## Eudoxus

Philosopher and mathematician of the 4th c. B.C., Eudoxus was an intimate friend of Plato. He is attributed with the first theory of hedonism in ethics, with the first astronomical theory in antiquity, and with several important mathematical discoveries.

Eudoxus was from Cnidus of the Propontis. According to Diogenes Laertius (*Lives of Eminent Philosophers* VIII), his arrival in Athens was motivated by his intention to listen to the teachings of the circle of Socrates, and the Sophists; on that occasion, he made friends with Plato. Diogenes reports that Eudoxus studied geometry under Archytas, and medicine under Philiston; and, like every other great Greek mathematician of the antiquity, he was registered with a study trip to Egypt.

Eudoxus is not considered a student of Plato, neither a typical member of the Academy. For all that, he was an equal member of the platonic circle. The relative chronology of him and Plato is a matter of dispute. The former contention that Eudoxus lived from 408 to 355 is today replaced by an estimate dating of his death after 347 - that is, posterior to Plato's death.

Eudoxus was acclaimed by his contemporaries for his mathematical and astronomical discoveries, as well as for his wisdom and character. Evidence for the latter is offered by Aristotle in his explanation of the reason why Eudoxus' hedonistic thesis became so convincing: "His arguments owed their acceptance however more to the excellence of his character than to their own merit. He had the reputation of being a man of exceptional temperance..." (*Nicomachean Ethics* 1172b12-13). During the last years of his life, Eudoxus returned to Cnidus, where he successfully pulled off the assignment of composing a new legislation system - a task that added to his reputation.

The only insights we have in Edoxus' philosophy, we gain from Aristotle. In the context of the platonic theory of Forms, Aristotle mentions that Eudoxus, like Anaxagoras before him, contented that "the admixture of white causes a thing to be white" (*Metaphysics* 991a15-17) - which means that he somehow posited the Forms into the things. The fact that Aristotle does not abide to this position suggests that Eudoxus had in mind a mix of Form and mater of the Anaxogorean sort.

Eudoxus submits a strong candidacy, in the history of philosophy, for having introduced the theory of hedonism, i.e., the thesis that pleasure is the ultimate good.

"That pleasure is the Good was held by Eudoxus, on the following grounds. He saw that all creatures, rational and irrational alike, seek to obtain it; but in every case (he argued) that which is desirable is good" (Aristotle,Nicomachean Ethics1172b9-12).

This unprecedented claim provoked intense debate among the members of the Academy. Plato's mature dialogue *Philebus* is construed as an account of this debate. Hedonism did not do well in the platonic circle, but influenced Aristotle's ethics. It was restored by Epicurus and his followers in the Hellenistic period.

Eudoxus is considered the greatest Greek mathematician after Archimedes. An extensive part of Euclids' *Elements* incorporates mathematical discoveries made by Eudoxus: the general theory of proportions in Book V; his findings on incommensurable quantities in Book X; and the method of exhaustion (his most significant contribution to mathematics) in Book XII. The mark of these discoveries is impressed on the platonic dialogues in whatever place Plato emphasizes on the close relation between mathematics and philosophy. So, if it is true that Eudoxus frequented the Academy, he must have been the best verification of the platonic precept "let no one ignorant of geometry come under my roof".

Of equal importance is Eudoxus' contribution to the grounding of astronomy - the only empirical branch of mathematical science that the Greeks developed. By the titles of some of his works (*Phaenomena*, *Mirror*, *Oktaeteris*) we can infer that he was preoccupied with astronomical observations, and that he tried to solve practical astronomical problems such as the design of a calendar - a pressing problem for Athens of the period between the end of the 5th and the dawn of the 4th c. B.C.

The theory of the homocentric spheres, that Eudoxus spelled out in his work *On Speeds*, is the first systematic attempt to explain the motions of the Sun, the Moon, and the five planets known in antiquity. His system consisted of a complex of 26 interconnected, homocentric spheres rotating in regular motions around the earth, thus currying the planets with them. This could be construed as Eudoxus' solution to the problem Plato had allegedly posed to mathematicians of his circle: "By the assumption of what uniform and orderly motions can the apparent motions of the planets be accounted for?" (quoted in Burket 1972, p. 329). A step was made by Plato in the *Timaeus* where he explains the observed irregular motion of the Sun as the synthesis of the two regular motion of the circles of the "Same" and the "Different". In the place of the platonic circles, Eudoxus posited three spheres for the Sun and the Moon, and four for the planets. This composition of the regular motions of the spheres not only provided an adequate account for most of the planetary phenomena, it could also predict the planetary motions. We do not know whether Eudoxus appraised his system as a mere mathematical conception, or if he truly believed that it represents the true nature of the heaven. The way Aristotle handles Eudoxus' system in his cosmology place the weight on the first assumption; for Aristotle integrates Eudoxus' system, but attempts to render it mechanically functional by supplementing it with an equal number of spheres (see *Metaphysics* 12.1073b17ff.).

Eudoxus' theory opened the way for the fast evolution of astronomy. It also reassured the platonic world-view according to which the heavens is the realm of complete uniformity and order.

- Lasserre, F.
*Die Fragmente des Eudoxus von Knidos*. Bερολίνο, 1966. - De Santillana, G. "Eudoxus and Plato.A Study in Chronology."
*Isis*32 (1949) - Dicks, D.R.
*Early Greek Astronomy to Aristotle*. Λονδίνο, 1970. - Fritz, K. von. "Die Ideenlehre des Eudoxos von Knidos und ihr Vehrhältnis zur platonischen Ideenlehre."
*Philologus*85 (1926) - Rackham, H.
*Aristotle in 23 Volumes*. London, 1934.