IN THE HIGHER GEO3IETET. 21 



y 2 T* -_, n \ &i A 



sin V - -j x J (ax X*}] ?-(x c) is equal to - 



d 4 J a 



X 7 X - X arc 90 ; and, by the quadrature of the circle, 



a 8 x 2 x a 



X arc SID., v -I : X J (a x X s ), is the area 



4 (2 4 



whose abscissa is x ; consequently the semicircle's area is 



X arc 90. But the areas of ellipses are to the corresponding 

 areas of the circles described on their transverse axes, as the 



T / g 



conjugate to the transverse ; therefore - X ( X arc sin. 



a \ 4 



J | X V (fl ^ X s ) ] is the area whose abscissa is 



a 4 ' J 



x, of a semi-ellipse, whose axes are a and b ; and consequently 



b a* 



- X -7- X arc 90 is the area of the semi-ellipse. Therefore 

 a 4 



I/ TVf 



the area A p M ^ (x c) is equal to of A M F B. But 



2 M -|- N 



- (x c) = ~ x (A p A c) = ~ X P c, is the triangle 

 c P M ; consequently, A p M c p M, or ACM, is equal to 



M -f- N 



X Ay . B; and ACM : AMFB :: M : M + N; or (dividendo) 

 ACM : c M F B : : M : N ; and the area of the ellipse is cut in 

 a given ratio by the line drawn through the focus. Q. E. D. 



Of this s-. luti it maybe remarked, that it does not assume 

 as a postulate the description of the cycloid ; but gives a 

 simple construction of that curve, flowing from a curious 

 property, by which it is related to a given circle. This 

 cycloid, too, gives, by its intersection with the ellipse, the 

 point required, directly, and not by a subsequent construc- 

 tion, as Sir Isaac Newton's does. I was induced to give the 

 demonstration, from a conviction that it is a good instance of 

 the superiority of modern over ancient analysis ; and in itself 

 perhaps no inelegant specimen of algebraic demonstration. 



