Also another symmetry in the theory of motion arises from the fact that rest in the points of ^S corresponds to uniform motion along a definite family of parallel straight lines in the space of a . We must note the three
Vi] Congruence 127
characteristics, (i) of the uniformity of the motion corresponding to any point of ^ along its correlated straight line in a, and (ii) of the equality in magnitude of the velocities along the various lines of a correlated to rest in the various points of ^3, 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 diffe- rent moments of one time-system. With this enlarged concept we say that a complete family of parallel levels in a time-system a is the complete family of levels in which the moments of a intersect the moments of ^S. This complete family of parallel levels is also evidently a family lying in the moments of the time-system /3. By introducing a third time-system y, 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 parallelo- grams 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.
128 THE CONCEPT OF NAi UKii L^n.
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 seg- ments 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 of a time-system ^ are moving with equal velocities in any other time-system a along parallel lines. Thus we can speak of the velocity in a due to the time-system ^ without specifying any particular point in j3. 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 parallelo-
grams 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 perpen- dicularity. 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,
Vi] Congruence 129
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 ^C be a pair of rectangular rects intersecting
r'mB and C, and AD and AE be another pair of rect-
angular rects intersecting r m D and E, then either D
or E lies in the segment EC 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 He 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 symmetry 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 a and ^ be the names of two time-systems. The directions of motion in the space of a due to rest in a point of ^ is called the ' ^-direction in a ' and the direction of motion in the space of ^ due to rest in a point of a is called the 'a -direction in j8.' Consider a motion in the space of a consisting of a certain velocity in the j8-direction of a and a certain velocity at right-angles to it. This motion represents rest in the space of another time-system —
130 THE CONCEPT OF NATUKii l^h.
A particular case of this axiom is that relative velocities are equal and opposite. Namely rest in a is represented in j8 by a velocity along the a -direction which is equal to the velocity along the j8- direction in a which repre- sents rest in ^.
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 a is congruent to the time-
unit of system /S, and the time-unit of system jS is
congruent to the time-unit of system y, then the time-
units of a and y are also congruent.
By means of these axioms formulae for the trans-
Vi] Congruence 131
formation of measurements made in one time-system
to measurements of the same facts of nature made in
another time-system can be deduced. These formulae
will be found to involve one arbitrary constant v^^hich
I vv^ill call k.
It is of the dimensions of the square of a velocity.
Accordingly four cases arise. In the first case k is
zero. This case produces nonsensical results in opposi-
tion to the elementary deliverances of experience. We
put this case aside.
In the second case k is infinite. This case yields the
ordinary formulae for transformation in relative motion,
namely those formulae vv^hich are to be found in every
elementary book on dynamics.
In the third case, k is negative. Let us call it — c^,
where c will be of the dimensions of a velocity. This
case yields the formulae of transformation which
Larmor discovered for the transformation of Maxwell's
equations of the electromagnetic field. These formulae
were extended by H. A. Lorentz, and used by Einstein
and Minkowski as the basis of their novel theory of
relativity. I am not now speaking of Einstein's more
recent theory of general relativity by which he deduces
his modification of the law of gravitation. If this be the
case which applies to nature, then c must be a close
approximation to the velocity of light in vacuo. Perhaps
it is this actual velocity. In this connexion 'in vacuo'
must not mean an absence of events, namely the absence
of the all-pervading ether of events. It must mean the
absence of certain types of objects.
In the fourth case, k is positive. Let us call it ¥■,
where A will be of the dimensions of a velocity. This gives
a perfectly possible type of transformation formulae,
132 THE CONCEPT OF NATURE ICH.
but not one which explains any facts of experience.
It has also another disadvantage. With the assumption
of this fourth case the distinction between space and
time becomes unduly blurred. The whole object of
these lectures has been to enforce the doctrine that
space and time spring from a common root, and that
the ultimate fact of experience is a space-time fact. But
after all mankind does distinguish very sharply between
space and time, and it is owing to this sharpness of
distinction that the doctrine of these lectures is some-
what of a paradox. Now in the third assumption this
sharpness of distinction is adequately preserved. There
is a fundamental distinction between the metrical pro-
perties of point -tracks and rects. But in the fourth
assumption this fundamental distinction vanishes.
Neither the third nor the fourth assumption can
agree with experience unless we assume that the
velocity c of the third assumption, and the velocity h
of the fourth assumption, are extremely large compared
to the velocities of ordinary experience. If this be the
case the formulae of both assumptions will obviously
reduce to a close approximation to the formulae of the
second assumption which are the ordinary formulae of
dynamical textbooks. For the sake of a name, I will
call these textbook formulae the 'orthodox' formulae.
There can be no question as to the general approxi-
mate correctness of the orthodox formulae. It would be
merely silly to raise doubts on this point. But the
determination of the status of these formulae is by no
means settled by this admission. The independence
of time and space is an unquestioned presupposition
of the orthodox thought which has produced the ortho-
dox formulae. With this presupposition and given the
Vi] Congruence
absolute points of one absolute space, the orthodox
formulae are immediate deductions. Accordingly,
these formulae are presented to our imaginations as
facts which cannot be otherwise, time and space being
what they are. The orthodox formulae have therefore
attained to the status of necessities which cannot be
questioned in science. Any attempt to replace these
formulae by others was to abandon the rdle of physical
explanation and to have recourse to mere mathematical
formulae.
But even in physical science difficulties have accumu-
lated round the orthodox formulae. In the first place
Maxwell's equations of the electromagnetic field are
not invariant for the transformations of the orthodox
formulae; whereas they are invariant for the trans-
formations of the formulae arising from the third of the
four cases mentioned above, provided that the velocity c
is identified with a famous electromagnetic constant
quantity.
Again the null results of the delicate experiments
to detect the earth's variations of motion through the
ether in its orbital path are explained immediately by
the formulae of the third case. But if we assume the
orthodox formulae we have to make a special and ar-
bitrary assumption as to the contraction of matter during
motion. I mean the Fitzgerald-Lorentz assumption.
Lastly Fresnel's coefficient of drag which represents
the variation of the velocity of light in a moving medium
is explained by the formulae of the third case, and
requires another arbitrary assumption if we use the
orthodox formulae.
It appears therefore that on the mere basis of physical explanation there are advantages in the formulae
134 THE CONCEPT OF NATURE [ch.