A14 Eutocius, Commentary on Archimedes' On the Sphere and Cylinder II (III. 84.12–88.2 Heiberg/Stamatis)

(See also the Verba Filiorum of the Banu Musa, Clagett 1964: 334–41)

The Solution of Archytas, as Eudemus reports it.

Let the two given straight lines be AΔ and Γ. It is then necessary to find two mean proportionals of AΔ and Γ. Let the circle ABΔZ be drawn around the greater AΔ, let AB, equal to Γ a, be fit into (the circle) and being extended let it meet the line, which is tangent to the circle and drawn from Δ, at Π. Let line BEZ be drawn parallel to Π ΔO, and let a right semicylinder be conceived on the semicircle ABΔ, and on AΔ a semicircle at right angles lying in the rectangle of the semicylinder. When this semicircle is rotated from Δ to B, while the endpoint A of the diameter remains fixed, it will cut the cylindrical surface in its rotation and will describe a line on it. And again, if, while AΔ remains fixed, the triangle AΠΔ is rotated in an opposite motion to that of the semicircle, it will make the surface of a cone with the line AΔ, which as it is rotated will meet the line on the cylinder in a point. At the same time the point B will also describe a semicircle on the surface of the cone. Let the moving semicircle have as its position ΔKA at the place where the lines meet, and let the triangle being rotated in the opposite direction have as its place Δo?ΛA, and let the point of intersection described above be K.

The ancient evidence unambiguously shows that the upsilon in Archytas' name is long. Herodian, the great Greek grammarian of the second century ad, explicitly says in two places that the upsilon in Ἀρχύτας is long (Hdn. Gr. iii. 1, p. 77.12 Lentz; iii. 2, p. 851.33; see also iii. 1, p. 57.9–10; 3.2, p. 654.27–28; 3.2, p. 656.15. See also [ps-?] Arcadius,De Accentibus 28.17 and Theognostus, Canones sive De Orthographia 244.2 and 249.5). Moreover, the name Archytas appears in poems by Bion of Borysthenes (335–245 bc) and Eratosthenes of Cyrene (285–194 bc) and in each case the upsilon is shown to be long by the meter.

Archytas of Tarentum fits the popular conception of a Pythagorean better than anyone in the Pythagorean tradition, including Pythagoras himself. He was a distinguished mathematician; indeed we know of no other Pythagorean who even approached the mathematical prowess, which Archytas displays in his stunning solution to the problem of doubling the cube, the so-called Delian problem (A14). He also showed the typical Pythagorean interest in the mathematics of music, but again his analysis of the music of his day is by far the most sophisticated piece of harmonic theory in the early Pythagorean tradition (A16). Finally, he was elected leader of his city-state, Tarentum, seven consecutive times and his accomplishments in the political sphere are more impressive and better documented than those in the legends about Pythagoras (see “Life, writings and reception” above). Thus, we have the Pythagorean whom some have seen as the model for Plato's philosopher king (Guthrie 1962: 333). It is true, nonetheless, that Archytas has received relatively little attention from scholars of the history of ancient philosophy, let alone the educated community as a whole. He might justly be labeled “the lost Pythagorean.” Pythagoras has, of course, garnered the most attention, in large part because of his enormous importance in later antiquity (see, e.g., O'Meara 1989). The legend of Pythagoras has recently been debunked to some extent, and a more accurate appreciation of his accomplishment has been achieved (Burkert 1972a, Huffman 1999a: 66–75).

PRE-ARISTOTELIAN VIEWS ON SLEEP AND DREAMS

‘Anyone who has a correct understanding of the signs that occur in sleep, will discover that they have great significance for everything.’ This is the opening sentence of the fourth book of the Hippocratic work On Regimen, a treatise dating probably from the first half of the fourth century bce and dealing with the interpretation of dreams from a medical point of view, that is, as signs pointing to the (future) state of the body of the dreamer. The passage reflects the general opinion in ancient Greece that dreams are of great importance as ‘signs’ (sēmeia) or ‘indications’ (tekmēria), not only of the physical constitution of the dreamer and of imminent diseases or mental disturbances befalling him/her, but also of divine intentions, of things that may happen in the future, things hidden to normal human understanding. Dreams played an important part in Greek divination and religion, especially in the healing cult of Asclepius, because they were believed to contain important therapeutic indications or even to bring about healing themselves. The belief in the divine origin of dreams and in their prophetic power was widespread, even among intellectuals. As a result, dreams were mostly approached with caution because of their ambiguous nature. The Greeks realised that dreams, while often presenting many similarities with daytime experiences, may at the same time be bizarre or monstrous.

Few areas in classical scholarship have seen such rapid growth as the study of ancient medicine. Over the last three decades, the subject has gained broad appeal, not only among scholars and students of Greek and Roman antiquity but also in other disciplines such as the history of medicine and science, the history of philosophy and ideas, (bio-)archaeology and environmental history, and the study of the linguistic, literary, rhetorical and cultural aspects of intellectual ‘discourse’. The popularity of the subject even extends beyond the confines of academic communities, and ancient medicine has proved to be an effective tool in the promotion of the public understanding of medicine and its history.

The reasons for these changes are varied and complex, and to do justice to all would require a much fuller discussion than I can offer here. In this introductory chapter, I will concentrate on what I perceive to be the most important developments and in so doing set out the rationale of the present collection of papers. Evidently, ancient medicine possesses remarkable flexibility in attracting interest from a large variety of people approaching the field from a broad range of disciplines, directions and backgrounds, for a number of different reasons and with a wide variety of expectations. The purpose of publishing these papers in the present form is to make them more easily accessible to this growing audience.