Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/224

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154
Book 1.
Mathematical Priciples
Principia - 1729 - Book 1, Section 10 - Banner.png

Section VII.


Concerning the rectilinear aſcent and deſcent of bodies.




Proposition XXXII. Problem XXIV.

Suppoſing that the centripetal force is reciprocally proportional to the ſquare of the diſtance of the places from the centre; it is required to define the ſpaces which a body, falling directly, deſcribes in given times.

Plate 15, Figure 1

Case 1. If the body does not fall perpendicularly it will (by cor. 1. prop. 13.) deſcribe ſome conic ſection whoſe focus is placed in the centre of force. Suppoſe that conic ſection to be ARPB (Pl. 15. Fig. 1.) and its focus S. And firſt, if the figure be an ellipsis; upon the greater axe thereof AB deſcribe the ſemi-circle ADB, and let the right line DPC paſs through the falling body,

making