THEORY OF APPROXIMATION, 75 



straight lines, triangles, circles, and other regular geo- 

 metrical figures ; to our science it is a matter of indif- 

 ference whether they do or do not exist, because in any 

 case they must be beyond our powers of appreciation. If 

 we submitted a perfect circle to the most rigorous scrutiny 

 and measurement, it is impossible that we should discover 

 whether it were perfect or not. Nevertheless in geometry 

 we argue concerning perfect rectilineal figures and curves, 

 and the conclusions apply to existing objects so far as we 

 can assure ourselves that they agree with the hypothetical 

 conditions of our reasoning. Now this is in reality all that 

 we can do in the most perfect of the sciences of nature. 



Doubtless in astronomy we meet with the nearest ap- 

 proximation to actual conditions. The law of gravity is 

 not a complex one in itself, and we believe it with much 

 probability to be exactly true ; but we cannot calculate 

 out in any one case its accurate results. The law asserts 

 that every particle of matter in the universe attracts every 

 other particle, with a force depending on the masses of the 

 particles and their distance. We cannot then know the 

 force acting on any one particle unless we know the masses 

 and distances and positions of all the other particles in the 

 universe. The physical astronomer has from the first 

 made a sweeping assumption, namely, that all the other 

 millions of existing systems exert no perturbing effects in 

 our planetary system, that is to say, no effects in the least 

 appreciable. Thus the problem becomes at once hypo- 

 thetical, because there is little doubt that gravitation be- 

 tween our sun and planets and other systems must exist 

 in some degree. But even when they consider the re- 

 lations of our planetary bodies inter se, all their processes 

 are grossly approximative. 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 



