Six mathematical gems from the history of distance geometry

This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron’s formula, Cauchy’s theorem on the rigidity of polyhedra, Cayley’s generalization of Heron’s formula to higher dimensions, Menger’s characterization of abstract semi-metric spaces, a result of Gödel on metric spaces on the sphere, and Schoenberg’s equivalence of distance and positive semidefinite matrices, which is at the basis of Multidimensional Scaling.