DOC. 52 GEOMETRY AND EXPERIENCE 219

GEOMETRY AND EXPERIENCE

243

seems

to promise

success

at

the

outset,

and

the

smaller the

radius

of the disc

in

proportion to

that

of the

sphere,

the

more promis-

ing

it

seems.

But

as

the

construction

progresses

it

becomes

more

and

more

patent

that

the

arrangement

of the

discs

in the

manner

indicated,

without

interruption,

is

not

possible, as

it

should

be

possible by

the

Euclidean

geometry

of

the

plane.

In

this

way creatures

which

cannot

leave the

spherical

surface,

and

cannot

even peep

out

from the

spherical

surface

into

three-

dimensional

space,

might discover,

merely by

experimenting

with

discs,

that

their

two-dimensional

“space”

is not

Euclidean,

but

spherical space.

From the latest results of the

theory

of

relativity

it

is probable

[33]

that

our

three-dimensional

space

is

also

approximately

spherical,

that

is,

that the

laws

of

disposition

of

rigid

bodies in

it

are

not

given by

Euclidean

geometry,

but

approximately

by spherical

geometry,

if

only

we

consider

parts

of

space

which

are

suffi-

ciently

extended. Now this

is

the

place

where the reader’s

imagination

boggles. “Nobody

can

imagine

this

thing,”

he cries

indignantly.

“It

can

be

said,

but

cannot

be

thought.

I

can

imagine

a

spherical

surface well

enough,

but

nothing

analogous

to

it

in

three

dimensions.”

We

must try

to surmount

this

barrier in

the mind, and the

patient

reader will

see

that it

is by

no means

a

particularly diffi-

cult

task. For this

purpose

we

will first

give our

attention

once

more

to

the

geometry

of two-dimensional

spherical

surfaces.

In the

adjoining

figure

let

K be the

spherical

surface,

touched

at

S

by a plane,

E,

which,

for

facility

of

presentation,

is

shown

in the

drawing

as a

bounded

surface. Let L be

a

disc

on

the

spherical

surface. Now

let

us imagine

that

at

the

point

N of the

FIG. 2