Eudoxus of Cnidus

Greek mathematician and astronomer

born c. 395–390 bc, Cnidus, Asia Minor [now in Turkey]

died c. 342–337 bc, Cnidus

Main

Greek mathematician and astronomer who substantially
advanced proportion theory, contributed to the
identification of constellations and thus to the development
of observational astronomy in the Greek world, and
established the first sophisticated, geometrical model of
celestial motion. He also wrote on geography and contributed
to philosophical discussions in Plato’s Academy. Although
none of his writings survive, his contributions are known
from many discussions throughout antiquity.

Life

According to the 3rd century ad historian Diogenes Laërtius
(the source for most biographical details), Eudoxus studied
mathematics with Archytas of Tarentum and medicine with
Philistion of Locri. At age 23 he attended lectures in
Athens, possibly at Plato’s Academy (opened c. 387 bc).
After two months he left for Egypt, where he studied with
priests for 16 months. Earning his living as a teacher,
Eudoxus then returned to Asia Minor, in particular to
Cyzicus on the southern shore of the Sea of Marmara, before
returning to Athens where he associated with Plato’s
Academy.

Aristotle preserved Eudoxus’s views on metaphysics and
ethics. Unlike Plato, Eudoxus held that forms are in
perceptible things. He also defined the good as what all
things aim for, which he identified with pleasure. He
eventually returned to his native Cnidus where he became a
legislator and continued his research until his death at age
53. Followers of Eudoxus, including Menaechmus and
Callippus, flourished in both Athens and in Cyzicus.

Mathematician

Eudoxus’s contributions to the early theory of proportions
(equal ratios) forms the basis for the general account of
proportions found in Book V of Euclid’s Elements (c. 300
bc). Where previous proofs of proportion required separate
treatments for lines, surfaces, and solids, Eudoxus provided
general proofs. It is unknown, however, how much later
mathematicians may have contributed to the form found in the
Elements. He certainly formulated the bisection principle
that given two magnitudes of the same sort one can
continuously divide the larger magnitude by at least halves
so as to construct a part that is smaller than the smaller
magnitude.

Similarly, Eudoxus’s theory of incommensurable magnitudes
(magnitudes lacking a common measure) and the method of
exhaustion (its modern name) influenced Books X and XII of
the Elements, respectively. Archimedes (c. 285–212/211 bc),
in On the Sphere and Cylinder and in the Method, singled out
for praise two of Eudoxus’s proofs based on the method of
exhaustion: that the volumes of pyramids and cones are
one-third the volumes of prisms and cylinders, respectively,
with the same bases and heights. Various traces suggest that
Eudoxus’s proof of the latter began by assuming that the
cone and cylinder are commensurable, before reducing the
case of the cone and cylinder being incommensurable to the
commensurable case. Since the modern notion of a real number
is analogous to the ancient notion of ratio, this approach
may be compared with 19th-century definitions of the real
numbers in terms of rational numbers. Eudoxus also proved
that the areas of circles are proportional to the squares of
their diameters.

Eudoxus is also probably largely responsible for the
theory of irrational magnitudes of the form a ± b (found in
the Elements, Book X), based on his discovery that the
ratios of the side and diagonal of a regular pentagon
inscribed in a circle to the diameter of the circle do not
fall into the classifications of Theaetetus of Athens (c.
417–369 bc). According to Eratosthenes of Cyrene (c. 276–194
bc), Eudoxus also contributed a solution to the problem of
doubling the cube—that is, the construction of a cube with
twice the volume of a given cube.

Astronomer

In two works, Phaenomena and Mirror, Eudoxus described
constellations schematically, the phases of fixed stars (the
dates when they are visible), and the weather associated
with different phases. Through a poem of Aratus (c. 315–245
bc) and the commentary on the poem by the astronomer
Hipparchus (c. 100 bc), these works had an enduring
influence in antiquity. Eudoxus also discussed the sizes of
the Sun, Moon, and Earth. He may have produced an eight-year
cycle calendar (Oktaëteris).

Perhaps Eudoxus’s greatest fame stems from his being the
first to attempt, in On Speeds, a geometric model of the
motions of the Sun, the Moon, and the five planets known in
antiquity. His model consisted of a complex system of 27
interconnected, geo-concentric spheres, one for the fixed
stars, four for each planet, and three each for the Sun and
Moon. Callippus and later Aristotle modified the model.
Aristotle’s endorsement of its basic principles guaranteed
an enduring interest through the Renaissance.

Eudoxus also wrote an ethnographical work (“Circuit of
the Earth”) of which fragments survive. It is plausible that
Eudoxus also divided the spherical Earth into the familiar
six sections (northern and southern tropical, temperate, and
arctic zones) according to a division of the celestial
sphere.

Assessment

Eudoxus is the most innovative Greek mathematician before
Archimedes. His work forms the foundation for the most
advanced discussions in Euclid’s Elements and set the stage
for Archimedes’ study of volumes and surfaces. The theory of
proportions is the first completely articulated theory of
magnitudes. Although most astronomers seem to have abandoned
his astronomical views by the middle of the 2nd century bc,
his principle that every celestial motion is uniform and
circular about the centre endured until the time of the
17th-century astronomer Johannes Kepler. Dissatisfaction
with Ptolemy’s modification of this principle (where he made
the centre of the uniform motion distinct from the centre of
the circle of motion) motivated many medieval and
Renaissance astronomers, including Nicolaus Copernicus
(1473–1543).

**Henry Ross Mendell**