D O C . 3 1 I D E A S A N D M E T H O D S 1 4 5
of coordinates, instead of two systems of parallel straight lines, two systems of ar-
bitrary sets of curves with the sole requirement that only one curve of each set
should go through each point of the surface. The curves of each set are numbered
such that 〈continuously〉 neighboring curves get neighboring numbers. Then, two
curves run through every point of the surface, one curve from each set, and their
numbers are called its “coordinates” . This Gaussian coordinate system
is—speaking graphically—nothing other than an arbitrarily deformed and
stretched planar Cartesian coordinate system. It is by virtue of this bending that
Gaussian coordinates no longer have any physical meaning whatsoever. They are
not more than a completely arbitrary numbering of points of the surface so that,
however, continuity is preserved.
Nevertheless, the distance between two neighboring points on the
surface is expressable in a certain law of the coordinate differences . After
all, an infinitesimal piece of surface can always be viewed in first-order approxi-
mation as a piece of a plane; and there exists for the element a locally Cartesian
coordinate system such that
.
If one replaces these locally Cartesian coordinates with Gaussian coordinates
, one obtains by a simple reflection
. (13)
The etc. are here functions of and which are determined, on the one
hand, by the choice of the Gaussian coordinates, and on the other hand also by the
choice of the surface whose geometrical laws we want to study. If the functions
etc. are known, then they also define the metric properties of the surface.
Equation (13) is the Gaussian generalization of the Euclidean-Pythagorean the-
orem. In case of a plane and for Cartesian coordinates ,
it takes the characteristic form
of the Euclidean plane.
For the entire consideration, it is immaterial that the surface considered is part
of the three-dimensional space and—when looked at in that space—is curved. Es-
sential is only that we have before us a two-dimensional continuum whose laws of
measurement in the infinitesimal are Euclidean but deviate from this norm in the
finite domain. The metric behavior of the infinitesimal and rigid little rods on a
x1, x2) (
ds P1, P2
dx1, dx2
X1, X2) (
ds2
dX1
2
dX2
2
+ =
x1, x2) (
ds2 g11dx1
2
2g12dx1dx2 g22dx2
2
+ + =
g11 x1 x2
[p. 31]
g11
g11 g22 1; g12 0) = = = (
ds2 dx1
2 dx22
+ =