THE FUTURE OF MATHEMATICS. 43 



preceded ours, the greatest masters would have an- 

 swered " Yes." To-day we are so familiar vvath this 

 notion that we can speak of it, even in a university 

 course, without exciting too much astonishment. 



But of what use can it be ? This is easy to see. In 

 the first place it gives us a very convenient language, 

 which expresses in very concise terms what the ordi- 

 nary language of analysis would state in long-winded 

 phrases. More than that, this language causes us to 

 give the same name to things which resemble one 

 another, and states analogies which it does not allow 

 us to forget. It thus enables us still to find our way 

 in that space which is too great for us, by calling to 

 our mind continually the visible space, which is only 

 an imperfect image of it, no doubt, but still an image. 

 Here again, as in all the preceding examples, it is 

 the analogy with what is simple that enables us to 

 understand what is complex. 



This geometry of more than three dimensions is 

 not a simple analytical geometry, it is not purely 

 quantitative, but also qualitative, and it is principally 

 on this ground that it becomes interesting. There is a 

 science called Geometry of Positioti, which has for its 

 object the study of the relations of position of the 

 different elements of a figure, after eliminating their 

 magnitudes. This geometry is purely qualitative ; its 

 theorems would remain true if the figures, instead of 

 being exact, were rudely imitated by a child. W^e can 

 also construct a Geometry of Positiojt of more than 

 three dimensions. The importance of Geometry of 

 Position is immense, and I cannot insist upon it too 

 much ; what Riemann, one of its principal creators, 

 has gained from it would be sufficient to demonstrate 



