1 4 4 D O C . 3 1 I D E A S A N D M E T H O D S
The abstraction of basic geometric concepts (straight lines, distances, etc.), from
objects of experience whose abstracted images they are, serves a systematic interest
but should not delude us; geometry, after all, is created in order to teach us about
the behavior of things of daily experience. If there were no practically rigid bodies
that can be brought into congruence with each other, we would not talk about con-
gruence of lengths, triangles, etc. It is obvious that geometry obtains meaning for
the physicist only by the fact that he associates natural things to these basic con-
cepts, e.g., to the concept of distance between two marks affixed to a practically
rigid body. In turn, Euclidean geometry becomes an experimental science in the
proper sense due to this association, just like mechanics. Its theorem then can be
confirmed or disproven in the same sense as those of mechanics. Our previously
found result, that Euclidean geometry does not obtain on a rotating circular disk, is
to be interpreted in this sense.
Euclidean geometry became important for physics primarily in the analytical
form we owe to Descartes. This transformation of Euclid’s system was possible
through the discovery of the Cartesian coordinate system whose physical interpre-
tation we have considered already. Its introduction (e.g., in plane geometry) is pos-
sible by covering the plane with a net of squares whose sides consist of identical
rodlets that can be brought into mutual congruence. That means one can, beginning
at an arbitrary point in the grid, completely characterize any other point in the grid
by two numbers (coordinates). Thus, coordinates have an immediate physical
meaning.
Gauss now posed the problem to establish in an analogous manner an analytical
geometry on an arbitrarily given curved surface. This requires, first of all, charac-
terizing all points on the surface by means of numbers (coordinates). But now it be-
came obvious that there is no such quadratic grid to define Cartesian coordinates.
The reason being that on a curved surface the laws for the placement of rigid little
rods are not provided by the Euclidean line segment geometry of the plane. Carte-
sian surface coordinates therefore do not exist on a curved surface (e.g., an ellip-
soidal or spherical surface).
However, what still exists on a curved surface is the distance of neighboring
points, as it can be measured by rigid measuring rods. Geometry on the curved sur-
face must be based upon this concept, except that the simple relation
, pertaining to Cartesian coordinate differentials and elementary
distance, is dropped on curved surfaces.
The possibility of introducing Cartesian coordinates in the plane is based on the
existence of a system of mutually orthogonal and parallel straight lines that can be
numbered in sequence of distance. However, a curved surface has no 〈completely
adequate〉 analog of parallel straight lines. Therefore, Gauss used for the definition
[p. 30]
ds
ds2 dx2 dy2 + =