88 LAMBERT— THE STRAIGHT LINE CONCEPT. [April 14, 



geometry of the plane with hyperbolic measurement. 1 The type 

 of surfaces with constant negative total curvature is the pseudo- 

 sphere of revolution, generated by revolving the tractrix about its 

 asymptote. 



If the constant total curvature of the curved surface is -f- a 2 , the 

 geometry of geodesies on the curved surface is identical with the 

 geometry of the plane with elliptic measurement. 1 The type of 

 curved surfaces with constant positive total curvature is the sphere. 

 It is important to note that the entire elliptic plane is represented 

 on the hemisphere. 



These statements show the reasonableness of using as equivalent 

 the terms elliptic space and space of positive curvature ; hyperbolic 

 space and space of negative curvature ; parabolic space and space 

 of zero curvature. 



CONTINUITY OF THE STRAIGHT LINE. 



It remains to examine the elemental structure of the straight line. 

 Adopting as definition of continuity the totality of all real num- 

 bers, is the totality of distances from a fixed point of the line to all 

 other points of the line continuous? This question must be an- 

 swered by establishing a correspondence between sets of numbers 

 and points and lines, that is by a system of analytic geometry. 



Let any pair of numbers (x, y) correspond to a point, any pair 

 of numbers (//, v) correspond to a straight line, and let the equa- 

 tion ux + vy -+- i = o denote that the point (x, y) is on the line 

 (u, 7>). The straight line is now determined by any two points 

 and two straight lines intersect in only one point, that is, the 

 straight line is the straight line of Euclid. If x, y and it, v are 

 any numbers of the totality of numbers obtained from unity by 

 applying a finite number of times the operations addition, subtrac- 

 tion, multiplication, division and taking the positive square root of 

 unity plus the square of any number previously determined, Hilbert 

 has proved that all the constructions of Euclid are possible. The 

 straight line, however, is clearly not continuous, for no transcen- 

 dental numbers occur in the totality of numbers represented on the 

 straight line. 



The continuity of the straight line is not a necessity of Euclid's 



1 Except for certain self-evident limitations due to the peculiarities of the 

 surface. 



