(arithmetic, geometry, harmonics, astronomy), which he regarded as akin (B1). He was the
first to solve the famous problem of doubling the cube (A14) by having found two mean
proportionals between two given lines (a:x = x:y = y:2a; x^3 = 2a^3 ). His remarkable stereo-
metrical construction that for the first time introduces movement in geometry employed the
intersection of the cone, the torus and the half-cylinder, which produced the necessary
curve. Archytas’ discoveries might have prompted his pupil E K to
develop a similar kinematic theory for the motion of the heavenly bodies. Archytas’ arith-
metic was closely related to harmonics. He demonstrated that between numbers in the ratio
(n+1): n there is no mean proportional (A19), hence the basic harmonic intervals, e.g. the
octave (2:1), the fourth (4:3) and the fifth (3:2) cannot be divided in half. His researches in
acoustics combined mathematics with empirical observations and experiments, though not
always with correct results: following H, he considered the pitch of a sound to
depend on the velocity of its propagation (B1); Ciancaglini questioned this standard inter-
pretation. These and other studies of Archytas (A16–17, B2) completed Pythagorean har-
monics, which was further advanced by the E S C. In astronomy,
contrary to the subsequently dominant scheme, he argued for an unlimited universe (A24).
In physics Archytas developed the mathematical approach characteristic for Pythago-
reanism: any motion occurs according to proportion (analogia). In “natural,” circular
motion it is “the proportion of equality,” for “it is the only motion that returns to itself” (A
23a), as in the circular motion of the heavenly bodies. The causes of mechanical motion are
the unequal and uneven (A23), e.g., unequal arms of the lever. The A
C M drew upon Archytas’ discoveries. There are grounds to regard him
as a founder of mechanics and, possibly, of optics (A1, 25).
Following P, Archytas was engaged in philosophical analysis of mathematics,
in particular of its epistemological potential (B1, 3–4). He taught that arithmetic promotes
consent and justice in the society and even improves morality (B3). P’s first trip to Italy
(388 BCE) started his long acquaintance with Archytas. Though their relationship was not
devoid of rivalry, it was Archytas’ intervention that made possible Plato’s return from his
trip to Surakousai (361 BCE), where he was kept by the tyrant Dionysius II. Archytas was an
important source of Plato’s knowledge of Pythagoreanism and stimulated many of his
general ideas: on the ruler-philosopher, on beneficial influence of mathematics on the soul,
on mathematical sciences as a threshold of dialectic, etc. Mathematics, in which Archytas
was the main expert in his generation, served as a model for Plato’s theory of ideas and for
Aristotle’s logic. Aristotle devoted two special works to Archytas’ philosophy (A13); the
Peripatetic A, whose father was close to Archytas, wrote his biography.
DK 47; Thesleff (1965); F. Krafft, Dynamische und statische Betrachtungsweise in der antiken Mechanik (1970);
van der Waerden (1979); Barker (1989); G.E.R. Lloyd, “Plato and Archytas in the Seventh Letter,”
Phronesis 35 (1990) 159–173; C.A. Ciancaglini, “L’acustica in Archita,” Maia 50 (1998) 213–251;
M. Burnyeat, “Archytas and optics,” Science in context 18 (2005) 35–53; C.A. Huffman, Archytas of
Tarentum: Pythagorean, Philosopher and Mathematician King (2005); Zhmud (2006).
Leonid Zhmud
Arrabaios (of Macedon?) (250 BCE – 25 CE)
A P. cites his “Pontic” recipe for blood-spitting, G, CMLoc 7.4
(13.83 K.): reduce bear-berry (arkou staphulos) by one third, boiling in rainwater. Known only
from Macedon, LGPN 4.48 (5th–2nd cc. BCE; cf. Errabaios, 4.127; see Krahe, Lexicon altil-
ARRABAIOS (OF MACEDON?)