I. Non-Euclidian 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, observation, induction and even imagination in the realms of higher mathematics. For instance, nothing surprises the layman—I use the word layman 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 mathematicians 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 always true. Thus, out of various endeavors to prove Euclid's "parallel" axiom, arose non-Euclidean geometry, the beginning of which is sometimes 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 Russian Lobatchevsky and the Hungarian Bolyai, who, about 1830, independently 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.