384 INDUCTION. 



that the square of any number is four times the square of half the 

 number, is true independently of any cause. The onlj laws of resem- 

 blance, therefore, which we are called upon to consider independently 

 of causation belong to the province of mathematics. 



§ 4. The same thing is evident with respect to the only remaining 

 one of our five categories, Order in Place. The order in place, of the 

 effects of a cause, is (like everything else belonging to the eftects) a 

 consequence of the laws of that cause. The order in place, or, as we 

 have termed it, the collocation, of the primeval causes is (as well as 

 their resemblance) in each instance an ultimate fact, in which no laws 

 or uniformities are traceable. The only remaining general propo- 

 sitions respecting order in place, and the only ones which have nothing 

 to do with causation, are some of the truths of geometry ; laws through 

 which we are able, from the order in place of certain points, lines, or 

 spaces, to infer the order in place of others which are connected with 

 the former in some known mode ; quite independently of the partic- 

 ular nature of those points, lines, or spaces, in any other respect than 

 position or magnitude, as well as independently of the physical cause 

 from which in any p'articular case they happen to derive their origin. 



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 necessity that this inquiry should occupy us long, as we have already 

 in the second Book, made considerable progress in it. We there re- 

 marked, that the directly inductive truths of mathematics are few in 

 number ; consisting of the axioms, together with certain propositions 

 ■concerning existence, tacitly involved in most of the so-called defi- 

 nitions. And we proved, at such length as makes any return to the 

 subject altogether superfluous, that these oi'iginal premisses, from 

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

 standing all appearances to the contrary, results of observation and ex- 

 perience ; founded, in short, on the evidence of the senses. That 

 things equal to the same thing are equal to anoXher, or that two straight 

 lines which have once intersected with one another continue to diverge, 

 are inductive truths ; resting indeed, like the law of universal causation, 

 only upon induction -per enumeratloncm simplicem ; upon the fact that 

 they have been perpetually found true and never once 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 attainable by the human faculties, so is this even more 

 evidently true of the general propositions to which we are now ad- 

 verting ; 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, in the case of the" law of causation, there were) instances 

 which were apparently, though not really, exceptions to them. Their 

 infallible truth was recognized 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 con- 

 sidered as truths recognized 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 



