Elementary Illustrations of the Differential and Integral Calculus

On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes

Excerpts

On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
And conversely it may be proved by any number of examples, that when an equation in which $a$ occurs has been deduced strictly on the supposition that $a$ is a line measured in one direction, a change of sign in $a$ will turn the equation into that which would have been deduced by the same reasoning, had we begun by measuring the line $a$ in the contrary direction.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
Hence the equation $ay + bx = ab$ belongs to all parts of the straight line $EF$, if we agree to consider $M''P''$ as negative, when $MP$ is positive, and $OM'$ as negative when $OM$ is positive.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
In this manner, if four points be taken similarly situated in the four angles, the numerical values of whose co-ordinates are $x = 4$ and $y = 6$, and if the co-ordinates of that point which lies in the angle $AOB$, are called $+4$ and $+6$; those of the points lying in the angle $BOC$ will be $-4$ and $+6$; in the angle $COD$ $-4$ and $-6$; and in the angle $DOE$ $+4$ and $-6$.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
The latter method is preferable, inasmuch as it enables us to contain, in one investigation, all the different cases of a problem.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
Thus, if $OE = 4$, and $OF = 5$, and $OM = 1$, we can determine $MP$ from the equation $ay + bx = ab$, or $4y + 5 = 20$, which gives $y$ or $MP = 3\frac{3}{4}$.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
And thus, if $y$ be any function of $x$, we can obtain a geometrical representation of the same, by making $y$ the ordinate, and $x$ the abscissa of a curve, every ordinate of which shall be the linear representation of the numerical value of the given function corresponding to the numerical value of the abscissa, the linear unit being a given line.

Equations

On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
x^{2} + y^{2} = r^{2}
A point whose co-ordinates are x and y lies on a circle of radius r centred at the origin O exactly when x squared plus y squared equals r squared.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
ay + bx = ab
The straight line EF, cutting the axes at distances a and b from O, satisfies ay + bx = ab for every one of its points, with the signs of x and y taken in the stated directions.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
ay - bx = ab
For a point P' on EF produced, with x taken as the signed co-ordinate, the relation is ay - bx = ab, which is the earlier equation with the sign of x changed.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
bx - ay = ab
For a point P'' on FE produced, the relation is bx - ay = ab, which is the earlier equation with the sign of y changed.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
x^{2} + b^{2}\, \frac{(a - x)^{2}}{a^{2}} = r^{2}
Substituting the value of y from the line equation into the circle equation gives this equation in x alone.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
(a^{2} + b^{2}) x^{2} - 2ab^{2}x + a^{2}(b^{2} - r^{2}) = 0
The abscissas x of the points where the line meets the circle are the roots of this quadratic equation in x.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
(a^{2} + b^{2}) y^{2} - 2a^{2}by + b^{2}(a^{2} - r^{2}) = 0
The ordinates y of the points where the line meets the circle are the roots of this quadratic equation in y.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
x = a\, \frac{b^{2} ± \sqrt{(a^{2} + b^{2})r^{2} - a^{2}b^{2}}}{a^{2} + b^{2}}
The abscissas of the two intersection points of the line and the circle, taking the upper or the lower sign throughout.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
y = b\, \frac{a^{2} \mp \sqrt{(a^{2} + b^{2})r^{2} - a^{2}b^{2}}}{a^{2} + b^{2}}
The ordinates of the two intersection points of the line and the circle, with the sign of the root taken opposite to that in the formula for x.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
(a^{2} + b^{2})r^{2} > a^{2}b^{2}
The line meets the circle in two points exactly when r is greater than the perpendicular from O to EF, whose length is ab over the square root of a squared plus b squared.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
(a^{2} + b^{2})r^{2} = a^{2}b^{2}
The two intersection points coincide and the straight line EF is a tangent to the circle.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
(a^{2} + b^{2})r^{2} < a^{2}b^{2}
The values of x and y are impossible and the straight line does not meet the circle.
This equation is in Algebraical Geometry (Algebraical Geometry)
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
(-x)^{2} = x^{2}
Squaring a negative value of x gives the same result as squaring the positive value, so the left branch of the curve mirrors the right.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
OE = a
The length OE, measured along the axis of x, is called a and is the intercept of the straight line EF on that axis.
On the Connexion of the Signs of Algebraical and the Directions of Geometrical Magnitudes
OF = b
The length OF, measured along the axis of y, is called b and is the intercept of the straight line EF on that axis.

Problems

No exercises in this chapter.