Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/95

 one to the other in the duplicate ratio of the ides AD, AE. Q.E.D.

The paces which a body decribes by any finite force urging it, whether that force is determined and immutable, or is continually augmented or continually diminihed, are in the very beginning of the motion one to the other in the duplicate ratio of the times.

Let the times be repreented by the lines AD, AE, and the velocities generated in those times be ordinates DB, EC. The paces described with these velocities will be as the areas ABD, ACE, decribed by thoe ordinates, that is, at the very beginning of the motion (by Lem. 9.) in the duplicate ratio of the times AD, AE.

And hence one may eaily infer, that the errors of the bodies decribing imilar parts of imilar figures in proportional times, are nearly in the duplicate ratio of the times in which they generated; if o be these errors are generated by any equal forces imilarly applied to the bodies, and meaured by the ditances of the bodies from those places of the imilar figures, at which, without the action of thoe forces, the bodies would have arrived in thoe proportional times.