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TEE POPULAR SCIENCE MONTELY 



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 ex- 

 pansive, 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, in- 

 struments more delicate and accurate than any that science has yet de- 

 vised. 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 out- 

 ward from this point, they will be found to diverge, spreading farther 

 apart without limit (Fig. 2). 



D 



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 

 space the internal angles of such a triangle, added together, always 

 equal two right angles, but here they fall short of two right angles, and 



