N*b**N*x**N + a**N"],["shape/d4df701d42",6,"factor: N*a**N + x**N"],["wentworth-first-steps-in-algebra-1894/ex-36/13",4,"Wentworth 1894, Exercise 36 (13)"],["form/00d286a643",5,"factor: 64*a**6 + x**9"],["wentworth-first-steps-in-algebra-1894/ex-36/14",4,"Wentworth 1894, Exercise 36 (14)"],["form/8f18559a6c",5,"factor: a**15 + 64*x**12"],["wentworth-first-steps-in-algebra-1894/ex-36/15",4,"Wentworth 1894, Exercise 36 (15)"],["form/416e199789",5,"factor: 8*a**6 + 27*x**15"],["wentworth-first-steps-in-algebra-1894/ex-36/16",4,"Wentworth 1894, Exercise 36 (16)"],["form/7655dfebce",5,"factor: 27*x**9 + 512"],["wentworth-first-steps-in-algebra-1894/ex-36/17",4,"Wentworth 1894, Exercise 36 (17)"],["form/0f15b76297",5,"factor: 64*x**3 + 343"],["wentworth-first-steps-in-algebra-1894/ex-36/18",4,"Wentworth 1894, Exercise 36 (18)"],["form/a7bd64d003",5,"factor: 27*x**3 + 125"],["wentworth-first-steps-in-algebra-1894/ex-37/1",4,"Wentworth 1894, Exercise 37 (1)"],["wentworth-first-steps-in-algebra-1894/ex-37/2",4,"Wentworth 1894, Exercise 37 (2)"],["wentworth-first-steps-in-algebra-1894/ex-37/3",4,"Wentworth 1894, Exercise 37 (3)"],["form/d83b2ed6a2",5,"factor: x**2 + 16*x + 64"],["wentworth-first-steps-in-algebra-1894/ex-37/4",4,"Wentworth 1894, Exercise 37 (4)"],["form/8d49d255e6",5,"factor: 25*a**2 + 10*a*x + x**2"],["wentworth-first-steps-in-algebra-1894/ex-37/5",4,"Wentworth 1894, Exercise 37 (5)"],["form/7524cabeb4",5,"factor: x**2 - 16*x + 64"],["wentworth-first-steps-in-algebra-1894/ex-37/6",4,"Wentworth 1894, Exercise 37 (6)"],["form/4551dc1ada",5,"factor: 25*a**2 - 10*a*x + x**2"],["wentworth-first-steps-in-algebra-1894/ex-37/8",4,"Wentworth 1894, Exercise 37 (8)"],["wentworth-first-steps-in-algebra-1894/ex-37/9",4,"Wentworth 1894, Exercise 37 (9)"],["wentworth-first-steps-in-algebra-1894/ex-37/10",4,"Wentworth 1894, Exercise 37 (10)"],["form/d7c784ed8a",5,"factor: 16*a**2 - 24*a*x + 9*x**2"],["wentworth-first-steps-in-algebra-1894/ex-37/11",4,"Wentworth 1894, Exercise 37 (11)"],["form/56fa7f4c88",5,"factor: 16*a**2 + 8*a*x + x**2"],["wentworth-first-steps-in-algebra-1894/ex-37/12",4,"Wentworth 1894, Exercise 37 (12)"],["form/fefab0ccbc",5,"factor: 16*a**2 - 8*a*x + x**2"],["wentworth-first-steps-in-algebra-1894/ex-37/13",4,"Wentworth 1894, Exercise 37 (13)"],["form/d5fc981483",5,"factor: 25*a**2 - 20*a*x + 4*x**2"],["wentworth-first-steps-in-algebra-1894/ex-37/14",4,"Wentworth 1894, Exercise 37 (14)"],["form/2d24e4f744",5,"factor: 100*x**2 + 20*x + 1"],["wentworth-first-steps-in-algebra-1894/ex-37/15",4,"Wentworth 1894, Exercise 37 (15)"],["form/813178b8dd",5,"factor: 49*x**2 - 28*x + 4"],["wentworth-first-steps-in-algebra-1894/ex-37/16",4,"Wentworth 1894, Exercise 37 (16)"],["form/269c2c94a0",5,"factor: 25*a**2 + 60*a*x + 36*x**2"],["wentworth-first-steps-in-algebra-1894/ex-37/17",4,"Wentworth 1894, Exercise 37 (17)"],["form/eab3b95e9d",5,"factor: 4*a**2 - 36*a*x + 81*x**2"],["wentworth-first-steps-in-algebra-1894/ex-37/18",4,"Wentworth 1894, Exercise 37 (18)"],["form/ee3bf754db",5,"factor: a**2*b**2 + 14*a*b*x**2 + 49*x**2"],["shape/1a16925647",6,"factor: N*a*b*x**N + N*x**N + a**N*b**N"],["wentworth-first-steps-in-algebra-1894/ex-38/1",4,"Wentworth 1894, Exercise 38 (1)"],["wentworth-first-steps-in-algebra-1894/ex-38/2",4,"Wentworth 1894, Exercise 38 (2)"],["wentworth-first-steps-in-algebra-1894/ex-38/3",4,"Wentworth 1894, Exercise 38 (3)"],["form/b956f0c97a",5,"factor: x**2 + 6*x + 5"],["wentworth-first-steps-in-algebra-1894/ex-38/4",4,"Wentworth 1894, Exercise 38 (4)"],["form/a4df71bde9",5,"factor: x**2 - 6*x + 5"],["wentworth-first-steps-in-algebra-1894/ex-38/5",4,"Wentworth 1894, Exercise 38 (5)"],["wentworth-first-steps-in-algebra-1894/ex-38/6",4,"Wentworth 1894, Exercise 38 (6)"],["form/367bfb9425",5,"factor: x**2 - 4*x - 5"],["wentworth-first-steps-in-algebra-1894/ex-38/7",4,"Wentworth 1894, Exercise 38 (7)"],["wentworth-first-steps-in-algebra-1894/ex-38/8",4,"Wentworth 1894, Exercise 38 (8)"],["wentworth-first-steps-in-algebra-1894/ex-38/9",4,"Wentworth 1894, Exercise 38 (9)"],["wentworth-first-steps-in-algebra-1894/ex-38/10",4,"Wentworth 1894, Exercise 38 (10)"],["form/ba9f03847f",5,"factor: x**2 - 3*x - 18"],["wentworth-first-steps-in-algebra-1894/ex-38/11",4,"Wentworth 1894, Exercise 38 (11)"],["form/8da71965e1",5,"factor: x**2 + 9*x + 14"],["wentworth-first-steps-in-algebra-1894/ex-38/12",4,"Wentworth 1894, Exercise 38 (12)"],["form/5ca340799c",5,"factor: x**2 - 9*x + 14"],["wentworth-first-steps-in-algebra-1894/ex-38/13",4,"Wentworth 1894, Exercise 38 (13)"],["wentworth-first-steps-in-algebra-1894/ex-38/16",4,"Wentworth 1894, Exercise 38 (16)"],["form/a31b960f4c",5,"factor: x**2 + x - 20"],["shape/6fd1081fa6",6,"factor: N + x + x**N"],["wentworth-first-steps-in-algebra-1894/ex-38/17",4,"Wentworth 1894, Exercise 38 (17)"],["form/bb9ad098b1",5,"factor: x**2 - 10*x + 21"],["wentworth-first-steps-in-algebra-1894/ex-38/18",4,"Wentworth 1894, Exercise 38 (18)"],["form/18532350a0",5,"factor: x**2 - 4*x - 21"],["wentworth-first-steps-in-algebra-1894/ex-38/19",4,"Wentworth 1894, Exercise 38 (19)"],["wentworth-first-steps-in-algebra-1894/ex-38/20",4,"Wentworth 1894, Exercise 38 (20)"],["form/516ffb65bb",5,"factor: x**2 - 15*x + 56"],["form/0be96af72f",5,"factor: x**2 + 4*x - 21"],["wentworth-first-steps-in-algebra-1894/ex-38/21",4,"Wentworth 1894, Exercise 38 (21)"],["form/8aec406f9e",5,"factor: x**2 - x - 56"],["wentworth-first-steps-in-algebra-1894/ex-38/22",4,"Wentworth 1894, Exercise 38 (22)"],["wentworth-first-steps-in-algebra-1894/ex-38/23",4,"Wentworth 1894, Exercise 38 (23)"],["form/034cf2d6f7",5,"factor: x**2 + 13*x + 30"],["wentworth-first-steps-in-algebra-1894/ex-38/24",4,"Wentworth 1894, Exercise 38 (24)"],["form/18437ef19f",5,"factor: x**2 + 7*x - 30"],["wentworth-first-steps-in-algebra-1894/ex-38/25",4,"Wentworth 1894, Exercise 38 (25)"],["form/bc40843490",5,"factor: x**2 - 7*x - 30"],["wentworth-first-steps-in-algebra-1894/ex-38/26",4,"Wentworth 1894, Exercise 38 (26)"],["form/89f351fb67",5,"factor: -6*a**2 + a*x + x**2"],["shape/a203414405",6,"factor: N*a**N + a*x + x**N"],["wentworth-first-steps-in-algebra-1894/ex-38/27",4,"Wentworth 1894, Exercise 38 (27)"],["form/69673b14c4",5,"factor: -6*a**2 - a*x + x**2"],["shape/4114ebb940",6,"factor: N*a**N - a*x + x**N"],["wentworth-first-steps-in-algebra-1894/ex-38/28",4,"Wentworth 1894, Exercise 38 (28)"],["wentworth-first-steps-in-algebra-1894/ex-38/29",4,"Wentworth 1894, Exercise 38 (29)"],["form/e9c0a24642",5,"factor: -4*a**2 - 3*a*x + x**2"],["wentworth-first-steps-in-algebra-1894/ex-38/30",4,"Wentworth 1894, Exercise 38 (30)"],["form/43eb44ed0a",5,"factor: a**2*x**2 - 2*a*x - 63"],["shape/36c13862ea",6,"factor: N*a*x + N + a**N*x**N"],["wentworth-first-steps-in-algebra-1894/ex-38/31",4,"Wentworth 1894, Exercise 38 (31)"],["form/180cc908ff",5,"factor: a**2 + 2*a*x - 63*x**2"],["shape/9f90e63203",6,"factor: N*a*x + N*x**N + a**N"],["wentworth-first-steps-in-algebra-1894/ex-38/32",4,"Wentworth 1894, Exercise 38 (32)"],["form/d020a62ab5",5,"factor: 20*a**2 - 9*a*x + x**2"],["wentworth-first-steps-in-algebra-1894/ex-38/33",4,"Wentworth 1894, Exercise 38 (33)"],["wentworth-first-steps-in-algebra-1894/ex-38/34",4,"Wentworth 1894, Exercise 38 (34)"],["wentworth-first-steps-in-algebra-1894/ex-38/35",4,"Wentworth 1894, Exercise 38 (35)"],["wentworth-first-steps-in-algebra-1894/ex-38/36",4,"Wentworth 1894, Exercise 38 (36)"],["form/dfe459d76f",5,"factor: 84*a**2 + 19*a*x + x**2"],["wentworth-first-steps-in-algebra-1894/ex-38/37",4,"Wentworth 1894, Exercise 38 (37)"],["form/1789de04be",5,"factor: a**2*x**2 - 23*a*b*x + 102*b**2"],["wentworth-first-steps-in-algebra-1894/ex-38/38",4,"Wentworth 1894, Exercise 38 (38)"],["form/dbf0255b68",5,"factor: 20*a**4 - 9*a**2*x**2 + x**4"],["wentworth-first-steps-in-algebra-1894/ex-38/39",4,"Wentworth 1894, Exercise 38 (39)"],["form/0066b057b4",5,"factor: a**4*x**4 - 24*a**2*b**2*x**2 + 143*b**4"],["shape/57c5460b87",6,"factor: N*a**N*b**N*x**N + N*b**N + a**N*x**N"],["wentworth-first-steps-in-algebra-1894/ex-38/40",4,"Wentworth 1894, Exercise 38 (40)"],["form/20790ff797",5,"factor: a**6*x**6 - 23*a**3*b**2*x**3 + 132*b**4"],["wentworth-first-steps-in-algebra-1894/ex-38/41",4,"Wentworth 1894, Exercise 38 (41)"],["form/662a6b7751",5,"factor: -96*a**2*b**2 - 20*a*b*x + x**2"],["shape/98f9045786",6,"factor: N*a*b*x + N*a**N*b**N + x**N"],["wentworth-first-steps-in-algebra-1894/ex-38/42",4,"Wentworth 1894, Exercise 38 (42)"],["wentworth-first-steps-in-algebra-1894/ex-38/43",4,"Wentworth 1894, Exercise 38 (43)"],["form/9961752f96",5,"factor: -96*a**2*b**2 - 10*a*b*x + x**2"],["wentworth-first-steps-in-algebra-1894/ex-38/44",4,"Wentworth 1894, Exercise 38 (44)"],["form/009bd1f398",5,"factor: -96*a**2*b**2 + 29*a*b*x + x**2"],["wentworth-first-steps-in-algebra-1894/ex-38/45",4,"Wentworth 1894, Exercise 38 (45)"],["form/13eca786e6",5,"factor: -96*a**2*b**2 - 46*a*b*x + x**2"],["wentworth-first-steps-in-algebra-1894/ex-38/46",4,"Wentworth 1894, Exercise 38 (46)"],["wentworth-first-steps-in-algebra-1894/ex-38/47",4,"Wentworth 1894, Exercise 38 (47)"],["wentworth-first-steps-in-algebra-1894/ex-38/48",4,"Wentworth 1894, Exercise 38 (48)"],["shape/2c71c910a7",6,"factor: N*b**N - a*b*x + a**N*x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/2",4,"Wentworth 1894, Exercise 39 (2)"],["wentworth-first-steps-in-algebra-1894/ex-39/3",4,"Wentworth 1894, Exercise 39 (3)"],["form/e4555c7b37",5,"factor: -a + x + (-a + x)**2"],["shape/ccfc9cb269",6,"factor: -a + x + (-a + x)**N"],["wentworth-first-steps-in-algebra-1894/ex-39/4",4,"Wentworth 1894, Exercise 39 (4)"],["form/78fd26340a",5,"factor: (a + x)**2 - 1"],["shape/81901f9add",6,"factor: (a + x)**N - 1"],["wentworth-first-steps-in-algebra-1894/ex-39/5",4,"Wentworth 1894, Exercise 39 (5)"],["wentworth-first-steps-in-algebra-1894/ex-39/6",4,"Wentworth 1894, Exercise 39 (6)"],["shape/98205e3b7a",6,"factor: N*a + N*a**N + x + x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/7",4,"Wentworth 1894, Exercise 39 (7)"],["form/76ebc900ba",5,"factor: -a**3 - a + x**3 + x"],["shape/824a6da387",6,"factor: -a - a**N + x + x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/8",4,"Wentworth 1894, Exercise 39 (8)"],["form/0999b5b6b1",5,"factor: 9*a**2 - 6*a*x + x**2"],["shape/e69781fb7a",6,"factor: N*a*x + N*a**N + x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/9",4,"Wentworth 1894, Exercise 39 (9)"],["form/8c9668a277",5,"factor: x**2 - x - 2"],["wentworth-first-steps-in-algebra-1894/ex-39/12",4,"Wentworth 1894, Exercise 39 (12)"],["form/a2aae96677",5,"factor: x**2 - 11*x - 26"],["wentworth-first-steps-in-algebra-1894/ex-39/13",4,"Wentworth 1894, Exercise 39 (13)"],["form/e5106a959b",5,"factor: a*b**2 + 3*a + b**2*x + 3*x"],["shape/6206b85f9e",6,"factor: N*a + N*x + a*b**N + b**N*x"],["wentworth-first-steps-in-algebra-1894/ex-39/14",4,"Wentworth 1894, Exercise 39 (14)"],["wentworth-first-steps-in-algebra-1894/ex-39/15",4,"Wentworth 1894, Exercise 39 (15)"],["form/cdd79cba73",5,"factor: x**2 - 7*x + 12"],["wentworth-first-steps-in-algebra-1894/ex-39/16",4,"Wentworth 1894, Exercise 39 (16)"],["wentworth-first-steps-in-algebra-1894/ex-39/17",4,"Wentworth 1894, Exercise 39 (17)"],["form/0a670839af",5,"factor: x**4 + 10*x**2 + 25"],["wentworth-first-steps-in-algebra-1894/ex-39/18",4,"Wentworth 1894, Exercise 39 (18)"],["form/17b1915c4c",5,"factor: x**2 - 18*x + 81"],["wentworth-first-steps-in-algebra-1894/ex-39/19",4,"Wentworth 1894, Exercise 39 (19)"],["form/87614db2f4",5,"factor: x**2 - 21*x + 110"],["wentworth-first-steps-in-algebra-1894/ex-39/20",4,"Wentworth 1894, Exercise 39 (20)"],["form/c749da5f64",5,"factor: x**2 + 19*x + 88"],["wentworth-first-steps-in-algebra-1894/ex-39/21",4,"Wentworth 1894, Exercise 39 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x)**N"],["wentworth-first-steps-in-algebra-1894/ex-39/28",4,"Wentworth 1894, Exercise 39 (28)"],["form/73501c1bb0",5,"factor: b**2 - (-a + x)**2"],["shape/f97b568605",6,"factor: b**N - (-a + x)**N"],["wentworth-first-steps-in-algebra-1894/ex-39/29",4,"Wentworth 1894, Exercise 39 (29)"],["shape/ea05f38307",6,"factor: N*x**N - (N*x - 1)**N"],["wentworth-first-steps-in-algebra-1894/ex-39/30",4,"Wentworth 1894, Exercise 39 (30)"],["form/be5e1cab49",5,"factor: -a**3 + 8*x**3"],["shape/0540b0ddde",6,"factor: N*x**N - a**N"],["wentworth-first-steps-in-algebra-1894/ex-39/31",4,"Wentworth 1894, Exercise 39 (31)"],["form/c7275666d9",5,"factor: -3*a*x**2 + x**3"],["shape/e00c6e05e4",6,"factor: N*a*x**N + x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/32",4,"Wentworth 1894, Exercise 39 (32)"],["form/85922afbcb",5,"factor: -27*a**3 + x**3"],["wentworth-first-steps-in-algebra-1894/ex-39/33",4,"Wentworth 1894, Exercise 39 (33)"],["form/1c5c63a6ca",5,"factor: x**2 + 3*x - 40"],["wentworth-first-steps-in-algebra-1894/ex-39/34",4,"Wentworth 1894, Exercise 39 (34)"],["wentworth-first-steps-in-algebra-1894/ex-39/35",4,"Wentworth 1894, Exercise 39 (35)"],["shape/b1c0a07fc0",6,"factor: N*x**N + 1"],["wentworth-first-steps-in-algebra-1894/ex-39/36",4,"Wentworth 1894, Exercise 39 (36)"],["form/0e7a9745e3",5,"factor: -9*a**4*x**2 + x**6"],["shape/0c7b78510d",6,"factor: N*a**N*x**N + x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/37",4,"Wentworth 1894, Exercise 39 (37)"],["form/255c291929",5,"factor: 2*a**2*x + 3*a*x**2 + x**3"],["shape/3f178aaf9f",6,"factor: N*a*x**N + N*a**N*x + x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/38",4,"Wentworth 1894, Exercise 39 (38)"],["form/961748647d",5,"factor: 3*a**2*x**2 + 4*a*x**3 + x**4"],["shape/e12200978a",6,"factor: N*a*x**N + N*a**N*x**N + x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/39",4,"Wentworth 1894, Exercise 39 (39)"],["form/06478a5725",5,"factor: 4*a**4 - 4*a**2*x + x**2"],["shape/1a26039994",6,"factor: N*a**N*x + N*a**N + x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/40",4,"Wentworth 1894, Exercise 39 (40)"],["form/ec7b735f87",5,"factor: 16*x**4 + 8*x**2 + 1"],["wentworth-first-steps-in-algebra-1894/ex-39/41",4,"Wentworth 1894, Exercise 39 (41)"],["form/f82f3b510a",5,"factor: -4*a**2*x**2 + 9*x**4"],["wentworth-first-steps-in-algebra-1894/ex-39/42",4,"Wentworth 1894, Exercise 39 (42)"],["form/838fe94031",5,"factor: -2*a**3*x - a**2*x**2 + a*x**3"],["shape/e398b0326c",6,"factor: N*a**N*x + a*x**N - a**N*x**N"],["wentworth-first-steps-in-algebra-1894/ex-39/43",4,"Wentworth 1894, Exercise 39 (43)"],["form/df691dc7af",5,"factor: x**4 - x**3 + 8*x - 8"],["shape/591f86077c",6,"factor: N*x + N"],["wentworth-first-steps-in-algebra-1894/ex-39/44",4,"Wentworth 1894, Exercise 39 (44)"],["form/c0c8377dbf",5,"factor: -a*b**3 - a*x**3 + b**3*x + x**4"],["shape/9c69ebab6f",6,"factor: -a*b**N - a*x**N + b**N*x + x**N"],["wentworth-first-steps-in-algebra-1894/ex-3/2",4,"Wentworth 1894, Exercise 3 (2)"],["form/9401ebf361",5,"evaluate: 8*a*b at a=7, b=5, c=3"],["wentworth-first-steps-in-algebra-1894/ex-3/3",4,"Wentworth 1894, Exercise 3 (3)"],["wentworth-first-steps-in-algebra-1894/ex-3/4",4,"Wentworth 1894, Exercise 3 (4)"],["form/c409f4dfb1",5,"evaluate: 2*a**2 at a=7, b=5, c=3"],["shape/de34e87970",6,"evaluate: N*a**N"],["wentworth-first-steps-in-algebra-1894/ex-3/5",4,"Wentworth 1894, Exercise 3 (5)"],["form/21bc2efeab",5,"evaluate: 3*a**3 at a=7, b=5, c=3"],["wentworth-first-steps-in-algebra-1894/ex-3/6",4,"Wentworth 1894, Exercise 3 (6)"],["form/f021572811",5,"evaluate: 2*a**4 at a=7, b=5, c=3"],["wentworth-first-steps-in-algebra-1894/ex-3/7",4,"Wentworth 1894, Exercise 3 (7)"],["form/53c0bd7cb8",5,"evaluate: 5*a*b at a=7, b=5, c=3"],["wentworth-first-steps-in-algebra-1894/ex-3/8",4,"Wentworth 1894, Exercise 3 (8)"],["form/ccd5d8e310",5,"evaluate: a*b*c at a=7, b=5, c=3"],["shape/b2baeab2cf",6,"evaluate: a*b*c"],["wentworth-first-steps-in-algebra-1894/ex-3/9",4,"Wentworth 1894, Exercise 3 (9)"],["wentworth-first-steps-in-algebra-1894/ex-3/11",4,"Wentworth 1894, Exercise 3 (11)"],["wentworth-first-steps-in-algebra-1894/ex-3/12",4,"Wentworth 1894, Exercise 3 (12)"],["form/ce9b0d85fd",5,"evaluate: a**2*b*c/7 at a=7, b=5, c=3"],["shape/32dd9c3d0d",6,"evaluate: N*a**N*b*c"],["wentworth-first-steps-in-algebra-1894/ex-3/13",4,"Wentworth 1894, Exercise 3 (13)"],["form/fd03b11444",5,"evaluate: 4*a*b*c**2 at a=5, b=2, c=0, x=1, y=3"],["shape/a32e06c858",6,"evaluate: N*a*b*c**N"],["wentworth-first-steps-in-algebra-1894/ex-3/14",4,"Wentworth 1894, Exercise 3 (14)"],["form/5fb4adaf24",5,"evaluate: 3*a*b**5*c**2 at a=5, b=2, c=0, x=1, y=3"],["shape/4b7d973bf0",6,"evaluate: N*a*b**N*c**N"],["wentworth-first-steps-in-algebra-1894/ex-3/15",4,"Wentworth 1894, Exercise 3 (15)"],["form/8713b9dcfe",5,"evaluate: 2*a*b**2*c at a=5, b=2, c=0, x=1, y=3"],["wentworth-first-steps-in-algebra-1894/ex-3/16",4,"Wentworth 1894, Exercise 3 (16)"],["form/64d520e8c3",5,"evaluate: 2*a**2*b**2*c**2*d**2 at a=5, b=2, c=0, x=1, y=3"],["shape/37f8bec8fd",6,"evaluate: N*a**N*b**N*c**N*d**N"],["wentworth-first-steps-in-algebra-1894/ex-3/17",4,"Wentworth 1894, Exercise 3 (17)"],["wentworth-first-steps-in-algebra-1894/ex-3/18",4,"Wentworth 1894, Exercise 3 (18)"],["form/f10ec1401d",5,"evaluate: 2*a*b*c**3*d**3 at a=5, b=2, c=0, x=1, y=3"],["shape/d4360360f5",6,"evaluate: N*a*b*c**N*d**N"],["wentworth-first-steps-in-algebra-1894/ex-3/19",4,"Wentworth 1894, Exercise 3 (19)"],["shape/2be931acb2",6,"evaluate: N*a*b*c*d*e"],["wentworth-first-steps-in-algebra-1894/ex-3/20",4,"Wentworth 1894, Exercise 3 (20)"],["form/4a3bc60cd3",5,"evaluate: 3*a*b*c**3*d**2 at a=5, b=2, c=0, x=1, y=3"],["wentworth-first-steps-in-algebra-1894/ex-3/21",4,"Wentworth 1894, Exercise 3 (21)"],["form/397908e1d7",5,"evaluate: 3*a*b**2*c*d**2 at a=5, b=2, c=0, x=1, y=3"],["shape/791a5aaf5f",6,"evaluate: N*a*b**N*c*d**N"],["wentworth-first-steps-in-algebra-1894/ex-40/5",4,"Wentworth 1894, Exercise 40 (5)"],["form/eac89d1f45",5,"hcf: (28*a**4, 63*b**4)"],["shape/c6875ecb40",6,"hcf: (N*a**N, N*b**N)"],["wentworth-first-steps-in-algebra-1894/ex-40/6",4,"Wentworth 1894, Exercise 40 (6)"],["form/56f9e8a244",5,"hcf: (54*a**2*b**2, 56*a**3*b**3)"],["shape/e75d7520aa",6,"hcf: (N*a**N*b**N, N*a**N*b**N)"],["wentworth-first-steps-in-algebra-1894/ex-40/7",4,"Wentworth 1894, Exercise 40 (7)"],["form/3df9e82876",5,"hcf: (a**3 + 27*b**3, a**3 + 3*a**2*b)"],["shape/b8cc2dd7ee",6,"hcf: (N*b**N + a**N, N*a**N*b + a**N)"],["form/893f68a769",5,"hcf: (a**2 - 9, a**2 + 3*a)"],["shape/b87edb7100",6,"hcf: (N + a**N, N*a + a**N)"],["wentworth-first-steps-in-algebra-1894/ex-40/9",4,"Wentworth 1894, Exercise 40 (9)"],["form/f6eb52bfae",5,"hcf: (8*a**3 + 1, 2*a*b**3 + b**3)"],["shape/27463d4c35",6,"hcf: (N*a**N + 1, N*a*b**N + 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(N*a + N + a**N, N*a + N + a**N)"],["wentworth-first-steps-in-algebra-1894/ex-40/15",4,"Wentworth 1894, Exercise 40 (15)"],["form/d9003aa911",5,"hcf: (a**2 - 10*a + 24, a**2 - 9*a + 18)"],["wentworth-first-steps-in-algebra-1894/ex-40/16",4,"Wentworth 1894, Exercise 40 (16)"],["form/8fc3c44583",5,"hcf: (a**3 + 1, a**2 - a + 1)"],["shape/9cf269726e",6,"hcf: (a**N + 1, -a + a**N + 1)"],["wentworth-first-steps-in-algebra-1894/ex-40/17",4,"Wentworth 1894, Exercise 40 (17)"],["form/a17bdf0961",5,"hcf: (a**2 - 4*a + 3, a**2 - 3*a + 2)"],["wentworth-first-steps-in-algebra-1894/ex-40/18",4,"Wentworth 1894, Exercise 40 (18)"],["shape/e8118d817c",6,"hcf: (N*a*b + N*b**N + a**N, N*a*b + a**N + b**N)"],["wentworth-first-steps-in-algebra-1894/ex-40/19",4,"Wentworth 1894, Exercise 40 (19)"],["form/069ddeb278",5,"hcf: (a**2 - 25, a**2 - 4*a - 5)"],["shape/7619f3cb7e",6,"hcf: (N + a**N, N*a + N + a**N)"],["wentworth-first-steps-in-algebra-1894/ex-40/20",4,"Wentworth 1894, Exercise 40 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0)"],["wentworth-first-steps-in-algebra-1894/ex-51/16",4,"Wentworth 1894, Exercise 51 (16)"],["form/0784decbd6",5,"solve: Eq(x/28 + 23/28, -7*x/3 + 38/3)"],["wentworth-first-steps-in-algebra-1894/ex-51/17",4,"Wentworth 1894, Exercise 51 (17)"],["form/1e340aece1",5,"solve: Eq(9*x/8 + 7/8, -x/2 + 31/2)"],["wentworth-first-steps-in-algebra-1894/ex-51/18",4,"Wentworth 1894, Exercise 51 (18)"],["form/47bd1f850f",5,"solve: Eq(-22*x/15 - 22/15, 0)"],["wentworth-first-steps-in-algebra-1894/ex-52/1",4,"Wentworth 1894, Exercise 52 (1)"],["form/58a96bcf66",5,"solve: Eq(11*x/21 - 1/21, 1)"],["wentworth-first-steps-in-algebra-1894/ex-52/2",4,"Wentworth 1894, Exercise 52 (2)"],["form/424fc08198",5,"solve: Eq(88*x/21 - 176/21, 0)"],["wentworth-first-steps-in-algebra-1894/ex-52/3",4,"Wentworth 1894, Exercise 52 (3)"],["form/af6b04c19e",5,"solve: Eq(-4*x/21 - 44/7, 0)"],["wentworth-first-steps-in-algebra-1894/ex-52/4",4,"Wentworth 1894, Exercise 52 (4)"],["form/bcf266b299",5,"solve: Eq(33*x/20 + 7/20, 65*x/36 + 7/36)"],["wentworth-first-steps-in-algebra-1894/ex-52/5",4,"Wentworth 1894, Exercise 52 (5)"],["form/c12c345cec",5,"solve: Eq(x/12 + 7/12, x/30 + 37/60)"],["wentworth-first-steps-in-algebra-1894/ex-52/6",4,"Wentworth 1894, Exercise 52 (6)"],["form/77fbd225d8",5,"solve: Eq(x**2 - 3*x/2 + (-x + 2)*(2*x - 1)/2 - 1/2, 0)"],["wentworth-first-steps-in-algebra-1894/ex-52/7",4,"Wentworth 1894, Exercise 52 (7)"],["form/5ed29eea75",5,"solve: Eq(13*x/6 - 13/4, -x/8 + 4/3)"],["wentworth-first-steps-in-algebra-1894/ex-52/8",4,"Wentworth 1894, Exercise 52 (8)"],["form/e1e62d55e2",5,"solve: Eq(x/2 + 1/6, 25/6)"],["wentworth-first-steps-in-algebra-1894/ex-52/9",4,"Wentworth 1894, Exercise 52 (9)"],["form/d5c16c4ff8",5,"solve: Eq(2*x/3 + 2/3, x/20 + 15/4)"],["wentworth-first-steps-in-algebra-1894/ex-52/10",4,"Wentworth 1894, Exercise 52 (10)"],["form/6c1689702b",5,"solve: Eq((2*x - 3)/(2*x - 1) + (5*x + 3)/(x - 1), 6)"],["shape/3bfbf74966",6,"solve: Eq((N*x + N)/(N*x - 1) + (N*x + N)/(x - 1), N)"],["wentworth-first-steps-in-algebra-1894/ex-52/11",4,"Wentworth 1894, Exercise 52 (11)"],["form/007d923d70",5,"solve: Eq(3*x/(4*x + 1) + 1, -x/(4*x - 2) + 2)"],["shape/0ed1f74b77",6,"solve: Eq(N*x/(N*x + 1) + 1, N - x/(N*x + N))"],["wentworth-first-steps-in-algebra-1894/ex-52/12",4,"Wentworth 1894, Exercise 52 (12)"],["shape/185f4b49c3",6,"solve: Eq(0, N*x/(N*x + 1) + 1)"],["wentworth-first-steps-in-algebra-1894/ex-52/13",4,"Wentworth 1894, Exercise 52 (13)"],["form/97e3392c6f",5,"solve: Eq(-(x - 1)/(x + 1) + (x + 1)/(2*x - 2), (-x**2 + 17)/(2*x**2 - 2))"],["shape/00fce99764",6,"solve: Eq(-(x - 1)/(x + 1) + (x + 1)/(N*x + N), (N - x**N)/(N*x**N + N))"],["wentworth-first-steps-in-algebra-1894/ex-53/1",4,"Wentworth 1894, Exercise 53 (1)"],["form/8585c0df0f",5,"solve: Eq(5*x/9 + 13/18 - (x + 2)/(x - 3), 5*x/9 - 4/9)"],["wentworth-first-steps-in-algebra-1894/ex-53/2",4,"Wentworth 1894, Exercise 53 (2)"],["form/436b4cb643",5,"solve: Eq(1/2, (x - 1)/(3*x - 4))"],["shape/4bde3e2f89",6,"solve: Eq(N, (x - 1)/(N*x + N))"],["wentworth-first-steps-in-algebra-1894/ex-53/3",4,"Wentworth 1894, Exercise 53 (3)"],["form/9f336b0857",5,"solve: Eq(11*x/14 - (11*x - 7)/(19*x + 7) - 6/7, 11*x/14 - 9/7)"],["wentworth-first-steps-in-algebra-1894/ex-5/5",4,"Wentworth 1894, Exercise 5 (5)"],["wentworth-first-steps-in-algebra-1894/ex-5/6",4,"Wentworth 1894, Exercise 5 (6)"],["shape/7cf28e6dee",6,"solve: False"],["wentworth-first-steps-in-algebra-1894/ex-53/4",4,"Wentworth 1894, Exercise 53 (4)"],["form/d56789787e",5,"solve: Eq(2*x/5 + (2*x - 3)/(17*x - 12) - 1/5, 2*x/5 - 3/10)"],["wentworth-first-steps-in-algebra-1894/ex-53/5",4,"Wentworth 1894, Exercise 53 (5)"],["form/0f8b4f1d34",5,"solve: Eq(11*x/7 - 13/7 - (13*x + 7)/(3*x + 7), 11*x/7 - 75/14)"],["form/575aaf110f",5,"solve: Eq(-5/9 + (6*x - 13)/(2*x + 3), 0)"],["shape/ad353bf462",6,"solve: Eq(N + 1, 0)"],["wentworth-first-steps-in-algebra-1894/ex-54/1",4,"Wentworth 1894, Exercise 54 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b**2))"],["shape/859f0d13a6",6,"solve: Eq(N*a/(a + b) + x/(a - b), b*x/(a**N - b**N))"],["wentworth-first-steps-in-algebra-1894/ex-55/1",4,"Wentworth 1894, Exercise 55 (1)"],["wentworth-first-steps-in-algebra-1894/ex-55/2",4,"Wentworth 1894, Exercise 55 (2)"],["form/b9233b8f2f",5,"solve: Eq(11*x/70, 11)"],["wentworth-first-steps-in-algebra-1894/ex-55/3",4,"Wentworth 1894, Exercise 55 (3)"],["form/1511128309",5,"solve: Eq(4*x/9, 16)"],["wentworth-first-steps-in-algebra-1894/ex-55/4",4,"Wentworth 1894, Exercise 55 (4)"],["form/535f82babc",5,"solve: Eq(x/6 + 1/2, 10)"],["wentworth-first-steps-in-algebra-1894/ex-56/none",4,"Wentworth 1894, Exercise 56 (None)"],["shape/caca07671a",6,"solve: Eq((N + x)/(N - x), N)"],["wentworth-first-steps-in-algebra-1894/ex-56/1",4,"Wentworth 1894, Exercise 56 (1)"],["form/d5feaeaed0",5,"solve: (Eq(x, 4*a + 5), Eq(a + x, 100))"],["shape/b112859555",6,"solve: (Eq(x, N*a + N), Eq(a + x, N))"],["wentworth-first-steps-in-algebra-1894/ex-56/2",4,"Wentworth 1894, 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(Eq(a, N*x), Eq(N + a, N*x + N))"],["wentworth-first-steps-in-algebra-1894/ex-57/3",4,"Wentworth 1894, Exercise 57 (3)"],["form/98b37fbec7",5,"solve: (Eq(a + x, 30), Eq(a + 5, x/3 + 5/3))"],["shape/0c426801cb",6,"solve: (Eq(a + x, N), Eq(N + a, N*x + N))"],["wentworth-first-steps-in-algebra-1894/ex-57/4",4,"Wentworth 1894, Exercise 57 (4)"],["form/fcf4d3507c",5,"solve: (Eq(a, 35), Eq(b, a/4), Eq(b + x, a/2 + x/2))"],["shape/b66ba6d6e3",6,"solve: (Eq(a, N), Eq(b, N*a), Eq(b + x, N*a + N*x))"],["wentworth-first-steps-in-algebra-1894/ex-57/5",4,"Wentworth 1894, Exercise 57 (5)"],["form/58cc44bd8a",5,"solve: (Eq(a, 60), Eq(b, 2*a/3), Eq(b - x, a/5 - x/5))"],["shape/508f89676c",6,"solve: (Eq(a, N), Eq(b, N*a), Eq(b - x, N*a + N*x))"],["wentworth-first-steps-in-algebra-1894/ex-5/8b",4,"Wentworth 1894, Exercise 5 (8b)"],["wentworth-first-steps-in-algebra-1894/ex-57/6",4,"Wentworth 1894, Exercise 57 (6)"],["form/3f68c1a4a3",5,"solve: (Eq(x, a/3), Eq(x - 4, a/4 - 1))"],["shape/429b47275f",6,"solve: (Eq(x, N*a), Eq(N + x, N*a - 1))"],["wentworth-first-steps-in-algebra-1894/ex-57/7",4,"Wentworth 1894, Exercise 57 (7)"],["form/efa99600a2",5,"solve: (Eq(a, 50), Eq(b, a/2), Eq(b + x, 2*a/3 + 2*x/3))"],["wentworth-first-steps-in-algebra-1894/ex-57/8",4,"Wentworth 1894, Exercise 57 (8)"],["form/58884d5483",5,"solve: (Eq(a, x/2), Eq(a - 10, x/4 - 5/2))"],["wentworth-first-steps-in-algebra-1894/ex-57/9",4,"Wentworth 1894, Exercise 57 (9)"],["wentworth-first-steps-in-algebra-1894/ex-58/1",4,"Wentworth 1894, Exercise 58 (1)"],["wentworth-first-steps-in-algebra-1894/ex-58/2",4,"Wentworth 1894, Exercise 58 (2)"],["form/c53db01a2e",5,"solve: Eq(1/x, 47/60)"],["wentworth-first-steps-in-algebra-1894/ex-58/3",4,"Wentworth 1894, Exercise 58 (3)"],["wentworth-first-steps-in-algebra-1894/ex-58/4",4,"Wentworth 1894, Exercise 58 (4)"],["form/a172716826",5,"solve: Eq(11/60 + 1/x, 1/4)"],["wentworth-first-steps-in-algebra-1894/ex-58/5",4,"Wentworth 1894, Exercise 58 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(18)"],["wentworth-first-steps-in-algebra-1894/ex-60/none",4,"Wentworth 1894, Exercise 60 (None)"],["wentworth-first-steps-in-algebra-1894/ex-60/1",4,"Wentworth 1894, Exercise 60 (1)"],["form/7b9a935fda",5,"solve: Eq(x/3, x/4 + 3)"],["wentworth-first-steps-in-algebra-1894/ex-60/2",4,"Wentworth 1894, Exercise 60 (2)"],["form/45f30d5e0a",5,"solve: Eq(15*x/2, 13*x/2 + 26)"],["wentworth-first-steps-in-algebra-1894/ex-60/3",4,"Wentworth 1894, Exercise 60 (3)"],["form/7fd4a76349",5,"solve: Eq(3*x/4, 6)"],["wentworth-first-steps-in-algebra-1894/ex-60/4",4,"Wentworth 1894, Exercise 60 (4)"],["form/09ab3403b5",5,"solve: Eq(x/40, x/30 - 2)"],["wentworth-first-steps-in-algebra-1894/ex-61/1",4,"Wentworth 1894, Exercise 61 (1)"],["wentworth-first-steps-in-algebra-1894/ex-61/2",4,"Wentworth 1894, Exercise 61 (2)"],["wentworth-first-steps-in-algebra-1894/ex-9/6",4,"Wentworth 1894, Exercise 9 (6)"],["form/291f7def59",5,"solve: Eq(3*x - 6, 2*x - 6)"],["wentworth-first-steps-in-algebra-1894/ex-61/3",4,"Wentworth 1894, Exercise 61 (3)"],["form/9b29617d90",5,"solve: (Eq(a, 3*b/4 + 90), Eq(5*a, 4*b))"],["shape/3817228c65",6,"solve: (Eq(a, N*b + N), Eq(N*a, N*b))"],["wentworth-first-steps-in-algebra-1894/ex-62/1",4,"Wentworth 1894, Exercise 62 (1)"],["form/8991095485",5,"solve: Eq(x, x/12 + 25)"],["wentworth-first-steps-in-algebra-1894/ex-62/2",4,"Wentworth 1894, Exercise 62 (2)"],["wentworth-first-steps-in-algebra-1894/ex-62/3",4,"Wentworth 1894, Exercise 62 (3)"],["wentworth-first-steps-in-algebra-1894/ex-9/7",4,"Wentworth 1894, Exercise 9 (7)"],["form/f1368e9445",5,"solve: Eq(8*x + 7, 4*x + 27)"],["wentworth-first-steps-in-algebra-1894/ex-9/8",4,"Wentworth 1894, Exercise 9 (8)"],["form/96906212da",5,"solve: Eq(5*x - 10, 3*x + 4)"],["wentworth-first-steps-in-algebra-1894/ex-62/4",4,"Wentworth 1894, Exercise 62 (4)"],["form/9d0db56c43",5,"solve: Eq(x, x/12 + 20)"],["wentworth-first-steps-in-algebra-1894/ex-62/5",4,"Wentworth 1894, 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(Eq(2*a + 7*x, 63), Eq(-a + 8*x, 3))"],["shape/f21f0da53d",6,"solve: (Eq(N*a + N*x, N), Eq(N*x - a, N))"],["wentworth-first-steps-in-algebra-1894/ex-65/8",4,"Wentworth 1894, Exercise 65 (8)"],["form/8b3dcbce78",5,"solve: (Eq(-4*a + 5*x, 7), Eq(3*a + 7*x, 70))"],["wentworth-first-steps-in-algebra-1894/ex-65/9",4,"Wentworth 1894, Exercise 65 (9)"],["form/7a189b1224",5,"solve: (Eq(21*a + x, 2), Eq(27*a + 2*x, 19))"],["wentworth-first-steps-in-algebra-1894/ex-65/10",4,"Wentworth 1894, Exercise 65 (10)"],["form/53fd25f148",5,"solve: (Eq(-12*a + 5*x, -2), Eq(-13*a + 6*x, -1))"],["shape/063d2b4d9a",6,"solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, -1))"],["wentworth-first-steps-in-algebra-1894/ex-65/11",4,"Wentworth 1894, Exercise 65 (11)"],["form/ed776842b9",5,"solve: (Eq(-5*a + 3*x, 5), Eq(a + 7*x, 265))"],["wentworth-first-steps-in-algebra-1894/ex-65/12",4,"Wentworth 1894, Exercise 65 (12)"],["wentworth-first-steps-in-algebra-1894/ex-65/13",4,"Wentworth 1894, Exercise 65 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N))"],["wentworth-first-steps-in-algebra-1894/ex-65/25",4,"Wentworth 1894, Exercise 65 (25)"],["form/f3349c37c9",5,"solve: (Eq(-5*a/12 - 5*x/12, 5*a/4), Eq(-13*a/12 - x/12, 3/2))"],["shape/9554891118",6,"solve: (Eq(N*a + N*x, N*a), Eq(N*a + N*x, N))"],["wentworth-first-steps-in-algebra-1894/ex-65/26",4,"Wentworth 1894, Exercise 65 (26)"],["wentworth-first-steps-in-algebra-1894/ex-66/1",4,"Wentworth 1894, Exercise 66 (1)"],["wentworth-first-steps-in-algebra-1894/ex-66/2",4,"Wentworth 1894, Exercise 66 (2)"],["wentworth-first-steps-in-algebra-1894/ex-66/3",4,"Wentworth 1894, Exercise 66 (3)"],["form/84927d9060",5,"solve: (Eq(-a + x, a/8 + 2), Eq(a + x, 36))"],["wentworth-first-steps-in-algebra-1894/ex-66/4",4,"Wentworth 1894, Exercise 66 (4)"],["form/dcd8651e36",5,"solve: (Eq(3*a + 4*x, 33), Eq(6*a + 5*x, 48))"],["wentworth-first-steps-in-algebra-1894/ex-66/5",4,"Wentworth 1894, Exercise 66 (5)"],["form/b498c57f61",5,"solve: (Eq(5*a + 4*x, 8), Eq(10*a + 7*x, 15))"],["wentworth-first-steps-in-algebra-1894/ex-66/6",4,"Wentworth 1894, Exercise 66 (6)"],["wentworth-first-steps-in-algebra-1894/ex-66/7",4,"Wentworth 1894, Exercise 66 (7)"],["form/7f7b092970",5,"solve: (Eq(7*a + 6*x, 1000), Eq(13*a + 11*x, 1844))"],["wentworth-first-steps-in-algebra-1894/ex-67/none",4,"Wentworth 1894, Exercise 67 (None)"],["wentworth-first-steps-in-algebra-1894/ex-67/1",4,"Wentworth 1894, Exercise 67 (1)"],["form/360a8269fc",5,"solve: (Eq((x + 2)/(a - 2), 1), Eq((a + x)/(a - 5), 5))"],["wentworth-first-steps-in-algebra-1894/ex-67/2",4,"Wentworth 1894, Exercise 67 (2)"],["form/1711e8b2b9",5,"solve: (Eq(x/(a + 1), 1/2), Eq((x + 2)/a, 3/5))"],["shape/f043861c08",6,"solve: (Eq(x/(a + 1), N), Eq((N + x)/a, N))"],["wentworth-first-steps-in-algebra-1894/ex-67/3",4,"Wentworth 1894, Exercise 67 (3)"],["form/665fb96fdb",5,"solve: (Eq(x/(a + 1), 1/7), Eq((x + 1)/a, 1/5))"],["shape/2ae7e11505",6,"solve: (Eq(x/(a + 1), N), Eq((x + 1)/a, N))"],["wentworth-first-steps-in-algebra-1894/ex-6/12",4,"Wentworth 1894, Exercise 6 (12)"],["wentworth-first-steps-in-algebra-1894/ex-6/13",4,"Wentworth 1894, Exercise 6 (13)"],["form/a4954b5794",5,"solve: Eq(x + 10, a)"],["shape/0a242c7afb",6,"solve: Eq(N + x, a)"],["wentworth-first-steps-in-algebra-1894/ex-67/4",4,"Wentworth 1894, Exercise 67 (4)"],["form/dbc7edf105",5,"solve: (Eq(2*x/(a - 1), 1/2), Eq((x + 1)/(2*a), 1/7))"],["shape/39e8e28189",6,"solve: (Eq(N*x/(a - 1), N), Eq(N*(x + 1)/a, N))"],["wentworth-first-steps-in-algebra-1894/ex-67/5",4,"Wentworth 1894, Exercise 67 (5)"],["wentworth-first-steps-in-algebra-1894/ex-68/1",4,"Wentworth 1894, Exercise 68 (1)"],["form/20c68dbb03",5,"solve: (Eq(x, 10*a + b), Eq(a + b, 9), Eq(10*a + b + 9, a + 10*b))"],["shape/800dbe8505",6,"solve: (Eq(x, N*a + b), Eq(a + b, N), Eq(N*a + N + b, N*b + a))"],["wentworth-first-steps-in-algebra-1894/ex-6/3a",4,"Wentworth 1894, Exercise 6 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N))"],["wentworth-first-steps-in-algebra-1894/ex-70/3",4,"Wentworth 1894, Exercise 70 (3)"],["form/c15585ec47",5,"solve: (Eq((x - 2)/(a - 1), 3/4), Eq((x + 2)/(a + 3), 4/5))"],["shape/cd077995c9",6,"solve: (Eq((N + x)/(a - 1), N), Eq((N + x)/(N + a), N))"],["wentworth-first-steps-in-algebra-1894/ex-72/15",4,"Wentworth 1894, Exercise 72 (15)"],["form/9127eace59",5,"solve: Eq(2*x**2 - 12*x, -10)"],["shape/bc7b78e92f",6,"solve: Eq(N*x + N*x**N, N)"],["shape/36eefe9a49",6,"solve: Eq(N - x + x**N, 0)"],["wentworth-first-steps-in-algebra-1894/ex-70/4",4,"Wentworth 1894, Exercise 70 (4)"],["form/7dffadc0d2",5,"solve: (Eq(30*a + 50*x, 74), Eq(50*a + 30*x, 70))"],["wentworth-first-steps-in-algebra-1894/ex-70/5",4,"Wentworth 1894, Exercise 70 (5)"],["wentworth-first-steps-in-algebra-1894/ex-71/3",4,"Wentworth 1894, Exercise 71 (3)"],["shape/b3bdcde203",6,"solve: Eq(N*x**N + N, N + x**N)"],["wentworth-first-steps-in-algebra-1894/ex-71/4",4,"Wentworth 1894, Exercise 71 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N*x**N)"],["wentworth-first-steps-in-algebra-1894/ex-73/8",4,"Wentworth 1894, Exercise 73 (8)"],["form/8b640c3c08",5,"solve: (Eq(a*x, 360), Eq(2*a + 2*x, 76))"],["shape/502f9058ef",6,"solve: (Eq(a*x, N), Eq(N*a + N*x, N))"],["wentworth-first-steps-in-algebra-1894/ex-79/8",4,"Wentworth 1894, Exercise 79 (8)"],["wentworth-first-steps-in-algebra-1894/ex-73/9",4,"Wentworth 1894, Exercise 73 (9)"],["shape/2dea1cce3e",6,"solve: (Eq(a, N + x), Eq(a*x, N))"],["wentworth-first-steps-in-algebra-1894/ex-73/10",4,"Wentworth 1894, Exercise 73 (10)"],["form/26ba6e4747",5,"solve: (Eq(a + x, 64), Eq(2*a - x**2, 8))"],["shape/ad6fe62ae8",6,"solve: (Eq(a + x, N), Eq(N*a - x**N, N))"],["wentworth-first-steps-in-algebra-1894/ex-74/1",4,"Wentworth 1894, Exercise 74 (1)"],["form/a6b3365e2a",5,"evaluate: 785"],["wentworth-first-steps-in-algebra-1894/ex-79/9",4,"Wentworth 1894, Exercise 79 (9)"],["wentworth-first-steps-in-algebra-1894/ex-79/10",4,"Wentworth 1894, Exercise 79 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1894, Exercise 77 (2)"],["wentworth-first-steps-in-algebra-1894/ex-77/3",4,"Wentworth 1894, Exercise 77 (3)"],["form/3ffd2dbe65",5,"evaluate: 3/64"],["wentworth-first-steps-in-algebra-1894/ex-77/4",4,"Wentworth 1894, Exercise 77 (4)"],["form/dc8611c737",5,"evaluate: 256"],["wentworth-first-steps-in-algebra-1894/ex-77/9",4,"Wentworth 1894, Exercise 77 (9)"],["form/92a9c8357f",5,"evaluate: 381/32"],["wentworth-first-steps-in-algebra-1894/ex-77/10",4,"Wentworth 1894, Exercise 77 (10)"],["form/63e6cdcf85",5,"evaluate: 255/16"],["wentworth-first-steps-in-algebra-1894/ex-77/11",4,"Wentworth 1894, Exercise 77 (11)"],["form/4d29b11b64",5,"evaluate: 511/4"],["wentworth-first-steps-in-algebra-1894/ex-77/12",4,"Wentworth 1894, Exercise 77 (12)"],["form/c62c758f05",5,"evaluate: 44"],["form/cc0c28ef2b",5,"evaluate: 65/54"],["wentworth-first-steps-in-algebra-1894/ex-77/14",4,"Wentworth 1894, Exercise 77 (14)"],["form/5c2469ff78",5,"evaluate: 127/100"],["wentworth-first-steps-in-algebra-1894/ex-77/15",4,"Wentworth 1894, Exercise 77 (15)"],["form/e931d13e0d",5,"evaluate: 819/10"],["wentworth-first-steps-in-algebra-1894/ex-77/16",4,"Wentworth 1894, Exercise 77 (16)"],["form/91499fa11e",5,"evaluate: 14641"],["wentworth-first-steps-in-algebra-1894/ex-78/1",4,"Wentworth 1894, Exercise 78 (1)"],["cap/other:polynomial square root",17,"other:polynomial square root"],["wentworth-first-steps-in-algebra-1894/ex-78/2",4,"Wentworth 1894, Exercise 78 (2)"],["wentworth-first-steps-in-algebra-1894/ex-78/3",4,"Wentworth 1894, Exercise 78 (3)"],["wentworth-first-steps-in-algebra-1894/ex-78/4",4,"Wentworth 1894, Exercise 78 (4)"],["wentworth-first-steps-in-algebra-1894/ex-78/5",4,"Wentworth 1894, Exercise 78 (5)"],["wentworth-first-steps-in-algebra-1894/ex-78/6",4,"Wentworth 1894, Exercise 78 (6)"],["wentworth-first-steps-in-algebra-1894/ex-79/3",4,"Wentworth 1894, Exercise 79 (3)"],["wentworth-first-steps-in-algebra-1894/ex-79/4",4,"Wentworth 1894, Exercise 79 (4)"],["wentworth-first-steps-in-algebra-1894/ex-79/5",4,"Wentworth 1894, Exercise 79 (5)"],["form/cdc70cc126",5,"evaluate: 16/5"],["wentworth-first-steps-in-algebra-1894/ex-79/11",4,"Wentworth 1894, Exercise 79 (11)"],["form/051bc4ec97",5,"evaluate: 5601"],["wentworth-first-steps-in-algebra-1894/ex-79/12",4,"Wentworth 1894, Exercise 79 (12)"],["form/017d5c10fb",5,"evaluate: 1234"],["wentworth-first-steps-in-algebra-1894/ex-79/13",4,"Wentworth 1894, Exercise 79 (13)"],["form/0a614a5fe0",5,"evaluate: sqrt(2)"],["wentworth-first-steps-in-algebra-1894/ex-79/14",4,"Wentworth 1894, Exercise 79 (14)"],["form/151758d3f2",5,"evaluate: sqrt(3)"],["wentworth-first-steps-in-algebra-1894/ex-79/15",4,"Wentworth 1894, Exercise 79 (15)"],["form/555344fae9",5,"evaluate: sqrt(5)"],["wentworth-first-steps-in-algebra-1894/ex-79/16",4,"Wentworth 1894, Exercise 79 (16)"],["wentworth-first-steps-in-algebra-1894/ex-79/17",4,"Wentworth 1894, Exercise 79 (17)"],["form/209cf3e3ab",5,"evaluate: sqrt(2)/2"],["wentworth-first-steps-in-algebra-1894/ex-79/18",4,"Wentworth 1894, Exercise 79 (18)"],["form/83858c9005",5,"evaluate: 3*sqrt(10)/10"],["wentworth-first-steps-in-algebra-1894/ex-79/19",4,"Wentworth 1894, Exercise 79 (19)"],["form/0615a5658a",5,"evaluate: sqrt(6)/3"],["shape/d58d1fd837",6,"evaluate: N*N**N"],["wentworth-first-steps-in-algebra-1894/ex-79/20",4,"Wentworth 1894, Exercise 79 (20)"],["form/3137cf1c06",5,"evaluate: sqrt(3)/2"],["wentworth-first-steps-in-algebra-1894/ex-79/21",4,"Wentworth 1894, Exercise 79 (21)"],["form/4bf72c8f22",5,"evaluate: 2*sqrt(5)/5"],["wentworth-first-steps-in-algebra-1894/ex-79/22",4,"Wentworth 1894, Exercise 79 (22)"],["wentworth-first-steps-in-algebra-1894/ex-7/1",4,"Wentworth 1894, Exercise 7 (1)"],["wentworth-first-steps-in-algebra-1894/ex-7/2",4,"Wentworth 1894, Exercise 7 (2)"],["shape/064e19a6ff",6,"solve: Eq(-b + x, a)"],["wentworth-first-steps-in-algebra-1894/ex-7/3",4,"Wentworth 1894, Exercise 7 (3)"],["form/bf8f095570",5,"solve: Eq(7*x, 28)"],["wentworth-first-steps-in-algebra-1894/ex-7/4",4,"Wentworth 1894, Exercise 7 (4)"],["wentworth-first-steps-in-algebra-1894/ex-7/5",4,"Wentworth 1894, Exercise 7 (5)"],["form/684e4b9f3a",5,"solve: Eq(a*b, x)"],["shape/684e4b9f3a",6,"solve: Eq(a*b, x)"],["wentworth-first-steps-in-algebra-1894/ex-7/6",4,"Wentworth 1894, Exercise 7 (6)"],["wentworth-first-steps-in-algebra-1894/ex-7/10",4,"Wentworth 1894, Exercise 7 (10)"],["wentworth-first-steps-in-algebra-1894/ex-7/11",4,"Wentworth 1894, Exercise 7 (11)"],["wentworth-first-steps-in-algebra-1894/ex-7/12",4,"Wentworth 1894, Exercise 7 (12)"],["wentworth-first-steps-in-algebra-1894/ex-7/13",4,"Wentworth 1894, Exercise 7 (13)"],["wentworth-first-steps-in-algebra-1894/ex-7/14",4,"Wentworth 1894, Exercise 7 (14)"],["wentworth-first-steps-in-algebra-1894/ex-80/2",4,"Wentworth 1894, Exercise 80 (2)"],["wentworth-first-steps-in-algebra-1894/ex-80/3",4,"Wentworth 1894, Exercise 80 (3)"],["wentworth-first-steps-in-algebra-1894/ex-80/4",4,"Wentworth 1894, Exercise 80 (4)"],["wentworth-first-steps-in-algebra-1894/ex-80/5",4,"Wentworth 1894, Exercise 80 (5)"],["wentworth-first-steps-in-algebra-1894/ex-80/6",4,"Wentworth 1894, Exercise 80 (6)"],["wentworth-first-steps-in-algebra-1894/ex-81/1",4,"Wentworth 1894, Exercise 81 (1)"],["form/4bc1d49582",5,"evaluate: 36"],["wentworth-first-steps-in-algebra-1894/ex-81/2",4,"Wentworth 1894, Exercise 81 (2)"],["form/b9a8f7204c",5,"evaluate: 35"],["wentworth-first-steps-in-algebra-1894/ex-81/3",4,"Wentworth 1894, Exercise 81 (3)"],["form/fa5006c44c",5,"evaluate: 45"],["wentworth-first-steps-in-algebra-1894/ex-81/4",4,"Wentworth 1894, Exercise 81 (4)"],["form/561f4aac4f",5,"evaluate: 65"],["wentworth-first-steps-in-algebra-1894/ex-81/5",4,"Wentworth 1894, Exercise 81 (5)"],["form/34c82f4cfd",5,"evaluate: 48"],["wentworth-first-steps-in-algebra-1894/ex-81/6",4,"Wentworth 1894, Exercise 81 (6)"],["form/750d193b9f",5,"evaluate: 637"],["wentworth-first-steps-in-algebra-1894/ex-81/7",4,"Wentworth 1894, Exercise 81 (7)"],["wentworth-first-steps-in-algebra-1894/ex-81/9",4,"Wentworth 1894, Exercise 81 (9)"],["wentworth-first-steps-in-algebra-1894/ex-81/10",4,"Wentworth 1894, Exercise 81 (10)"],["wentworth-first-steps-in-algebra-1894/ex-81/11",4,"Wentworth 1894, Exercise 81 (11)"],["form/0e6e187121",5,"evaluate: 617/50"],["wentworth-first-steps-in-algebra-1894/ex-81/12",4,"Wentworth 1894, Exercise 81 (12)"],["form/23b530df34",5,"evaluate: 49/4"],["wentworth-first-steps-in-algebra-1894/ex-81/13",4,"Wentworth 1894, Exercise 81 (13)"],["form/8e09680f9a",5,"evaluate: 10**(1/3)/10"],["wentworth-first-steps-in-algebra-1894/ex-81/14",4,"Wentworth 1894, Exercise 81 (14)"],["form/c5d44b46a1",5,"evaluate: 50**(1/3)/10"],["wentworth-first-steps-in-algebra-1894/ex-81/15",4,"Wentworth 1894, Exercise 81 (15)"],["wentworth-first-steps-in-algebra-1894/ex-81/16",4,"Wentworth 1894, Exercise 81 (16)"],["wentworth-first-steps-in-algebra-1894/ex-81/17",4,"Wentworth 1894, Exercise 81 (17)"],["form/c17c33babb",5,"evaluate: 10**(1/3)"],["wentworth-first-steps-in-algebra-1894/ex-81/18",4,"Wentworth 1894, Exercise 81 (18)"],["form/f492b61de8",5,"evaluate: 87**(1/3)"],["wentworth-first-steps-in-algebra-1894/ex-81/19",4,"Wentworth 1894, Exercise 81 (19)"],["form/619b69fa81",5,"evaluate: 2**(2/3)*5**(1/3)/2"],["shape/2f5c7854d0",6,"evaluate: N*N**(2*N)"],["wentworth-first-steps-in-algebra-1894/ex-81/20",4,"Wentworth 1894, Exercise 81 (20)"],["form/ad0056ba32",5,"evaluate: 2050**(1/3)/10"],["wentworth-first-steps-in-algebra-1894/ex-81/21",4,"Wentworth 1894, Exercise 81 (21)"],["wentworth-first-steps-in-algebra-1894/ex-81/22",4,"Wentworth 1894, Exercise 81 (22)"],["form/ff0b0f1113",5,"evaluate: 2**(1/3)*3**(2/3)/3"],["wentworth-first-steps-in-algebra-1894/ex-81/23",4,"Wentworth 1894, Exercise 81 (23)"],["form/58f10ddf3f",5,"evaluate: 6**(1/3)/2"],["wentworth-first-steps-in-algebra-1894/ex-81/24",4,"Wentworth 1894, Exercise 81 (24)"],["form/06b6a2183c",5,"evaluate: 33**(2/3)/11"],["wentworth-first-steps-in-algebra-1894/ex-8/2",4,"Wentworth 1894, Exercise 8 (2)"],["wentworth-first-steps-in-algebra-1894/ex-8/3",4,"Wentworth 1894, Exercise 8 (3)"],["wentworth-first-steps-in-algebra-1894/ex-8/4",4,"Wentworth 1894, Exercise 8 (4)"],["wentworth-first-steps-in-algebra-1894/ex-8/5",4,"Wentworth 1894, Exercise 8 (5)"],["wentworth-first-steps-in-algebra-1894/ex-8/6",4,"Wentworth 1894, Exercise 8 (6)"],["wentworth-first-steps-in-algebra-1894/ex-8/16",4,"Wentworth 1894, Exercise 8 (16)"],["wentworth-first-steps-in-algebra-1894/ex-8/7",4,"Wentworth 1894, Exercise 8 (7)"],["wentworth-first-steps-in-algebra-1894/ex-8/9",4,"Wentworth 1894, Exercise 8 (9)"],["wentworth-first-steps-in-algebra-1894/ex-8/10a",4,"Wentworth 1894, Exercise 8 (10a)"],["wentworth-first-steps-in-algebra-1894/ex-8/10b",4,"Wentworth 1894, Exercise 8 (10b)"],["wentworth-first-steps-in-algebra-1894/ex-8/11",4,"Wentworth 1894, Exercise 8 (11)"],["wentworth-first-steps-in-algebra-1894/ex-8/12",4,"Wentworth 1894, Exercise 8 (12)"],["wentworth-first-steps-in-algebra-1894/ex-8/13",4,"Wentworth 1894, Exercise 8 (13)"],["wentworth-first-steps-in-algebra-1894/ex-8/14",4,"Wentworth 1894, Exercise 8 (14)"],["wentworth-first-steps-in-algebra-1894/ex-8/15",4,"Wentworth 1894, Exercise 8 (15)"],["wentworth-first-steps-in-algebra-1894/ex-8/17",4,"Wentworth 1894, Exercise 8 (17)"],["wentworth-first-steps-in-algebra-1894/ex-8/18",4,"Wentworth 1894, Exercise 8 (18)"],["wentworth-first-steps-in-algebra-1894/ex-8/19",4,"Wentworth 1894, Exercise 8 (19)"],["wentworth-first-steps-in-algebra-1894/ex-8/20",4,"Wentworth 1894, Exercise 8 (20)"],["wentworth-first-steps-in-algebra-1894/ex-9/2",4,"Wentworth 1894, Exercise 9 (2)"],["wentworth-first-steps-in-algebra-1894/ex-9/3",4,"Wentworth 1894, Exercise 9 (3)"],["form/ed14270019",5,"solve: Eq(3*x + 4, x + 10)"],["wentworth-first-steps-in-algebra-1894/ex-8/21",4,"Wentworth 1894, Exercise 8 (21)"],["wentworth-first-steps-in-algebra-1894/ex-8/22",4,"Wentworth 1894, Exercise 8 (22)"],["wentworth-first-steps-in-algebra-1894/ex-8/23",4,"Wentworth 1894, Exercise 8 (23)"],["wentworth-first-steps-in-algebra-1894/ex-9/11",4,"Wentworth 1894, Exercise 9 (11)"],["form/046180a1f1",5,"solve: Eq(2*x + 3, -2*x + 19)"],["wentworth-first-steps-in-algebra-1894/ex-9/12",4,"Wentworth 1894, Exercise 9 (12)"],["form/04e166c5be",5,"solve: Eq(19*x - 3, 2*x + 14)"],["wentworth-first-steps-in-algebra-1894/ex-9/13",4,"Wentworth 1894, Exercise 9 (13)"],["form/32b3500049",5,"solve: Eq(7*x - 70, 5*x - 20)"],["wentworth-first-steps-in-algebra-1894/ex-9/14",4,"Wentworth 1894, Exercise 9 (14)"],["form/652de3fb87",5,"solve: Eq(2*x - 22, -2*x + 108)"],["wentworth-first-steps-in-algebra-1894/ex-9/16",4,"Wentworth 1894, Exercise 9 (16)"],["wentworth-first-steps-in-algebra-1894/ex-9/17",4,"Wentworth 1894, Exercise 9 (17)"],["form/abc05e907b",5,"solve: Eq(33*x - 70, 3*x + 20)"],["wentworth-first-steps-in-algebra-1894/ex-9/18",4,"Wentworth 1894, Exercise 9 (18)"],["form/8940635e6d",5,"solve: Eq(7*x + 10, 17)"],["wentworth-first-steps-in-algebra-1894/ex-9/19",4,"Wentworth 1894, Exercise 9 (19)"],["form/9e81dbdfec",5,"solve: Eq(7*x - 2, 47)"],["wentworth-first-steps-in-algebra-1894/ex-9/20",4,"Wentworth 1894, Exercise 9 (20)"],["form/42c1649aec",5,"solve: Eq(3*x - 6, -2*x + 59)"],["form/ec23fd6dd5",5,"solve: Eq(x - 8, 4)"],["wentworth-first-steps-in-algebra-1894/ex-9/22",4,"Wentworth 1894, Exercise 9 (22)"],["form/05c8e31dba",5,"solve: Eq(2*x - 11, x + 3)"],["wentworth-first-steps-in-algebra-1894/ex-9/23",4,"Wentworth 1894, Exercise 9 (23)"],["form/846088f18e",5,"solve: Eq(6*x - 16, 3*x + 5)"],["wentworth-first-steps-in-algebra-1894/ex-9/24",4,"Wentworth 1894, Exercise 9 (24)"],["form/86b5ac33b8",5,"solve: Eq(3*x + 7, 2*x + 12)"],["wentworth-first-steps-in-algebra-1894/ex-9/25",4,"Wentworth 1894, Exercise 9 (25)"],["form/37e7d5f6b0",5,"solve: Eq(2*x - 4, x - 4)"],["wentworth-first-steps-in-algebra-1894/ex-9/26",4,"Wentworth 1894, Exercise 9 (26)"],["form/f62eae54d0",5,"solve: Eq(3*x + 3, -x + 63)"],["form/3092d980fd",5,"solve: Eq(x - 7, -x + 14)"],["shape/2210930e95",6,"solve: Eq(N + x, N - x)"],["wentworth-first-steps-in-algebra-1894/ex-9/28",4,"Wentworth 1894, Exercise 9 (28)"],["form/30f00e3f19",5,"solve: Eq(x**2 - 2*x - 3, x**2 - 3*x + 1)"],["shape/28dbba8561",6,"solve: Eq(N*x + N + x**N, N*x + x**N + 1)"],["wentworth-first-steps-in-algebra-1894/ex-9/29",4,"Wentworth 1894, Exercise 9 (29)"],["form/9c50955b0a",5,"solve: Eq(x + 7, 10)"],["wentworth-first-steps-in-algebra-1894/ex-9/30",4,"Wentworth 1894, Exercise 9 (30)"],["form/4afd37a501",5,"solve: Eq(9*x + 2, 5*x + 18)"],["wentworth-first-steps-in-algebra-1894/ex-9/31",4,"Wentworth 1894, Exercise 9 (31)"],["form/a55da11b68",5,"solve: Eq(2*x**2 + 3*x - 5, 2*x**2 - 7*x - 1)"],["shape/8dc02af54b",6,"solve: Eq(N*x + N*x**N + N, N*x + N*x**N - 1)"],["wentworth-first-steps-in-algebra-1894/ex-9/32",4,"Wentworth 1894, Exercise 9 (32)"],["wentworth-first-steps-in-algebra-1894/ex-9/33",4,"Wentworth 1894, Exercise 9 (33)"],["form/ca74f2df54",5,"solve: Eq(15*x - 9, 3*x + 99)"],["wentworth-first-steps-in-algebra-1894/ex-9/34",4,"Wentworth 1894, Exercise 9 (34)"],["form/d90485d6f3",5,"solve: Eq(3*x + 1, 2*x + 8)"],["shape/2b84e2632b",6,"solve: Eq(N*x + 1, N*x + N)"],["wentworth-first-steps-in-algebra-1894/ex-9/35",4,"Wentworth 1894, Exercise 9 (35)"],["form/3cab3c90c2",5,"solve: Eq(12*x - 60, -3*x + 15)"],["wentworth-first-steps-in-algebra-1894/ex-9/36",4,"Wentworth 1894, Exercise 9 (36)"],["form/eaa0d375ad",5,"solve: Eq(2*x + 18, 30)"],["wentworth-first-steps-in-algebra-1894/ex-9/37",4,"Wentworth 1894, Exercise 9 (37)"],["form/87cf4e424e",5,"solve: Eq(2*x + 6, -6*x + 54)"],["dickson-theory-of-equations-1922/ex-page100/3",4,"Dickson 1922, Exercise Page100 (3)"],["form/0b2c51290b",5,"solve: Eq(x, tan(x/2))"],["shape/078b716886",6,"solve: Eq(x, tan(N*x))"],["dickson-theory-of-equations-1922/ex-page100/4",4,"Dickson 1922, Exercise Page100 (4)"],["form/11091e0ffd",5,"solve: Eq(sin(x), x/2)"],["shape/9efb2285df",6,"solve: Eq(sin(x), N*x)"],["dickson-theory-of-equations-1922/ex-page100/5",4,"Dickson 1922, Exercise Page100 (5)"],["form/219d801a4b",5,"solve: Eq(x**3, 15*x**3/2 - 15*(x - 1)**3/2)"],["shape/4a523edb16",6,"solve: Eq(x**N, N*x**N + N*(x - 1)**N)"],["dickson-theory-of-equations-1922/ex-page100/6",4,"Dickson 1922, Exercise Page100 (6)"],["dickson-theory-of-equations-1922/ex-page100/13",4,"Dickson 1922, Exercise Page100 (13)"],["dickson-theory-of-equations-1922/ex-page100/7",4,"Dickson 1922, Exercise Page100 (7)"],["form/f8a7f321d6",5,"solve: Eq(x*tan(x), 1)"],["shape/f8a7f321d6",6,"solve: Eq(x*tan(x), 1)"],["dickson-theory-of-equations-1922/ex-page100/8",4,"Dickson 1922, Exercise Page100 (8)"],["form/ac5c4e1896",5,"solve: Eq(tan(x), x)"],["shape/ac5c4e1896",6,"solve: Eq(tan(x), x)"],["dickson-theory-of-equations-1922/ex-page100/9",4,"Dickson 1922, Exercise Page100 (9)"],["form/29ce7794e9",5,"solve: Eq(tan(x), 2*x/(-x**2 + 2))"],["shape/ee38be4baf",6,"solve: Eq(tan(x), N*x/(N - x**N))"],["dickson-theory-of-equations-1922/ex-page100/10i",4,"Dickson 1922, Exercise Page100 (10i)"],["dickson-theory-of-equations-1922/ex-page100/14",4,"Dickson 1922, Exercise Page100 (14)"],["form/f87f2e363b",5,"solve: Eq(x - 253*pi*sin(x)/3240, 2992337*pi/1620000)"],["shape/a6b0eff7f0",6,"solve: Eq(pi*N*sin(x) + x, pi*N)"],["dickson-theory-of-equations-1922/ex-page100/15",4,"Dickson 1922, Exercise Page100 (15)"],["form/c8705609d5",5,"solve: Eq(524662411096539147*x/4882812500000000000 + 174887470365513049/244140625000000000, 2*x/25 + 1)"],["shape/5968cc4007",6,"solve: Eq(N*x + N, N*x + 1)"],["dickson-theory-of-equations-1922/ex-page100/16",4,"Dickson 1922, Exercise Page100 (16)"],["form/a0680e8a82",5,"solve: Eq(x**3 - 29*x - 48, 0)"],["dickson-theory-of-equations-1922/ex-page100/17",4,"Dickson 1922, Exercise Page100 (17)"],["form/5cfba3878a",5,"solve: Eq((3000 - 3000/(x + 1)**3)/x, 8000)"],["shape/c13438214a",6,"solve: Eq((N*(x + 1)**N + N)/x, N)"],["dickson-theory-of-equations-1922/ex-page102/1",4,"Dickson 1922, Exercise Page102 (1)"],["form/c803f852a8",5,"solve: (Eq(8*a + x, 53), Eq(-a + 8*x, 34))"],["shape/e83a8f3b28",6,"solve: (Eq(N*a + x, N), Eq(N*x - a, N))"],["dickson-theory-of-equations-1922/ex-page102/2",4,"Dickson 1922, Exercise Page102 (2)"],["form/2a71b83ad3",5,"solve: (Eq(4*a + 3*x, 10), Eq(a + 4*x, 9))"],["dickson-theory-of-equations-1922/ex-page102/3",4,"Dickson 1922, Exercise Page102 (3)"],["form/dbe041fab5",5,"solve: (Eq(a*x + b*c, a**2), Eq(-a*c + b*x, a*b))"],["shape/432224af5a",6,"solve: (Eq(a*x + b*c, a**N), Eq(-a*c + b*x, a*b))"],["dickson-theory-of-equations-1922/ex-page106/1",4,"Dickson 1922, Exercise Page106 (1)"],["dickson-theory-of-equations-1922/ex-page106/2",4,"Dickson 1922, Exercise Page106 (2)"],["dickson-theory-of-equations-1922/ex-page106/3",4,"Dickson 1922, Exercise Page106 (3)"],["dickson-theory-of-equations-1922/ex-page106/4",4,"Dickson 1922, Exercise Page106 (4)"],["dickson-theory-of-equations-1922/ex-page10/1",4,"Dickson 1922, Exercise Page10 (1)"],["dickson-theory-of-equations-1922/ex-page10/2",4,"Dickson 1922, Exercise Page10 (2)"],["dickson-theory-of-equations-1922/ex-page10/3",4,"Dickson 1922, Exercise Page10 (3)"],["dickson-theory-of-equations-1922/ex-page10/4",4,"Dickson 1922, Exercise Page10 (4)"],["dickson-theory-of-equations-1922/ex-page10/5",4,"Dickson 1922, Exercise Page10 (5)"],["dickson-theory-of-equations-1922/ex-page10/6",4,"Dickson 1922, Exercise Page10 (6)"],["dickson-theory-of-equations-1922/ex-page10/7",4,"Dickson 1922, Exercise Page10 (7)"],["dickson-theory-of-equations-1922/ex-page10/8",4,"Dickson 1922, Exercise Page10 (8)"],["dickson-theory-of-equations-1922/ex-page10/9",4,"Dickson 1922, Exercise Page10 (9)"],["dickson-theory-of-equations-1922/ex-page112/1",4,"Dickson 1922, Exercise Page112 (1)"],["dickson-theory-of-equations-1922/ex-page112/2",4,"Dickson 1922, Exercise Page112 (2)"],["dickson-theory-of-equations-1922/ex-page112/3",4,"Dickson 1922, Exercise Page112 (3)"],["form/c81265656a",5,"evaluate: -3"],["dickson-theory-of-equations-1922/ex-page112/4",4,"Dickson 1922, Exercise Page112 (4)"],["form/3f28409398",5,"evaluate: -8"],["dickson-theory-of-equations-1922/ex-page112/5",4,"Dickson 1922, Exercise Page112 (5)"],["cap/core.matrix",17,"core.matrix"],["dickson-theory-of-equations-1922/ex-page115/2",4,"Dickson 1922, Exercise Page115 (2)"],["form/b6a1c24ad6",5,"solve: (Eq(a + b + c, 0), Eq(a + 2*b + 3*c, -1), Eq(a + 3*b + 6*c, 0))"],["shape/c9fa00212e",6,"solve: (Eq(a + b + c, 0), Eq(N*b + N*c + a, -1), Eq(N*b + N*c + a, 0))"],["dickson-theory-of-equations-1922/ex-page115/3",4,"Dickson 1922, Exercise Page115 (3)"],["form/17e40dccc4",5,"solve: (Eq(a - 2*b + c, 12), Eq(a + 2*b + 3*c, 48), Eq(6*a + 4*b + 3*c, 84))"],["shape/44efb1e15e",6,"solve: (Eq(N*b + a + c, N), Eq(N*b + N*c + a, N), Eq(N*a + N*b + N*c, N))"],["dickson-theory-of-equations-1922/ex-page115/4",4,"Dickson 1922, Exercise Page115 (4)"],["form/29edd8990d",5,"solve: (Eq(-2*a + 3*c, -1), Eq(3*a - 2*b, 7), Eq(3*b - 2*c, 6))"],["shape/3163bead1d",6,"solve: (Eq(N*a + N*c, -1), Eq(N*a + N*b, N), Eq(N*b + N*c, N))"],["dickson-theory-of-equations-1922/ex-page115/5",4,"Dickson 1922, Exercise Page115 (5)"],["dickson-theory-of-equations-1922/ex-page115/6",4,"Dickson 1922, Exercise Page115 (6)"],["form/1965c9f42e",5,"solve: (Eq(-2*a + 2*b - c + 3*d, 4), Eq(-a + b + 7*c + d, 2), Eq(-a + 4*b - 3*c + 2*d, 5), Eq(3*a + 3*b + 5*c - 5*d, 0))"],["shape/242c8ba885",6,"solve: (Eq(N*a + N*b + N*d - c, N), Eq(N*c - a + b + d, N), Eq(N*b + N*c + N*d - a, N), Eq(N*a + N*b + N*c + N*d, 0))"],["dickson-theory-of-equations-1922/ex-page115/7",4,"Dickson 1922, Exercise Page115 (7)"],["dickson-theory-of-equations-1922/ex-page119/1",4,"Dickson 1922, Exercise Page119 (1)"],["dickson-theory-of-equations-1922/ex-page119/2",4,"Dickson 1922, Exercise Page119 (2)"],["dickson-theory-of-equations-1922/ex-page119/3",4,"Dickson 1922, Exercise Page119 (3)"],["dickson-theory-of-equations-1922/ex-page119/4",4,"Dickson 1922, Exercise Page119 (4)"],["dickson-theory-of-equations-1922/ex-page119/5a",4,"Dickson 1922, Exercise Page119 (5a)"],["dickson-theory-of-equations-1922/ex-page119/5b",4,"Dickson 1922, Exercise Page119 (5b)"],["dickson-theory-of-equations-1922/ex-page119/5c",4,"Dickson 1922, Exercise Page119 (5c)"],["form/4f0c6d90a8",5,"solve: (Eq(a*x + b + c, a - 3), Eq(a*b + c + x, -2), Eq(a*c + b + x, -2))"],["shape/4efdc9bee8",6,"solve: (Eq(a*x + b + c, N + a), Eq(a*b + c + x, N), Eq(a*c + b + x, N))"],["dickson-theory-of-equations-1922/ex-page119/6a",4,"Dickson 1922, Exercise Page119 (6a)"],["form/0a37fba9e6",5,"solve: (Eq(e + f + x, 1), Eq(a*x + b*e + c*f, d), Eq(a**2*x + b**2*e + c**2*f, d**2))"],["shape/6d8afc8223",6,"solve: (Eq(e + f + x, 1), Eq(a*x + b*e + c*f, d), Eq(a**N*x + b**N*e + c**N*f, d**N))"],["dickson-theory-of-equations-1922/ex-page119/6b",4,"Dickson 1922, Exercise Page119 (6b)"],["dickson-theory-of-equations-1922/ex-page120/4",4,"Dickson 1922, Exercise Page120 (4)"],["dickson-theory-of-equations-1922/ex-page119/6c",4,"Dickson 1922, Exercise Page119 (6c)"],["dickson-theory-of-equations-1922/ex-page120/5",4,"Dickson 1922, Exercise Page120 (5)"],["dickson-theory-of-equations-1922/ex-page121/1",4,"Dickson 1922, Exercise Page121 (1)"],["dickson-theory-of-equations-1922/ex-page121/2",4,"Dickson 1922, Exercise Page121 (2)"],["form/4bff85fc57",5,"solve: (Eq(7*b + c + x, 1), Eq(-b + 3*c + 2*x, 2), Eq(5*b - 5*c + 3*x, a), Eq(-3*b + 2*c + 4*x, 1))"],["shape/7c99f795ff",6,"solve: (Eq(N*b + c + x, 1), Eq(N*c + N*x - b, N), Eq(N*b + N*c + N*x, a), Eq(N*b + N*c + N*x, 1))"],["dickson-theory-of-equations-1922/ex-page121/3",4,"Dickson 1922, Exercise Page121 (3)"],["dickson-theory-of-equations-1922/ex-page121/4",4,"Dickson 1922, Exercise Page121 (4)"],["form/911535098f",5,"solve: (Eq(3*a + 2*x, 12), Eq(-5*a + 4*x, 2), Eq(-7*a + 10*x, 16))"],["shape/fcc61495c6",6,"solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N), Eq(N*a + N*x, N))"],["dickson-theory-of-equations-1922/ex-page121/5",4,"Dickson 1922, Exercise Page121 (5)"],["dickson-theory-of-equations-1922/ex-page121/6",4,"Dickson 1922, Exercise Page121 (6)"],["cap/other:rank",17,"other:rank"],["dickson-theory-of-equations-1922/ex-page126/1",4,"Dickson 1922, Exercise Page126 (1)"],["form/11e111ad0f",5,"solve: (Eq(a*x + b*e + c*f, d), Eq(a**2*x + b**2*e + c**2*f, d**2), Eq(a**4*x + b**4*e + c**4*f, d**4))"],["shape/204f689936",6,"solve: (Eq(a*x + b*e + c*f, d), Eq(a**N*x + b**N*e + c**N*f, d**N), Eq(a**N*x + b**N*e + c**N*f, d**N))"],["dickson-theory-of-equations-1922/ex-page126/2",4,"Dickson 1922, Exercise Page126 (2)"],["dickson-theory-of-equations-1922/ex-page126/3",4,"Dickson 1922, Exercise Page126 (3)"],["form/1e8bff0dec",5,"factor: -a**2*b + a**2*c + a*b**2 - a*c**2 - b**2*c + b*c**2"],["shape/8a1a852200",6,"factor: a*b**N - a*c**N - a**N*b + a**N*c + b*c**N - b**N*c"],["dickson-theory-of-equations-1922/ex-page126/4",4,"Dickson 1922, Exercise Page126 (4)"],["cap/other:proof",17,"other:proof"],["dickson-theory-of-equations-1922/ex-page126/5",4,"Dickson 1922, Exercise Page126 (5)"],["form/72b690d169",5,"factor: a**3 - 3*a*b*c + b**3 + c**3"],["shape/ef3d299bc2",6,"factor: N*a*b*c + a**N + b**N + c**N"],["dickson-theory-of-equations-1922/ex-page126/6",4,"Dickson 1922, Exercise Page126 (6)"],["form/97d4f61df3",5,"factor: a**4 - 2*a**2*b**2 - 2*a**2*c**2 - 2*a**2*d**2 + 8*a*b*c*d + b**4 - 2*b**2*c**2 - 2*b**2*d**2 + c**4 - 2*c**2*d**2 + d**4"],["shape/cb6aa4658e",6,"factor: N*a*b*c*d + N*a**N*b**N + N*a**N*c**N + N*a**N*d**N + N*b**N*c**N + N*b**N*d**N + N*c**N*d**N + a**N + b**N + c**N + d**N"],["dickson-theory-of-equations-1922/ex-page126/7",4,"Dickson 1922, Exercise Page126 (7)"],["form/fa2481bb76",5,"factor: a**4 - 4*a**2*b*d - 2*a**2*c**2 + 4*a*b**2*c + 4*a*c*d**2 - b**4 + 2*b**2*d**2 - 4*b*c**2*d + c**4 - d**4"],["shape/ff72bb8ba3",6,"factor: N*a*b**N*c + N*a*c*d**N + N*a**N*b*d + N*a**N*c**N + N*b*c**N*d + N*b**N*d**N + a**N - b**N + c**N - d**N"],["dickson-theory-of-equations-1922/ex-page126/8",4,"Dickson 1922, Exercise Page126 (8)"],["dickson-theory-of-equations-1922/ex-page126/9",4,"Dickson 1922, Exercise Page126 (9)"],["dickson-theory-of-equations-1922/ex-page126/10",4,"Dickson 1922, Exercise Page126 (10)"],["dickson-theory-of-equations-1922/ex-page126/11",4,"Dickson 1922, Exercise Page126 (11)"],["dickson-theory-of-equations-1922/ex-page126/12",4,"Dickson 1922, Exercise Page126 (12)"],["form/847b6aaee2",5,"solve: Eq(a*b*c + a*b*x + a*c*x + b*c*x, 0)"],["shape/847b6aaee2",6,"solve: Eq(a*b*c + a*b*x + a*c*x + b*c*x, 0)"],["dickson-theory-of-equations-1922/ex-page133/1",4,"Dickson 1922, Exercise Page133 (1)"],["dickson-theory-of-equations-1922/ex-page133/2",4,"Dickson 1922, Exercise Page133 (2)"],["dickson-theory-of-equations-1922/ex-page133/3",4,"Dickson 1922, Exercise Page133 (3)"],["dickson-theory-of-equations-1922/ex-page133/4",4,"Dickson 1922, Exercise Page133 (4)"],["dickson-theory-of-equations-1922/ex-page133/5",4,"Dickson 1922, Exercise Page133 (5)"],["dickson-theory-of-equations-1922/ex-page133/6",4,"Dickson 1922, Exercise Page133 (6)"],["dickson-theory-of-equations-1922/ex-page136/1",4,"Dickson 1922, Exercise Page136 (1)"],["dickson-theory-of-equations-1922/ex-page133/7",4,"Dickson 1922, Exercise Page133 (7)"],["dickson-theory-of-equations-1922/ex-page133/8",4,"Dickson 1922, Exercise Page133 (8)"],["dickson-theory-of-equations-1922/ex-page133/9",4,"Dickson 1922, Exercise Page133 (9)"],["dickson-theory-of-equations-1922/ex-page133/10",4,"Dickson 1922, Exercise Page133 (10)"],["dickson-theory-of-equations-1922/ex-page133/11",4,"Dickson 1922, Exercise Page133 (11)"],["dickson-theory-of-equations-1922/ex-page136/2",4,"Dickson 1922, Exercise Page136 (2)"],["dickson-theory-of-equations-1922/ex-page136/3a",4,"Dickson 1922, Exercise Page136 (3a)"],["dickson-theory-of-equations-1922/ex-page133/12",4,"Dickson 1922, Exercise Page133 (12)"],["dickson-theory-of-equations-1922/ex-page133/13",4,"Dickson 1922, Exercise Page133 (13)"],["dickson-theory-of-equations-1922/ex-page133/14",4,"Dickson 1922, Exercise Page133 (14)"],["dickson-theory-of-equations-1922/ex-page133/15",4,"Dickson 1922, Exercise Page133 (15)"],["dickson-theory-of-equations-1922/ex-page133/16",4,"Dickson 1922, Exercise Page133 (16)"],["dickson-theory-of-equations-1922/ex-page136/3b",4,"Dickson 1922, Exercise Page136 (3b)"],["dickson-theory-of-equations-1922/ex-page141/1",4,"Dickson 1922, Exercise Page141 (1)"],["dickson-theory-of-equations-1922/ex-page133/17",4,"Dickson 1922, Exercise Page133 (17)"],["dickson-theory-of-equations-1922/ex-page136/4",4,"Dickson 1922, Exercise Page136 (4)"],["dickson-theory-of-equations-1922/ex-page136/5a",4,"Dickson 1922, Exercise Page136 (5a)"],["dickson-theory-of-equations-1922/ex-page136/5b",4,"Dickson 1922, Exercise Page136 (5b)"],["dickson-theory-of-equations-1922/ex-page136/5c",4,"Dickson 1922, Exercise Page136 (5c)"],["dickson-theory-of-equations-1922/ex-page136/5d",4,"Dickson 1922, Exercise Page136 (5d)"],["dickson-theory-of-equations-1922/ex-page13/2",4,"Dickson 1922, Exercise Page13 (2)"],["form/a3b371a50b",5,"evaluate: x**3 - 3*x**2 + 6*x - 5 at x=3"],["shape/d6a81e74b1",6,"evaluate: N*x + N*x**N + N + x**N"],["dickson-theory-of-equations-1922/ex-page13/3",4,"Dickson 1922, Exercise Page13 (3)"],["dickson-theory-of-equations-1922/ex-page13/4",4,"Dickson 1922, Exercise Page13 (4)"],["dickson-theory-of-equations-1922/ex-page13/5",4,"Dickson 1922, Exercise Page13 (5)"],["dickson-theory-of-equations-1922/ex-page13/6",4,"Dickson 1922, Exercise Page13 (6)"],["dickson-theory-of-equations-1922/ex-page13/7",4,"Dickson 1922, Exercise Page13 (7)"],["dickson-theory-of-equations-1922/ex-page13/8",4,"Dickson 1922, Exercise Page13 (8)"],["cap/other:symmetric_function_reduction",17,"other:symmetric_function_reduction"],["dickson-theory-of-equations-1922/ex-page141/2",4,"Dickson 1922, Exercise Page141 (2)"],["dickson-theory-of-equations-1922/ex-page13/9",4,"Dickson 1922, Exercise Page13 (9)"],["dickson-theory-of-equations-1922/ex-page13/10",4,"Dickson 1922, Exercise Page13 (10)"],["dickson-theory-of-equations-1922/ex-page13/11",4,"Dickson 1922, Exercise Page13 (11)"],["dickson-theory-of-equations-1922/ex-page140/1",4,"Dickson 1922, Exercise Page140 (1)"],["dickson-theory-of-equations-1922/ex-page140/2",4,"Dickson 1922, Exercise Page140 (2)"],["dickson-theory-of-equations-1922/ex-page140/3",4,"Dickson 1922, Exercise Page140 (3)"],["form/0bbb9a7129",5,"solve: Eq(5*b**2*x - 5*b*x**3 + x**5, a)"],["shape/f16feb96ab",6,"solve: Eq(N*b*x**N + N*b**N*x + x**N, a)"],["dickson-theory-of-equations-1922/ex-page140/4",4,"Dickson 1922, Exercise Page140 (4)"],["dickson-theory-of-equations-1922/ex-page141/5",4,"Dickson 1922, Exercise Page141 (5)"],["cap/other:symmetric_function_identity_proof",17,"other:symmetric_function_identity_proof"],["dickson-theory-of-equations-1922/ex-page141/6",4,"Dickson 1922, Exercise Page141 (6)"],["dickson-theory-of-equations-1922/ex-page141/7",4,"Dickson 1922, Exercise Page141 (7)"],["dickson-theory-of-equations-1922/ex-page142/2",4,"Dickson 1922, Exercise Page142 (2)"],["dickson-theory-of-equations-1922/ex-page142/3",4,"Dickson 1922, Exercise Page142 (3)"],["dickson-theory-of-equations-1922/ex-page142/4",4,"Dickson 1922, Exercise Page142 (4)"],["dickson-theory-of-equations-1922/ex-page142/5",4,"Dickson 1922, Exercise Page142 (5)"],["dickson-theory-of-equations-1922/ex-page142/6",4,"Dickson 1922, Exercise Page142 (6)"],["dickson-theory-of-equations-1922/ex-page142/7",4,"Dickson 1922, Exercise Page142 (7)"],["dickson-theory-of-equations-1922/ex-page142/8",4,"Dickson 1922, Exercise Page142 (8)"],["dickson-theory-of-equations-1922/ex-page142/9",4,"Dickson 1922, Exercise Page142 (9)"],["dickson-theory-of-equations-1922/ex-page142/10",4,"Dickson 1922, Exercise Page142 (10)"],["cap/other:newton_identities",17,"other:newton_identities"],["dickson-theory-of-equations-1922/ex-page142/11",4,"Dickson 1922, Exercise Page142 (11)"],["dickson-theory-of-equations-1922/ex-page142/12i",4,"Dickson 1922, Exercise Page142 (12i)"],["dickson-theory-of-equations-1922/ex-page142/12ii",4,"Dickson 1922, Exercise Page142 (12ii)"],["dickson-theory-of-equations-1922/ex-page153/1",4,"Dickson 1922, Exercise Page153 (1)"],["dickson-theory-of-equations-1922/ex-page142/13i",4,"Dickson 1922, Exercise Page142 (13i)"],["cap/cas.matrix",17,"cas.matrix"],["dickson-theory-of-equations-1922/ex-page142/13ii",4,"Dickson 1922, Exercise Page142 (13ii)"],["dickson-theory-of-equations-1922/ex-page142/14",4,"Dickson 1922, Exercise Page142 (14)"],["dickson-theory-of-equations-1922/ex-page152/1",4,"Dickson 1922, Exercise Page152 (1)"],["form/efb395fbde",5,"solve: (Eq(a*x, 5*a), Eq(-a**2 + x**2, 9))"],["shape/f180fa0a9f",6,"solve: (Eq(a*x, N*a), Eq(-a**N + x**N, N))"],["dickson-theory-of-equations-1922/ex-page152/2",4,"Dickson 1922, Exercise Page152 (2)"],["form/702b0c75c7",5,"solve: (Eq(b**2 + x**2, 25), Eq(a*(b**2 - 25) + x**2 + x*(3*a - 3), 0))"],["shape/81fc83c90d",6,"solve: (Eq(b**N + x**N, N), Eq(a*(N + b**N) + x*(N*a + N) + x**N, 0))"],["dickson-theory-of-equations-1922/ex-page152/3",4,"Dickson 1922, Exercise Page152 (3)"],["dickson-theory-of-equations-1922/ex-page152/4",4,"Dickson 1922, Exercise Page152 (4)"],["form/0a7a782c26",5,"solve: Eq(x**6 + 3*x**4 + 32*x**3 + 67*x**2 + 32*x + 65, 0)"],["cap/cas.solve.complex",17,"cas.solve.complex"],["shape/60186cbfe5",6,"solve: Eq(N*x + 3*N*x**N + N + x**N, 0)"],["dickson-theory-of-equations-1922/ex-page153/2",4,"Dickson 1922, Exercise Page153 (2)"],["dickson-theory-of-equations-1922/ex-page153/3",4,"Dickson 1922, Exercise Page153 (3)"],["form/c9be8ef62e",5,"solve: Eq(x**4 - 6*x**3 + 13*x**2 - 14*x + 6, 0)"],["dickson-theory-of-equations-1922/ex-page153/4",4,"Dickson 1922, Exercise Page153 (4)"],["form/3d4c902902",5,"solve: Eq(a*x + b + x**3, 0)"],["shape/5eb048deb2",6,"solve: Eq(a*x + b + x**N, 0)"],["dickson-theory-of-equations-1922/ex-page153/5",4,"Dickson 1922, Exercise Page153 (5)"],["shape/f326be78c9",6,"solve: Eq(N*x + 2*N*x**N + N + x**N, 0)"],["dickson-theory-of-equations-1922/ex-page153/6",4,"Dickson 1922, Exercise Page153 (6)"],["dickson-theory-of-equations-1922/ex-page153/7",4,"Dickson 1922, Exercise Page153 (7)"],["dickson-theory-of-equations-1922/ex-page153/8",4,"Dickson 1922, Exercise Page153 (8)"],["dickson-theory-of-equations-1922/ex-page153/9",4,"Dickson 1922, Exercise Page153 (9)"],["dickson-theory-of-equations-1922/ex-page15/1",4,"Dickson 1922, Exercise Page15 (1)"],["dickson-theory-of-equations-1922/ex-page15/2",4,"Dickson 1922, Exercise Page15 (2)"],["dickson-theory-of-equations-1922/ex-page15/3",4,"Dickson 1922, Exercise Page15 (3)"],["dickson-theory-of-equations-1922/ex-page153/10",4,"Dickson 1922, Exercise Page153 (10)"],["dickson-theory-of-equations-1922/ex-page153/11",4,"Dickson 1922, Exercise Page153 (11)"],["dickson-theory-of-equations-1922/ex-page15/4",4,"Dickson 1922, Exercise Page15 (4)"],["dickson-theory-of-equations-1922/ex-page15/5",4,"Dickson 1922, Exercise Page15 (5)"],["dickson-theory-of-equations-1922/ex-page15/6",4,"Dickson 1922, Exercise Page15 (6)"],["form/b341a05f62",5,"solve: Eq(x**4 - 2*x**3 - 12*x**2 + 10*x + 3, 0)"],["dickson-theory-of-equations-1922/ex-page15/7",4,"Dickson 1922, Exercise Page15 (7)"],["form/aa8f3392db",5,"identity: (2*x**4 - x**3 - 6*x**2 + 4*x - 8)/(x**2 - 4)"],["shape/6e289c92d6",6,"identity: (N*x + 2*N*x**N + N - x**N)/(N + x**N)"],["dickson-theory-of-equations-1922/ex-page15/8",4,"Dickson 1922, Exercise Page15 (8)"],["form/e969cdc0dd",5,"identity: (x**4 - 3*x**3 + 3*x**2 - 3*x + 2)/(x**2 - 3*x + 2)"],["shape/15f7876bce",6,"identity: (N*x + 2*N*x**N + N + x**N)/(N*x + N + x**N)"],["dickson-theory-of-equations-1922/ex-page17/1",4,"Dickson 1922, Exercise Page17 (1)"],["dickson-theory-of-equations-1922/ex-page17/2",4,"Dickson 1922, Exercise Page17 (2)"],["dickson-theory-of-equations-1922/ex-page15/9",4,"Dickson 1922, Exercise Page15 (9)"],["dickson-theory-of-equations-1922/ex-page17/3",4,"Dickson 1922, Exercise Page17 (3)"],["dickson-theory-of-equations-1922/ex-page17/4",4,"Dickson 1922, Exercise Page17 (4)"],["dickson-theory-of-equations-1922/ex-page17/5",4,"Dickson 1922, Exercise Page17 (5)"],["dickson-theory-of-equations-1922/ex-page17/6",4,"Dickson 1922, Exercise Page17 (6)"],["dickson-theory-of-equations-1922/ex-page17/7",4,"Dickson 1922, Exercise Page17 (7)"],["dickson-theory-of-equations-1922/ex-page19/1",4,"Dickson 1922, Exercise Page19 (1)"],["form/7de7810eec",5,"identity: (x - 3)*(x - 2)*(x - 1)"],["shape/945e80f218",6,"identity: (N + x)**2*(x - 1)"],["dickson-theory-of-equations-1922/ex-page19/2",4,"Dickson 1922, Exercise Page19 (2)"],["form/41782ffb85",5,"identity: (x - 2)**2*(x + 2)**2"],["shape/ae34a66783",6,"identity: (N + x)**(2*N)"],["dickson-theory-of-equations-1922/ex-page19/3",4,"Dickson 1922, Exercise Page19 (3)"],["form/961f8437ab",5,"solve: Eq(x**4 - 6*x**3 + 13*x**2 - 12*x + 4, 0)"],["dickson-theory-of-equations-1922/ex-page19/4",4,"Dickson 1922, Exercise Page19 (4)"],["dickson-theory-of-equations-1922/ex-page19/5",4,"Dickson 1922, Exercise Page19 (5)"],["form/5c7cd43abc",5,"solve: Eq(4*x**3 - 16*x**2 - 9*x + 36, 0)"],["shape/ad3c31ad4a",6,"solve: Eq(N*x + 2*N*x**N + N, 0)"],["dickson-theory-of-equations-1922/ex-page19/6",4,"Dickson 1922, Exercise Page19 (6)"],["form/02481c7fbf",5,"solve: Eq(x**3 - 9*x**2 + 23*x - 15, 0)"],["dickson-theory-of-equations-1922/ex-page19/7",4,"Dickson 1922, Exercise Page19 (7)"],["form/8c9dfc3dfd",5,"solve: Eq(x**4 - 6*x**3 + 12*x**2 - 10*x + 3, 0)"],["dickson-theory-of-equations-1922/ex-page19/8",4,"Dickson 1922, Exercise Page19 (8)"],["form/d3a035605d",5,"solve: Eq(x**3 - 14*x**2 - 84*x + 216, 0)"],["dickson-theory-of-equations-1922/ex-page19/9",4,"Dickson 1922, Exercise Page19 (9)"],["form/b20fefbdf7",5,"solve: Eq(x**3 - 3*x**2 - 13*x + 15, 0)"],["dickson-theory-of-equations-1922/ex-page19/10",4,"Dickson 1922, Exercise Page19 (10)"],["shape/43d89acfdb",6,"solve: Eq(N*x + N*x**N + N + x**N, 0)"],["dickson-theory-of-equations-1922/ex-page19/11",4,"Dickson 1922, Exercise Page19 (11)"],["dickson-theory-of-equations-1922/ex-page19/12",4,"Dickson 1922, Exercise Page19 (12)"],["dickson-theory-of-equations-1922/ex-page19/13a",4,"Dickson 1922, Exercise Page19 (13a)"],["dickson-theory-of-equations-1922/ex-page19/13b",4,"Dickson 1922, Exercise Page19 (13b)"],["dickson-theory-of-equations-1922/ex-page19/13c",4,"Dickson 1922, Exercise Page19 (13c)"],["dickson-theory-of-equations-1922/ex-page23/2",4,"Dickson 1922, Exercise Page23 (2)"],["dickson-theory-of-equations-1922/ex-page19/14",4,"Dickson 1922, Exercise Page19 (14)"],["dickson-theory-of-equations-1922/ex-page19/15",4,"Dickson 1922, Exercise Page19 (15)"],["form/3ba12962d9",5,"solve: Eq(x**3 - 28*x + 48, 0)"],["dickson-theory-of-equations-1922/ex-page20/1",4,"Dickson 1922, Exercise Page20 (1)"],["form/dd4991948a",5,"solve: Eq(x**3 - 3*x**2 - 6*x - 20, 0)"],["dickson-theory-of-equations-1922/ex-page20/2",4,"Dickson 1922, Exercise Page20 (2)"],["form/09e0922b86",5,"solve: Eq(x**4 - 4*x**3 + 5*x**2 - 2*x - 2, 0)"],["dickson-theory-of-equations-1922/ex-page20/3",4,"Dickson 1922, Exercise Page20 (3)"],["dickson-theory-of-equations-1922/ex-page20/4",4,"Dickson 1922, Exercise Page20 (4)"],["dickson-theory-of-equations-1922/ex-page20/5",4,"Dickson 1922, Exercise Page20 (5)"],["dickson-theory-of-equations-1922/ex-page20/6",4,"Dickson 1922, Exercise Page20 (6)"],["dickson-theory-of-equations-1922/ex-page20/7",4,"Dickson 1922, Exercise Page20 (7)"],["form/702d5978f3",5,"solve: Eq(x**3 - x**2*(sqrt(3) + 4) + x*(5 + 4*sqrt(3)) - 5*sqrt(3), 0)"],["shape/1ccebfd432",6,"solve: Eq(N*N**N + x*(N*N**N + N) - x**N*(N + N**N) + x**N, 0)"],["dickson-theory-of-equations-1922/ex-page20/8",4,"Dickson 1922, Exercise Page20 (8)"],["dickson-theory-of-equations-1922/ex-page20/9",4,"Dickson 1922, Exercise Page20 (9)"],["dickson-theory-of-equations-1922/ex-page20/10",4,"Dickson 1922, Exercise Page20 (10)"],["form/0b0a5e56ac",5,"solve: Eq(x**4 - 2*x**3 - 5*x**2 - 6*x + 2, 0)"],["dickson-theory-of-equations-1922/ex-page20/11",4,"Dickson 1922, Exercise Page20 (11)"],["dickson-theory-of-equations-1922/ex-page20/12",4,"Dickson 1922, Exercise Page20 (12)"],["dickson-theory-of-equations-1922/ex-page20/13",4,"Dickson 1922, Exercise Page20 (13)"],["dickson-theory-of-equations-1922/ex-page23/3",4,"Dickson 1922, Exercise Page23 (3)"],["dickson-theory-of-equations-1922/ex-page23/4",4,"Dickson 1922, Exercise Page23 (4)"],["dickson-theory-of-equations-1922/ex-page23/5",4,"Dickson 1922, Exercise Page23 (5)"],["dickson-theory-of-equations-1922/ex-page23/6",4,"Dickson 1922, Exercise Page23 (6)"],["dickson-theory-of-equations-1922/ex-page23/7",4,"Dickson 1922, Exercise Page23 (7)"],["dickson-theory-of-equations-1922/ex-page25/1",4,"Dickson 1922, Exercise Page25 (1)"],["form/89cd2a6f14",5,"solve: Eq(x**3 + 8*x**2 + 13*x + 6, 0)"],["dickson-theory-of-equations-1922/ex-page25/2",4,"Dickson 1922, Exercise Page25 (2)"],["form/8cc7e24b13",5,"solve: Eq(x**3 - 5*x**2 - 2*x + 24, 0)"],["dickson-theory-of-equations-1922/ex-page25/3",4,"Dickson 1922, Exercise Page25 (3)"],["form/9e84dca217",5,"solve: Eq(x**3 - 10*x**2 + 27*x - 18, 0)"],["dickson-theory-of-equations-1922/ex-page25/4",4,"Dickson 1922, Exercise Page25 (4)"],["form/75402f2748",5,"solve: Eq(x**4 + 4*x**3 + 8*x + 32, 0)"],["dickson-theory-of-equations-1922/ex-page25/5",4,"Dickson 1922, Exercise Page25 (5)"],["dickson-theory-of-equations-1922/ex-page27/1",4,"Dickson 1922, Exercise Page27 (1)"],["dickson-theory-of-equations-1922/ex-page27/2",4,"Dickson 1922, Exercise Page27 (2)"],["form/88e09114be",5,"solve: Eq(x**3 - 9*x**2 - 24*x + 216, 0)"],["dickson-theory-of-equations-1922/ex-page27/3",4,"Dickson 1922, Exercise Page27 (3)"],["form/c745e3691e",5,"solve: Eq(x**4 - 23*x**3 + 187*x**2 - 653*x + 936, 0)"],["dickson-theory-of-equations-1922/ex-page27/4",4,"Dickson 1922, Exercise Page27 (4)"],["form/db11f4253a",5,"solve: Eq(x**5 + 47*x**4 + 423*x**3 + 140*x**2 + 1213*x - 420, 0)"],["dickson-theory-of-equations-1922/ex-page27/5",4,"Dickson 1922, Exercise Page27 (5)"],["form/c7cf5baa5e",5,"solve: Eq(x**5 - 34*x**3 + 29*x**2 + 212*x - 300, 0)"],["dickson-theory-of-equations-1922/ex-page28/1",4,"Dickson 1922, Exercise Page28 (1)"],["form/46e1b2b7e1",5,"solve: Eq(x**4 - 40*x**3/3 + 130*x**2/3 - 40*x + 9, 0)"],["dickson-theory-of-equations-1922/ex-page28/2",4,"Dickson 1922, Exercise Page28 (2)"],["form/55a3e2f6e9",5,"solve: Eq(6*x**3 - 11*x**2 + 6*x - 1, 0)"],["shape/9c0e092161",6,"solve: Eq(N*x + 2*N*x**N - 1, 0)"],["dickson-theory-of-equations-1922/ex-page28/3",4,"Dickson 1922, Exercise Page28 (3)"],["form/9dd1dabfa3",5,"solve: Eq(108*x**3 - 270*x**2 - 42*x + 1, 0)"],["shape/4714fd9719",6,"solve: Eq(N*x + 2*N*x**N + 1, 0)"],["dickson-theory-of-equations-1922/ex-page28/4",4,"Dickson 1922, Exercise Page28 (4)"],["form/5bda650a4f",5,"solve: Eq(32*x**3 - 6*x - 1, 0)"],["dickson-theory-of-equations-1922/ex-page28/5",4,"Dickson 1922, Exercise Page28 (5)"],["form/821a5288e2",5,"solve: Eq(96*x**3 - 16*x**2 - 6*x + 1, 0)"],["dickson-theory-of-equations-1922/ex-page28/6",4,"Dickson 1922, Exercise Page28 (6)"],["form/996b526b3c",5,"solve: Eq(24*x**3 - 2*x**2 - 5*x + 1, 0)"],["dickson-theory-of-equations-1922/ex-page28/7",4,"Dickson 1922, Exercise Page28 (7)"],["form/789680dfb7",5,"solve: Eq(x**3 - x**2/2 - 2*x + 1, 0)"],["shape/f28355929e",6,"solve: Eq(N*x + N*x**N + x**N + 1, 0)"],["dickson-theory-of-equations-1922/ex-page28/8",4,"Dickson 1922, Exercise Page28 (8)"],["form/7671e92861",5,"solve: Eq(x**3 - 2*x**2/3 + 3*x - 2, 0)"],["dickson-theory-of-equations-1922/ex-page28/9",4,"Dickson 1922, Exercise Page28 (9)"],["dickson-theory-of-equations-1922/ex-page28/10",4,"Dickson 1922, Exercise Page28 (10)"],["dickson-theory-of-equations-1922/ex-page28/11",4,"Dickson 1922, Exercise Page28 (11)"],["dickson-theory-of-equations-1922/ex-page2/5",4,"Dickson 1922, Exercise Page2 (5)"],["form/3e80a28437",5,"identity: 2*sqrt(3) + 8"],["shape/f94c8d9aae",6,"identity: N*N**N + N"],["dickson-theory-of-equations-1922/ex-page2/6",4,"Dickson 1922, Exercise Page2 (6)"],["form/7e3d3955d1",5,"identity: (2 - I)*(3 + sqrt(5)*I)/5"],["shape/4a8c3eb716",6,"identity: N*(N - I)*(N + I*N**N)"],["dickson-theory-of-equations-1922/ex-page2/7",4,"Dickson 1922, Exercise Page2 (7)"],["form/17ac9470c9",5,"identity: (5*a + 3)/(-3*a + 2)"],["dickson-theory-of-equations-1922/ex-page2/8",4,"Dickson 1922, Exercise Page2 (8)"],["form/b89c38de38",5,"identity: (a*c + b)/(-a*c + b)"],["shape/b89c38de38",6,"identity: (a*c + b)/(-a*c + b)"],["dickson-theory-of-equations-1922/ex-page2/9",4,"Dickson 1922, Exercise Page2 (9)"],["dickson-theory-of-equations-1922/ex-page2/10",4,"Dickson 1922, Exercise Page2 (10)"],["dickson-theory-of-equations-1922/ex-page2/11",4,"Dickson 1922, Exercise Page2 (11)"],["dickson-theory-of-equations-1922/ex-page2/12",4,"Dickson 1922, Exercise Page2 (12)"],["dickson-theory-of-equations-1922/ex-page2/13",4,"Dickson 1922, Exercise Page2 (13)"],["dickson-theory-of-equations-1922/ex-page2/14",4,"Dickson 1922, Exercise Page2 (14)"],["form/87902ac93f",5,"solve: Eq(x**2, 60*a - 11)"],["shape/225824c74e",6,"solve: Eq(x**N, N*a + N)"],["dickson-theory-of-equations-1922/ex-page2/15",4,"Dickson 1922, Exercise Page2 (15)"],["form/4ccfff38c5",5,"solve: Eq(x**2, -12*a + 5)"],["dickson-theory-of-equations-1922/ex-page2/16",4,"Dickson 1922, Exercise Page2 (16)"],["form/85656f266a",5,"solve: Eq(x**2, a*(2*b**2 - 2*c**2) + 4*b*c)"],["shape/86eecea83c",6,"solve: Eq(x**N, N*b*c + a*(N*b**N + N*c**N))"],["dickson-theory-of-equations-1922/ex-page30/1",4,"Dickson 1922, Exercise Page30 (1)"],["form/bbce29f79d",5,"solve: Eq(x**2 - 5*x + 4, 0)"],["dickson-theory-of-equations-1922/ex-page30/2",4,"Dickson 1922, Exercise Page30 (2)"],["form/65305ad133",5,"solve: Eq(x**2 + 5*x + 4, 0)"],["dickson-theory-of-equations-1922/ex-page30/3",4,"Dickson 1922, Exercise Page30 (3)"],["form/842e4cc521",5,"solve: Eq(x**2 + 5*x - 4, 0)"],["dickson-theory-of-equations-1922/ex-page30/4",4,"Dickson 1922, Exercise Page30 (4)"],["form/9a0b8275f6",5,"solve: Eq(x**2 - 5*x - 4, 0)"],["dickson-theory-of-equations-1922/ex-page30/5",4,"Dickson 1922, Exercise Page30 (5)"],["form/820c89e455",5,"solve: Eq(x**2 - 4*x + 4, 0)"],["dickson-theory-of-equations-1922/ex-page30/6",4,"Dickson 1922, Exercise Page30 (6)"],["form/38c209ab62",5,"solve: Eq(x**2 - 3*x + 4, 0)"],["dickson-theory-of-equations-1922/ex-page40/1",4,"Dickson 1922, Exercise Page40 (1)"],["dickson-theory-of-equations-1922/ex-page40/2",4,"Dickson 1922, Exercise Page40 (2)"],["cap/other:ruler_compass_construction",17,"other:ruler_compass_construction"],["dickson-theory-of-equations-1922/ex-page40/3",4,"Dickson 1922, Exercise Page40 (3)"],["dickson-theory-of-equations-1922/ex-page40/4",4,"Dickson 1922, Exercise Page40 (4)"],["dickson-theory-of-equations-1922/ex-page40/5",4,"Dickson 1922, Exercise Page40 (5)"],["dickson-theory-of-equations-1922/ex-page44/6",4,"Dickson 1922, Exercise Page44 (6)"],["dickson-theory-of-equations-1922/ex-page40/6",4,"Dickson 1922, Exercise Page40 (6)"],["form/5215327065",5,"solve: Eq(x**5 - 7*x**4 + x**3 - x**2 + 7*x - 1, 0)"],["dickson-theory-of-equations-1922/ex-page40/7",4,"Dickson 1922, Exercise Page40 (7)"],["dickson-theory-of-equations-1922/ex-page40/8",4,"Dickson 1922, Exercise Page40 (8)"],["dickson-theory-of-equations-1922/ex-page40/9",4,"Dickson 1922, Exercise Page40 (9)"],["dickson-theory-of-equations-1922/ex-page40/10",4,"Dickson 1922, Exercise Page40 (10)"],["cap/other:ruler_compass_impossibility",17,"other:ruler_compass_impossibility"],["form/91ae10aeda",5,"solve: Eq(x**5 - 4*x**4 + x**3 + x**2 - 4*x + 1, 0)"],["shape/8fbfc8945f",6,"solve: Eq(N*x + N*x**N + 3*x**N + 1, 0)"],["dickson-theory-of-equations-1922/ex-page44/7",4,"Dickson 1922, Exercise Page44 (7)"],["form/caf73c6552",5,"solve: Eq(2*x**6 - 5*x**5 + 4*x**4 - 4*x**2 + 5*x - 2, 0)"],["shape/004fe2196f",6,"solve: Eq(N*x + 4*N*x**N + N, 0)"],["dickson-theory-of-equations-1922/ex-page40/11",4,"Dickson 1922, Exercise Page40 (11)"],["dickson-theory-of-equations-1922/ex-page40/12",4,"Dickson 1922, Exercise Page40 (12)"],["dickson-theory-of-equations-1922/ex-page40/13",4,"Dickson 1922, Exercise Page40 (13)"],["dickson-theory-of-equations-1922/ex-page40/14",4,"Dickson 1922, Exercise Page40 (14)"],["dickson-theory-of-equations-1922/ex-page40/15",4,"Dickson 1922, Exercise Page40 (15)"],["dickson-theory-of-equations-1922/ex-page44/8",4,"Dickson 1922, Exercise Page44 (8)"],["form/0d0e452875",5,"solve: Eq(x**5 + 1, 31*(x + 1)**5)"],["shape/e3053c8fb0",6,"solve: Eq(x**N + 1, N*(x + 1)**N)"],["dickson-theory-of-equations-1922/ex-page66/1",4,"Dickson 1922, Exercise Page66 (1)"],["dickson-theory-of-equations-1922/ex-page40/16",4,"Dickson 1922, Exercise Page40 (16)"],["dickson-theory-of-equations-1922/ex-page44/1",4,"Dickson 1922, Exercise Page44 (1)"],["dickson-theory-of-equations-1922/ex-page44/2",4,"Dickson 1922, Exercise Page44 (2)"],["dickson-theory-of-equations-1922/ex-page44/3",4,"Dickson 1922, Exercise Page44 (3)"],["cap/other:gauss_periods",17,"other:gauss_periods"],["dickson-theory-of-equations-1922/ex-page44/4",4,"Dickson 1922, Exercise Page44 (4)"],["dickson-theory-of-equations-1922/ex-page44/5",4,"Dickson 1922, Exercise Page44 (5)"],["form/8d4d45ef7e",5,"solve: Eq(x**4 + 4*x**3 - 3*x**2 + 4*x + 1, 0)"],["shape/9485913dcc",6,"solve: Eq(N*x + 2*N*x**N + x**N + 1, 0)"],["dickson-theory-of-equations-1922/ex-page46/1",4,"Dickson 1922, Exercise Page46 (1)"],["form/0d06dd3d8a",5,"solve: Eq(x**3 - 18*x + 35, 0)"],["dickson-theory-of-equations-1922/ex-page48/1",4,"Dickson 1922, Exercise Page48 (1)"],["dickson-theory-of-equations-1922/ex-page46/2",4,"Dickson 1922, Exercise Page46 (2)"],["form/25f01b4ea2",5,"solve: Eq(x**3 + 6*x**2 + 3*x + 18, 0)"],["dickson-theory-of-equations-1922/ex-page46/3",4,"Dickson 1922, Exercise Page46 (3)"],["form/091ef9bfd9",5,"solve: Eq(x**3 - 2*x + 4, 0)"],["dickson-theory-of-equations-1922/ex-page46/4",4,"Dickson 1922, Exercise Page46 (4)"],["form/d91e20750c",5,"solve: Eq(28*x**3 + 9*x**2 - 1, 0)"],["shape/e1ecf9b584",6,"solve: Eq(2*N*x**N - 1, 0)"],["dickson-theory-of-equations-1922/ex-page48/2",4,"Dickson 1922, Exercise Page48 (2)"],["form/93d0f59f4e",5,"evaluate: 13068"],["dickson-theory-of-equations-1922/ex-page48/3",4,"Dickson 1922, Exercise Page48 (3)"],["form/0583fec3dc",5,"evaluate: 0"],["shape/0583fec3dc",6,"evaluate: 0"],["dickson-theory-of-equations-1922/ex-page48/4",4,"Dickson 1922, Exercise Page48 (4)"],["dickson-theory-of-equations-1922/ex-page48/5",4,"Dickson 1922, Exercise Page48 (5)"],["dickson-theory-of-equations-1922/ex-page49/1",4,"Dickson 1922, Exercise Page49 (1)"],["form/fe1fa32b4c",5,"solve: Eq(x**3 - 15*x + 4, 0)"],["dickson-theory-of-equations-1922/ex-page49/2",4,"Dickson 1922, Exercise Page49 (2)"],["form/aa40e64a03",5,"solve: Eq(x**3 - 2*x - 1, 0)"],["shape/9c0f8b803a",6,"solve: Eq(N*x + x**N - 1, 0)"],["dickson-theory-of-equations-1922/ex-page49/3",4,"Dickson 1922, Exercise Page49 (3)"],["form/bb008f8335",5,"solve: Eq(x**3 - 7*x + 7, 0)"],["dickson-theory-of-equations-1922/ex-page49/4",4,"Dickson 1922, Exercise Page49 (4)"],["dickson-theory-of-equations-1922/ex-page49/5",4,"Dickson 1922, Exercise Page49 (5)"],["dickson-theory-of-equations-1922/ex-page49/6",4,"Dickson 1922, Exercise Page49 (6)"],["form/015fc5cd03",5,"solve: Eq(x**3 + 3*x**2 - 2*x - 5, 0)"],["form/6576c5d6ed",5,"solve: Eq(x**3 + x**2 - 2*x - 1, 0)"],["shape/ffe7cb0f64",6,"solve: Eq(N*x + 2*x**N - 1, 0)"],["dickson-theory-of-equations-1922/ex-page51/1",4,"Dickson 1922, Exercise Page51 (1)"],["form/552901293d",5,"solve: Eq(x**4 - 8*x**3 + 9*x**2 + 8*x - 10, 0)"],["dickson-theory-of-equations-1922/ex-page51/2",4,"Dickson 1922, Exercise Page51 (2)"],["form/3c56e52a16",5,"solve: Eq(x**4 - 2*x**3 - 7*x**2 + 8*x + 12, 0)"],["dickson-theory-of-equations-1922/ex-page51/3",4,"Dickson 1922, Exercise Page51 (3)"],["form/9f405e2a0f",5,"solve: Eq(x**4 - 3*x**2 + 6*x - 2, 0)"],["dickson-theory-of-equations-1922/ex-page51/4",4,"Dickson 1922, Exercise Page51 (4)"],["form/7891384315",5,"solve: Eq(x**4 - 2*x**2 - 8*x - 3, 0)"],["dickson-theory-of-equations-1922/ex-page51/5",4,"Dickson 1922, Exercise Page51 (5)"],["form/cd44a2ad28",5,"solve: Eq(x**4 - 10*x**2 - 20*x - 16, 0)"],["dickson-theory-of-equations-1922/ex-page54/1",4,"Dickson 1922, Exercise Page54 (1)"],["form/e24f2bcf02",5,"solve: (Eq(a, x**2), Eq(a*x + a - 4*x + 6, 0))"],["shape/bc7d031a93",6,"solve: (Eq(a, x**N), Eq(N*x + N + a*x + a, 0))"],["dickson-theory-of-equations-1922/ex-page54/2",4,"Dickson 1922, Exercise Page54 (2)"],["cap/other:discriminant",17,"other:discriminant"],["dickson-theory-of-equations-1922/ex-page54/3",4,"Dickson 1922, Exercise Page54 (3)"],["dickson-theory-of-equations-1922/ex-page54/4",4,"Dickson 1922, Exercise Page54 (4)"],["form/53458116b1",5,"solve: Eq(x**3 - 27*x + 54, 0)"],["dickson-theory-of-equations-1922/ex-page54/5",4,"Dickson 1922, Exercise Page54 (5)"],["dickson-theory-of-equations-1922/ex-page54/6",4,"Dickson 1922, Exercise Page54 (6)"],["cap/other:quartic_resolvent",17,"other:quartic_resolvent"],["dickson-theory-of-equations-1922/ex-page54/7",4,"Dickson 1922, Exercise Page54 (7)"],["dickson-theory-of-equations-1922/ex-page54/8",4,"Dickson 1922, Exercise Page54 (8)"],["dickson-theory-of-equations-1922/ex-page59/1",4,"Dickson 1922, Exercise Page59 (1)"],["dickson-theory-of-equations-1922/ex-page59/2",4,"Dickson 1922, Exercise Page59 (2)"],["dickson-theory-of-equations-1922/ex-page59/3a",4,"Dickson 1922, Exercise Page59 (3a)"],["dickson-theory-of-equations-1922/ex-page59/3b",4,"Dickson 1922, Exercise Page59 (3b)"],["dickson-theory-of-equations-1922/ex-page59/4",4,"Dickson 1922, Exercise Page59 (4)"],["form/c41c48c4a9",5,"solve: Eq(x**4 + x**3 - x - 2, 0)"],["shape/1607cbe5cb",6,"solve: Eq(N - x + 2*x**N, 0)"],["form/55420160e7",5,"solve: Eq(x**3 - 2*x - 5, 0)"],["cap/core.table",17,"core.table"],["dickson-theory-of-equations-1922/ex-page59/5",4,"Dickson 1922, Exercise Page59 (5)"],["dickson-theory-of-equations-1922/ex-page59/6",4,"Dickson 1922, Exercise Page59 (6)"],["dickson-theory-of-equations-1922/ex-page59/7",4,"Dickson 1922, Exercise Page59 (7)"],["dickson-theory-of-equations-1922/ex-page59/8",4,"Dickson 1922, Exercise Page59 (8)"],["dickson-theory-of-equations-1922/ex-page59/9a",4,"Dickson 1922, Exercise Page59 (9a)"],["form/40a044cd02",5,"differentiate: x**6 + 5*x**4"],["shape/828975f0da",6,"differentiate: N*x**N + x**N"],["dickson-theory-of-equations-1922/ex-page59/9b",4,"Dickson 1922, Exercise Page59 (9b)"],["form/584802c74c",5,"differentiate: 2*x**5 - 7*x**3 + x"],["dickson-theory-of-equations-1922/ex-page59/10",4,"Dickson 1922, Exercise Page59 (10)"],["dickson-theory-of-equations-1922/ex-page62/1",4,"Dickson 1922, Exercise Page62 (1)"],["dickson-theory-of-equations-1922/ex-page62/2",4,"Dickson 1922, Exercise Page62 (2)"],["dickson-theory-of-equations-1922/ex-page62/3",4,"Dickson 1922, Exercise Page62 (3)"],["dickson-theory-of-equations-1922/ex-page62/4",4,"Dickson 1922, Exercise Page62 (4)"],["dickson-theory-of-equations-1922/ex-page62/5",4,"Dickson 1922, Exercise Page62 (5)"],["dickson-theory-of-equations-1922/ex-page62/6",4,"Dickson 1922, Exercise Page62 (6)"],["dickson-theory-of-equations-1922/ex-page64/1",4,"Dickson 1922, Exercise Page64 (1)"],["dickson-theory-of-equations-1922/ex-page64/2",4,"Dickson 1922, Exercise Page64 (2)"],["dickson-theory-of-equations-1922/ex-page64/3",4,"Dickson 1922, Exercise Page64 (3)"],["dickson-theory-of-equations-1922/ex-page64/4",4,"Dickson 1922, Exercise Page64 (4)"],["dickson-theory-of-equations-1922/ex-page64/5",4,"Dickson 1922, Exercise Page64 (5)"],["dickson-theory-of-equations-1922/ex-page66/2",4,"Dickson 1922, Exercise Page66 (2)"],["dickson-theory-of-equations-1922/ex-page66/3",4,"Dickson 1922, Exercise Page66 (3)"],["dickson-theory-of-equations-1922/ex-page64/6",4,"Dickson 1922, Exercise Page64 (6)"],["dickson-theory-of-equations-1922/ex-page66/4",4,"Dickson 1922, Exercise Page66 (4)"],["dickson-theory-of-equations-1922/ex-page66/5",4,"Dickson 1922, Exercise Page66 (5)"],["dickson-theory-of-equations-1922/ex-page66/6",4,"Dickson 1922, Exercise Page66 (6)"],["dickson-theory-of-equations-1922/ex-page66/7",4,"Dickson 1922, Exercise Page66 (7)"],["dickson-theory-of-equations-1922/ex-page66/8",4,"Dickson 1922, Exercise Page66 (8)"],["form/7a38ff549d",5,"solve: Eq(-a - b*x + x**4 - x**2, 0)"],["shape/271f01b5ef",6,"solve: Eq(-a - b*x, 0)"],["dickson-theory-of-equations-1922/ex-page6/3b",4,"Dickson 1922, Exercise Page6 (3b)"],["dickson-theory-of-equations-1922/ex-page66/9",4,"Dickson 1922, Exercise Page66 (9)"],["dickson-theory-of-equations-1922/ex-page66/10",4,"Dickson 1922, Exercise Page66 (10)"],["dickson-theory-of-equations-1922/ex-page6/1",4,"Dickson 1922, Exercise Page6 (1)"],["dickson-theory-of-equations-1922/ex-page6/2a",4,"Dickson 1922, Exercise Page6 (2a)"],["form/87b373788c",5,"solve: Eq(x**3, -27)"],["dickson-theory-of-equations-1922/ex-page6/2b",4,"Dickson 1922, Exercise Page6 (2b)"],["form/924b7032b2",5,"solve: Eq(x**3, -a)"],["shape/35bed45115",6,"solve: Eq(x**N, -a)"],["dickson-theory-of-equations-1922/ex-page6/2c",4,"Dickson 1922, Exercise Page6 (2c)"],["form/6cc9d16fbe",5,"solve: Eq(x**3, exp(2*pi*a/3))"],["shape/55f4bec68b",6,"solve: Eq(x**N, exp(pi*N*a))"],["dickson-theory-of-equations-1922/ex-page6/3a",4,"Dickson 1922, Exercise Page6 (3a)"],["form/ed9733e351",5,"solve: Eq(x**2, a)"],["shape/7e1a45c900",6,"solve: Eq(x**N, a)"],["dickson-theory-of-equations-1922/ex-page6/3c",4,"Dickson 1922, Exercise Page6 (3c)"],["form/5789678982",5,"solve: Eq(x**2, exp(2*pi*a/3))"],["dickson-theory-of-equations-1922/ex-page6/4",4,"Dickson 1922, Exercise Page6 (4)"],["dickson-theory-of-equations-1922/ex-page6/5",4,"Dickson 1922, Exercise Page6 (5)"],["dickson-theory-of-equations-1922/ex-page6/6",4,"Dickson 1922, Exercise Page6 (6)"],["dickson-theory-of-equations-1922/ex-page6/7",4,"Dickson 1922, Exercise Page6 (7)"],["dickson-theory-of-equations-1922/ex-page6/8",4,"Dickson 1922, Exercise Page6 (8)"],["dickson-theory-of-equations-1922/ex-page74/1",4,"Dickson 1922, Exercise Page74 (1)"],["dickson-theory-of-equations-1922/ex-page74/2",4,"Dickson 1922, Exercise Page74 (2)"],["dickson-theory-of-equations-1922/ex-page74/3",4,"Dickson 1922, Exercise Page74 (3)"],["dickson-theory-of-equations-1922/ex-page74/4",4,"Dickson 1922, Exercise Page74 (4)"],["dickson-theory-of-equations-1922/ex-page74/5",4,"Dickson 1922, Exercise Page74 (5)"],["dickson-theory-of-equations-1922/ex-page74/6",4,"Dickson 1922, Exercise Page74 (6)"],["dickson-theory-of-equations-1922/ex-page74/7",4,"Dickson 1922, Exercise Page74 (7)"],["dickson-theory-of-equations-1922/ex-page74/8",4,"Dickson 1922, Exercise Page74 (8)"],["dickson-theory-of-equations-1922/ex-page74/9",4,"Dickson 1922, Exercise Page74 (9)"],["dickson-theory-of-equations-1922/ex-page74/10",4,"Dickson 1922, Exercise Page74 (10)"],["dickson-theory-of-equations-1922/ex-page74/11",4,"Dickson 1922, Exercise Page74 (11)"],["dickson-theory-of-equations-1922/ex-page74/12",4,"Dickson 1922, Exercise Page74 (12)"],["dickson-theory-of-equations-1922/ex-page78/1",4,"Dickson 1922, Exercise Page78 (1)"],["cap/other:sturm_sequence",17,"other:sturm_sequence"],["dickson-theory-of-equations-1922/ex-page78/2",4,"Dickson 1922, Exercise Page78 (2)"],["dickson-theory-of-equations-1922/ex-page79/1",4,"Dickson 1922, Exercise Page79 (1)"],["dickson-theory-of-equations-1922/ex-page74/13",4,"Dickson 1922, Exercise Page74 (13)"],["dickson-theory-of-equations-1922/ex-page74/14",4,"Dickson 1922, Exercise Page74 (14)"],["dickson-theory-of-equations-1922/ex-page74/15",4,"Dickson 1922, Exercise Page74 (15)"],["dickson-theory-of-equations-1922/ex-page74/16",4,"Dickson 1922, Exercise Page74 (16)"],["dickson-theory-of-equations-1922/ex-page79/2",4,"Dickson 1922, Exercise Page79 (2)"],["dickson-theory-of-equations-1922/ex-page79/3",4,"Dickson 1922, Exercise Page79 (3)"],["dickson-theory-of-equations-1922/ex-page79/4",4,"Dickson 1922, Exercise Page79 (4)"],["dickson-theory-of-equations-1922/ex-page79/5",4,"Dickson 1922, Exercise Page79 (5)"],["dickson-theory-of-equations-1922/ex-page79/6",4,"Dickson 1922, Exercise Page79 (6)"],["dickson-theory-of-equations-1922/ex-page79/7",4,"Dickson 1922, Exercise Page79 (7)"],["dickson-theory-of-equations-1922/ex-page79/8",4,"Dickson 1922, Exercise Page79 (8)"],["dickson-theory-of-equations-1922/ex-page79/9",4,"Dickson 1922, Exercise Page79 (9)"],["dickson-theory-of-equations-1922/ex-page83/1",4,"Dickson 1922, Exercise Page83 (1)"],["dickson-theory-of-equations-1922/ex-page83/2",4,"Dickson 1922, Exercise Page83 (2)"],["form/aeeb18b721",5,"solve: Eq(x**4 - 5*x**3 + 9*x**2 - 7*x + 2, 0)"],["dickson-theory-of-equations-1922/ex-page83/3",4,"Dickson 1922, Exercise Page83 (3)"],["form/ddf2ddfafc",5,"solve: Eq(x**4 + 2*x**3 - 3*x**2 - 4*x + 4, 0)"],["dickson-theory-of-equations-1922/ex-page83/4",4,"Dickson 1922, Exercise Page83 (4)"],["form/1defe0056e",5,"solve: Eq(x**4 - x**2 - 2*x + 2, 0)"],["dickson-theory-of-equations-1922/ex-page85/1",4,"Dickson 1922, Exercise Page85 (1)"],["dickson-theory-of-equations-1922/ex-page85/2",4,"Dickson 1922, Exercise Page85 (2)"],["dickson-theory-of-equations-1922/ex-page85/3",4,"Dickson 1922, Exercise Page85 (3)"],["dickson-theory-of-equations-1922/ex-page85/4",4,"Dickson 1922, Exercise Page85 (4)"],["dickson-theory-of-equations-1922/ex-page85/5",4,"Dickson 1922, Exercise Page85 (5)"],["dickson-theory-of-equations-1922/ex-page85/6",4,"Dickson 1922, Exercise Page85 (6)"],["dickson-theory-of-equations-1922/ex-page89/1",4,"Dickson 1922, Exercise Page89 (1)"],["form/4898066d8c",5,"solve: Eq(x**3 + 2*x + 20, 0)"],["dickson-theory-of-equations-1922/ex-page89/2",4,"Dickson 1922, Exercise Page89 (2)"],["dickson-theory-of-equations-1922/ex-page89/3",4,"Dickson 1922, Exercise Page89 (3)"],["dickson-theory-of-equations-1922/ex-page89/4",4,"Dickson 1922, Exercise Page89 (4)"],["form/f53d1c3b55",5,"solve: Eq(x**4 + 4*x**3 - 35*x**2/2 - 18*x + 117/2, 0)"],["dickson-theory-of-equations-1922/ex-page89/5",4,"Dickson 1922, Exercise Page89 (5)"],["form/45a4401bfd",5,"solve: Eq(x**4 - 11727*x + 40385, 0)"],["dickson-theory-of-equations-1922/ex-page89/6",4,"Dickson 1922, Exercise Page89 (6)"],["form/186a60549d",5,"solve: Eq(x**3, 10)"],["dickson-theory-of-equations-1922/ex-page89/7",4,"Dickson 1922, Exercise Page89 (7)"],["dickson-theory-of-equations-1922/ex-page89/9",4,"Dickson 1922, Exercise Page89 (9)"],["dickson-theory-of-equations-1922/ex-page89/10",4,"Dickson 1922, Exercise Page89 (10)"],["dickson-theory-of-equations-1922/ex-page89/11",4,"Dickson 1922, Exercise Page89 (11)"],["form/9c6e79337b",5,"solve: Eq(x**3 + 3*x**2 + x - 6, 0)"],["shape/4a2cc9272d",6,"solve: Eq(N*x**N + N + x + x**N, 0)"],["dickson-theory-of-equations-1922/ex-page89/12",4,"Dickson 1922, Exercise Page89 (12)"],["form/206bc7248f",5,"solve: (Eq(a, x**2 - x), Eq(a**2 + x**2, 9))"],["shape/6ad4239e3f",6,"solve: (Eq(a, -x + x**N), Eq(a**N + x**N, N))"],["dickson-theory-of-equations-1922/ex-page89/13",4,"Dickson 1922, Exercise Page89 (13)"],["form/82f85bdfec",5,"solve: Eq(x**3 - x - 9, 0)"],["form/8611f5fc93",5,"evaluate: 997**(1/3)/5"],["dickson-theory-of-equations-1922/ex-page89/14",4,"Dickson 1922, Exercise Page89 (14)"],["form/2157fb20b5",5,"solve: Eq(x**3 - 3*x**2 + 1, 0)"],["shape/185d374116",6,"solve: Eq(N*x**N + x**N + 1, 0)"],["dickson-theory-of-equations-1922/ex-page89/15",4,"Dickson 1922, Exercise Page89 (15)"],["form/c29f767d1f",5,"evaluate: cos(pi/9)"],["shape/fbfbf49240",6,"evaluate: cos(pi*N)"],["dickson-theory-of-equations-1922/ex-page89/16",4,"Dickson 1922, Exercise Page89 (16)"],["form/40b2e09689",5,"solve: Eq((x + 6)*(x + 8)*(x + 10), 780)"],["shape/665134fa60",6,"solve: Eq((N + x)**3, N)"],["dickson-theory-of-equations-1922/ex-page89/17",4,"Dickson 1922, Exercise Page89 (17)"],["form/b49bd8222a",5,"solve: Eq(x**3 - 28*x + 56, 0)"],["dickson-theory-of-equations-1922/ex-page89/18",4,"Dickson 1922, Exercise Page89 (18)"],["form/0cc68a95ea",5,"solve: Eq(2700*(x + 1)**3, 1000*x + 1000*(x + 1)**2 + 2000)"],["shape/0049138722",6,"solve: Eq(N*(x + 1)**N, N*x + N*(x + 1)**N + N)"],["dickson-theory-of-equations-1922/ex-page89/19",4,"Dickson 1922, Exercise Page89 (19)"],["form/78bb868d01",5,"solve: Eq(3500*(x + 1)**4, 1000*x + 1000*(x + 1)**3 + 1000*(x + 1)**2 + 2000)"],["shape/8d858e58be",6,"solve: Eq(N*(x + 1)**N, N*x + 2*N*(x + 1)**N + N)"],["dickson-theory-of-equations-1922/ex-page89/20",4,"Dickson 1922, Exercise Page89 (20)"],["form/5fcab03b7e",5,"solve: Eq(2500*(x + 1)**3, 1000*x + 1000*(x + 1)**2 + 2000)"],["dickson-theory-of-equations-1922/ex-page94/2",4,"Dickson 1922, Exercise Page94 (2)"],["dickson-theory-of-equations-1922/ex-page96/1",4,"Dickson 1922, Exercise Page96 (1)"],["dickson-theory-of-equations-1922/ex-page96/2",4,"Dickson 1922, Exercise Page96 (2)"],["form/e065566ed9",5,"solve: Eq(x**3 - 2*x**2 - 2, 0)"],["dickson-theory-of-equations-1922/ex-page96/3",4,"Dickson 1922, Exercise Page96 (3)"],["dickson-theory-of-equations-1922/ex-page96/4",4,"Dickson 1922, Exercise Page96 (4)"],["dickson-theory-of-equations-1922/ex-page96/5",4,"Dickson 1922, Exercise Page96 (5)"],["dickson-theory-of-equations-1922/ex-page98/1",4,"Dickson 1922, Exercise Page98 (1)"],["dickson-theory-of-equations-1922/ex-page98/2",4,"Dickson 1922, Exercise Page98 (2)"],["dickson-theory-of-equations-1922/ex-page98/3",4,"Dickson 1922, Exercise Page98 (3)"],["form/581013f9ea",5,"solve: Eq(2*x - log(x), 9)"],["shape/97ab8bd1a7",6,"solve: Eq(N*x - log(x), N)"],["dickson-theory-of-equations-1922/ex-page98/4",4,"Dickson 1922, Exercise Page98 (4)"],["form/35231ce8d7",5,"solve: Eq(3*x - log(x), 9)"],["dickson-theory-of-equations-1922/ex-page98/5",4,"Dickson 1922, Exercise Page98 (5)"],["form/e6f72e6b10",5,"solve: Eq(sin(pi*x/180)/2 + sin(pi*x/90), 16/25)"],["shape/4134179fc3",6,"solve: Eq(N*sin(pi*N*x) + sin(pi*N*x), N)"],["dickson-theory-of-equations-1922/ex-page98/6",4,"Dickson 1922, Exercise Page98 (6)"],["form/b4ed85b4ce",5,"solve: Eq(x - sin(x)/2, pi/4)"],["dickson-theory-of-equations-1922/ex-page98/7",4,"Dickson 1922, Exercise Page98 (7)"],["dickson-theory-of-equations-1922/ex-page98/8",4,"Dickson 1922, Exercise Page98 (8)"],["form/654ee3467d",5,"solve: Eq(sin(pi*x/180) + sin(pi*x/90), 6/5)"],["shape/98ee013d11",6,"solve: Eq(2*sin(pi*N*x), N)"],["dickson-theory-of-equations-1922/ex-page98/9",4,"Dickson 1922, Exercise Page98 (9)"],["form/cbd5fed18b",5,"solve: Eq(sin(x), x - 2)"],["shape/daf56cbab0",6,"solve: Eq(sin(x), N + x)"],["dickson-theory-of-equations-1922/ex-page98/10",4,"Dickson 1922, Exercise Page98 (10)"],["form/3b684d814e",5,"solve: Eq(x, 3*log(x))"],["shape/89754520bf",6,"solve: Eq(x, N*log(x))"],["dickson-theory-of-equations-1922/ex-page99/2",4,"Dickson 1922, Exercise Page99 (2)"],["dickson-theory-of-equations-1922/ex-page99/3",4,"Dickson 1922, Exercise Page99 (3)"],["form/77f8d6384f",5,"solve: Eq(x**4 - 3*x**2 - 6*x, 2)"],["shape/1a6fc54ba7",6,"solve: Eq(N*x + N*x**N + x**N, N)"],["dickson-theory-of-equations-1922/ex-page99/4",4,"Dickson 1922, Exercise Page99 (4)"],["form/fa75c52766",5,"solve: Eq(x**4 - 4*x**3 + 11*x**2 - 14*x + 10, 0)"],["dickson-theory-of-equations-1922/ex-page99/5",4,"Dickson 1922, Exercise Page99 (5)"],["form/e03e6cf8aa",5,"solve: Eq(x**4 - 4*x**3 + 9*x**2 - 16*x + 20, 0)"],["dickson-theory-of-equations-1922/ex-page9/1",4,"Dickson 1922, Exercise Page9 (1)"],["dickson-theory-of-equations-1922/ex-page9/2",4,"Dickson 1922, Exercise Page9 (2)"],["dickson-theory-of-equations-1922/ex-page9/3",4,"Dickson 1922, Exercise Page9 (3)"],["dickson-theory-of-equations-1922/ex-page9/4",4,"Dickson 1922, Exercise Page9 (4)"],["form/a0b6a6215b",5,"solve: Eq(x**5, -1)"],["shape/14bb12d2c2",6,"solve: Eq(x**N, -1)"],["dickson-theory-of-equations-1922/ex-page9/5",4,"Dickson 1922, Exercise Page9 (5)"],["dickson-theory-of-equations-1922/ex-page9/6",4,"Dickson 1922, Exercise Page9 (6)"],["hardy-course-of-pure-mathematics-1921/ex-i/1",4,"Hardy 1921, Exercise I (1)"],["hardy-course-of-pure-mathematics-1921/ex-app-i/6",4,"Hardy 1921, Exercise App-I (6)"],["hardy-course-of-pure-mathematics-1921/ex-app-i/11",4,"Hardy 1921, Exercise App-I (11)"],["hardy-course-of-pure-mathematics-1921/ex-app-i/12",4,"Hardy 1921, Exercise App-I (12)"],["hardy-course-of-pure-mathematics-1921/ex-i/2",4,"Hardy 1921, Exercise I (2)"],["hardy-course-of-pure-mathematics-1921/ex-ii/2",4,"Hardy 1921, Exercise II (2)"],["hardy-course-of-pure-mathematics-1921/ex-ii/3",4,"Hardy 1921, Exercise II (3)"],["hardy-course-of-pure-mathematics-1921/ex-iii/1",4,"Hardy 1921, Exercise III (1)"],["hardy-course-of-pure-mathematics-1921/ex-iv/4",4,"Hardy 1921, Exercise IV (4)"],["hardy-course-of-pure-mathematics-1921/ex-ix/1",4,"Hardy 1921, Exercise IX (1)"],["hardy-course-of-pure-mathematics-1921/ex-ix/2",4,"Hardy 1921, Exercise IX (2)"],["hardy-course-of-pure-mathematics-1921/ex-l/1b",4,"Hardy 1921, Exercise L (1b)"],["form/97c23a7bce",5,"integrate: (x + 1)/((x**2 + 4)*sqrt(x**2 + 9))"],["shape/668cbb7962",6,"integrate: (N + x**N)**N*(x + 1)/(N + x**N)"],["hardy-course-of-pure-mathematics-1921/ex-l/9b",4,"Hardy 1921, Exercise L (9b)"],["form/13e3791508",5,"integrate: (x - 1)/((2*x**2 - 6*x + 5)*sqrt(7*x**2 - 22*x + 19))"],["shape/e55f8ea4ba",6,"integrate: (x - 1)*(N*x + N*x**N + N)**N/(N*x + N*x**N + N)"],["hardy-course-of-pure-mathematics-1921/ex-lii/1b",4,"Hardy 1921, Exercise LII (1b)"],["form/ea6d4ad59b",5,"integrate: cos(a*x)*cos(b*x)"],["shape/ea6d4ad59b",6,"integrate: cos(a*x)*cos(b*x)"],["form/9bdddab7c3",5,"integrate: cos(x)**2"],["shape/f70ec022b5",6,"integrate: cos(x)**N"],["form/0dcbd1f55b",5,"integrate: sin(x)**3"],["hardy-course-of-pure-mathematics-1921/ex-li/2h",4,"Hardy 1921, Exercise LI (2h)"],["form/9364850dd5",5,"integrate: sin(3*x)**2*cos(2*x)**3"],["shape/9c4dec76b1",6,"integrate: sin(N*x)**N*cos(N*x)**N"],["hardy-course-of-pure-mathematics-1921/ex-lii/1d",4,"Hardy 1921, Exercise LII (1d)"],["hardy-course-of-pure-mathematics-1921/ex-liii/1b",4,"Hardy 1921, Exercise LIII (1b)"],["form/f2b21d89c6",5,"integrate: tan(x)"],["shape/f2b21d89c6",6,"integrate: tan(x)"],["form/b84960f39f",5,"integrate: cot(x)"],["hardy-course-of-pure-mathematics-1921/ex-liii/2d",4,"Hardy 1921, Exercise LIII (2d)"],["hardy-course-of-pure-mathematics-1921/ex-liv/1",4,"Hardy 1921, Exercise LIV (1)"],["hardy-course-of-pure-mathematics-1921/ex-liv/13b",4,"Hardy 1921, Exercise LIV (13b)"],["hardy-course-of-pure-mathematics-1921/ex-liv/13c",4,"Hardy 1921, Exercise LIV (13c)"],["hardy-course-of-pure-mathematics-1921/ex-liv/15",4,"Hardy 1921, Exercise LIV (15)"],["todhunter-spherical-trigonometry-1886/ex-v/2",4,"Todhunter 1886, Exercise V (2)"],["todhunter-spherical-trigonometry-1886/ex-ix/7",4,"Todhunter 1886, Exercise IX (7)"],["todhunter-spherical-trigonometry-1886/ex-iv/17",4,"Todhunter 1886, Exercise IV (17)"],["todhunter-spherical-trigonometry-1886/ex-ix/8",4,"Todhunter 1886, Exercise IX (8)"],["todhunter-spherical-trigonometry-1886/ex-v/4",4,"Todhunter 1886, Exercise V (4)"],["todhunter-spherical-trigonometry-1886/ex-v/9",4,"Todhunter 1886, Exercise V (9)"],["todhunter-spherical-trigonometry-1886/ex-v/14",4,"Todhunter 1886, Exercise V (14)"],["todhunter-spherical-trigonometry-1886/ex-v/18",4,"Todhunter 1886, Exercise V (18)"],["todhunter-spherical-trigonometry-1886/ex-vi/1",4,"Todhunter 1886, Exercise VI (1)"],["todhunter-spherical-trigonometry-1886/ex-vii/1",4,"Todhunter 1886, Exercise VII (1)"],["todhunter-spherical-trigonometry-1886/ex-vii/8",4,"Todhunter 1886, Exercise VII (8)"],["todhunter-spherical-trigonometry-1886/ex-vii/9",4,"Todhunter 1886, Exercise VII (9)"],["todhunter-spherical-trigonometry-1886/ex-viii/8",4,"Todhunter 1886, Exercise VIII (8)"],["todhunter-spherical-trigonometry-1886/ex-viii/14",4,"Todhunter 1886, Exercise VIII (14)"],["todhunter-spherical-trigonometry-1886/ex-xi/6",4,"Todhunter 1886, Exercise XI (6)"],["todhunter-spherical-trigonometry-1886/ex-xii/12",4,"Todhunter 1886, Exercise XII (12)"],["cap/other:solid_geometry",17,"other:solid_geometry"],["todhunter-spherical-trigonometry-1886/ex-xiii/14",4,"Todhunter 1886, Exercise XIII (14)"],["todhunter-spherical-trigonometry-1886/ex-xv/13",4,"Todhunter 1886, Exercise XV (13)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-19-22/20",4,"Macfarlane 1906, Exercise Probs-19-22 (20)"],["todhunter-spherical-trigonometry-1886/ex-xv/17",4,"Todhunter 1886, Exercise XV (17)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-10-18/11",4,"Macfarlane 1906, Exercise Probs-10-18 (11)"],["form/9bff9b5837",5,"evaluate: 6000000*sqrt(3)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-10-18/12",4,"Macfarlane 1906, Exercise Probs-10-18 (12)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-10-18/13",4,"Macfarlane 1906, Exercise Probs-10-18 (13)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-10-18/14",4,"Macfarlane 1906, Exercise Probs-10-18 (14)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-10-18/15",4,"Macfarlane 1906, Exercise Probs-10-18 (15)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-10-18/16",4,"Macfarlane 1906, Exercise Probs-10-18 (16)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-19-22/21",4,"Macfarlane 1906, Exercise Probs-19-22 (21)"],["form/52d7bbc680",5,"solve: (Eq(x, 11*sqrt(2)/2), Eq(240*pi*a, 11*sqrt(2)/2))"],["shape/83c2c3a0f0",6,"solve: (Eq(x, N*N**N), Eq(pi*N*a, N*N**N))"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-19-22/22",4,"Macfarlane 1906, Exercise Probs-19-22 (22)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-1-9/2",4,"Macfarlane 1906, Exercise Probs-1-9 (2)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-1-9/3",4,"Macfarlane 1906, Exercise Probs-1-9 (3)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-1-9/4",4,"Macfarlane 1906, Exercise Probs-1-9 (4)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-1-9/5",4,"Macfarlane 1906, Exercise Probs-1-9 (5)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-1-9/6",4,"Macfarlane 1906, Exercise Probs-1-9 (6)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-1-9/7",4,"Macfarlane 1906, Exercise Probs-1-9 (7)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-1-9/8",4,"Macfarlane 1906, Exercise Probs-1-9 (8)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-1-9/9",4,"Macfarlane 1906, Exercise Probs-1-9 (9)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-23-27/24",4,"Macfarlane 1906, Exercise Probs-23-27 (24)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-23-27/25",4,"Macfarlane 1906, Exercise Probs-23-27 (25)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-23-27/26",4,"Macfarlane 1906, Exercise Probs-23-27 (26)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-23-27/27",4,"Macfarlane 1906, Exercise Probs-23-27 (27)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37/30",4,"Macfarlane 1906, Exercise Probs-28-37 (30)"],["form/d60031cd73",5,"evaluate: 56"],["cap/other:vector_dot_product",17,"other:vector_dot_product"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37/31",4,"Macfarlane 1906, Exercise Probs-28-37 (31)"],["form/fb18c02d24",5,"evaluate: 326"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37/32",4,"Macfarlane 1906, Exercise Probs-28-37 (32)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37/33",4,"Macfarlane 1906, Exercise Probs-28-37 (33)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37/34",4,"Macfarlane 1906, Exercise Probs-28-37 (34)"],["cap/other:vector_cross_product",17,"other:vector_cross_product"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37/35",4,"Macfarlane 1906, Exercise Probs-28-37 (35)"],["cap/other:vector_direction_angle_notation",17,"other:vector_direction_angle_notation"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37/36",4,"Macfarlane 1906, Exercise Probs-28-37 (36)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-28-37/37",4,"Macfarlane 1906, Exercise Probs-28-37 (37)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-38-43/38",4,"Macfarlane 1906, Exercise Probs-38-43 (38)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-38-43/39",4,"Macfarlane 1906, Exercise Probs-38-43 (39)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-38-43/40",4,"Macfarlane 1906, Exercise Probs-38-43 (40)"],["form/f1bc01eeaa",5,"evaluate: -431250"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-38-43/41",4,"Macfarlane 1906, Exercise Probs-38-43 (41)"],["form/71ae4461a1",5,"evaluate: 571/6"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-38-43/42",4,"Macfarlane 1906, Exercise Probs-38-43 (42)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-44-47/45",4,"Macfarlane 1906, Exercise Probs-44-47 (45)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-44-47/46",4,"Macfarlane 1906, Exercise Probs-44-47 (46)"],["cap/other:central_axis_vector",17,"other:central_axis_vector"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-44-47/47",4,"Macfarlane 1906, Exercise Probs-44-47 (47)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-48-53/50",4,"Macfarlane 1906, Exercise Probs-48-53 (50)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-48-53/51",4,"Macfarlane 1906, Exercise Probs-48-53 (51)"],["cap/other:versor_series",17,"other:versor_series"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-48-53/52",4,"Macfarlane 1906, Exercise Probs-48-53 (52)"],["cap/other:spherical_trig_theorem",17,"other:spherical_trig_theorem"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-48-53/53",4,"Macfarlane 1906, Exercise Probs-48-53 (53)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-54-62/56",4,"Macfarlane 1906, Exercise Probs-54-62 (56)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-54-62/57",4,"Macfarlane 1906, Exercise Probs-54-62 (57)"],["cap/other:exponential_theorem",17,"other:exponential_theorem"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-54-62/58",4,"Macfarlane 1906, Exercise Probs-54-62 (58)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-54-62/59",4,"Macfarlane 1906, Exercise Probs-54-62 (59)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-54-62/60",4,"Macfarlane 1906, Exercise Probs-54-62 (60)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-54-62/61",4,"Macfarlane 1906, Exercise Probs-54-62 (61)"],["macfarlane-vector-analysis-quaternions-1906/ex-probs-54-62/62",4,"Macfarlane 1906, Exercise Probs-54-62 (62)"],["thompson-calculus-made-easy-1914/ex-i/1",4,"Thompson 1914, Exercise I (1)"],["thompson-calculus-made-easy-1914/ex-i/2",4,"Thompson 1914, Exercise I (2)"],["form/7c5a88be3e",5,"differentiate: x**(-3/2)"],["thompson-calculus-made-easy-1914/ex-i/3",4,"Thompson 1914, Exercise I (3)"],["shape/ed65e19e4d",6,"differentiate: x**(N*a)"],["thompson-calculus-made-easy-1914/ex-i/4",4,"Thompson 1914, Exercise I (4)"],["form/ec76d67e59",5,"differentiate: x**(12/5)"],["cap/cas.derive",17,"cas.derive"],["thompson-calculus-made-easy-1914/ex-i/6",4,"Thompson 1914, Exercise I (6)"],["form/e047ad6c6e",5,"differentiate: (x**(-5))**(1/3)"],["shape/2b85717254",6,"differentiate: (x**N)**N"],["thompson-calculus-made-easy-1914/ex-i/7",4,"Thompson 1914, Exercise I (7)"],["form/a5c7e144d7",5,"differentiate: Abs(x)**(-8/5)"],["shape/54221fdf4d",6,"differentiate: Abs(x)**N"],["thompson-calculus-made-easy-1914/ex-i/8",4,"Thompson 1914, Exercise I (8)"],["form/677f165bd9",5,"differentiate: 2*x**a"],["shape/8d95d0ea17",6,"differentiate: N*x**a"],["thompson-calculus-made-easy-1914/ex-i/10",4,"Thompson 1914, Exercise I (10)"],["form/c5f85653b0",5,"differentiate: (x**(-a))**(1/b)"],["shape/c5f85653b0",6,"differentiate: (x**(-a))**(1/b)"],["thompson-calculus-made-easy-1914/ex-ii/2",4,"Thompson 1914, Exercise II (2)"],["form/0741c6daad",5,"differentiate: -a + 13*x**(3/2)"],["shape/10f4218c5b",6,"differentiate: N*x**N - a"],["thompson-calculus-made-easy-1914/ex-ii/3",4,"Thompson 1914, Exercise II (3)"],["form/1476ba6085",5,"differentiate: sqrt(a) + 12*sqrt(x)"],["shape/32af61a48b",6,"differentiate: N*x**N + a**N"],["thompson-calculus-made-easy-1914/ex-ii/4",4,"Thompson 1914, Exercise II (4)"],["form/012aeadeaf",5,"differentiate: sqrt(a)*sqrt(x)"],["shape/6ea34ccb33",6,"differentiate: a**N*x**N"],["thompson-calculus-made-easy-1914/ex-ii/5",4,"Thompson 1914, Exercise II (5)"],["form/87f2758279",5,"differentiate: (a*x**c - 1)/b"],["shape/87f2758279",6,"differentiate: (a*x**c - 1)/b"],["thompson-calculus-made-easy-1914/ex-ii/6",4,"Thompson 1914, Exercise II (6)"],["form/8cf0e85118",5,"differentiate: 59*x**2/50 + 112/5"],["thompson-calculus-made-easy-1914/ex-ii/7",4,"Thompson 1914, Exercise II (7)"],["form/da7eb45816",5,"differentiate: a*(3*x/250000 + 1)"],["shape/276e7db1da",6,"differentiate: a*(N*x + 1)"],["thompson-calculus-made-easy-1914/ex-iii/1d",4,"Thompson 1914, Exercise III (1d)"],["shape/3685cc1415",6,"differentiate: N*x**N + N"],["cap/core.units",17,"core.units"],["thompson-calculus-made-easy-1914/ex-ii/8",4,"Thompson 1914, Exercise II (8)"],["form/615c7dcf19",5,"evaluate: a*x**b at V=80, a=0.5e-10, b=6"],["shape/7fde08f75a",6,"evaluate: a*x**b"],["form/7768cd073b",5,"evaluate: a*x**b at V=100, a=0.5e-10, b=6"],["form/5da9fd32f9",5,"evaluate: a*x**b at V=120, a=0.5e-10, b=6"],["cap/cas.subst",17,"cas.subst"],["thompson-calculus-made-easy-1914/ex-ii/91",4,"Thompson 1914, Exercise II (91)"],["form/f067b0f0f0",5,"differentiate: sqrt(b*c/d)/(sqrt(pi)*a*x)"],["shape/17e86330a7",6,"differentiate: pi**N*(b*c/d)**N/(a*x)"],["thompson-calculus-made-easy-1914/ex-ii/92",4,"Thompson 1914, Exercise II (92)"],["thompson-calculus-made-easy-1914/ex-ii/93",4,"Thompson 1914, Exercise II (93)"],["shape/0c8d35a038",6,"differentiate: pi**N*(c*d/x)**N/(a*b)"],["thompson-calculus-made-easy-1914/ex-ii/94",4,"Thompson 1914, Exercise II (94)"],["thompson-calculus-made-easy-1914/ex-ii/10",4,"Thompson 1914, Exercise II (10)"],["thompson-calculus-made-easy-1914/ex-ii/11a",4,"Thompson 1914, Exercise II (11a)"],["shape/7a0a9857ee",6,"differentiate: pi*N*x"],["cap/cas.limit",17,"cas.limit"],["cap/core.const",17,"core.const"],["thompson-calculus-made-easy-1914/ex-ii/11b",4,"Thompson 1914, Exercise II (11b)"],["shape/1305b2cc26",6,"differentiate: pi*x**N"],["thompson-calculus-made-easy-1914/ex-ii/11c",4,"Thompson 1914, Exercise II (11c)"],["form/fcba4cddca",5,"differentiate: pi*a*x"],["shape/fcba4cddca",6,"differentiate: pi*a*x"],["thompson-calculus-made-easy-1914/ex-ii/11d",4,"Thompson 1914, Exercise II (11d)"],["form/241b3aeaca",5,"differentiate: pi*a*x**2/3"],["shape/77365fc362",6,"differentiate: pi*N*a*x**N"],["thompson-calculus-made-easy-1914/ex-ii/11e",4,"Thompson 1914, Exercise II (11e)"],["form/8050bbf689",5,"differentiate: 4*pi*x**2"],["shape/45cff8a011",6,"differentiate: pi*N*x**N"],["thompson-calculus-made-easy-1914/ex-iii/1c",4,"Thompson 1914, Exercise III (1c)"],["thompson-calculus-made-easy-1914/ex-ii/11f",4,"Thompson 1914, Exercise II (11f)"],["form/ab1406c092",5,"differentiate: 4*pi*x**3/3"],["thompson-calculus-made-easy-1914/ex-ii/12",4,"Thompson 1914, Exercise II (12)"],["form/7f9abb11da",5,"differentiate: a*(-3*b/250000 + 3*x/250000 + 1)/pi"],["shape/e05a6c82fd",6,"differentiate: a*(N*b + N*x + 1)/pi"],["thompson-calculus-made-easy-1914/ex-iii/2",4,"Thompson 1914, Exercise III (2)"],["thompson-calculus-made-easy-1914/ex-iii/3",4,"Thompson 1914, Exercise III (3)"],["form/2fa69640d9",5,"differentiate: (x - I)*(x + I)"],["shape/2fa69640d9",6,"differentiate: (x - I)*(x + I)"],["thompson-calculus-made-easy-1914/ex-iii/5",4,"Thompson 1914, Exercise III (5)"],["shape/467726d778",6,"differentiate: (N + x)**2"],["thompson-calculus-made-easy-1914/ex-iii/6",4,"Thompson 1914, Exercise III (6)"],["form/6edba0f6e4",5,"differentiate: 13709*x*(22601*x**2/500 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log(a*x*exp(x))"],["thompson-calculus-made-easy-1914/ex-xii/16",4,"Thompson 1914, Exercise XII (16)"],["form/11d308d15d",5,"differentiate: log(a*x)**3"],["shape/1f58a15caa",6,"differentiate: log(a*x)**N"],["thompson-calculus-made-easy-1914/ex-xiii/1",4,"Thompson 1914, Exercise XIII (1)"],["thompson-calculus-made-easy-1914/ex-xiii/41",4,"Thompson 1914, Exercise XIII (41)"],["thompson-calculus-made-easy-1914/ex-xiii/2",4,"Thompson 1914, Exercise XIII (2)"],["shape/fd124c672a",6,"evaluate: N/log(N)"],["form/846f0758fd",5,"evaluate: 24*log(100)/log(2)"],["thompson-calculus-made-easy-1914/ex-xiii/3",4,"Thompson 1914, Exercise XIII (3)"],["form/203977f91b",5,"evaluate: log(5/4)/600"],["shape/110ec0fabc",6,"evaluate: N"],["thompson-calculus-made-easy-1914/ex-xiii/42",4,"Thompson 1914, Exercise XIII (42)"],["thompson-calculus-made-easy-1914/ex-xiii/43",4,"Thompson 1914, Exercise XIII (43)"],["thompson-calculus-made-easy-1914/ex-xiii/5a",4,"Thompson 1914, Exercise XIII 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exp(x)*sin(x)**2"],["shape/1e45fcbcef",6,"differentiate: exp(x)*sin(x)**N"],["thompson-calculus-made-easy-1914/ex-xiv/102",4,"Thompson 1914, Exercise XIV (102)"],["form/60a940a54e",5,"differentiate2: exp(x)*sin(x)**2"],["shape/66c5682d0e",6,"differentiate2: exp(x)*sin(x)**N"],["thompson-calculus-made-easy-1914/ex-xiv/13",4,"Thompson 1914, Exercise XIV (13)"],["form/032f052fb3",5,"differentiate: sin((2*x + 3)**(23/10))"],["thompson-calculus-made-easy-1914/ex-xiv/111",4,"Thompson 1914, Exercise XIV (111)"],["thompson-calculus-made-easy-1914/ex-xiv/112",4,"Thompson 1914, Exercise XIV (112)"],["shape/12e6cad457",6,"differentiate: sin((N*x + N)**N)"],["thompson-calculus-made-easy-1914/ex-xiv/113",4,"Thompson 1914, Exercise XIV (113)"],["thompson-calculus-made-easy-1914/ex-xiv/12i",4,"Thompson 1914, Exercise XIV (12i)"],["shape/0a514f051d",6,"differentiate: 1/cos(x)"],["thompson-calculus-made-easy-1914/ex-xiv/14",4,"Thompson 1914, Exercise XIV 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sqrt(3)*sqrt(1/cos(x))*tan(x)"],["shape/46420217dc",6,"differentiate: N**N*(1/cos(x))**N*tan(x)"],["shape/38ea463c74",6,"differentiate: N*a*x**N + N*a + N*a**N*x + N*x**N"],["thompson-calculus-made-easy-1914/ex-xv/1b",4,"Thompson 1914, Exercise XV (1b)"],["form/66ef5140c8",5,"differentiate: -2*a**3*x + a**3/3 - 2*a*x**2 + x/3"],["shape/c78da91a58",6,"differentiate: N*a*x**N + N*a**N*x + N*a**N + N*x"],["thompson-calculus-made-easy-1914/ex-xix/9",4,"Thompson 1914, Exercise XIX (9)"],["form/e5482e3ac9",5,"integrate: (5*x + 1)/(x**2 + x - 2)"],["shape/dec24f1661",6,"integrate: (N*x + 1)/(N + x + x**N)"],["thompson-calculus-made-easy-1914/ex-xix/10",4,"Thompson 1914, Exercise XIX (10)"],["form/b9370fe4f6",5,"integrate: (x**2 - 3)/(x**3 - 7*x + 6)"],["shape/00b8a56e44",6,"integrate: (N + x**N)/(N*x + N + x**N)"],["thompson-calculus-made-easy-1914/ex-xix/11",4,"Thompson 1914, Exercise XIX 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2*x - 2*log(a) - log(x)"],["shape/0ef8cedaa7",6,"extremum: N*x + N*log(a) + a - log(x)"],["thompson-calculus-made-easy-1914/ex-xv/11y",4,"Thompson 1914, Exercise XV (11y)"],["form/3e828cff47",5,"extremum: 2*a + x - log(a) - 2*log(x)"],["shape/c28a9fb51b",6,"extremum: N*a + N*log(x) + x - log(a)"],["thompson-calculus-made-easy-1914/ex-xv/121",4,"Thompson 1914, Exercise XV (121)"],["form/7ca0d9a63b",5,"extremum: 3*2**(2/3)*a**(2/3)/sin(x)**(1/3)"],["shape/ed11a6b43b",6,"extremum: N*N**N*a**N*sin(x)**N"],["thompson-calculus-made-easy-1914/ex-xvii/2",4,"Thompson 1914, Exercise XVII (2)"],["thompson-calculus-made-easy-1914/ex-xvii/3",4,"Thompson 1914, Exercise XVII (3)"],["form/8f403b501b",5,"integrate: x**3/a"],["shape/a97cfb6671",6,"integrate: x**N/a"],["thompson-calculus-made-easy-1914/ex-xv/122",4,"Thompson 1914, Exercise XV (122)"],["thompson-calculus-made-easy-1914/ex-xvi/3",4,"Thompson 1914, Exercise XVI (3)"],["form/fec22c05db",5,"evaluate: log(13/10)"],["shape/30a21d61ab",6,"evaluate: log(N)"],["form/659eac3389",5,"integrate: x/4"],["thompson-calculus-made-easy-1914/ex-xvi/4b",4,"Thompson 1914, Exercise XVI (4b)"],["form/83d40b1696",5,"integrate: cos(x)"],["thompson-calculus-made-easy-1914/ex-xvii/4",4,"Thompson 1914, Exercise XVII (4)"],["form/1c563c5733",5,"integrate: a + x**2"],["shape/23fa07454a",6,"integrate: a + x**N"],["thompson-calculus-made-easy-1914/ex-xvii/5",4,"Thompson 1914, Exercise XVII (5)"],["form/7440f4a504",5,"integrate: 5/x**(7/2)"],["thompson-calculus-made-easy-1914/ex-xvii/6",4,"Thompson 1914, Exercise XVII (6)"],["form/3245be5891",5,"integrate: 4*x**3 + 3*x**2 + 2*x + 1"],["shape/122c1b3482",6,"integrate: N*x + 2*N*x**N + 1"],["thompson-calculus-made-easy-1914/ex-xvii/7",4,"Thompson 1914, Exercise XVII (7)"],["shape/4a74e021c1",6,"integrate: N*a*x + N*b*x**N + N*c*x**N"],["thompson-calculus-made-easy-1914/ex-xvii/8",4,"Thompson 1914, Exercise XVII (8)"],["form/5166826a6f",5,"integrate: (a + x**2)/(a + x)"],["shape/a923256e1a",6,"integrate: (a + x**N)/(a + x)"],["thompson-calculus-made-easy-1914/ex-xvii/9",4,"Thompson 1914, Exercise XVII (9)"],["form/cd318349cf",5,"integrate: (x + 3)**3"],["shape/08ded38c3e",6,"integrate: (N + x)**N"],["thompson-calculus-made-easy-1914/ex-xvii/10",4,"Thompson 1914, Exercise XVII (10)"],["form/c81006f98a",5,"integrate: (-a + x)*(x + 2)"],["shape/4d6a4ba89c",6,"integrate: (N + x)*(-a + x)"],["thompson-calculus-made-easy-1914/ex-xvii/11",4,"Thompson 1914, Exercise XVII (11)"],["thompson-calculus-made-easy-1914/ex-xvii/12",4,"Thompson 1914, Exercise XVII (12)"],["thompson-calculus-made-easy-1914/ex-xvii/13",4,"Thompson 1914, Exercise XVII (13)"],["shape/2c4b80597d",6,"integrate: cos(a*x)**N"],["thompson-calculus-made-easy-1914/ex-xvii/14",4,"Thompson 1914, Exercise XVII (14)"],["form/93360f7717",5,"integrate: sin(x)**2"],["shape/ffe42642bb",6,"integrate: sin(x)**N"],["thompson-calculus-made-easy-1914/ex-xvii/15",4,"Thompson 1914, Exercise XVII 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6))/6"],["shape/da6f4878b2",6,"evaluate: N*Integral(N + x + x**N, (x, 0, N))"],["cap/cas.defint",17,"cas.defint"],["cap/core.integ.num",17,"core.integ.num"],["thompson-calculus-made-easy-1914/ex-xviii/2",4,"Thompson 1914, Exercise XVIII (2)"],["form/ba59d5754d",5,"evaluate: Integral(2*a*sqrt(x), (x, 0, a))"],["shape/3623f5feea",6,"evaluate: Integral(N*a*x**N, (x, 0, a))"],["thompson-calculus-made-easy-1914/ex-xviii/3a",4,"Thompson 1914, Exercise XVIII (3a)"],["shape/8a3343f8b2",6,"evaluate: Integral(sin(x), (x, 0, pi))"],["thompson-calculus-made-easy-1914/ex-xviii/3b",4,"Thompson 1914, Exercise XVIII (3b)"],["form/f3c5961945",5,"evaluate: Integral(sin(x), (x, 0, pi))/pi"],["shape/f3c5961945",6,"evaluate: Integral(sin(x), (x, 0, pi))/pi"],["thompson-calculus-made-easy-1914/ex-xviii/4a",4,"Thompson 1914, Exercise XVIII (4a)"],["form/f9faae4e93",5,"evaluate: Integral(sin(x)**2, (x, 0, pi))"],["shape/43a2f0071d",6,"evaluate: Integral(sin(x)**N, (x, 0, pi))"],["thompson-calculus-made-easy-1914/ex-xviii/4b",4,"Thompson 1914, Exercise XVIII (4b)"],["form/b045fe4ce6",5,"evaluate: Integral(sin(x)**2, (x, 0, pi))/pi"],["shape/73d2d682a5",6,"evaluate: Integral(sin(x)**N, (x, 0, pi))/pi"],["thompson-calculus-made-easy-1914/ex-xviii/5a",4,"Thompson 1914, Exercise XVIII (5a)"],["shape/854dbbd1ae",6,"evaluate: Integral(N*x**N, (x, 0, 1))"],["thompson-calculus-made-easy-1914/ex-xviii/5b",4,"Thompson 1914, Exercise XVIII (5b)"],["form/dfa9cba5ad",5,"evaluate: Integral(-x**(5/2) + x**2, (x, 0, 1))"],["shape/d06f0b6a51",6,"evaluate: Integral(0, (x, 0, 1))"],["thompson-calculus-made-easy-1914/ex-xviii/6",4,"Thompson 1914, Exercise XVIII (6)"],["form/729ad56a63",5,"evaluate: Integral(pi*b**2*x**2/a**2, (x, 0, a))"],["shape/dabde37bdc",6,"evaluate: Integral(pi*a**N*b**N*x**N, (x, 0, a))"],["thompson-calculus-made-easy-1914/ex-xviii/7",4,"Thompson 1914, Exercise XVIII (7)"],["form/110e93772c",5,"evaluate: Integral(x**3 - log(x), (x, 0, 1))"],["thompson-calculus-made-easy-1914/ex-xviii/8",4,"Thompson 1914, Exercise XVIII (8)"],["thompson-calculus-made-easy-1914/ex-xviii/9a",4,"Thompson 1914, Exercise XVIII (9a)"],["form/26d41e36d5",5,"evaluate: Integral(pi*sin(x)**2, (x, 0, pi))"],["shape/0919a5828e",6,"evaluate: Integral(pi*sin(x)**N, (x, 0, pi))"],["thompson-calculus-made-easy-1914/ex-xviii/9b",4,"Thompson 1914, Exercise XVIII (9b)"],["form/fda9abc739",5,"evaluate: Integral(2*pi*sqrt(cos(x)**2 + 1)*sin(x), (x, 0, pi))"],["shape/bb5f74b7b8",6,"evaluate: Integral(pi*N*(cos(x)**N + 1)**N*sin(x), (x, 0, pi))"],["cap/core.hyp",17,"core.hyp"],["thompson-calculus-made-easy-1914/ex-xviii/10a",4,"Thompson 1914, Exercise XVIII (10a)"],["form/f0ad6f0a2c",5,"evaluate: Integral(a/x, (x, 1, a))"],["shape/f0ad6f0a2c",6,"evaluate: Integral(a/x, (x, 1, a))"],["thompson-calculus-made-easy-1914/ex-xviii/10b",4,"Thompson 1914, Exercise XVIII (10b)"],["form/a75647e912",5,"evaluate: Integral(a/x, (x, 1, a))/(a - 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3)))/3"],["shape/32bfcb14fb",6,"evaluate: N*N**N*Integral((N*x + N + x**N)**N, (x, 0, N))**N"],["thompson-calculus-made-easy-1914/ex-xviii/13a",4,"Thompson 1914, Exercise XVIII (13a)"],["form/293ab2d110",5,"evaluate: sqrt(2)*sqrt(Integral((a*sin(x) + a*sin(3*x))**2, (x, 0, 2*pi)))/(2*sqrt(pi))"],["shape/0a8e241aaa",6,"evaluate: pi**N*N*N**N*Integral((a*sin(x) + a*sin(N*x))**N, (x, 0, pi*N))**N"],["thompson-calculus-made-easy-1914/ex-xviii/13b",4,"Thompson 1914, Exercise XVIII (13b)"],["form/e0b08f6752",5,"evaluate: Integral(a*sin(x) + a*sin(3*x), (x, 0, 2*pi))/(2*pi) at A_1=2"],["shape/fed8427e82",6,"evaluate: N*Integral(a*sin(x) + a*sin(N*x), (x, 0, pi*N))/pi"],["thompson-calculus-made-easy-1914/ex-xviii/14a",4,"Thompson 1914, Exercise XVIII (14a)"],["form/72eee77fa6",5,"evaluate: Integral(171*exp(21*x/100)/50, (x, 2, 8))"],["shape/acf6fd49e7",6,"evaluate: Integral(N*exp(N*x), (x, N, N))"],["thompson-calculus-made-easy-1914/ex-xviii/14b",4,"Thompson 1914, Exercise XVIII (14b)"],["form/a7b9ae769b",5,"evaluate: Integral(171*exp(21*x/100)/50, (x, 2, 8))/6"],["shape/87d9674b6f",6,"evaluate: N*Integral(N*exp(N*x), (x, N, N))"],["thompson-calculus-made-easy-1914/ex-xviii/15",4,"Thompson 1914, Exercise XVIII (15)"],["thompson-calculus-made-easy-1914/ex-xviii/16",4,"Thompson 1914, Exercise XVIII 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