September 16, 1904.] 



SCIENCE. 



367 



play in these theorems? This question in- 

 terested Hilbert and under his inspiration 

 M. Dehn has made it the subject of a thesis 

 which I can not pass over in silence. The 

 conclusions of Dehn show that without the 

 axiom of Archimedes the theorems of 

 Legendre are no longer true. 



It is still true that if one triangle has 

 the sum of its angles equal to (or greater 

 than) (or less than) two right angles the 

 same is true of all the others. It is still 

 true that if this sum is less than two right 

 angles, one can draw many parallels to a 

 straight through one point. It is true 

 that if it is greater than two right angles, 

 the postulatum of Euclid is false, and that 

 if it is equal to two right angles it is im- 

 possible that two straights always meet, 

 but the other theorems of Legendre are not 

 true. 



There exists a plane geometry where, the 

 sum of the angles of a triangle being 

 greater than two right angles, one can draw 

 to a straight through one point an infinity 

 of parallels (so I call straights which do 

 not meet) ; this is the non-Legendrean 

 geometry. 



A geometry exists where the sum of the 

 angles is equal to two right angles, and 

 where one can draw to a straight through 

 one point an infinity of parallels. This 

 is the semi-Euclidean geometry. 



It will suffice to explain here what this 

 latter is, the former being altogether 

 analogous. For this it is necessary to re- 

 turn to what I have said of the non- 

 Archimedean geometry. I have explained 

 how the non- Archimedean plane is deduced 

 from the ordinary plane by the adjunction 

 of new points ; how for deducing a non- 

 Archimedean straight D^ from the ordi- 

 nary straight D,,, it is necessary to add 

 to it: 



1. On the one hand, an infinity of new 

 points between every two demi-straights 

 8' and S" of which the totality forms D^. 



2. On the other hand, an infinity of new 

 points to the right of all the ordinary 

 points of Dg, and an infinity of new points 

 to the left of all the ordinary points of D^. 

 Well, retain the new points of the first sort, 

 that is to say, those which are at a finite 

 distance, and suppress the new points of 

 the second sort, that is to say, those which 

 are at an infinite distance. 



Then let D be any straight and A any 

 point ; then there will be an infinity of 

 straights passing through A and which do 

 not meet D, those, namely, which would 

 have met it in one of the new points of the 

 second sort, if these points had not been 

 suppressed. However, all the theorems of 

 Euclid remain and every rotation or every 

 translation -^ill transform into itself the 

 non- Archimedean plane so mutilated. 



It seems that here is a contradiction with 

 the results of the article just cited: 'Ueber 

 eine neue Begriindung. * * * ' 



If, as Hilbert has shown, the geometry 

 of Lobachevski can be deduced from his 

 postulate without the intervention of the 

 axiom of Archimedes, how can there be a 

 geometry semi-Euclidean, that is to say a 

 geometry where the theorems of Euclid 

 accord with the postulate of Lobachevski? 



It seems that this difficulty springs from 

 this, that the enunciation of the postulate 

 is not the same in the two cases. 



Dehn assumes that through a point one 

 can draw an infinity of straights which do 

 not meet a given straight, and an infinity 

 of straights which meet it. 



The first form an ensemble E^, the others 

 form' an ensemble Eo. Hilbert supposes, 

 in addition, that there exists a limiting 

 straight which appertains to the ensemble 

 E^, and such that every straight comprised 

 betAveen this limit straight and a straight 

 of E^ appertains likewise to E^. It is this 

 limit straight which Hilbert considers as • 

 the parallel properly so called. 



In the geometry of Dehn this parallel 



