Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/227

 making right angles with the axis; and drawing DS, PS, the area ASD will be proportional to the area ASP, and therefore alo to the time. The axis AB till remaining the ame, let the breadth of the ellipis be perpetually diminihed, and the area ASD will always remain proportional the time. Suppoe that breadth to be diminihed in infinitum; and the orbit APB in that cae coinciding with the axis AB, and the focus S with the extreme point of the axis B, the body will decend in the right line AC, and the area ABD will become proportional to the time. Wherefore the pace AC will be given which the body decribes in a given time by its perpendicular fall from the place A, if the area ABD is taken proportional to the time, and from the point D, the right line DC is let fall perpendicularly on the right line AB. Q. E. I.

If the figure RPB is an hyperbola, (Fig. 2.) on the ame principal diameter AB decribe the rectangular hyperbola BED; and becaue the areas CSP, CBfP, SPfB, are everally to the everal areas CSD, CBED, SDEB in the given ratio of the heights CP, CD; and the area SPfB is proportional to the time in which the body P will move through the arc PfB, the area SDEB will be alo proportional to that time. Let the latus rectum of the hyperbola RPB be diminihed in infinitum, the latus tranverum remaining the eme; and the arc PB will come to coincide with the right line CB, and the focus S with the vertex B, and the right line SD with the right line BD. And there fire the area BDEB will be proportional to the time in which the body C, by its