On the hypotheses which lie at the foundation of geometry on Superphysics

Application to Space

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§ 1. By means of these inquiries into the determination of the measurerelations of an n-fold extent the conditions may be declared which are necessary and sufficient to determine the metric properties of space, if we assume the independence of line-length from position and expressibility of the lineelement as the square root of a quadric differential, that is to say, flatness in the smallest parts.

Measure-relations

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2. Measure-relations of which a manifoldness of n dimensions is capable on the assumption that lines have a length independent of position, and consequently that every line may be measured by every other.

Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude, we come to the second of the problems proposed above, viz. the study of the measure-relations of which such a manifoldness is capable, and of the conditions which suffice to determine them. These measure-relations can only be studied in abstract notions of quantity, and their dependence on one another can only be represented by formulæ. On certain assumptions, however, they are decomposable into relations which, taken separately, are capable of geometric representation; and thus it becomes possible to express geometrically the calculated results. In this way, to come to solid ground, we cannot, it is true, avoid abstract considerations in our formulæ, but at least the results of calculation may subsequently be presented in a geometric form. The foundations of these two parts of the question are established in the celebrated memoir of Gauss, Disqusitiones generales circa superficies curvas.

Plan of the Investigation

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On the Hypotheses which lie at the Bases of Geometry.

Bernhard Riemann

Geometry:

assumes both the notion of space and the first principles of constructions in space.
gives definitions of them which are merely nominal, while the true determinations appear as axioms.

The relation of these assumptions remains consequently in darkness.