xxi.] THEOEY OF APPROXIMATION. 459 



that every particle of matter in the universe attracts every 

 other particle, with a force depending on the masses of 

 the particles and their distances. We cannot know the 

 force acting on any particle unless we know the masses 

 and distances and positions of all other particles in the 

 universe. The physical astronomer has made a sweeping 

 assumption, namely, that all the millions of existing 

 systems exert no perturbing effects on our planetary 

 system, that is to say, no effects in the least appreciable. 

 The problem at once becomes hypothetical, because there 

 is little doubt that gravitation between our sun and planets 

 and other systems does exist. Even when they consider 

 the relations of our planetary bodies inter se, all their 

 processes are only approximate. In the first place they 

 assume that each of the planets is a perfect ellipsoid, 

 with a smooth surface and a homogeneous interior. That 

 this assumption is untrue every mountain and valley, every 

 sea, every mine affords conclusive evidence. If astronomers 

 are to make their calculations perfect, they must not only 

 take account of the Himalayas and the Andes, but must 

 calculate separately the attraction of every hill, nay, of 

 every ant-hill. So far are they from having considered 

 any local inequality of the surface, that they have not yet 

 decided upon the general form of the earth ; it is still a 

 matter of speculation whether or not the earth is an ellip- 

 soid with three unequal axes. If, as is probable, the globe 

 is irregularly compressed in some directions, the calcula- 

 tions of astronomers will have to be repeated and refined, 

 in order that they may approximate to the attractive 

 power of such a body. If we cannot accurately learn the 

 form of our own earth, how can we expect to ascertain 

 that of the moon, the sun, and other planets, in some of 

 which probably are irregularities of greater proportional 

 amount ? 



In a further way the science of physical astronomy is 

 merely approximate and hypothetical. Given homogeneous 

 ellipsoids acting upon each other according to the law of 

 gravity, the best mathematicians have never and perhaps 

 never will determine exactly the resulting movements. 

 Even when three bodies simultaneously attract each other 

 the complication of effects is so great that only approxi- 

 mate calculations can be made. Astronomers have not 



