*THE POPULAR SCIENCE MONTHLY*

In 1854, the German Riemann^{[1]} discovered another geometry. This geometry is based on the hypothesis that through a given point *no* straight line can be drawn parallel to a given straight line. Thus we have the three geometries: Euclid's (or the parabolic geometry), in which the "parallel axiom" holds, Lobatchevsky's (or the hyperbolic geometry), Riemann's (or the elliptic geometry). As is now well established, all three geometries are consistent with reality: Euclid's is true for a plane (a surface of zero curvature); Riemann's is true for a spherical surface (a two-dimensional space of constant positive curvature); Lobatchevsky's is true on the so-called pseudo-spherical surface of indefinite extent (a two-dimensional space of constant negative curvature). This pseudo-spherical surface is a saddle-shaped surface, like the inner surface of a solid ring.

It is to be noted that the straight line of one geometry is not the straight line of another, but in all three geometries it is the *shortest* distance between two points. Such straightest lines are "geodetic" lines. It will perhaps be evident now why in a sense the discovery of the non-Euclidean geometries was a stepping-stone to the consideration of hyperspace; though we should bear in mind that the two conceptions are entirely distinct, neither one being dependent upon the other. The logical conception of non-Euclidean geometry is far more difficult than the abstract notion of the fourth dimension. The study of the results arrived at by Lobatchevsky, Bolyai, Riemann, Beltrami and others forced men to think of "spaces," and it is hardly too much to say that the stimulus thus given to "high thinking" of this nature gave rise to the hypothetical acceptance of a fourth (or any higher) dimensional space.^{[2]}

II. The Fourth Dimension

I come now to the consideration of hyperspace, which is space of any dimension above three, but for convenience and simplicity I shall confine myself mainly to fourth dimensional space.

To get any clear notion of the fourth dimension, one must make up his mind to exercise much patience, perhaps reading and re-reading many times articles by various authors. In this exposition of the subject, I would warn the reader against supposing that any attempt is here made to convince him of the possibility of the existence of fourth dimensional space. He is not even asked to *believe in* a material space

- ↑ Riemann, "Ueber die Hypothesen seiche der Geometrie zu Grunde liegen," first read in 1854.
- ↑ Lobatschevsky's "The Theory of Parallels" and Bolyai's "The Science Absolute of Space" were translated into English by George Bruce Halsted and first appeared in
*Scientiæ Baccalaureus,*a journal published for a short time by the Missouri School of Mines. By this and other publications Professor Halsted did much to popularize non-Euclidean geometry. Perhaps the most available short treatise on the subject in America is Professor Henry P. Manning's "Non-Euclidean Geometry."