266 ATTRACTION OF BODIES. 



the attraction being in the whole particles of the sphere; 

 and where the bodies move within the spherical surface, 

 what was demonstrated of concentric motion in those curves, 

 or where the centre of the curve is that of the attracting 

 forces, applies to the case of the sphere's centre being that of 

 attraction. For in the former case the centripetal force 

 decreases as the square of the distance increases ; and in the 

 latter case thatforce increases as the distance increases. Thus it 

 is to be observed, that in the two cases of attraction decreasing 

 inversely as the squares of the central distance (the case of 

 gravitation beyond the surface of bodies), and of attraction 

 increasing directly with the central distance (the case of 

 gravitation within the surface), the same law of attraction 

 prevails with respect to the corpuscular action of the spheres 

 as regulates the mutual action of those spheres and their 

 motions in revolution. But this identity of the law of attrac- 

 tion is confined to these two cases. 



Having laid down the law of attraction for these more 

 remarkable cases, instead of going through others where the 

 operation of attraction is far more complicated, Sir Isaac 

 Newton gives a general method for determining the attraction 

 whatever be the proportions between the force and the dis- 

 tance. This method is marked by all the geometrical ele- 

 gance of the author's other solutions ; and though it depends 

 upon quadratures, it is not liable to the objections in practice 

 which we before found to lie against a similar method applied 

 to the finding of orbits and forces ; for the results are easily 

 enough obtained, and in convenient forms. 



If A E B is the sphere whose attraction upon the point P 

 it is required to determine, whatever be the proportion 

 according to which that attraction varies with the distance, 

 and only supposing equal particles of A E B to have equal 

 attractive forces ; then from any point E describe the circle 

 E F, and another ef infinitely near, and draw E D, e d ordi- 

 nates to the diameter A B. The sphere is composed of small 

 concentric hollow spheres E e/F ; and its whole attraction is 

 equal to the sum of their attractions. Now that attraction 



