It is a different matter when the distance has to be judged from the railway line.  Here the following method suggests itself.  If we call A^1 and B^1 the two points on the train whose distance apart is required, then both of these points are moving with the velocity v along the embankment.  In the first place we require to determine the points A and B of the embankment which are just being passed by the two points A^1 and B^1 at a particular time t —­ judged from the embankment.  These points A and B of the embankment can be determined by applying the definition of time given in Section 8.  The distance between these points A and B is then measured by repeated application of thee measuring-rod along the embankment.

A priori it is by no means certain that this last measurement will supply us with the same result as the first.  Thus the length of the train as measured from the embankment may be different from that obtained by measuring in the train itself.  This circumstance leads us to a second objection which must be raised against the apparently obvious consideration of Section 6.  Namely, if the man in the carriage covers the distance w in a unit of time —­ measured from the train, —­ then this distance —­ as measured from the embankment —­ is not necessarily also equal to w.

  Notes

*) e.g. the middle of the first and of the hundredth carriage.

THE LORENTZ TRANSFORMATION

The results of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity (Section 7) has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows: 

(1) The time-interval (time) between two events is independent of the condition of motion of the body of reference.

(2) The space-interval (distance) between two points of a rigid body is independent of the condition of motion of the body of reference.

If we drop these hypotheses, then the dilemma of Section 7 disappears, because the theorem of the addition of velocities derived in Section 6 becomes invalid.  The possibility presents itself that the law of the propagation of light in vacuo may be compatible with the principle of relativity, and the question arises:  How have we to modify the considerations of Section 6 in order to remove the apparent disagreement between these two fundamental results of experience?  This question leads to a general one.  In the discussion of Section 6 we have to do with places and times relative both to the train and to the embankment.  How are we to find the place and time of an event in relation to the train, when we know the place and time of the event with respect to the railway embankment ?  Is there a thinkable answer to this question of such a nature that the law of transmission of light in vacuo does not contradict the principle of relativity