Mr. Brougham's general Theorems 
x ( ~ x arc . sin. \/ — -j- ■ a y s/ ax — .r 2 j — ~ {^x — c) 
is equal to ~ x x x arc . qo° ; and, by the quadrature 
of the circle, — x arc . sin. — + ^-=: - x s/ ax — x\ is the 
4 a 4 
area whose abscissa is x; consequently, the semicircle's area is 
— x arc . qo°. But the areas of ellipses are, to the corresponding 
areas of the circles described on their transverse axes, as the 
conjugate to the transverse; therefore — x x arc . sin. ^ ~ 
4- x V a x — x\ is the area whose abscissa is x, of a 
semi-ellipse whose axes are a and b; and, consequently* 
x — x arc . go 0 is the area of the semi-ellipse. Wherefore, 
the area APM — (x — c) is equal to of AMFB. But 
(x — c) | = “ x AP — AC = x PC) is the triangle 
CPM; consequently, APM — - CPM, or ACM, is equal to - 
x AMFB; and ACM : AMFB :: M : M N; or ( dividendo ) 
ACM : CMFB : : 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 solution it may be 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, 
whereby 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 construction, as Sir I. Newton's does. 
I was induced to give the demonstration, from a conviction that 
4t is a good instance of the superiority of modern over ancient 
