216 DOC. 52 GEOMETRY AND EXPERIENCE

240

CONTRIBUTIONS TO SCIENCE

that

part

of

the

universe which

is

accessible

to

our

observation.

This

hope

is

illusory.

The distribution of the visible

stars

is

extremely irregular,

so

that

we on no

account may venture to

[p.

130]

set

the

average density

of

star-matter

in the universe

equal

to,

let

us

say,

the

average density

in the

Galaxy.

In

any

case,

how-

ever

great

the

space

examined

may

be,

we

could

not

feel

con-

vinced that

there

were

any

more stars

beyond

that

space.

So

it

seems

impossible to

estimate the

average density.

But there

is

another

road,

which

seems

to

me more prac-

ticable, although

it

also

presents

great

difficulties. For if

we

inquire

into

the deviations of the

consequences

of the

general

theory

of

relativity

which

are

accessible

to

experience,

from the

consequences

of the Newtonian

theory,

we

first of

all find

a

deviation

which manifests itself in close

proximity to

gravitat-

ing

mass,

and

has been confirmed

in

the

case

of

the

planet

Mer-

cury.

But if the universe

is

spatially

finite,

there

is

a

second

deviation

from the Newtonian

theory,

which,

in

the

language

of

the Newtonian

theory, may

be

expressed

thus: the

gravita-

tional

field

is

such

as

if it

were

produced, not

only by

the

ponderable

masses,

but in

addition

by

a

mass-density

of

negative

sign,

distributed

uniformly throughout

space.

Since this ficti-

tious

mass-density

would have

to

be

extremely

small,

it would

be

noticeable

only

in

very

extensive

gravitating

systems.

Assuming

that

we

know,

let

us say,

the statistical

distribution

and

the

masses

of the

stars

in the

Galaxy,

then

by

Newton’s law

we can

calculate the

gravitational

field

and the

average

velocities

which

the

stars must

have,

so

that the

Galaxy

should

not

col-

lapse

under the

mutual attraction

of its

stars,

but should

main-

tain

its actual

extent.

Now if the actual velocities of the

stars-

which

can

be

measured-were smaller than the calculated

velocities,

we

should

have

a

proof

that the

actual attractions

at

great

distances

are

smaller than

by

Newton’s law. From such

a

deviation it

could be

proved

indirectly

that the universe

is

finite. It would

even

be

possible to

estimate its

spatial

dimen-

sions.

[31]

Can

we

visualize

a

three-dimensional universe

which is

finite,

yet

unbounded?

[30]

[29]