Just as non-transference of direction is caused by the curvature, or non-Euclideanism, of space (which we might also call direction-curvature), and disappears with the vanishing of this curvature, so in an analogous manner the non-transference of length is caused by a new type of curvature (called Weylian curvature or distance-curvature), and disappears completely when this type of curvature is non-existent. In a general way we may call a space a Weyl space when it possesses a non-vanishing distance-curvature; the appellation classical spaces will be reserved for all the categories of space which we have studied thus far (Euclidean, Riemannian, Lobatchewskian, semi-Euclidean), since in all these spaces a Weylian or distance curvature is non-existent.
We may note that just as ordinary non-Euclideanism, or direction-curvature, transcended our choice of mesh-system, so now, in similar fashion, the Weylian characteristics—the non-transference of length, and, again, the presence of distance-curvature—transcend our choice of a gauge-system.
Now we have seen that the Euclideanism or non-Euclideanism of a manifold was ascribed to the metrical field which it contained. In a similar way we must assume that the Weylian characteristics must also be ascribed to the metrical field. However, the metrical field due to the
’s which has been discussed so far, is incapable of accounting for Weylian peculiarities. It is necessary to complete the description of the field in order to render it accountable for these new characteristics. When this is done, it is found that in a four-dimensional continuum, in addition to the ten
’s which suffice to describe the metrical field of a four-dimensional classical continuum such as space-time, it is necessary to add four new quantities which we may call the four
’s. The generalised metrical field then contains fourteen separate magnitudes in place of ten as in Einstein’s space-time. Just as the direction-curvature of space characterising non-Euclideanism at a point could be expressed by a tensor (the Riemann-Christoffel tensor), built up from the changes in values between the