Archytas of Tarentum Biography

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World of Mathematics on Archytas of Tarentum

Archytas of Tarentumwas a Greek mathematician of the Pythagorean school who formulated the harmonic mean and was the first to integrate mathematics and mechanics. He also developed an ingenious geometric solution for the ancient Greek problem of doubling the cube. A contemporary of the famous Greek philosopher Plato, Archytas was also famous in his own time as a philosopher, statesman, and military leader.

Only a few fragments identified as the work of Archytas have survived. As a result, most of what we know about his life and work comes from ancient Greek writers, such as Aristotleand Proclus. Archytas was probably born in Tarentum (now Taranto, Italy) around 428 B.C., possibly into an aristocratic family. Nothing else is known about his early life.

In the beginning of the fourth century B.C., Dionysius the Elder, a tyrant of Syracuse, had driven the Pythagoreansout of most southern Italy's cities. Tarentum was the last city in the region where the Pythagorean school of philosophy and mathematics maintained a strong presence in education and politics. A close friend of Plato, Archytas may have been his chief teacher of Pythagorean science and philosophy. More importantly, Archytas saved this great philosopher's life by obtaining a pardon for him through a letter he wrote to Dionysius the Younger, who wanted to execute Plato for subversive activities.

Archytas was a powerful statesman and an influential leader in Tarentum and throughout the Greek city-states. Immensely popular, he served as general of his city's citizen army for seven years, despite a law forbidding anyone to hold the position for more than one year. The confederation of Hellenic cities of Magna Grecia also appointed him commander, with full autocratic authority, over the confederation's armies. Troops under his command reportedly were never defeated in battle. According to some accounts, Archytas eventually gave up his command because of envious detractors, and his troops were immediately captured. Archytas was also greatly admired for his virtues and noble character, which included a love of children and the just and kind treatment of his slaves.

Makes Major Contributions in Mathematics and Geometry

Archytas' creativity and ingeniousness were grounded in his recognition of how the sciences interconnect, especially the disciplines of mathematics, geometry, music, and astronomy. For example, he was the first to apply mathematics to the realm of mechanics and wrote a systematic treatise on the subject. He invented the simple pulley and screw and wrote on the mathematical basis of astronomy. Through his work in the theory of means and proportions he differentiated three basic means: the arithmetic mean, the geometric mean, and the harmonic mean. Seven other means were eventually added by Archytas and others. In his theory of music, Archytas developed numerical ratios representing intervals of the tetrachord on which he based his three musical scales: the enharmonic, the chromatic, and the diatonic. Claudius Ptolemythe astronomer credited Archytas as the most important Pythagorean to delve into the theory of music.

According to Proclus, Archytas also increased the number of theorems in geometry, developing them into a systematic body of knowledge, and influenced many other Greek mathematicians. Archytas is credited with most of the geometry contained in Book VIII of Euclid's Elements, which served as the primary textbook of elementary geometry and logic for more than a thousand years.

Archytas' most famous mathematical achievement was to provide an elegant geometric solution to the Delian problem, known as duplicating or doubling the cube, or enlarging a cube according to a given ratio. Although Plato had complained that the Greeks knew little about three-dimensional geometry, Archytas exhibited a comprehensive knowledge of this area in his remarkable solution to the problem. By inventing a new type of three-dimensional curve through the intersection of a cylinder, a cone, and a torus (or doughnut shape), he was able to find the two mean proportionals (or geometric means) between two lines, a method first proposed by Hippocrates of Chios for doubling the cube.

Although he was a pioneer in mathematics and geometry, Archytas has been criticized for not following his contemporaries' lead in applying clear and logical explanations for his theories. His inability to cope well with the logical aspects of his work also affected other interests. For example, in the largest fragment of his extant works, Archytas proposed a complicated theory of sound. He correctly theorized that faster motion produces higher sounds or notes. However, based solely on empirical observations without the application of mathematical theories, Archytas wrongly concluded that higher sounds reach the listener faster than lower sounds.

Archytas' philosophy was probably based on his training as a Pythagorean, leading him to hold a strong belief in numbers as a mystical and basic part of nature. In a small fragment of one of his works, Archytas also reveals that he believed the universe was infinite in extent. Although Aristotle wrote three books on the philosophy of Archytas, none of them have survived the passing of time.

While Archytas is rightly remembered today for his seminal contributions in mathematics and geometry, he should also be admired as a man of action who applied his keen intellect to affairs of state. A successful army general, Archytas was an influential leader who helped forge alliances among Greek city-states to provide greater protection against foreign powers. Archytas also had his playful side; his interest in mechanics led him to create two mechanical devices: a mechanical wooden pigeon that could fly and, according to Aristotle, a type of rattle to amuse and occupy infants. In an ode, the famous Greek poet Horacerecounts Archytas' death by shipwreck in the Adriatic Sea.