DEMONSTRATION, AND NECESSARY TRUTHS. 339 



therefore, which comprises the twofold conception of 

 unconditional truth and perfect accuracy, is not an 

 attribute of all mathematical truths, but of those only 

 which relate to pure Number, as distinguished from 

 Quantity in the more enlarged sense ; and only so long 

 as we abstain from supposing that the numbers are a 

 precise index to actual quantities. The certainty 

 usually ascribed to the conclusions of geometry, and 

 even to those of mechanics, is nothing whatever but 

 certainty of inference. We can have full assurance 

 of particular results under particular suppositions, but 

 we can not have the same assurance that these sup- 

 positions are accurately true, nor that they include 

 all the data which may exercise an influence over the 

 result in any given instance. 



4. It appears, therefore, that the method of all 

 Deductive Sciences is hypothetical. They proceed by 

 tracing the consequences of certain assumptions ; 

 leaving for separate consideration whether the assump- 

 tions are true or not, and if not exactly true, whether 

 they are a sufficiently near approximation to the truth. 

 The reason is obvious. Since it is only in questions 

 of pure number that the assumptions are exactly true, 

 and even there, only so long as no conclusions except 

 purely numerical ones are to be founded upon them ; 

 it must, in all other cases of deductive investigation, 

 form a part of the inquiry, to determine how much 

 the assumptions want of being exactly true in the case 

 in hand. This is generally a matter of observation, 

 to be repeated in every fresh case ; or if it has to be 

 settled by argument instead of observation, may 

 require, in every different case, different evidence, and 

 present every degree of difficulty from the lowest to 

 the highest. But the other part of the process 



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