Proposition LIV. Problem XXXVI.
Granting the quadratures of curvilinear figures, it is required to find the times, in which bodies by means of any centripetal force will deſcend or aſcend in any curve lines deſcribed in a plane paſſing through the centre of force.
Let the body deſcend from any place S (Pl. 20. Fig. 4.) and move in any curve STtR, given in a plane paſſing through the centre of force C. Join CS, and let it be divided into innumerable equal parts, and let Dd be one of thoſe parts. From the centre C, with the intervals CD, Cd, let the circles DT, dt be deſcribed, meeting the curve line STtR in T and t. And becauſe the law of centripetal force is given, and alſo the altitude CS from which the body at firſts fell; there will be given the velocity of the body in any other altitude CT (by prop. 39.) But the time in which the body deſcribes the lineola Tt is as the length of that lineola, that is. as the fecant of the angle tTC directly, and the velocity inverſely. Let the ordinate DN, proportional to this time, be made perpendicula to the right line CS at the point D, an becauſe Dd is given, the rectangle Dd x DN that is, the area DNnd, will be proportional to the ſame time. Therefore if PNn be a curve line in which the point N is perpetually found, and its
