382 TEE POPULAR SCIENCE MONTHLY 



In 1854, the German Riemann^ 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 

 geometiy), Riemann's (or the elliptic geometry). As is now well estab- 

 lished, all three geometries are consistent with reality : Euclid's is true 

 for a plane (a surface of zero curvature) ; Eiemann's is true for a 

 spherical surface (a two-dimensional space of constant positive curva- 

 ture) ; Lobatchevsky's is true on the so-called pseudo-spherical surface 

 of indefinite extent (a two-dimensional space of constant negative curva- 

 ture). 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 considera- 

 tion of hyperspace; though we should bear in mind that the two con- 

 ceptions 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, Eiemann, 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.^ 



II. The Foueth 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 sub- 

 ject, 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 Gnmde 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 Scicntice Baccalaureus, a journal published for a short time by 

 the Missouri School of Mines. By this and other publications Professor Ilalsted 

 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." 



