Page:Popular Science Monthly Volume 78.djvu/568

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he might set out in exactly the same direction as the road runs, or he might incline a very little toward it; in either case he would never meet it. This is equivalent to saying that Lobachevski's space is more expansive, more generously given, and if the reader will follow me a step further, I may say that this space becomes roomier increasingly with every step that the man takes forward.

It costs nothing to imagine ourselves entering this domain of Lobachevski; indeed, for aught we know, we may be actually in it now. And it will cost us no more to imagine our expedition equipped with instruments for measuring angles and for drawing straight lines, instruments more delicate and accurate than any that science has yet devised. A series of experiments may then be carried out to illustrate the properties of this hyperbolic region, I shall limit the narrative to some of the results we could obtain:

A. Parallels, really straight lines that never meet, have a point of nearest approach to each other, but if followed in either direction outward from this point, they will be found to diverge, spreading farther apart without limit (Fig. 2).

PSM V78 D568 Euclid postulate of two parallel lines.png

Fig. 2. The lines AB and CD are straight, as may be seen by viewing the figure from one side with the eye close to the paper. They are In the same plane and will never meet. Yet by an optical illusion we here obtain within a small compass the same appearance as would be furnished by two parallels to an eye located In Lobachevski's space and capable of surveying a tremendous stretch of the parallels from a very great distance. The diagram probably has no reference whatever to non-Euclidean geometry. It elucidates mental, not physical, phenomena.

B. If two perpendiculars are erected on a Lobachevski parallel, they will spread away from each other, becoming farther apart the farther we extend them outward from the base line.

C. With this base line and the two perpendiculars, we might think we had three sides of a rectangle, but no—for after making three of the corners right angles, the fourth must needs be an acute angle. A true rectangle is impossible (Fig. 3).

D. We can, however, draw a straight-sided triangle. In Euclidean apace the internal angles of such a triangle, added together, always equal two right angles, but here they fall short of two right angles, and