,
, which may be regarded as the co-ordinates of a rectangular system of co-ordinates, and which distinguish it from all other points; a continuous variation of these three numbers enables us to specify every single point of space in turn. The assemblage of points in space represents, in Riemann's notation, "a multiply extended magnitude" (an
-fold manifoldness or manifold) between the single elements (points) of which a continuous transition is possible. We are familiar with diverse continuous manifolds, e.g. the system of colours, of tones and various others. A feature which is common to all of them is that, in order to specify a single element out of the entire manifold (to define a particular point, a particular colour, or a particular tone), a characteristic number of magnitude-determinations, i.e. co-ordinates, is required: this characteristic number is called the dimensions of the respective manifold. Its value is three for space, two for a plane, one for a line. The system of colours is a continuous manifold of the dimension three, corresponding to the three "primary" colours, red, green, and violet, by mixing which in due proportions every colour can be produced.
But the assumption of continuity for the transition from one element to another in the same manifold, and the determination of the dimensions of the latter, does not give us any information about the possibility of comparing limited parts of the same manifold with one another, e.g. about the possibility of comparing two tones with one another or two single colours; i.e. nothing has yet been stated about the metric relations (measure-conditions) of the manifold, about the nature of the scale, according to which measurements can be undertaken within the manifold. In order to be able to do this, we must allow experience to give us the facts from which to establish the metric (measure-) laws which hold for each particular manifold (space-points, colours, tones) under various physical conditions; these metric laws will be different according to the set of empirical facts chosen for this purpose.[3]
In the case of the manifold of space-points, experience has taught us that finite rigid point-systems can be freely moved in space without altering their form or dimensions; the conception of "congruence" which has been derived from this fact, has become a vital factor for a measure-determination.[4] It sets us the problem of building up a mathematical expression from the numbers
