3^4 THE MATHEMATICAL PRINCIPLES [BOOK 11. correspondent particles to continue to describe like figures. These things will be so (by Cor. 1 and S, Prop. IV.; Book 1), if those centres are at rest but if they are moved, yet by reason of the similitude of the translations, their situations among the particles of the system will remain similar , so that the changes introduced into the figures described by the particles will still be similar. So that the motions of correspondent and similar par ticles will continue similar till their first meeting with each other ; and thence will arise similar collisions, and similar reflexions; which will again beget similar motions of the particles among themselves (by what was just now shown), till they mutually fall upon one another again, and so on ad infinitum. COR. 1. Hence if any two bodies, which are similar and in like situations to the correspondent particles of the systems, begin to move amongst them in like manner and in proportional times, and their magnitudes and densi ties be to each other as the magnitudes and densities of the corresponding particles, these bodies will continue to be moved in like manner and in proportional times: for the case of the greater parts of both systems and of the particles is the very same. COR. 2. And if all the similar and similarly situated parts of both sys tems be at rest among themselves ; and two of them, which are greater than the rest, and mutually correspondent in both systems, begin to move in lines alike posited, with any similar motion whatsoever, they will excite similar motions in the rest of the parts of the systems, and will continue to move among those parts in like manner and in proportional times ; and will therefore describe spaces proportional to their diameters. PROPOSITION XXXIII. THEOREM XXVII. The same things biting supposed, I say, that the greater parts of the systems are resisted in a ratio compounded of the duplicate ratio of their velocities, and the duplicate ratio of their diameters, and Ihe sim ple ratio of the density of the parts of the systems. For the resistance arises partly from the centripetal or centrifugal, forces with which the particles of the system mutually act on each other, partly from the collisions and reflexions of the particles and the greater parts. The resistances of the first kind are to each other as the whole motive forces from which they arise, that is, as the whole accelerative forces and the quantities of matter in corresponding parts ; that is (by the sup position), as the squares of the velocities directly, and the distances of the corresponding particles inversely, and the quantities of matter in the cor respondent parts directly : and therefore since the distances of the parti cles in one system are to the correspondent distances of the particles of the ;ther S3 the diameter of one particle or part in *he former system to the