262 OPERATIONS SUBSIDIARY TO INDUCTION. 



be reduced to a technical rule ; and such, in fact, the rules of 

 the calculus are. But if the symbols represent any other 

 things than mere numbers, let us say even straight or curve 

 lines, we have then to apply theorems of geometry not true of 

 all lines without exception, and to select those which are true 

 of the lines we are reasoning about. And how can we do this 

 unless we keep completely in mind what particular lines these 

 are ? Since additional geometrical truths may be introduced 

 into the ratiocination in any stage of its progress, we cannot 

 suffer ourselves, during even the smallest part of it, to use the 

 names mechanically (as we use algebraical symbols) without 

 an image annexed to them. It is only after ascertaining that 

 the solution of a question concerning lines can be made to 

 depend on a previous question concerning numbers, or in 

 other words after the question has been (to speak technically) 

 reduced to an equation, that the unmeaning signs become 

 available, and that the nature of the facts themselves to which 

 the investigation relates can be dismissed from the mind. Up 

 to the establishment of the equation, the language in which 

 mathematicians carry on their reasoning does not differ in 

 character from that employed by close reasoners on any other 

 kind of subject. 



I do not deny that every correct ratiocination, when 

 thrown into the syllogistic shape, is conclusive from the mere 

 form of the expression, provided none of the terms used be 

 ambiguous ; and this is one of the circumstances which have 

 led some writers to think that if all names were so judiciously 

 constructed and so carefully defined as not to admit of any 

 ambiguity, the improvement thus made in language would 

 not only give to the conclusions of every deductive science 

 the same certainty with those of mathematics, but would 

 reduce all reasonings to the application of a technical form, 

 and enable their conclusiveness to be rationally assented to 

 after a merely mechanical process, as is undoubtedly the case 

 in algebra. But, if we except geometry, the conclusions of 

 which are already as certain and exact as they can be made, 

 there is no science but that of number, in which the practical 



