60 ORIGIN AND SIGNIFICANCE OF 



We can even go a step further, and infer how the 

 objects in a pseudospherical world, were it possible to 

 enter one, would appear to an observer, whose eye- 

 measure and experiences of space had been gained like 

 ours in Euclid's space. Such an observer would con- 

 tinue to look upon rays of light or the lines of vision 

 as straight lines, such as are met with in flat space, 

 and as they really are in the spherical representation 

 of pseudospherical space. The visual image of the 

 objects in pseudospherical space would thus make the 

 same impression upon him as if he were at the centre 

 of Beltrami's sphere. He would think he saw the 

 most remote objects round about him at a finite 

 distance, 1 let us suppose a hundred feet off. But as 

 he approached these distant objects, they would dilate 

 before him, though more in the third dimension than 

 superficially, while behind him they would contract. 

 He would know that his eye judged wrongly. If he 

 saw two straight lines which in his estimate ran 

 parallel for the hundred feet to his world's end, he 

 would find on following them that the farther he 

 advanced the more they diverged, because of the 

 dilatation of all the objects to which he approached. 

 On the other hand, behind him, their distance would 

 seem to diminish, so that as he advanced they would 



1 The reciprocal of the square of this distance, expressed in 

 negative quantity, would be the measure of curvature of the pseudo- 

 spherical space. 



