Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/116

 meeting in D and E: And the right lines TD, VE produced, will meet in S the centre required.

For the perpendiculars let fall from the centre S on the tangents PT, QT, are reciprocally as the velocities of the bodies in the points P and Q (by cor. 1. prop. 1.) and therefore, by conruction, as the perpendiculars AP, BQ directly; that is, as the perpendiculars let fall from the point D on the tangents. Whence it is eay to infer, that the points S, D, T, are in one right line. And by the like argument the points S, E, V are alo in one right line; and therefore the centre S is in the point where the right lines TD, VE meet. Q. E. D.

In a pace void of reiŧance, if a body revolves in any orbit about an immoveable centre, and in the leat time decribes any arc jut then nacent; and the vered ine of that arc is uppofed to be drawn, biecting the chord, and produced paing through the centre of force: the centripetal force in the middle of the arc, will be as the vered ine directly and the quare of the time inverely.

For the vered ine in a given time is as the force (by cor. 4. prop. 1.) and augmenting the time in any ratio, becaue the arc will be augmented in the ame ratio, the vered ine will be augmented in the duplicate of that ratio, (by cor. 2 and 3. lem. 2.) and therefore is as the force and the quare of the time. Subduct on both ides the duplicate ratio of the time, and the force