The Duplicators, Part I: Eutocius’ Collection of Cube Duplications - Eratosthenes

Biographical note: Eratosthenes of Cyrene (ca 276 BCE–ca 194 BCE). Eratosthenes is known for a variety of mathematical topics, including the so-called sieve (related to prime number theory) and a relatively accurate estimate for the circumference of the Earth. He was a polymath but not “best” in any field, and supposedly was nicknamed “Beta” for this reason. Read more about Eratosthenes at MacTutor.

(Heiberg 84.13) Let there be two unequal straight lines, ΑΕ and ΔΘ, between which it is required to find two means in continuous proportion. Let ΑΕ be placed at right angles to some straight line ΕΘ, and on ΕΘ, let three parallelograms be set up, ΑΖ, ΖΙ, and ΙΘ, and let the diameters ΑΖ, ΛΗ, and ΙΘ be drawn in each: hence these diameters will be parallel.

Above: Figure 13.1: Eratosthenes’ First Diagram. This diagram is static. The point F is unlabeled in the manuscripts.
In the next diagram, it is movable, and I label it here for continuity.

When the middle parallelogram ΖΙ remains fixed, let the parallelogram ΑΖ be slid along the top surface of the middle one, and ΙΘ along the bottom surface of the middle one, as in the second diagram, until the points Α, Β, Γ, and Δ are in a straight line. Let the line through the points Α, Β, Γ, and Δ be drawn, and let it intersect ΕΘ (having been extended) at Κ.

Above: Eratosthenes’ Second Diagram. The objective with this diagram is to have the points Β and Γ on the red line ΑΔ. This is accomplished by moving the doors (by moving points Α and F). The point Δ is also moveable; adjusting it changes the ratio ΑΕ : ΔΘ. By default, ΑΕ is twice ΔΘ. I have added a few labels to this diagram. First, the point F, which is unlabelled in the manuscripts. Second, the manuscripts only label the points Λ and Ι (and their counterparts, Λ2 and Ι2) in the first diagram. For completeness, I include them here; when the doors are moved to their starting positions, the Λ points coincide and the Ι points coincide. Similarly for the points Ζ and Η: while they are also labeled in the second diagram, their counterparts Z₁ and Η₁ are not. (Note that Eratosthenes’ First Diagram corresponds to a static version of this diagram.)

Indeed, in the parallels ΑΕ and ΖΒ, we have by Elements VI.4, \begin{equation} \tag{97} \text{ ΑΚ : ΚΒ = ΕΚ : ΚΖ;} \end{equation}and in the parallels ΑΖ and ΒΗ,  \begin{equation} \tag{98} \text{ΑΚ : ΚΒ = ΖΚ : ΚΗ. } \end{equation}Therefore [by Elements V.11]  \begin{equation} \tag{99} \text{ ΑΚ : ΚΒ = ΕΚ : ΚΖ = ΚΖ : ΚΗ.} \end{equation}Again, in the parallels ΒΖ and ΓΗ, \begin{equation} \tag{100} \text{ ΒΚ : ΚΓ = ZK : ΚΗ;} \end{equation}and in the parallels ΒΗ and ΓΘ, \begin{equation} \tag{101} \text{ ΒΚ : ΚΓ = ΗΚ : ΚΘ.} \end{equation}Therefore \begin{equation} \tag{102} \text{ΒΚ : ΚΓ = ΖΚ : ΚΗ = ΗΚ : ΚΘ. } \end{equation}But \begin{equation} \tag{103} \text{ΖΚ : ΚΗ = ΕΚ : ΚΖ; } \end{equation}and therefore \begin{equation} \tag{104} \text{ ΕΚ : ΚΖ = ΖΚ : ΗΚ = ΗΚ : ΚΘ.} \end{equation}

But \begin{equation} \tag{105} \text{ΕΚ : ΚΖ = ΑΕ : ΒΖ,} \end{equation}and \begin{equation} \tag{106} \text{ ΖΚ : ΚΗ = ΒΖ : ΓΗ,} \end{equation}and \begin{equation} \tag{107} \text{ ΗΚ : ΚΘ = ΓΗ : ΔΘ;} \end{equation}therefore \begin{equation} \tag{108} \text{ ΑΕ : ΒΖ = ΒΖ : ΓΗ = ΓΗ : ΔΘ.} \end{equation}Therefore both ΒΖ and ΓΗ have been found as two mean proportionals between ΑΕ and ΔΘ.