intervals exactly in half, and because “canonic theory,” on the other hand, denies that such
a division is mathematically possible in the first place. For this conclusion Panaitios relies on
several premises: (1) that the intervals in music can be shown to correspond to certain
mathematical ratios, which he demonstrates by means of a brief canonic division; (2)
that from this division it is evident that the ratio corresponding to the tone (tonos) is 9:8; (3)
that the 9:8 ratio cannot receive a geometric mean expressible in a ratio of whole numbers
(relying on a proof attributed by B to A and spelled out at E
S C prop. 3). From this he concludes that the tone cannot be divided into
two equal intervals (cf. Sect. can. prop. 16), and that the term he ̄mitonion is consequently as
much a misuse of language as the term he ̄mionos (mule, lit. “half-ass”).
The fragment is also noteworthy for its mention of sympathetic vibration of strings.
The phenomenon was noted by other ancient authors (A, the A
C P, A Q), but Panaitios is the only extant author
to connect it with the discovery of the concord-ratios.
Düring (1932); RE 18.3 (1949) 440–441 (#6), K. Ziegler; Barker (1989); Mathiesen (1999).
David Creese
Panaitios of Rhodes (Lindos) (ca 150 – 109 BCE)
Born ca 185 BCE, student of D B and A T and
successor of Antipatros as head of the Stoa from 129– 109 BCE. Active in Rome from the
140s onwards, he later divided his time between Rome and Athens, and his ethics, informed
by upper-class Roman interests, shows a marked practicality. He was a member of Scipio
Africanus the Younger’s circle (cf. C Somn.). Panaitios is generally taken to mark the
division between the “early” and “middle” Stoa. He emphasizes the role of an individual’s
own nature and dispositions in contrast to the earlier Stoics’ grounding of natural-law
ethics in a “universal nature.” His ethics underpins much of Cicero’s discussion in the
De officiis. Panaitios was also more eclectically influenced than most earlier Stoics, using
(among other targets of earlier Stoic critique) both P and A as authorities
and sources. In particular he preferred an eternal kosmos to the traditional Stoic con-
flagration. He was also, according to Cicero, the only Stoic to reject astrology, and his
arguments against astrology and divination are a source for some of the discussion in
Cicero, Div. 2.88. S (Q.Nat. 7.30.2) reports that Panaitios thought comets were “false,”
rather than “real,” stars.
Ed.: M. van Straaten, Panaetii Rhodii fragmenta (1952), with a rather over-enthusiastic idea of what
should count as a “fragment.”
Daryn Lehoux
Pandrosion, and anonymous students (ca 285 – 320 CE)
The female teacher of mathematics to whom P addresses the tract forming the first
part of what later became the third book of the Mathematical Collection (1.30–131 Hultsch), a
long and skillful response to a challenge set to him by (at least) three of Pandrosion’s
students, seeking his opinion about some geometrical constructions (30.17–22). The first
one (1.32), a clever (though erroneous) construction, perhaps derived from E’
mesolabe ̄ (Knorr 1989: 63–69) and was meant to find two geometrical means between two
given lines. The second one (68.17–25), seemingly not fully understood by Pappos himself, is
PANAITIOS OF RHODES (LINDOS)