falling deſcribes the ver ſmall line DE is as that line directly and the velocity V inverſely, and the force will be as the increment I of the velocity directly and the time inverſely, and therefore if we take the firſt ratio's when thoſe quantities are juſt naſcent as , that is as the length DF. Therefore a force proportional to DF or EG will cauſe the body to deſcend with a velocity that is as the right line whoſe power is the area ABGE. Q. E. D.
Moreover ſince the time, in which a very ſmall line DE of a given length may be deſcribed, is as the velocity inverſely, and therefore alſo inverſely as a right line whoſe ſquare is equal to the area ABFD; and ſince the line DL, and by conſequence the naſcent area DLME, will be as the ſame right line inverſely: the time will be as the area DLME, and the ſum of all the times will be as the ſum of all the area's; that is (by cor. lem. 4.) the whole time in which the line AE is deſcribed will be as the whole area ATVME. Q. E. D,
Cor. 1. Let P be the place from whence a body ought to fall, ſo as that when urged by any known uniform centripetal force (ſuch as gravity is vulgarly ſuppoſed to be) it may acquire in the place D a velocity, equal to the velocity which another body, falling by any force whatever, hath acquired in that place D. In the perpendicular DF let there be taken DR, which may be to DF as that uniform force to the other force in the place D. Compleat the rectangle PDRQ, and cut off the area ABFD equal
