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

The title of this paper is a nod to Underwood Dudley’s book The Trisectors [1994], but the purpose is rather different. Whereas Dudley exposes the myriad false trisections, my paper shows the myriad true cube duplications. This has one major benefit to me over Dudley: since the authors of the cube-duplications are long dead, and their methods are sound, I do not expect to be sued [Dilworth v. Dudley 1996]. 

It is well known to mathematicians and students with knowledge of abstract algebra that both of these problems are impossible with straightedge and compass, as is the problem of squaring a circle. This, I reckon, was at least suspected by the Greeks: see, for example, the humorous reference to squaring the circle in Aristophanes’ Birds [2000, lines 999–1012]. By allowing tools beyond just a straightedge and compass, the problems of trisecting the angle and duplicating the cube are solvable. (Squaring the circle, is, alas, not.) 

Fortunately for us as historians, Greek mathematicians found a number of solutions to the problem by instead solving a related one of finding two mean proportionals; I’ll detail the relationship in the next section. These solutions are incredibly diverse: some make use of mechanical instruments, others involve motion of figures in three dimensions, still others make interesting uses of the properties of conic sections. These are recorded by two ancient authors: Pappus, in Book III of the Collection, and Eutocius, in his commentary on Archimedes’ work On the Sphere and the Cylinder. While Eutocius lived later than Pappus, his collection contains more of the duplications, and so I have chosen to start this work with him. In future work, I plan to translate the relevant passages of Pappus and remark on the relationship between Eutocius’ and Pappus’ presentations of the methods that they both present. 

The text of this paper has two major parts: a translation of the relevant passages in Eutocius, along with my own notes, commentary, and diagrams. The Greek text is that of Heiberg’s edition [Eutocius 1891], recently reprinted by Cambridge [2013], and available in machine-readable format in the Thesaurus Linguae Graecae [Pantelia 2014]. Page and line numbers from Heiberg’s edition are given in parentheses at the start of some paragraphs, for those readers wishing to look at the corresponding Greek text. Selections of this text (and also many others) are also available in a two-volume set from the Loeb Classical Library [Thomas 1939a, 1939b]. These versions are especially helpful for those with limited (or no) proficiency with Greek, but who still want to see the Greek: all Loeb volumes have facing page style translation along with the original language text (Greek in our case, but the collection also features a vast array of Latin works). 

Netz has done a translation of both Archimedes’ text [2004] and Eutocius’ commentary [2004]. In particular, his work also has copious commentary about the diagrams in the manuscript tradition. The diagrams I give here are heavily influenced by Netz and Heiberg, though I have made some changes in order to make use of GeoGebra or to improve clarity. For example, the manuscript diagrams represent conic sections as circular arcs, but I have represented them in true conic-section form. Also, I have added parts to diagrams when it seemed helpful; for example, a circle mentioned in an argument but not drawn in a figure. The captions to each figure detail what changes or additions I have made. 

Readers of Convergence will recognize the relation of this paper with a recent one by Cawthorne and Green, Cubes, Conic Sections, and Crockett Johnson [2014]. In it, the authors give an overview of the cube duplication problem, discuss some of Johnson’s biography, and provide an analysis of Johnson's relevant painting. Readers might also recall that Albrecht Dürer knew of both the problem and at least some of the ancient solutions. His work The Painter’s Manual [1977] contains adaptations of the solutions to the problem due to Sporus and Plato, along with a variation on the tale Eratosthenes gives for the history of the problem. He also gives a method for drawing conchoid lines, similar in spirit to the method Eutocius ascribes to Nicomedes, but does not give the cube duplication that makes use of the conchoid line. The 1538 edition adds another method which resembles in some way both Descartes’ mesolabe and Eratosthenes’ sliding plates. Ishizu [2009] gives a very interesting discussion of Dürer’s motivation for solving the problem as it relates to a particular engraving, Melancolia (1514). 

As we will see in the History of the Problem section, solutions to the cube duplication problem were relevant also in engineering. Eratosthenes explicitly mentions the need for the solution if one is constructing artillery. Two other solutions appear in artillery manuals written by Heron and Philon; in the case of Heron, Eutocius cites Heron’s artillery manual as his source (though, it should be noted, the texts are not identical).