Proposition VI. Theorem IV.
Homogeneous and equal ſpherical bodies, oppos'd by reſiſtances that are in the duplicate ratio of the velocities, and moving on by their innate force only, will, in times which are reciprocally as the velocities at the beginning, deſcribe equal ſpaces, and loſe parts of their velocities proportional to the wholes.
To the rectangular aſymptotes CD, CH deſcribe any hyperbola BbEe, cutting the perpendiculars AB, ab, DE, de, in B, b, E, e; let the initial velocities be expounded by the perpendiculars AB, DE, and the times by the lines Aa, Dd. Therefore as Aa is to Dd, ſo (by the hypotheſis) is DE to AB, and ſo (from the nature of the hyperbola) is CA to CD; and, by compoſition, ſo is Ca to Cd. Therefore the areas ABba, DEed, that is, the ſpaces deſcribed, are equal among themſelves, and the firſt velocities AB, DE are proportional to the laſt ab, de; and therefore, by diviſion, proportional to the parts of the velocities loſt. AB—ab, DE—de. Q.E.D.
Proposition VII. Theorem V.
If ſpherical bodies are reſiſted in the duplicate ratio of their velocities, in times which are as the firſt motions directly and the firſt reſiſtances inverſely, they will loſe parts of their motions proportional to the wholes, and will deſcribe ſpaces proportional to thoſe times and the firſt velocities conjunctly.
For the parts of the motions loſt are as the reſiſtances and times conjunctly. Therefore, that thoſe parts may be
