188 THE MATHEMATICAL PRINCIPLES [BOOK 1 moment as the length TX, that is (because CV\ WV, and TX, TW proportional to them are given), as the length TW, that is (by Cor. 1, Prop. XLIX) as the length of the arc of the cycloid TR. If there fore two pendulums APT, Apt, be unequally drawn aside from the perpendicular AR, and let fall together, their accelerations will be always as the arcs to be de scribed TR, tR. But the parts described at the beginning of the motion are as the accelerations, thai is, as the wholes that are to be described at the be ginning, and therefore the parts which remain to be described, and the subsequent accelerations proportional to those parts, are also as the wholes, and so on. Therefore the accelerations, and consequently the velocities generated, and the parts described with those velocities, and the parts to be described, are always as the wholes ; and therefore the parts to be described preserving a given ratio to each other will vanish together, that is, the two bodies oscillating will arrive together at the perpendicular AR. And since on the other hand the ascent of the pendulums from the lowest place R through the same cycloidal arcs with a retrograde motion, is retarded in the several places they pass through by the same forces by which their de scent was accelerated : it is plain that the velocities of their ascent and de scent through the same arcs are equal, and consequently performed in equal times ; and, therefore, since the two parts of the cycloid RS and RQ lying on either side of the perpendicular are similar and equal, the two pendu lums will perform as well the wholes as the halves of their oscillations in the same times. Q.E.D. ' COR. The force with which the body T is accelerated or retarded in any place T of the cycloid, is to the whole weight of the same body in the highest place S or Q, as the arc of the cycloid TR is to the arc SR or QR PROPOSITION LIL PROBLEM XXXIV. To define the velocities of the pendulums in the several places, and the times in which both the entire oscillations, and the several parts of them are performed. About any centre G, with the interval GH equal to the arc of the cycloid RS, describe a semi-circle HKM bisected by the semi-diameter GK. And if a centripe tal force proportional to the distance of the places from the centre tend to the centre G, and it be in the peri meter HIK equal to the centripetal force in the perime ter of the globe Q,OS tending towards its centre, and at the same time that the pendulum T is let fall from the highest place S, a body, as L, is let fall from H to G ; then because th<