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SCIENCE. 



[X. S. Vol. XX. No. 507. 



angular in B, C, D, and having, conse- 

 quently, angle DAB acute ; E — a point 

 situated between D and A, F — a point 

 situated between B and A, such that 

 CE = BA,CF = DA. 



"We have (1) BF^DE. (2) CE is 

 asymptote (Lobaehevskian parallel) to BA, 

 CF to DA. (3) The perpendiculars let 

 fall on CB, CD from points equidistant 

 from C taken on CE, CF intersect on the 

 diagonal CA. 



"3. Regular spheric and non-Euclidean 

 polygons (List No. 9.— M. Barbarin has 

 found constructions simple and novel for 

 the regular polygons of 3, 6, 5, 10, 15 

 sides, applicable at the same time in 

 Euclidean geometry and in non-Euclidean 

 geometry of Riemann and of Lobachevski, 

 for the sphere and for the plane. 



"4. The fifth hook of Metageometry 

 (List No. 4, ch. IV., pp. 94-99, No. 5, No. 

 7, pp. 37-41).— M. Barbarin calls fifth 

 book of 'Metageometry' that which corre- 

 sponds to the fifth book of the 'Elements 

 of Legendre' or to the eleventh of Euclid. 



"He makes an elementary exposition of 

 it more complete than does any of his pre- 

 decessors ; here and there it could have been 

 intuitive if he had depended more on the 

 asymptotic property of Lobaehevskian 

 parallels. There is room to cite in this 

 work the two following theorems: (1) That 

 a right angle may be projected upon a 

 plane into a right angle, it is necessary and 

 sufficient that the projector of the vertex 

 be nomial to the plane and to one of the 

 sides of the angle. 



"(2) Two Riemannean straights not 

 situated in the same plane have two com- 

 mon normals; if these normals are equal, 

 the two straights are equidistant. 



"Descriptive non-Euclidean geometry 

 rests on the first proposition. 



"From the second, it results that there 

 exist, in Riemannean geometry, skew 

 squares and rectangles having four right 



angles and surfaces equidistant from a 

 straight with rectilinear generators. 



"5. Coordinates; geometry of n dimen- 

 sions (List No. 1, No. 4, § II., pp. 14-28, 

 § IV., pp. 84-101).— In his 'Studies in non- 

 Euclidean Analytic Geometry,' M. Bar- 

 barin has been led to certain new develop- 

 ments of the theory of coordinates, and, 

 consequently, to expound by the calculus, 

 the fundamental properties of the straight 

 and of the plane, of angles and of distances, 

 of the circle and of the sphere. 



"This is, in the main, under analytic 

 form, the complement of his other studies 

 on the 'Elements.' 



"The memoir of pure analysis, entitled 

 ' General Geometry of Spaces ' is a general- 

 ization of the formulas of Euclidean or 

 non-Euclidean geometry of three dimen- 

 sions relatively to the straight, to the plane, 

 to the triangle and to the trihedrals and 

 the trigonometric relations relative to them, 

 when one considers a variety of n dimen- 

 sions. The author shows, in particular, 

 that for such varieties there exists also a 

 limit-case that we may call Euclidean 

 geometry of n dimensions. 



"III. Conies and quadrics (List No. 2, 

 No. 4 ; the essential part of 2 is reproduced 

 in 4). The 'Studies in non-Euclidean 

 Analytic Geometry' constitute M. Barbar- 

 in 's largest work. It is devoted, for the 

 major part (§111., pp. 29-84; § V., pp. 

 101-139), to a classification of conies and 

 quadrics more complete than that of his 

 predecessors, without having any recourse 

 to the Cayleyan geometry. 



"A. (7omcs.— The author first reduces to 

 its most simple forms the general equation 

 of the second degree or rather a ternary 

 quadratic form. 



"In Riemannean geometry he finds only 

 two kinds of curves: imaginary conic and 

 ellipse, the latter having the circle as 

 variety. 



"The Lobaehevskian geometry is much 



