Menaechmus

Menaechmus was an ancient Greek mathematician who discovered conic sections while investigating the problem of doubling the cube.

Doubling the cube: The problem of constructing a cube with twice the volume of a given cube using classical geometric tools. This construction problem prompted Menaechmus’s search for a geometric solution.
Two mean proportionals: Two quantities inserted between given magnitudes so that all four form a continued proportion. Finding these means between a cube’s edge and twice its edge solves the doubling problem.
Plato: An Athenian philosopher whose dialogues shaped ancient Greek thought and education. A later anecdote connects Menaechmus’s geometric work with Plato’s disapproval of mechanical solutions.
Conics: The mathematical study of curves produced by cutting a cone with a plane. Menaechmus’s construction helped establish conics as objects of systematic mathematical investigation.
Conic section: A curve formed by intersecting a plane with a cone. Menaechmus’s key discovery was that these curves could solve cubic construction problems.
Intersection of a parabola and a hyperbola: A pair of conic curves whose intersection can encode a solution to a cubic equation. Menaechmus used such intersections to obtain the proportional means needed for doubling the cube.
Eutocius of Ascalon: A sixth-century Greek commentator on ancient mathematical works, including those of Archimedes and Apollonius. His commentary preserves accounts of early solutions to doubling the cube associated with Menaechmus.
Cissoid of Diocles: A cubic curve introduced by Diocles and used to construct a cube root geometrically. It shows how the cube-doubling challenge continued to spur new curves and constructions after Menaechmus.
Parabola: A conic section whose points are equidistant from a fixed point and a fixed line. A parabola supplies one of the curves used in Menaechmus’s cube-doubling construction.
Cube root: A number which, when multiplied by itself three times, gives a specified value. The side of a doubled cube must be the original side multiplied by the cube root of two.

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Eudoxus of CnidusRelated: He belonged to the same mathematical milieu and studied under Eudoxus, according to ancient sources.