Eudoxus of Cnidus
Artist's depiction of Eudoxus, from a
(410 or 408 BC – 355 or 347 BC)
was a Greek astronomer
, scholar and student of Plato
. Since all his own works are lost, our knowledge
of him is obtained from secondary sources, such as Aratus
's poem on astronomy
Theodosius of Bithynia's
may be based on a work of Eudoxus.
His name "Eudoxus" means "good opinion" or of "good fame" (in
Εὔδοξος), from eu
= opinion or belief or fame). It is analogous
to the latin name "Benedictus" (Benedict, Benedetto).
Eudoxus's father Aeschines of Cnidus loved to watch stars at night.
first travelled to Tarentum to study
with Archytas, from whom he learned
While in Italy,
Eudoxus visited Sicily, where he studied medicine with Philiston
Around 387 BC, at the age of 23, he traveled with the physician
, who according to Diogenes
Laertius some believed was his lover, to Athens to study with the
followers of Socrates
. He eventually became
the pupil of Plato
, with whom he studied for
several months, but due to a disagreement they had a falling out.
Eudoxus was quite poor and could only afford an apartment at the
Piraeus. To attend Plato's lectures, he walked the seven miles
(11 km) each direction, each day. Due to his poverty, his
friends raised funds sufficient to send him to Heliopolis
, Egypt to pursue his study of
astronomy and mathematics. He lived there 16 months. From Egypt, he then
traveled north to Cyzicus, located on the
south shore of the Sea of Marmara, and the Propontis.
He traveled south to the court of Mausolus
. During his travels he gathered many
students of his own.
Around 368 BC, he returned to Athens with his students. Eudoxus
eventually returned to his native Cnidus, where he served in the
city assembly. While in Cnidus, he built an observatory and
continued writing and lecturing on theology, astronomy and
meteorology. He had one son, Aristagoras, and three daughters,
Actis, Philtis and Delphis.
In mathematical astronomy his fame is due to the introduction of
the astronomical globe
, and his early
contributions to understanding the movement of the planets
His work on proportions
insight into numbers
; it allows rigorous
treatment of continuous quantities and not just whole numbers
or even rational numbers
. When it was revived by
and others in
the 16th century, it became the basis for quantitative work in
science for a century, until it was replaced by the algebraic
methods of Descartes
Eudoxus rigorously developed Antiphon
's method of exhaustion
, which was used in
a masterly way by Archimedes
. This method
is a precursor to the integral
An algebraic curve
(the Kampyle of Eudoxus
) is named after
craters on Mars and the Moon are named in his honor.
- a2x4 = b4(x2 +
The Pythagoreans had discovered that the diagonal of a square does
not have a common unit of measurement with the sides of the square;
this is the famous discovery that the square root of 2 cannot be
expressed as the ratio of two integers. This discovery had heralded
the existence of incommensurable quantities beyond the integers and
rational fractions, but at the same time it threw into question the
idea of measurement and calculations in geometry as a whole. For
example, Euclid provides an elaborate proof of the Pythagorean
theorem, by using addition of areas instead of the much simpler
proof from similar triangles, which relies on ratios of line
Ancient Greek mathematicians calculated not with quantities and
equations as we do today, but instead they used proportionalities
to express the relationship between quantities. Thus the ratio of
two similar quantities was not just a numerical value, as we think
of it today; the ratio of two similar quantities was a primitive
relationship between them.
Eudoxus was able to restore confidence in the use of
proportionalities by providing an astounding definition for the
meaning of the equality between two ratios. This definition of
proportion forms the subject of Euclid's Book V.
In Definition 5 of Euclid's Book V we read:
Let us clarify it by using modern-day notation. If we take four
, and d
then the first and second have a ratio a/b; similarly the third and
fourth have a ratio c/d.
Now to say that a/b = c/d we do the following:For any two arbitrary
, form the
first and third; likewise form the equimultiples
of the second and
If it happens that m
, then we must also have m
.If it happens that m
, then we must also have
. Finally, if it
happens that m
, then we
must also have m
Notice that the definition depends on comparing the similar
, and the
similar quantities m
and does not depend on the existence of a common unit of measuring
The complexity of the definition reflects the deep conceptual and
methodological innovation involved. It brings to mind the famous
fifth postulate of Euclid
concerning parallels, which is more extensive and complicated in
its wording than the other postulates.
The Eudoxian definition of proportionality uses the quantifier,
"for every ..." to harness the infinite and the infinitesimal, just
as do the modern epsilon-delta
of limit and continuity.
Additionally, the Archimedean
stated as definition 4 of Euclid's book V is
originally due not to Archimedes but to Eudoxus.
In ancient Greece
, astronomy was a
branch of mathematics; astronomers sought to create geometrical
models that could imitate the appearances of celestial motions.
Identifying the astronomical work of Eudoxus as a separate category
is therefore a modern convenience. Some of Eudoxus' astronomical
texts whose names have survived include:
- Disappearances of the Sun, possibly on eclipses
- Oktaeteris (Ὀκταετηρίς), on an eight-year lunisolar
cycle of the calendar
- Phaenomena (Φαινόμενα) and Entropon
(Ἔντροπον), on spherical
astronomy, probably based on observations made by Eudoxus in
Egypt and Cnidus
- On Speeds, on planetary motions
We are fairly well informed about the contents of
, for Eudoxus' prose text was the basis for a
poem of the same name by Aratus
quoted from the text of Eudoxus in his
commentary on Aratus.
Eudoxan planetary models
A general idea of the content of On Speeds
can be gleaned
from Aristotle's Metaphysics
8, and a commentary by Simplicius
(6th century CE) on De caelo
, another work
by Aristotle. According to a story reported by Simplicius, Plato
posed a question for Greek astronomers: "By the assumption of what
uniform and orderly motions can the apparent motions of the planets
be accounted for?" (quoted in Lloyd 1970, p. 84). Plato
proposed that the seemingly chaotic wandering motions of the
planets could be explained by combinations of uniform circular
motions centered on a spherical Earth, apparently a novel idea in
the 4th century.
In most modern reconstructions of the Eudoxan model, the Moon is
assigned three spheres:
- The outermost rotates westward once in 24 hours, explaining
rising and setting.
- The second rotates eastward once in a month, explaining the
monthly motion of the Moon through the zodiac.
- The third also completes its revolution in a month, but its
axis is tilted at a slightly different angle, explaining motion in
latitude (deviation from the ecliptic), and
the motion of the lunar nodes.
The Sun is also assigned three spheres. The second completes its
motion in a year instead of a month. The inclusion of a third
sphere implies that Eudoxus mistakenly believed that the Sun had
motion in latitude.
The five visible planets (Venus
, and Saturn
assigned four spheres each:
- The outermost explains the daily motion.
- The second explains the planet's motion through the
- The third and fourth together explain retrogradation, when a planet
appears to slow down, then briefly reverse its motion through the
zodiac. By inclining the axes of the two spheres with respect to
each other, and rotating them in opposite directions but with equal
periods, Eudoxus could make a point on the inner sphere trace out a
figure-eight shape, or hippopede.
Importance of Eudoxan system
, a Greek astronomer of the 4th
century, added seven spheres to Eudoxus' original 27 (in addition
to the planetary spheres, Eudoxus included a sphere for the fixed
stars). Aristotle described both systems, but insisted on adding
"unrolling" spheres between each set of spheres to cancel the
motions of the outer set. Aristotle was concerned about the
physical nature of the system; without unrollers, the outer motions
would be transferred to the inner planets.
A major flaw in the Eudoxan system is its inability to explain
changes in the brightness of planets as seen from Earth. Because
the spheres are concentric, planets will always remain at the same
distance from Earth. This problem was pointed out in Antiquity by
Autolycus of Pitane
responded by introducing the deferent and epicycle
, which caused a
planet to vary its distance. However, Eudoxus' importance to
is considerable, as
he was the first to attempt a mathematical explanation of the
, in The Nicomachean Ethics
Eudoxus an argument in favor of hedonism
that is, that pleasure is the ultimate good that activity strives
for. According to Aristotle, Eudoxus put forward the following
arguments for this position:
- All things, rational and irrational, aim at pleasure; things
aim at what they believe to be good; a good indication of what the
chief good is would be the thing that most things aim at.
- Similarly, pleasure's opposite − pain − is universally avoided,
which provides additional support for the idea that pleasure is
universally considered good.
- People don't seek pleasure as a means to something else, but as
an end in its own right.
- Any other good that you can think of would be better if
pleasure were added to it, and it is only by good that good can be
- Of all of the things that are good, happiness is peculiar for
not being praised, which may show that it is the crowning
- Diogenes Laertius; VIII.87
- largely in book ten
- this particular argument is referenced in book one