Diokle ̄s (ca 200 – 175 BCE)
Mathematician, probably lived in Arkadia
for a while, as he mentions (in the intro-
duction to his one surviving work) another
mathematician visiting him there. Frag-
ments of Diokle ̄s had long been known
from E’ commentary on
A’ On the Sphere and Cylinder, but
until Toomer’s 1976 publication of the
then-newly-found Arabic translation of
Diokle ̄s’ On Burning Mirrors, no full text
survived. On Burning Mirrors treats gener-
ally, but not exclusively, the geometry of
conic sections. Diokle ̄s presents some orig-
inal work on focal properties of para-
bolas, including the construction of a
parabola with any given focal length (lead-
ing Toomer to attribute to Diokle ̄s the first
construction of a parabola from focus and
directrix). Diokle ̄s next turns to a problem
posed by Archime ̄de ̄s: dividing a sphere
such that the two segments bear a given
ratio to each other, which Diokle ̄s solves by
means of intersecting an ellipse and an
hyperbola. Finally, he addresses the classic
question of doubling the cube, solved in
two ways. His first solution employs the
intersection of two parabolas and his second a cissoid, to find two mean proportionals
between two given magnitudes. Cf. “D” (probably later).
Ed.: Toomer (1976); Rashed (2000) 3–151.
W.R. Knorr, “The Geometry of Burning Mirrors in Antiquity,” Isis 74 (1983) 53–73.
Daryn Lehoux
Diokle ̄s of Karustos (400 – 300 BCE)
Life. Son of Arkhidamos, one of the most celebrated physicians and medical writers in
antiquity: regularly described by ancient sources as a Dogmatist (fr.2: C, and
later). He is sometimes said to be H’ pupil or follower (frr.3, 40), to have
come second in age and fame only to Hippokrate ̄s (fr.4), and was called a younger Hip-
pokrate ̄s by the Athenians (fr.3). Since Diokle ̄s is thought to have written the first system-
atic handbook on anatomy (fr.17), Wellmann dated him to the first half of the 4th c. BCE.
W. Jaeger’s later date, ca 340 – 260, making Diokle ̄s A’s younger contemporary
(and his pupil), rests on controvertible evidence (e.g. a probably spurious dietetic letter
transmitted by P A, fr.183a). Thus, it is only certain that Diokle ̄s lived
after Hippokrate ̄s and presumably somewhat earlier than H and E-
. His relationship to his predecessors and Aristotle or the Peripatos remains obscure:
the Diokle ̄s quoted in T’ On Stones (fr.239a) is not necessarily the Karustian,
Diokle ̄s’ Cissoid. Given: circle ABDG with per-
pendicular diameters AB and DG, where arc
DZ=ZH=ΗΘ=DN=NS=SO, and KZ, LH and
ΜΘ are perpendicular to AB. Line DPQRB is a
cissoid, and ZK and KB are two mean proportion-
als between AK and KP, so that AK : ZK = ZK :
KB = KB: KP. We can then (if we like) work out
that for the special case where AK is twice the
length of KP, then a cube whose sides equal line
KP will be doubled by building a cube whose sides
Diokle ̄ s’ Cissoid © Lehoux and Massie
DIOKLE ̄S OF KARUSTOS