418 THE COURSE OF MATHEMATICAL STUDIES. 



mind distinctly apprehends as such, the contradictory is seen to 

 be inconceivable ; this inconceivableness of the contradictory 

 being ex parte mentis the criterion of necessary truth. Never- 

 theless, although when we think of any simple proposition in 

 arithmetic or geometry, we perceive not merely that it is true, 

 but that it must of necessity be so, this is nowise the case with 

 respect to all demonstrated or demonstrable results. The in- 

 tuition, so to speak, of the ablest mathematician is confined 

 within a narrower circle than that of the truths which he can 

 prove. He may satisfy himself of the cogency of each step of 

 the demonstration, and yet the essence of the conclusion the 

 fundamental principle of its truth remains unseen. The on is 

 manifest, but the Sm obscure ; and consequently a proposition 

 contradictory to that to which he has been led does not appear 

 to him an absurdity, but simply an untruth. It might, for what 

 he sees, have been true, though he knows that actually it is not, 

 and thus while he is aware that his conclusion is true neces- 

 sarily, yet still it seems as if it were so only contingently and 

 as a matter of fact, the demonstration appearing assensum con- 

 stringere, non rem. In a word, his conception of the matter is 

 still imperfect. But between this state of mind and that which 

 is produced by the contemplation of any elementary proposition, 

 there is no fixed or definite boundary. Every one who has 

 really studied mathematics must remember cases in which, after 

 long and patient thought, the reason of the truth of a propo- 

 sition, with the demonstration of which he may have been 

 acquainted for years, has seemed to dawn on him ; the propo- 

 sition thenceforth becoming, as it were, a part of his own mind 

 a matter about which he is no more capable of doubting than 

 about the primary conceptions of form and magnitude. The 

 mind thus brought into nearer, if not immediate, contact with 

 necessary truth is conscious of its own development ; and herein, 

 I believe, resides the special benefit to be derived from the study 

 of mathematics, a benefit, that is, distinct from the exercise of 

 patience and attention which it undoubtedly requires, but which 

 is required also in other pursuits. The study of mathematics is 

 especially valuable, not because it gives the Student practice in 

 ratiocination but because it enlarges the sphere of his intuition, 

 by giving him distinct and conscious possession of truths which 

 lay hid in his conceptions of figure, number, and the like. But 



