tre S at the given interval SG, take GA to . If that ratio is the ſame as of the number 2 to 1 the point A is infinitely remote; in which caſe a parabola is to be deſcribed with any latus rectum to the vertex S, and axis SG; as appears by prop. 34. But if that ratio is leſs or greater than ratio of 2 to 1, in the former caſe a circle, in the latter a rectangular hyperbola, is to be deſcribed on the diameter SA; as appears by prop. 33. Then about the centre S. with an interval equal to half the latus rectum, deſcribe the circle HkK, and at the place G of the aſcending or deſcending body, and at any other place C, erect the perpendiculars GI, CD; meeting the conic ſection or circle in I and D. Then joining SI, SD, let the ſectors HSK, HSk be made equal to the ſegments SEIS, SEDS, and by prop. 35. the body G will deſcribe the ſpace GC in the ſame time in which the body K may deſcribe the arc Kk. Q. E. F.
Proposition XXXVIII. Theorem XII.
Supposing that the centripetal force is proportional to the altitude or diſtance of places from the centre, I ſay, that the times and velocities of falling bodies, and the ſpaces which they deſcribe, are reſpectively proportional to the arcs, and the right and verſed ſines of the arcs. Pl. 17. Fig. 1.
Suppoſe the body to fall from any place A in the right line AS; and about the centre of force S
