the remaining part of the ſame thread PT which has not yet touched the ſemi-cycloid continuing ſtraight. Then will the weight T oſcillate in the given cycloid QRS. Q. E. F.
For let the thread P meet the cycloid QRS in T and the circle QOS in V and let CV be drawn; and to the rectilinear part of the thread PT from the extreme points P and T let there be erected the perpendiculars BP. TW, meeting the right line CV in B and W. It is evident from the conſtruction and generation of the ſimilar figures AS, SR, that thoſe perpendiculars PB, TW, cut off from CV the lengths VB, VW equal to the diameters of the wheels OA, OR. Therefore TP is to VP (which is double the fine of the angle VBP when BV is radius) as BW to BV, or AQ + OP to AO, that is (ſince CA and CO, CO and CR, and by diviſion AO and OR are proportional) as CA + CO to CA; or, if BV be biſected in E, as 2 CE to CB. Therefore (by cor. 1. prop. 49) the length of the rectilinear part of the thread PT is always equal to the arc of the cycloid PS, and the whole thread APT is always equal to the half of the cycloid APS, that is (by cor. 2. prop. 49.) to the length AR. And therefore contrary-wiſe, if the ſtring remain always equal to the length AR the point T will always move in given cycloid QRS. Q. E. D.
Cor. The ſtring AR is equal to the ſemi-cycloid AS, and therefore has the ſame ratio to AC the ſemi-diameter of the exterior globe as the like ſemi-cycloid SR has to CO the ſemi-diameter of the interior globe.
