Spherical Trigonometry, for the Use of Colleges and Schools
Index of terms
Every concept, theorem, method and person the concept graph places in a chapter of this book, with the chapters that teach it and the other books on the shelf that teach it too.
Concepts (85)
- ambiguous case
Taught in: Solution of Right-angled Triangles, Solution of Oblique-Angled Triangles, Numerical Solution of Spherical Triangles
Also in: Elements of Plane Trigonometry, Plane Geometry
- angular coordinates on a sphere
Taught in: On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Miscellaneous Propositions
- angular radius
Taught in: Circumscribed and Inscribed Circles, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
- approximation
Taught in: On certain approximate Formul\ae, Geodetical Operations
Also in: A Course of Pure Mathematics, An Introduction to Mathematics, Calculus Made Easy, Elementary Illustrations of the Differential and Integral Calculus, Mathematical Recreations and Essays, Plane Geometry
- area
Taught in: On certain approximate Formul\ae, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Polyhedrons
Also in: A Course of Pure Mathematics, Calculus Made Easy, Elementary Illustrations of the Differential and Integral Calculus, Mathematical Recreations and Essays, Plane Geometry, Vector Analysis and Quaternions
- associated triangles
Taught in: Circumscribed and Inscribed Circles
- axis of a circle on a sphere
Taught in: Great and Small Circles
- base line
Taught in: Geodetical Operations
Also in: Elements of Plane Trigonometry
- chord
Taught in: On certain approximate Formul\ae
Also in: A Course of Pure Mathematics, Elementary Illustrations of the Differential and Integral Calculus, Plane Geometry
- chordal triangle
Taught in: On certain approximate Formul\ae, Geodetical Operations
- circle
Taught in: Great and Small Circles, Miscellaneous Propositions
Also in: A Course of Pure Mathematics, An Introduction to Mathematics, Elementary Illustrations of the Differential and Integral Calculus, First Course in the Theory of Equations, Mathematical Recreations and Essays, Plane Geometry, Solid Geometry with Problems and Applications
- circular measure
Taught in: Spherical Triangles, Spherical Geometry, Area of a Spherical Triangle. Spherical Excess, On certain approximate Formul\ae, Geodetical Operations, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
Also in: A Course of Pure Mathematics, Elements of Plane Trigonometry
- circular parts
Taught in: Solution of Right-angled Triangles
- circumscribed circle
Taught in: Circumscribed and Inscribed Circles, Miscellaneous Propositions
Also in: A Course of Pure Mathematics, Plane Geometry
- cosine
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Right-angled Triangles, On certain approximate Formul\ae, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Numerical Solution of Spherical Triangles
Also in: A Course of Pure Mathematics, An Introduction to Mathematics, Calculus Made Easy, Elementary Illustrations of the Differential and Integral Calculus, Elements of Plane Trigonometry, Solid Geometry with Problems and Applications, Vector Analysis and Quaternions
- cotangent
Taught in: Solution of Right-angled Triangles, On certain approximate Formul\ae, Numerical Solution of Spherical Triangles
Also in: A Course of Pure Mathematics, Calculus Made Easy, Elements of Plane Trigonometry
- diagonal
Taught in: Polyhedrons
Also in: Elementary Illustrations of the Differential and Integral Calculus, Mathematical Recreations and Essays, Plane Geometry
- diameter
Taught in: Great and Small Circles, On certain approximate Formul\ae
Also in: Plane Geometry, Solid Geometry with Problems and Applications
- dodecahedron
Taught in: Polyhedrons
- edge
Taught in: Polyhedrons, Miscellaneous Propositions
Also in: Mathematical Recreations and Essays, Solid Geometry with Problems and Applications
- equilateral triangle
Taught in: On certain approximate Formul\ae
Also in: A Course of Pure Mathematics, Plane Geometry
- escribed circle
Taught in: Miscellaneous Propositions
Also in: Plane Geometry
- excircle
Taught in: Circumscribed and Inscribed Circles, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Miscellaneous Propositions
Also in: Elements of Plane Trigonometry
- existence conditions of a triangle
Taught in: Solution of Right-angled Triangles, Solution of Oblique-Angled Triangles
- face
Taught in: Polyhedrons, Miscellaneous Propositions
Also in: Mathematical Recreations and Essays, Solid Geometry with Problems and Applications
- figure of the Earth
Taught in: Geodetical Operations
- geodesy
Taught in: Geodetical Operations
Also in: Elements of Plane Trigonometry
- great circle
Taught in: Great and Small Circles, Spherical Triangles, Spherical Geometry, Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, On certain approximate Formul\ae, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Polyhedrons, Miscellaneous Propositions
Also in: Mathematical Recreations and Essays, Solid Geometry with Problems and Applications
- hexahedron
Taught in: Polyhedrons
- horizontal angle
Taught in: Geodetical Operations
Also in: Elements of Plane Trigonometry
- hypotenuse
Taught in: Solution of Right-angled Triangles
Also in: A Course of Pure Mathematics, Elementary Illustrations of the Differential and Integral Calculus, Plane Geometry
- icosahedron
Taught in: Polyhedrons
- inclination of two lines
Taught in: Polyhedrons
Also in: Elements of Plane Trigonometry
- inscribed circle
Taught in: Circumscribed and Inscribed Circles, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Miscellaneous Propositions
Also in: Elements of Plane Trigonometry, Plane Geometry
- isosceles spherical triangle
Taught in: Spherical Geometry, Solution of Oblique-Angled Triangles
- locus
Taught in: Miscellaneous Propositions
Also in: A Course of Pure Mathematics, An Introduction to Mathematics, Plane Geometry, Solid Geometry with Problems and Applications
- logarithm
Taught in: Numerical Solution of Spherical Triangles
Also in: A Course of Pure Mathematics, Calculus Made Easy, Elementary Illustrations of the Differential and Integral Calculus, Elements of Plane Trigonometry, Mathematical Recreations and Essays
- lune
- mechanical equilibrium
Taught in: Miscellaneous Propositions
Also in: Treatise on Thermodynamics
- nine points circle
Taught in: On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
- observational error
Taught in: Geodetical Operations, On small variations in the parts of a Spherical Triangle
- octahedron
Taught in: Polyhedrons
- parallelepiped
Taught in: Polyhedrons
Also in: Solid Geometry with Problems and Applications, Vector Analysis and Quaternions
- perpendicular
Taught in: On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
Also in: Elementary Illustrations of the Differential and Integral Calculus, Plane Geometry, Solid Geometry with Problems and Applications
- plane angle
Taught in: Polyhedrons
Also in: Plane Geometry, Solid Geometry with Problems and Applications
- plane triangle
Taught in: On certain approximate Formul\ae, Geodetical Operations, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
Also in: Treatise on Thermodynamics
- plane trigonometry
Taught in: Geodetical Operations, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
Also in: An Introduction to Mathematics
- polar triangle
Taught in: Spherical Geometry, Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Oblique-Angled Triangles, Circumscribed and Inscribed Circles, On small variations in the parts of a Spherical Triangle, Miscellaneous Propositions, Numerical Solution of Spherical Triangles
- pole of a circle
Taught in: Great and Small Circles, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Miscellaneous Propositions
- polygon
Taught in: Polyhedrons
Also in: Elementary Illustrations of the Differential and Integral Calculus, Plane Geometry, Vector Analysis and Quaternions
- polyhedral angle
Taught in: Spherical Triangles, Spherical Geometry, Polyhedrons, Miscellaneous Propositions
- polyhedron
Taught in: Polyhedrons, Miscellaneous Propositions
- polyhedron inscribed in a sphere
Taught in: Polyhedrons
- primitive triangle
Taught in: Spherical Geometry
- quadrant
Taught in: Great and Small Circles, Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Right-angled Triangles, Solution of Oblique-Angled Triangles, On certain approximate Formul\ae, On small variations in the parts of a Spherical Triangle, Miscellaneous Propositions
Also in: An Introduction to Mathematics, Plane Geometry, Solid Geometry with Problems and Applications, Vector Analysis and Quaternions
- quadrantal triangle
Taught in: Arcs drawn to fixed points on the Surface of a Sphere
- re-entrant angle
Taught in: Miscellaneous Propositions
Also in: Plane Geometry
- regular polyhedron
Taught in: Polyhedrons, Miscellaneous Propositions
- right spherical triangle
Taught in: Solution of Right-angled Triangles, Solution of Oblique-Angled Triangles, Numerical Solution of Spherical Triangles
- rotation
Taught in: Miscellaneous Propositions
Also in: A Course of Pure Mathematics, Vector Analysis and Quaternions
- same affection
Taught in: Solution of Right-angled Triangles
- secondary to a great circle
Taught in: Great and Small Circles
- side
Taught in: Spherical Triangles, Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Right-angled Triangles, On small variations in the parts of a Spherical Triangle, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
Also in: Plane Geometry
- sine
Taught in: Great and Small Circles, Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Right-angled Triangles, On certain approximate Formul\ae, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Numerical Solution of Spherical Triangles
Also in: A Course of Pure Mathematics, An Introduction to Mathematics, Calculus Made Easy, Elementary Illustrations of the Differential and Integral Calculus, Elements of Plane Trigonometry, Mathematical Recreations and Essays, Solid Geometry with Problems and Applications, Vector Analysis and Quaternions
- sine of a solid angle
Taught in: Miscellaneous Propositions
- small circle
Taught in: Great and Small Circles, Circumscribed and Inscribed Circles, On certain approximate Formul\ae, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Miscellaneous Propositions
- small variation
Taught in: On small variations in the parts of a Spherical Triangle
- sphere
Taught in: Great and Small Circles
Also in: A Course of Pure Mathematics, An Elementary Treatise on Electricity, Mathematical Recreations and Essays, Solid Geometry with Problems and Applications
- sphere inscribed in a polyhedron
Taught in: Polyhedrons
- spherical angle
Taught in: Great and Small Circles, Spherical Triangles, Spherical Geometry, Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Right-angled Triangles, Circumscribed and Inscribed Circles, On small variations in the parts of a Spherical Triangle, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
- spherical distance
Taught in: Great and Small Circles, Miscellaneous Propositions
- spherical polygon
Taught in: Spherical Triangles, Area of a Spherical Triangle. Spherical Excess
- spherical triangle
Taught in: Spherical Triangles, Spherical Geometry, Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Right-angled Triangles, Solution of Oblique-Angled Triangles, Circumscribed and Inscribed Circles, Area of a Spherical Triangle. Spherical Excess, On certain approximate Formul\ae, Geodetical Operations, On small variations in the parts of a Spherical Triangle, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Polyhedrons, Miscellaneous Propositions
Also in: Solid Geometry with Problems and Applications, Vector Analysis and Quaternions
- spherical trigonometry
Taught in: Spherical Triangles, Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Right-angled Triangles, On certain approximate Formul\ae, Geodetical Operations, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
Also in: Mathematical Recreations and Essays, Vector Analysis and Quaternions
- spheroid
Taught in: Geodetical Operations
- station
Taught in: Geodetical Operations
Also in: Elements of Plane Trigonometry
- supplementary angles
Taught in: Spherical Geometry
Also in: Elements of Plane Trigonometry
- surface of a polyhedron
Taught in: Polyhedrons
- symmetrical spherical triangles
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Area of a Spherical Triangle. Spherical Excess
- symmetry
Taught in: Miscellaneous Propositions
Also in: Plane Geometry
- tangent function
Taught in: Solution of Right-angled Triangles, Numerical Solution of Spherical Triangles
Also in: A Course of Pure Mathematics, Calculus Made Easy, Elementary Illustrations of the Differential and Integral Calculus, Elements of Plane Trigonometry, Mathematical Recreations and Essays, Solid Geometry with Problems and Applications
- temperature
Taught in: Geodetical Operations
Also in: An Elementary Treatise on Electricity, Mathematical Recreations and Essays, Treatise on Thermodynamics
- tetrahedron
Taught in: Polyhedrons
Also in: Solid Geometry with Problems and Applications, Vector Analysis and Quaternions
- triangle
Taught in: Geodetical Operations
Also in: A Course of Pure Mathematics, An Introduction to Mathematics, Elements of Plane Trigonometry, Plane Geometry
- vertex
Taught in: Polyhedrons, Miscellaneous Propositions
Theorems (48)
- angles opposite equal sides of a spherical triangle are equal
Taught in: Spherical Geometry
- arc from pole of great circle to circumference is a quadrant
Taught in: Great and Small Circles
- arc of a small circle compared with arc of a great circle
Taught in: Great and Small Circles
- arc through midpoints of two sides of a spherical triangle
Taught in: Miscellaneous Propositions
- area of a lune
- area of a sphere
- area of a spherical polygon
- area of a spherical triangle
Taught in: Area of a Spherical Triangle. Spherical Excess, Geodetical Operations
- Cagnoli's theorem
- Cauchy's network theorem
Taught in: Miscellaneous Propositions
- concurrence of altitudes
Taught in: On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
- concurrence of altitudes of a spherical triangle
Taught in: On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Miscellaneous Propositions
- concurrence of arcs through a point of a spherical triangle
Taught in: Miscellaneous Propositions
- cosine formula for a side of a spherical triangle
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Oblique-Angled Triangles, On small variations in the parts of a Spherical Triangle
- cosine formula for an angle of a spherical triangle
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Oblique-Angled Triangles
- cotangent formulae for a spherical triangle
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
- Delambre's analogies
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
- Euler's theorem
Taught in: Miscellaneous Propositions
- five regular polyhedra
Taught in: Polyhedrons
- greater angle lies opposite greater side in a spherical triangle
Taught in: Spherical Geometry
- greater side lies opposite greater angle in a spherical triangle
Taught in: Spherical Geometry
- half-angle formulae for a spherical triangle
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Oblique-Angled Triangles, Numerical Solution of Spherical Triangles
- half-side formulae for a spherical triangle
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle
- law of sines
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Oblique-Angled Triangles
Also in: Elements of Plane Trigonometry
- Legendre's theorem
Taught in: On certain approximate Formul\ae, Geodetical Operations
- Lhuilier's theorem
- Napier's analogies
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Oblique-Angled Triangles, On the connexion of Formul\ae\ in Plane and Spherical Trigonometry, Numerical Solution of Spherical Triangles
- Napier's Rules of Circular Parts
Taught in: Solution of Right-angled Triangles
- plane section of a sphere is a circle
Taught in: Great and Small Circles
- points of a circle on a sphere are equidistant from its pole
Taught in: Great and Small Circles
- polar triangle relation is symmetric
Taught in: Spherical Geometry
- polyhedral formula
Taught in: Polyhedrons
- power of a point
Taught in: On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
- right spherical triangle relations
Taught in: Solution of Right-angled Triangles, Numerical Solution of Spherical Triangles
- sides opposite equal angles of a spherical triangle are equal
Taught in: Spherical Geometry
- spherical angle measured by arc of great circle
Taught in: Great and Small Circles
- sum of angles of a spherical triangle lies between two and six right angles
Taught in: Spherical Geometry
- sum of cosines to fixed points varies as the cosine of one arc
Taught in: Arcs drawn to fixed points on the Surface of a Sphere
- sum of cosines to the solid angles of a regular polyhedron is zero
Taught in: Arcs drawn to fixed points on the Surface of a Sphere
- sum of plane angles of a polyhedron
Taught in: Polyhedrons
- sum of sides of a spherical triangle is less than a great circle
Taught in: Spherical Geometry
- sum of squared cosines to the solid angles of a regular polyhedron is one third of their number
Taught in: Arcs drawn to fixed points on the Surface of a Sphere
- sum of squared cosines to the vertices of a quadrantal triangle is one
Taught in: Arcs drawn to fixed points on the Surface of a Sphere
- sum of the angles of a triangle
Taught in: Geodetical Operations
Also in: An Introduction to Mathematics, Plane Geometry
- sum of two sides of a spherical triangle exceeds the third
Taught in: Spherical Geometry
- two great circles bisect each other
Taught in: Great and Small Circles
- volume of a parallelepiped
Taught in: Polyhedrons
Also in: Solid Geometry with Problems and Applications, Vector Analysis and Quaternions
- volume of a tetrahedron
Taught in: Polyhedrons, Miscellaneous Propositions
Methods (24)
- approximate solution of a spherical triangle
Taught in: On certain approximate Formul\ae, Geodetical Operations
- Carnot's method
Taught in: Polyhedrons
- deducing plane trigonometry from spherical trigonometry
Taught in: On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
- expansion in powers
Taught in: On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
- finding the angular radius of the circumscribed circle of a spherical triangle
Taught in: Circumscribed and Inscribed Circles
- finding the angular radius of the inscribed circle of a spherical triangle
Taught in: Circumscribed and Inscribed Circles
- General Roy's rule
Taught in: Geodetical Operations
- limiting case
Taught in: On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
- neglecting higher-order small quantities
Taught in: On small variations in the parts of a Spherical Triangle
Also in: Calculus Made Easy, Elementary Illustrations of the Differential and Integral Calculus
- numerical solution of a spherical triangle
Taught in: Numerical Solution of Spherical Triangles
- reduction to the horizon
Taught in: Geodetical Operations
- solving a right spherical triangle
Taught in: Numerical Solution of Spherical Triangles
- solving a right-angled triangle
Taught in: Solution of Right-angled Triangles, Solution of Oblique-Angled Triangles
Also in: Elements of Plane Trigonometry
- solving a spherical triangle from its three sides
Taught in: Numerical Solution of Spherical Triangles
- solving a triangle
Taught in: Solution of Right-angled Triangles, Solution of Oblique-Angled Triangles
Also in: Elements of Plane Trigonometry
- solving a triangle from its three angles
Taught in: Solution of Oblique-Angled Triangles
- solving a triangle from its three sides
Taught in: Solution of Oblique-Angled Triangles
Also in: Elements of Plane Trigonometry
- solving a triangle from two angles and the included side
Taught in: Solution of Oblique-Angled Triangles
- solving a triangle from two angles and the side opposite one of them
Taught in: Solution of Oblique-Angled Triangles
- solving a triangle from two sides and the angle opposite one of them
Taught in: Solution of Oblique-Angled Triangles
- solving a triangle from two sides and the included angle
Taught in: Solution of Oblique-Angled Triangles
Also in: Elements of Plane Trigonometry
- solving an oblique spherical triangle
Taught in: Numerical Solution of Spherical Triangles
- triangulation
Taught in: Geodetical Operations
Also in: An Introduction to Mathematics
- verification
Taught in: Numerical Solution of Spherical Triangles
Also in: Elementary Illustrations of the Differential and Integral Calculus, The First Steps in Algebra
Quantities (5)
- force
Taught in: Miscellaneous Propositions
Also in: An Elementary Treatise on Electricity, An Introduction to Mathematics, Calculus Made Easy, Elementary Illustrations of the Differential and Integral Calculus, Mathematical Recreations and Essays, Vector Analysis and Quaternions
- radius
Taught in: Great and Small Circles, Geodetical Operations
Also in: An Introduction to Mathematics, Elementary Illustrations of the Differential and Integral Calculus, Elements of Plane Trigonometry, Plane Geometry, Solid Geometry with Problems and Applications
- semi-perimeter
Taught in: Circumscribed and Inscribed Circles
Also in: Elements of Plane Trigonometry
- spherical excess
Taught in: Area of a Spherical Triangle. Spherical Excess, On certain approximate Formul\ae, Geodetical Operations, Miscellaneous Propositions
- volume
Taught in: Polyhedrons
Also in: An Introduction to Mathematics, Mathematical Recreations and Essays, Solid Geometry with Problems and Applications, Treatise on Thermodynamics, Vector Analysis and Quaternions
Units (2)
- degree of angle
Taught in: Spherical Triangles
Also in: Calculus Made Easy, Elementary Illustrations of the Differential and Integral Calculus, Elements of Plane Trigonometry, Plane Geometry
- radian
Taught in: Spherical Triangles
Also in: Calculus Made Easy, Elements of Plane Trigonometry, First Course in the Theory of Equations, Vector Analysis and Quaternions
People (7)
- Adrien Marie Legendre
Taught in: On certain approximate Formul\ae
Also in: Mathematical Recreations and Essays, Solid Geometry with Problems and Applications
- Augustin-Louis Cauchy
Taught in: Miscellaneous Propositions
Also in: A Course of Pure Mathematics, First Course in the Theory of Equations, Mathematical Recreations and Essays
- Delambre
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Geodetical Operations
- Euclid
Taught in: On the connexion of Formul\ae\ in Plane and Spherical Trigonometry
Also in: A Course of Pure Mathematics, Elements of Plane Trigonometry, Mathematical Recreations and Essays
- George Everest
Taught in: Geodetical Operations
- John Napier
Taught in: Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle, Solution of Right-angled Triangles
Also in: A Course of Pure Mathematics, Calculus Made Easy, Elements of Plane Trigonometry
- Leonhard Euler
Taught in: Miscellaneous Propositions
Also in: Elementary Illustrations of the Differential and Integral Calculus, First Course in the Theory of Equations, Mathematical Recreations and Essays, Treatise on Thermodynamics