366 



SCIENCE. 



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



tional to the difference between the pro- 

 jection of the generatrix on the axis and 

 this generatrix multiplied by the cosine of 

 the angle which it makes with the axis if it 

 meets it, or of the normal common to the 

 generatrix and axis. He has thence de- 

 duced a general formula for volumes de- 

 composed into infinitesimal spindles of 

 revolution. 



' ' By means of this formula he has been 

 able to reach a number of known results; 

 especially he has been able to make an 

 advance in the very difficult question of 

 the volume of the tetrahedron, which, as 

 is known, has arrested Gauss, Bolyai and 

 Lobaehevski and all their successors. 



"M. Barbarin finds an expression for 

 the volume of the tetrahedron where are 

 introduced naturally the products of the 

 edges by the corresponding dihedrals. To 

 achieve the solution of the question, it is 

 requisite to find, under a finite form, cer- 

 tain functions relative to the faces which 

 present themselves in the calculations under 

 the form of integrals or complicated series. 



"2. Geodesic lines of tubes and pseudo- 

 spheres (List, No. 4, pp. 139-164).— In the 

 last section of his ' Studies in non-Euclidean 

 Analytic Geometry' the author has reached 

 one of the most beautiful theorems of meta- 

 geometry. 



"It has been known, since Lobaehevski 

 and Bolyai, that the characteristic geom- 

 etry of orispheres is Euclidean; since Bel- 

 trami, that of the Euclidean pseudosphere 

 is Lobachevskian ; finally it is evident that 

 that of the sphere is Riemannean. 



"The theorem of Barbarin (it is to be 

 hoped that it will retain this name) com- 

 prises and generalizes in the most unex- 

 pected manner these particular proposi- 

 tions. Here it is in its most condensed 

 form : Each of the three spaces, Euclidean, 

 Lobachevskian, Riemannean, contains sur- 

 faces of constant cxirvalure of which the 



geodesic lines have the metric properties of 

 the straights of the three spaces. 



"These surfaces are the spheres (char- 

 acteristic geometry, Riemannean) ; the 

 tubes or surfaces equidistant from a 

 straight, it being possible for the distance 

 to be infinite, which gives the orispheres 

 (characteristic geometry, Euclidean) ; 

 finally the pseudospheres, that is to say the 

 surfaces of revolution which have for 

 meridians a tractrix or line of equal tan- 

 gents (characteristic geometry, Lobachev- 

 skian ) . 



"The property of the surfaces equidis- 

 tant from a straight is almost evident and 

 has been found also by Whitehead; but 

 the existence of Lobachevskian tractriees 

 and pseudospheres and above all of Rie- 

 mannean and the properties of their 

 geodesies were not suspected before M. 

 Barbarin. 



"The curvature of the pseudospheres is 

 negative in Riemannean space as in Euclid- 

 ean space ; it is negative, nul or positive in 

 Lobachevskian space. 



"V. Resume and Conclusion. — Non- 

 Euclidean geometry owes to M. Bai-barin 

 (1) fundamental properties of the plane 

 trirectangular quadrilateral; (2) the 

 discovery of Riemannean equidistant 

 straights; (3) the complete classification 

 of non-Euclidean conies and quadrics; (4) 

 the most intuitive formula that we know 

 for the determination of volumes, with a 

 remarkable application to the tetrahedron; 

 (5) finally and above all the beautiful 

 general theorem cited above on the geodesies 

 of tubes and pseudospheres, in the three 

 geometries. 



' ' All these resiilts have been obtained by 

 the direct study of the fignires without bor- 

 rowing anything from the Cayleyan geom- 

 etry. 



' ' If Lobaehevski should come back to the 

 world, he would recos'nize in M. Barbarin 



