Eudoxus of Cnidus

Another article concerns Eudoxus of Cyzicus.

Eudoxus of Cnidus (Greek Εύδοξος) (410 or 408 BC - 355 or 347 BC) was a Greek astronomer, mathematician, physician, scholar and friend 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 Sphaerics may be based on a work of Eudoxus.

He was a pupil in mathematics of Archytas in Athens. 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 shows tremendous insight into 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 1500s, 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. The work of Eudoxus and Archimedes as precursors of calculus was only exceeded in mathematical sophistication and rigour by Indian Mathematician Bhaskara and later by Newton.

An algebraic curve (the Kampyle of Eudoxus) is named after him

a²x⁴ = b⁴(x² + y²).

Also, craters on Mars and the Moon are named in his honor.

Astronomy

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, on an eight-year lunisolar cycle of the calendar
• Phaenomena and Entropon, 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 Phaenomena, for Eudoxus' 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 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.

Eudoxus rose to the challenge by assigning to each planet a set of nested concentric spheres. By tilting the axes of the spheres, and by assigning each a different period of revolution, he was able to approximate the celestial "appearances."

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, Mercury, Mars, Jupiter, and Saturn) are assigned four spheres each:

• The outermost explains the daily motion.
• The second explains the planet's motion through the zodiac.
• 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

Callippus, 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. Astronomers responded by introducing the deferent and epicycle, which caused a planet to vary its distance. However, Eudoxus' importance to Greek astronomy is considerable, as he was the first to attempt a mathematical explanation of the planets.

References

• Evans, James (1998). The History and Practice of Ancient Astronomy. Oxford University Press. ISBN 0-19-509539-1.
• Huxley, GL (1980). Eudoxus of Cnidus p. 465-7 in: the Dictionary of Scientific Biography, volume 4.
• Lloyd, GER (1970). Early Greek Science: Thales to Aristotle. W.W. Norton.

External links

• Models of Planetary Motion—Eudoxus
• The Universe According to Eudoxus (Java applet)
• Eudoxus of Cnidus
• O'Connor, John J., and Edmund F. Robertson. "Eudoxus of Cnidus". MacTutor History of Mathematics archive.
• Application of mathematical principles associated with Eudoxus

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