GEOMETRICAL AXIOMS. 61 



appear always to diverge more and more. But two 

 straight lines which from his first position seemed to 

 converge to one and the same point of the background 

 a hundred feet distant, would continue to do this 

 however far he went, and he would never reach their 

 point of intersection. 



Now we can obtain exactly similar images of our 

 real world, if we look through a large convex lens of 

 corresponding negative focal length, or even through a 

 pair of convex spectacles if ground somewhat prisma- 

 tically to resemble pieces of one continuous larger lens. 

 With these, like the convex mirror, we see remote ob- 

 jects as if near to us, the most remote appearing no 

 farther distant than the focus of the lens. In going 

 about with this lens before the eyes, we find that the 

 objects we approach dilate exactly in the manner I 

 have described for pseudospherical space. Now any one 

 using a lens, were it even so strong as to have a focal 

 length of only sixty inches, to say nothing of a hun- 

 dred feet, would perhaps observe for the first moment 

 that he saw objects brought nearer. But after going 

 about a little the illusion would vanish, and in spite 

 of the false images he would judge of the distances 

 rightly. We have every reason to suppose that what 

 happens in a few hours to any one beginning to wear 

 spectacles would soon enough be experienced in pseu- 

 dospherical space. In short, pseudospherical space 



