SPACE AND GEOMETRY 63 



shear ; it is like the movement of a pack of cards 

 when the top slides and the bottom remains sta- 

 tionary. The geometry of shear rotation defined 

 by such a transformation lacks the fullness and 

 complexity of either the Euclidean or the 

 asymptotic varieties. It might for this reason 

 be called a "degenerate" geometry. 



Thus by taking liberties successively with 

 two of Euclid's postulates it is possible to ob- 

 tain the four additional types of geometry 

 sho\\Ti at either side of the central Euclidean 

 geometry in the following tabulation: 



No No No Negative Positive 



curvature curvature curvature curvature curvature 



Asymptotic Shear Circular Circular Circular 



rotation rotation rotation rotation rotation 



The two on the right are the older non-Euclid- 

 ean geometries, those on the left the newer ones. 

 There may be other branches of mathematics 

 which deserve to be called geometries, but these 

 five, together with their hybrids, constitute the 

 great family of Euclididag. 



Since I do not wish to tax your patience dur- 

 ing these chapters by discussing matters of a 

 technical character, tliis is about all that I 

 should say regarding the non-EucKdean geome- 



