xxi.] THEORY OF APPROXIMATION. 463 



be modified by every new series of experiments, and it may 

 not improbably be shown that the assumed law can never 

 be made to agree exactly with the results of experiment. 



Philosophers have not always supposed that the law of 

 gravity was exactly true. Newton, though he had the 

 highest confidence in its truth, admitted that there were 

 motions in the planetary system which he could not 

 reconcile with the law. Euler and Clairaut who were, 

 with D'Alembert, the first to apply the full powers of 

 mathematical analysis to the theory of gravitation as ex- 

 plaining the perturbations of the planets, did not think 

 the law sufficiently established to attribute all discrepancies 

 to the errors of calculation and observation. They did 

 not feel certain that the force of gravity exactly obeyed 

 the well-known rule. The law might involve other powers 

 of the distance. It might be expressed in the form 



and the coefficients a and c might be so small that those 

 terms would become apparent only in very accurate 

 comparisons with fact. Attempts have been made to 

 account for difficulties, by attributing value to such 

 neglected terms. Gauss at one time thought the even 

 more fundamental principle of gravity, that the force 

 is dependent only on mass and distance, might not 

 be exactly true, and he undertook accurate pendulum 

 experiments to test this opinion. Only as repeated 

 doubts have time after time been resolved in favour of 

 the law of Newton, has it been assumed as precisely 

 correct. But this belief does not rest on experiment or 

 observation only. The calculations of physical astronomy, 

 however accurate, could never show that the other terms 

 of the above expression were absolutely devoid of value. 

 It could only be shown that they had such slight value 

 as never to become apparent. 



There are, however, other reasons why the law is pro- 

 bably complete and true as commonly stated. Whatever 

 influence spreads from a point, and expands uniformly 

 through space, will doubtless vary inversely in intensity 

 as the square of the distance, because the area over which 

 it is spread increases as the square of the radius. This 

 part of the law of gravity may be considered as due to 



