We have now seen, in as interactive a manner as possible, the wealth of solutions offered by Greek mathematicians to solve the problem of finding two mean proportionals, and hence to solve the problem of duplicating a cube. The variety of approaches has always fascinated me, and I continue to look in awe at Archytas’ construction.

My plans for future work on this topic include looking at the duplication techniques recorded by other authors. Pappus, in Book III of the *Collection*, gives four methods. The methods presented there are a subset of those given by Eutocius: Pappus gives the constructions of Eratosthenes, Nicomedes, and Heron. For Nicomedes, though, he omits Eutocius’ preparatory materials on the construction and properties of the conchoid line. Pappus also gives his own method. As for Heron and Philon, each gives a construction in their artillery manuals, and describe the context of how finding mean proportionals is related to the construction of artillery. I also plan to look in more detail at the methods Dürer gives in his *Painter’s Manual*.

I also plan to direct attention to the mechanical methods of solving the problem. As I’ve presented them here, their “mechanical” nature is somewhat subdued, and they are really more abstract mechanical thought-experiments than actual machines. But it seems clear to me from Eutocius’ assessment of the utility of Philon’s construction that he at least imagined actually using a ruler, rather than the mathematical abstraction of one. Similarly, his description of Plato’s construction and Nicomedes’ conchoid lines are full of clearly mechanical terms: dove-tail grooves, knobs that slide in grooves, etc. I suspect that at least some of the mathematicians involved in the story had constructed (or commissioned) these machines. Eratosthenes himself went so far as to dedicate an altar and included on it his mesolabe; however, there is no archaeological record of it or of any of the other devices. That said, reconstruction of these machines should prove to further enrich our understanding and appreciation of the ancient techniques.

Readers interested in reading more about the problem, its history, and the textual issues are invited to consult the following works.

- Knorr's two books,
*The Ancient Tradition of Geometric Problems*[1993 reprint] and the later*Textual Studies in Ancient and Medieval Geometry*[1989]. The latter includes very thorough textual comparison between the methods given by Eutocius and those preserved in other authors. This includes Arabic traditions, such as Diocles’ work, which is not extant in Greek. - Saito’s paper,
*Doubling the Cube: A new interpretation of its significance in early Greek geometry*[1995]. The paper deals with technical details of justifying Hippocrates’ reduction of the cube duplication problem to finding two mean proportionals. - More philosophically-minded readers will no doubt be interested in the connection of Plato to the story. While the solution Eutocius attributes to Plato is almost certainly not genuinely due to
*the*Plato, mathematical themes certainly play prominently in the Platonic corpus. Kouremenos’ paper, “The tradition of the Delian problem and its origins in the Platonic corpus” [2011], discusses the lack of historicity of the “legend” of the Delians needing to actually duplicate a cubical temple, despite its repetition in various forms in authors from Plato to Eutocius to Dürer.