288 THE MATHEMATICAL PRINCIPLES [_BoOK 11 arc PQ is the fourth part of the decrement of the arc PR. Whence also if the area QSr be taken equal to the area PSQ, the decrement of the arc PQ will be equal to half the lineola Rr ; and therefore the force of resist ance and the centripetal force are to each other as the lineola jRrandTQ which they generate in the same time. Because the centripetal force with which the body is urged in P is reciprocally as SP2, and (by Lem. X, Book I) the lineola TQ, which is generated by that force, is in a ratio compounded of the ratio of this force and the duplicate ratio of the time in which the arc PQ, is described (for in this case I neglect the resistance, as being infinitely less than the centripetal force), it follows that TQ X SP2, that is (by the last Lemma), fPQ2 X SP, will be in a duplicate ra tio of the time, and therefore the time is as PQ, X v/SP ; and the velo city of the body, with which the arc PQ is described in that time, as PQ 1 — -«p or , that is, in the subduplicate ratio of SP reciprocally. And, by a like reasoning, the velocity with which the arc QR is described, is in the subduplicate ratio of SQ reciprocally. Now those arcs PQ and QR are as the describing velocities to each other ; that is, in the subdu plicate ratio of SQ to SP, or as SQ to x/SP X SQ; and, because of the equal angles SPQ, SQ?', and the equal areas PSQ, QSr, the arc PQ is to the arc Qr as SQ to SP. Take the differences of the proportional conse quents, and the arc PQ will be to the arc Rr as SQ to SP — VSP X ~SQ~, or ^VQ. For the points P and Q coinciding, the ultimate ratio of SP — X SQ to