104 NEWTON S PRINCIPIA. 



and reaction are equal and opposite, by the third of the 

 Laws of Motion originally stated, it is evident that the 

 case of a fixed centre cannot exist when the attraction, 

 which we call the centripetal force, proceeds from a body 

 placed in the centre, unless, indeed, some counteracting 

 force shall fix this body to one point ; for if no force exists 

 but the mutual action of the two bodies, the central body 

 must be acted upon by the one which moves round it, and 

 its position must be affected by this action. Hence, for 

 example, if there were only two heavenly bodies, M and 

 E, and the one, M, moved round the other, E, by a pro 

 jectile force originally impressed upon it, the other, E, 

 would also move round M, unless the mass of the latter 

 body was infinitely small, and its attraction, proportional 

 to this mass, could not sensibly affect the larger body. 

 Again, if two bodies, the one moving round the other, 

 both together move round a third, S, the action of this 

 third will affect the motions of the other two relatively 

 to each other. Thus each smaller system will be affected, 

 both as to the motions and orbits of the bodies composing 

 it, by the action of the body in the common centre of the 

 whole; and they will also be affected by the action of 

 the bodies in the other systems, having the same com 

 mon centre. The inquiry, therefore, divides itself into two 

 branches ; first, the difference between the motions which 

 we have hitherto been considering when the centre was 

 fixed, and the actual motions of the system, as that of 

 the moon and earth revolving round each other with a 

 moveable centre ; secondly, the still more important dif 

 ference between the motions already considered, and the 

 actual motions, which difference is caused by the mutual 

 actions of the whole bodies on each, and varies both the 

 motions and the orbits of all. 



i. Suppose two bodies mutually attracting each other 



