DOC. 52 GEOMETRY AND EXPERIENCE 213

GEOMETRY

AND

EXPERIENCE

237

physics.

But

it

is

my

conviction

that in the

present

stage

of

development

of theoretical

physics

these

concepts

must

still be

employed

as

independent

concepts;

for

we are

still far from

[16]

possessing

such certain

knowledge

of the

theoretical

principles

of atomic

structure

as

to

be able

to construct

solid bodies and

clocks

theoretically

from

elementary

concepts.

[17]

Further,

as

to

the

objection

that there

are no

really

rigid

bodies

in

nature,

and that therefore the

properties predicated

of

rigid

bodies do

not apply

to

physical

reality-this

objection

is

by no means so

radical

as

might

appear

from

a

hasty

examina-

tion.

For

it

is not

a

difficult task

to

determine the

physical

state

of

a

measuring-body

so

accurately

that its

behavior

relative

to

other

measuring-bodies

shall be

sufficiently

free from

ambiguity

to

allow

it

to

be

substituted

for the

“rigid”

body.

It

is to

meas-

uring-bodies

of this

kind

that

statements

about

rigid

bodies

must

be

referred.

All

practical geometry

is

based

upon

a

principle

which

is

[18]

accessible

to experience,

and which

we

will

now try

to

realize.

Suppose

two

marks have been

put

upon

a

practically-rigid

body.

A

pair

of

two

such marks

we

shall call

a

tract.

We

imagine

two

practically-rigid

bodies,

each with

a

tract

marked

out

on

it.

These

two tracts

are

said

to

be

“equal to

one

another” if the

marks of the

one

tract

can

be

brought

to

coincide

permanently

with

the marks of the other. We

now assume

that:

If

two tracts

are

found

to

be

equal

once

and

anywhere, they

are

equal always

and

everywhere.

[19]

Not

only

the

practical geometry

of

Euclid,

but also its

nearest

generalization,

the

practical

geometry

of Riemann, and there-

with the

general

theory

of

relativity, rest upon

this

assumption.

[20]

Of the

experimental

reasons

which

warrant

this

assumption

I

will

mention

only one.

The

phenomenon

of the

propagation

of

[p.

128]

light

in

empty space

assigns

a

tract, namely,

the

appropriate path

of

light, to

each

interval

of local

time,

and

conversely.

Thence it

follows that the above

assumption

for

tracts must

also

hold

good

for

intervals

of clock-time in the

theory

of

relativity.

Conse-

quently

it

may

be

formulated

as

follows: if

two

ideal

clocks

are

going at

the

same

rate at any

time and

at any

place

(being