must be taken as standing for the inertial mass of an amount of energy
.
[Note 6] (p. 29). The facts that every pair of points (point-pair) in space have the same magnitude-relation (viz. the same expression for the mutual distance between them) and that with the aid of this relation, every point-pair can be compared with every other, constitute the characteristic feature which distinguishes space from the remaining continuous manifolds which are known to us. We measure the mutual distance between two points on the floor of a room, and the mutual distance between two points which he vertically above one another on the wall, with the same measuring-scale, which we thus apply in any direction at pleasure. This enables us to "compare" the mutual distance of a point-pair on the floor with the mutual distance of any other pair of points on the wall.
In the system of tones, on the contrary, quite different conditions prevail. The system of tones represents a manifold of two dimensions, if one distinguishes every tone from the remaining tones by its pitch and its intensity. It is, however, not possible to compare the "distance" between two tones of the same pitch but different intensity (analogous to the two points on the floor) with the "distance" between two tones of different pitch but equal intensity (analogous to the two points on the wall). The measure-conditions are thus quite different in this manifold.
In the system of colours, too, the measure-relations have their own peculiarity. The dimensions of the manifold of colours are the same as those of space, as each colour can be produced by mixing the three "primary" colours. But there is no relation between two arbitrary colours, which would correspond to the distance between two points in space. Only when a third colour is derived by mixing these two, does one obtain an equation between these three colours similar to that which connects three points in space lying in one straight line.
These examples, which are borrowed from Helmholtz's essays, serve to show that the measure-relations of a continuous manifold are not already given in its definition as a continuous manifold. nor by fixing its dimensions. A continuous manifold generally allows of various measure-relations. It is only experience which enables us to derive the measure-laws which are valid for each particular manifold. The fact, discovered by experience, that the dimensions of bodies are independent of their particular position and motion, led to the laws of Euclidean geometry where congruence is the deciding factor in comparing various portions of space. These questions have been exhaustively treated by Helmholtz in various essays. References:—
Riemann, "Über die Hypothesen, welche der Geometrie zugrunde liegen" (1854). Newly published and annotated by H. Weyl, Berlin, 1919.
Helmholtz. "Ueber die tatsächlichen Grundlagen der Geometrie," Wiss. Abh. 2, S. 10.
Helmholtz. "Ueber die Tatsachen, welche der Geometrie zugrunde liegen," Wiss. Abh. 2, S. 618.