I have no complete theory of physical measurements to offer, but it seemed desirable to illustrate how difficult it is to say precisely what measurement means in an advanced science such as physics. We have certain postulates, such as "lengths which are equal to the same length are equal to one another," but actual measurements, when made with sufficient accuracy, are not found to verify these postulates. Therefore we invent physical laws to save the postulates. With each fresh law it becomes more difficult to say exactly what we do mean when, e.g., we give the wave-length of a certain line in the spectrum of hydrogen in terms of the metre. (This is particularly odd in view of the fact that these wave-lengths are given to more significant figures than can be warranted by the operations applicable to the standard metre itself, whose length is only known, in comparison with other lengths, to a very moderate degree of approximation.) In physical theory, measurement should rest upon an integration of the formula for . But in physical practice the of that formula can only be determined by means of measurements. Thus the only thing we seem warranted in saying[Pg 94] is this: It is possible to correct the results of actual measurements according to certain known rules, in such a way that the corrected lengths shall satisfy such postulates as Euclid's first axiom; when this is done, we find, by means of physical theory, that all electrons have the same size. But this is not, considered empirically, at all a simple fact. And considered as a statement in the deductive theory it probably has a good meaning, but one which demands much elucidation. Until this is forthcoming, all use of numbers as measures of physical quantities in theoretical physics raises problems, since we do not know what, in theoretical physics, replaces the operation of measurement as conducted in the laboratory and in daily life. The theory of length-measurement raises problems which bring us naturally to Weyl's relativistic theory of electromagnetism, which we must now briefly consider. FOOTNOTES: [27] See his essay in Science, Religion, and Reality, edited by Needham, 1925. [28] Cf. Mathematical Theory of Relativity, §§ 52, 54, 66. [29] Eddington, op. cit. p. 115. [30] Op. cit., § 66, pp. 152-155. [Pg 95] CHAPTER X WEYL'S THEORY 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).