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

Biographical note: Apollonius of Perga (ca 262 BCE–ca 190 BCE) is perhaps most well known for his treatise on conic sections, the Conics, wherein the terms parabola, hyperbola, and ellipse were used for the first time (the curves themselves were studied earlier, but by different names). The work significantly extends previous knowledge of conic sections, and contains many original results. His other achievements include results in mathematical astronomy and a method for approximating \(\pi.\) Some of his works do not survive, including the last book (Book VIII) of the Conics. Read more about Apollonius at MacTutor.

(Heiberg 64.16) Let there be two given straight lines ΒΑ and ΑΓ, between which it is necessary to find two mean proportionals, ΒΑΓ containing a right angle at Α. And with center Β and radius ΑΓ, let the perimeter of a circle ΚΘΛ be drawn. Again, with center Γ and radius ΑΒ, let the perimeter of a circle ΜΘΝ be drawn, and let it cut ΚΘΛ at Θ, and let ΘΑ, ΘΒ, and ΘΓ be joined.

Above: Apollonius’ Diagram. Netz notes that the manuscripts have arcs for the circles instead of full circles (ΝΘΜ and ΚΛΘ). I have added the red circle. It has center Ξ and variable radius, which here is controlled by the slider bar. It is desired that the blue line ΔΕ pass through Θ: this will occur by adjusting the slider bar to an appropriate value.

Therefore ΒΓ is a parallelogram, and ΘΑ is its diameter. Let ΘΑ be bisected at Ξ. With center Ξ, let a circle be drawn, cutting the segments ΑΒ and ΑΓ (having been extended) at Δ and Ε, so that ΔΕ is in straight line with Θ. This will happen^[16] when a ruler is moved around Θ, cutting ΑΔ and ΑΕ, and being guided until the segments from Ξ to Δ and Ε become equal.

For when this has occurred, the thing being sought will be: for the construction is the same as those given by Heron and Philon, and it is clear that the same proof will suffice. 

[16] Aorist optative + ἄν: Smyth 1828.