moveable in p, ſimilar and equal to the bodies S and P. Then let the right lines PR and pr touch the curves PQ and pq in P and p, and produce CQ and sq to R and r. And becauſe the figures CPRQ, sprq are ſimilars, RQ will be to rq as CP to sp, and therefore in a given ratio. Hence if the force with which the body P is attracted towards the body S, and by conſequence towards the intermediate point the centre C, were to the force with which the body p is attracted towards the centre s, in the ſame given ratio; theſe forces would in equal times attract the bodies from the tangents PR, pr to the arcs PQ, pq, through the intervals proportional to them RQ, rq; and therefore this laſt force (tending to s) would make the body p revolve in the curve pqv, which would become ſimilar to the curve PQV, in which the firſt force obliges the body P to revolve; and their revolutions would be compleated in the ſame times. But becauſe thoſe forces are not to each other in the ratio of CP to sp, but (by reaſon of the ſimilarity and equality of the bodies S and s, P and p, and the equality of the diſtances SP, sp) mutually equal; the bodies in equal times will be equally drawn from the tangents; and therefore that the body p may be attracted through the greater interval rq, there is required a greater time, which will be in the ſubduplicate ratio of the intervals; becauſe by lemma 10, the ſpaces deſcribed at the very beginning of the motion are in a duplicate ratio of the times. Suppoſe then the velocity of the body p to be to the velocity of the body P in a ſubduplicate ratio of the diſtance sp to the diſtance CP, ſo that the arcs pq, PQ, which are in a ſimple proportion to each other,
Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/303
