Eudoxus of Cnidus

Greek astronomer and mathematician (c. 400 bc–350 bc)

Born in Cnidus, which is now in Turkey, Eudoxus is reported as having studied mathematics under Archytas, a Pythagorean. He also studied under Plato and in Egypt. Although none of his works have survived they are quoted extensively by Hipparchus. Eudoxus was the first astronomer who had a complete understanding of the celestial sphere. It is only this understanding that reveals the irregularities of the movements of the planets that must be taken into account in giving an accurate description of the heavens. For Eudoxus the Earth was at rest and around this center 27 concentric spheres rotated. The outermost sphere carried the fixed stars, each of the planets required four spheres, and the Sun and the Moon three each. All these spheres were necessary to account for the daily and annual relative motions of the heavenly bodies. He also described the constellations and the changes in the rising and setting of the fixed stars in the course of a year.

In mathematics, Eudoxus is thought to have contributed the theory of proportion to be found in Book V of Euclid – the importance of this being its applicability to irrational as well as rational numbers. The method of exhaustion in Book XII is also attributed to Eudoxus. This tackled in a mathematical way for the first time the difficult problem of calculating an area bounded by a curve.

The astronomer, mathematician, and physician Eudoxus of Cnidus (ca. 408-ca. 355 B.C.) was the first Greek astronomer to properly apply mathematics to astronomy.

Eudoxus was born in Cnidus, a Greek colony in Asia Minor, into a family of physicians; he studied at the medical school there. At the age of 23 he went to Athens as an assistant to a doctor. He attended lectures at the Academy, recently founded by Plato. On returning to Cnidus, Eudoxus completed his studies.

A few years later Eudoxus went to Egypt with another doctor. He studied the heavens from an observatory at Heliopolis on the Nile. His astronomical observations appear in his Phaenomena, but apparently this book did not contain theories such as those he was later to expound. The book locates the constellations relative to each other and to imaginary lines on the celestial sphere. Much space is given to a compilation of lists of stars which rise above or fall below the horizon at the beginning of each month.

During Eudoxus' 14 months in Egypt, one of his objectives was to produce a satisfactory calendar. His skill at making the detailed observations required for a good calendar is probably a result of his medical training, for although the teaching of medicine in Eudoxus' time may not have been very strong in the area of cures, it did emphasize the detailed description of symptoms.

Astronomical Theory

On returning to Asia Minor, Eudoxus established his own school in Cyzicus. While here he wrote On Speeds, his most important astronomical work, in which he expounded his theory of the motions of the stars, sun, moon, and planets. The recent discovery of the spherical shape of the earth may have inspired Eudoxus' hypothesis of homocentric planetary spheres. According to this theory, the motion of a planet can be explained by imagining that the planet is attached to the equator of a sphere; this sphere, with the center of the earth as its center, rotates uniformly about its polar axis. The poles are implanted in a second sphere that is concentric with the first; the second sphere also rotates uniformly about its polar axis, which is at a fixed angle to the axis of the first sphere. This relationship continues successively to other spheres.

If the sun, for example, is imagined as fixed on the equator of one sphere, then rotations of the spheres, with appropriate speed and direction, will give the path of the sun. The fixed stars are imagined to be on the largest concentric sphere, which rotates about the polar axis of the earth. Altogether 27 spheres are required to picture the motions of all the important bodies. Although this theory did not explain all the observable planetary motions, it was accurate enough to cause many of Eudoxus' successors to assume that only minor modifications would be needed to make it more accurate.

Contribution to Mathematics

It is fairly certain that Eudoxus' principal contributions to mathematics were his theory of proportions and his method of exhaustion. Both of these appear in Euclid's collection of geometrical theorems, the Elements, and are fundamental in the work of later mathematicians.

Eudoxus' theory of proportions is concerned with the ratio of magnitudes. One problem in describing the theory is that many of the theorems appear to be very obvious formulas. A typical example is the following (using modern terminology): given the positive numbers a, b, and c, if a is greater than b then a/c is greater than b/c. Only a person who has investigated the not so obvious way in which such simple properties are proved would begin to appreciate the significance of Eudoxus' work. Another source of possible difficulty for the modern reader of Eudoxus' theory of proportions is that, although it was valid for irrational numbers, to the Greeks "numbers" meant the natural numbers. Thus Eudoxus used more general "magnitudes," which were represented by lengths of line segments.

Eudoxus' method of exhaustion was a rigorous way of calculating areas and volumes; it puts him closer to modern mathematics than any of his other works. Archimedes quotes two theorems which were considered to be true before Eudoxus' time but which were first proved by Eudoxus: the volume of a pyramid is one-third the volume of a prism with the same base and height; and the volume of a cone is one-third the volume of a cylinder with the same base and height.

Later Life

Eudoxus moved next to Athens, but his stay there was short. The rulers of Cnidus had been overthrown and a democracy established. The people sent a request to Eudoxus to write a constitution for a new government. He returned to Cnidus and composed the legislation. He made his home there for the rest of his life, continuing his teaching and establishing an astronomical observatory. He revised some of his earlier writings and composed a description of his travels in seven books entitled Circuit of the Earth.

Eudoxus also had a reputation as a philosopher. According to Aristotle, Eudoxus held pleasure to be the chief good, for all creatures sought it and all attempted to escape its opposite, pain. Also, according to Eudoxus, pleasure was an end in itself and not a relative good. But Eudoxus was not an immoderate hedonist, for Aristotle, who may have known Eudoxus personally, gives a picture of him that is quite the contrary: "His arguments about pleasure carried conviction more on account of the perfection of his character than through their contents. Eudoxus passed indeed for a man of remarkable moderation. Again he did not seem to embrace these arguments as being a friend of pleasure, but because he regarded them as conforming to the truth."

Further Reading

Brief accounts of Eudoxus' life and work are given in Thomas L. Heath, A Manual of Greek Mathematics (1931), and in Bartel L. van der Waerden, Science Awakening (1950; trans. 1954). A number of interesting theories involving Eudoxus are discussed in François Lasserre, The Birth of Mathematics in the Age of Plato (1964). An authoritative description of Eudoxus' astronomical theory is given by Otto Neugebauer in The Exact Sciences in Antiquity (1952; 2d ed. 1957). For the place of Eudoxus' mathematical contribution in the development of the modern calculus see Carl B. Boyer, The History of the Calculus and Its Conceptual Development (1939; rev. ed. 1949).

Eudoxus, of Cnidus (c.400–c.350 BC), outstanding mathematician and astronomer in the half-century before Euclid, a younger contemporary of Plato and an associate of his in the Academy. His writings are lost but their contents are fairly well known from later writers. He is largely responsible for the discovery of the theory of proportion between incommensurables (covered by Euclid in Elements, book 5) and, in showing that a proportion could be established between any magnitudes, for removing the check given to geometry when irrational numbers were discovered. In solid geometry he showed that the volume of a pyramid was one third of that of a prism of equal height on the same base, and correspondingly that the volume of a cone was one third that of its containing cylinder. In astronomy his work marks the beginning of a new epoch, for he was the first to give a mathematical account of the movements of the heavenly bodies that tried to explain the fact that the planets occasionally appear to interrupt their movement eastward (relative to the fixed stars) along the zodiac, and for a time to move backwards from east to west (retrogression). Eudoxus suggested that the paths of the sun and moon and planets were produced by the circular movements of concentric spheres, rotating at different speeds and about different axes with the earth at rest at their common centre. The system showed great mathematical skill and managed to account fairly successfully for a wide variety of phenomena. It was modified by Callipus to remove the most obvious inconsistencies with observable fact, and in this form it was accepted by Aristotle. Eudoxus' astronomical calendar of the risings and settings of the constellations formed the basis of Aratus' astronomical poem the Phaenomena.

(c. 408-c. 355 bc) Greek pupil of Plato's, and outstanding geometer and cosmologist. Eudoxus invented a general theory of proportion able to handle incommensurables, also a mathematical theory of the limit enabling avoidance of infinitesimals. He constructed a mathematical model of the heavens, based on advanced mathematics but poor observational data.

Eudoxus of Cnidus (yūdŏk'səs, nī'dəs) , 408?–355? B.C., Greek astronomer, mathematician, and physician. From the accounts of various ancient writers, he appears to have studied with Plato in Athens, spent some time in Heliopolis, Egypt, founded a school in Cyzicus, and spent his later years in Cnidus, where he had an observatory. It is claimed that he calculated the length of the solar year, indicating a calendar reform like that made later by Julius Caesar, and that he was the discoverer of some parts of geometry included in the work of Euclid. He was the first Greek astronomer to explain the movements of the planets in a scientific manner. His system involved a number of concentric spheres supporting the planets in their paths. Some scientists still held this belief at the time of Copernicus.

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