1 4 6 D O C . 3 1 I D E A S A N D M E T H O D S
curved surface are—when seen two-dimensionally—non-Euclidean even though
the little rod, in three-dimensional space, is supposed to play the role of a length in
the sense of Euclidean geometry in three dimensions.
It is to be well noted that Euclidean or non-Euclidean behavior is not a property
of the surface itself but rather one of certain measuring rodlets in reference to the
surface. The disk-type surface considered, e.g., sub 19), is non-Euclidean for co-
rotating measurement rodlets, but Euclidean for non-corotating ones. Geometric
statements always refer to the possibilities for the placement of rigid bodies.
Riemann extended these considerations to manifolds of three or more dimen-
sions. This is possible without difficulty if one considers that the entire analysis
given above is independent 〈and makes no use〉 of the assumption that the surface
considered is part of a three-dimensional Euclidean space. So let there be a three-
dimensional space with the property, in reference to rigid rodlets of a certain kind,
that the latter can only be arranged according to the laws of Euclidean geometry in
the infinitesimal, but not so in the finite domain. Then there are (in the finite) no
coordinates by means of which the elementary distance of neighboring points could
be represented with rodlets in accordance with the formula
.
However, if one characterizes the individual points of the space—under preserva-
tion of continuity—arbitrarily by means of three “Gaussian coordinates,” then the
distance squared expresses itself in the form
(14)
The quantities , etc., which themselves depend upon the coordinates ,
completely describe the possibilities of placing “rigid” measuring rodlets into the
space considered. The metric properties of all structures definable by our rodlets
within this space must be derivable from them by pure calculation.
The theoretical formulas 〈and results〉 obtainable in this manner are of great gen-
erality in their formulation insofar as they hold for arbitrary coordinates and not
just for a certain choice of coordinates (Cartesian coordinates) like those of Euclid-
ean geometry. If one introduces instead of the original Gaussian variables
arbitrary functions of these quantities as new coordinates,
one finds exactly corresponding formulas. Theorems formulated with the use of
ds2
dx1
2
dx2
2
dx3
2
+ + =
ds2 g11dx1
2
g22dx2
2
g33dx3
2
+
+ 2g13dx1dx3 2g23dx2dx3.
+ +2g12dx1dx2
+
=
[p. 32]
g11 x1, x2, x3
x1, x2, x3 x′1, x′2, x′3