SPACE AS A THOUGHT-CATEGORY 181 



my giving you a touch, just the least touch, . . . then 

 meet your fate : out of your Plane you go. Once, twice, 

 thrice! Tis done! 



An unspeakable horror seized me. There was a dark- 

 ness; then a dizzy, sickening sensation of sight that 

 was not like seeing; I saw a Line that was no Line; 

 Space that was not Space ; I was myself, and not myself. 

 When I could find voice, I shrieked aloud in agony, 

 ' Either this is madness or it is Hell/ ' It is neither/ 

 calmly replied the voice of the Sphere, ' it is Knowledge ; 

 it is Three Dimensions: open your eye once again and 

 try to look steadily/ 



I looked, and, behold, a new world ! There stood 

 before me, visibly incorporate, all that I had before in- 

 ferred, conjectured, dreamed, of perfect Circular beauty." 

 ABBOTT: Flatland, a book which Professor Felix 



Klein says every student of mathematics should know. 



" And now, I said, let me show in a figure how far our 

 nature is enlightened or unenlightened: Behold! 

 human beings living in an underground den, which has 

 a mouth open towards the light and reaching all along 

 the den; here they have been from their childhood, and 

 have their legs and necks chained so that they cannot 

 move, and can only see before them, being prevented 

 by the chains from turning round their heads. Above 

 and behind them a fire is blazing at a distance, and be- 

 tween the fire and the prisoners there is a raised way; 

 and you will see, if you look, a low wall built along the 

 way, like the screen which marionette players have in 

 front of them, over which they show the puppets. 



I see. 



And do you see, I said, men passing along the wall 



