Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/284

 the remaining part of the ame thread PT which has not yet touched the emi-cycloid continuing traight. Then will the weight T ocillate 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 contruction and generation of the imilar figures AS, SR, that thoe 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 $$\scriptstyle \frac 12$$ BV is radius) as BW to BV, or AQ + OP to AO, that is (ince CA and CO, CO and CR, and by diviion AO and OR are proportional) as CA + CO to CA; or, if BV be biected 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-wie, if the tring remain always equal to the length AR the point T will always move in given cycloid QRS. Q. E. D.

The tring AR is equal to the emi-cycloid AS, and therefore has the ame ratio to AC the emi-diameter of the exterior globe as the like emi-cycloid SR has to CO the emi-diameter of the interior globe.