Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/14

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To define the motion of a body which, in a imilar medium, acends or decends in a right line, and is reted in the ratio of its velocity, and acted upon by an uniform force of gravity.

The body acending, let the gravity be expounded by any given rectangle BACH; and the reistance of the medium, at the beginning of the acent, by the rectangle BADE, taken on the contrary ide of the right line AB. Through the point B, with the rectangular aymptotes AC, CH, decribe an Hyperbola, cutting the perpendiculars DE, de, in G, g; and the body acending will in the time DGgd decribe the pace EGge; in the time DGBA, the pace of the whole acent EGB; in the time ABKI, the pace of decent BFK; and in the time IKIQ the pace of decent KFfk; and the velocities of the bodies (proportional to the reitance of themedium) in thee periods of time, will be ABED, ABed, o, ABFI, ABfi repectively; and the greatest velocity which the body can acquire by decending, will be BACH.

For let the rectangle BACH be reolved into innumerable rectangles Ak, Kl, Lm, Mn, &c. which hall be as the increments of the velocities produced in o many equal times; then will o, Ak, Al, Am, An, &c. be as the whole velocities, and therefore (by uppoition) as the reitances 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 reitance in the beginning of the econd time; then from the force of gravity ubduct the reitances, and ABHC, KkHC, LlHC, MmHC, &c. will be as the abolute forces with which the body is acted upon in the beginning of each of the times, and therefore