As to the precise character of the non-Euclideanism which is revealed, we may leave this to later chapters and to the competing essayists. We need only point out here that it will not necessarily be restricted to the matter of parallelism. The parallel postulate is of extreme interest to us for two reasons; first because historically it was the means by which the possibilities and the importance of non-Euclidean geometry were forced upon our attention; and second because it happens to be the immediate ground of distinction between Euclidean geometry and two of the most interesting alternatives. But Euclidean geometry is characterized, not by a single postulate, but by a considerable number of postulates. We may attempt to omit any one of these so that its ground is not specifically covered at all, or to replace any one of them by a direct alternative. We might conceivably do away with the superposition postulate entirely, and demand that figures be proved equivalent, if at all, by some more drastic test. We might do away with the postulate, first properly formulated by Hilbert, on which our ideas of the property represented in the word "between" depend. We might do away with any single one of the Euclidean postulates, or with any combination of two or more of them. In some cases this would lead to a lack of categoricity and we should get no geometry at all; in most cases, provided we brought a proper degree of astuteness to the formulation of alternatives for the rejected postulates, we should get a perfectly good system of non-Euclidean geometry: one realized, if at all, by other elements than the Euclidean point, line and plane, and one whose elements behave toward one another differently from the Euclidean point, line and plane. Merely to add definiteness to this chapter, I annex here the statement that in the geometry which Einstein builds up as more nearly representing the true external world than does Euclid's, we shall dispense with Euclid's (implicit) assumption, underlying his (explicitly stated) superposition postulate, to the effect that the act of moving things about does not affect their lengths. We shall at the same time dispense with his parallel postulate. And we shall add a fourth dimension to his three--not, of course, anything in the nature of a fourth Euclidean straight line perpendicular, in Euclidean space, to three lines that are already perpendicular to each other, but something quite distinct from this, whose nature we shall see more exactly in the next chapter. If the present chapter has made it clear that it is proper for us to do this, and has prevented anyone from supposing that the results of doing it must be visualized in a Euclidean space of three dimensions or of any number of dimensions, it will have served its purpose. VI THE SPACE-TIME CONTINUUM Minkowski's World of Events, and the Way It Fits Into Einstein's Structure BY THE EDITOR, EXCEPT AS NOTED