Mr. T. S. Davies on the Demonstration of certain Formulce. 115 



under such a variety of circumstances, as to be enabled to de- 

 duce from experiment alone the due corrections for tempera- 

 ture, pressure, moisture, and the direction and velocity of the 

 wind, independent of all theory. , 



XV. Thotights on the Demo7istration of certain Formula;. By 



T. S. Davies, Esq. 

 A S upon the truth of the binomial theorem has been supposed 

 ■^^ to rest the truth of almost every thing we know in the 

 higher branches of analysis, it was very natural for the most 

 profound powers of investigation to be employed in effecting 

 a satisfactory demonstration of it. The theorem, however, in 

 its simplest case was discovered by induction; its extension 

 was also by i?iductio?i ,■ and the clearest, indeed the only proof 

 we even now possess of its truth is altogether inductive. On 

 this latter topic I am aware that great names are against me : 

 but are they not also opposed to each other ? Many have at- 

 tempted to perform this task, but each has easily discovered 

 some fallacy in the other's proofs : and, indeed, I doubt 

 whether there be any writer who has not felt a latent suspicion 

 of the logical accuracy even of that perfect one of all — his own. 

 It becomes, then, an important course of inquiry which in- 

 vestigates the causes of these failures ; and perhaps a few very 

 simple considerations may lead us to believe that the nature 

 of the evidence upon which this and other theorems rest, is of 

 a character essentially different from that which is commonly 

 supposed to appertain to mathematical truth. It will thence 

 appear probable that the failures have not arisen from want 

 of skill, but from aiming at a species of proof of which the 

 subject does not admit. Nor can the perspicuity of our pro- 

 cesses and the facility of acquisition fail to be materially im- 

 proved by a reversal of our common mode of proceeding, 

 should that procedure be founded upon an erroneous isstimate 

 of the objects to be attained and the proper steps of attain- 

 ment. 



The first great principle we should ever bear in mind is, that 

 all the " laws of number " are mere deductions from a com- 

 parison instituted between a great variety of separate cases, or 

 else deductions from the results thus obtained. Such, indeed, 

 is the origin of our belief of the numerical fact, that m times 

 n is equal to n times /»; 



that (x ± 1/r = ^'" ± mx^'-^y + ; 



that a" fx = f{x + nh)-~n.'P[x + n — l.h] +...; 



P 2 tllMt 



