ry of the quantitative relations between two time-systems when
the times and lengths in the two systems are measured in congruent
units.

The axiom can be explained as follows: Let {alpha} and {beta} be the names
of two time-systems. The directions of motion in the space of {alpha} due
to rest in a point of {beta} is called the '{beta}-direction in {alpha}'
and the direction of motion in the space of {beta} due to rest in a
point of {alpha} is called the '{alpha}-direction in {beta}.' Consider a
motion in the space of {alpha} consisting of a certain velocity in the
{beta}-direction of {alpha} and a certain velocity at right-angles to
it. This motion represents rest in the space of another time-system--call
it {pi}. Rest in {pi} will also be represented in the space of {beta} by
a certain velocity in the {alpha}-direction in {beta} and a certain
velocity at right-angles to this {alpha}-direction. Thus a certain
motion in the space of {alpha} is correlated to a certain motion in the
space of {beta}, as both representing the same fact which can also be
represented by rest in {pi}. Now another time-system, which I will name
{sigma}, can be found which is such that rest in its space is
represented by the same magnitudes of velocities along and perpendicular
to the {alpha}-direction in {beta} as those velocities in {alpha}, along
and perpendicular to the {beta}-direction, which represent rest in {pi}.
The required axiom of kinetic symmetry is that rest in {sigma} will be
represented in {alpha} by the same velocities along and perpendicular to
the {beta}-direction in {alpha} as those velocities in {beta} along and
perpendicular to the {alpha}-direction which represent rest in {pi}.

A particular case of this axiom is that relative velocities are equal
and opposite. Namely rest in {alpha} is represented in {beta} by a
velocity along the {alpha}-direction which is equal to the velocity
along the {beta}-direction in {alpha} which represents rest in {beta}.

Finally the sixth axiom of congruence is that the relation of congruence
is transitive. So far as this axiom applies to space, it is superfluous.
For the property follows from our previous axioms. It is however
necessary for time as a supplement to the axiom of kinetic symmetry. The
meaning of the axiom is that if the time-unit of system {alpha} is
congruent to the time-unit of system {beta}, and the time-unit of system
{beta} is congruent to the time-unit of system {gamma},