THE FUTURE OF MATHEMATICS. 31 



unity, that enables us to obtain a clear comprehension 

 of the whole as well as of the parts. But that is 

 also precisely what causes it to give a large return ; 

 and in fact the more we see this whole clearly and 

 at a single glance, the better we shall perceive the 

 analogies with other neighbouring objects, and con- 

 sequently the better chance we shall have of guessing 

 the possible generalizations. Elegance may result 

 from the feeling of surprise caused by the un- 

 looked-for occurrence together of objects not habitu- 

 ally associated. In this, again, it is fruitful, since it 

 thus discloses relations till then unrecognized. It is 

 also fruitful even when it only results from the con- 

 trast between the simplicity of the means and the 

 complexity of the problem presented, for it then causes 

 us to reflect on the reason for this contrast, and gener- 

 ally shows us that this reason is not chance, but is to 

 be found in some unsuspected law. Briefly stated, the 

 sentiment of mathematical elegance is nothing but the 

 satisfaction due to some conformity between the solu- 

 tion we wish to discover and the necessities of our 

 mind, and it is on account of this very conformity 

 that the solution can be an instrument for us. This 

 aesthetic satisfaction is consequently connected with 

 the economy of thought. Again the comparison with 

 the Erechtheum occurs to me, but I do not wish to 

 serve it up too often. 



It is for the same reason that, when a somewhat 

 lengthy calculation has conducted us to some simple 

 and striking result, we are not satisfied until we have 

 shown that we might have foreseen, if not the whole 

 result, at least its most characteristic features. Why 

 is this ? What is it that prevents our being contented 



