FUNCTIONS AND VALUE OF THE SYLLOGISM. 213 



of the particular circle ABC ; or at least would be so, if the 

 facts precisely accorded with our assumptions. The enuncia- 

 tion, as it is called, that is, the general theorem which stands 

 at the head of the demonstration, is not the proposition 

 actually demonstrated. One instance only is demonstrated : 

 hut the process by which this is done, is a process which, 

 when we consider its nature, we perceive might be exactly 

 copied in an indefinite number of other instances ; in every 

 instance which conforms to certain conditions. The con- 

 trivance of general language furnishing us with terms which 

 connote these conditions, we are able to assert this indefinite 

 multitude of truths in a single expression, and this expression 

 is the general theorem. By dropping the use of diagrams, and 

 substituting, in the demonstrations, general phrases for the 

 letters of the alphabet, we might prove the general theorem 

 directly, that is, we might demonstrate all the cases at once ; 

 and to do this we must, of course, employ as our premises, the 

 axioms and definitions in their general form. But this only 

 means, that if we can prove an individual conclusion by assum- 

 ing an individual fact, then in whatever case we are warranted 

 in making an exactly similar assumption, we may draw an 

 exactly similar conclusion. The definition is a sort of notice 

 to ourselves and others, what assumptions we think ourselves 

 entitled to make. And so in all cases, the general propositions, 

 whether called definitions, axioms, or laws of nature, which we 

 lay down at the beginning of our reasonings, are merely 

 abridged statements, in a kind of short-hand, of the parti- 

 cular facts, which, as occasion arises, we either think we may 

 proceed on as proved, or intend to assume. In any one de- 

 monstration it is enough if we assume for a particular case 

 suitably selected, what by the statement of the definition or 

 principle we announce that we intend to assume in all cases 

 which may arise. The definition of the circle, therefore, is to 

 one of Euclid's demonstrations, exactly what, according to 

 Stewart, the axioms are ; that is, the demonstration does not 

 depend on it, but yet if we deny it the demonstration fails. The 

 proof does not rest on the general assumption, but on a similar 

 assumption confined to the particular case : that case, however, 



