Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/23

 Sect. II as CK to CA, and, by diviion, 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 ince AK and ABxCA are given, AB—Kk will be as ABxKk; and latly, when AB and Kk coincide, as AB2. And, by the like reaoning, Kk—Ll, Ll—Mm, &c. will be as Kk2, Ll2, &c. Therefore the quares of the lines AB, Kk, Ll, Mm, &c. are as their differences; and therefore, ince the quares of the velocities were hewn above to be as their differences, the progreion of both will be alike. This being demontrated, it follows alo that the areas decribed by thee lines are in a like progreion with the paces decribed by thee velocities. Therefore if the velocity at the beginning of the firt time AK be expounded by the line AB, and the velocity at the beginning of the econd time KL by the line Kk and the length decribed in the firt time by the area AKkB; all the following velocities will be expounded by the following lines Ll, Mm, &c. and the lengths decribed, by the areas Kl, Lm, &c. And, by compoition, if the whole time be expounded by AM, the um of its parts, the whole length decribed will be expounded by AMmB the um of its parts. Now conceive the time AM to be divided into the parts AK, KL, LM, &c. o that CA, CK, CL, CM, &c. may be in a geometrical progreion; and thoe parts will be in the ame progreion, and the velocities AB, Kk, Ll, Mm, &c. will be in the ame progreion inverly, and the paces decribed Ak, Kl, Lm, &c. will be equal. Q.E.D.

Hence it appears, that if the time be expounded by any part AD of the aymptote. 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 pace decribed, by the adjacent hyperbolic area ABGD; and