If a body is resisted in the duplicate ratio of its velocity, and moves by its innate force only through a similar medium; and the times be taken in a geometrical progression, proceeding from less to greater terms: I say, that the velocities at the beginning of each of the times are in the same geometrical progression inversely; and that the spaces are equal, which are described in each of the times. For since the resistance of the medium is proportional to the square of the velocity, and the decrement of the velocity is proportional to the resistance: if the time be divided into innumerable equal particles, the squares of the velocities at the beginning of each of the times will be proportional to the differences of the same velocities. Let those particles of time be AK, KL, LM, &c., taken in the right line CD; and erect the perpendiculars AB, Kk, Ll, Mm, &c., meeting the hyperbola BklmG, described with the centre C, and the rectangular asymptotes CD, CH, in B, k, l, m, &c.; then AB will be to Kk 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 × Kk to AB × CA. Therefore since AK and AB × CA are given, AB - Kk will be as AB × Kk; 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,[Pg 259] 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 inversely, and the spaces described Ak, Kl, Lm, &c., will be equal. Q.E.D.