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
- Diogenes Laertius, Vitae
philosophorum, viii.83.
- Aulus
Gellius, "Attic Nights", Book X, 12.9 at LacusCurtius
- ARCHYTAS OF TARENTUM, Technology Museum of Thessaloniki,
Macedonia, Greece
- Modern rocketry
- Automata history
- Vitruvius,
De architectura, vii.14.
- Thomas Nelson Winter, " The Mechanical Problems in the Corpus of Aristotle,"
DigitalCommons@University of Nebraska - Lincoln, 2007.
- Eutocius,
commentary on Archimedes' On the sphere and
cylinder.
- 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.
External links
Further reading
- Carl A. Huffman, "Archytas of Tarentum", Cambridge University
Press, 2005, ISBN 0521837464