Section 2= The motions of bodies that are resisted in the duplicate ratio of the velocities

Proposition 5 Theorem 3

If a body is resisted in the duplicate ratio of its velocity, and moves by innate force only through a similar medium; and the times be taken in a geometrical progression, proceeding from less to greater terms = 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 is to the resist velocity proportional ance = 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.

KL, LM, Let those particles of time be AK, &c., taken in the right line CD; and erect the perpendiculars AB, Kk, L/, &c., meeting the hyperbola BklmG, described with the Mm, centre C, and the rectangular asymptotes then will be to in B, kj I, m, (fee. CD, CH. AB Kk as CK Kk to Kk as AK to CA, and, by division, AB C to AK as Kk an(1 alternate ^ AB to ^ C ARIMT and therefore as AB X Kk to AB X CA. to CA Kk will be as AB Therefore since AK and AB X CA are given,* AB X Kk ; and, lastly, when AB and KA* coincide, as AB 2 And, by the like (fee. will be as Kk 2 LI Therefore the reasoning, KAr-U, J-M/??,

squares of the lines AB, KA", L/, Mm, (fee., 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 follows also that the areas described by these lines are in a like progres Therefore if the velo sion with the spaces described by these velocities. it city at the beginning of the first time AK be expounded by the line AB,SEC.

and the velocity at the beginning of the second time KL by the line K& and the length described in the hrst time by the area AKA*B, all the fol lowing velocities will be expounded by the following lines \J. Mm, .fee. and the lengths described, by the areas K/, mi. &c. And, by compo sition, if the whole time be expounded by AM, the sum of its parts, the I whole length described will be expounded by AM/ftB the sum of to be divided into the parts AK, KL, Now conceive the time AM its parts. LM, (fee CA, CK, CL, CM, may be in a geometrical progression and those parts will be in the same progression, and the velocities AB, K/r, L/, Mm, (fee., will be in the same progression inversely, and the spaces de so that (fee. ; Ak, K/, Lw, (fee., will be equal. Q,.E.D.

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 scribed COR. DG ABGD and the whole space described by the adjacent hyperbolic and the space which any body can describe in the same time AD, with the first velocity AB, in a non-resisting medium, by the rectan ordinate area gle

AB X AD. COR in a resisting medium is given, by in a non- the space described with the uniform velocity to the rectangle X AD. resisting medium, as the hyperbolic area COR. 3. The resistance of the medium is also given, by making it equal, taking Hence the space described 2. AB it to ABGD AB in the very beginning of the motion, to an uniform centripetal force, which could generate, in a body falling through a non-resisting medium, the ve AB in the time AC. For if BT be drawn touching the hyperbola and meeting the asymptote in T, the right line will be equal to and will the time in which the first AC, resistance, uniformly con express take the whole AB tinned, may away velocity locity AT in B. COR. And 4. thence is also given the proportion of this resistance to the ay other given centripetal force. force of gravity, or COR. 5. And, vice versa, if there is given the proportion of the resist- nee to any given centripetal force, the time is also given, in which c AC ; centripetal force equal to the resistance AB and thence CH may generate any velocity as the B. given point through which the hyperbola, having for its asymptotes, is to be described as also the space ABGD, which a body, by beginning its motion with that velocity AB, can describe in any is CD time ; : AD. in a similar resisting medium.