But it is obvious that we cannot, by ordinary mechanics, consider the earth as being accelerated in all directions, which we should have to do, apparently, to account for the fact that the gravitational pull is always toward the center. [It is obvious that we cannot explain gravitation by assuming that the earth's surface is continually moving outward with an accelerated velocity.]227 So Einstein found that, as long as we treat the problem by Euclid's geometry, we cannot reach a satisfactory solution. But he found that to the four-dimensional space made up of the three ordinary dimensions of space, together with the time-dimension which we have already mentioned in discussing the special theory, may be attributed a peculiar geometry, the nature of which departs more and more from Euclidean geometry as we approach a gravitational body, and the net result of which is to make possible the universal correspondence of gravitation and acceleration. This modification of the geometry of space is often spoken of as the "curvature of space," an expression which is puzzling, especially as the space which is "curved" is four-dimensional time-space. But we can get an idea of what is meant by considering figures, triangles say, drawn on the surface, of a sphere. These triangles, although drawn on a surface, will not have the same properties as triangles drawn on flat paper--their three angles will not together equal right angles. They will be non-Euclidean. This is only a rough analogy, but we can see that the curvature of the surface causes a departure from Euclidean geometry for plane figures, and consequently the departure from Euclidean laws extended to four dimensions may be referred to as caused by "curvature of space." It is difficult to imagine a lump of matter affecting the geometry of the space round it. Once more we must use a rough illustration. Imagine a very hot body, and that, knowing nothing of its properties, we have to measure up the space round it with metal measuring-rods. The nearer we are to the body, the longer the rods will become, owing to the expansion of the metal. When we measure out a square, one side of which is nearer the body than the opposite side, its angles will not be right angles. If we knew nothing of the laws of heat we should say that the body had made the space round it non-Euclidean.