along a certain path, and as unequal if we displaced our unit rod along some other path. In other words, length is non-transferable. In view of the indeterminateness that surrounds the comparison of lengths in different places, we must confine ourselves to the comparison of lengths at any one place or at points separated by infinitesimal intervals. We therefore agree to consider that at every point in space there is fixed a rod which is to serve as unit of length when we measure lengths situated by its side. The totality of these unit rods constitutes what is known as a gauge-system. These gauges may be selected arbitrarily; and it is not meant to imply that they are necessarily of equal length with respect to one another, since we have seen that comparison of lengths in different places is ambiguous. Their sole utility is to serve as standards of measurement for lengths situated by their sides, and thus to enable us to compare lengths situated at the same point.
A gauge-system is the natural extension of a mesh-system, with which we are already acquainted. Just as we may vary the shape of the mesh-system, so now we may vary the nature of the gauge-system.
Let us endeavour to ascertain the significance of this non-transference of length as relating to the nature of space itself. In particular, let us suppose that two rods coincide at a point
; then, while one rod remains fixed at
, the other is made to follow a closed curve, returning finally to its starting point. Owing to the difference in routes followed by the two rods, one having remained fixed and the other been made to describe a curve, the two rods will no longer coincide when we place them side by side after the completion of the circuit.
This non-transference of length is very similar to what is known as the non-transference of direction which holds for the non-Euclidean spaces that we have studied so far. Thus, consider a curved surface, a sphere—for example. If at a point
on the sphere two rods coincide in direction, and if, while one rod remains fixed, the other is displaced from point to point along a closed curve, and displaced in such a way that for each successive infinitesimal displacement the rod maintains the same direction, it would be erroneous to suppose that when the second rod had been returned to its starting point its direction would still coincide with that of the first rod. A distinct discrepancy would be found to exist between the directions of the two rods, and the size of this discrepancy would depend on the area contained within the closed curve which the second rod had been made to follow over the surface. This discrepancy has nothing to do with our choice of a mesh-system; it transcends this choice completely. On the spherical surface we are discussing, it would vary with the closed curve described by the rod; but if we had operated on a plane sheet of paper instead of on a surface curved like a sphere, the discrepancy would never have arisen at all. Its existence is due, therefore, to the curvature, or non-Euclideanism, of the surface, and its magnitude may serve as a means of determining the precise intensity of this non-Euclideanism around every point of the surface.