$ i8a, 183] Universal Gravitation 229 



density throughout, different shells being, however, of 

 different densities. For example, the result is true for a 

 hollow indiarubber ball as well as for a solid one, but 

 is not true for a sphere made up of a hemisphere of wood 

 and a hemisphere of iron fastened together. 



183. The law of gravitation being thus provisionally 

 established, the great task which lay before Newton, and 

 to which he devotes the greater part of the first and third 

 books of the Principia, was that of deducing from it and 

 the " laws of motion " the motions of the various members 

 of the solar system, and of shewing, if possible, that the 

 motions so calculated agreed with those observed. If this 

 were successfully done, it would afford a verification of the 

 most delicate and rigorous character of Newton's principles. 



The conception of the solar system as a mechanism, each 

 member of which influences the motion of every other 

 member in accordance with one universal law of attraction, 

 although extremely simple in itself, is easily seen to give rise 

 to very serious difficulties when it is proposed actually to 

 calculate the various motions. If in dealing with the 

 motion of a planet such as Mars it were possible to regard 

 Mars as acted on only by the attraction of the sun, and to 

 ignore the effects of the other planets, then the problem 

 would be completely solved by the propositions which 

 Newton established in 1679 ( 175), and by their means the 

 position of Mars at any time could be calculated with any 

 required degree of accuracy. But in the case which 

 actually exists the motion of Mars is affected by the forces 

 with which all the other planets (as well as the satellites) 

 attract it, and these forces in turn depend on the position of 

 Mars (as well as upon that of the other planets) and hence 

 upon the motion of Mars. A problem of this kind in all 

 its generality is quite beyond the powers of any existing 

 mathematical methods. Fortunately, however, the mass 

 of even the largest of the planets is so very much less than 

 that of the sun, that the motion of any one planet is only 

 slightly affected by the others ; and it may be regarded as 

 moving very nearly as it would move if the other planets 

 did not exist, the effect of these being afterwards allowed 

 for as producing disturbances or perturbations in its path. 

 Although even in this simplified form the problem of the 



