MATHEMATICAL DISCOVERY. 59 



It may appear surprising that sensibility should 

 be introduced in connexion with mathematical de- 

 monstrations, which, it would seem, can only interest 

 the intellect. But not if we bear in mind the feeling 

 of mathematical beauty, of the harmony of numbers 

 and forms and of geometric elegance. It is a real 

 aesthetic feeling that all true mathematicians recognize, 

 and this is truly sensibility. 



Now, what are the mathematical entities to which 

 we attribute this character of beauty and elegance, 

 which are capable of developing in us a kind of 

 aesthetic emotion ? Those whose elements are har- 

 moniously arranged so that the mind can, without 

 effort, take in the whole without neglecting the details. 

 This harmony is at once a satisfaction to our JEsthetic 

 requirements, and an assistance to the mind which 

 it supports and guides. At the same time, by setting 

 before our eyes a well-ordered whole, it gives us 

 a presentiment of a mathematical law. Now, as I 

 have said above, the only mathematical facts worthy 

 of retaining our attention and capable of being useful 

 are those which can make us acquainted with a 

 mathematical law. Accordingly we arrive at the 

 following conclusion. The useful combinations arc 

 precisely the most beautiful, I mean those that can 

 most charm that special sensibility that all mathe- 

 maticians know, but of which laymen arc so ignorant 

 that they are often tempted to smile at it. 



What follows, then ? Of the very large number of 

 combinations which the subliminal ego bh'ndly forms, 

 almost all are without interest and without utility. 

 pjut, for that very reason, they are without action on 

 the ajsthetic sensibility ; the consciousness will never 



