DEMONSTRATION, AND NECESSARY TRUTHS. 337 



science resembles geometry in the further circum- 

 stance, that some of its inductions are not exactly 

 true ; and that the peculiar certainty ascribed to it, 

 on account of which its propositions are called Neces- 

 sary Truths, is fictitious and hypothetical, being true 

 in no other sense than that those propositions neces- 

 sarily follow from the hypothesis of the truth of 

 premisses 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 the definitions of the various 

 numbers, in the improper or geometrical sense of the 

 word Definition ; and secondly, the two following 

 axioms : The sums of equals are equal, The differ- 

 ences of equals are equal. These two are sufficient ; 

 for the corresponding propositions respecting unequals 

 may be proved from these, by the process well known 

 to mathematicians under the name of reductio ad 

 absurdum. 



These axioms, and likewise the so-called defini- 

 tions, are, as already shown, results of induction; true 

 of all objects whatever, and, as it may seem, exactly 

 true, without any hypothetical assumption of unquali- 

 fied truth where an approximation 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 sci- 

 ences in this, that the absolute certainty which is 

 predicable of its demonstrations is independent of all 

 hypothesis. 



On more accurate investigation, however, it will 

 be found that, even in this case, there is one hypo- 

 VOL. i. z 



