COR. 1. If the velocity be expounded by the length GD, the space described will be as the hyperbolic area DESR. COR. 2. And if the point G be assumed any how, the point G will be found, by taking GR to GD as the velocity at the beginning to the velocity after any space RSED is described. The point G being given, the space is given from the given velocity: and the contrary. COR. 3. Whence since (by Prop. XI) the velocity is given from the given[Pg 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 XIII. THEOREM X. Supposing that a body attracted downwards by an uniform gravity ascends 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 diameters, 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 DF draw the indefinite line BAP, parallel to the semi-diameter DF. In that 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 + 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, Book II, Elem.), as DP2. Then the area DPQ, which is proportional to PQ, is as DP2, and the area DTV, which is to the area DPQ as DT2 to DP2, is as the given quantity DT2. Therefore the area EDT decreases uniformly according to the rate of the future time, by subduction of given particles DTV, and is therefore proportional to the time of the whole ascent. Q.E.D.