Prop VIII. (fig. 10.) If from any point F of the hyperbola, there is drawn to the transverse diameter, AR, a right line or dinately applied to it F G ; and from the extremity of the diameter there is drawn AH perpendicular to it, and equal to the latus rectum ; the square of the ordinate shall be equal to the rectangle applied to the latus rectum, being of the breadth of the abscissa between the ordinate and the vertex, and which exceeds it by a figure like and alike situated to that which is contained under the diameter and the latus rectum.
For join BH, and from the point G let there be drawn GM parallel to All, and meeting BH in M, and through Id let there be drawn MN parallel to AB, meet ing All in N, and let the rectangles MNHO, BAHP, be completed. Then since the rectangle AGB is to the square of GF, as AB is to AH, i. e. as GB is to GM, i e. as the rectangle A G B is to the rectangle" AGM; A G B shall be to the square of GF, as the same AGB to the rectangle AGM: wherefore the square of G F is equal to the rectangle A G M, which is applied to the latus rectum, AH, having the breadth AG, and exceeds the rectangle HA GO by the rectangle MNHO, like to BAHP ; from which ex. cess the name of hyperbola was given to this curve by Apolhedus.
Prob. 1. Au easy method to describe the hyperbola, fig. 11. having the trans verse diameter, 11 E, and the foci N n given. From N, at any distance, as N F, strike an arch ; and with the same open ing of the compasses with one foot in E, the vertex, set off EG equal to NF in the axis continued ; then with the distance GD, and one foot in n, the other focus, cross the former arch in F. So F is a
point in the hyperbola : and by this me thod repeated may be found any other point f, further on, and.as many more as you please.
An asymptote being taken for a diame ter ; divided into equal parts, and through all the divisions, which form so many abscisses continually increasing equally, t• ordinates to the curve being drawn paral lel to the other symptote ; the absciss es will represent an infinite series of na tural numbers, and the corresponding hyperbolic or asymptotic spaces will re present the series of logarithms of the same number. Hence different hyperbo las will furnish different series of loga rithms; so that to determine any particu lar series of logarithms, choice must be made of some particular hyperbola. Now the most simple of all hyperbolas is the equilateral one, i. e. that whose asymp totes make a right angle between them selves.
Equilateral hyperbola is that wherein the conjugate axes are equal.
Apollonian hyperbola is the common hyperbola, or the hyperbola of the first kind : thus called in contradistinction to the hyperbolas of the higher kinds, or infinite hyperbolas : for the hyperbo la of the first kind, or order, has two asymptotes ; that of the second order has three ; that of the third four, &c.