*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 discovered 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 enclosure 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.