SPACE AND GEOMETRY 66 



Finally at the right we see the asymptotic rota- 

 tion, in which OX and OY, retaining their 

 perpendicularity, move scissorwise toward the 

 singular line OL, which we have called the 

 asymptote. 



The latter process is shown also in Figure 5, 

 where the perpendicular lines OA and OA' go 

 by rotation into OB and OB', and these into 

 OC and OC, and such rotation can be repeated 

 indefinitely while the pair of perpendiculars ap- 

 proach nearer and nearer to the singular line 

 OL. It is thus impossible to rotate a line such as 

 OA into a line such as OA', for these two classes 

 of lines are permanently separated by the sin- 

 gular lines or asymptotes (which are the broken 

 lines of the figure) . And here we may point out 

 a very remarkable property of these singular 

 lines. Since OA may be rotated to OB and then 

 to OC, and so on without limit, always keeping 

 the same length, the terminus approaches the 

 singular line beyond any point which we can 

 mark off upon it, and we thus reach the conclu- 

 sion that any interval marked off upon one of 

 these singular lines is of zero length. 



As we proceed to develop the asymptotic ge- 

 ometry we are struck by the observation that 

 almost every theorem of Euclid has its counter- 



