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In order to attain the greatest possible clearness, let us return to `
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our example of the railway carriage supposed to be travelling `
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uniformly. We call its motion a uniform translation ("uniform" because `
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it is of constant velocity and direction, " translation " because `
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although the carriage changes its position relative to the embankment `
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yet it does not rotate in so doing). Let us imagine a raven flying `
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through the air in such a manner that its motion, as observed from the `
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embankment, is uniform and in a straight line. If we were to observe `
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the flying raven from the moving railway carriage. we should find that `
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the motion of the raven would be one of different velocity and `
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direction, but that it would still be uniform and in a straight line. `
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Expressed in an abstract manner we may say : If a mass m is moving `
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uniformly in a straight line with respect to a co-ordinate system K, `
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then it will also be moving uniformly and in a straight line relative `
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to a second co-ordinate system K1 provided that the latter is `
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executing a uniform translatory motion with respect to K. In `
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accordance with the discussion contained in the preceding section, it `
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follows that: `
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If K is a Galileian co-ordinate system. then every other co-ordinate `
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system K' is a Galileian one, when, in relation to K, it is in a `
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condition of uniform motion of translation. Relative to K1 the `
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mechanical laws of Galilei-Newton hold good exactly as they do with `
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respect to K. `
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We advance a step farther in our generalisation when we express the `
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tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate `
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system devoid of rotation, then natural phenomena run their course `
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with respect to K1 according to exactly the same general laws as with `
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respect to K. This statement is called the principle of relativity (in `
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the restricted sense). `
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As long as one was convinced that all natural phenomena were capable `
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of representation with the help of classical mechanics, there was no `
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need to doubt the validity of this principle of relativity. But in `
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view of the more recent development of electrodynamics and optics it `
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became more and more evident that classical mechanics affords an `
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insufficient foundation for the physical description of all natural `
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phenomena. At this juncture the question of the validity of the `
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principle of relativity became ripe for discussion, and it did not `
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appear impossible that the answer to this question might be in the `
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negative. `
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Nevertheless, there are two general facts which at the outset speak `
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very much in favour of the validity of the principle of relativity. `
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Even though classical mechanics does not supply us with a sufficiently `
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broad basis for the theoretical presentation of all physical `
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phenomena, still we must grant it a considerable measure of " truth," `
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since it supplies us with the actual motions of the heavenly bodies `
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with a delicacy of detail little short of wonderful. The principle of `
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relativity must therefore apply with great accuracy in the domain of `
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mechanics. But that a principle of such broad generality should hold `
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with such exactness in one domain of phenomena, and yet should be `
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invalid for another, is a priori not very probable. `
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We now proceed to the second argument, to which, moreover, we shall `
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return later. If the principle of relativity (in the restricted sense) `
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does not hold, then the Galileian co-ordinate systems K, K1, K2, etc., `
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which are moving uniformly relative to each other, will not be `
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equivalent for the description of natural phenomena. In this case we `
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should be constrained to believe that natural laws are capable of `
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being formulated in a particularly simple manner, and of course only `
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on condition that, from amongst all possible Galileian co-ordinate `
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systems, we should have chosen one (K[0]) of a particular state of `
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motion as our body of reference. We should then be justified (because `
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of its merits for the description of natural phenomena) in calling `
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this system " absolutely at rest," and all other Galileian systems K " `
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in motion." If, for instance, our embankment were the system K[0] then `
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our railway carriage would be a system K, relative to which less `
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simple laws would hold than with respect to K[0]. This diminished `
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simplicity would be due to the fact that the carriage K would be in `
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motion (i.e."really")with respect to K[0]. In the general laws of `
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nature which have been formulated with reference to K, the magnitude `
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and direction of the velocity of the carriage would necessarily play a `
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part. We should expect, for instance, that the note emitted by an `
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organpipe placed with its axis parallel to the direction of travel `
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would be different from that emitted if the axis of the pipe were `
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placed perpendicular to this direction. `
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Now in virtue of its motion in an orbit round the sun, our earth is `
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comparable with a railway carriage travelling with a velocity of about `
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30 kilometres per second. If the principle of relativity were not `
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valid we should therefore expect that the direction of motion of the `
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earth at any moment would enter into the laws of nature, and also that `
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physical systems in their behaviour would be dependent on the `
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orientation in space with respect to the earth. For owing to the `
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alteration in direction of the velocity of revolution of the earth in `
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the course of a year, the earth cannot be at rest relative to the `
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hypothetical system K[0] throughout the whole year. However, the most `
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careful observations have never revealed such anisotropic properties `
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in terrestrial physical space, i.e. a physical non-equivalence of `
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different directions. This is very powerful argument in favour of the `
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principle of relativity. `
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THE THEOREM OF THE `
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ADDITION OF VELOCITIES `
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EMPLOYED IN CLASSICAL MECHANICS `
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Let us suppose our old friend the railway carriage to be travelling `
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along the rails with a constant velocity v, and that a man traverses `
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the length of the carriage in the direction of travel with a velocity `
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w. How quickly or, in other words, with what velocity W does the man `
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advance relative to the embankment during the process ? The only `
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possible answer seems to result from the following consideration: If `
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the man were to stand still for a second, he would advance relative to `
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the embankment through a distance v equal numerically to the velocity `
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of the carriage. As a consequence of his walking, however, he `
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traverses an additional distance w relative to the carriage, and hence `
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also relative to the embankment, in this second, the distance w being `
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numerically equal to the velocity with which he is walking. Thus in `
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total be covers the distance W=v+w relative to the embankment in the `
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second considered. We shall see later that this result, which `
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expresses the theorem of the addition of velocities employed in `
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classical mechanics, cannot be maintained ; in other words, the law `
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that we have just written down does not hold in reality. For the time `
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being, however, we shall assume its correctness. `
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THE APPARENT INCOMPATIBILITY OF THE `
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LAW OF PROPAGATION OF LIGHT WITH THE `
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PRINCIPLE OF RELATIVITY `
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There is hardly a simpler law in physics than that according to which `
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light is propagated in empty space. Every child at school knows, or `
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believes he knows, that this propagation takes place in straight lines `
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with a velocity c= 300,000 km./sec. At all events we know with great `
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exactness that this velocity is the same for all colours, because if `
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this were not the case, the minimum of emission would not be observed `
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simultaneously for different colours during the eclipse of a fixed `
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star by its dark neighbour. By means of similar considerations based `
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on observa- tions of double stars, the Dutch astronomer De Sitter was `
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also able to show that the velocity of propagation of light cannot `
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depend on the velocity of motion of the body emitting the light. The `
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assumption that this velocity of propagation is dependent on the `
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direction "in space" is in itself improbable. `
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In short, let us assume that the simple law of the constancy of the `
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velocity of light c (in vacuum) is justifiably believed by the child `
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at school. Who would imagine that this simple law has plunged the `
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conscientiously thoughtful physicist into the greatest intellectual `
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difficulties? Let us consider how these difficulties arise. `
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Of course we must refer the process of the propagation of light (and `
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indeed every other process) to a rigid reference-body (co-ordinate `
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system). As such a system let us again choose our embankment. We shall `
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imagine the air above it to have been removed. If a ray of light be `
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sent along the embankment, we see from the above that the tip of the `
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ray will be transmitted with the velocity c relative to the `
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embankment. Now let us suppose that our railway carriage is again `
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travelling along the railway lines with the velocity v, and that its `
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direction is the same as that of the ray of light, but its velocity of `
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course much less. Let us inquire about the velocity of propagation of `
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the ray of light relative to the carriage. It is obvious that we can `
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here apply the consideration of the previous section, since the ray of `
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light plays the part of the man walking along relatively to the `
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carriage. The velocity w of the man relative to the embankment is here `
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replaced by the velocity of light relative to the embankment. w is the `
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required velocity of light with respect to the carriage, and we have `
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`
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w = c-v. `
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The velocity of propagation ot a ray of light relative to the carriage `
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thus comes cut smaller than c. `
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But this result comes into conflict with the principle of relativity `
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set forth in Section V. For, like every other general law of `
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nature, the law of the transmission of light in vacuo [in vacuum] `
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must, according to the principle of relativity, be the same for the `
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railway carriage as reference-body as when the rails are the body of `
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reference. But, from our above consideration, this would appear to be `
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impossible. If every ray of light is propagated relative to the `
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embankment with the velocity c, then for this reason it would appear `
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that another law of propagation of light must necessarily hold with `
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respect to the carriage -- a result contradictory to the principle of `
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relativity. `
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In view of this dilemma there appears to be nothing else for it than `
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to abandon either the principle of relativity or the simple law of the `
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propagation of light in vacuo. Those of you who have carefully `
`
followed the preceding discussion are almost sure to expect that we `
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should retain the principle of relativity, which appeals so `
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convincingly to the intellect because it is so natural and simple. The `
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law of the propagation of light in vacuo would then have to be `
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replaced by a more complicated law conformable to the principle of `
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relativity. The development of theoretical physics shows, however, `
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that we cannot pursue this course. The epoch-making theoretical `
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investigations of H. A. Lorentz on the electrodynamical and optical `
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phenomena connected with moving bodies show that experience in this `
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domain leads conclusively to a theory of electromagnetic phenomena, of `
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which the law of the constancy of the velocity of light in vacuo is a `
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necessary consequence. Prominent theoretical physicists were theref `
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ore more inclined to reject the principle of relativity, in spite of `
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the fact that no empirical data had been found which were `
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contradictory to this principle. `
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At this juncture the theory of relativity entered the arena. As a `
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