"356 



SCIENCE. 



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



all the axioms, is not susceptible of any ex- 

 tension." 



In speaking of the non-Archimedean 

 geometry, the 'Review' made no mention 

 of Veronese. The 'Report,' however, says: 

 "In this conception, so audacious, Hilbert 

 had had a precursor. In his foundations 

 of geometry Veronese had had an analogous 

 idea. Chapter VI. of his introduction is 

 the development of a veritable arithmetic 

 and of a veritable geometry non- Archimed- 

 ean where the transfinite numbers of Can- 

 tor play a preponderant role. Neverthe- 

 less, by the elegance and the simplicity of 

 his exposition, by the depth of his philo- 

 sophic views, by the advantage he has ob- 

 tained from the fundamental idea, Hilbert 

 has made the new geometry his own. ' ' 



The 'Report' is in 39 pages. The in- 

 corporation of the already published 

 'Review' stops with page 25. The last 

 fourteen pages are entirely new, as fol- 

 lows: The memoir we have just analyzed 

 puts in evidence the importance of the new 

 non-Archimedean geometry. It discusses 

 the role of the axiom of Archimedes in 

 geometric reasoning; and the principal 

 result of this discussion may be summed up 

 thiis : If we abandon this axiom and retain 

 only the axioms of the first four groups, 

 the essential results of Euclidean geometry 

 are not altered; but this is not so if one 

 retains only the projective axioms [assump- 

 tions of association] and those of order 

 [betweenness] , together with the postu- 

 latum of Euclid, but abandons at the same 

 time the axiom of Archimedes and the 

 metric axioms [assumptions of congru- 

 ence] ; we come then to the non-Pascalian 

 geometry. 



Then comes the question, does this that 

 we have just said of the Euclidean geom- 

 etry remain true of the Lobaehevskian 1 



In other words, if we preserve only the 

 axioms of the first three groups (projective, 

 of order and metric) and replace the postu- 



latum of Euclid by that of Lobachevski, 

 shall we arrive at the fundamental theorems 

 of Lobachevski without using the axiom of 

 Archimedes? 



This is the question that Hilbert has 

 settled in his article 'Ueber eine neue 

 Begriindung der Bolyai-Lobatschefsky- 

 schen Geometric.' 



He answers it affirmatively and shows in 

 particular that there always exists a com- 

 mon perpendicular to two straights of the 

 plane which do not meet without being 

 parallel. 



I would call attention to the statement 

 of the postulate of Lobachevski: "If & is 

 any straight of the plane and A a point 

 not situated on this straight, there pass 

 always through A two demi-straights 

 [rays], a^ and a,, which are not in the pro- 

 longation one of the other and which do 

 not cut the straight h, while every semi- 

 straight passing through A and situated in 

 the angle formed by a^ and o, meets 6." 



"It is these two demi-straights, a^ and O2, 

 which have received the name of parallels." 

 They do not meet the straight h, but they 

 serve as limit to the angle wherein are 

 found the straights which meet h and to the 

 angle wherein are found the straights 

 which do not meet h. 



I would signalize an elegant theory of 

 what might be called the points at infinity 

 of the Lobaehevskian plane, and of which 

 the laws are the same as those of the addi- 

 tion and the multiplication of real num- 

 bers. One may draw thence a very simple 

 and very suggestive exposition of the non- 

 Euclidean geometry. 



The origin of our acquaintance with the 

 theory of parallels is found in the theorems 

 of Legendre which establish a necessary 

 correlation between the sum of the angles 

 of a triangle and the choice between the 

 three geometries, Euclidean, Lobaehevskian, 

 Riemannean. 



What role does the axiom of Archimedes 



