THE FOUR EXPERIMENTAL METHODS. 447 



the dependence of the effect upon the cause, that could be 

 founded on those variations. Examples are not wanting of 

 such mistakes. " The formulae," says Sir John Herschel,* 

 " which have been empirically deduced for the elasticity of 

 steam, (till very recently,) and those for the resistance of 

 fluids, and other similar subjects," when relied on beyond the 

 limits of the observations from which they were deduced, " have 

 almost invariably failed to support the theoretical structures 

 which have been erected on them." 



In this uncertainty, the conclusion we may draw from the 

 concomitant variations of a and A, to the existence of an 

 invariable and exclusive connexion between them, or to the 

 permanency of the same numerical relation between their 

 variations when the quantities are much greater or smaller 

 than those which we have had the means of observing, cannot 

 be considered to rest on a complete induction. All that in 

 such a case can be regarded as proved on the subject of causa- 

 tion is, that there is some connexion between the two pheno- 

 mena ; that A, or something which can influence A, must be 

 one of the causes which collectively determine a. We may, 

 however, feel assured that the relation which we have observed 

 to exist between the variations of A and a, will hold true in all 

 cases which fall between the same extreme limits ; that is, 

 wherever the utmost increase or diminution in which the result 

 has been found by observation to coincide with the law, is not 

 exceeded. 



The four methods which it has now been attempted to de- 

 scribe, are the only possible modes of experimental inquiry 

 of direct induction a posteriori, as distinguished from deduc- 

 tion : at least, I know not, nor am able to imagine, any 

 others. And even of these, the Method of Residues, as we 

 have seen, is not independent of deduction ; though, as it 

 also requires specific experience, it may, without impro- 

 priety, be included among methods of direct observation and 

 experiment. 



These, then, with such assistance as can be obtained from 



* Discourse on the Study of Natural Philosophy, p. 179. 



