D O C . 3 1 3 O N C U R R E N T S T AT E O F F I E L D T H E O R Y 3 0 1 On a purely formal level, the basis of this theory can be characterized as follows: If one draws two line elements (or elementary vectors) P P ' and P P '' starting from one point P on the continuum, with the coordinates (d ) and (d''x1, …, d''x4), then their ratios should have an objective meaning that can be cal- culated from the quadratic form given in (2). Through this quadratic form, which is defined up to a factor λ, the directional relations (angles) between two vectors that begin at the same point P are also fixed. In contrast, the ratio of two line ele- ments or vectors that begin at two separate points in the continuum can be ascribed no real meaning, neither to the ratio of their lengths nor to that of their directions. It is astonishing that for a continuum that has such a paucity of structural properties, a theory of invariants can be constructed that exhibits such a richness of forms, so that it even becomes a candidate for a representation of the physical properties of space.[9] The next metric continuum, which is richer in content, is that of Riemann. In the Riemannian continuum, not only the length ratio of two vectors that start at the same point P is attributed reality, but also to that of two vectors that start at points P and P ' at a finite distance from each other. This leads mathematically to the result that the quantity (3) has a particular value for every line element (up to an unimportant factor that is in- dependent of the x ν ). The fact that the mathematicians concerned themselves first with a continuum thus structured is historically understandable. Every surface that is embedded in a three-dimensional Euclidean space is a two-dimensional Rieman- nian continuum. It is well known that Gauss treated the theory of surfaces from this point of view Riemann then generalized the concept by recognizing that its em- bedding in a Euclidean space is not essential, and that the essential considerations of the theory can be carried to arbitrarily many dimensions. It is also well known that Riemann already had the idea that the continuum of our world of spatial expe- rience could have such a metric structure. The equations of the gravitational field in general relativity are the simplest coordinate-invariant differential equations to which the g μν of a Riemannian con- tinuum can be subjected, where the g μν themselves (or the quantity ds) describe both the metric relations within the spatio-temporal continuum and also the gravi- tational field. The general theory of relativity would be a complete theory from the point of view of logical unity if the electromagnetic field could also be represented in terms of the g μν within that theory. It was clear from the outset that this is not the case the theory was forced to introduce a logically independent linear form, φ i d xi, where the φ i are supposed to play the role of the electromagnetic potentials. Thus, the electromagnetic field was in a certain sense only externally grafted onto the theory, accompanied by the loss of the unified character of the theoretical 'x 1 , d 'x 2 d 'x 3 d 'x 4   ds 2 g  dx  dx  = [p. 130]