[8] _e.g._ the middle of the first and of the hundredth carriage.
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 _ and _B _ 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 _ and _B _ 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 VIII.
The distance between these points A and B is then measured by repeated
application of the 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 VI. 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_.
XI.
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 VII) 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.
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