with the
points of the continuum so that we have simply
ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.
In this case relations hold in the four-dimensional continuum which
are analogous to those holding in our three-dimensional measurements.
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.
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