474 INDUCTION. 



quantity, and in such a manner that we can trace the 

 numerical relation 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 

 precautions, 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 anni- 

 hilated, 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 simulta- 

 neously ; 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 (to use a mathematical phrase) a function 

 not of A alone but of A and something else : its 

 changes will 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 

 to the variations of A. In that case, when A dimi- 

 nishes, a will seem to approach not towards zero, but 

 towards some other limit : and when the series of 

 variations is such as to indicate what that limit is, if 

 constant, or the law of its variation if variable, the 

 limit will exactly measure how much of a is the effect 

 of some other and independent cause, and the re- 

 mainder will be the effect of A (or of the cause of A). 



These conclusions, however, must not be drawn 

 without certain precautions. In the first place, the 

 possibility of drawing them at all, manifestly supposes 

 that we are acquainted not only with the variations, 

 but with the absolute quantities, both of A and a. If 



