any the ſame right line CO, ſhould deſcribe in a free ſpace any curve line ST; the area AOP would always proportional to the time.
Proposition LVI. Problem XXXVII.
Granting the quadratures of curvilinear figures and ſuppoſing that there are given both the law of centripetal force tending to a given centre, and the curve ſupeſficies whoſ axis paſſes through that centre; it if required to find the trajectory which a body will deſcribe in that ſuperificies, when going of from a given place with a given velocity, and in a given direction in that ſuperficies.
The laſt conſtruction remaining, let the body T go from the given place S (Pl. 20. Fig. 6.) in the direction of a line given by poſition, and turn into the trajectory ſought STR whoſe orthographic projection in the plane BLO is AP. And from the given velocity of the body in the altitude SC, its velocity in any other altitude TC will be alſo given. With that velocity in a given moment of time let the body deſcribe the particle Tt of its trajectory, and let Pp be the projection of that particle deſcribed in the plane AOP. Join Op, and a little circle being deſcribed upon the curve ſuperficies about the centre T with the interval Tt, let the projection of that little circle in the plane
