An essay on the foundations of geometry. Bertrand Russell

far more clearly than any of his predecessors, the axioms which distinguish spatial quantity from other quantities with which mathematics is conversant. But by the assumption, from the start, that space can be regarded as a quantity, he has been led to state the problem as: What sort of magnitude is space? rather than: What must space be in order that we may be able to regard it as a magnitude at all? He does not realise, either—indeed in his day there were few who realized—that an elaborate Geometry is possible which does not deal with space as a quantity at all. His definition of space as a species of manifold, therefore, though for analytical purposes it defines, most satisfactorily, the nature of spatial magnitudes, leaves obscure the true ground for this nature, which lies in the nature of space as a system of relations, and is anterior to the possibility of regarding it as a system of magnitudes at all.

      But to proceed with the mathematical development of Riemann's ideas. We have seen that he declared measurement to consist in a superposition of the magnitudes to be compared. But in order that this may be a possible means of determining magnitudes, he continues, these magnitudes must be independent of their position in the manifold (p. 259). This can occur, he says, in several ways, as the simplest of which, he assumes that the lengths of lines are independent of their position. One would be glad to know what other ways are possible: for my part, I am unable to imagine any other hypothesis on which magnitude would be independent of place. Setting this aside, however, the problem, owing to the fact that measurement consists in superposition, becomes identical with the determination of the most general manifold in which magnitudes are independent of place. This brings us to Riemann's other fundamental conception, which seems to me even more fruitful than that of a manifold.

      20. (2) Measure of curvature. This conception is due to Gauss, but was applied by him only to surfaces; the novelty in Riemann's dissertation was its extension to a manifold of n dimensions. This extension, however, is rather briefly and obscurely expressed, and has been further obscured by Helmholtz's attempts at popular exposition. The term curvature, also, is misleading, so that the phrase has been the source of more misunderstanding, even among mathematicians, than any other in Pangeometry. It is often forgotten, in spite of Helmholtz's explicit statement[20], that the "measure of curvature" of an n-dimensional manifold is a purely analytical expression, which has only a symbolic affinity to ordinary curvature. As applied to three-dimensional space, the implication of a four-dimensional "plane" space is wholly misleading; I shall, therefore, generally use the term space-constant instead[21]. Nevertheless, as the conception grew, historically, out of that of curvature, I will give a very brief exposition of the historical development of theories of curvature.

      Just as the notion of length was originally derived from the straight line, and extended to other curves by dividing them into infinitesimal straight lines, so the notion of curvature was derived from the circle, and extended to other curves by dividing them into infinitesimal circular arcs. Curvature may be regarded, originally, as a measure of the amount by which a curve departs from a straight line; in a circle, which is similar throughout, this amount is evidently constant, and is measured by the reciprocal of the radius. But in all other curves, the amount of curvature varies from point to point, so that it cannot be measured without infinitesimals. The measure which at once suggests itself is, the curvature of the circle most nearly coinciding with the curve at the point considered. Since a circle is determined by three points, this circle will pass through three consecutive points of the curve. We have thus defined the curvature of any curve, plane or tortuous; for, since any three points lie in a plane, such a circle can always be described.

      If we now pass to a surface, what we want is, by analogy, a measure of its departure from a plane. The curvature, as above defined, has become indeterminate, for through any point of the surface we can draw an infinite number of arcs, which will not, in general, all have the same curvature. Let us, then, draw all the geodesics joining the point in question to neighbouring points of the surface in all directions. Since these arcs form a singly infinite manifold, there will be among them, if they have not all the same curvature, one arc of maximum, and one of minimum curvature[22]. The product of these maximum and minimum curvatures is called the measure of curvature of the surface at the point under consideration. To illustrate by a few simple examples: on a sphere, the curvatures of all such lines are equal to the reciprocal of the radius of the sphere, hence the measure of curvature everywhere is the square of the reciprocal of the radius of the sphere. On any surface, such as a cone or a cylinder, on which straight lines can be drawn, these have no curvature, so that the measure of curvature is everywhere zero—this is the case, in particular, with the plane. In general, however, the measure of curvature of a surface varies from point to point.

      Gauss, the inventor of this conception[23], proved that, in order that two surfaces may be developable upon each other—i.e. may be such that one can be bent into the shape of the other without stretching or tearing—it is necessary that the two surfaces should have equal measures of curvature at corresponding points. When this is the case, every figure which is possible on the one is, in general, possible on the other, and the two have practically the same Geometry[24]. As a corollary, it follows that a necessary condition, for the free mobility of figures on any surface, is the constancy of the measure of curvature[25]. This condition was proved to be sufficient, as well as necessary, by Minding[26].

      21. So far, all has been plain sailing—we have been dealing with purely geometrical ideas in a purely geometrical manner—but we have not, as yet, found any sense of the measure of curvature, in which it can be extended to space, still less to an n-dimensional manifold. For this purpose, we must examine Gauss's method, which enables us to determine the measure of curvature of a surface at any point as an inherent property, quite independent of any reference to the third dimension.

      The method of determining the measure of curvature from within is, briefly, as follows: If any point on the surface be determined by two coordinates, u, v, then small arcs of the surface are given by the formula

      ds² = Edu² + 2Fdu dv + Gdv²,

      where E, F, G are, in general, functions of u, v.[27] From this formula alone, without reference to any space outside the surface, we can determine the measure of curvature at the point u, v, as a function of E, F, G and their differentials with respect to u and v. Thus we may regard the measure of curvature of a surface as an inherent property, and the above geometrical definition, which involved a reference to the third dimension, may be dropped. But at this point a caution is necessary. It will appear in Chap. III. (§ 176), that it is logically impossible to set up a precise coordinate system, in which the coordinates represent spatial magnitudes, without the axiom of Free Mobility, and this axiom, as we have just seen, holds on surfaces only when the measure of curvature is constant. Hence our definition of the measure of curvature will only be really free from reference to the third dimension, when we are dealing with a surface of constant measure of curvature—a point which Riemann entirely overlooks. This caution, however, applies only in space, and if we take the coordinate system as presupposed in the conception of a manifold, we may neglect the caution altogether—while remembering that the possibility of a coordinate system in space involves axioms to be investigated later. We can thus see how a meaning might be found, without reference to any higher dimension, for a constant measure of curvature of three-dimensional space, or for any measure of curvature of an n-dimensional manifold in general.

      22. Such a meaning is supplied by Riemann's dissertation, to which, after this long digression, we can now return. We may define a continuous manifold as any continuum of elements, such that a single element is defined by n continuously variable magnitudes. This definition does not really include space, for coordinates in space do not define a point, but its relations to the origin, which is itself arbitrary. It includes, however, the analytical conception of space with which Riemann deals, and may, therefore, be allowed to stand for the moment. Riemann then assumes that the difference—or distance, as it may be loosely called—between any two elements is comparable, as regards magnitude, to the difference between any other two. He assumes further, what it is Helmholtz's merit to have proved, that the difference ds between two consecutive elements can