At this stage we must mention the premonitions of Riemann on the subject of the metrical field of real space. Riemann did not attribute this structure of space to the presence of some invisible medium, the ether, possessing a structure of its own. According to him the origin of the metrical field should be sought elsewhere. He felt that the metrical field of space should be compared to a magnetic or an electric field pervading space. And just as a magnetic field exists in the space surrounding a magnet, Riemann searched for the physical cause of the metrical field. With characteristic boldness, he found it in the matter of the universe; the metrical field thus became a species of material field. If Riemann’s ideas are accepted we can understand how a redistribution of the star matter in the universe, altering as it would the lay of the metrical field, would produce deformations in the shape of a given body and variations in the paths of light rays. As Weyl tells us, a spherical ball of clay compressed into any other form might again be made to appear spherical were all the matter in the universe to be redistributed in a suitable way. It would also follow that were all the matter in the universe to be annihilated, and as a result the metrical field to vanish, space (assuming that any physical meaning were left the term) would become completely amorphous, just like mathematical space; light rays would not know where to move, all geodesics having disappeared; and were one lone material body to be introduced into otherwise empty space, it would not know what shape to take. Without the metrical field, physical space would be unthinkable. Still further important consequences follow from this matter-moulding [Pg 58] hypothesis of Riemann. Prior to these views, the principle of sufficient reason appeared to imply that physical space would always turn out to be homogeneous—the same in all places. This does not necessarily mean Euclidean, for, as we know, Riemann’s and Lobatchewski’s geometries also correspond to homogeneous spaces; but all varying degrees of non-Euclideanism from place to place were thought to be excluded a priori. Of the vast realm of possible types of geometries or spaces discovered by Riemann, only the homogeneous types survived; a situation which Weyl finds appropriately expressed by the classical line: “Parturiunt montes, nascetur ridiculus mus.” But with the new views advocated by Riemann the situation changes entirely; for now the texture, structure or geometry of space is defined by the metrical field, itself produced by the distribution of matter. Any non-homogeneous distribution of matter would then entail a variable structure or geometry for space from place to place.