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 conſruction, as the perpendiculars AP, BQ directly; that is, as the perpendiculars let fall from the point D on the tangents. Whence it is eaſy to infer, that the points S, D, T, are in one right line. And by the like argument the points S, E, V are alſo in one right line; and therefore the centre S is in the point where the right lines TD, VE meet. Q. E. D.
Proposition VI. Theorem V.
In a ſpace void of reſiſŧance, if a body revolves in any orbit about an immoveable centre, and in the leaſt time deſcribes any arc juſt then naſcent; and the verſed ſine of that arc is ſuppofed to be drawn, biſecting the chord, and produced paſſing through the centre of force: the centripetal force in the middle of the arc, will be as the verſed ſine directly and the ſquare of the time inverſely.
For the verſed ſine in a given time is as the force (by cor. 4. prop. 1.) and augmenting the time in any ratio, becauſe the arc will be augmented in the ſame ratio, the verſed ſine will be augmented in the duplicate of that ratio, (by cor. 2 and 3. lem. 2.) and therefore is as the force and the ſquare of the time. Subduct on both ſides the duplicate ratio of the time, and the force
