more spheres rotating oppositely with the synodic period of the planet, one with poles on
the equator of the second sphere, and the last, carrying the planet, with poles at an angle to
this. The motivation for the extra two spheres is uncertain, although the spheres for Mer-
cury and Venus certainly provided elongations from the Sun. Other suggested phenomena
include retrograde motion, invisibility periods, and latitudes.
On the other hand, the Sun and Moon only required one additional sphere each. Here
the second and third sphere of the Sun moved east/west, although the purpose of the third
sphere is not known, except that it provided a latitudinal motion for the Sun, rightly criti-
cized by H two centuries later. The best understood model is that of the Moon.
The outer sphere moved with the daily motion, while the second moved west/east with the
longitudinal motion plus the latitudinal motion, and the third moved oppositely with the
latitudinal motion, but at an angle perhaps of^1 / 12 or^1 / 15 of a right angle to the second
sphere. Here the latitudinal motion is, e.g., 1 cycle per interval from downward node to
downward node, while the longitudinal motion is 1 cycle per interval between conjunctions
with the same star (the zodiacal motion).
Eudoxos’ other work in astronomy includes notably the systematic organization of fixed
stars, which he described in two works, the Mirror and later the Phainomena, which survive in
A’ versification and Hipparkhos’ commentary on Aratos. In the first work the ratio of
longest day to shortest night for Greece is given as 5 : 3, and in the other as 12 : 7, as also in
P O. His division of the year, consisting of three seasons of 91 days and
autumn of 92 days, makes the seasons as equal as possible and would seem to be an explicit
rejection of M and E. Eudoxos may have endorsed an oktaete ̄ris calendar
(cf. K), also a rejection of Meto ̄n and Eukte ̄mo ̄n, but the attribution of the
book Oktaete ̄ris to him was doubted in antiquity (esp. by Eratosthene ̄s, cf. A,
Introduction 19). Traces of his parape ̄gma, possibly erected on De ̄los, survive in G
and P.
Eudoxos also wrote an extensive geography in seven books, The Circuit of the Earth, and
may have been the first to divide the Earth into regions according to projections of celestial
circles: equatorial, tropical, arctic and antarctic. He held that the inhabited world, from
India to the Iberian Peninsula, was twice as long as it was wide. The work included
ethnologies.
In other matters, Eudoxos defined the good as “what all things aim at” and identified it
with pleasure (Aristotle, Nicomachean Ethics 1.1.12 and 10.2–3). He also thought that Forms
were immanent in things (cf. Aristotle, Metaphyics A.9).
Ed.: Fr. Lasserre, Die Fragmente des Eudoxos von Knidos (1966).
Knorr (1975); Neugebauer (1975) 596, 620–621, 662, 675–689; W.R. Knorr, “Archimedes and the
Pre-Euclidean Proportion Theory,” AIHS 28 (1978) 183–244; I. Mueller, Philosophy of Mathematics and
Deductive Structure in Euclid’s Elements (1981); A.C. Bowen and B.R. Goldstein, “A New View of Early
Greek Astronomy,” Isis 74 (1983) 330–340; R.M. Dancy, Two Studies in the Early Academy (1991);
Henry Mendell, “The Trouble with Eudoxus,” in P. Suppes, J. Moravcsik, and Henry Mendell, edd.,
Ancient and Medieval Traditions in the Exact Sciences: Essays in Memory of Wilbur Knorr (2001) 59–138;
Henry Mendell, “Reflections on Eudoxus, Callippus and their Curves: Hippopedes and Callip-
popedes,” Centaurus 40 (1998) 177–275; I. Yavetz, “On the Homocentric Spheres of Eudoxus,”
AHES 51 (1998) 221–278.
Henry Mendell
EUDOXOS OF KNIDOS