Archytas of Tarentum
c. 428-350? B.C.
Greek Mathematician
His achievements as a mathematician by themselves give Archytas of Tarentum distinction: not only was he first to integrate mathematics and mechanics, but he formulated the harmonic mean as a method for solving the problem of doubling the cube. But Archytas's achievements went beyond the realm of mathematics: he was not only a philosopher, but a great statesman and military leader. As a man, too, Archytas gained admiration for his acts of kindness, one of which was destined quite literally to change history.
Probably born at Tarentum (now Taranto, Italy), Archytas grew up during a time when Greek states still dominated the Mediterranean. Among the most powerful of those was Syracuse in Sicily, whose leader, the tyrant Dionysius the Elder (c. 430-367 B.C.), had begun to conquer parts of mainland Italy. He had thus driven most Pythagoreans from the southern part of the Italian Peninsula, and the school at Tarentum was the last local institution dedicated to the ideas of Pythagoras (c. 580-c. 500 B.C.).
It is possible that while teaching at the school in Tarentum, Archytas may have had for a pupil a man destined to become one of the most influential thinkers of all time: Plato (427-347 B.C.), who was almost his exact contemporary. In any case, Archytas played a key role in the history of Western thought by saving the great philosopher from execution by Dionysius the Younger, who succeeded his father in 367 B.C.
Plato had gone to Sicily as tutor for the young tyrant, but the latter had proven more interested in drinking and revelry than in learning. Looking for an excuse to dismiss his teacher, Dionysius had readily believed a story by a jealous official that Plato's teaching was part of an Athenian plot to gain control of Sicily. Only a letter from Archytas saved Plato's life at a time when most of his greatest works remained unwritten.
But this was far from Archytas's only contribution to thought. By applying mathematical principles to the use of pulley and screw, he became the first to wed mathematics to mechanics, and he later wrote a treatise on the marriage of the two disciplines. He also discussed the theory of means, and differentiated between an arithmetic, geometric, and harmonic means. Though he did not develop the last concept, Archytaswas the one who gave the harmonic mean—previously called a "subcontrary"—its name.
Archytas of Tarentum. (Bettmann/Corbis. Reproduced with permission.)The investigations of the harmonic mean arose in the course of Archytas's work on the problem of doubling a cube. This was a challenge much harder than it seemed at first, because if one simply doubled all sides of the cube, the resulting shape would be not twice, but eight times as large as the original. Archytas's innovation lay not in his use of two mean proportionals to deal with this question, but in his application of a semicircle rotating in three-dimensional space as a way of finding the harmonic mean. Thus he was among the first to introduce movement into geometry.
He was also credited with adding greatly to the number of known geometrical theorems, and has been credited with most of the ideas contained in Book VII of Euclid's (c. 325-c. 250 B.C.) highly influential Elements. In line with Pythagorean principles, Archytas was particularly interested in the application of mathematics to music, and discussed numerical ratios between notes. Scholars have often criticized him, however, for not always presenting clear and consistent explanations of his findings.
Part of the explanation for this may well be that Archytas was distracted. A highly influential figure in Magna Grecia, as Greek-controlled Italy was called, he was elected commander-in-chiefby a confederation of city-states, and reputedly went on to an unbroken series of military victories. He also wrote philosophical works, but few of those writings have survived—nor have three works by Aristotle (384-322 B.C.) on the philosophy of Archytas.
Archytas was also widely admired for his nobility and kindness, which he displayed in particular by his gentle treatment of his slaves and his fondness for babies and small children. The latter led him to apply his interest in mathematics and mechanics to the creation of two toys: a flying mechanical pigeon, and a rattle for amusing babies. According to the poet Horace (65-8 B.C.), Archytas died in a shipwreck on the Adriatic Sea.
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