Septembee 16, 1904.] 



SGIENCE. 



363 



ture is constant. Thus we easily come 

 back again to the sphere. 



"After this analysis, all commentary is 

 useless. One sees at how many different 

 points of view Hilbert has placed himself, 

 how profound is his analysis. 



' ' His works mark an epoch and he seems 

 entirely worthy of the Lobachevski prize. 

 — Poincare. " 



The 'Report on the works of Monsieur 

 Barbarin, professor of higher mathematics 

 at the Lyceum of Bordeaux, relative to the 

 non-Euclidean geometry,' is by Professor 

 Mansion, of Ghent, as follows: 



"I. List of the Works of M. Barbann. 

 — M. Barbarin has published, from 1898 

 to 1902, the following memoirs and works 

 relative to the non-Euclidean geometry. 



"1. Geometric generale des espaces {As- 

 sociation frangaise pour I'avancement des 

 sciences. Congres de Nantes, 1898, pp. 

 111-132). 



"2. Proprietes angulaires des cercles 

 focaux dans les coniques {Ibid., 1898, pp. 

 132-139). 



"3. Constructions spheriques a la regie 

 et au eompas {Mathesis, 1899, pp. 57-60; 

 81-85). 



' ' 4. Etudes de geometrie analytique non- 

 euclidienne {Memoires courounnes et autres 

 Memoires publics par I'Academie royale 

 de Belgique, 1900, t. LX., 167 pp. in 8°. 

 This memoir was presented to the Royal 

 Academy of Belgium, the fourth of Decem- 

 ber, 1897). 



"5. Le cinquieme livre de la Meta- 

 geometrie {Mathesis, 1901, pp. 177-191). 



"6. Les cosegments et les volumes en 

 geometrie non euclidienne (Extrait des 

 memoires de Bordeaux, 1901; 20 pp. in 8°). 



"7. La geometrie non euclidienne. Paris, 

 Naud, February, 1902 (collection scientia, 

 79 pp. in 12°). 



"8. Bilateres et trilateres en Metageom- 

 etrie (Mathesis, 1902, pp. 187-193). 



"9. Polygones reguliers spheriques et 

 non euclidiens {Le MMematiche pure ed 

 applicate, 1902, t. IL, pp. 137-145). 



We will now analyse these works, class- 

 ing the results found by M. Barbarin un- 

 der three heads : 



"Elementary geometry, conies and quad- 

 rics, infinitesimal geometry. 



"II. Elementary geometry.— In his little 

 book entitled ' The non-Euclidean geometry ' 

 (List No. 7), M. Barbarin expounds the 

 first principles of the geometry, especially 

 after Saecheri, Bolyai and Lobachevski 

 and, among the moderns, DeTilly, Gerard, 

 Mansion. But, besides, he makes known, 

 whether in this little book or in divers 

 special notes, results which are his own. 



"1. Bilaterals and trilaterals (List No. 

 8).— The author proves in an elementary 

 manner, witliout recourse to analysis, that 

 the locus of points equidistant from two 

 straights is a straight; that the bisectors, 

 the medians and the altitudes of a trilateral 

 are copunetal (meet in the same point) 

 [real at a finite or infinite distance, or 

 ideal] , even if the vertices of the trilateral 

 are all or in part reals at infinity or ideals. 



"He deduces from the theorem on the 

 three altitudes a novel construction of the 

 normal common to two Lobaehevskian 

 straights which only meet at an ideal point. 



"2. Fundamental constructions (List 

 No. 3, No. 4, § I., pp. 5-14; No. 7, pp. 46- 

 49).— M. Barbarin gives the means of con- 

 structing, with the ruler and the compasses, 

 a right-angled triangle or a trirectangular 

 quadrilateral given two elements and 

 thence deduces all the fundamental con- 

 structions of the non-Euclidean geometry. 

 "He depends, in these constructions, on 

 the old theorems of Lobachevski and 

 Bolyai and on three new theorems which 

 seem to have escaped these illustrious geom- 

 eters. Here they are, in Lobaehevskian 

 geometry : 



"Let ABCD be a quadrilateral trirect- 



