be proportional to the wholes, the reſiſtance and time conjunctly ought to be as the motion. Therefore the time will be as the motion directly and the reſiſtance inverſely. Wherefore the particles of the times being taken in that ratio, the bodies will always loſe parts of their motions proportional to the wholes, and therefore will retain velocities always proportional to their firſt velocities. And becauſe of the given ratio of the velocities, they will always deſcribe ſpaces, which are as the firſt velocities and the times conjunctly. Q. E. D.
Cor. 1. Therefore if bodies equally ſwift are reſiſted in a duplicate ratio of their diameters: Homogeneous globes moving with any velocities whatſoever, by deſcribing ſpaces proportional to their diameters, will loſe parts of their motions proportional to the wholes. For the motion of each globe will be as its velocity and maſs conjunctly, that is, as the velocity and the cube of its diameter; the reſiſtance (by fupoſition) will be as the ſquare of the diameter and the ſquare of the velocity conjunctly; and the time (by this propoſition) is in the former ratio directly and in the latter inverſely, that is, as the diameter directly and the velocity inverſely; and therefore the ſpace, which is proportional to the time and velocity, is as the diameter.
Cor. 2. If bodies equally ſwift are reſiſted in a ſeſquiplicate ratio of their diameters: Homogeneous globes, moving with any velocities whatſoever, by deſcribing ſpaces that are in a ſeſquiplicate ratio of the diameters, will loſe parts of their motions proportional to the whales.
Cor. 3. And univerſally, if equally ſwift bodies are reſiſted in the ratio of any power of the diameters: the ſpaces, in which homogeneous globes, moving with any velocity whatſoever, will loſe parts of their motions proportional to the wholes, will be as the cubes of the diameters applied to that power. Let thoſe diameters
