—such as e.g. the fourth root of a homogeneous differential expression of the fourth degree in these variables—which could provide a measure for the length of the line-element (vide [Note 9]). But at present there is no ground for abandoning the simplest general expression for the line-element (viz. that of the second degree), and adopting more complicated functions. Within the range (of fulfilment) of the two postulates, which we have imposed upon every description of physical events, the former expression for
satisfies all requirements. Nevertheless, it must never be forgotten that the choice of an analytical expression for the line-element always contains a hypothetical factor; and it is the duty of the physicist to remain fully conscious of this fact at all times, without being in any way prejudiced. It is for this reason that Riemann closes his essay with the following remarks, which impress one particularly with their great importance for the present time:[7]
[7]B. Riemann, Über die Hypothesen, welche der Geometrie zugrunde liegen. New edn., annotated by H. Weyl, Berlin: Springer & Co., 1919.
"The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space. In this question, which we may still regard as belonging to the doctrine of space, is found the application of the remark made above; that in a discrete[8] manifold, the principle or character of its metric relations is already given in the notion of the manifold, whereas in a continuous manifold this ground has to be found elsewhere, i.e. has to come from outside.
"Either, therefore, the reality which underlies space must form a discrete[9] manifold, or we must seek the ground of its metric relations (measure-conditions) outside it, in binding forces which act upon it.
"A decisive answer to these questions can be obtained only by starting from the conception of phenomena which has hitherto been justified by experience, and of which Newton laid the foundation, and then making in this conception the successive changes required by facts which admit of no explanation on the old theory; researches of this kind, which commence with general notions, cannot be other than useful in preventing the work from being hampered by too narrow views, and in keeping progress in the knowledge of the inter-connections of things from being checked by traditional prejudices.
"This carries us over into the sphere of another science, that of physics, into which the character and purpose of the present discussion will not allow us to enter."