146 INDUCTION. 



in short, on the evidence of the senses. That things equal to 

 the same thing are equal to one another, and that two straight 

 lines which have once intersected one another continue to 

 diverge, are inductive truths; resting, indeed, like the law of 

 universal causation, only on induction per enumerationem 

 simplicem; on the fact that they have been perpetually per- 

 ceived to he true, and never once found to he false. But, as 

 we have seen in a recent chapter that this evidence, in the 

 case of a law so completely universal as the law of causation, 

 amounts to the fullest proof, so is this even more evidently 

 true of the general propositions to which we are now advert- 

 ing ; because, as a perception of their truth in any individual 

 case whatever, requires only the simple act of looking at the 

 objects in a proper position, there never could have been in 

 their case (what, for a long period, there were in the case of 

 the law of causation) instances which were apparently, though 

 not really, exceptions to them. Their infallible truth was 

 recognised from the very dawn of speculation ; and as their 

 extreme familiarity made it impossible for the mind to conceive 

 the objects under any other law, they were, and still are, 

 generally considered as truths recognised by their own evi- 

 dence, or by instinct. 



5. There is something which seems to require explana- 

 tion, in the fact that the immense multitude of truths (a mul- 

 titude still as far from being exhausted as ever) comprised in 

 the mathematical sciences, can be elicited from so small a 

 number of elementary laws. One sees not, at first, how it is 

 that there can be room for such an infinite variety of true 

 propositions, on subjects apparently so limited. 



To begin with the science of number. The elementary or 

 ultimate truths of this science are the common axioms con- 

 cerning equality, namely, "Things which are equal to the 

 same thing are equal to one another," and " Equals added to 

 equals make equal sums," (no other axioms are required,*) 



* The axiom, "Equals subtracted from equals leave equal differences, 

 maybe demonstrated from the two axioms in the text. If A = a and]B = 6 



