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SCIENCE 



[N. S Vol. XLII. No. 1089 



the level of conunon thought; and that the 

 superstructure, quickly transcending the 

 power of imagination to follow it, ascends 

 higher and higher, ever keeping open to 

 the sky; he knows that the manifold 

 chambers — each of them a mansion in 

 itself^are all of them connected in won- 

 drous ways, together constituting a fit lab- 

 oratory and dwelling for the spirit of men 

 of genius. He has assumed the task of 

 presenting a vision of it that shall be 

 worthy of a world-exposition. Can he 

 keep the obligation? He wishes to show 

 that the life and work of pure mathema- 

 ticians are human life and work: he de- 

 sires to show that these toilers and dwell- 

 ers in the chambers of pure thought are 

 representative men. He would exhibit the 

 many-chambered house to the thronging 

 multitudes of his fellow men and women; 

 he would lead them into it; he would con- 

 duct them from chamber to chamber by 

 the curiously winding corridors, passing 

 now downward, now upward, by delicate 

 passage-ways and subtle stairs; he would 

 show them that the wondrous castle is not 

 a dead or static aiSair like a structure of 

 marble or steel, but a living architecture, a 

 living mansion of life, human as their own ; 

 he would show them the mathetic spirit at 

 work, how it is ever weaving, tirelessly 

 weaving, fabrics of beauty, finer than 

 gossamer yet stronger than cables of steel ; 

 he would show them how it is ever enlarg- 

 ing its habitation, deepening its founda- 

 tions, expanding more and more and ele- 

 vating the superstructure; and, what is 

 even more amazing, how it perpetually per- 

 forms the curious miracle of permanence 

 combined with change, transforming, that 

 is, the older portions of the edifice without 

 destroying it, for the structure is eternal: 

 in a word, he would show them a vision of 

 the whole, and he would do it in a way to 

 make them perceive and feel that, in thus 



beholding there a partial and progressive 

 attainment of the higher ideals of man, 

 they were but gazing upon a partial and 

 progressive realization of their own appe- 

 titions and dreams. 



That is what he would do. But how? 

 Mengenlehre, Zahlenlehre, algebras of 

 many kinds, countless geometries of count- 

 less infinite spaces, function theories, 

 transformations, invariants, groups and 

 the rest — how can these with all their 

 structural finesse, with their heights and 

 depths and limitless ramifications, with 

 their labyrinthine and interlocking modern 

 developments — I will not say how can they 

 be presented in the measure and scale of a 

 great exposition — but how is it possible in 

 one hour to give laymen even a glimpse 

 of the endless array? Nothing could be 

 more extravagant or more absurd than such 

 an undertaking. Compared with it, the 

 American traveler's hope of being able to 

 see Rome in a single forenoon was a most 

 reasonable expectation. But it is worth 

 while trying to realize how stupendous the 

 absurdity is. 



It is evident that our would-be delineator 

 must compromise. He can not expound, 

 he can not exhibit, he can not even deline- 

 ate the doctrines whose human worth he 

 would thus disclose to his fellow men and 

 women. The fault is neither his nor theirs. 

 It must be imputed to the nature of things. 

 But he need not, therefore, despair and he 

 need not surrender. The method he has 

 proposed — the method of exposition — that 

 indeed he must abandon as hopeless, but 

 not his aim. He is addressing men and 

 women who are no doubt without his spe- 

 cial knowledge and his special discipline, as 

 he in his turn is without theirs, but who are 

 yet essentially like himself. He would have 

 them as fellows and comrades persuaded 

 of the dignity of his Fach: he would have 

 them feel that it is also theirs; he would 



