Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/233



The ame things uppoed, I ay that the area of the figure DES, decribed by the indefinite radius SD, is equal to the area which a body with a radius equal to half the latus rectum of the figure DES, by uniformly revolving about the centre S, may be decribed in the ame time.. Pl. 16. Fig. 1.

For uppoe a body C in the mallet moment of time decribes in falling the infinitely little line Cc, while another body K uniformly revolving about the centre S in the circle OKk, decribes the arc Kk. Erect the perpendiculars CD, cd, meeting the figure DES in D. Join SD, Sd, SK, Sk, and draw Dd meeting the axis AS in T, and thereon let fall the perpendicular SY.

. If the figure DES is a circle or a rectangular hyperbola, biect its tranvere diameter AS in O, and SO will be half the latus rectum. And becaue TC is to TD as Cc to Dd, and TD to TS as CD to ST; ex æquo TC will be to TS, as CD x Cc to ST x Dd. But (by cor. 1. prop. 33) TC is to TS as AC to AO, to wit, if in the coalecence of the points D, d, the ultimate ratio's of the lines are taken. Wherefore AC is to AO or SK as CD x Cc to ST x Dd. Farther, the velocity of the decending body in C is to the velocity of a body decribing a circle about