THEORY OF APPROXIMATION. 81 



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 gravita- 

 tion as explaining the perturbations of the planets, did 

 not treat the law as sufficiently established to attribute 

 all discrepancies to the errors of calculation and obser- 

 vation. In short, they did not feel certain that the force 

 of gravity exactly obeyed the well known rule. The 

 law might have involved other powers of the distance. 

 It might have been expressed, for example, in the form 



a b c 



' + D + & + W + ' 



and the coefficients a and c might have been so small 

 that those terms would only become apparent in very 

 accurate comparisons with fact. Attempts have been 

 made from time to time to account for difficulties, by 

 attributing value to such neglected terms. Gauss at 

 one time thought that the even more fundamental prin- 

 ciple 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 these repeated doubts have been time 

 after time 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 general 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 theoretical reasons why the 

 law is probably complete and true as commonly stated. 

 Whatever influence or power spreads from a point, and 

 expands uniformly through space, will doubtless vary in- 

 versely in intensity as the square of the distance, simply 

 because the area over which it is spread increases as the 



VOL. II. G 



