Proposition XV. Theorem VII.
The ſame things being ſuppoſed, I ſay that the periodic times in ellipſes are in the ſeſsquiplicate ratio of their greater axes.
For the leſſer axe is a mean proportional between the greater axe and the latus rectum; and therefore the rectangle under the axes is in the ratio compounded of the ſubduplicate ratio of the latus rectum and the ſeſquiplicate ratio of the greater axe. But this rectangle (by cor. prop. 14) is in a ratio compounded of the ſubduplicate ratio of the latus rectum and the ratio of the periodic time. Subduct from both ſides the ſubduplicate ratio of the latus rectum, and there will remain the ſeſquiplicate ratio of the greater axe, equal to the ratio of the periodic time. Q. E. D.
Cor. Therefore the periodic times in ellipſes are the ſame as in circles whoſe diameters are equal to the greater axes of the ellipſes.
Proposition XVI. Theorem VIII.
The ſame things being ſuppoſed, and right lines being drawn to the bodies that ſhall touch the orbits, and perpendiculars being let fall on the tangents from the common focus: I ſay that the velocities of the bodies are in a ratio compounded of the ratio of the perpendiculars inverſely, and the ſubduplicate
