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DOC. 52 GEOMETRY AND EXPERIENCE

242

CONTRIBUTIONS TO SCIENCE

side of

any

square

has the side of

another

square

adjacent

to

it.

The construction

is

never

finished;

we

can always

go

on laying

squares-if

their

laws

of

disposition

correspond to

those

of

plane

figures

of

Euclidean

geometry.

The

plane

is

therefore infinite

in relation

to

the

cardboard

squares. Accordingly we

say

that

the

plane

is

an

infinite continuum

of

two

dimensions,

and

space

an

infinite continuum

of three dimensions. What

is

here

meant

by

the

number

of

dimensions,

I think I

may assume

to

be

known.

Now

we

take

an example

of

a

two-dimensional

continuum

which

is finite,

but

unbounded. We

imagine

the surface of

a

large globe

and

a

quantity

of small

paper discs,

all of the

same

size.

We

place one

of the discs

anywhere on

the surface

of the

globe.

If

we move

the disc about,

anywhere

we

like,

on

the

surface of the

globe,

we

do

not

come

upon

a

boundary

any-

where

on

the

journey.

Therefore

we

say

that

the

spherical

sur-

face of the

globe

is

an

unbounded continuum.

Moreover,

the

spherical

surface

is

a

finite

continuum.

For if

we

stick

the

paper

discs

on

the

globe,

so

that

no

disc

overlaps

another, the surface

of the

globe

will

finally

become

so

full that there

is

no room

for

another

disc.

This

means exactly

that

the

spherical

surface of

the

globe

is

finite in relation

to

the

paper

discs.

Further,

the

spherical

surface

is

a

non-Euclidean

continuum

of

two

dimen-

sions,

that

is to

say,

the laws of

disposition

for the

rigid

figures

lying

in it

do

not agree

with

those of the

Euclidean

plane.

This

can

be shown

in

the

following

way.

Take

a

disc and surround

it

in

a

circle

by

six

more

discs,

each of which

is

to

be surrounded

in

turn by

six

discs,

and

so on.

If this

construction

is

made

on

a

plane

surface,

we

obtain

an

uninterrupted arrangement

in

which

there

are

six discs

touching

every

disc

except

those

which

lie

on

the outside. On

the

spherical

surface

the

construction

also

FIG.

1