562 THE POPULAR SCIENCE MONTHLY 



A. No parallels are possible; all great circles (straightest lines) 

 must somewhere meet. 



B. Euclid's axiom that: Through any two points only one straight 

 can pass, is, in most cases, correct; but when the points chosen are 

 diametrically opposed, as are the poles of the earth, then an infinity of 

 great circles cross at such points, and any two of these lines enclose 

 a space. 



C. Concerning triangles, the internal angles are together always 

 equal to more than two right angles, the excess increasing with the size 

 of the triangle. Rectangles are impossible. Two plane figures (except 

 circles) can not be similar in shape unless equal in size. 



D. The Riemannian space is not infinite in extent, but returns into 

 itself. It is, however, boundless ; one could never come to the end of it. 

 With eyes adapted to enormous distances a creature looking in any 

 direction might see the back of its own head. 



E. On the hypothesis that our own universe is of this nature, " a 

 finite number of our common building bricks," as Halsted says, 

 " might be written down which might be more than our universe could 

 contain." And if our earth should increase in bulk, at last the lower 

 surface would advance upon us from above, and, reaching us, would 

 fill the whole universe. 



It is commonly supposed that these peculiarities of Riemannian 

 space are easier to conceive than are the results at which Lobachevski 

 arrives, but this is probably not the case. Long ago, Beltrami discov- 

 ered that the whole space of Lobachevski, notwithstanding that it is 

 infinite in extent in all directions, can be conceived as packed within a 

 hollow globe of finite radius. Imagine, therefore, a great sphere of a 

 hundred yards' radius, with a door leading into it. Looking in, let us 

 suppose that we can discover a railroad track on a trestle extending 

 from the doorway diametrically across to the other side, and a small 

 man — an inch high — standing between the rails and at the exact center 

 of the sphere. Nothing about this view suggests anything but ordinary 

 space to us; it is only for the little being at the center that this en- 

 closure constitutes Lobachevski's universe. 



Another assumption is now to be granted: let the man dwindle in 

 size whenever he moves out of the center toward the shell of the sphere. 

 Growing less and less, he would have no size at all upon reaching the 

 shell, but he could never reach it, for the length of his stride would 

 lessen in proportion to his lessening stature. To him, therefore, the 

 sphere is infinite in extent. Likewise all other objects — the boards on 

 the footpath, the foot-rule and the keys in his pocket, and the pocket 

 itself — ^have their sizes determined by their location within the sphere, 

 let only the rails of the track be continuously parallel both according 

 to his and our own notion of things, and also according to Euclid's 

 definition. 



