_ds_2 = _g_11_du_2 + 2_g_12_du dv_ + _g_22_dv_2, where _g_11, _g_12, _g_22, are magnitudes which depend in a perfectly definite way on _u_ and _v_. The magnitudes _g_11, _g_12 and _g_22, determine the behaviour of the rods relative to the _u_-curves and _v_-curves, and thus also relative to the surface of the table. For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring-rods, but only in this case, it is possible to draw the _u_-curves and _v_-curves and to attach numbers to them, in such a manner, that we simply have: _ds_2 = _du_2 + _dv_2 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 themselves clearly until this construction is extended over a considerable portion of the surface of the table.