Eudoxus of Cnidus

Greek geometer

• Born: c. 390 b.c.e.
• Birthplace: Cnidus, Asia Minor (now in Turkey)
• Died: c. 337 b.c.e.
• Place of death: Cnidus, Asia Minor (now in Turkey)

Eudoxus and his disciples resolved classical difficulties in the fields of geometry and geometric astronomy. Their approach became definitive for later research in these fields.

Early Life

As for so many ancient figures, little is known about the life of Eudoxus (yew-DAHK-suhs) of Cnidus. If one follows the account of the ancient biographer Diogenes Laertius (c. 250 c.e.), Eudoxus first visited Athens at age twenty-three to study medicine and philosophy. He soon returned home to Cnidus, however, and from there, joining the company of the Cnidian physician Chrysippus, he moved on to Egypt, where for more than a year he studied among the priests and engaged in astronomical investigations. Later, as he traveled and lectured in the wider Aegean area (specifically, Cyzicus and the Propontis), he built up a following and thus returned to Athens a man of considerable distinction. His main subsequent activity seems to have centered on Cnidus, where he was honored as a lawgiver. His renown extended to many areas, including astronomy, geometry, medicine, geography, and philosophy.

There is disagreement over his dates. The ancient chronicler Apollodorus sets Eudoxus’s prime activity in 368-365 b.c.e. In general, the prime means age forty; if that holds here, Eudoxus’s birth would be set c. 408. There is reason for doubt, however, because this early a date conflicts with other biographical data. G. L. Huxley favors c. 400; G. de Santillana and others argue for c. 390. Eudoxus is reported to have died at the age of fifty-three; the corresponding date would be 355, 347, or 337.

Life’s Work

None of Eudoxus’s writings survives, but fragments cited by ancient authors offer a reasonable impression of their diversity and significance. His principal efforts were in the areas of mathematics and astronomy, the former best represented in portions of Euclid’s Stoicheia (c. 300 b.c.e.; The Elements of Geometrie of the Most Auncient Philosopher Euclide of Megara, 1570, commonly known as the Elements), the latter in astronomical discussions of the fourth-century cosmology of Aristotle and the ancient commentaries on it.

According to Archimedes (287-212 b.c.e.), Eudoxus was the first to set out a rigorous proof of the theorems that any pyramid equals one-third of the associated prism (that is, having the same height and base as the pyramid), and any cone equals one-third of the associated cylinder. Eudoxus also appears to have proved two other theorems: that circles are as the squares of their diameters and that spheres are as the cubes of their diameters. The proofs of these four theorems constitute the main part of book 17 of Elements, and the technique used there is likely to derive from Eudoxus.

To take the circle theorem as an example, one could imagine a regular polygon having so many sides that it seems practically indistinguishable from a circle. As two such polygons (with equally numerous sides) are proportional to the squares of their diameters, the same could be supposed for the corresponding circles. Presumably, an argument of this sort was assumed by geometers who used the circle theorem in the time before Eudoxus. In the strict sense, however, the reasoning would be invalid, for only by an infinite process can rectilinear figures eventuate in the circle.

In the Eudoxean scheme, one assumes the stated proportion to be false: If two circles are not in the ratio of the squares of their diameters, then one can construct two similar regular polygons, one inscribed in each circle, and one can make the difference between the polygon and its circle so small that the polygon is found to be simultaneously greater and less than a specified amount. Because that is clearly impossible, the stated theorem must be true. (This indirect manner for proving theorems on curved figures is often called, if somewhat misleadingly, the “method of exhaustion.”)

A key move in this proof is making the polygon sufficiently close to the circle. To this end, one observes that as the number of sides is doubled, the difference between the polygon and the circle is reduced by more than half. The procedure of successively bisecting a given quantity will eventually make it less than any preassigned amount, however, as Euclid proves in Elements. According to Archimedes, however, it seems that Eudoxus took this bisection principle as an axiom.

The notion of proportion itself runs into a similar difficulty with the infinite. As long as quantities are related to one another in terms of whole or fractional numbers (for example, if one area is 7⁄5; of another area), their ratios can be specified from these same numbers (that is, the ratio of the one area to the other will be 7 to 5). Yet what if no such numbers exist? For example, it was found, a century or so before Eudoxus, that the diagonal and side of a square cannot equal a ratio of whole numbers. (In modern terms, one calls the associated number √2 “irrational”; its decimal equivalent 1.414 . . . will be nonterminating and nonrepeating.) Only by means of some form of infinite sequence can “commensurable” quantities (those whose ratio is expressible by two integers) equal the ratio of incommensurable ones. Geometers in the generation before Eudoxus had pursued the study of incommensurables with considerable interest, but Eudoxus was the first to see how the theorems on ratios could be rigorously proved when their terms were incommensurable.

It is usually supposed that Eudoxus’s approach was identical to that given by Euclid in book 5 of Elements. Other writers, in particular Archimedes, however, knew of a different technique of proportions that seems more like what Eudoxus would have proposed. By this technique, one first establishes the stated theorem for the case of commensurable quantities. For the incommensurable case, one uses an indirect argument: If the proportionality does not hold, one can find commensurable terms whose ratio differs by less than a specified amount from the ratio of the given incommensurable terms—this is done by successively bisecting one of the givens until it is less than the difference between two others; when the commensurable case of the theorem (already proved) is applied, a certain term will be found to be simultaneously greater and less than another. Because that is impossible, the theorem must be true.

The defining notions of the proportion theory in Euclid’s book 5 can be derived as a simple modification of this technique, for the role that the intermediate commensurable terms play in it is assumed by the Euclidean definition of proportion: that A:B = C:D means that for all positive integers m, n, if mA > nB, then also mC > nD; the same holds true if = or < is substituted for >. Proofs in this Euclidean manner do not require a division into commensurable and incommensurable cases, nor do they make use of the bisection principle; in general, they are rather easier to set up than in the alternative technique. It is thus possible to see Euclid’s approach as an intended refinement of the Eudoxean.

In either the Eudoxean or Euclidean form, this manner of proportion theory can be made to correspond to the modern definition of real number, as formulated by the German mathematician Julius Wilhelm Richard Dedekind. In each example, the real term (possibly irrational) is considered to separate all the rationals into those greater and those less than it.

It seems likely that Eudoxus also contributed to the study of incommensurable lines. His predecessor Theaetetus (d. c. 369) had shown that if the squares of two lines are commensurable with each other but do not have the ratio of square integers, then the lines themselves will be incommensurable with each other; further, if two such lines A, B are taken, the lines A � B will be incommensurable with them, not only as lines but also in square (lines of this latter type were called alogoi, literally, “without ratio”). The further study of the alogoi lines, as collected in book 10 of Euclid’s Elements, divided into twelve classes all the alogoi formed as the square roots of R(A � B), where R is a unit line, and A and B are commensurable with each other in square only. Presumably, Eudoxus and his followers played a part in this investigation.

Eudoxus’s efforts are rooted in a concern for logical precision in geometry, and this interest may reflect his close association with the Platonic Academy at Athens. Two anecdotes (of questionable historicity) celebrate this connection. The first explains how Eudoxus came to be involved in seeking the cube duplication, the so-called Delian problem. To allay a plague, the citizens of Delos were commanded by the oracle to double a cube-shaped altar. When their attempts failed, they sent to Plato, who directed his mathematical associates Archytas, Menaechmus, and Eudoxus to solve it. When they did so, however, Plato criticized their efforts for being too mechanical. The solutions of a dozen different ancient geometers are known, but that of Eudoxus has not been preserved. It supposedly employed “curved lines” of some sort, and reconstructions have been proposed.

In a second story, Plato is said to have posed to Eudoxus the problem of “saving the phenomena” of planetary motion on the restriction to uniform circular motion. An account of Eudoxus’s scheme is transmitted by Simplicius of Cilicia (sixth century c.e.) in his commentary on Aristotle’s De caelo (before 335 b.c.e.; On the Heavens, 1777). From this account a reconstruction was worked out by the Italian historian of astronomy Giovanni Virginio Schiaparelli in 1875. The Eudoxean system reproduces the apparent motion of a planet by combining the rotations of a set of homocentric spheres. The planet is set on the equator of a uniformly rotating sphere. If a second sphere is set about the first, rotating with equal speed to the first but in the opposite direction and having its axis inclined, then the planet will trace out an eight-shaped curve (which the ancients called the hippopede, or horse fetter), so as to complete the full double loop once for every full revolution of the spheres. One superimposes over this a third spherical rotation, corresponding to the general progress of the planet in the ecliptic, and finally over this a fourth rotation, corresponding to the daily rotation of the whole heaven. In this way, each of the five planets requires four spheres, while the sun and the moon each take three.

Schiaparelli’s exposition thus revealed the ingenuity of Eudoxus’s scheme for reproducing geometrically the seemingly erratic forward and backward (retrograde) motion of the planets. Nevertheless, the model proves unsuccessful in some respects: Because the planets do not vary in distance from the earth (the center of their spheres), Eudoxus cannot account for their variable brightness or for asymmetries in the shape of their retrograde paths. Even worse, the values that Eudoxus had to assign for the rotations of the spheres do not produce retrogrades for Mars or Venus, and the sun and the moon are given uniform motions, contrary to observation. Apparently, the latter two defects were recognized, for Eudoxus’s follower Callippus introduced seven additional spheres (two each for sun and moon, one each for Mercury, Venus, and Mars) to make the needed corrections.

The Eudoxean-Callippean scheme is enshrined in Aristotle’s Metaphysica (c. 335-323 b.c.e.; Metaphysics, 1801), in which it serves as the mathematical basis of a comprehensive picture of the entire physical cosmos. Doubtless, Eudoxus proposed his geometric model without specific commitments on physical and cosmological issues. Nevertheless, it suited well the basic Aristotelian principles—for example, that the cosmos separates into two spherical realms, the celestial and, at its center, the terrestrial, and that the natural motions of matter in the central realm (for example, earthy substances moving in straight lines toward the center of the cosmos) differ from those in the outer (where motion is circular, uniform, and eternal). Ironically, these Aristotelian principles persisted in later cosmology, even after astronomers had switched from the homocentric spheres to eccentrics, epicycles, and other geometric devices.

Eudoxus also produced works of a descriptive and empirical sort in astronomy and geography. His Phaenomena (fourth century b.c.e.; phenomena) and Enoptron (fourth century b.c.e.; mirror) recorded observations of the stars—the basis, one would suppose, of a systematic almanac of celestial events (for example, solstices and equinoxes, lunar phases, heliacal risings of stars). He adopted, as Diogenes and others report, an oktaeteris, or eight-year calendar cycle. As known to later authors, an oktaeteris is one of the cycles found to reconcile the solar year of 365.25 days with the period of the moon’s phases (somewhat over 29.5), by parsing out the 2,922 days in eight years into ninety-nine lunar periods (fifty-one of thirty days and forty-eight of twenty-nine). However, it is unclear whether this was the arrangement used by Eudoxus. His geographical treatise, the Gēs periodos (fourth century b.c.e.; circuit of the Earth), systematically described the lands and peoples of the known world, from Asia in the east to the western Mediterranean region. A connection with his astronomical studies can be seen in the use of the ratio of longest to shortest periods of daylight for designating the latitudes of places.

Significance

However interesting Eudoxus’s contributions to calendarics, geography, and philosophy may be, they are secondary to his achievement in mathematics, for he may justly be viewed as the most significant geometer in the pre-Euclidean period. He advanced the study of incommensurables, introduced a new technique for generalizing the theory of proportion, and made exact the theory of limits with his new method of “exhaustion.” Remarkable for the logical precision of his proofs, Eudoxus here set the standard against which even the foremost of the later geometers, such as Euclid and Archimedes, measured their own efforts.

Eudoxus’s influence on geometric astronomy is more subtle. Already, early in the third century b.c.e., astronomers had discarded his system of homocentric spheres in their pursuit of viable geometric models for the planetary motions. If the shortcomings of Eudoxus’s model were evident, however, it nevertheless defined for later astronomers the essence of their task: to represent the planetary phenomena by means of uniform circular motion. Eudoxus’s success thus remains implicit in the later development of astronomy, from Apollonius and Hipparchus to Ptolemy.

Bibliography

Charles, David. Aristotle on Meaning and Essence. Oxford, England: Clarendon Press, 2000. In this work on Aristotle, Charles is critical of his subject’s analysis of Eudoxus and Callippus’s astronomical conclusions.

De Santillana, G. “Eudoxus and Plato: A Study in Chronology.” In Reflections on Men and Ideas. Cambridge, Mass.: MIT Press, 1968. A revised chronology of Eudoxus’s life is argued on the basis of a detailed examination of the ancient biographical data and collateral historical evidence.

Knorr, W. R. The Ancient Tradition of Geometric Problems. Boston: Birkhäuser, 1986. Chapter 3 considers Eudoxus’s studies of exhaustion and cube duplication, discussed in the wider context of pre-Euclidean geometry.

Neugebauer, O. A History of Ancient Mathematical Astronomy. New York: Springer-Verlag, 1975. All facets of Eudoxus’s contributions to astronomy are covered; particularly detailed is the discussion of his planetary models. Includes an index.

Waerden, B. L. van der. Science Awakening. 4th ed. Princeton Junction, N.J.: Scholar’s Bookshelf, 1988. The author provides a brief, insightful review of Eudoxus’s mathematical work.