264 DOC. 71 PRINCETON LECTURES
PRE-RELATIVITY PHYSICS
with their relation
to
the
solid bodies of
experience,
then
it
is easy
to
say
what
we
mean by
the
three-dimensionality
of
space;
to
each
point
three
numbers,
x1, x2, x3
(co-ordi-
nates), may
be
associated,
in such
a way
that
this associa-
tion
is
uniquely reciprocal,
and that
x1, x2,
and
x3
vary
continuously
when
the
point
describes
a
continuous
series
of
points
(a
line).
It
is
assumed in
pre-relativity
physics
that the
laws of
the
configuration
of
ideal
rigid
bodies
are
consistent with
Euclidean
geometry.
What
this
means may
be
expressed
as
follows:
Two
points
marked
on a
rigid body
form
an
interval.
Such
an
interval
can
be
oriented
at
rest,
rela-
tively
to
our
space
of
reference,
in
a
multiplicity
of
ways.
If, now,
the
points
of this
space
can
be
referred
to
co-ordi-
nates
x1, x2, x3,
in such
a way
that
the differences of the
co-ordinates,
Ax1, Ax2, Ax3,
of the
two
ends of
the interval
furnish the
same sum
of
squares,
(1) s2
=
Ax12
+
Ax22
+
Ax32
for
every
orientation
of
the
interval,
then
the
space
of
reference
is
called
Euclidean,
and the co-ordinates Car-
tesian.* It
is
sufficient,
indeed,
to
make
this
assumption
[6]
in
the limit
for
an infinitely
small interval.
Involved
in this
assumption
there
are some
which
are
rather
less
special,
to
which
we
must
call
attention
on
account
of
their fundamental
significance.
In the
first
place,
it
is
assumed
that
one can
move
an
ideal
rigid body
in
an
arbitrary
manner.
In the
second
place,
it
is
assumed
that the behaviour
of ideal
rigid
bodies
towards orienta-
[7]
tion
is
independent
of the
material
of
the
bodies
and their
changes
of
position,
in the
sense
that
if
two
intervals
can
*
This relation
must
hold for
an
arbitrary
choice of the
origin
and of the
direction
(ratios
Ax1
:
Ax2
:
Ax3)
of the interval.
[4]