708 



SCIENCE. 



[N. S. Vol. XIV. No. 358- 



IV. ' General geometry in the plane and 

 in space.' Eesume of the principal general 

 propositions. 



V. ' Trigonometry. ' Elementary demon- 

 stration, after Gerard and Mansion, of the 

 formulas for triangles and quadrilaterals. 



VI. ' Measurement of areas and volumes.' 



VII. ' The contradictors of the non- 

 Euclidean geometry.' The principal ob- 

 jections made against the non-Euclidean 

 geometry. Answers to be made thereto. 



VIII. * Physical geometry. ' How we 

 might attempt to find out if the physical 

 world is not Euclidean ; how angles and 

 distances could be measured with a much 

 greater approximation, for example, angles 

 with an error much less than y^^ of a 

 second. 



A brief article by Professor Barbarin, 

 ' On the utility of studying non-Euclidean 

 geometry,' which appears in the May (1901) 

 number of Professor Cristoforo Alasia's new 

 Italian journal Le Matematiche, shows that 

 Hoiiel had reached the weighty insight 

 which we have quoted from Study, namely, 

 that knowledge of non-Euclidean geometry 

 is essential for any mastery of Euclidean 

 geometry. 



Says Barbarin : 



" I. The question of the source of the 

 theory of parallels has been one of the 

 most interesting scientific preoccupations of 

 the century ; it has made to flow torrents 

 of books, and given theme to remarkable 

 works. Thanks to the theorems of 

 Legendre, to the discoveries of the two 

 Bolyai, of Lobachevski and of Eiemann ; 

 thanks to the original researches of Bel- 

 trami and of Sophus Lie, of Poincar^, 

 Flye Ste. Marie, Klein, De Tilly, etc. , we 

 cannot any more be mistaken as to the true 

 scope of the celebrated proposition which 

 bears the name of Postulate of Euclid. 



"1°. This is not in any way contained 

 in the classic definitions of the straight and 

 the plane. 



"2°. This is, among three hypotheses 

 equally admissible, and which cannot all 

 be rejected, only the most simple. 



" Is it perhaps chance alone which gave 

 to the great Greek geometer the choice 

 of his system of geometry ? or did he per- 

 ceive, at least in part, the difficulties and 

 the greater theoretic complication of the 

 other two? We shall never know with 

 certainty. 



"But in the presence of his work, so 

 perfect and so rigorous, one thing, how- 

 ever, appears not to be doubtful : the place 

 which he assigned to his proposition, the 

 enunciation which he gave of it, attest 

 that this proposition had to his eyes only 

 the value of an hypothesis ; otherwise he 

 would have formulated it in other terms 

 and would have attempted to demonstrate 

 it. 



" The ideas of Lobachevski and of Eie- 

 mann were diffused only very slowly. They 

 were so, above all, thanks to the transla- 

 tions of Hoiiel. 



" This scientist, whose activity and power 

 of work were prodigious, could not resist 

 the desire to master all the European lan- 

 guages, with the aim of being able to read 

 in their original text, and then make known 

 to his contemporaries the most celebrated 

 mathematical works. 



" He admired Lobachevski, whom hesur- 

 named the modern Euclid, and in his course 

 professed at the scientific faculty of Bor- 

 deaux, he did not let pass any occasion to 

 put him in evidence. 



' ' II. Hoiiel was persuaded that the 

 knowledge of the non-Euclidean geometry 

 is indispensable for thoroughly mastering 

 the mechanism of the Euclidean geometry. 



" Despite its paradoxical form, this idea 

 is most just. 



" General geometry or metageometry con- 

 tains in fact a great number of propositions 

 common to all the systems, and which ought 

 to be enunciated in the same terms in each 



