264 



INDUCTION. 



effect, the Method of Concomitant 

 "Variations does not, however, pre- 

 suppose this as an axiom. It only 

 requires the converse proposition, 

 that anything on whose modifications, 

 modifications of an effect are invari- 

 ably consequent, must be the cause 

 (or connected with the cause) of that 

 effect ; a proposition, the truth of 

 which is evident ; for if the thing 

 itself had no influence on the effect, 

 neither could the modifications of the 

 thing have any influence. If the 

 stars have no power over the fortunes 

 of mankind, it is implied in the very 

 terms that the conjunctions or oppo- 

 sitions of different stars can have no 

 such power. 



Although the most striking applica- 

 tions of the Method of Concomitant 

 Variations take place in the cases 

 in which the Method of Difference, 

 strictly so called, is impossible, its 

 use is not confined to those cases ; it 

 may often usefully follow after the 

 Method of Difference, to give addi- 

 tional precision to a solution which 

 that has found . When by the Method 

 of Difference it has first been ascer- 

 tained that a certain object produces 

 a certain effect, the Method of Con- 

 comitant Variations may be usefully 

 called in to determine according to 

 what law the quantity or the different 

 relations of the effect follow those 

 of the cause. 



§ 7. The case in which this method 

 admits of the most extensive employ- 

 ment is that in which the variations 

 of the cause are variations of quan- 

 tity. Of such variations we may in 

 general affirm with safety that they 

 will be attended not only with varia- 

 tions, but with similar variations of 

 the effect : the proposition, that more 

 of the cause is followed by more of 

 the effect, being a corollary from 

 the principle of the Composition of 

 Causes, which, as we have seen, is the 

 general rule of causation ; cases of 

 the opposite description, in which 

 causes change their properties on 

 being conjoined with one another. 



being, on the contrary, special and ex- 

 ceptional. Suppose, then, that when 

 A changes in quantity, a also changes 

 in quantity, and in such a manuer 

 that we can trace the numerical rela- 

 tion which the changes of the one bear 

 to such changes of the other as take 

 place within our limits of observation. 

 We may then, with certain precau- 

 tions, safely conclude that the same 

 numerical relation will hold beyond 

 those limits. If, for instance, we find 

 that when A is double, a is double ; 

 that when A is treble or quadruple, 

 a is treble or quadruple ; we may 

 conclude that if A were a half or a 

 third, a would be a half or a third ; 

 and finally, that if A were annihi- 

 lated, a would be annihilated ; and 

 that a is wholly the effect of A, or 

 wholly the effect of the same cause 

 with A And so with any other 

 numerical relation according to which 

 A and a would vanish simultaneously ; 

 as, for instance, if a were proportional 

 to the square of A. If, on the other 

 hand, a is not wholly the effect of A, 

 but yet varies when A varies, it is 

 probably a mathematical function not 

 of A alone, but of A and something 

 else ; its changes, for example, may 

 be such as would occur if part of it 

 remained constant, or varied on some 

 other principle, and the remainder 

 varied in some numerical relation ta 

 the variations of A. In that case, 

 when A diminishes, a will be seen to 

 approach not towards zero, but to- 

 wards some other limit ; and when 

 the series of variations is such as to 

 indicate what that limit is, if con- 

 stant, or the law of its variation if 

 variable, the limit will exactly mea- 

 sure how much of a is the effect of 

 some other and independent cause, 

 and the remainder will be the effect 

 of A (or of the cause of A). 



These conclusions, however, must 

 not be drawn without certain precau- 

 tions. In the first place, the possi- 

 bility of drawing them at all mani- 

 festly supposes that we are acquainted 

 not only with the variations, but with 

 the absolute quantities both of A and 



