thereof proportional to the centripetal force with which the body tends towards the centre C; TM a right line perpendicular to the curve ſuperficies; TI a part thereof proportional to the force of preſſure with which the body urges the ſuperficies, and therefore with which it is again repelled by the ſuperficies towards M; PTF a right line parallel to the axis and paſſing through the body, and GF, IH right lines let fall perpendicularly from the points G and I upon that parallel PHTF. I ſay now that the area AOP, deſcribed by the radius OP from the beginning of the motion is proportional to the time. For the force TG (by cor. 2. of the laws of motion) is reſolved into the forces TF, FG; and the force TI into the forces TH, HI; but the forces TF, TH acting in the direction of the line PF perpendicular to the plane AOP, introduce no change in the motion of the body but in a direction perpendicular to that plane. Therefore its motion ſo far as it has the ſame direction with the poſition of the plane, that is, the motion of the point P, by which the projection AP of the trajectory is deſcribed in that plane, is the ſame as if the forces TF, TH were taken away, and the body were acted on by the forces FG, HI alone; that is, the ſame as if the body were to deſcribe in the plane AOP the curve AP by means of a centripetal force tending to the centre O, and equal to the ſum of the forces FG and HI. But with ſuch a force as that (by prop. 1.) the area AOP will be deſcribed proportional to the time. Q. E. D.
Cor. By the ſame reaſoning if a body, acted on by forces tending to two or more centres in
