444 INDUCTION. 



then, that when A changes in quantity, a also changes in 

 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 annihilated, 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 prin- 

 ciple, and the remainder varied in some numerical relation to 

 the variations of A. In that case, when A diminishes, a will 

 be seen 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 remainder 

 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 ac- 

 quainted not only with the variations, but with the absolute 

 quantities both of A and a. If we do not know the total 

 quantities, we cannot, of course, determine the real numerical 

 relation according to which those quantities vary. It is there- 

 fore an error to conclude, as some have concluded, that because 

 increase of heat expands bodies, that is,, increases the dis- 

 tance between their particles, therefore the distance is wholly 

 the effect of heat, and that if we could entirely exhaust the 



