SEC. IL] OF NATURAL PHILOSOPHY. 107 Co.*. 1. Hence if the one body L, by a radius drawn to the other body T, describes areas proportional to the times ; and from the whole force, by which the firr.t body L is urged (whether that force is simple, or, according to Cor. 2 of the Laws, compounded out of several forces), we subduct (by the same Cor.) that whole accelerative force by which the other body is urged ; the who_e remaining force by which the first body is urged will tend to the ( ther body T, as its centre. COR. 2. And, if these areas are proportional to the times nearly, the re maining force will tend to the other body T nearly. COR. 3. And vice versa, if the remaining force tends nearly to the other body T, those areas will be nearly proportional to the times. COR. 4. If the body L, by a radius drawn to the other body T, describes areas, which, compared with the times, are very unequal ; and that other body T be either at rest, or moves uniformly forward in a right line : the action of the centripetal force tending to that other body T is either none at all, or it is mixed and compounded with very powerful actions of other forces : and the whole force compounded of them all, if they are many, is directed to another (immovable or moveaJble) centre. The same thing ob tains, when the other body is moved by any motion whatsoever ; provided that centripetal force is taken, wrhich remains after subducting that whole force acting upon that other body T. SCHOLIUM. Because the equable description of areas indicates that a centre is re spected by that force with which the body is most affected, and by which it is drawn back from its rectilinear motion, and retained in its orbit ; why may we not be allowed, in the following discourse, to use the equable de scription of areas as an indication of a centre, about which all circular motion is performed in free spaces ? PROPOSITION IV. THEOREM IV. The centripetal forces of bodies, which by equable 'motions describe differ ent circles, tend to the centres of the same circles ; and are one to tJie other as the squares of t/ie arcs described in equal times applied to the radii of the circles. These forces tend to the centres of the circles (by Prop. II., and Cor. 2, Prop. L), and are one to another as the versed sines of the least arcs de scribed in equal times (by Cor. 4, Prop. I.) ; that is, as the squares of the same arcs applied to the diameters of the circles (by Lem. VII.) ; and there fore since those arcs are as arcs described in any equal times, and the diame ers ace as the radii, the forces will be as the squares of any arcs descr bed in the same time applied to the radii of the circles. Q.E.D. ^OR. 1. Therefore, since those arcs are as the velocities of the bodies.