COROL. Hence if to the rectangular asymptotes AC, CH, the hyperbola BG is described, and AB, DG be drawn perpendicular to the asymptote AC, and both the velocity of the body, and the resistance of the medium, at the very beginning of the motion, be expressed by any given line AC, and, after some time is elapsed, by the indefinite line DC; the time may be expressed by the area ABGD, and the space described in that time by the line AD. For if that area, by the motion of the point D, be uniformly increased in the same manner as the time, the right line DC will decrease in a geometrical ratio in the same manner as the velocity; and the parts of the right line AC, described in equal times, will decrease in the same ratio. PROPOSITION III. PROBLEM I. To define the motion of a body which, in a similar medium, ascends or descends in a right line, and is resisted in the ratio of its velocity, and acted upon by an uniform force of gravity. The body ascending, let the gravity be expounded by any given rectangle BACH; and the resistance of the medium, at the beginning of the ascent, by the rectangle BADE, taken on the contrary side of the right line AB. Through the point B, with the rectangular asymptotes AC, CH, describe an hyperbola, cutting the perpendiculars DE, de, in[Pg 253] G, g; and the body ascending will in the time DGgd describe the space EGge; in the time DGBA, the space of the whole ascent EGB; in the time ABKI, the space of descent BFK; and in the time IKki the space of descent KFfk; and the velocities of the bodies (proportional to the resistance of the medium) in these periods of time will be ABED, ABed, O, ABFI, ABfi respectively; and the greatest velocity which the body can acquire by descending will be BACH.