Under these conditions, the _u_-curves and _v_-curves are straight
lines in the sense of Euclidean geometry, and they are perpendicular to
each other. Here the Gaussian coordinates are simply Cartesian ones. It
is clear that Gauss co-ordinates are nothing more than an association
of two sets of numbers with the points of the surface considered, of
such a nature that numerical values differing very slightly from each
other are associated with neighbouring points in space.
So far, these considerations hold for a continuum of two dimensions.
But the Gaussian method can be applied also to a continuum of three,
four or more dimensions. If, for instance, a continuum of four
dimensions be supposed available, we may represent it in the following
way. With every point of the continuum, we associate arbitrarily four
numbers, _x_1, _x_2, _x_3, _x_4, which are known as co-ordinates.
Adjacent points correspond to adjacent values of the coordinates. If a
distance _ds_ is associated with the adjacent points _P_ and _P _, this
distance being measurable and well defined from a physical point of
view, then the following formula holds:
_ds_2 = _g_11_dx_12 + 2_g_12_dx_1_dx_2 . . . . + _g_44_dx_42,
where the magnitudes _g_11, etc., have values which vary with the
position in the continuum. Only when the continuum is a Euclidean one
is it possible to associate the co-ordinates _x_1 . . _x_4. with the
points of the continuum so that we have simply
_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.
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 _ds_2 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
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