other symmetry in the theory of motion arises from the fact that
rest in the points of {beta} corresponds to uniform motion along a
definite family of parallel straight lines in the space of {alpha}. We
must note the three characteristics, (i) of the uniformity of the motion
corresponding to any point of {beta} along its correlated straight line
in {alpha}, and (ii) of the equality in magnitude of the velocities
along the various lines of {alpha} correlated to rest in the various
points of {beta}, and (iii) of the parallelism of the lines of this
family.

We are now in possession of a theory of parallels and a theory of
perpendiculars and a theory of motion, and from these theories the
theory of congruence can be constructed. It will be remembered that a
family of parallel levels in any moment is the family of levels in which
that moment is intersected by the family of moments of some other
time-system. Also a family of parallel moments is the family of moments
of some one time-system. Thus we can enlarge our concept of a family of
parallel levels so as to include levels in different moments of one
time-system. With this enlarged concept we say that a complete family of
parallel levels in a time-system {alpha} is the complete family of
levels in which the moments of {alpha} intersect the moments of {beta}.
This complete family of parallel levels is also evidently a family lying
in the moments of the time-system {beta}. By introducing a third
time-system {gamma}, parallel rects are obtained. Also all the points of
any one time-system form a family of parallel point-tracks. Thus there
are three types of parallelograms in the four-dimensional manifold of
event-particles.

In parallelograms of the first type the two pairs of parallel sides are
both of them pairs of rects. In parallelograms of the second type one
pair of parallel sides is a pair of rects and the other pair is a pair
of point-tracks. In parallelograms of the third type the two pairs of
parallel sides are both of them pairs of point-tracks.

The first axiom of congruence is that the opposite sides of any
parallelogram are congruent. This axiom enables us to compare the
lengths of any two segments either respectively on parallel rects or on
the same rect. Also it enables us to compare the lengths of any two
segments either respectively on parallel point-tracks or on the same
point-track. It follows from this axiom that two objects at rest in any
two points o