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]