DEMONSTRATION, AND NECESSARY TRUTHS. 289 



exactly true ; and that the peculiar certainty ascribed to it, 

 on account of which its propositions are called Necessary 

 Truths, is fictitious and hypothetical, being true in no other 

 sense than that those propositions legitimately follow from the 

 hypothesis of the truth of premises which are avowedly mere 

 approximations to truth. 



3. The inductions of arithmetic are of two sorts : first, 

 those which we have just expounded, such as One and one are 

 two, Two and one are three, &c., which may be called ihe 

 definitions of the various numbers, in the improper or geome- 

 trical sense of the word Definition ; and secondly, the two fol- 

 lowing axioms : The sums of equals are equal, The differences 

 of equals are equal. These two are sufficient ; for the corre- 

 sponding propositions respecting unequals may be proved from 

 these, by a reductio ad absurdum. 



These axioms, and likewise the so-called definitions, are, 

 as has already been said, results of induction ; true of all ob- 

 jects whatever, and, as it may seem, exactly true, without the 

 hypothetical assumption of unqualified truth where an approxi- 

 mation to it is all that exists. The conclusions, therefore, it 

 will naturally be inferred, are exactly true, and the science of 

 number is an exception to other demonstrative sciences in this, 

 that the categorical certainty which is predicable of its demon- 

 strations is independent of all hypothesis. 



On more accurate investigation, however, it will be found 

 that, even in this case, there is one hypothetical element in the 

 ratiocination. In all propositions concerning numbers, a con- 

 dition is implied, without which none of them would be true ; 

 and that condition is an assumption which may be false. The 

 condition, is that 1 = 1; that all the numbers are numbers of 

 the same or of equal units. Let this be doubtful, and not one 

 of the propositions of arithmetic will hold true. How can we 

 know that one pound and one pound make two pounds, if one 

 of the pounds may be troy, and the other avoirdupois ? They 

 may not make two pounds of either, or of any weight. How 

 can we know that a forty-horse power is always equal to itself, 

 unless we assume that all horses are of equal strength ? It is 

 VOL. i. 19 



