38 



NEWTON S PKINCIPIA. 



sines of the arcs A M and a m are proportional to the 

 squares of the small arcs. Hence if the distances of two 



bodies from their respective centres of force be D, d, the 

 deflecting force in any points A and a being as the versed 

 sines, those forces are as A M 2 : a m 2 ; and from hence 

 follows generally in all curves, that which has been demon 

 strated respecting motion in circular orbits. 



The planets then and their satellites being known by Kep 

 ler s laws to move in elliptical orbits, and to describe round 

 the sun in one focus areas proportional to the times by their 

 radii vectores drawn to that focus, and it being further 

 found by those laws that the squares of their periodic times 

 are as the cubes of the mean distances from the focus^ 

 they are by these propositions of Sir Isaac Newton which 

 we have been considering, shown to be deflected from the 

 tangent of their orbit, and retained in their paths, by a 

 force acting inversely as the squares of the distances from 

 the centre of motion. 



But another important corollary is also derived from 

 the same proposition. If the projectile or tangential force 

 in the direction A T ceases (next figure), the body, 

 instead of moving in any arc A N, is drawn by the 

 same centripetal force in the straight line A S. Let A n 

 be the part of A S, through which the body falls by the 

 force of gravity, in the same time that it would take to 

 describe the arc A N. Let A M be the infinitely small 



