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  • Archytas of Tarentum: Pythagorean, Philosopher, and Mathematician-King
  • Patrick Lee Miller
Carl Huffman , Archytas of Tarentum: Pythagorean, Philosopher, and Mathematician-King. Cambridge-New York: Cambridge University Press, 2005. Pp. xv + 665. Cloth, $180.00.

Archytas of Tarentum (b. 435–10, d. 360–50) has in some ages been considered a major philosopher. He was one of the three most important early Pythagoreans—along with Philolaus and Pythagoras himself—and so his reputation has risen and fallen with the popularity of this movement. When Pythagoreanism captivated some Romans at the end of the Republic, they championed him as a native talent; much later, some medieval Europeans inflated him into one of the great wise men of the ancient world. But Pythagoreanism has ebbed in recent centuries, and Archytas is now known only to specialists in ancient philosophy, and even then only by acquaintance from a few passages of Plato and Aristotle.

The most vivid of these is in the Seventh Letter, which tells how Archytas saved Plato's life from every teacher's worst nightmare: the student-turned-captor, Dionysius II. Aristotle's extant writings mention Archytas only three times, but among his lost writings there were three books about the Archytan philosophy—more than he wrote about any other single figure. Huffman is the first modern scholar to follow Aristotle's lead, devoting an entire work to this philosophy. Although he begins his book by claiming to "have no illusions of having produced that mythical beast, 'the definitive edition'" (xii), anyone intrepid enough to read its more than six-hundred pages may come away ready to believe in Cerberus, Charybdis, and the Chimaera; for once you believe in one mythical beast, why not believe in others too?

At the heart of Huffman's book lies a dense explanation—one that could have been leavened by more diagrams—of Archytas's doubling of the cube. Meno's slave famously tries to "double the square," but cannot succeed without the help of Socratic interrogation and an implicit version of the Pythagorean theorem. When it came to doubling the cube, a far more difficult problem, Archytas became the first to accomplish the feat, elegantly exploiting the notions of mean and proportion. Likewise, in mathematical harmonics he showed that a mean proportional could not be produced by the division of a "superparticular" ratio (a ratio of the form n + 1 : n). And yet notions of ratio and proportion animated more than just his work in geometry and harmonics. Seeing everything as a manifestation of proportion was, for Archytas, practicing "logistic." This science he considered supreme less for its theoretical primacy than for its practical power. By making proportional analogies, rulers could apply abstract sciences to concrete matters, where genuine wisdom is manifest. Such wisdom is accessible to all, according to Archytas, so the rulers should be the people themselves.

He thus justified the democracy of his native Tarentum, despite the aristocratic tendencies of other Pythagoreans. As is well known, Plato considered dialectic the supreme science, practiced only by the very few who enjoy privileged access to invisible Forms. He even went so far as to ridicule Archytas in Republic (530c–d) for attending too slavishly to sensible harmonies. But Archytas not only championed democracy, he led one seven times during the period when it was among the most powerful city-states in the Greek world. He thus rises from the pages of Huffman's book as a dramatic counterpoint to Plato, not to mention an unacknowledged influence on both him and Aristotle. Does he emerge as [End Page 165] their peer? Not quite. Huffman's introductory essays show Archytas embellishing—deftly, but embellishing nonetheless—the schematic philosophy of his teacher, Philolaus. (Who better to make this assessment than the author of the equally magisterial Philolaus of Croton?) Applying Philolaus's notions of unlimited and limiters to definition, for example, he practiced a technique that was imitated by Aristotle.

Archytas's best work seems to have been in the technical subjects of geometry and harmonics, but two of his arguments reveal a sharp polemical imagination. One is ethical and concludes that nothing is more hostile to intellect, and thus evil, than bodily pleasure...


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