78 



NEWTON S PRINCIPIA. 



radius is ^ d ; in the case of a circle, it is to that of a body 



moving in a circle whose radius is d, as A/ D d : 



A/ D. And the like proportion subsists in the case of the 

 hyperbola. 



Further, a rule is thus deduced for determining, con 

 versely, the time of descent, the place being given. A 

 circle is to be described on A S = D, as the diameter, 

 and another from S the centre, towards which the body 



falls, with the radius . P being the point to which it 

 2i 



has fallen, if the area S X B be taken equal to S C A, 

 the time taken to fall through A P is equal to the time 



in which the body would move uniformly from B to X. 

 Hence the periodic times being in the sesquiplicate ratio 



of the distances (t = d ^) and because 2 2 = 2 A/ 2, the 

 time taken to fall through the whole distance to the centre 

 is to the periodic time of a body revolving at twice that 

 distance round the same centre as 1 to 4 V 2 ; and thus 

 we can calculate the time (supposing the planetary orbits 

 to be circular) which any one would take to fall in a 

 straight line to the sun, or any satellite to its principal 

 planet, if the projectile motion were suddenly to cease. 



