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--- abstract: | 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.\ [**Keywords**]{}: Euler’s conjecture, Cayley-Menger determinants, Multidimensional scaling, Euclidean Distance Matrix bibliography: - 'ibmer.bib' --- [Six mathematical gems from the history of Distance Geometry]{} - [*CNRS LIX, École Polytechnique, F-91128 Palaiseau, France*]{}\ Email:[<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>](<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>) - [*IMECC, University of Campinas, 13081-970, Campinas-SP, Brazil*]{}\ Email:[<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>](<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>) Introduction {#s:intro} ============ Distance Geometry (DG) is the study of geometry with the basic entity being distance (instead of lines, planes, circles, polyhedra, conics, surfaces and varieties). As with most everything else, it all began with the Greeks, specifically Heron, or Hero, of Alexandria sometime between 150BC and 250AD, who showed how to compute the area of a triangle given its side lengths [@heron]. After a hiatus of almost two thousand years, we reach Arthur Cayley’s: the first paper of volume I of his [*Collected Papers*]{}, dated 1841, is about the relationships between the distances of five points in space [@cayley1841]. The gist of what he showed is that a tetrahedron can only exist in a plane if it is flat (in fact, he discussed the situation in one more dimension). This yields algebraic relations on the side lengths of the tetrahedron. Hilbert’s influence on foundations and axiomatization was very strong in the 1930s [*Mitteleuropa*]{} [@hilbert]. This pushed many people towards axiomatizing existing mathematical theories [@tarskied]. Karl Menger, a young professor of geometry at the University of Vienna and an attendee of the Vienna Circle, proposed in 1928 a new axiomatization of metric spaces using the concept of distance and the relation of congruence, and, using an extension of Cayley’s algebraic machinery (which is now known as Cayley-Menger determinant), generalized Heron’s theorem to compute the volume of arbitrary $K$-dimensional simplices using their side lengths [@menger28]. The Vienna Circle was a group of philosophers and mathematicians which convened in Vienna’s [*Reichsrat café*]{} around the nineteen-thirties to discuss philosophy, mathematics and, presumably, drink coffee. When the meetings became excessively politicized, Menger distanced himself from it, and organized instead a seminar series, which ran from 1929 to 1937 [@mengerK]. A notable name crops up in the intersection of Menger’s geometry students, the Vienna Circle participants, and the speakers at Menger’s [*Kolloquium*]{}: Kurt Gödel. Most of the papers Gödel published in the Kolloquium’s proceedings are about logic and foundations,[^1] but two, dated 1933, are about the geometry of distances on spheres and surfaces. The first [@mengerK 18 Feb. 1932, p. 198] answers a question posed at a previous seminar by Laura Klanfer, and shows that a set $X$ of four points in any metric space, congruent to four non-coplanar