DUPLICATION
, is the doubling of a quantity, or multiplying it by 2, or adding it to itself.
Duplication of a Cube, is finding out the side of a cube that shall be double in solidity to a given cube: which is a celebrated problem, much cultivated by the ancient geometricians, about 2000 years ago.
It was first proposed by the oracle of Apollo at Delphos; which, being consulted about the manner of stopping a plague then raging at Athens, returned for answer, that the plague should cease when Apollo's altar, which was cubical, should be doubled. Upon this, they applied themselves in good earnest, to seek | the duplicature of the cube, which from thence was called the Delian problem.
This problem cannot be effected geometrically, as it requires the solution of a cubic equation, or requires the finding of two mean proportionals, viz, between the side of the given cube and the double of the same, the first of which two mean proportionals is the side of the double cube, as was first observed by Hippocrates of Chios. For, let a be the side of the given cube, and z the side of the double cube sought; then it is z^{3} = 2a^{3}, or a^{2} : z^{2} :: z : 2a; so that, if a and z be the first and 2d terms of a set of continued proportionals, then a^{2} : z^{2} is the ratio of the square of the 1st to the square of the 2d, which, it is known, is the same as the ratio of the 1st term to the 3d, or of the 2d to the 4th, that is of z to 2a; therefore z being the 2d term, 2a will be the 4th. So that z, the side of the cube sought, is the 2d of four terms in continued proportion, the 1st and 4th being a and 2a, that is, the side of the double cube is the first of two mean proportionals between a and 2a.
Eutocius, in his Commentaries on Archimedes, gives several ways of performing this by the mesolabe. In Pappus too are found three different ways; the first according to Archimedes, the second according to Hero, and the 3d by an instrument invented by Pappus, which gives all the proportions required. The sieur de Comiers has likewise published a demonstration of the same problem, by means of a compass with three legs. But all these methods are only mechanical. See Valerius Maximus, lib. 8; also Eutocius's Com. on lib. 2. Archimedes de Sphæra & Cylindro; and Pappus, lib. 3, prop. 5, & lib. 4, prop. 22.