slab, provided that the ends of one of our rods could be made to
coincide with these two points; for how else should we define the
distance without our proceeding being in the highest measure grossly
arbitrary? The method of Cartesian coordinates must then be discarded,
and replaced by another which does not assume the validity of Euclidean
geometry for rigid bodies.[20] The reader will notice that the
situation depicted here corresponds to the one brought about by the
general postulate of relativity (Section XXIII).
[20] Mathematicians have been confronted with our problem in the
following form. If we are given a surface (_e.g._ an ellipsoid) in
Euclidean three-dimensional space, then there exists for this surface
a two-dimensional geometry, just as much as for a plane surface. Gauss
undertook the task of treating this two-dimensional geometry from
first principles, without making use of the fact that the surface
belongs to a Euclidean continuum of three dimensions. If we imagine
constructions to be made with rigid rods _in the surface_ (similar to
that above with the marble slab), we should find that different laws
hold for these from those resulting on the basis of Euclidean plane
geometry. The surface is not a Euclidean continuum with respect to the
rods, and we cannot define Cartesian co-ordinates _in the surface_.
Gauss indicated the principles according to which we can treat the
geometrical relationships in the surface, and thus pointed out the way
to the method of Riemann of treating multi-dimensional, non-Euclidean
_continuum_. Thus it is that mathematicians long ago solved the formal
problems to which we are led by the general postulate of relativity.
XXV.
GAUSSIAN CO-ORDINATES
image033
According to Gauss, this combined analytical and geometrical mode of
handling the problem can be arrived at in the following way. We imagine
a system of arbitrary curves (see Fig. 4) drawn on the surface of the
table. These we designate as _u_-curves, and we indicate each of them
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