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.
XXVI.
THE SPACE-TIME CONTINUUM OF THE SPECIAL 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 XVII. 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 another, which is moving
uniformly with reference to the first, the equations of the Lorentz
transformation are valid. These last form the basis for the derivation
of deductions from the special theory of relativity, and in themselves
they are nothing more than the expression of the universal validity of
the law of transmission of light for all Galileian systems of
reference.
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