362 



SCIENCE. 



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



it been impossible to use it, we could have 

 been sure of this only after having con- 

 structed a special pseudogeometry. 



"As it is, two points A and B being 

 given, Hilbert constructs the middle of the 

 sect AB, then the middle of the two sects 

 MA and MB and so on. He thus obtains 

 an infinity of points which form an en- 

 semble E ; he considers the derivative of 

 this ensemble E, that is to say the assemb- 

 lage of the limiting-points of E, points 

 such that in any interval containing one 

 there are an infinity of points of E. He 

 shows that this derivative is a continuous 

 line, and it is this line that he calls the 

 straight [die wahre Gerade] . 



' ' The fundamental principles of the ordi- 

 nary Euclidean or non-Euclidean geometry 

 may then be easily established and in par- 

 ticular the metric axioms. 



"It is impossible not to be struck by the 

 contrast between the point of view where 

 Hilbert places himself here and that which 

 he had adopted in his 'Festschrift.' In 

 this 'Festschrift' the axioms of continuity 

 occupy the last rank and the grand affair 

 was to know what geometry became when 

 one threw them aside. Here on the con- 

 trary it is continuity which is the point of 

 departure and Hilbert is especially pre- 

 occupied to' see what one gets from con- 

 tinuity alone, joined to the notion of the 

 group. 



"It remains for us to speak of a memoir 

 entitled 'Flachen von konstanter Kriim- 

 mung. ' 



"It is known that Beltrami has shown 

 that there are in ordinary space surfaces 

 which are the image of the Lobachevskian 

 plane, namely the surfaces of constant neg- 

 ative curvature ; we know what an impulse 

 this discovery gave to the non-Euclidean 

 geometry. But is it possible to represent 

 the whole entire Lobachevskian plane on 

 a surface of Beltrami without singular 

 point? 



"Hilbert demonstrates that it is not; he 

 founds his proof on the following theorems 

 relative to the Beltrami surfaces. 



"A quadrilateral formed of asymptotic 

 lines has its opposite sides equal. 



"The surface of a polygon formed of 

 asymptotic lines is proportional to the 

 spherical excess and it is the same with the 

 surface of a polygon formed of geodesic 

 lines; only in the first ease the spherical 

 excess is positive, in the second case it is 

 negative. 



"The author shows then that on a Bel- 

 trami surface without singular point one 

 can not trace a closed asymptotic line ; that 

 an asymptotic line can neither cut itself, 

 nor cut another asymptotic line in more 

 than one point. Every other hypothesis 

 would lead to polygons of which the spher- 

 ical excess would be nul. Thence it follows 

 as a consequence that if such a surface 

 corrfS[K)nds point for point to the noti- 

 Euclideau plane, this correspondence must 

 be biuniform. But then in evaluating the 

 total surface starting from the area of the 

 polygon formed of asymptotic lines or from 

 the area of the geodesic polygon we find in 

 the first case that this total surface is finite, 

 in the second that it is infinite. This con- 

 tradiction demonstrates the theorem enun- 

 ciated. 



"In that which concerns the surfaces of 

 positive constant curvature, to which the 

 geometry of Riemann corresponds, Hilbert 

 demonstrates that aside from the sphere 

 there is no other closed surface of this sort. 

 In fact, if we consider a portion of surface 

 of constant positive curvature, the maxi- 

 mum of the great radius of curvature can 

 not be attained m the interior of this por- 

 tion, but only on its contour. This is a 

 proposition entirely analogous to a well- 

 known theorem relative to the potential. 



"It follows thence immediately that if 

 the surface is closed, the maximum can be 

 nowhere attained and the radius of curva- 



