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 co-ordinates. If a distance $d s$ 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: $d s^{2} = g_{11} {d x_{1}}^{2} + 2 g_{12} d x_{1} d x_{2} + \dots + g_{44} {d x_{4}}^{2},$ 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 $d s^{2} = {d x_{1}}^{2} + {d x_{2}}^{2} + {d x_{3}}^{2} + {d x_{4}}^{2} .$ 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 $d s^{2}$ 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.

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 co-ordinates) 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 co-ordinates)

which differ by an indefinitely small amount are assigned to adjacent points. The Gaussian co-ordinate 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.