parts of the revolving orbit up, pk; and let the diſtance of the points P and K be ſuppoſed of the utmoſt ſmallneſs. Let fall a perpendicular kr from the point k to the right line pC, and produce it to m, ſo that mr may be to kr as the angle VCp to the angle VCP. Becauſe the altitudes of the bodies, PC and pC, KC and kC, are always equal, it is manifeſt: that the increments or decrements of the lines PC and pC are always equal; and therefore if each of the ſeveral motions of the bodies in the places P and p be reſolved 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, tranſverſe to the former, hath a direction perpendicular to the lines PC and pC; the motions towards the centre will be equal, and the tranſverſe motion of the body p will be to the tranſverſe 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 ſame 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 ſomewhere in the line mkr, which, paſſing through the point k, is perpendicular to the line pC; and by its tranſverſe motion, will acquire a diſtance from the line pC, 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
Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/257
