FUNCTIONS AND VALUE OF THE SYLLOGISM. 257 



with other propositions of a similar kind, some of 

 which when generalized are called definitions, and 

 others axioms, we prove that a certain conclusion is 

 true, not of all circles, but of the particular circle 

 ABC; or at least would be so, if the facts precisely 

 accorded with our assumptions. The enunciation, as 

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

 at the head of the demonstration, is not the propo- 

 sition actually demonstrated. One instance only is 

 demonstrated : but 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 contrivance of 

 general language furnishing us with terms which con- 

 note 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 

 premisses, the axioms and definitions in their general 

 form. But this only means, that if we can prove an 

 individual conclusion by assuming 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 particular 

 VOL. i. s 



