266 



INDUCTION. 



dictions of effects ; but it is not 

 peculiarly applicable to the Method 

 of Concomitant Variations. The un- 

 certainty, however, of which I am 

 about to speak is characteristic of 

 that method, especially in the cases 

 in which the extreme limits of our 

 observation are very narrow in com- 

 parison with the possible variations 

 in the quantities of the pheno- 

 mena. Any one who has the slightest 

 acquaintance with mathematics is 

 aware that very different laws of 

 variation may produce numerical re- 

 sults which differ but slightly from 

 one another within narrow limits ; 

 and it is often only when the absolute 

 amounts of variation are considerable 

 that the difference between the re- 

 sults given by one law and by another 

 becomes appreciable. When, there- 

 fore, such variations in the quantity 

 of the antecedents as we have the 

 means of observing are small in com- 

 parison with the total quantities, there 

 is much danger lest we should mis- 

 take the numerical law, and be led 

 to miscalculate the variations which 

 would take place beyond the limits ; 

 a miscalculation which would vitiate 

 any conclusion respecting the de- 

 pendence 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 elasticity 

 of steam, (till very recently,) and those 

 for the resistance of fluids, and other 

 similar subjects," when relied on be- 

 yond 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 con- 

 nection between them, or to the per- 

 manency of the same numerical rela- 

 tion between their variations when 



* Discourse on the Study of Natural Philo- 

 tophy, p. 179. 



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 in- 

 duction. All that in such a case can 

 be regarded as proved on the subject 

 of causation is, that there is some 

 connection 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 describe are the 

 only possible modes of experimental 

 inquiry — of direct induction a pos- 

 teriori, as distinguished from deduc- 

 tion : at least, I know not, nor am 

 able to imagine, any others. And 

 even of these, the Method of Resi- 

 dues, as we have seen, is not inde- 

 pendent of deduction ; though, as it 

 also requires specific experience, it 

 may, without impropriety, be included 

 among methods 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. Be- 

 fore proceeding to point out certain 

 circumstances by which the employ- 

 ment 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, ac- 

 cordingly, will form the subject of 

 the succeeding chapter. 



