42 



NEWTON S PRINCIPIA. 



nexion between its corollaries, and the Theory of Gravi 

 tation. 



The versed sine of the half of any evanescent arc (or 

 sagitta of the arc) of a curve in which a body revolves, 

 was proved to be as the centripetal force, and as the 

 square of the times ; or as F x T 2 . Therefore the 

 force F is directly as the versed sine, and inversely as the 

 square of the time. From this it follows that the central 

 force may be measured in several ways. The arc being 

 Q C, we are to measure the central force in its middle 

 point P. Then the areas being as the times ; twice 

 the triangle S P Q, or Q L x S P is as T 



in 



the last expression ; and, therefore, Q R being parallel 



to L P, the central force at P is as 



O 



Q p2 



O -L X 



So if S Y be the perpendicular upon the tangent P Y, 

 because P K and the arc P Q, evanescent, coincide, twice 

 the triangle S P Q is equal to S Y x Q P ; and the 



