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uniformly. We call its motion a uniform translation ("uniform" because
it is of constant velocity and direction, " translation " because
although the carriage changes its position relative to the embankment
yet it does not rotate in so doing). Let us imagine a raven flying
through the air in such a manner that its motion, as observed from the
embankment, is uniform and in a straight line. If we were to observe
the flying raven from the moving railway carriage. we should find that
the motion of the raven would be one of different velocity and
direction, but that it would still be uniform and in a straight line.
Expressed in an abstract manner we may say : If a mass m is moving
uniformly in a straight line with respect to a co-ordinate system K,
then it will also be moving uniformly and in a straight line relative
to a second co-ordinate system K1 provided that the latter is
executing a uniform translatory motion with respect to K. In
accordance with the discussion contained in the preceding section, it
follows that:

If K is a Galileian co-ordinate system. then every other co-ordinate
system K' is a Galileian one, when, in relation to K, it is in a
condition of uniform motion of translation. Relative to K1 the
mechanical laws of Galilei-Newton hold good exactly as they do with
respect to K.

We advance a step farther in our generalisation when we express the
tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate
system devoid of rotation, then natural phenomena run their course
with respect to K1 according to exactly the same general laws as with
respect to K. This statement is called the principle of relativity (in
the restricted sense).

As long as one was convinced that all natural phenomena were capable
of representation with the help of classical mechanics, there was no
need to doubt the validity of this principle of relativity. But in
view of the more recent development of electrodynamics and optics it
became more and more evident that classical mechanics affords an
insufficient foundation for the physical description of all natural
phenomena. At this juncture the question of the validity of the
principle of relativity became ripe for discussion, and it did not
appear impossible that the answer to this question might be in the
negative.

Nevertheless, there are two general facts which at the outset speak
very much in favour of the validity of the principle of relativity.
Even though classical mechanics does not supply us with a sufficiently
broad basis for the theoretical presentation of all physical
phenomena, still we must grant it a considerable measure of " truth,"
since it supplies us with the actual motions of the heavenly bodies
with a delicacy of detail little short of wonderful. The principle of
relativity must therefore apply with great accuracy in the domain of
mechanics. But that a principle of such broad generality should hold
with such exactness in one domain of phenomena, and yet should be
invalid for another, is a priori not very probable.

We now proceed to the second argument, to which, moreover, we shall
return later. If the principle of relativity (in the restricted sense)
does not hold, then the Galileian co-ordinate systems K, K1, K2, etc.,
which are moving uniformly relative to each other, will not be
equivalent for the description of natural phenomena. In this case we
should be constrained to believe that natural laws are capable of
being formulated in a particularly simple manner, and of course only
on condition that, from amongst all possible Galileian co-ordinate
systems, we should have chosen one (K[0]) of a particular state of
motion as our body of reference. We should then be justified (because
of its merits for the description of natural phenomena) in calling
this system " absolutely at rest," and all other Galileian systems K "
in motion." If, for instance, our embankment were the system K[0] then
our railway carriage would be a system K, relative to which less
simple laws would hold than with respect to K[0]. This diminished
simplicity would be due to the fact that the carriage K would be in
motion (i.e."really")with respect to K[0]. In the general laws of
nature which have been formulated with reference to K, the magnitude
and direction of the velocity of the carriage would necessarily play a
part. We should expect, for instance, that the note emitted by an
organpipe placed with its axis parallel to the direction of travel
would be different from that emitted if the axis of the pipe were
placed perpendicular to this direction.

Now in virtue of its motion in an orbit round the sun, our earth is
comparable with a railway carriage travelling with a velocity of about
30 kilometres per second. If the principle of relativity were not
valid we should therefore expect that the direction of motion of the
earth at any moment would enter into the laws of nature, and also that
physical systems in their behaviour would be dependent on the
orientation in space with respect to the earth. For owing to the
alteration in direction of the velocity of revolution of the earth in
the course of a year, the earth cannot be at rest relative to the
hypothetical system K[0] throughout the whole year. However, the most
careful observations have never revealed such anisotropic properties
in terrestrial physical space, i.e. a physical non-equivalence of
different directions. This is very powerful argument in favour of the
principle of relativity.



THE THEOREM OF THE
ADDITION OF VELOCITIES
EMPLOYED IN CLASSICAL MECHANICS


Let us suppose our old friend the railway carriage to be travelling
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