Cube duplication, also called the Delian problem, is one of the geometric problems of antiquity which asks, given the length of an edge of a cube, that a second cube be constructed having double the volume of the first. The only tools allowed for the construction are the classic (unmarked) straightedge and compass.
The problem appears in a Greek legend which tells how the Athenians, suffering under a plague, sought guidance from the Oracle at Delos as to how the gods could be appeased and the plague ended. The Oracle advised doubling the size of the altar to the god Apollo. The Athenians therefore built a new alter twice as big as the original in each direction and, like the original, cubical in shape (Wells, 1986, p. 33). However, as the Oracle (notorious for ambiguity and double-speaking in his prophecies) had advised doubling the size (i.e., volume), not linear dimension (i.e., scale), the new altar was actually eight times as big as the old one. As a result, the gods remained unappeased and the plague continued to spread unabated. The reasons for the dissatisfaction of the gods under these circumstances is not entirely clear, especially since eight times the volume of original altar was a factor of four greater than actually requesting. It can therefore only be assumed that Greek gods were unusually ticklish on the subject of "altar"-ations being performed to their exact specifications.
Under these restrictions, the problem cannot be solved because the Delian constant 
(the required ratio of sides of the original cube
and that to be constructed) is not a Euclidean number.
However, the impossibility of the construction required nearly 2000 years, with the
first proof constructed by Descartes in 1637. The problem can be solved, however,
using a Neusis construction.
