DOC. 42 SPECIAL AND GENERAL RELATIVITY
337
Euclidean
and
Non-Euclidean
Continuum
93
has
one
rod
in
common
with the
first.
We
proceed
in
like
manner
with each of these
squares
until
finally
the whole
marble slab
is
laid
out
with
squares.
The
arrangement
is such,
that
each side of
a
square belongs to two squares
and each
[51]
corner
to
four
squares.
[52]
It
is
a
veritable wonder
that
we can carry
out
this business
without
getting
into
the
greatest
difficulties. We
only
need
to
think
of the
following.
If
at any moment
three
squares meet at
a
corner,
then
two
sides of the fourth
square
are
already
laid,
and,
as
a
consequence,
the
arrangement
of the
remaining
two
sides of the
square
is
already
completely
determined. But
I
am
now
no longer
able
to adjust
the
quadrilateral
so
that
its
di-
agonals
may
be
equal.
If
they are equal
of
their
own
accord,
then this
is
an
especial
favour
of
the marble
slab
and
of
the
little
rods,
about which I
can
only
be
thankfully surprised.
We
must
needs
experience many
such
surprises
if the
construc-
tion
is to
be successful.
If
everything
has
really gone smoothly,
then
I
say
that
the
points
of the marble slab constitute
a
Euclidean continuum
with
respect to
the little
rod,
which
has
been used
as
a
"dis-
tance"
(line-interval).
By
choosing
one corner
of
a
square
as
"origin,"
I
can
characterize
every
other
corner
of
a
square
with
reference
to
this
origin by
means
of
two
numbers.
I
only
need
state
how
many
rods
I
must
pass
over
when,
starting
from
the
origin,
I
proceed
towards
the
"right"
and then
"upwards,"
in
order
to
arrive
at
the
corner
of the
square
under
consideration.
These
two
numbers
are
then the "Cartesian co-ordinates"
of
this
corner
with
reference
to
the "Cartesian co-ordinate
sys-
tem"
which
is
determined
by
the
arrangement
of
little
rods.