560 THE POPULAR SCIENCE MONTHLY 



Fig. 4 will do good service if it enables us to understand what people 

 mean when they assert profoundly that "non-Euclidean space is 

 curved." We ought to discard this misleading, though very prevalent, 

 expression, for it is as confusing as to say that space is straight, or 

 cold, or pink. It certainly sounds absurd to call a straight line curved 

 under any circumstances, and so it is, so long as we confine our think- 

 ing to any one kind of space. But in carrying lines over from one 

 space to another, there is this change of emphasis. For example, the 

 parallels of Lobachevski, when transferred into Euclidean space, cease 

 to be parallels and become, as shown in Fig. 4, hyperbolic curves. 



fig. 4. 



Contrariwise, the parallels of Euclid, transferred into Lobachevski's 

 space, retain merely their secondary property of equidistance, and pass 

 under the name of equidistantials, since they are no longer true par- 

 allels, nor even straight, but rather they are very long curves. 



We read in the Arabian Mghts of the magical carpet of Tangu, 

 which could be made to fly incredible distances by wishing it to do so. 

 Imagination can furnish us with a similar carpet that will flit from one 

 realm of space to another. Suppose then that our carpet is being woven 

 at a non-Euclidean factory. It should be pliable, but it must not 

 stretch, and it must possess truly princely size, having leagues upon 

 leagues of surface. When spread out, it must lie perfectly flat and 



