258 REASONING. 



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

 may proceed upon as proved, or intend to assume. 

 In any one demonstration 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 upon it, but yet if we deny it the 

 demonstration fails. The proof does not rest upon 

 the general assumption, but upon a similar assump- 

 tion confined to the particular case : that case, how- 

 ever, being chosen as a specimen or paradigm of the 

 whole class of cases included in the theorem, there 

 can be no ground for making the assumption in that 

 case which does not exist in every other ; and if you 

 deny the assumption as a general truth, you deny the 

 right to make it in the particular instance. 



There are, undoubtedly, the most ample reasons 

 for stating both the principles and the theorems in 

 their general form, and these will be explained pre- 

 sently, so far as explanation is requisite. But, that 

 an unpractised learner, even in making use of one 

 theorem to demonstrate another, reasons rather from 

 particular to particular than from the general propo- 

 sition, is manifest from the difficulty he finds in 

 applying a theorem to a case in which the configura- 

 tion of the diagram is extremely unlike that of the 

 diagram by which the original theorem was demon- 

 strated. A difficulty which, except in cases of unusual 

 mental power, long practice can alone remove, and 

 removes chiefly by rendering us familiar with all the 

 configurations consistent with the general conditions 

 of the theorem. 



