37 



Now if w = |, T : f.: R^ : r*, or T 2 : f- :: B* : r* ; in 



other words the squares of the periodic times are as the 

 cubes of the distances from the centre, which is the law 

 discovered by Kepler from observation actually to prevail 

 in the case of the planets. And as he also showed from ob 

 servation that they describe equal areas in equal times by 

 their radii vectores drawn to the sun, it follows from the 

 fundamental proposition, first, that they are deflected from 

 the tangents of their orbits by a power tending towards the 

 sun; and then it folio vf ^secondly 9 from the last deduction re 

 specting it, (namely, the proportion of F : f - - : -59) that 



this central force acts inversely as the squares of the 

 distances, always supposing the bodies to move in cir 

 cular orbits, to which our demonstration has hitherto 

 been confined.* 



The extension, however, of the same important pro 

 position to the motion of bodies in other curves is easily 

 made, that is to the motion of bodies in different parts 

 of the same curve or in curves which are similar. For 

 in evanescent portions of the same curve, the osculating 

 circle, or circle which has the same curvature at any point, 

 coincides with the curve at that point; and if a line is 

 drawn to the extremity of that circle s diameter, A M B 

 and a m b may be considered as triangles ; and as they are 

 right angled at M and ??z, A M- is equal to A P x A B and 

 a m 2 to a pxa 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 



* We shall afterwards show, from other considerations, that this sesqui- 

 plicate proportion only holds true on the supposition of the bodies all 

 moving without exerting any action on each other, when we come to con 

 sider Laplace s theorems on elliptical motion. 



D 3 



