Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/303

 moveable in p, imilar 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 becaue the figures CPRQ, sprq are imilars, 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 conequence towards the intermediate point the centre C, were to the force with which the body p is attracted towards the centre s, in the ame given ratio; thee 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 lat force (tending to s) would make the body p revolve in the curve pqv, which would become imilar to the curve PQV, in which the firt force obliges the body P to revolve; and their revolutions would be compleated in the ame times. But becaue thoe forces are not to each other in the ratio of CP to sp, but (by reaon of the imilarity and equality of the bodies S and s, P and p, and the equality of the ditances 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 ubduplicate ratio of the intervals; becaue by lemma 10, the paces decribed at the very beginning of the motion are in a duplicate ratio of the times. Suppoe then the velocity of the body p to be to the velocity of the body P in a ubduplicate ratio of the ditance sp to the ditance CP, o that the arcs pq, PQ, which are in a imple proportion to each other,