214 DOC. 52 GEOMETRY AND EXPERIENCE
238
CONTRIBUTIONS TO SCIENCE
then
in
immediate
proximity to
each
other),
they
will
always go
at
the
same
rate,
no
matter
where
and
when
they are again
compared
with
each other
at
one
place.
If
this
law
were
not
valid for natural
clocks,
the
proper
frequencies
for the
separate
atoms
of the
same
chemical element would
not
be
in
such
exact
[21]
agreement
as experience
demonstrates.
The
existence of
sharp
spectral
lines
is
a convincing
experimental proof
of
the
above-
mentioned
principle
of
practical geometry.
This,
in
the last
[22]
analysis,
is
the
reason
which enables
us
to
speak
meaningfully
of
a
Riemannian
metric of the four-dimensional
space-time
con-
tinuum.
According
to
the view advocated
here,
the
question
whether
this continuum has
a
Euclidean,
Riemannian,
or
any
other
structure is
a
question
of
physics
proper
which
must
be answered
by experience,
and
not
a
question
of
a
convention
to
be chosen
on
grounds
of
mere expediency.
Riemann’s
geometry
will
hold
if the
laws
of
disposition
of
practically-rigid
bodies
approach
those of Euclidean
geometry
the
more
closely
the smaller
the
dimensions of the
region
of
space-time
under consideration.
It
is true
that this
proposed
physical
interpretation
of
geome-
try
breaks down when
applied immediately to
spaces
of sub-
molecular
order of
magnitude.
But
nevertheless,
even
in
ques-
tions
as
to
the
constitution
of
elementary
particles,
it
retains
part
of
its
significance.
For
even
when it
is
a
question
of describ-
ing
the electrical
elementary
particles
constituting matter,
the
attempt
may
still be made
to
ascribe
physical
meaning
to
those
field
concepts
which have been
physically
defined for the
pur-
pose
of
describing
the
geometrical
behavior
of bodies which
are
large
as
compared
with
the
molecule.
Success alone
can
decide
as
to
the
justification
of such
an
attempt,
which
postulates physi-
cal
reality
for
the fundamental
principles
of
Riemann’s
geome-
try
outside of the domain of their
physical
definitions. It
might
possibly turn
out
that this
extrapolation
has
no
better
warrant
than
the
extrapolation
of the
concept
of
temperature to
parts
of
[24] a
body
of molecular
order
of
magnitude.
It
appears
less problematical
to
extend
the
concepts
of
prac-
tical
geometry
to
spaces
of cosmic
order
of
magnitude.
It
might,
[23]
[p.
129]