f a time-system {beta} are moving with equal velocities in any
other time-system {alpha} along parallel lines. Thus we can speak of the
velocity in {alpha} due to the time-system {beta} without specifying any
particular point in {beta}. The axiom also enables us to measure time in
any time-system; but does not enable us to compare times in different
time-systems.

The second axiom of congruence concerns parallelograms on congruent
bases and between the same parallels, which have also their other pairs
of sides parallel. The axiom asserts that the rect joining the two
event-particles of intersection of the diagonals is parallel to the rect
on which the bases lie. By the aid of this axiom it easily follows that
the diagonals of a parallelogram bisect each other.

Congruence is extended in any space beyond parallel rects to all rects
by two axioms depending on perpendicularity. The first of these axioms,
which is the third axiom of congruence, is that if ABC is a triangle
of rects in any moment and D is the middle event-particle of the base
BC, then the level through D perpendicular to BC contains A when
and only when AB is congruent to AC. This axiom evidently expresses
the symmetry of perpendicularity, and is the essence of the famous pons
asinorum expressed as an axiom.

The second axiom depending on perpendicularity, and the fourth axiom of
congruence, is that if r and A be a rect and an event-particle in the
same moment and AB and AC be a pair of rectangular rects intersecting r
in B and C, and AD and AE be another pair of rectangular rects
intersecting r in D and E, then either D or E lies in the segment BC and
the other one of the two does not lie in this segment. Also as a
particular case of this axiom, if AB be perpendicular to r and in
consequence AC be parallel to r, then D and E lie on opposite sides of B
respectively. By the aid of these two axioms the theory of congruence
can be extended so as to compare lengths of segments on any two rects.
Accordingly Euclidean metrical geometry in space is completely
established and lengths in the spaces of different time-systems are
comparable as the result of definite properties of nature which indicate
just that particular method of comparison.

The comparison of time-measurements in diverse time-systems requires two
other axioms. The first of these axioms, forming the fifth axiom of
congruence, will be called the axiom of 'kinetic symmetry.' It expresses
the symmet