30 SCIENCE AND METHOD. 



An algebraical formula which gives us the solution 

 of a type of numerical problems, if we finally replace 

 the letters by numbers, is the simple example which 

 occurs to one's mind at once. Thanks to the formula, 

 a single algebraical calculation saves us the trouble of 

 a constant repetition of numerical calculations. But 

 this is only a rough example ; every one feels that 

 there are analogies which cannot be expressed by a 

 formula, and that they are the most valuable. 



If a new result is to have any value, it must unite 

 elements long since known, but till then scattered 

 and seemingly foreign to each other, and suddenly 

 introduce order where the appearance of disorder 

 reigned. Then it enables us to see at a glance each 

 of these elements in the place it occupies in the whole. 

 Not only is the new fact valuable on its own account, 

 but it alone gives a value to the old facts it unites. 

 Our mind is frail as our senses are ; it would lose 

 itself in the complexity of the world if that complexity 

 were not harmonious ; like the short-sighted, it would 

 only see the details, and would be obliged to forget 

 each of these details before examining the next, 

 because it would be incapable of taking in the whole. 

 The only facts worthy of our attention are those 

 which introduce order into this complexity and so 

 make it accessible to us. 



Mathematicians attach a great importance to the 

 elegance of their methods and of their results, and 

 this is not mere dilettantism. What is it that gives 

 us the feeling of elegance in a solution or a demonstra- 

 tion ? It is the harmony of the different parts, their 

 symmetry, and their happy adjustment ; it is, in a 

 word, all that introduces order, all that gives them 



