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.