SEC. IJ.J OF NATURAL PHILOSOPHY. 265 as ABMI, IMNK, KNOL, (fee., and the absolute forces AC, 1C, KC, LC, (fee., will be in a geometrical progression. Q,.E.D. And by a like rea soning, in the ascent of the body, taking, on the contrary side of the point A, the equal areas AB?m, i/nnk, knol, (fee., it will appear that the absolute forces AC. iG, kC, 1C, (fee., are continually proportional. Therefore if all the spaces in the ascent and descent are taken equal, all the absolute forces 1C, kC, iC, AC, 1C, KC, LC, (fee., will be continually proportional. Q,.E.D. COR. 1. Hence if the space described be expounded by the hyperbolic area ABNK, the force of gravity, the velocity of the body, and the resist ance of the medium, may be expounded by the lines AC, AP, and AK re spectively • and vice versa. COR. 2. And the greatest velocity which the body can ever acquire in an infinite descent will be expounded by the line AC. COR. 3. Therefore if the resistance of the medium answering to any given velocity be known, the greatest velocity will be found, by taking it to that given velocity in a ratio subduplicate of the ratio which the force of gravity bears to that known resistance of the medium. PROPOSITION IX. THEOREM VII. Supposing ivhat is above demonstrated, I say, that if the tangents of t-he angles of the sector of a circle, and of an hyperbola, be taken propor tional to the velocities, the radius being of a fit magnitude, all the time of the ascent to the highest place icill be as the sector of the circle, and all the time of descending from the highest place as the sector of t/ie hyperbola. To the right line AC, which ex presses the force of gravity, let AD drawn perpendicular and equal. From the centre D with the semi-diameter AD describe as well the cmadrant A^E -t of a circle, as the rectangular hyper bola AVZ, whose axis is AK, principal vertex A, and asymptote DC. Let Dp, DP be drawn ; and the circular sector AtD will be as all the time of the as cent to the highest place ; and the hy perbolic sector ATD as all the time of descent from the highest place; ii BO be that the tangents Ap, AP of those sectors be as the velocities. CASE 1. Draw Dvq cutting off the moments or least particles tDv and ^ ?, described in the same time, of the sector ADt and of the triangle AD/?. Since those particles (because of the common angle D) are in a du plicate ratio of the sides, the particle tDv will be as — -^-^-— , that is