THE FOUR EXPERIMENTAL METHODS. 291 



has the slightest acquaintance with mathematics, is aware that very differ- 

 ent laws of variation may produce numerical results which differ but slight- 

 ly from one another within narrow limits; and it is often only when the 

 absolute amounts of variation are considerable, that the difference between 

 the results given by one law and by another becomes appreciable. When, 

 therefore, such variations in the quantity of the antecedents as we have the 

 means of observing are small in comparison with the total quantities, there 

 is much danger lest we should mistake the numerical law, and be led to 

 miscalculate the variations M'hich would take place beyond the limits; a 

 miscalculation which would vitiate any conclusion respecting the depend- 

 ence of the effect upon the cause, that could be founded on those varia- 

 tions. Examples are not wanting of such mistakes. " The formulae," says 

 Sir John Herschel,* " which have been empirically deduced for the elas- 

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

 other similar subjects," when relied on beyond the limits of the observa- 

 tions from which they were deduced, " have almost invariably failed to sup- 

 port 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 con- 

 nection 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, can not be consider- 

 ed to rest on a complete induction. All that in such a case can be regard- 

 ed as proved on the subject of causation is, that there is some connection 

 between the two phenomena; that A, or something which can influence A, 

 must be one of the causes which collectively determine a. We may, how- 

 ever, feel assured that the relation which we have observed to exist be- 

 tween the variations of A and a, will hold true in all cases which fall be- 

 tween 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 describe, are the 

 only possible modes of experimental inquiry — of direct induction a 2'>oste- 

 riori, as distinguished from deduction : 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 impropriety, be included among meth- 

 ods of direct observation and experiment. 



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

 compose the available resources of the human mind for ascertaining the 

 laws of the succession of phenomena. Before proceeding to point out cer- 

 tain circumstances by which the employment of these methods is subjected 

 to an immense increase of complication and of difficulty, it is expedient to 

 illustrate the use of the methods, by suitable examples drawn from actual 

 physical investigations. These, accordingly, will form the subject of the 

 succeeding chapter. 



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



