The Duplicators, Part I: Eutocius' Collection of Cube Duplications - Finding One Mean Proportional

Euclid, in Elements VI.13, gives the construction for finding a single mean proportional. That is, given magnitudes Α and Γ, Euclid finds an intermediate magnitude Β so that \begin{equation} \tag{1} \text{A : B :: B : Γ.} \end{equation}

He does this by placing Α and Γ in straight line with each other, and then describing a semicircle with their combined length as its diameter. Extending a perpendicular from the point where they meet to the semicircle gives the length Β.

Above: Euclid's Elements VI.13. Lengths A and Γ can be adjusted by moving the points at the ends of the diameter formed by A and Γ .

Euclid also describes the concept of duplicate and triplicate ratios, in the definitions to Book V. From this definition, and Elements VI.19 cor.^[1] we conclude that if  \begin{equation} \tag{2} \text{ A : B :: B : Γ,} \end{equation}then \begin{equation} \tag{3} \text{A : Γ :: figure on A : figure on B,} \end{equation}where the figures are similar. For example, if we define the figures to be squares then \begin{equation} \tag{4} \text{A : Γ :: sq.(A) : sq.(B).} \end{equation}If we make the ratio Α : Γ the same as 1 : 2, then the square on B will be twice the square on A, and hence we have doubled the square. So if our goal was to double the square, this sequence of propositions gives us the method. But it also gives us more: since we only assumed that Α : Γ is as 1 : 2 at the very end, the problem we have solved is more general. 

[1] A brief note about the term “corollary” seems appropriate. In Greek, the term Euclid uses is πόρισμα, “porism.”  The sense of the word is what we mean with the word “corollary.”  Since “porism” may seem overly archaic to readers (especially those who have not taken Greek), I will use “corollary,” or the abbreviation “cor.,” for this term.