234 



CENTRAL FORCES; 



through which the body falls by the force of gravity, in the 

 same time that it would take to describe the arc AN. Let 

 A M be the infinitely small arc described in an instant ; and 

 A P its versed sine. It was before shown, in the corollaries 

 to the first proposition, 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 centripetal 

 force at every instant, and uniformly accelerates the descend- 

 ing 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*, 

 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 APM, AB: AM::AM:AP 



AM 2 



and A P = . Substituting this in the former proportion, 



-A. -L> 



AM 2 A M 2 



we have ^- : A n : : A M 2 : A N 2 , or An : A N 2 : : -j-^- : 



A M*, that is : : 1 : A B. Therefore A N 2 = A n x A B, or 

 the arc described, is a mean proportional between the dia- 

 meter 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. 



