326 ANlSrUAL REPORT SMITHSONIAN INSTITUTION, 1961 



Figure 1. 



do not meet at any "point" (therefore, if they do not meet inside r). 

 It is clear that if I is any "line" and A is any "point" not on that line, 

 then there are an infinite number of "lines" through A which are 

 "parallel" to I; in figure 1 all the "lines" in the sector between AP 

 and J.^' are "parallel" to I. 



So far, we have managed to contradict one of the axioms of Euclid- 

 ean geometry ; but this is not of interest unless we can show that the 

 other concepts and axioms of Euclidean geometry can be defined and 

 retain their truth. The difficulties are the definitions of "distance" 

 and "angle" ; these definitions are more technical, and that of "angle" 

 depends on a knowledge of complex logarithms; but the reader who 

 does not know about these may be consoled by the remark that it is 

 not essential that he understand the details of these definitions. 



The "distance" between two points B and G is defined as follows: 

 If BG meets at P and Q^ the "distance" from 5 to 6^ is 



u-n^)) 1 „ PBjQG 



^^ -^^^ PC^ 



where PB and so on are the ordinary Euclidean (signed) distances. 

 As G tends to P or Q, its "distance" from B becomes infinite, so that 

 the "length" of a line is infinite as in ordinary geometry. It is easy 

 to check that if B, G, D are collinear, then ''BG''+''GD''=''BD,'' so 

 that these "distances" add in the usual manner. It is possible, though 



