198 POPULAR SCIENCE MONTHLY 



sketch : nevertheless he has not hesitated to publish it ; and he would 

 probably believe he finds in it, if not a rigorous demonstration, at least 

 a kind of moral certainty. A logician would have rejected with horror 

 such a conception, or rather he would not have had to reject it, because 

 in his mind it would never have originated. 



Again, permit me to compare two men, the honor of French science, 

 who have recently been taken from us, but who both entered long ago 

 into immortality. I speak of M. Bertrand and M. Hermite. They 

 were scholars of the same school at the same time; they had the same 

 education, were under the same influences ; and yet what a difference ! 

 Not only does it blaze forth in their writings; it is in their teaching, 

 in their way of speaking, in their very look. In the memory of all 

 their pupils these two faces are stamped in deathless lines ; for all who 

 have had the pleasure of following their teaching, this remembrance is 

 still fresh; it is easy for us to evoke it. 



While speaking, M. Bertrand is always in motion; now he seems 

 in combat with some outside enemy, now he outlines with a gesture of 

 the hand the figures he studies. Plainly he sees and he is eager to 

 paint, this is why he calls gesture to his aid. With M. Hermite, it is 

 just the opposite; his eyes seem to shun contact with the world; it is 

 not without, it is within he seeks the vision of truth. 



Among the German geometers of this century, two names above 

 all are illustrious, those of the two scientists who have founded the 

 general theory of functions, Weierstrass and Eiemann. Weierstrass 

 leads everything back to the consideration of series and their analytic 

 transformations; to express it better, he reduces analysis to a sort of 

 prolongation of arithmetic; you may turn through all his books with- 

 out finding a figure. Eiemann, on the contrary, at once calls geometry 

 to his aid; each of his conceptions is an image that no one can forget, 

 once he has caught its meaning. 



More recently, Lie was an intuitionalist ; this might have been 

 doubted in reading his books, no one could doubt it after talking with 

 him; you saw at once that he thought in pictures. Madame Kova- 

 levski was a logician. 



Among our students we notice the same differences ; some prefer to 

 treat their problems ' by analysis,' others ' by geometry.' The first are 

 incapable of ' seeing in space,' the others are quickly tired of long 

 calculations and become perplexed. 



The two sorts of minds are equally necessary for the progress of 

 science; both the logicians and the intuitionalists have achieved great 

 things that others could not have done. Who would venture to say 

 whether he preferred that Weierstrass had never written or that there 

 had never been a Eiemann? Analysis and synthesis have then both 



