220

DOC. 52 GEOMETRY AND EXPERIENCE

244

CONTRIBUTIONS TO SCIENCE

spherical

surface, diametrically

opposite to

S,

there

is

a

lumi-

nous

point,

throwing

a

shadow L' of the disc L

upon

the

plane

E.

Every

point

on

the

sphere

has its shadow

on

the

plane.

If

the disc

on

the

sphere

K

is moved,

its shadow

L'

on

the

plane

E

also

moves.

When the disc L

is

at S,

it

almost

exactly

coincides

with its shadow. If

it

moves on

the

spherical

surface

away

from

S

upwards,

the disc shadow

L'

on

the

plane

also

moves

away

from

S

on

the

plane

outwards,

growing

bigger

and

bigger.

As

the disc L

approaches

the luminous

point

N,

the shadow

moves

off

to infinity,

and becomes

infinitely great.

Now

we

put

the

question:

what

are

the

laws

of

disposition

of the disc-shadows L'

on

the

plane

E?

Evidently they

are ex-

actly

the

same

as

the

laws

of

disposition

of the

discs

L

on

the

spherical

surface. For

to

each

original

figure on

K there

is

a

corresponding

shadow

figure

on

E. If

two

discs

on

K

are

touch-

ing,

their shadows

on

E

also

touch. The

shadow-geometry on

the

plane agrees

with the

disc-geometry on

the

sphere.

If

we

call

the disc-shadows

rigid

figures,

then

spherical geometry

holds

good

on

the

plane

E with

respect to

these

rigid

figures.

In

par-

ticular, the

plane is

finite with

respect

to

the

disc-shadows,

since

only

a

finite number of the shadows

can

find

room on

the

plane.

At this

point

somebody

will

say,

“That

is

nonsense.

The

disc-

shadows

are

not rigid

figures.

We have

only to

move

a

two-foot

rule about

on

the

plane

E

to

convince ourselves

that

the

shadows

constantly

increase in

size

as they move away

from

S

on

the

plane

toward

infinity.”

But what if the two-foot

rule

were

to

behave

on

the

plane

E in the

same

way

as

the

disc-shadows L'?

It would then be

impossible to

show that the shadows increase

in

size

as

they

move away

from

S;

such

an

assertion would then

no longer

have

any meaning

whatever.

In

fact the

only

objec-

tive assertion that

can

be made

about

the disc-shadows

is

just

this,

that

they are

related in

exactly

the

same

way as are

the

rigid

discs

on

the

spherical

surface in the

sense

of Euclidean

geome-

try.

We

must

carefully

bear in mind that

our statement as

to

the

growth

of the

disc-shadows,

as

they

move

away

from

S

toward

infinity,

has in

itself

no objective meaning,

as

long

as we are