Proposition LI. Theorem XVIII.
If a centripetal force tending on all ſides to the centre C of a globe (Pl. 19. Fig. 4.) be in all places as the diſtance of the place from the centre, and by this force alone acting upon it, the body T oſcillate (in the manner above deſcrid) in the perimeter of the cycloid QRS; I ſay that all the oſcillations bow unequal ſoever in themſeves will be performed in equal times.
For upon the tangent TW infinitely produced let fall the perpendicular CX and join CT. Becauſe the centripetal force with which the body T is impelled towards C is as the diſtance CT; let this (by cor. 2. of the laws) be reſolved into the parts CX, TX of which CX impelling the body directly from P ſtretches the thread PT and by the reſiſtance the thread makes to it is totally employed, producing no other effect; but the other part TX, impelling the body tranſverſely or towards X directly accelerates the motion in the cycloid. Then it is plain that the acceleration of the body, proportional to this accelerating force, will be every moment as the length TX, that is, (becauſe CV, WV, and TX, TW proportional to them are given) a the length TW that is (by cor. 1. prop. 49.) as the length of the arc of the cycloid TR. If
