Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/26

16 be proportional to the wholes, the reitance and time conjunctly ought to be as the motion. Therefore the time will be as the motion directly and the reitance inverely. Wherefore the particles of the times being taken in that ratio, the bodies will always loe parts of their motions proportional to the wholes, and therefore will retain velocities always proportional to their firt velocities. And becaue of the given ratio of the velocities, they will always decribe paces, which are as the firt velocities and the times conjunctly. Q. E. D.

Therefore if bodies equally wift are reited in a duplicate ratio of their diameters: Homogeneous globes moving with any velocities whatoever, by decribing paces proportional to their diameters, will loe parts of their motions proportional to the wholes. For the motion of each globe will be as its velocity and mas conjunctly, that is, as the velocity and the cube of its diameter; the reitance (by fupoition) will be as the quare of the diameter and the quare of the velocity conjunctly; and the time (by this propoition) is in the former ratio directly and in the latter inverely, that is, as the diameter directly and the velocity inverely; and therefore the pace, which is proportional to the time and velocity, is as the diameter.

If bodies equally wift are reited in a equiplicate ratio of their diameters: Homogeneous globes, moving with any velocities whatoever, by decribing paces that are in a equiplicate ratio of the diameters, will loe parts of their motions proportional to the whales.

And univerally, if equally wift bodies are reited in the ratio of any power of the diameters: the paces, in which homogeneous globes, moving with any velocity whatoever, will loe parts of their motions proportional to the wholes, will be as the cubes of the diameters applied to that power. Let thoe diameters