Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/141



The ame things being uppoed, I ay that the periodic times in ellipes are in the esquiplicate ratio of their greater axes.

For the leer 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 ubduplicate ratio of the latus rectum and the equiplicate ratio of the greater axe. But this rectangle (by cor. prop. 14) is in a ratio compounded of the ubduplicate ratio of the latus rectum and the ratio of the periodic time. Subduct from both ides the ubduplicate ratio of the latus rectum, and there will remain the equiplicate ratio of the greater axe, equal to the ratio of the periodic time. Q. E. D.

Therefore the periodic times in ellipes are the ame as in circles whoe diameters are equal to the greater axes of the ellipes.

The ame things being uppoed, and right lines being drawn to the bodies that hall touch the orbits, and perpendiculars being let fall on the tangents from the common focus: I ay that the velocities of the bodies are in a ratio compounded of the ratio of the perpendiculars inverely, and the ubduplicate