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SCIENCE. 



[N. S. Vol. XX. No. 507. 



matter farthez', but his. -spaces were all as- 

 sumed by him to be liuiriber-manifolds. 



A new progress remained to be accom- 

 plished, and the honor of this belongs to 

 Hilbert. It is important, however, to say 

 a word of the works which have prepared 

 and rendered possible this advance. 



Since the time of Lobachevski, mathe- 

 matical thought has undergone a profound 

 evolution, not alone in geometry, but in 

 arithmetic and analysis. The notion of 

 number has been made more clear and pre- 

 cise; at the same time it has received 

 diverse generalizations. The most precious 

 for the analyst is that which results from 

 the introduction of neomonics, which 

 modern mathematicians could not now dis- 

 pense with. 



Again, many Italian geometers, such as 

 Peano and Padoa, have created a pasi- 

 graphie, that is to say, a sort of universal 

 algebra where all reasonings ai-e replaced 

 by symbols or formulas. 



Finally must be cited the book of 

 Veronese on the foundations of geometry, 

 where the author applies for the first time 

 to geometry the transfinite njumbers of 

 Georg Cantor. 



In 1899 Hilbert published a memoir en- 

 titled 'Grundlagen der Geometric,' full of 

 ideas the most original. Moreover, this 

 was not the first time he had occupied him- 

 self with analogous questions, witness his 

 letter of 1894 to Felix Klein: 'Ueber die 

 gerade Linie als kiirzeste Verbindung 

 zweier Punkte.' He has since published 

 in divers journals a series of articles en- 

 titled: 'On the theorem of the equality of 

 the basal angles in. the isosceles triangle'; 

 'New founding of the Bolyai-Lobachevski 

 geometry'; 'On the foundations of geom- 

 etry'; 'On surfaces of constant Gaussian 

 curvature. ' 



All these articles have been united in a 

 second edition of his .jubilee memoir; and 

 I must add that this second edition contains 



a series of improvements and additions 

 which greatly augment its value. 



It is, therefore, this second edition that 

 we Avill follow in our analysis; but we will 

 join with it, on the one hand, other works 

 of Hilbert, siich as his article 'Ueber den 

 Zahlbegrilf' and his Paris address on the 

 mathematical problems of the future, and, 

 on the other hand, many theses written by 

 his scholars, under his direct inspiration, 

 and which consequently aid us in compre- 

 hending his thought. The principal are : 

 'Ueber die Geometrieen in denen die 

 Geraden die kiirzesten sind' by Georg 

 Hamel, and 'Die Legendre 'schen Satze 

 ueber die Winkelsumme im Dreieck' by M. 

 Dehn. 



"Hilbert commences," continues Poin- 

 care, "by establishing the complete list of 

 axioms, straining not to forget one. 



"Is his list final? It is permitted to 

 believe so, since it seems to have been drawn 

 up with care.'' 



So says Poincare for the fourth time. 

 But if Hilbert 's receiving the Lobachevski 

 prize depended on his list of axioms being 

 'definitive,' it could not be given to him. 

 A young pupil of my own, R. L. Moore, by 

 a charmingly simple proof has abolished 

 the ugliest of the list, and Hilbert has 

 already acknowledged the redundancy. 

 Another point Hilbert himself changed in 

 the French translation of his 'Festschrift' 

 by Laugel. Poincare had said: "The 

 axiom that the sect AB is congruent to the 

 inverse sect BA (which implies the sym- 

 metry of space) is not identical with those 

 which are explicitly stated. I do not know 

 whether it could be logically deduced 

 from them ; I believe it could. ' ' 



In his 'Report,' Poincare now says: 

 "An important point is not here treated; 

 the list of axioms should be completed by 

 saying that the sect AB is congruent to the 

 inverse sect BA. 



"This axiom implies the symmetry of 



