as CK to CA, and, by diviſion, AB—Kk; to Kk as AK to CA, and, alternately, AB—Kk to AK as Kk to CA, and therefore as AB x Kk, to AB x CA. Therefore ſince AK and ABxCA are given, AB—Kk will be as ABxKk; and laſtly, when AB and Kk coincide, as AB2. And, by the like reaſoning, Kk—Ll, Ll—Mm, &c. will be as Kk2, Ll2, &c. Therefore the ſquares of the lines AB, Kk, Ll, Mm, &c. are as their differences; and therefore, ſince the ſquares of the velocities were ſhewn above to be as their differences, the progreſſion of both will be alike. This being demonſtrated, it follows alſo that the areas deſcribed by theſe lines are in a like progreſſion with the ſpaces deſcribed by theſe velocities. Therefore if the velocity at the beginning of the firſt time AK be expounded by the line AB, and the velocity at the beginning of the ſecond time KL by the line Kk and the length deſcribed in the firſt time by the area AKkB; all the following velocities will be expounded by the following lines Ll, Mm, &c. and the lengths deſcribed, by the areas Kl, Lm, &c. And, by compoſition, if the whole time be expounded by AM, the ſum of its parts, the whole length deſcribed will be expounded by AMmB the ſum of its parts. Now conceive the time AM to be divided into the parts AK, KL, LM, &c. ſo that CA, CK, CL, CM, &c. may be in a geometrical progreſſion; and thoſe parts will be in the ſame progreſſion, and the velocities AB, Kk, Ll, Mm, &c. will be in the ſame progreſſion inverſly, and the ſpaces deſcribed Ak, Kl, Lm, &c. will be equal. Q.E.D.
Cor. 1. Hence it appears, that if the time be expounded by any part AD of the aſymptote. and the velocity in the beginning of the time by the ordinate AB; the velocity at the end of the time will be expounded by the ordinate DG; and the whole ſpace deſcribed, by the adjacent hyperbolic area ABGD; and
