LAW OP THE UNIVERSE. 



233 



circle's diameter, A M B and a in b may be considered as 

 triangles ; and as they are right angled at M and m, A M 2 is 

 equal to A P x A B and a m* to a p x a b ; and where the 



curvature is the same as in corresponding points of similar 

 curves, those squares are proportional to the lines A P, or 

 a p ; or those versed sines of the arcs A M and a m are pro- 

 portional 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 

 demonstrated 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, 



