48 ORIGIN AND SIGNIFICANCE OP 



as a short expression for a complex relation, from the 

 one case in which the quantity designated admits, of 

 sensible representation. 



Now whenever the value of this measure of curva- 

 ture in any space is everywhere zero, that space every- 

 where conforms to the axioms of Euclid ; and it may be 

 called a flat (homaloid) space in contradistinction to 

 other spaces, analytically constructible, that may be 

 c ailed curved, because their measure of curvature has a 

 value other than zero. Analytical geometry may be as 

 completely and consistently worked out for such spaces 

 as ordinary geometry can for our actually existing 

 homaloid space. 



If the measure of curvature is positive we have 

 spherical space, in which straightest lines return upon 

 themselves and there are no parallels. Such a space 

 would, like the surface of a sphere, be unlimited but 

 not infinitely great. A constant negative measure of 

 curvature on the other hand gives pseudo-spherical 

 space, in which straightest lines run out to infinity, and 

 a pencil of straightest lines may be drawn, in any 

 flattest surface, through any point which does not inter- 

 sect another given straightest line in that surface. 



Beltrami * has rendered these last relations imagin- 

 able by showing that the points, lines, and surfaces of 

 a pseudospherical space of three dimensions, can be so 

 1 Teoria fondamentale, $ 



