HYP 



HYP 



that is, as C B l +C N 1 is to C D 1 or 

 GN 1 . 



Prop. VI. (fig. 9.) It is another pro- 

 perty of the hyperbola, that the asymp- 

 totes, D d, E e, do never absolutdy meet 

 with the curve. See ASTMPTOTE. 



Prop. VII. If through any point F (fig. 

 9.) of the hyperbola, there is drawn a 

 right line 1 F L parallel to the second 

 axis, and meeting the asymptotes in I 

 and L; the rectangle contained under 

 the right lines which are intercepted be- 

 tween the asymptotes and the hyper- 

 bola, is equal to the square of the half of 

 the second axis, that is, C B 1 = I F L 

 = I HL. 



Prop VIII. (fig. 10.) If from any point 

 F of the hyperbola, there is drawn to the 

 transverse diameter, AB, a right line or- 

 dinjitely 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 lo 

 the latus rectum, being of the breadth of v 

 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 

 lutiis rectum. 



For join BH, and from the point G let 

 there be drawn GM parallel to AH, and 

 meeting BH in M, and through M let 

 there be drawn MN parallel to AB, meet- 

 ing AH 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; AGB 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 AGM, 

 which is applied to tke latus rectum, AH, 

 having the breadth AG, and exceeds the 

 rectangle H A G O by the rectangle 

 MNHO, like to BAHP ; from which ex. 

 cess the name of hyperbola was given to 

 this curve by Apollqnius. 



Prob. 1. Aii easy method to describe 

 the hyperbola, fig. 11. having the trans- 

 verse diameter, D 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/, 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, y 

 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 hyperbc- 

 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. 



HYPERBOLE, in rhetoric, a figure, 

 whereby the truth and reality of things 

 are excessively either enlarged or di- 

 minished. See RHETORIC. 



HYPERBOLIC, or hyperbolical, some- 

 thing relating either to an hyperbole, or 

 an hyperbola. 



HYPERBOLIC cylindroid, is a solid fi- 

 gure, whose generation is given by Sir 

 Christopher Wren, in the " Philosophical 

 Transactions." Thus, two opposite hy- 

 perbolas being joined by the transverse 

 axis, and through the centre a right line 

 being drawn at right angles to that axis ; 

 and about that, as an axis, the hyperbo- 

 las being supposed to revolve ; by such 

 revolution, a body will be generated, 

 which is called the hyperbolic cylindroid, 

 whose bases, and all sections parallel to 

 them, will be circles. In a subsequent 

 transaction, the same author applies it 

 to the grinding of hyperbolical glasses : 

 affirming that they must be formed this 

 way, or not at all. Hyperbolic leg of a 

 curve, is that which approaches infi- 

 nitely near to some asymptote. Sir Isaac 

 Newton reduces all curves, both of the 

 first and higher kinds, into those with 

 hyperbolic legs, and those with parabolic 



