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But there are two classes of experimental facts hitherto obtained which can be represented in the Maxwell-Lorentz theory only by the introduction of an auxiliary hypothesis, which in itself—i.e. without making use of the theory of relativity—appears extraneous. 2 But no person whose mode of thought is logical can rest satisfied with this condition of things. He asks: “How does it come that certain reference-bodies (or their states of motion) are given priority over other reference-bodies (or their states of motion)? What is the reason for this preference? In order to show clearly what I mean by this question, I shall make use of a comparison. 2 Every description of the scene of an event or of the position of an object in space is based on the specification of the point on a rigid body (body of reference) with which that event or object coincides. This applies not only to scientific description, but also to everyday life. If I analyse the place specification “Trafalgar Square, London,” 2 I arrive at the following result. The earth is the rigid body to which the specification of place refers; “Trafalga This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to a system of co-ordinates, the scene of any event will be determined (for the main part) by the specification of the lengths of the three perpendiculars or co-ordinates (x, y, z) which can be dropped from the scene of the event to those three plane surfaces. The lengths of these three perpendiculars can be determined by a series of manipulations with rigid measuring-rods performed according to the rules and methods laid down by Euclidean geometry. 8 We can now appreciate why that argument is not convincing, which we brought forward against the general principle of relativity at the end of Section XVIII. It is certainly true that the observer in the railway carriage experiences a jerk forwards as a result of the application of the brake, and that he recognises in this the non-uniformity of motion (retardation) of the carriage. But he is compelled by nobody to refer this jerk to a “real” acceleration (retardation) of the carriage. He might also interpret his experience thus: “My body of reference (the carriage) remains permanently at rest. With reference to it, however, there exists (during the period of application of the brakes) a gravitational field which is directed forwards and which is variable with respect to time. Under the influence of this field, the embankment together with the earth moves non-uniformly in such a manner that their original velocity in the backwards direction is continuously reduced.” In one of the most notable of these attempts Michelson devised a method which appears as though it must be decisive. Imagine two mirrors so arranged on a rigid body that the reflecting surfaces face each other. A ray of light requires a perfectly definite time T to pass from one mirror to the other and back again, if the whole system be at rest with respect to the æther. It is found by calculation, however, that a slightly different time T' is required for this process, if the body, together with the mirrors, be moving relatively to the æther. And yet another point: it is shown by calculation that for a given velocity v with reference to the æther, this time T' is different when the body is moving perpendicularly to the planes of the mirrors from that resulting when the motion is parallel to these planes. Although the estimated difference between these two times is exceedingly small, Michelson and Morley performed an experiment involving interference in which this difference should have been clearly detectable. But the experiment gave a negative result—a fact very perplexing to physicists. Lorentz and FitzGerald rescued the theory from this difficulty by assuming that the motion of the body relative to the æther produces a contraction of the body This hypothesis, which is not justifiable by any electrodynamical facts, supplies us then with that particular law of motion which has been confirmed with great precision in recent years. 4 But there are two classes of experimental facts hitherto obtained which can be represented in the Maxwell-Lorentz theory only by the introduction of an auxiliary hypothesis, which in itself—i.e. without making use of the theory of relativity—appears extraneous. 2 From this consideration we see that it will be advantageous if, in the description of position, it should be possible by means of numerical measures to make ourselves independent of the existence of marked positions (possessing names) on the rigid body of reference. In the physics of measurement this is attained by the application of the Cartesian system of co-ordinates. 7 But no person whose mode of thought is logical can rest satisfied with this condition of things. He asks: “How does it come that certain reference-bodies (or their states of motion) are given priority over other reference-bodies (or their states of motion)? What is the reason for this preference? In order to show clearly what I mean by this question, I shall make use of a comparison. 2 Moreover, according to this equation the time difference t' of two events with respect to K' does not in general vanish, even when the time difference t of the same events with reference to K vanishes. Pure “space-distance” of two events with respect to K results in “time-distance” of the same events with respect to K'. But the discovery, of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact ±ò[Óis recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. 1 In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude This hypothesis, which is not justifiable by any electrodynamical facts, supplies us then with that particular law of motion which has been confirmed with great precision in recent years. 4