Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/295



Granting the quadratures of curvilinear figures, it is required to find the times, in which bodies by means of any centripetal force will decend or acend in any curve lines decribed in a plane paing through the centre of force.

Let the body decend from any place S (Pl. 20. Fig. 4.) and move in any curve STtR, given in a plane paing through the centre of force C. Join CS, and let it be divided into innumerable equal parts, and let Dd be one of thoe parts. From the centre C, with the intervals CD, Cd, let the circles DT, dt be decribed, meeting the curve line STtR in T and t. And becaue the law of centripetal force is given, and alo the altitude CS from which the body at firts 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 decribes the lineola Tt is as the length of that lineola, that is. as the fecant of the angle tTC directly, and the velocity inverely. Let the ordinate DN, proportional to this time, be made perpendicula to the right line CS at the point D, an becaue Dd is given, the rectangle Dd x DN that is, the area DNnd, will be proportional to the ame time. Therefore if PNn be a curve line in which the point N is perpetually found, and its