Proposition LII. Problem XXXIV.
To define the velocities of the pendulums in the ſeveral places, and the times in which both the entire oſcillation, and the ſeveral parts of them are performed.
About any centre G (Pl. 20. Fig. 1.) 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 centripetal force proportional to the diſtance of the places from the centre tend to the centre G, and it be in the perimeter HIK equal to the centripetal force in the perimeter of the globe QOS 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 the forces which act upon the bodies are equal at the beginning, and always proportional to the ſpaces to be deſcribed TR, LG, and therefore if TR and LG are equal, are alſo equal in the places T and L, it is plain that thoſe bodies deſcribe at the beginning equal ſpaces ST, HL, and therefore are ſtill acted upon equally, and continue to deſcribe equal ſpaces. Therefore by prop. 38; the time in which the body deſcribes the arc ST is to the time of one oſcillation, as the arc HI the time in which the
