DEMONSTRATION, AND NECESSARY TRUTHS. 83 



peared to be experimental truths, the definitions, as they are 

 incorrectly called, in those sciences, were found by us to be 

 generalizations from experience which are not even, accurately 

 speaking, truths ; being propositions in which, while we assert 

 of some kind of object, some property or properties which 

 observation shows to belong to it, we at the same time deny 

 that it possesses any other properties, though in truth other 

 properties do in every individual instance accompany, and in 

 almost all instances modify, the property thus exclusively 

 predicated. The denial, therefore, is a mere fiction, or suppo- 

 sition, made for the purpose of excluding the consideration of 

 those modifying circumstances, when their influence is of too 

 trifling amount to be worth considering, or adjourning it, when 

 important, to a more convenient moment. 



From these considerations it would appear that Deductive 

 or Demonstrative Sciences are all, without exception, Induc- 

 tive Sciences ; that their evidence is that of experience ; but 

 that they are also, in virtue of the peculiar character of one 

 indispensable portion of the general formulae according to 

 which their inductions are made, Hypothetical Sciences. Their 

 conclusions are only true on certain suppositions, which are, 

 or ought to be, approximations to the truth, but are seldom, 

 if ever, exactly true ; and to this hypothetical character is to 

 be ascribed the peculiar certainty, which is supposed to be 

 inherent in demonstration. 



What we have now asserted, however, cannot be received 

 as universally true of Deductive or Demonstrative Sciences, 

 until verified by being applied to the most remarkable of all 

 those sciences, that of Numbers; the theory of the Calculus; 

 Arithmetic and Algebra. It is harder to believe of the doc- 

 trines of this science than of any other, either that they are 

 not truths a priori, but experimental truths, or that their 

 peculiar certainty is owing to their being not absolute but only 

 conditional truths. This, therefore, is a case which merits 

 examination apart ; and the more so, because on this subject 

 we have a double set of doctrines to contend with ; that of the 

 a priori philosophers on one side ; and on the other, a theory 

 the most opposite to theirs, which was at one time, very gene- 



