[Note: This discussion should be supplemented with the sections on authenticity in the commentary on each of the genuine fragments and testimonia and by the more detailed discussion of specific pseudo-Archytan treatises in the appendix on spurious writings.]

In the case of most ancient authors, if a medieval manuscript or an ancient source ascribes a given text to that author, we assume that the text is genuine until proven otherwise. There are of course works ascribed in the ancient tradition to prominent authors, such as Plato and Aristotle, which modern scholars with good reason regard as spurious. The number of genuine works of Plato and Aristotle, however, far outnumber works judged to be spurious. This situation is almost completely reversed in the Pythagorean tradition. Thesleff's collection of spurious Pythagorean texts (1965) runs to some 245 pages. Out of the forty-four authors listed in Thesleff, in the case of only two, Archytas and Philolaus, are most modern scholars willing to agree that even some authentic fragments also survive, although there are some reliable testimonia about a few others (e.g. Hippasus and Eurytus). In the case of Archytas there are approximately 45 pages and 1,200 lines of almost certainly spurious texts collected in Thesleff, in contrast to the 7 pages and 100 lines of text in DK, which most scholars have accepted as authentic. Thus, in terms of number of lines, the amount of authentic material is less than 10 percent of the amount of spurious material.

The last book devoted to Archytas was published over 160 years ago (Gruppe 1840). Even that work was not really a study of Archytas' thought but rather an unsuccessful attempt to argue that no authentic fragments of Archytas had survived from antiquity. It is not an exaggeration to say, then, that there has never been a book-length study of Archytas of Tarentum. There have not even been many shorter treatments. Erich Frank gave Archytas a fairly prominent role in his reconstruction of early Pythagoreanism (1923), but that reconstruction was eccentric and has been largely rejected by scholars. Essentially the only commentary has been that in Italian by Maria Timpanaro Cardini, as part of a three-volume commentary on all the Pythagoreans (1958–64). In recent years there have been a few important articles and sections of larger works dealing with isolated aspects of Archytas' work, notably his harmonic theory (e.g. Barker 1989, 1994; Bowen 1982; Cambiano 1998 and Lloyd 1990), but to say that Archytas has been neglected would be an understatement. Nonetheless, Archytas is one of the three most important figures in ancient Pythagoreanism (along with Pythagoras himself and Philolaus); we cannot hope to understand ancient Pythagoreanism without understanding Archytas. He was also an important philosopher, mathematician and political leader in his own right. Most scholarship on Greek philosophy during the first half of the fourth century has been devoted to Plato and the Academy.

SOURCES

(The original texts and translations of the testimonia for Archytas' life are found in Part Three, Section One)

Archytas did not live the life of a philosophical recluse. He was the leader of one of the most powerful Greek city-states in the first half of the fourth century bc. Unfortunately he is similar to most important Greek intellectuals of the fifth and fourth centuries bc, in that we have extremely little reliable information about his activities. This dearth of information is all the more frustrating since we know that Aristoxenus wrote a biography of Archytas, not long after his death (A9). Two themes bulk large in the bits of evidence that do survive from that biography and from other evidence for Archytas' life. First, there is Archytas' connection to Plato, which, as we will see, was more controversial in antiquity than in most modern scholarship. The Platonic Seventh Letter, whose authenticity continues to be debated, portrays Archytas as saving Plato from likely death, when Plato was visiting the tyrant Dionysius II at Syracuse in 361 bc. Second, for Aristoxenus, Archytas is the paradigm of a successful leader. Elected general (stratēgos) repeatedly, he was never defeated in battle; as a virtuous, kindly and democratic ruler, he played a significant role in the great prosperity of his native Tarentum, located on the heel of southern Italy.

The standard collection of all the pseudo-Archytan writings is Thesleff (1965). New editions and commentaries on individual treatises have appeared since Thesleff and are cited below. In a few cases, where Thesleff has not provided the text, I provide enough of the text to reveal its nature. All of the following treatises are, in my judgment, surely spurious except On Law and Justice, where the evidence is more complicated. In the case of most treatises, I have given the main arguments for spuriousness. On the pseudo-Archytan treatises in general see Burkert 1972b, Centrone 1990 and 1994b, Moraux 1984: 605–83, and Thesleff 1972 and 1961.

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.