Eudoxus of Cnidus

Last updated
Eudoxus of Cnidus
Bornc.400 BC [1]
Diedc.350 BC [1]
Knidos, Asia Minor
Known for Kampyle of Eudoxus
Concentric spheres
Scientific career

Eudoxus of Cnidus ( /ˈjuːdəksəs/ ; Ancient Greek : Εὔδοξος ὁ Κνίδιος, Eúdoxos ho Knídios; c. 408 – c.355 BC [1] [2] ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his works are lost, though some fragments are preserved in Hipparchus' commentary on Aratus's poem on astronomy. [3] Sphaerics by Theodosius of Bithynia may be based on a work by Eudoxus.



Eudoxus was born and died in Cnidus (also spelled Knidos), [2] which was a city on the southwest coast of modern day Turkey. The years of Eudoxus' birth and death are not fully known but the range may have been c. 408 – c.355 BC, [1] [2] or c. 390 – c.337 BC. His name Eudoxus means "honored" or "of good repute" (εὔδοξος, from eu "good" and doxa "opinion, belief, fame"). It is analogous to the Latin name Benedictus.

Eudoxus's father, Aeschines of Cnidus, loved to watch stars at night. Eudoxus first traveled to Tarentum to study with Archytas, from whom he learned mathematics. While in Italy, Eudoxus visited Sicily, where he studied medicine with Philiston.

At the age of 23, he traveled with the physician Theomedon—who (according to Diogenes Laërtius) some believed was his lover [4] —to Athens to study with the followers of Socrates. He eventually attended lectures of Plato and other philosophers 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 7 miles (11 km) in 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 for 16 months. From Egypt, he then traveled north to Cyzicus, located on the south shore of the Sea of Marmara, the Propontis. He traveled south to the court of Mausolus. During his travels he gathered many students of his own.[ citation needed ]

Around 368 BC, Eudoxus returned to Athens with his students. According to some sources,[ citation needed ] around 367 he assumed headship (scholarch) of the Academy during Plato's period in Syracuse, and taught Aristotle.[ citation needed ] He 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 concentric spheres, and his early contributions to understanding the movement of the planets.

His work on proportions shows insight into real numbers; it allows rigorous treatment of continuous quantities and not just whole numbers or even rational numbers. When it was revived by Tartaglia and others in the 16th century, it became the basis for quantitative work in science, and inspired the work of Richard Dedekind. [5]

Craters on Mars and the Moon are named in his honor. An algebraic curve (the Kampyle of Eudoxus) is also named after him.


Eudoxus is considered by some to be the greatest of classical Greek mathematicians, and in all Antiquity second only to Archimedes. [6] Eudoxus was probably the source for most of book V of Euclid's Elements. [7] He rigorously developed Antiphon's method of exhaustion, a precursor to the integral calculus which was also used in a masterly way by Archimedes in the following century. In applying the method, Eudoxus proved such mathematical statements as: areas of circles are to one another as the squares of their radii, volumes of spheres are to one another as the cubes of their radii, the volume of a pyramid is one-third the volume of a prism with the same base and altitude, and the volume of a cone is one-third that of the corresponding cylinder. [8]

Eudoxus introduced the idea of non-quantified mathematical magnitude to describe and work with continuous geometrical entities such as lines, angles, areas and volumes, thereby avoiding the use of irrational numbers. In doing so, he reversed a Pythagorean emphasis on number and arithmetic, focusing instead on geometrical concepts as the basis of rigorous mathematics. Some Pythagoreans, such as Eudoxus's teacher Archytas, had believed that only arithmetic could provide a basis for proofs. Induced by the need to understand and operate with incommensurable quantities, Eudoxus established what may have been the first deductive organization of mathematics on the basis of explicit axioms. The change in focus by Eudoxus stimulated a divide in mathematics which lasted two thousand years. In combination with a Greek intellectual attitude unconcerned with practical problems, there followed a significant retreat from the development of techniques in arithmetic and algebra. [8]

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 (Elements I.47), by using addition of areas and only much later (Elements VI.31) a simpler proof from similar triangles, which relies on ratios of line segments.

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:

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

By using modern-day notation, this is clarified as follows. If we take four quantities: a, b, c, and d, then the first and second have a ratio ; similarly the third and fourth have a ratio .

Now to say that we do the following: For any two arbitrary integers, m and n, form the equimultiples m·a and m·c of the first and third; likewise form the equimultiples n·b and n·d of the second and fourth.

If it happens that m·a > n·b, then we must also have m·c > n·d. If it happens that m·a = n·b, then we must also have m·c = n·d. Finally, if it happens that m·a < n·b, then we must also have m·c < n·d.

Notice that the definition depends on comparing the similar quantities m·a and n·b, and the similar quantities m·c and n·d, and does not depend on the existence of a common unit of measuring these quantities.

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 definitions of limit and continuity.

Additionally, the Archimedean property stated as definition 4 of Euclid's book V is originally due not to Archimedes but to Eudoxus. [9]


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's astronomical texts whose names have survived include:

We are fairly well informed about the contents of Phaenomena, for Eudoxus's prose text was the basis for a poem of the same name by Aratus. Hipparchus 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 XII, 8, and a commentary by Simplicius of Cilicia (6th century AD) 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 BC.

In most modern reconstructions of the Eudoxan model, the Moon is assigned three spheres:

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.

Animation depicting Eudoxus's model of retrograde planetary motion. The two innermost homocentric spheres of his model are represented as rings here, each turning with the same period but in opposite directions, moving the planet along a figure-eight curve, or hippopede. Animated Hippopede of Eudoxus.gif
Animation depicting Eudoxus's model of retrograde planetary motion. The two innermost homocentric spheres of his model are represented as rings here, each turning with the same period but in opposite directions, moving the planet along a figure-eight curve, or hippopede.
Eudoxus's model of planetary motion. Each of his homocentric spheres is represented here as a ring which rotates on the axis shown. The outermost (yellow) sphere rotates once per day; the second (blue) describes the planet's motion through the zodiac; the third (green) and fourth (red) together move the planet along a figure-eight curve (or hippopede) to explain retrograde motion. Eudoxus' Homocentric Spheres.png
Eudoxus's model of planetary motion. Each of his homocentric spheres is represented here as a ring which rotates on the axis shown. The outermost (yellow) sphere rotates once per day; the second (blue) describes the planet's motion through the zodiac; the third (green) and fourth (red) together move the planet along a figure-eight curve (or hippopede) to explain retrograde motion.

The five visible planets (Venus, Mercury, Mars, Jupiter, and Saturn) are assigned four spheres each:

Importance of Eudoxan system

Callippus, a Greek astronomer of the 4th century, added seven spheres to Eudoxus's 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 Eudoxian 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. Astronomers responded by introducing the deferent and epicycle, which caused a planet to vary its distance. However, Eudoxus's importance to astronomy and in particular to Greek astronomy is considerable.


Aristotle, in the Nicomachean Ethics , [10] attributes to 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:

  1. 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.
  2. Similarly, pleasure's opposite—pain—is universally avoided, which provides additional support for the idea that pleasure is universally considered good.
  3. People don't seek pleasure as a means to something else, but as an end in its own right.
  4. 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 increased.
  5. Of all of the things that are good, happiness is peculiar for not being praised, which may show that it is the crowning good. [11]

See also

Related Research Articles

Euclid Greek mathematician, inventor of axiomatic geometry

Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I. His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.

Celestial sphere Imaginary sphere of arbitrarily large radius, concentric with the observer

In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, which may be centered on Earth or the observer. If centered on the observer, half of the sphere would resemble a hemispherical screen over the observing location.

Ratio Relationship between two numbers of the same kind

In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six. Similarly, the ratio of lemons to oranges is 6∶8 and the ratio of oranges to the total amount of fruit is 8∶14.

The cosmological model of concentricspheres, developed by Eudoxus, Callippus, and Aristotle, employed celestial spheres all centered on the Earth. In this respect, it differed from the epicyclic and eccentric models with multiple centers, which were used by Ptolemy and other mathematical astronomers until the time of Copernicus.


Archytas was an Ancient Greek philosopher, mathematician, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder of mathematical mechanics, as well as a good friend of Plato.

Celestial spheres elements of some cosmological models

The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus, and others. In these celestial models, the apparent motions of the fixed stars and planets are accounted for by treating them as embedded in rotating spheres made of an aetherial, transparent fifth element (quintessence), like jewels set in orbs. Since it was believed that the fixed stars did not change their positions relative to one another, it was argued that they must be on the surface of a single starry sphere.

Speusippus Ancient Greek philosopher

Speusippus was an ancient Greek philosopher. Speusippus was Plato's nephew by his sister Potone. After Plato's death, c. 348 BC, Speusippus inherited the Academy, near age 60, and remained its head for the next eight years. However, following a stroke, he passed the chair to Xenocrates. Although the successor to Plato in the Academy, Speusippus frequently diverged from Plato's teachings. He rejected Plato's Theory of Forms, and whereas Plato had identified the Good with the ultimate principle, Speusippus maintained that the Good was merely secondary. He also argued that it is impossible to have satisfactory knowledge of any thing without knowing all the differences by which it is separated from everything else.

Fixed stars Astronomical bodies that appear not to move relative to each other in the night sky

The fixed stars compose the background of astronomical objects that appear not to move relative to each other in the night sky compared to the foreground of Solar System objects that do. Generally, the fixed stars are taken to include all stars other than the Sun. Nebulae and other deep-sky objects may also be counted among the fixed stars.


Hippasus of Metapontum was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this. However, the few ancient sources which describe this story either do not mention Hippasus by name or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer.

Autolycus of Pitane

Autolycus of Pitane was a Greek astronomer, mathematician, and geographer. The lunar crater Autolycus was named in his honour.

Greek mathematics Mathematics of Ancient Greeks

Greek mathematics refers to mathematics texts written during and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by Greek culture and the Greek language. The word "mathematics" itself derives from the Ancient Greek: μάθημα, romanized: máthēmaAttic Greek: [má.tʰɛː.ma]Koine Greek: [ˈma.θ], meaning "subject of instruction". The study of mathematics for its own sake and the use of generalized mathematical theories and proofs is an important difference between Greek mathematics and those of preceding civilizations.

The history of science in early cultures covers protoscience in ancient history, prior to the development of science in the Middle Ages. In prehistoric times, advice and knowledge was passed from generation to generation in an oral tradition. The development of writing enabled knowledge to be stored and communicated across generations with much greater fidelity. Combined with the development of agriculture, which allowed for a surplus of food, it became possible for early civilizations to develop and spend more of their time devoted to tasks other than survival, such as the search for knowledge for knowledge's sake.

History of science in classical antiquity

The history of science in classical antiquity encompasses both those inquiries into the workings of the universe aimed at such practical goals as establishing a reliable calendar or determining how to cure a variety of illnesses and those abstract investigations known as natural philosophy. The ancient peoples who are considered the first scientists may have thought of themselves as natural philosophers, as practitioners of a skilled profession, or as followers of a religious tradition. The encyclopedic works of Aristotle, Archimedes, Hippocrates, Hipparchus, Galen, Ptolemy, Euclid, and others spread throughout the world. These works and the important commentaries on them were the wellspring of science.

Callippus was a Greek astronomer and mathematician.

The unmoved mover or prime mover is a concept advanced by Aristotle as a primary cause or "mover" of all the motion in the universe. As is implicit in the name, the unmoved mover moves other things, but is not itself moved by any prior action. In Book 12 of his Metaphysics, Aristotle describes the unmoved mover as being perfectly beautiful, indivisible, and contemplating only the perfect contemplation: self-contemplation. He equates this concept also with the active intellect. This Aristotelian concept had its roots in cosmological speculations of the earliest Greek pre-Socratic philosophers and became highly influential and widely drawn upon in medieval philosophy and theology. St. Thomas Aquinas, for example, elaborated on the unmoved mover in the Quinque viae.

Ancient Greek astronomy

Greek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the ancient Greek, Hellenistic, Greco-Roman, and Late Antiquity eras. It is not limited geographically to Greece or to ethnic Greeks, as the Greek language had become the language of scholarship throughout the Hellenistic world following the conquests of Alexander. This phase of Greek astronomy is also known as Hellenistic astronomy, while the pre-Hellenistic phase is known as Classical Greek astronomy. During the Hellenistic and Roman periods, much of the Greek and non-Greek astronomers working in the Greek tradition studied at the Musaeum and the Library of Alexandria in Ptolemaic Egypt.

This is a timeline of ancient Greek mathematicians.

Dynamics of the celestial spheres

Ancient, medieval and Renaissance astronomers and philosophers developed many different theories about the dynamics of the celestial spheres. They explained the motions of the various nested spheres in terms of the materials of which they were made, external movers such as celestial intelligences, and internal movers such as motive souls or impressed forces. Most of these models were qualitative, although a few of them incorporated quantitative analyses that related speed, motive force and resistance.

Wilbur Richard Knorr was an American historian of mathematics and a professor in the departments of philosophy and classics at Stanford University. He has been called "one of the most profound and certainly the most provocative historian of Greek mathematics" of the 20th century.

Pythagorean astronomical system

An astronomical system positing that the Earth, Moon, Sun and planets revolve around an unseen "Central Fire" was developed in the 5th century BC and has been attributed to the Pythagorean philosopher Philolaus. The system has been called "the first coherent system in which celestial bodies move in circles", anticipating Copernicus in moving "the earth from the center of the cosmos [and] making it a planet". Although its concepts of a Central Fire distinct from the Sun, and a nonexistent "Counter-Earth" were erroneous, the system contained the insight that "the apparent motion of the heavenly bodies" was due to "the real motion of the observer". How much of the system was intended to explain observed phenomena and how much was based on myth and religion is disputed. While the departure from traditional reasoning is impressive, other than the inclusion of the 5 visible planets, very little of the Pythagorean system is based on genuine observation. In retrospect, Philolaus's views are "less like scientific astronomy than like symbolical speculation."


  1. 1 2 3 4 Blackburn, Simon (2008). The Oxford Dictionary of Philosophy (revised 2nd ed.). Oxford, United Kingdom: Oxford University Press. ISBN   9780199541430 . Retrieved 30 November 2020.
  2. 1 2 3 O'Connor, J. J.; Robertson, E. F. "Eudoxus of Cnidus". University of St Andrews . Retrieved 30 November 2020.
  3. Lasserre, François (1966) Die Fragmente des Eudoxos von Knidos (de Gruyter: Berlin)
  4. Diogenes Laertius; VIII.87
  5. Milenko Nikolić (2012) "The ancient idea of real number in Eudoxus' theory of ratios", page 226, and "The analogy between Eudoxus' theory of ratios and Dedekind's theory of cut", page 238 in For Jan Struik, Cohen-Stachel-Wartofsky editors, Springer books
  6. Calinger, Ronald (1982). Classics of Mathematics. Oak Park, Illinois: Moore Publishing Company, Inc. p. 75. ISBN   0-935610-13-8.
  7. Ball 1908, p. 54.
  8. 1 2 Morris Kline, Mathematical Thought from Ancient to Modern Times Oxford University Press, 1972 pp. 48–50
  9. Knopp, Konrad (1951). Theory and Application of Infinite Series (English 2nd ed.). London and Glasgow: Blackie & Son, Ltd. p.  7.
  10. Largely in Book Ten.
  11. This particular argument is referenced in Book One.