The Duplicators, Part I: Eutocius' Collection of Cube Duplications - Finding Two Mean Proportionals

An immediate generalization to finding one mean proportional comes in Elements, Book XI, on solids. First, we make a definition. Suppose we have magnitudes Α, Β, Γ, and Δ. Then we say that these magnitudes are in continuous proportion if \begin{equation} \tag{5} \text{A : B :: B : Γ :: Γ : Δ.} \end{equation}

In Elements XI.33 cor., Euclid shows that if we have four magnitudes in continuous proportion, then \begin{equation} \tag{6} \text{A : Δ :: parallelepiped solid on A : parallelepiped solid on B.} \end{equation}

If we let Α : Δ be the same as 1 : 2, and take the parallelepipedal solids as cubes, then \begin{equation} \tag{7} \text{1 : 2 :: A : Δ :: cube(A) : cube(B).} \end{equation}This is all well and good, except unlike the single mean proportional, we do not (yet) know how to find Β or Γ given only Α and Δ.

Eratosthenes (next section) details that it was Hippocrates of Chios who realized the connection of the duplication problem with the problem of finding two mean proportionals. Again, the latter problem is more general, since we need not assume that Α : Δ is as 1 : 2.