Proposition III. Problem I.
To define the motion of a body which, in a ſimilar medium, aſcends or deſcends in a right line, and is reſted in the ratio of its velocity, and acted upon by an uniform force of gravity.
The body aſcending, let the gravity be expounded by any given rectangle BACH; and the reſistance of the medium, at the beginning of the aſcent, by the rectangle BADE, taken on the contrary ſide of the right line AB. Through the point B, with the rectangular aſymptotes AC, CH, deſcribe an Hyperbola, cutting the perpendiculars DE, de, in G, g; and the body aſcending will in the time DGgd deſcribe the ſpace EGge; in the time DGBA, the ſpace of the whole aſcent EGB; in the time ABKI, the ſpace of deſcent BFK; and in the time IKIQ the ſpace of deſcent KFfk; and the velocities of the bodies (proportional to the reſiſtance of theſmedium) in theſe periods of time, will be ABED, ABed, o, ABFI, ABfi reſpectively; and the greatest velocity which the body can acquire by deſcending, will be BACH.
For let the rectangle BACH be reſolved into innumerable rectangles Ak, Kl, Lm, Mn, &c. which ſhall be as the increments of the velocities produced in ſo many equal times; then will o, Ak, Al, Am, An, &c. be as the whole velocities, and therefore (by ſuppoſition) as the reſiſtances of the medium in the beginning of each of the equal times. Make AC to AK, or ABHC to ABkK as the force of gravity to the reſiſtance in the beginning of the ſecond time; then from the force of gravity ſubduct the reſiſtances, and ABHC, KkHC, LlHC, MmHC, &c. will be as the abſolute forces with which the body is acted upon in the beginning of each of the times, and therefore
