 Because the formula for \(ds\) was written down subject to only a few restrictions, there is no reason to believe that a Riemannian geometry is defined with respect to an antecedent Euclidean geometry. There is no claim that an \(n\)-dimensional Riemannian geometry is to be obtained by a map from an \(n\)-dimensional subset of some Euclidean \(m\)-dimensional space. This means that geometry can be done without reference to any Euclidean geometry. Euclidean geometry is no longer epistemologically prior to any study of other geometries. The reign of Euclid was—theoretically—over.