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The Duplicators, Part I: Eutocius' Collection of Cube Duplications - Notation

Colin B. P. McKinney (Wabash College)

For readability, I’ve “translated” some of the mathematical statements into what I think is an abbreviated form. Table 1.1 gives a brief list. 

Table 1.1: Summary of Notation

tri.(ABΓ) The triangle with vertices A, B, Γ.
sq.(AB) The square with side AB.
rect.(AB,ΓΔ) The rectangle with sides AB, ΓΔ.
rect.(AΓ) The rectangle about the diagonal AΓ.
rect.(ABΓ) The rectangle with sides AB and BΓ.
quad.(AΓ) The quadrilateral about the diagonal AΓ.
pllg.(AB,BΔ) The parallelogram with sides AB, BΔ.
pllg.(AB,ΓΔ) The parallelogram with sides equal to AB, ΓΔ.
pllg.(AΓ) The parallelogram about the diagonal AΓ.
A : B :: Γ : Δ As A is to B, so too is Γ to Δ.
dup.(A : B) The duplicate ratio of A : B.
(A : B) comp. (Γ : Δ) The ratio compounded from the ratios A : B and Γ : Δ
arc(AB) The arc from A to B (or through both).

One downside to this approach is that it does over-distill some of the Greek phrasing, but it makes for more readable English. For example, Eutocius often makes statements within statements: he might say something like “Therefore A is to B, that is C, as D is to E.” This can be cumbersome to translate, especially when trying to use equation-style formatting for readability. In a situation such as this, I have tended to overly notate, and render the statement as \begin{align} \text{A : B} & = \text{A : C} \\\\ & = \text{D : E.} \end{align}

A general point: Greek texts do not label points in the same way that we tend to today. By this I mean that Greek texts freely interchange ΑΒ and ΒΑ, or seemingly randomly order the vertices of a triangle ΑΒΓ, ΒΑΓ, ΒΓΑ, etc. As I work frequently with the Greek texts, the practice carries over both to my own translations (where it is justified) and to my supplementary notes and commentary (where it is not). This, I think, points out how critical it is for a reader, be he ancient or modern, to have a diagram at hand. 

References to propositions in the Elements [2003] or the Conics [2013] are given as I found helpful; for the Elements, I have also hyperlinked to David Joyce’s online edition ([8]). The reader should note, however, that these references are almost never explicitly made in the Greek text. 

In most of the sections that follow, my commentary is included either in a box or using footnotes.  This will very clearly delineate when I am speaking versus when Eutocius (or another ancient voice) is speaking.  An example of a commentary box is given below:

I suppose you could say that I am thinking inside the box.

Colin B. P. McKinney (Wabash College), "The Duplicators, Part I: Eutocius' Collection of Cube Duplications - Notation," Convergence (April 2016), DOI:10.4169/convergence20160401

The Duplicators, Part I: Eutocius's Collection of Cube Duplications