Septembee 16, 1904.] 



SCIENCE. 



35a 



of Archimedes. In addition the new 

 Axiom der Nachbarschaft is used.] 



"I come to speak now of an article en- 

 titled 'Ueber die gerade Linie als kiirzeste 

 Verbindung zweier Punkte' that I can not 

 separate from a thesis on the same subject, 

 written by Hamel under the inspiration 

 of Hilbert. Here we are less far from 

 home; not only is there no question of re- 

 nouncing the axiom of Archimedes, but we 

 encounter only analytic functions which 

 may be differentiated and integrated. 



"Suppose that one has defined straights 

 in the ordinary fashion and that one admits 

 the projective axioms, those of order and 

 the theorems of Desargues and Pascal. De- 

 fine now the length of an arc of any curve ; 

 it is not necessary to choose this definition 

 so as to satisfy the metric axioms, that is 

 to say so as to render possible the move- 

 ment of a rigid figure. 



"Is it possible so to make this choice that 

 the straight line shall be the shortest path 

 from one point to an other? 



"The definition of the straight is not 

 changed, but that of the circle is in a very 

 large degree arbitrary ; it is only necessary 

 that all the circles which have their center 

 on a straight and which pass through a 

 point of this straight have at this point the 

 same tangent. The problem permits an 

 infinity of solutions. 



"Minkowski, for an arithmetic purpose, 

 has developed one of them where all the 

 circles are curves similar to each other in 

 the ordinary sense of the word. Hilbert, 

 from 1894, had given another of them 

 which may be thus characterized: We con- 

 sider a connected closed curve which will 

 serve as fundamental curve. Let Z> be a 

 straight, M a point of this straight ; all the 

 circles which have their center on D and 

 which pass through M have the same 

 tangent T, and this tangent, when the point 

 M describes the straight i>, pivots around 

 a fixed point which is the intersection of 



two tangents to C at the points where this, 

 curve is cut by the straight D. 



"Finally Hamel has in his thesis given 

 the general solution of the question, but 

 this solution is too complicated to be ex- 

 pounded in few words. 



"I arrive at an important memoir of 

 Hubert's which is entitled 'Grundlagen 

 der Geometric,' which bears, consequently, 

 the same title as his 'Festschrift,' but 

 where, however, he places himself at a point 

 of view altogether different. 



"In his 'Festschrift,' in fact, as one sees 

 by the preceding analysis, the relations be- 

 tween the notion of space and the notion 

 of group, as they result from the works of 

 Lie, are left to one side or relegated to a 

 secondary part. The general properties 

 of groups do not appear in the list of his 

 fundamental axioms. 



"This is not so in the memoir of which 

 we are about to speak. Hilbert assumes 

 a plane about which he makes the following 

 hypotheses : 



"1. The points of this plane correspond 

 one to one to the points of the ordinary 

 plane or to a part of these points. Thus 

 each point of the new plane has its corre- 

 spondent in the ordinary plane ; but there 

 may be on the ordinary plane points which 

 have no correspondent on the new plane. 



' ' The new plane has, therefore, so to say, 

 less points than the ordinary plane, which 

 is the contrary of that which happened for 

 the non- Archimedean plane. The points of 

 the ordinary plane which have correspond- 

 ents on the new plane are called Bild- 

 punkte. The ensemble of Bildpunkte 

 forms on the ordinary plane a region which 

 Hilbert assumes continuous and connected 

 in such fashion that around each point of 

 this region one can describe a circle of 

 radius sufficiently small to be contained in 

 this region and that one can go from one 

 point to the other of the region, in follow- 



