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

Colin B. P. McKinney (Wabash College)

Biographical Note: Plato (ca 428 BCE–ca 348 BCE). Best known as a philosopher, mathematics features prominently in several of his dialogues, e.g. Theaetaetus, Meno, and Republic. It is unlikely that the device and method described here are actually due to Plato, for these reasons: we do not know of Plato actively engaging in mathematical research (though associates of his did), and the mechanical nature of the construction seems at odds with Plato’s philosophy of mathematics. It is somewhat strange to think that Eutocius would not have found the attribution dubious, unless there is another Plato who did mathematics (about whom we know nothing beyond this reference), or the attribution is meant in an ironic fashion. Read more about Plato at MacTutor.

(Heiberg 56.14) Given two straight lines, to find two mean proportionals in continuous proportion.

For let the two given straight lines be ΑΒ, ΒΓ, at right angles to each other, between which it is required to find two mean proportionals. Let them be extended in straight lines to Δ and Ε, and let a right angle ΖΗΘ be constructed; and in one side ΖΗ, let there be a cross-bar ΚΛ, being in a groove in the side ΖΗ, so that when moved, it remains parallel to ΗΘ.

This will be, if also another cross-bar ΘΜ is imagined attached to ΘΗ, parallel to ΖΗ: for having fashioned dove-tail[7] shaped grooves in the top sides of ΖΗ and ΘΜ, and having attached knobs to ΚΛ in the aforementioned grooves, the movement of ΚΛ will always be parallel to ΗΘ[8].

So having fashioned these things,[9] let one side of the angle, say ΗΘ, be placed so that it touches Γ; 

Above: Eutocius’ Diagram for Plato’s Device, First Constraint. The apparatus he describes is shown in red.
The points K and H are movable.

At this point, the apparatus has one constraint in terms of its placement over the figure: The side ΗΘ is placed so that Γ is on it. This still permits a considerable amount of freedom for the apparatus to be moved.

It is worth mentioning that the scale of the instrument needs to be reasonable given the scale of the to lines AB and ΒΓ. However, the sides ΑΒ and ΒΓ could easily be scaled up or down to accommodate the use of a particular size tool.

and let both the angle and the side ΚΛ be moved until the point Η is on the side ΒΔ and the side ΗΘ is touching Γ, 

Above: Eutocius’ Diagram for Plato’s Device, Second Constraint.
The points Z, K, and H are movable.

This is the second constraint introduced. No longer can the apparatus rotate freely about Γ. It is limited in the extents of its motion, since now the point Η must be somewhere on the line ΒΔ. At one extreme, when Η is at Β, the entire apparatus is oriented with the bottom edge ΗΘ coincident with ΒΓ, and the left edge ΖΗ coincident with ΑΒ. At the other extreme, it is situated somewhat diagonally, with the bottom edge of the apparatus ΗΘ connecting Δ and Γ. 

and the bar ΚΛ touches the side ΒΕ at Κ, 

Above: Eutocius’ Diagram for Plato’s Device, Third Constraint.
The points Z and H are movable.

Now the point Κ must be on the line segment ΒΕ. This links the motion of the second cross-bar ΚΛ with the bottom edge of the apparatus ΗΘ. At one extreme, we have points Η and Κ coinciding at Β, and the second cross-bar ΚΛ is coincident with the bottom of the apparatus ΗΘ and with the line ΒΓ. At the other extreme, K coincides with E, while H coincides with Δ. 

and the remaining side at Α, so that, as the diagram shows,[10] the right angle has position as ΓΔΕ, and the bar ΚΛ has the position which ΕΑ has: 

Above: Eutocius’ Diagram for Plato’s Device, Fourth Constraint.
This is the finished configuration.

The apparatus is now fully constrained. At this point, Δ and E are dynamically redefined. The positions of Δ and Ε were essentially arbitrary: they needed only to be in straight line with ΑΒ and ΒΓ, respectively. For this reason, we can redefine Δ and Ε to be where Η and Κ are, respectively. This sort of game is common enough in Greek mathematics. 

for having arranged everything thus, the prescribed thing will be. For since the angles at Δ and Ε are right, 

\begin{equation} \tag{8} \text{ΓB : BΔ :: ΔB : BE :: EB : BA.} \end{equation}

We can view this result as coming from two applications of Elements VI.13. If we imagine two semicircles ΑΕΔ and ΕΔΓ, then in each we can apply VI.13. For ΑΕΔ, this yields  \begin{equation} \tag{9} \text{ ΔB : BE :: EB : BA,} \end{equation} and for ΕΔΓ, it yields \begin{equation} \tag{10} \text{ΓB : BΔ :: ΔB : BE.} \end{equation} Hence the claim.

[7] Netz takes this word as “axe-shaped”; I’ve chosen “dove-tail” because of the connotation in woodworking.

[8]  Netz takes this differently, as the knobs Κ and Λ being parallel to the ruler HΘ, on the grounds that the manuscripts have a plural article rather than Heiberg’s singular one. On this point, I’m more inclined to agree with Heiberg. 

[9]  i.e. the two rods, the grooves, etc. 

[10]  Eutocius is referring to his diagram here, which is a sub-set of Eutocius' Diagram for Plato's Device, Fourth Constraint (the finished configuration). Eutocius’ figure has only the points Α, Β, Γ, Δ, Ε, and the lines ΕΓ, ΑΔ, ΑΕ, ΕΔ, and ΔΓ. The manuscripts apparently have a second figure showing the apparatus, and a complete study of these diagrams is a project for future work. 

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

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