290 REASONING. 



certain that 1 is always equal in number to 1 ; and where the 

 mere numher 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 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 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 whenever, from equality or inequality of 

 number, equality or inequality in any other respect is to be 

 inferred, arithmetic carried into such inquiries becomes as hy- 

 pothetical 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 mathematical certainty, 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 dis- 

 tinguished 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 

 given instance. 



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

 tive 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 



