THE FUTURE OF MATHEMATICS. 39 



What more would you have ? says the engineer ; and 

 yet, in spite of everything, we are not satisfied, for 

 we should have liked to be able to predict the con- 

 vergence. And why ? Because if we had known 

 how to predict it in the one case, we should know 

 how to predict it in another. We have been success- 

 ful, it is true, but that is little in our eyes if we have 

 no real hope of repeating our success. 



In proportion as the science develops, it becomes 

 more difficult to take it in in its entirety. Then an 

 attempt is made to cut it in pieces and to be satisfied 

 with one of these pieces — in a word, to specialize. Too 

 great a movement in this direction would constitute 

 a serious obstacle to the progress of the science. As 

 I have said, it is by unexpected concurrences between 

 its different parts that it can make progress. Too 

 much specializing would prohibit these concurrences. 

 Let us hope that congresses, such as those of Heidel- 

 berg and Rome, by putting us in touch with each 

 other, will open up a view of our neighbours' territory, 

 and force us to compare it with our own, and so 

 escape in a measure from our own little village. In 

 this way they will be the best remedy against the 

 danger I have just noted. 



But I have delayed too long over generalities ; it 

 is time to enter into details. 



Let us review the different particular sciences which 

 go to make up mathematics ; let us see what each of 

 them has done, in what direction it is tending, and 

 what we may expect of it. If the preceding views 

 are correct, we should see that the great progress of 

 the past has been made when two of these sciences 

 have been brought into conjunction, when men have 



