SECTION II.
Of the Motion of Bodies that are reſiſted in the duplicate ratio of their Velocities.
PROPOSITION V. THEOREM Ill.
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 iinverſely; and that the ſpaces are equal, which are deſcribed in each of the times.
For ſince the reſitance 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
