i9°5-] 



LAMBERT— THE STRAIGHT LINE CONCEPT. 



The chief attraction of hyperbolic geometry lies in the fact that 

 one has the power to see the whole of hyperbolic space and to 

 direct geometric constructions from a vantage point outside of this 

 space. For example, to draw a common perpendicular to two straight 

 lines, not intersecting and not parallel, connect by a straight line the 

 poles of the given straight lines with respect to the absolute. This 

 problem has been solved by Hilbert by methods such as a being 

 living in hyperbolic space would be obliged to use. It is a simple 

 matter to determine directly from the expression for distance that 

 the locus of points in the hyperbolic plane equidistant from a given 

 straight line is an ellipse tangent to the absolute where the given 

 line meets the absolute. 



GEOMETRY ON SURFACES OF CONSTANT TOTAL CURVATURE. 



If R l and R 2 are the maximum and minimum radii of curvature 

 of the normal sections of a curved surface at any point of the sur- 

 face, the reciprocal of the product of R x and R 2 is called the Gaus- 

 sian or total curvature of the surface at this point. The geometry 

 of geodesies on surfaces whose total curvature is constant has strik- 

 ing analogies to plane Euclidean geometry. Euclid's definition 

 of a straight line as a line which lies in the same manner with 

 respect to all the points in the line, and his definition of a plane 

 as a surface which lies in the same manner with respect to all 

 straight lines in the plane, when taken in connection with Euclid's 

 "Common Notions" implies the congruent displacement of a 

 straight line into itself, that is the displacement of the straight line 

 into itself such that any two points of the line may be made to coin- 

 cide with any other two points of the line provided the distance 

 between the first pair of points equals the distance between the 

 second pair ; and the congruent displacement of the plane into 

 itself, that is the displacement of the plane into itself such that any 

 portion of the plane bounded by straight lines may be made to 

 coincide with any other portion provided the two portions are 

 bounded by straight lines of equal length and the corresponding 

 angles are equal. Now surfaces whose total curvature is constant 

 and geodesies on these surfaces also possess this property of con- 

 gruent displacement, provided displacement is suitably defined. 



If the constant total curvature of the curved surface is — a' 1 , the 

 geometry of geodesies on the curved surface is identical with the 



