Page:Popular Science Monthly Volume 78.djvu/570

This page has been proofread, but needs to be validated.
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 thinking 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.

PSM V78 D570 Non euclidean representation of curved space.png

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

We read in the Arabian Nights 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