fulfils all these requirements; but it is by no means the only function of this kind.
[Note 9] (p. 29). But the expression of the fourth degree for the fine element would not permit of any geometrical interpretation of the formula, such as is possible with the expression
which latter may be regarded as a general case of Pythagoras' theorem.
[Note 10] (p. 30). By a "discrete" manifold we mean one in which no continuous transition of the single elements from one to another is possible, but each element to a certain extent represents an independent entity. The aggregate of all whole numbers, for instance, is a manifold of this type, or the aggregate of all planets in our solar system, etc., and many other examples may be found; and indeed all finite aggregates in the theory of aggregates are such discrete manifolds. "Measuring," in the case of discrete manifolds, is performed merely by "counting," and does not present any special difficulties, as all manifolds of this type are subject to the same principle of measurement. When Riemann then proceeds to say: "Either, therefore, the reality which underlies space must form a discrete manifold, or we must seek the ground of its metric relations outside it, in binding forces which act upon it," he only wishes to hint at a possibility, which is at present still remote, but which must, in principle, always be left open. In just the last few years a similar change of view has actually occurred in the case of another manifold which plays a very important part in physics, viz. "energy"; the meaning of the hint Riemann gives will become clearer if we consider this example.
Up till a few years ago, the energy which a body emanates by radiation was regarded as a continuously variable quantity: and attempts were therefore made to measure its amount at any particular moment by means of a continuously varying sequence of numbers. The researches of Max Planck have, however, led to the view that this energy is emitted in "quanta," and that therefore the "measuring" of its amount is performed by counting the number of "quanta." The reality underlying radiant energy, according to this, is a discrete and not a continuous manifold. If we now suppose that the view were gradually to take root that, on the one hand, all measurements in space only have to do with distances between ether-atoms; and that, on the other hand, the distances of single ether-atoms from one another can only assume certain definite values, all distances in space would be obtained by "counting" these values, and we should have to regard space as a discrete manifold.
[Note 11] (p. 32). C. Neumann. "Ueber die Prinzipien der Galilei-Newtonschen Theorie," Leipzig 1870, S. 18.
[Note 12] (p. 32). H. Streintz. "Die physikalischen Grundlagen der Mechanik," Leipzig, 1883.
[Note 13] (p. 33). A. Einstein. "Annalen der Physik," 4 Folge, Bd. 17, S. 891.
[Note 14] (p. 35). Minkowski was the first to call particular attention to this deduction of the special principle of relativity.