# Archytas: Map

### Map showing all locations mentioned on Wikipedia article:

Archytas ( ; 428–347 BC) was an Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato.

## Life and work

Archytas was born in Tarentum, Magna Graecia (now Italy) and was the son of Mnesagoras or Histiaeus. For a while, he was taught by Philolaus, and was a teacher of mathematics to Eudoxus of Cnidus. Archytas and Eudoxus' student was Menaechmus.

Archytas

Archytas is believed to be the founder of mathematical mechanics. As only described in the writings of Aulus Gellius five centuries after him, he was reputed to have designed and built the first artificial, self-propelled flying device, a bird-shaped model propelled by a jet of what was probably steam, said to have actually flown some 200 meters. This machine, which its inventor called The Pigeon, may have been suspended on a wire or pivot for its flight. Archytas also wrote some lost works, as he was included by Vitruvius in the list of the twelve authors of works of mechanics. Thomas Winter has suggested that the pseudo-Aristotelian Mechanical Problems is an important mechanical work by Archytas, not lost after all, but misattributed.

Archytas introduced the concept of a harmonic mean, important much later in projective geometry and number theory. According to Eutocius, Archytas solved the problem of doubling the cube in his manner with a geometric construction. Hippocrates of Chios before, reduced this problem to finding mean proportionals. Archytas' theory of proportions is treated in book VIII of Euclid's Elements, where is the construction for two proportional means, equivalent to the extraction of the cube root. According to Diogenes Laertius, this demonstration, which uses lines generated by moving figures to construct the two proportionals between magnitudes, was the first in which geometry was studied with concepts of mechanics. The Archytas curve, which he used in his solution of the doubling the cube problem, is named after him.

Politically and militarily, Archytas appears to have been the dominant figure in Tarentum in his generation, somewhat comparable to Pericles in Athens a half-century earlier. The Tarentines elected him strategos, 'general', seven years in a row – a step that required them to violate their own rule against successive appointments. He was allegedly undefeated as a general, in Tarentine campaigns against their southern Italian neighbors. The Seventh Letter of Plato asserts that Archytas attempted to rescue Plato during his difficulties with Dionysius II of Syracuse. In his public career, Archytas had a reputation for virtue as well as efficacy. Some scholars have argued that Archytas may have served as one model for Plato's philosopher king, and that he influenced Plato's political philosophy as expressed in The Republic and other works (i.e., how does a society obtain good rulers like Archytas, instead of bad ones like Dionysus II?).

Archytas drowned in a shipwreck in the sea of Mattinata. His body lay unburied on the shore till a sailor humanely cast a handful of sand on it. Otherwise, he would have had to wander on this side the Styx for a hundred years, such the virtue of a little dust, munera pulveris, as Horace calls it.

The crater Archytas on the Moon is named in his honour.

## The Archytas Curve

The Archytas Curve is created by placing a semicircle (with a diameter of d) on the diameter of one of the two circles of a cylinder (which also has a diameter of d) such that the plane of the semicircle is at right angles to the plane of the circle and then rotating the semicircle about one of its ends in the plane of the cylinder's diameter. This rotation will cut out a portion of the cylinder forming the Archytas Curve.[33321]

Another, less mathematical, way of thinking of this construction is that the Archytas Curve is basically the result of cutting out a torus formed by toating a hemisphere of diameter d out of a cylinder also of diameter d. A cone can go through the same procedures also producing the Archytas Curve. Archytas used his curve to determine the construction of a cube with a volume of half of that of a given cube.

## Notes

1. Diogenes Laertius, Vitae philosophorum, viii.83.
2. Aulus Gellius, "Attic Nights", Book X, 12.9 at LacusCurtius
3. ARCHYTAS OF TARENTUM, Technology Museum of Thessaloniki, Macedonia, Greece
4. Modern rocketry
5. Automata history
6. Vitruvius, De architectura, vii.14.
7. Thomas Nelson Winter, " The Mechanical Problems in the Corpus of Aristotle," DigitalCommons@University of Nebraska - Lincoln, 2007.
8. Eutocius, commentary on Archimedes' On the sphere and cylinder.
9. Plato blamed Archytas for his contamination of geometry with mechanics (Plutarch, Questionum convivialium libri iii, 718E-F): And therefore Plato himself dislikes Eudoxus, Archytas, and Menaechmus for endeavoring to bring down the doubling the cube to mechanical operations; for by this means all that was good in geometry would be lost and corrupted, it falling back again to sensible things, and not rising upward and considering immaterial and immortal images, in which God being versed is always God.