However, the Gauss treatment for ds2 which we have given above is not always
possible. It is only possible when sufficiently small regions of the continuum
under consideration may be regarded as Euclidean continua. For example, this
obviously holds in the case of the marble slab of the table and local variation of
temperature. The temperature is practically constant for a small part of the slab,
and thus the geometrical behaviour of the rods is almost as it ought to be
according to the rules of Euclidean geometry. Hence the imperfections of the
construction of squares in the previous section do not show themselves clearly
until this construction is extended over a considerable portion of the surface of
the table.
We can sum this up as follows: Gauss invented a method for the mathematical
treatment of continua in general, in which " size-relations " (" distances "
between neighbouring points) are defined. To every point of a continuum are
assigned as many numbers (Gaussian coordinates) as the continuum has
dimensions. This is done in such a way, that only one meaning can be attached to
the assignment, and that numbers (Gaussian coordinates) which differ by an
indefinitely small amount are assigned to adjacent points. The Gaussian
coordinate system is a logical generalisation of the Cartesian co-ordinate system.
It is also applicable to non-Euclidean continua, but only when, with respect to
the defined "size" or "distance," small parts of the continuum under
consideration behave more nearly like a Euclidean system, the smaller the part
of the continuum under our notice.
THE SPACE-TIME CONTINUUM OF THE SPEICAL THEORY
OF RELATIVITY CONSIDERED AS A EUCLIDEAN
CONTINUUM
We are now in a position to formulate more exactly the idea of Minkowski,
which was only vaguely indicated in Section 17. In accordance with the special
theory of relativity, certain co-ordinate systems are given preference for the
description of the four-dimensional, space-time continuum. We called these "
Galileian co-ordinate systems." For these systems, the four co-ordinates x, y, z, t,
which determine an event or — in other words, a point of the four-dimensional
continuum — are defined physically in a simple manner, as set forth in detail in
the first part of this book. For the transition from one Galileian system to