KEMAINING LAWS OF NATURE. 429 



It thus appears that mathematics is the only department of science into 

 the methods of which it still remains to inquire. And there is the less ne- 

 cessity that this inquiry should occupy us long, as we have already, in the 

 Second Book, made considerable progress in it. We there remarked, that 

 the directly inductive truths of mathematics are few in number; consist- 

 ing of the axioms, together with certain propositions concerning existence, 

 tacitly involved in most of the so-called definitions. And we gave what 

 appeared conclusive reasons for affirming that these original premises, from 

 which the remaining truths of the science are deduced, are, notwithstand- 

 ing all appearances to the contraiy, results of observation and experience ; 

 founded, 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 simplicemj on the fact that they have been perpetually 

 perceived to be true, and never once found to be false. But, as we have 

 seen in a recent chapter that this evidence, in the case of a law so complete- 

 ly 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 

 adverting ; 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 appar- 

 ently, though not really, exceptions to them. Their infallible truth was 

 recognized from the very dawn of speculation ; and as their extreme famil- 

 iarity made it impossible for the mind to conceive the objects under any 

 other law, they were, and still are, generally considered as truths recog- 

 nized by their own evidence, or by instinct. 



§ 5. There is something which seems to require explanation, in the fact 

 that the immense multitude of truths (a multitude 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 concerning 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),* to- 

 gether with the definitions of the various numbers. Like other so-called 

 definitions, these are composed of two things, the explanation of a name, 

 and the assertion of a fact ; of which the latter alone can form a first prin- 



* The axiom, " Equals subtracted from equals leave equal differences," may be demon- 

 strated from the two axioms in the text. If A=a and B=6, A— B=a— 6. For if not, let A 

 — B=a— 6-l-c. Then since B =6, adding equals to equals, A— a+c. ButA=a. Therefore 

 a = a-\-c, which is impossible. 



This proposition having been demonstrated, we may, by means of it, demonstrate the fol- 

 lowing: "If equals be added to unequals, the sums are unequal." If A=a and B not=6, 

 A-t-B is not=a-|-6. For suppose it to be so. Then, since A=a and A+B=a-}-6, sub- 

 tracting equals from equals, 'B — b; which is contrary to the hypothesis. 



So again, it may be proved that two things, one of which is equal and the other unequal to 

 a third thing, are unequal to one another. If A=a and A not=B, neither is a=B. For 

 suppose it to be equal. Then since A=a and a=B, and since things equal to the same 

 thing are equal to one another A=B ; which is contrary to the hypothesis. 



