If the continuum with which we have to do is one in which the "distance" between two elements possesses significance, and if it turns out that the invariant expression for this distance is not the Pythagorean one, but one indicating the non-Euclideanism of our continuum, we say that this continuum has a "curvature." This means that, if we interpret the elements of our continuum as points in space (which of course we may properly do) and if we then try to superpose this point-continuum upon a Euclidean continuum, it will not "go"; we shall be caught in some such absurdity as trying to force a sphere into coincidence with a plane. And of course if it won't go, the only possible reason is that it is curved or distorted, like the sphere, in such a way as to prevent its going. It is unfortunate that the visualizing of such curvature requires the visualizing of an additional dimension for the curved continuum to curve into; so that while we can picture a curved surface easily enough, we can't picture a curved three-space or four-space. But that is a barrier to visualization alone, and in no sense to understanding. OUR WORLD OF FOUR DIMENSIONS It will be observed that we have now a much broader definition of non-Euclideanism than the one which served us for the investigation of Euclid's parallel postulate. If we may at pleasure accept this postulate or replace it by another and different one, we may presumably do the same for any other or any others of Euclid's postulates. The very statement that the distance between elements of the continuum shall possess significance, and shall be measurable by considering a path in the continuum which involves other elements, is an assumption. If we discard it altogether, or replace it by one postulating that some other joint property of the elements than their distance be the center of interest, we get a non-Euclidean geometry. So for any other of Euclid's postulates; they are all necessary for a Euclidean system, and in the absence of any one of them we get a non-Euclidean system.