Such discussions might have appeared to be merely academic a few years ago; and non-Euclidean geometry, though of vast philosophical interest, might have seemed devoid of any practical importance. But to-day, thanks to Einstein, we have definite reasons for believing that ultra-precise observation of nature has revealed our natural geometry arrived at with solids and light rays to be slightly non-Euclidean and to vary from place to place. So although the non-Euclidean geometers never suspected it (with the exception of Gauss, Riemann and Clifford), our real world happens to be one of the dream-worlds whose possible existence their mathematical genius foresaw. Now, all these investigations initiated by attempts to prove the correctness of the parallel postulate led mathematicians to further discoveries. A more thorough study of Euclid’s axioms and postulates proved them to be inadequate for the deduction of Euclid’s geometry. Euclid himself had never been embarrassed by the incompleteness of his basic premises, for the simple reason that although he failed to express the missing postulates explicitly, he appealed to them implicitly in the course of his demonstrations. The great German mathematician Hilbert and others succeeded in filling the gap by stating explicitly a complete system of postulates for Euclidean and non-Euclidean geometries alike. Among the postulates missing in Euclid’s list was the celebrated postulate of Archimedes, according to which, by placing an indefinite number of equal lengths end to end along a line, we should eventually pass any point arbitrarily selected on the line. Hilbert, by denying this postulate, just as Lobatchewski and Riemann had denied Euclid’s parallel postulate, succeeded in constructing a new geometry known [Pg 38] as non-Archimedean. It was perfectly consistent but much stranger than the classical non-Euclidean varieties. Likewise, it was proved possible to posit a system of postulates which would yield Euclidean or non-Euclidean geometries of any number of dimensions; hence, so far as the rational requirements of the mind were concerned, there was no reason to limit geometry to three dimensions. Incidentally, we see to what rigour of analysis and to what profound introspection the mathematical mind must submit; for the implicit postulates appealed to unconsciously by Euclid are so inconspicuous that it is only owing to the dialectics of modern mathematicians that their presence was finally disclosed and the deficiency remedied by their explicit statement.