t7o 



REASONING. 



the categorical 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 hypothetical 

 element in the ratiocination. In all 

 propositions concerning numbers, a 

 condition 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 

 alwajrs equal to itself, unless we 

 assume that all horses are of equal 

 strength ? It is certain that i is 

 always equal in number to i ; and 

 where the mere number of objects, 

 or of the parts of an object, without 

 supposing them to be equivalent in 

 any other respect, is all that is 

 material, the conclusions of arith- 

 metic, so far as they go to that alone, 

 are true without mixture of hypo- 

 thesis. There are such cases in 

 statistics ; as, for instance, an inquiry 

 into the amount of the population of 

 any country. It is indifferent to that 

 inquiry whether they are grown people 

 or children, strong or weak, tall or 

 short ; the only thing we want to 

 ascertain is their number. But when- 

 ever, from equality or inequality of 

 number, equality or inequality in any 

 other respect is to be inferred, arith- 

 metic carried into such inquiries be- 

 comes as hypothetical a science as 

 geometry. All units must be assumed 

 to be equal in that other respect ; and 

 this is never accurately true, for one 

 actual pound weight is not exactly 

 equal to another, nor one measured 

 mile's length to another ; a nicer 

 balance, or more accurate measuring 



instruments, would always detect some 

 difference. 



What is commonly called mathe- 

 matical certainty, therefore, which 

 comprises the twofold conception of 

 unconditional truth and perfect accu- 

 racy, is not an attribute of all mathe- 

 matical truths, but of those only 

 which relate to pure Number, af 

 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 

 cannot have the same assurance that 

 these suppositions are accurately true, 

 nor that they include all the data 

 which may exercise an influence over 

 the result in any g\ven 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 con- 

 sideration whether the assumptions 

 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 immerical ones are to be 

 founded on 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 observa- 

 tion, 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 — namely, 

 to determine what else may be con- 



