10 PHILOSOPHICAL fRANSACTIONS. [aNNO 1703. 



ralue of which therefore is —^- — , as found above : therefore it is a fourth 



proportional to the three quantities before exhibited. This being granted, he 

 evidently demonstrates, that lf is the least of all the lines that can be drawn 



froin the point l to the ellipsis. Moreover, because it is q : g —■ r ::f:f —-. 



Therefore the rectangle^— -^ ci y f — -is the same rectangle which 



De la Hire calls his specimen. But this specimen, according to his definition, is 

 a rectangle like to the rectangle that constitutes the difference between the 

 square of the transverse axis and the figure, that is, the rectangle gq — qr, 

 being besides applied to the right line bd or y. Now that the rectangle Jf — 



— has all these conditions, is very evident. It may be observed, that it follows, 



from the value of n before found, that w >> -. For n = - + ^ • There* 



fore qn -{- rx =z ^ qr -\- qx. But because 9 >► r, it is qx >- rx, and therefore 

 qn >► ^qry and w > -J-r. 



When the point p (as was just now observed in the parabola) falls on the 

 vertex a. the minimum is determined in the axis : and because of x Vanishing, 

 we shall have n = -^r. Then assuming any point d between a and l, if ad be 



equal to any r, by comparison there arises pl^ — al'^ = .rx — — . And this 



is theor. 3, lib. 7, of De la Hire's Conies. For because it is q : q —- r : : x : x 



— — , it appears that the rectangle xx is the exemplar, but applied to 



the absciss x : and therefore this is the adequate measure of the defect of the 

 square of the least line, from the square of any other right line drawn from the 

 same point to the curve. And this is what he demonstrates in the place above 

 cited. 



Now the theorems belonging to the lesser or conjugate axis of the ellipsis, 

 (for hitherto we have insisted on the greater or transverse axis) are determined 

 just in the same manner. For now let ak, or half the lesser axis, be ^, k the 

 parameter -, and the point l is now supposed to be placed beyond the centre, on 

 the other side of gk. By proceeding as before, we shall find al or n = 



— + a: — — , and the subnormal, dl = - — — . That is, c : r : : - — 



»' 2 ' c2 2 



dp : J — — ; and therefore, drawing pl, it will be the greatest of all the lines 

 that can be drawn from the point l to the ellipsis; and lp' — le* = 

 ?=• —. ^ = to the rectangle, which is the exemplar, applied to bd or /. For 



