NEWTON S PRINCIPIA. 



39 



arc described in an instant ; and A P its versed sine. 

 It was before shown, in the corollaries to the first pro 

 position, that the centripetal force in A is as A P, and 

 the body would move by that force through A P, in the 

 same time in which it describes the arc A M. Now the 

 force of gravity being one which operates like the centri 

 petal force at every instant, and uniformly accelerates the 

 descending body, the spaces fallen through will be as the 

 squares of the times. Therefore, if A n is the space 

 through which the body falls in the same time that it 



would describe A N, A P is to A n as the square of the 

 time taken to describe A M to the square of the time of 

 describing A N, or as A M 2 : A N 2 , the motion being 

 uniform in the circular arc. But A M, the nascent arc, 

 is equal to its chord, and A M B being a right angled 

 triangle as well as A P M, A B : A M :: A M : A P and 



AM 2 



A P = . -p . Substituting this in the former proportion, 



AM 2 



we 



have 



AB 



n:: A M 2 : A N 2 , or A n : AN 5 



AM 2 , that is :: 1 : AB. Therefore AN 2 

 i 



= A n x A B, or the arc described, is a mean propor 

 tional between the diameter of the orbit, and the space 

 through which the body would fall by gravity alone, in 

 the same time in which it describes the arc. 



D 4 



