338 REASONING. 



thetical element in the ratiocination. In all proposi- 

 tions 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 always 

 equal to itself, unless we assume that all horses are of 

 equal strength ? It is certain that 1 is always equal 

 in number to 1; and where the mere number of 

 objects, or of the parts of an object, without suppos- 

 ing them to be equivalent in any other respect, is all 

 that is material, the conclusions of arithmetic, so far 

 as they go to that alone, are true without mixture of 

 hypothesis. There are a few such cases ; as, for 

 instance, an inquiry into the amount of 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 whenever, from equality or ine- 

 quality of number, equality or inequality in any other 

 respect is to be inferred, arithmetic carried into such 

 inquiries becomes as hypothetical a science as geo- 

 metry. All units must be assumed to be equal in 

 that other respect ; and this is never precisely true, 

 for one pound weight is not exactly equal to another, 

 nor one mile's length to another; a nicer balance, or 

 more accurate measuring instruments, would always 

 detect some difference. 



What is commonly called mathematical certainty, 



