V. THE SOLUTION OF THE DIFFICULTY.

We have thus seen that
according to the facts,
the velocity of light
IS ALWAYS THE SAME,
whether the source of light
is stationary or moving
(See FACT (2) in Chapter IV.),
and whether the velocity of light
is measured
relatively to the medium in which it travels,
or relatively to a moving observer
(See Chapter IV.)

Let us express these facts algebraically,
for two observers, K and K',
who are moving with uniform velocity
relative to each other,
thus:
K writes x = ct,             (6)
and K' writes x' = ct',     (7)
both using
THE SAME VALUE FOR
THE VELOCITY OF LIGHT,
namely c,
and each using
his own measurements of
length, x and x',
and time, t and t', respectively.

It is assumed that
at the instant when
the rays of light start on their path,
K and K' are at the SAME place,
and the rays of light
radiate out from that place
in all directions.

Now according to equation (6),
K, who is unaware of his motion through the ether
(since he cannot measure it),
may claim he is at rest,
and that in time t,
K' must have moved to the right,
as shown in the figure below;
and that, in the meantime,
the light,
which travels out in all directions from K,
has reached points at
the distance ct from K,
and hence
all points on the circumference
of the circle having the radius ct.

But K' claims that he is the one
who has remained stationary,
and that K, on the contrary,
has moved TO THE LEFT;
furthermore that the light travels out
from K' as a center,
instead of from K!
And this is what he means
when he says
x' = ct'.
How can they both be right?

We may be willing
not to take sides
in their controversy regarding the question as to
which one has moved --
K' to the right or K to the left --
because either leads to the same result.
But what about the circles?
They cannot possibly have both K and K'
as their centers!
One of them must be right and the other wrong.
This is another way of stating
the APPARENT CONTRADICTION BETWEEN
FACTS (1) and (2) (see Ch. IV.).

Now, at last, we are ready
for the explanation.

Although K claims that
at the instant when
the light has reached C (see figure above),
it has also reached
the point A, on the other side,
still,
WE MUST REMEMBER THAT
when K says
two events happen simultaneously
(namely the arrival of the light at C and A),
K' DOES NOT AGREE
THAT THEY ARE SIMULTANEOUS (see Ch. III.).
So that when
K' says that
the arrival of the light at C and B
(rather than at C and A)
ARE SIMULTANEOUS,
his statement
DOES NOT CONTRADICT THAT OF K,
since K and K'
DO NOT MEAN THE SAME THING
WHEN THEY SAY
"SIMULTANEOUS:"
for
K's clocks at C and A
do not agree with K''s clocks at C and A.
Thus when the light arrives at A,
the reading of K's clock there
is exactly the same as that of K's clock at C
(K having set all clocks in his system
by the method of signals described in Ch. III.),
while
K''s clock at A,
when the light arrives there,
reads a LATER TIME than his clock at C
when the light arrived at C, --
so that K' maintains that
the light reaches A
LATER than it reaches C,
and NOT at the SAME instant,
as K claims.

Hence we see that
they are not really contradicting each other,
but they are merely using
two different systems of clocks,
such that
the clocks in each system
agree with each other alright,
but the clocks in the one system
have NOT been set
in agreement with the clocks
in the other system (see Ch. III.).

That is,
if we take into account
the inevitable necessity of
using signals
in order to set clocks which are
at a distance from each other,
and that the arrival of the signals
at their destinations
are influenced by
our state of motion,
of which we are not aware (Ch. III.)
it becomes clear that
THERE IS NO REAL CONTRADICTION HERE,
but only a difference of description
due to INEVITABLE differences
in the setting of
various systems of clocks.

We now see
in a general qualitative way,
that the situation is
not at all mysterious or unreasonable,
as it seemed to be at first.
But we must now find out
whether these considerations,
when applied QUANTITATIVELY,
actually agree with the experimental facts.

And now a pleasant surprise awaits us.