, which are assigned to two definite points in space, and which we may imagine as the end-points of a rigid measuring rod, such that this expression may be regarded as a measure of the distance between them, that is, as an expression for the length of the rod, and may be introduced as such into the formulae expressing physical laws.
The equations of physical laws, which—in order to fulfil the conditions of continuity—must be differential laws, contain only the distances
, of infinitely near points, so-called line-elements. We must, therefore, inquire whether our two postulates of [§ 2] have any influence upon the analytical expression for the line-element
, and, if so, which expression for the latter is compatible with both. Riemann demands of a line-element in the first place that it can be compared in respect to its length with every other line-element independently of its position and direction. This is a distinguishing characteristic of the metric conditions ("measure relations") prevalent in space; in practice it denotes that the rods must be freely movable. This peculiarity does not exist, for instance, in the manifold of tones or in that of colours (vide [Note 7]). Riemann formulates this condition in the words, "that lines must have a length independent of their position and that every line is to be measurable by means of any other." He then discovers that: if
,
