can treat the geometrical relationships in the surface, and thus pointed out the
way to the method of Riemman of treating multi-dimensional, non-Euclidean
continuum. Thus it is that mathematicians long ago solved the formal problems
to which we are led by the general postulate of relativity.
GAUSSIAN CO-ORDINATES
According to Gauss, this combined analytical and geometrical mode of handling
the problem can be arrived at in the following way. We imagine a system of
arbitrary curves (see Fig. 4) drawn on the surface of the table. These we
designate as u-curves, and we indicate each of them by means of a number. The
Curves u= 1, u= 2 and u= 3 are drawn in the diagram. Between the curves u= 1
and u= 2 we must imagine an infinitely large number to be drawn, all of which
correspond to real numbers lying between 1 and 2. fig. 04 We have then a system
of u-curves, and this "infinitely dense" system covers the whole surface of the
table. These u-curves must not intersect each other, and through each point of the
surface one and only one curve must pass. Thus a perfectly definite value of u
belongs to every point on the surface of the marble slab. In like manner we
imagine a system of v-curves drawn on the surface. These satisfy the same
conditions as the u-curves, they are provided with numbers in a corresponding
manner, and they may likewise be of arbitrary shape. It follows that a value of u
and a value of v belong to every point on the surface of the table. We call these
two numbers the co-ordinates of the surface of the table (Gaussian co-ordinates).
For example, the point P in the diagram has the Gaussian co-ordinates u= 3, v=
- Two neighbouring points P and P1 on the surface then correspond to the co-
ordinates
P: u,v
P1: u + du, v + dv,
where du and dv signify very small numbers. In a similar manner we may
indicate the distance (line-interval) between P and P1, as measured with a little
rod, by means of the very small number ds. Then according to Gauss we have
ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2