making right angles with the axis; and drawing DS, PS, the area ASD will be proportional to the area ASP, and therefore alſo to the time. The axis AB ſtill remaining the ſame, let the breadth of the ellipſis be perpetually diminiſhed, and the area ASD will always remain proportional the time. Suppoſe that breadth to be diminiſhed in infinitum; and the orbit APB in that caſe coinciding with the axis AB, and the focus S with the extreme point of the axis B, the body will deſcend in the right line AC, and the area ABD will become proportional to the time. Wherefore the ſpace AC will be given which the body deſcribes 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.
Case 2. If the figure RPB is an hyperbola, (Fig. 2.) on the ſame principal diameter AB deſcribe the rectangular hyperbola BED; and becauſe the areas CSP, CBfP, SPfB, are ſeverally to the ſeveral 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 alſo proportional to that time. Let the latus rectum of the hyperbola RPB be diminiſhed in infinitum, the latus tranſverſum remaining the ſeme; 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
