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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