Three Problems Of Antiquity . Doubling the cube and Constructing a regular heptagon. All three, in algebraic terms, reduce to an algebraic equation of degree three.
Cubic equations possess a pertinent property which constitutes the contents of a lemma below.
After proving the lemma, I shall derive cubic equations for the three problems and show that they satisfy the conditions of the lemma. Then either one of its roots is rational or none of its roots is constructible. The lemma then asserts that the equation has no constructible roots. Assume on the contrary that the equation has a constructible root. Let u denote the third root.