322
DOC.
71
PRINCETON LECTURES
THE GENERAL THEORY
Plane
geometry
may
be based
upon
the
concept
of
the
[83]
distance
ds,
between
two
infinitely near points.
The
concept
of this
distance
is physically significant
because
the
distance
can
be
measured
directly
by means
of
a
rigid
measuring
rod.
By a
suitable
choice of
Cartesian
co-
ordinates this distance
may
be
expressed by
the formula
ds2
=
dx12
+
dx22.
We
may
base
upon
this
quantity
the
concepts
of
the
straight
line
as
the
geodesic
(ôfds
=
0),
the
interval,
the
circle,
and
the
angle, upon
which the Euclidean
plane geometry
is
built. A
geometry
may
be
developed
upon
another
continuously
curved
surface,
if
we
observe
that
an infinitesimally
small
portion
of the
surface
may
be
regarded
as
plane,
to
within
relatively
infinitesimal
quanti-
ties.
There
are
Cartesian
co-ordinates,
X1,
X2,
upon
such
a
small
portion
of the
surface,
and the
distance
between
two points,
measured
by
a
measuring rod,
is given
by
ds2
=
dX12
+
dX22
If
we
introduce
arbitrary
curvilinear
co-ordinates,
x1, x2,
on
the
surface,
then
dX1, dX2,
may
be
expressed
linearly
in
terms
of
dx1,
dx2.
Then
everywhere upon
the
surface
we
have
{1}
ds2
+
g11dx12
+
2g12dx1dx2
+
g22dx22
where
g11, g12, g22
are
determined
by
the
nature
of
the
surface and
the
choice of
co-ordinates;
if these
quantities
are
known,
then it
is
also known how networks of
rigid
rods
may
be
laid
upon
the
surface.
In other
words,
the
geometry
of surfaces
may
be based
upon
this
expression
for
ds2
exactly
as
plane
geometry is
based
upon
the
corre-
sponding
expression.
There
are
analogous
relations in the four-dimensional
space-time
continuum
of
physics.
In the immediate
[62]