164 THE MATHEMATICAL PRINCIPLES [BOOK 1. PROPOSITION XXXVI. PROBLEM XXV. To determine the times of the descent of a body falling from place A. Upon the diameter AS, the distance of the body from the centre at the beginning, describe the semi-circle ADS, as likewise the semi-circle OKH equal thereto, about the centre S. From any place C of the body erect the ordinate CD. ° Join SD, and make the sector OSK equal to the area ASD. It is evident (by Prop. XXXV) that the body in falling will describe the space AC in the same time in which another body, uniformly revolving about the centre S, may describe the arc OK. Q.E.F. M a given PROPOSITION XXXVII. PROBLEM XXVI. To define the times of the ascent or descent of a body projected upwards or downwards from a given place. Suppose the body to go oif from the given place G, in the direction of the line GS, with any velocity. In the duplicate ratio of this velocity to the uniform velocity in a circle, with which the body may revolve about \ H D the centre S at the given interval SG, take GA to £AS. 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 ver tex S, and axis SG ; as appears by Prop. XXXIV. But if that ratio is less or greater than the 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. XXXIII. Then about the centre S, with an interval equal to half the latus rectum, describe the circle H/vK ; 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, HS& be made equal to the segments SEIS, SEDS. and (by Prop. XXXV) the body G will describe