September 16, 1904.] 



SCIENCE. 



361 



ZaJilenmannigfaltigkeit in the sense of Lie. 

 This is, nevertheless, a step in advance, and 

 besides Hilbert analyzes better than any 

 one had done it before him the idea of 

 Zahlenmannigfaltigkeit, and gives hints 

 which may become the germ of an axiom- 

 atic theory of analysis situs. 



' ' I can here only summarize the general 

 march of Hilbert 's ideas. 



"He shows first that the points of a circle 

 can be arranged in a determined circular 

 order and that this order is not altered by 

 rotations; then he shows that this order 

 falls into the same type of order as the cor- 

 responding order of the ordinary circle, 

 that is to say, into the type of the con- 

 tinuous. Thence he deduces this conse- 

 quence that the circle is a continuous closed 

 curve, because it must correspond point for 

 point to the ordinary circle. 



"One sees then that if a rotation does 

 not displace one point of a circle, it will 

 not displace any other point of this circle. 

 Thence one can deduce that if a rotation 

 does not displace one point different from 

 its center, it will not displace any of the 

 points of the plane and will reduce to 

 identity. From this results finally that 

 the group of rotations around a point M 

 has the same structure as the group of 

 ordinary rotations. 



' ' One sees at the same time that there is 

 no movement which leaves fixed two points 

 of the plane, and that we can pass by rota- 

 tions from one point of the plane to any 

 other point whatsoever of the plane. 



"All these demonstrations are extremely 

 delicate ; they require the repeated employ- 

 ment of the theorems of Cantor. 



"This is to say that they are perforce 

 very long and that the goal which one per- 

 ceives immediately and which one thinks 

 to touch can be attained only by long 

 efforts. 



' ' The greatest step is then accomplished ; 

 now we know that our group derives from 



certain subgroups, those of rotations, of 

 which we know the structure, and this 

 structure makes these subgroups fall into 

 the category of Lie's continuous groups. 



"We have, therefore, only a few diffi- 

 culties still to vanquish, but Hilbert wishes 

 first to define the straight and he has done 

 it in a very original fashion. 



"He rejects first the projective defini- 

 tions of the straight which require consid- 

 erations foreign to his premises. On the 

 other hand, his geometry is a plane geom- 

 etry. 



"If we may use space of three dimen- 

 sions, the theory of groups leads us nat- 

 urally to a very simple definition of the 

 straight, considered as axis of rotation; 

 but here we can not use this, since we can 

 not go out of the plane. 



"Hilbert follows wholly another way. 

 Let there be two points, A and B; define 

 the middle of these points, that is to say 

 the center of a rotation which changes A 

 into B and B into A. Hilbert begins by 

 demonstrating that two points have always 

 a middle and have only one. It is here 

 that comes in an hypothesis which above 

 must have astonished the reader; we have 

 supposed that the last axiom (which one 

 states in an abridged fashion in saying 

 that the group of movements is a closed 

 system) is applicable not only if one en- 

 visages two points Af, and iJ„, but also if 

 we consider three points. We have, there- 

 fore, made an hypothesis more restrictive 

 than if we had limited ourselves to the 

 consideration of two points A^, and B^. 

 Was this restriction really necessary? 



"It is in this part of the theory that it 

 plays its role. We consider an infinity of 

 points jBj, B,, • • • B„, and the middles 

 M^, M^, •■• M„, ■ • • of the sects AB^, AB^, 

 • • • AB„ ; when n increases indefinitely, B 

 tends toward B and If „ toward M and we 

 make use of the hypothesis in question to 

 show that M is the middle of AB. Had 



