DOC. 52 GEOMETRY AND EXPERIENCE 221

GEOMETRY

AND

EXPERIENCE

245

unable

to compare

the disc-shadows

with Euclidean

rigid

bodies

which

can

be moved

about

on

the

plane

E. In

respect

of the

laws of

disposition

of the shadows

L',

the

point S

has

no special

privileges

on

the

plane any

more

than

on

the

spherical

surface.

The

representation given

above of

spherical

geometry

on

the

plane is important

for

us,

because

it

readily

allows itself

to

be

transferred

to

the three-dimensional

case.

Let

us imagine a

point S

of

our space,

and

a

great

number

of small

spheres,

L', which

can

all be

brought

to

coincide

with

one

another. But these

spheres are

not to

be

rigid

in the

sense

of Euclidean

geometry;

their

radius

is to

increase

(in

the

sense

of Euclidean

geometry)

when

they

are

moved

away

from

S to-

ward

infinity;

it

is to

increase

according to

the

same

law

as

the

radii

of the disc-shadows L'

on

the

plane.

After

having

gained

a

vivid mental

image

of the

geometrical

behavior

of

our

L'

spheres,

let

us assume

that in

our

space

there

are no

rigid

bodies

at

all

in

the

sense

of Euclidean

geometry,

but

only

bodies

having

the

behavior

of

our

L'

spheres.

Then

we

shall have

a

clear

picture

of

three-dimensional

spherical space,

or,

rather

of

three-dimensional

spherical geometry.

Here

our

spheres must

be called

“rigid”

spheres.

Their

increase in

size

as

they depart

from

S

is

not to

be detected

by

measuring

with

[34]

measuring-rods, any more

than

in the

case

of the disc-shadows

on

E,

because the standards

of

measurement

behave in the

same

way

as

the

spheres. Space

is homogeneous,

that

is

to

say,

the

same spherical configurations are possible

in

the

neighborhood

of

every point.*

Our

space

is finite, because,

in

consequence

of

the

“growth”

of the

spheres, only a

finite

number

of them

can

find

room

in

space.

In this

way,

by using

as a

crutch the

practice

in

thinking

and

visualization

which Euclidean

geometry gives us,

we

have

ac-

quired

a

mental

picture

of

spherical

geometry.

We

may

without

difficulty

impart

more

depth

and

vigor

to

these ideas

by

carry-

ing

out special imaginary

constructions. Nor would

it be

diffi-

cult

to represent

the

case

of what

is

called

elliptical geometry

in

[35]

*

This

is

intelligible

without calculation-but

only

for the two-dimensional

case-if

we

revert once

more

to

the

case

of the disc

on

the surface

of

the

sphere.