THE MATHEMATICAL PRINCIPLES [BOOK II. the particles of time will be as the same velocity. Therefore the veloci ties are proportional to their differences, and therefore (by Lem. 1, Book II) continually proportional. Therefore if out of an equal number of par ticles there be compounded any equal portions of time, the velocities at the beginning of those times will be as terms in a continued progression, which are taken by intervals, omitting every where an equal number of interme diate terms. But the ratios of these terms are compounded of the equaj ratios of the intermediate terms equally repeated, and therefore are equal Therefore the velocities, being proportional to those terms, are in geomet rical progression. Let those equal particles of time be diminished, and their number increased in infinitum, so that the impulse of resistance may become continual ; and the velocities at the beginnings of equal times, al ways continually proportional, will be also in this case continually pro portional. Q.E.D. CASE 2. And, by division, the differences of the velocities, that is, the parts of the velocities lost in each of the times, are as the wholes ; but the spaces described in each of the times are as the lost parts of the velocities (by Prop. 1, Book I), and therefore are also as the wholes. Q.E.D. TT COROL. Hence if to the rectangular asymptotes AC, CH, the hyperbola BG is described, and AB, DG be drawn per pendicular to the asymptote AC, and both the velocity of . the body, and the resistance of the medium, at the very be ginning 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 uniform ly increased in the same manner as the time, the right line DC will de crease 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 expound ed by any given rectangle BACH ; and the resist ance of the medium, at the beginning of the ascent, by the rectangle BADE, taken on the contrary side Jfl e B^l