THE FUTURE OF MATHEMATICS. 33 



cians of that day were willing to assume what we 

 explain by prolix dissertations. This does not mean 

 that they did not see it at all, but they passed it 

 over too hastily, and, in order to see it clearly, they 

 would have had to take the trouble to state it. 



Only, is it always necessary to state it so many 

 times ? Those who were the first to pay special 

 attention to exactness have given us reasonings that 

 we may attempt to imitate ; but if the demonstrations 

 of the future are to be constructed on this model, 

 mathematical works will become exceedingly long, 

 and if I dread length, it is not only because I am 

 afraid of the congestion of our libraries, but because 

 I fear that as they grow in length our demonstrations 

 will lose that appearance of harmony which plays such 

 a u.scful part, as I have just explained. 



It is economy of thought that we should aim at, 

 and therefore it is not sufficient to give models to 

 be copied. We must enable those that come after 

 us to do without the models, and not to repeat a 

 previous reasoning, but summarize it in a few lines. 

 And this has already been done successfully in certain 

 cases. For instance, there was a whole class of reason- 

 ings that resembled each other, and were found every- 

 where ; they were perfectly exact, but they were long. 

 One day some one thought of the term " uniformity of 

 convergence," and this term alone made them useless ; 

 it was no longer neces.sary to repeat them, since they 

 could now be assumed. Thus the hair-splitters can 

 render us a double service, first by teaching us to 

 do as they do if necessary, but more especially by 

 enabling us as often as possible not to do as they 

 do, and \et make no sacrifice of exactness. 



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