REMAINING LAWS OF NATURE. 



599 



particular 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 consider- 

 able progress in it. We there re- 

 marked that the directly inductive 

 truths of mathematics are few in 

 number, consisting of the axioms, to- 

 gether with certain propositions con- 

 cerning existence, tacitly involved in 

 most of the so-called definitions. And 

 we gave what appeared conclusive 

 reasons for affirming that these ori- 

 ginal premises, from which the re- 

 maining truths of the science are 

 deduced, are, notwithstanding all ap- 

 pearances to the contrary, results of 

 observation and experience, founded, 

 in short, on the evidence of the senses. 

 That things equal to the same thing 

 jue equal to one another, and that 

 two straight lines which have once 

 intersected one another continue to 

 diverge, are inductive truths, rest- 

 ing, indeed, like the law of universal 

 causation, only on induction per enu- 

 merationem simpliceni, 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 completely uni- 

 versal 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 ad- 

 verting ; because, as a perception of 

 their truth, in any individual case 

 whatever, requires only the simple 

 act of looking at the objects ia a 

 proper position, there never could 

 liave 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 impos- 



sible for the mind to conceive the 

 objects under any other law, they 

 were, and still are, generally con- 

 sidered as truths recognised by theiv 

 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 num- 

 ber. 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 otheraxioms are required,*) 

 together with the definitions of the 

 various numbers. L'lce other so-called 

 definitions, these i»i« composed of two 

 things, the explanation of a name and 

 the assertion of a fact : of which the 

 latter alone can form a first principle 

 or premise of a science. The fact 



* The axiom, " Equals subtracted from 

 equals leave equal differences," may be 

 demonstrated 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 H- c. Then 

 since B=6, adding equals to equals, 

 A = a -f- c. But A = a. Therefore a = « 

 + c; which is impossible. 



This proposition having been demon- 

 strated, we may, by means of it, demon- 

 strate the following: " If equals be added 

 to unequals, the sums are unequal." If 

 A = o and B not = 6, A -I- B is not =a + b. 

 For suppose it to be so. Then, since A = o 

 and A -t- B = a -t- 6, subtracting equals from 

 equals, B = 6 / 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 iti 

 contrary to the hypothesis. 



