THE FOURTH DIMENSION 381 



THE FOURTH DIMENSION 



BY PEOFESSOE SAMUEL M. BARTON 

 UNIVEESITT OF THE SOUTH, SEWANEE, TBNN. 



I. Non-Euclidean Geometry 



AS non-Euclidean geometry has not become popular enough, to find 

 a place in the ordinary college curriculum, and as its discovery 

 preceded any serious consideration of space of more dimensions than 

 three, it seems to me that, before taking up hyperspace proper, it would 

 be well at least to mention the non-Euclidean geometries. 



The mathematics of the college student is largely deductive, and 

 he but faintly realizes the important part played by intuition, observa- 

 tion, induction and even imagination in the realms of higher mathe- 

 matics. For instance, nothing surprises the layman — I use the word lay- 

 man as including all who have not made some special study of higher 

 mathematics — so much as to hear for the first time that the famous 

 axiom of Euclid, namely : If two lines are cut by a third, and the sum 

 of the interior angles on the same side of the cutting line is less than 

 two right angles, the lines will meet on that side when sufficiently 

 produced, is not necessarily true. And yet in very early times mathe- 

 maticians began to doubt the truth of this axiom as it did not seem to 

 be, like the rest, a simple elementary fact. The great geometer Legendre 

 and other mathematicians attempted to give a proof of this so-called 

 axiom, but without success. At last, to make a long story short, some 

 mathematicians began to believe that this proposition was not only not 

 self-evident, but was not capable of proof, and moreover that an equally 

 consistent geometry could be built up on the supposition that it is not 

 alwaj^s true. Thus, out of various endeavors to prove Euclid's " parallel " 

 axiom, arose non-Euclidean geometry, the beginning of which is some- 

 times attributed to Gauss; but as he did not publish anything on the 

 subject, it is impossible to say what his ideas were. In the greatness 

 of his heart, he generously gave full credit to Bolyai for his independent 

 discoveries. 



All honor, however, is due to two remarkable men, the Eussian 

 Lobatchevsky and the Hungarian Bolyai, who, about 1830, independ- 

 ently of each other, showed the denial of Euclid's parallel axiom led to 

 a system of two-dimensional geometry as self-consistent as Euclid's. 

 This new geometry is based on the assumption that through a given 

 point a number of straight lines can be drawn parallel to a given straight 

 line. 



