The Duplicators, Part I: Eutocius' Collection of Cube Duplications - History of the Problem, According to Eratosthenes

(Heiberg 88.4) Eratosthenes to King Ptolemy, greetings!

They say that one of the ancient tragedy writers introduced Minos, who was building a tomb to Glaucos, and upon hearing that the tomb was to be a hundred feet in each dimension, said:

You have described a small part of a royal tomb:
Let it be doubled, but no less fine;
Quickly! Double each side of the tomb!

He seems to have been in error: for when the sides are doubled, the plane figure becomes quadrupled, and the solid figure octupled. This was examined by geometers, too, in what way one could double a given solid figure while keeping the same shape: they called this problem that of duplicating the cube, since assuming a cubical figure, they sought to double it. Having been at a loss for a long time, Hippocrates of Chios conceived that if, between two straight lines, with the larger double the smaller, two mean proportionals were found taken in continuous proportion, then the cube will be doubled. Hence he changed the problem into a different problem, but one no less challenging. 

After a while, they say that the Delians, having been exhorted by an oracle to double some altar, fell upon the exact same problem; they sent messengers to the esteemed geometers who were with Plato in the Academy, asking them to solve their problem. Of those industrious men who devoted themselves to finding two mean proportionals, Archytas of Tarentum is said to have done so by means of half-cylinders, and Eudoxus by so-called kampyles. But as it happened, all of them wrote in a technical sense, and so it was not possible to execute any of them by hand. Except, perhaps, in some small way, that of Menaechmus,^[2] but even still it is difficult. But we have devised a mechanical method, by which we are capable of finding not just two mean proportionals between two given magnitudes, but any number of means that might be demanded! 

Having found this, we will be able to, generally speaking, transform a given solid figure which is contained by parallelograms into a cube, or change the form of one solid into another, both to make it similar and, increasing it, to maintain this similarity, be it for temples or altars. We will also be able to do this both for liquid and dry measures, for instance the μετρητής^[3] or the μέδιμνος;^[4] we will be able to convert a vessel into a cube, and by measuring its sides, determine the capacity in liquid or dry measures.

This idea will be useful to those desiring to augment ballistas or catapults, for it is necessary to increase both the thickness and magnitude of the aperture, the washers, and the sinew.^[5] If the power is desired to be proportionally increased, it is not possible to do so without finding these mean proportionals. I have written to you about both the proof and construction of the aforementioned machines.^[6]

[2] Netz [2013] takes this differently, translating that Menaechmus did it via the shortness, but there is no clue as to what this term might mean.

[3] A unit of liquid measurement; approximately 9 gallons [Liddell and Scott 1940].

[4] A unit of solid measurement, for grain; approximately 12 gallons [Liddell and Scott 1940].

[5] These terms (aperture, washers, sinew) are technical, and part of a missile-throwing type of catapult. Eratosthenes uses them freely, implying the intended audience (King Ptolemy, if the letter is to be believed) was well-acquainted with the terms. I leave further discussion on the problem and its connection to military texts for future work.

[6] This separate material seems to not be extant, though technical descriptions do appear in other texts. See, for example, [Marsden 1971].