SPACE 649 



lines of which remain of a constant length. Such a movement would 

 appear absurd, on the other hand, to animals without thickness living 

 on a surface of variable curvature. These surfaces of constant curva- 

 ture are of two kinds. The curvature of some is positive, and they 

 may be deformed so as to be applied to a sphere. The geometry of 

 these surfaces is therefore reduced to spherical geometry namely, 

 Riemann's. The curvature of others is negative. Beltrami has shown 

 that the geometry of these surfaces is identical with that of Lobat- 

 schewsky. Thus the two-dimensional geometries of Riemann and 

 Lobatschewsky are connected with Euclidean geometry. 



Interpretation of Non-Euclidean Geometries. Thus vanishes the 

 objection so far as two-dimensional geometries arc concerned. It 

 would be easy to extend Beltrami's reasoning to three dimensional 

 geometries, and minds which do not recoil before space of four dimen- 

 sions will see no difficulty in it; but such minds are few in number. 

 I prefer, then, to proceed otherwise. Let us consider a certain plane, 

 which I shall call the fundamental plane, and let us construct a kind 

 of dictionary by making a double series of terms written in two col- 

 umns, and corresponding each to each, just as in ordinary diction- 

 aries the Avords in two languages which have the same signification 

 correspond to one another : 



Space The portion of space situated above the 



fundamental plane. 



Plane Sphere cutting orthogonally the funda- 

 mental plane. 



Line Circle cutting orthogonally the funda- 

 mental plane. 



Sphere Sphere. 



Circle Circle. 



Angle Angle. 



Distance between 



two points Logarithm of the anharmonic ratio of these 



two points and of the intersection of the 

 fundamental plane with the circle pass- 

 ing through these two points and cutting 

 it orthogonally. 

 Etc. Etc. 



Let us now take Lobatschewsky's theorems and translate them by 

 the aid of this dictionary, as we would translate a German text with 

 the aid of a German-French dictionary. We shall then obtain the the- 

 orems of ordinary geometry. For instance, Lobatschewsky's theo- 

 rem : " The sum of the angles of a triangle is less than two right 

 angles," may be translated thus : " If a curvilinear triangle has for 

 its sides arcs of circles which if produced would cut orthogonally the 

 fundamental plane, the sum of the angles of this curvilinear triangle 



