SEC. Ilt.1 Or NATURAL PHILOSOPHY. 281 time, and (by this Prop.) the space is given from the given velocity ; the space will be given from the given time : and the contrary. PROPOSITION XKI. THEOREM X. Supposing that a body attracted downwards by an uniform gravity as cends or descends in a right line; and that the same is resisted partly in the ratio of its velocity, and partly in the duplicate ratio thereof: I say, that, if right lines parallel to the diameters of a circle and an hyperbola, be drawn through the ends of the, conjugate diame ters, and the velocities be as some segments of those parallels drawn from a given point, the times will be as the sectors of the, areas cut off by right lines drawn from the centre to the ends of the segments ; and the contrary. CASE 1. Suppose first that the body is ascending, and from the centre D, with any semi-diameter DB, describe a quadrant BETF of a circle, and through the end B of the semi-diameter DB draw the indefi nite line BAP, parallel to the semi-diameter DF. In chat line let there be given the point A, and take the segment AP proportional to the velocity. And since one part of the resistance is as the velocity, and another part as the square of the velocity, let the whole resistance be as AP2 -f 2BAP. Join DA, DP, cutting the circle in E and T, and let the gravity be expounded by DA2, so that the gravity shall be to the resistance in P as DA2 to AP2+2BAP ; and the time of the whole ascent will be as the sector EDT of the circle. For draw DVQ,, cutting off the moment PQ, of the velocity AP, and the moment DTV of the sector DET answering to a given moment of time ; and that decrement PQ, of the velocity will be as the sum of the forces of gravity DA2 and of resistance AP2 + 2BAP, that is (by Prop. XII BookII,Elem.),asDP*. Then the arsa DPQ, which is proportional to PQ: is as DP2, and the area DTV, which is to the area DPQ, as DT2 to DP2, it as the given quantity DT2. Therefore the area EDT decreases uniformly according to the rate of the future time, by subduction of given particles DT V, and is therefore proportional to the time of the whole ascent. Q..E.D. CASE 2. If the velocity in the ascent of the body be expounded by the length AP as before, and the resistance be made as AP2 -f- 2BAP,and if the force of grav ity be less than can be expressed by DA2 ; take BD of such a length, that AB2 — BD2 maybe proportional to the gravity, and let DF be perpendicular and equal F O