288 A Short History of Astronomy [Cn. xi. 



observation to be nearly spherical in form, and the rest were 

 generally supposed to be so also. 



Newton had shewn, with a considerable degree of proba- 

 bility, that these bodies attracted one another according to 

 the law of gravitation ; and there was no reason to suppose 

 that they exerted any other important influence on one 

 another's motions.* 



The problem which presented itself, and which may con- 

 veniently be called Newton's problem, was therefore : 



Given these 18 bodies, and their positions and motions 

 at any time, to deduce from their mutual gravitation by 

 a process of mathematical calculation their positions and 

 motions at any other time ; and to shew that these agree 

 with those actually observed. 



Such a calculation would necessarily involve, among other 

 quantities, the masses of the several bodies ; it was evidently 

 legitimate to assume these at will in such a way as to make 

 the results of calculation agree with those of observation, 

 If this were done successfully the masses would thereby be 

 determined. In the same way the commonly accepted 

 estimates of the dimensions of the solar system and of the 

 shapes of its members might be modified in any way not 

 actually inconsistent with direct observation. 



The general problem thus formulated can fortunately be 

 reduced to somewhat simpler ones. 



Newton had shewn (chapter ix., 182) that an ordinary 

 sphere attracted other bodies and was attracted by them, 

 as if its mass were concentrated at its centre ; and that the 

 effects of deviation from a spherical form became very 

 small at a considerable distance from the body. Hence, 

 except in special cases, the bodies of the solar system could 

 be treated as spheres, which could again be regarded as 

 concentrated at their respective centres. It will be con- 

 venient for the sake of brevity to assume for the future 

 that all " bodies " referred to are of this sort, unless the 

 contrary is stated or implied. The effects of deviations 

 from spherical form could then be treated separately 



* Some other influences are known e.g. the sun's heat causes 

 various motions of our air and water, and has a certain minute effect 

 on the earth's rate of rotation, and presumably produces similar 

 effects on other bodies. 



