Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/257

 parts of the revolving orbit up, pk; and let the ditance of the points P and K be uppoed of the utmot mallnes. Let fall a perpendicular kr from the point k to the right line pC, and produce it to m, o that mr may be to kr as the angle VCp to the angle VCP. Becaue the altitudes of the bodies, PC and pC, KC and kC, are always equal, it is manifet: that the increments or decrements of the lines PC and pC are always equal; and therefore if each of the everal motions of the bodies in the places P and p be reolved into two, (by cor. 2. of the laws of motion) one of which is directed towards the center, or according to the lines PC, pC, and the other, tranvere to the former, hath a direction perpendicular to the lines PC and pC; the motions towards the centre will be equal, and the tranvere motion of the body p will be to the tranvere motion of the body P, as the angular motion of the line pC to the angular motion of the line PC; that is, as the angle VCp to the angle VCP. Therefore at the ame time that the body P, by both its motions, comes to the point K, the body p, having an equal motion towards the centre, will be equally moved from p towards C, and therefore that time being expired, it will be found omewhere in the line mkr, which, paing through the point k, is perpendicular to the line pC; and by its tranvere motion, will acquire a ditance from the line pC, that will be to the ditance which the other body P acquires from the line PC, as the tranvere motion of the body p, to the tranvere motion of the other body P. Therefore ince kr is equal to the ditance which the body P acquires from the line PC, and mr is to kr as the angle