# Eudoxus of Cnidus

# Eudoxus of Cnidus

(*b*. Cnidus, ca. 400 b.c.; *d*. Cnidus, *ca*. 347 b.c.)

*astronomy, mathematics*.

A scholar and scientist of great eminence, Eudoxus, son of a certain Aischines, contributed to the development of astronomy, mathematics, geography, and philosophy, as well as providing his native city with laws. As a young man he studied geometry with Archytas of Tarentum, from whom he may well have taken his interest in number theory and music; in medicine he was instructed by the physician Philiston; and his philosophical inquiries were stimulated by Plato, whose lectures he attended as an impecunious student during his first visit to Athens. Later his friends in Cnidus paid for a visit to Egypt, where he seems to have had diplomatic dealings with King Nekhtanibef II on behalf of Agesilaus II of Sparta.

Eudoxus spent more than a year in Egypt, some of the time in the company of the priests at Heliopolis. He was said to have composed his *Oktaeteris*, or eight-year calendric cycle, during his sojourn with them. Next he settled at Cyzicus in northwestern Asia Minor and founded a school. He also visited the dynast Mausolus in Caria. A second visit to Athens, to which he was followed by some of his pupils, brought a closer association with Plato, but it is not easy to determine mutual influences in their thinking on ethical and scientific matters. It is unlikely that Plato had any influence upon the development of Eudoxian planetary theory or much upon the Cnidian’s philosophical doctrine of forms, which recalls Anaxagoras; but it is possible that Plato’s *Philebos* was written with the Eudoxian view of *hedone* (that pleasure, correctly understood, is the highest good) in mind.

Back in Cnidus, Eudoxus lectured on theology, cosmology, and meteorology, wrote textbooks, and enjoyed the respect of his fellow citizens. In mathematics his thinking lies behind much of Euclid’s *Elements*, especially books V, VI, and XII. Eudoxus investigated mathematical proportion, the method of exhaustion, and the axiomatic method—the “Euclidean” presentation of axioms and propositions may well have been first systematized by him. The importance of his doctrine of proportion lay in its power to embrace incommensurable quantities.

It is difficult to exaggerate the significance of the theory, for it amounts to a rigorous definition of real number. Number theory was allowed to advance again, after the paralysis imposed on it by the Pythagorean discovery of irrationals, to the inestimable benefit of all subsequent mathematics. Indeed, as T. L. Heath declares (*A History of Greek Mathematics*, I [Oxford, 1921], 326–327), “The greatness of the new theory itself needs no further argument when it is remembered that the definition of equal ratios in Eucl. V, Def. 5 corresponds exactly to the modern theory of irrationals due to Dedekind, and that it is word for word the same as Weierstrass’s definition of equal numbers.”

Eudoxus also attacked the so-called “Delian problem,” the traditional one of duplicating the cube; that is, he tried to find two mean proportions in continued proportion between two given quantities. His strictly geometrical solution is lost, and he may also have constructed an apparatus with which to describe an approximate mechanical solution; an epigram ascribed to Eratosthenes (who studied the works of Eudoxus closely) refers to his use of “lines of a bent form” in his solution to the Delian problem: the “organic” demonstration may be meant here. Plato is said to have objected to the use by Eudoxus (and by Archytas) of such devices, believing that they debased pure or ideal geometry. Proclus mentions “general theorems” of Eudoxus; they are lost but may have embraced all concepts of magnitude, the doctrine of proportion included. Related to the treatment of proportion (as found in *Elements* V) was his method of exhaustion, which was used in the calculation of the volume of solids. The method was an important step toward the development of integral calculus.

Archimedes states that Eudoxus proved that the volume of a pyramid is one-third the volume of the prism having the same base and equal height and that the volume of a cone is one-third the volume of the cylinder having the same base and height (these propositions may already have been known by Democritus, but Eudoxus was, it seems, the first to prove them). Archimedes also implies that Eudoxus showed that the areas of circles are to each other as the squares on their respective diameters and that the volumes of spheres to each other are as the cubes of their diameters. All four propositions are found in *Elements XII*, which closely reflects his work. Eudoxus is also said to have added to the first three classes of mathematical mean (arithmetic, geometric, and harmonic) two more, the subcontraries to harmonic and to geometric, but the attribution to him is not quite certain.

Perhaps the most important, and certainly the most influential, part of Eudoxus’ lifework was his application of spherical geometry to astronomy. In his book *On Speeds* he expounded a system of geocentric, homocentric rotating spheres designed to explain the irregularities in the motion of planets as seen from the earth. Eudoxus may have regarded his system simply as an abstract geometrical model, but Aristotle took it to be a description of the physical world and complicated it by the addition of more spheres; still more were added by Callippus later in the fourth century b.c. By suitable combination of spheres the periodic motions of planets could be represented approximately, but the system is also, as geometry, of intrinsic merit because of the hippopede, or “horse fetter,” an eight-shaped curve, by which Eudoxus represented a planet’s apparent motion in latitude as well as its retrogradation.

Eudoxus’ model assumes that the planet remains at a constant distance from the center, but in fact, as critics were quick to point out, the planets vary in brightness and hence, it would seem, in distance from the earth. Another objection is that according to the model, each retrogradation of a planet is identical with the previous retrogradation in the shape of its curve, which also is not in accord with the facts. So, while the Eudoxian system testified to the geometrical skill of its author, it could not be accepted by serious astronomers as definitive, and in time the theory of epicycles was developed. But, partly through the blessing of Aristotle, the influence of Eudoxus on popular astronomical thought lasted through antiquity and the Middle Ages. In explaining the system, Eudoxus gave close estimates of the synodic periods of Saturn, Jupiter, Mars, Mercury, and Venus (hence the title of the book, *On Speeds*). Only the estimate for Mars is seriously faulty, and here the text of Simplicius, who gives the values, is almost certainly in error (Eudoxus, frag. 124 in Lasserre).

Eudoxus was a careful observer of the fixed stars, both during his visit to Egypt and at home in Cnidus, where he had an observatory. His results were published in two books, the *Enoptron* (“Mirror”) and the *Phaenomena*. The works were criticized, in the light of superior knowledge, by the great astronomer Hipparchus two centuries later, but they were pioneering compendia and long proved useful. Several verbatim quotations are given by Hipparchus in his commentary on the astronomical poem of Aratus, which drew on Eudoxus and was also entitled *Phaenomena*. A book by Eudoxus called *Disappearances of the Sun* may have been concerned with eclipses, and perhaps with risings and settings as well. The statement in the *Suda Lexicon* that he composed an astronomical poem may result from a confusion with Aratus, but a genuine *Astronomia* in hexameters, in the Hesiodic tradition, is a possibility. A calendar of the seasonal risings and settings of constellations, together with weather signs, may have been included in the *Oktaeteris*. His observational instruments included sundials (Vitruvius, *De architectura* 9.8.1).

Eudoxus’ knowledge of spherical astronomy must have been helpful to him in the geographical treatise *Ges periodos* (“Tour [Circuit] of the Earth”). About 100 fragments survive; they give some idea of the plan of the original work. Beginning with remote Asia, Eudoxus dealt systematically with each part of the known world in turn, adding political, historical, and ethnographic detail and making use of Greek mythology. His method is comparable with that of such early Ionian logographers as Hecataeus of Miletus. Egypt was treated in the second book, and Egyptian religion, about which Eudoxus could write with authority, was discussed in detail. The fourth book dealt with regions to the north of the Aegean, including Thrace. In the sixth book he wrote about mainland Hellas and, it seems, North Africa. The discussion of Italy in the seventh book included an excursus on the customs of the Pythagoreans, about whom Eudoxus may have learned much from his master Archytas of Tarentum (Eudoxus himself is sometimes called a Pythagorean).

It is greatly to be deplored that not a single work of Eudoxus is extant, for he was obviously a dominant figure in the intellectual life of Greece in the age of Plato and Aristotle (the latter also remarked on the upright and controlled character of the Cnidian, which made people believe him when he said that pleasure was the highest good).

## BIBLIOGRAPHY

The biography of Eudoxus in Diogenes Laertius 8.86–8.90 is anecdotal but not worthless. The fragments have been collected, with commentary, in F. Lasserre’s book Die *Fragmente des Eudoxos von Knidos* (Berlin, 1966). Eudoxian parts of Euclid’s *Elements* are discussed by T. L. Heath in his edition of that work, 2nd ed., 3 vols. (Cambridge, 1926). The mathematical properties of the hippopede have been much studied; see especially O. Neugebauer, *The Exact Sciences in Antiquity*, 2nd ed. (Providence, R.I 1., 1957), 182–183. On the Ges periodos, see F. Gisinger, *Die Erdbeschreibung des Eudoxos von Knidos* (Leipzig-Berlin, 1921). A chronology of Eudoxus’ life and travels is G. Huxley, “Eudoxian Topics,” in *Greek, Roman and Byzantine Studies*, **4** (1963), 83–96.

See also Oskar Becker, “Eudoxos-Studien,” in *Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik*, Abt. B, Studien, **2** (1933), 311–333, 369–387, and 3 (1936), 236–244, 370–410; Hans Künsberg, *Der Astronom, Mathematiker und Geograph Eudoxos von Knidos*, 2 pts. (Dinkelsbühl, 1888–1890); and G. Schiaparelli, *Scritti sulla storia della astronomia antica*, II (Bologna, 1926), 2–112.

G. L. Huxley

# Eudoxus of Cnidus

# Eudoxus of Cnidus

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

# Eudoxus of Cnidus

**c. 408-c. 355 b.c.**

**Greek Astronomer and Mathematician**

First to apply mathematics properly in the study of astronomy, Eudoxus also contributed directly to mathematical study with his theory of proportion and his method of exhaustion. In addition, he worked as a medical doctor, and gained renown as a philosopher and political writer. He also wrote a seven-volume description of his travels entitled *Circuit of the Earth.*

Born in the Greek colony of Cnidus in Asia Minor, Eudoxus came from a long line of physicians, and trained as doctor. By the age of 23, he had moved to Athens to work as a physicians' assistant, and while in the great city attended lectures by Plato (427-347 b.c.) at the latter's Academy. He then returned to Cnidus, where he completed his studies before going to Egypt with another physician.

It was there, at the observatory in Heliopolis along the Nile, that Cnidus discovered his second calling. As a physician, Eudoxus was accustomed to making detailed observations, one of the few areas in which ancient medicine excelled, and the record of data he compiled at Heliopolis was quite thorough. This he presented in the *Phaenomena,* a rather straightforward astronomical study containing lists of stars that rise or fall below the horizon at the beginning of each month, as well as locations for all constellations relative to one another.

Returning to Asia Minor with the new course of his career set, Eudoxus founded a school in the town of Cyzicus. There he wrote *On Speeds,* a much more important work than *Phaenomena* in which he presented a new theory of the motion made by the Sun, Moon, and planets. Given the spherical shape of Earth, Eudoxus imagined a series of concentric spheres around it, and eventually developed a description of 27 spheres necessary for picturing the movement of all known bodies.

The idea of the concentric spheres seems obvious today: hence people think of a planet's orbit around the Sun—though it is actually elliptical rather than circular—as taking place along the equator of an imaginary sphere. But in Eudoxus's time, this idea, which would eventually be represented physically in a variety of astronomical instruments, was far from obvious.

His theory of proportions and his method of exhaustion would later find their way into the *Elements* of Euclid (c. 325-c. 250 b.c.). According to Eudoxus, if a given object is larger than a second one, then its ratio to a third object will also be larger than the ratio of the second to the third—which again is a seemingly self-evident point from the perspective of the twenty-first century a.d., but was not so in the fourth century b.c. As for Eudoxus's method of exhaustion, Archimedes (c. 287-212 b.c.) wrote that he gave the first proofs of two propositions already known at the time: that the volume of a pyramid is one-third that of a prism with the same base and height, and that the same relationship is the case for a cone and cylinder.

Eudoxus moved to Athens, but when the people of Cnidus overthrew the oligarchy there and established a democracy, they asked him to come back and write a constitution for the new state. He then made his way home, where after completing his work for the government he founded an observatory and school. According to Aristotle (384-322 b.c.), Eudoxus also made a reputation for himself as a philosopher.

**JUDSON KNIGHT**

# Eudoxus of Cnidus

# Eudoxus of Cnidus

**Circa 390-circa 340 b.c.e.**

Astronomer

**Diverse Scholar.** An outstanding mathematician, astronomer, geographer, and philosopher, Eudoxus hailed from the Dorian city of Cnidus in Asia Minor; he was a pupil of Archytas, a Pythagorean philosopher from Tarentum, in geometry and Philistion, a physician from southern Italy, in medicine. He studied in Athens, where he knew Plato. At some point he visited Egypt, where he learned astronomy from the priests. On astronomy he lectured in Cyzicus and in various courts before returning to Athens. Eudoxus is credited with inventing the theory of proportions in geometry, as well as working on limits. He pioneered in mathematical explanations for planetary motion, general observations of constellations, and work on developing calendars.

## Sources

R. M. Dancy, *Two Studies in the Early Academy* (Albany: State University of New York Press, 1991).

G. J. Toomer, “Eudoxus”, in *The Oxford Classical Dictionary*, edited by Simon Hornblower and Antony Spawforth, third edition (Oxford: Oxford University Press, 1999), pp. 565–566.

# Eudoxus of Cnidus

# Eudoxus of Cnidus

**400?-347? b.c.**

Greek astronomer and mathematician who wrote on astronomy, mathematics, geography, philosophy, and drafted laws for his native town. He attended some of Plato's lectures and established his own school, where he lectured on astronomy, theology, and meteorology. Only fragments of his writings survive. His lost books include *On Velocities,* which influenced the planetary theory of Aristotle, and *Tour of the Earth,* which was quoted heavily by later geographers.

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