174 THE MATHEMATICAL PRINCIPLES [BOOK J. »C, that will be to the distance which the other body P acquires from the line PC as the transverse motion of the body p to the transverse motion of the other body P. Therefore since kr is equal to the distance which the body P acquires from the line PC, and mr is to kr as the angle VC/? to the angle VCP, that is, as the transverse motion of the body p to the transverse motion of the body P, it is manifest that the body p, at the ex piration of that time, will be found in the place m. These things will be so, if the bodies jo and P are equally moved in the directions of the lines pC and PC, and are therefore urged with equal forces in those directions. I: ut if we take an angle pCn that is to the angle pCk as the angle VGj0 to the angle VCP, and nC be equal to kG, in that case the body p at the expiration of the time will really be in n ; and is therefore urged with a greater force than the body P, if the angle nCp is greater than the angle kCp, that is, if the orbit npk, move either in cmiseqnentia, or in antecedenticij with a celerity greater than the double of that with which the line CP moves in conseqnentia ; and with a less force if the orbit moves slower in antecedent-la. And ihj difference of the forces will be as the interval mn of the places through which the body would be carried by the action of that difference in that given space of time. About the centre C with the interval Cn or Ck suppose a circle described cutting the lines mr, tun pro duced in s and £, and the rectangle mn X nit will be equal to the rectan*//? n ^* */?? ^ •"•le mk X ins, and therefore mn will be equal to — — . But since mt the triangles pCk, pCn, in a given time, are of a given magnitude, kr and mr. a id their difference mk, and their sum ms, are reciprocally as the al titude pC, and therefore the rectangle mk X ms is reciprocally as the square of the altitude pC. But, moreover, mt is directly as