THE theory to be considered in this chapter is, from a geometrical point of view, a natural generalization of Einstein's arbitrariness of co-ordinates; from a physical point of view, it fits electromagnetism into the deductive system, which Einstein's theory does not do. The theory is due to Hermann Weyl, and will be found in his Space, Time, Matter (1922).

The puzzles about measurement considered at the end of Chapter IX. naturally suggest the point of view from which Weyl starts. As he says: "The same certainty that characterizes the relativity of motion accompanies the principle of the relativity of magnitude" (op. cit. p. 283). Measurement is a comparison of lengths, and Weyl suggests that, when lengths in different places are to be compared, the result may depend upon the route pursued in passing from the one place to the other. Lengths at the same place (i.e. having one end identical), if small, he regards as directly comparable; also he assumes continuity in the changes accompanying transportation. This is not the sum-total of his assumptions, nor the most general way of stating them; but before we can state them adequately certain explanations are necessary.

Reduced to its simplest terms, the conception used by Weyl may be expressed as follows. Given a vector at a point, what are we to mean by the statement that a vector at another point is equal to it? There must be some element of convention in our definition; let us therefore, as a first step, set up a unit of length in each place, and see what limitations it is desirable to impose on our initial arbitrariness.

There is, to begin with, an assumption which is made almost tacitly, and that is, that we can recognize something in one place as the "same" vector as something at another place. We may perhaps take this sameness as being merely analytical: the two are the same function of the co-ordinates at their respective places. I do not think this is all that is meant, since a vector is supposed to have some physical significance; but if more is meant, it is not clear how it is to be defined. We will therefore assume that, given a function of the co-ordinates which is a vector, we shall regard the same function of other values of the co-ordinates as the "same" vector at another place.

We next have to define "parallel displacement." This may be defined in various ways. Perhaps the most graphic description is to say that it is displacement along a geodesic (Eddington, op. cit. p. 71). Another definition is that it is a displacement such that the "covariant derivative" vanishes, the covariant derivative of a vector with respect to being defined as , where: For the definition of , see the beginning of Chapter IX. In the tensor calculus, covariant differentiation takes the place of ordinary differentiation for many purposes, since the covariant derivative of a tensor is a tensor, whereas the ordinary derivative is in general not a tensor. We assume that our units of length in different places are so chosen that, when a small displacement is moved to a neighbouring place by parallel displacement, the change in the measure of its length is small, and is proportional to its length. We assume, in short, that the ratio of the increase of length to the initial length for a change of co-ordinates () is: So that () form a vector, .

Now it is possible to express Maxwell's equations in terms of a vector which may be identified with the above vector. Hence it is possible to regard electromagnetic phenomena as explained by the variation of what is taken as the unit as we pass from point to point. I shall not attempt to explain the theory, as it would in any case be necessary to read a full account in order to grasp its significance.

Here, perhaps even more than elsewhere in relativity theory, it is difficult to disentangle the conventional elements from those having physical significance. On the face of it, it might seem as though we were attempting to account for actual physical phenomena by means of a mere convention as to choice of units. But this, of course, is not what is meant. The way the unit is assigned in different places is called by Eddington the "gauge-system": this is only partially arbitrary, and is in part the representation of the physical state of the world. This has to do with the fact that vectors are not purely analytical expressions, but also correspond to physical facts. It would seem, however, that the theory has not yet been expressed with the logical purity that is to be desired, chiefly because it is not prefaced by any clear account of what is to be understood by "measurement"—or, what comes to much the same thing from the standpoint of theory, what we are to mean when we talk of "moving" a vector, whether by parallel displacement or in any other way. To "move" something, we must be able to recognize some identity between things in different places. Perhaps all this is quite clear in the minds of competent exponents of the theory, but if so they have not succeeded in conveying their thoughts without loss of clarity to readers who have not their background. When Eddington says: "Take a displacement at and transfer it by parallel displacement to an infinitely near point " (p. 200), I find myself wondering how, exactly, the displacement is to preserve its identity throughout the transfer, and the only answer suggested by the accompanying formulæ is that the identity is that of an algebraic expression in terms of the co-ordinates. This, however, is clearly insufficient.

Professor Eddington, after expounding Weyl's theory, proceeds to generalize it, and some of his accompanying elucidations are relevant to our present difficulties. Thus he says (p. 217):

He proceeds to a generalized theory in which, at first, length is purely conventional, for comparisons at a point as well as for comparisons between different points. This generalized theory does not seem to involve the same kind of difficulties as those which have been troubling us. The following passage, for example, states the matter with great clearness (p. 226):

Here we have a logically satisfactory theoretical basis for a metric. We may suppose that, as a matter of fact, there are important properties of groups of four points which are "parallelogramical," and that actual physical measurement is an approximate method of discovering which groups have this property. We shall find certain laws approximately fulfilled by rough-and-ready measurements, and fulfilled with increasing accuracy as we introduce refinements into the process of measurement. Consider, for example, Euclid's first axiom: Things which are equal to the same thing are equal to one another. Presumably Euclid regarded this as a logically necessary proposition, and so do people who are engaged in the practice of measurement. If two lengths each equal to a metre are found to be not equal to each other, the plain man assumes that there must be a mistake somewhere. We are therefore continually redefining the actual operations of measurement with a view to verifying Euclid's first axiom as nearly as possible. But with the above-quoted definition of equality of length the first axiom becomes a substantial proposition, namely: If is a parallelogram, and likewise , then is a parallelogram. If this proposition is true, then it is theoretically possible to define measurement in such a way that two lengths each equal to a metre shall always be equal to each other. What is called "accuracy" is, speaking generally, an attempt to obtain a result conformable with some ideal standard supposed to be logical but in fact physical. What do we mean by saying that a length has been "wrongly" measured? Whatever result we obtain from measuring a given length, the result represents a fact in the world. But in what we call a "wrong" measurement, the fact ascertained is complex and of small universality. If the observer has simply misread a scale, the fact ascertained involves reference to his psychology. If he has neglected a physical correction—e.g. for the temperature of his measure—the fact refers only to a measurement carried out with that particular apparatus on that particular occasion. In relativity theory we have another set of what might be called "inaccurate" measurements—e.g. measurements of the masses of -particles or -particles emitted from radio-active bodies must be corrected for their motion relative to the observer before they acquire any general significance. It is always the search for simple relations which enter into general laws that governs successive refinements. But the existence of such relations (where they do exist) is an empirical fact, so that much that seems prima facie to be logically necessary is really contingent. On the other hand, the number of premisses in a deductive system which has to agree with an empirical science can, by logical skill, be diminished to an extent which may be astonishing. Of this, the theory of relativity is a very remarkable example. The theory is a combination of two diverse elements: on the one hand, new experimental data; on the other, a new logical method. It must be regarded as a happy accident that the two appeared together; if the right kind of theoretical genius had not happened to be forthcoming, we might have had to be content for a long time with patched-up hypotheses such as the FitzGerald contraction. As it is, the combination of experiment and theory has produced one of the supreme triumphs of human genius.