346

DOCS.

355,

356

JUNE

1917

from

the

improbability

that

all

of

the

light rays

emitted

from

the

sun

intersect

one

another

again

at

the

antipodal

point.-In

this

elliptic

space, however,

there

are

no

negative

parallaxes.[11]

For

the

parallax is

n = a

cot

r/R

(better,

tan

n

=

sina/R

.cotr/R).

Also in

system

B

it would have to be assumed

that

r

1/2nrR.

In

this

system,

the

parallax

has

a

minimum

n0

= a/R.

The

establishment of

a

[lower]

limit

for

R

from

the

nonexistence of

negative

parallaxes is

thus

also eliminated.

With

cordial

regards, yours,

W. de Sitter.

356. To

Willem

de

Sitter

[Berlin,]

22

June

1917

Dear de

Sitter,

Your detailed

letter

pleased

me

very

much because

I

see

how

deeply you

have

been

thinking

about the

problem

of

mutual

interest

to

us.

I

have

just

a

few

comments to add.[1]

1)

My opinion

is

not

that the

sphere

must

approximate

the

world

well.

The

system

could

actually

be

quite irregularly curved,

also

on a

large scale,

that

is,

it

could

relate

to

the

spherical

world

like

a

potato’s

surface

to

a

sphere’s

surface.

Large

parts

of it could

then

really

be void

(without

matter).

The

sphere

only

serves

to

show,

through

an

idealization,

that

a

spatially

closed

(finite)

system

is

possible.[2]

If,

therefore,

coordinated

Milky Way systems really

do

exist

(a

view

which,

as

far

as

I

know,

not all astronomers

share),

there

does

not

have

to

be

matter

in

the

space

between

them, by any

means.

It

is not

necessary

to

assume

that

matter

in

any

form

other than that

of the

stars

exists.

However,

it

is

necessary

to

assume

that the

world

is

considerably bigger

than the

Milky

Way’s

104

light-y[ears].

2)

I

do

not

quite

know

what

you

mean

by finitude

and

boundedness;

I

think

of

it

as

follows.

Between two

points,

A

and

B,

there

are

space-like

geodetic

paths;

the

length of

the shortest

of these

paths is

lAB.

Now,

if

a

number

G

exists such

that

for

any

choice of A

and

B

lAB

G,

I call

the

world

(spatially)

finite.

It

is

probably always possible

to

regard

such

a

continuum

as

closed

in

my

sense.

The

naturally

measured volume

is

then

finite

(strictly speaking,

one can

only

talk about the

world

being

closed if it

is “static”).

3)

When

I

say

that

your

world has

a

preferred center,[3]

I

mean

that the

points

are

not

of

equal

value, apart

from in

their

coordinate

system

arrangement.