The Duplicators, Part I: Eutocius’ Collection of Cube Duplications - Diocles, in On Burning Mirrors

Biographical note: Diocles (ca 240 BCE–ca 180 BCE). In addition to his cube duplication method using the cissoid curve, Diocles also was the first to prove the focal property of a parabola. This result is of immense practical value in the modern world, for everything from flashlights to satellite dishes. On Burning Mirrors, other than the loose excerpt provided here by Eutocius, survives only in Arabic. Read more about Diocles at MacTutor.

(Heiberg 66.9) In a circle, let two diameters ΑΒ and ΓΔ be drawn at right angles, and let two equal arcs ΕΒ and ΒΖ be cut off on each side of B, and through Ζ, let ΖΗ be drawn parallel to ΑΒ, and let ΔΕ be joined. I say that ΖΗ and ΗΔ are two mean proportionals between ΓΗ and ΗΘ. 

Above: This diagram is not present in the manuscripts, but I include it for clarity. This is a sub-diagram of the sketch below.

For let ΕΚ be drawn through Ε parallel to ΑΒ: therefore ΕΚ is equal to ΖΗ, and ΚΓ is equal to ΗΔ. For this will be clear, once the straight lines from Λ to Ε and Ζ are drawn: for the angles ΓΛΕ and ΖΛΔ are made equal and the angles at Κ and Η are right. Therefore all the sides and angles are equal^[17] since ΛΕ is equal to ΛΖ: and therefore the remaining segment ΓΚ is equal to the remaining segment ΗΔ. So since [by Elements VI.2], \begin{equation} \tag{25} \text{ΔK : KE = ΔH : HΘ} \end{equation} and \begin{equation} \tag {26} \text{ΔK : KE = EK : KΓ} \end{equation}

[by Elements VI.8 since angle ΓΕΔ is right], therefore ΕΚ is a mean proportional between ΔΚ and ΚΓ. Therefore \begin{equation} \tag{27} \text{ΔK : KE = EK : KΓ = ΔH : HΘ.} \end{equation}Also, ΔΚ is equal to ΓΗ, ΚΕ is equal to ΖΗ, and ΚΓ is equal to ΗΔ: therefore \begin{equation} \tag{28} \text{ΓH : HZ = ZH : HΔ = ΔH : HΘ.} \end{equation}If two [different] equal arcs ΜΒ and ΒΝ are chosen on each side of Β, and through Ν parallel to ΑΒ the line ΝΞ is drawn, and ΔΜ is joined, then again, ΝΞ and ΞΔ will be two mean proportionals between ΓΞ and ΞΟ.

So similarly, producing successive parallels between Β and Δ, and cutting arcs from Β in the direction of Γ, these arcs equal to the ones cut off by the parallels, and joining lines from Δ to the points so created, such as ΔΕ and ΔΜ, the parallels drawn between Β and Δ will be cut [by the lines such as ΔΕ and ΔΜ] at some points, which in the diagram are the points O and Θ. Through these points, placing a ruler joining them, we will then have a certain curve drawn in the circle, on which, if some point be chosen at random, and through this point a line be drawn parallel to ΛΒ, the line joined (from the point on the circle cut off and the diameter) and the one it cuts off of the diameter (on the Δ side) will be [two^[18]] mean proportionals between the cut off part of the diameter (on the Γ side) and the and the part of the [parallel] from the point on the curve to the diameter ΓΔ.^[19]

Above: Diocles’ First Diagram. Diocles has defined a series of points, such as Θ and Ο. The locus described is shown in red, and is a segment of the famous Cissoid of Diocese. Point E generates the locus.

Having prepared these things in advance, let there be two straight lines Α and Β, between which it is necessary to find two mean proportionals. And let there be a circle, in which two diameters ΓΔ and ΕΖ are drawn at right angles. And in the circle, let the curve ΔΘΖ be drawn through the successive points, as has been said before. Let it be made [using Elements VI.12] that \begin{equation} \tag{29} \text{A : B = ΓH : HK,} \end{equation}and having joined and extended ΓΚ, let it cut the curved line at Θ. And through Θ, let ΛΜ be drawn parallel to ΕΖ. 

Therefore ΜΛ and ΛΔ are two mean proportionals between ΓΛ and ΛΘ [by above]. And since [by Elements VI.2] \begin{equation} \tag{30} \text{ΓΛ : ΛΘ = ΓH : HK} \end{equation}and \begin{equation} \tag{31} \text{ΓH : HK = A : B,} \end{equation}if we insert means Ν and Ξ between Α and B in the same ratio as ΓΛ, ΛΜ, ΛΔ, and ΛΘ, then Ν and Ξ will have been taken as two mean proportionals: the very thing it was necessary to find. 

Diocles’ Second Diagram. The intersection point Θ was difficult to define in GeoGebra: since GeoGebra cannot find the intersection of a locus and a line, I had to do a conic approximation of the cissoid. The intersection of this conic and the line ΛΜ is defined as Θ, which is sufficiently close to the intersection point of ΛΜ and the cissoid for visual and computational purposes. The ratios referenced are numerically computed at the side, to demonstrate approximate accuracy (rounded to two decimal places, for cleanliness).

[17] In Greek, πάντα ἄρα πᾶσιν, lit. “each to each.” Equality is not explicitly mentioned, but is implied by the previous sentences. Compare the language of Euclid. 

[18] The number two is not explicitly stated here, but it is clear from context. 

[19] This passage was a bit difficult to render in a readable way that also was faithful to the Greek. What Eutocius is trying to say is that, as in the previous paragraph, we can continue to generate a series of points like Ο and Ξ. The ruler is used to connect these successive points, producing, as it were, a piecewise linear approximation of a curve. The full curve would, of course, require an infinite number of applications of this procedure. The property of certain lines being mean proportionals between others is true regardless of the point chosen on the curve, which has been demonstrated by the previous paragraph for O and Ξ. In some sense this is backwards of the usual arrangement of a Greek proof: normally the general claim is stated first, as the protasis, and the specific claim (with reference to a specific diagram) is referenced second.