The duplication of the cube is a traditional problem of Mathématiques. It is a geometrical problem , belonging to the Three major problems of Antiquity, with the Quadrature of the circle and the Trisection of the angle. This problem consists in building a cube, whose volume is twice larger than a given cube, using a rule and of a compass. That thus amounts multiplying the edge of the cube by $\backslash \; sqrt2$.

The problem has its origin in a legend reported by Eratosthène in the Platonicien and by Théon of Smyrna in its Arithmétique . The Déliens, victim of an epidemic of plague, required of the oracle Delphes how to put an end to this epidemic. The answer of oracle was that it was necessary to double the furnace bridge devoted to Apollon, furnace bridge whose form was a perfect cube. The architects went to find Plato to know how to make. This last answered them that the god did not certainly need a double furnace bridge, but that it made them reproach, via oracle, to neglect the geometry.

The question interested many mathematicians, for example Hippias d' Élis, Archytas de Tarente, Ménechme, Eudoxe de Cnide. Several solutions were proposed by intersection of Coniques or intersection of space figures, but no solution planes was not found with the only use of the rule and the compass.

In 1837, Pierre-Laurent Wantzel establishes a theorem giving the form of the equations of the soluble problems to the rule and the compass. It shows that $\backslash \; sqrt2$ is not constructible. The duplication of the cube is thus impossible to realize.

The relation between the geometrical construction and the algebraic theory is developed in constructible Nombre. The algebraic demonstrations are in quadratic Tour of extension.

See too

• Mésolabe
• Quadrature of the circle
• Trisection of the angle

References