# Eudoxus (1), of Cnidus, mathematician

Eudoxus of Cnidus, (*c.*390–*c.* 340 bce) was an outstanding mathematician and did important work in astronomy and geography; he was versatile in ‘philosophy’ in general. According to the not entirely trustworthy ancient biographical tradition (see especially Diog. Laert. 8. 86 ff.), he was a pupil of Archytas in geometry and of Philistion in medicine; he came to Athens to hear the Socratics when about 23, later spent time in Egypt studying astronomy with the priests, then lectured in Cyzicus and the Propontis, visited the court of Mausolus, and finally returned to teach at Athens, where he was acquainted with Plato; he drew up laws for Cnidus, and died aged 52.

In geometry he invented the general theory of proportion, applicable to incommensurable as well as commensurable magnitudes, found in Euclid bk. 5 (scholion in Heiberg, * Euclidis Opera* 5. 280). This greatly helped to assure the primacy of geometry in Greek mathematics. He also developed the method of approach to the limit (misnamed ‘method of exhaustion’ in modern works) which became the standard way of avoiding infinitesimals in ancient mathematics, He was thus able to prove that cone and pyramid are one-third of the cylinder and prism respectively with the same base and height (Archim.,

*Method*pref.). Of his solution to the problem of doubling the cube nothing certain is known.

In astronomy he was the first Greek to construct a mathematical system to explain the apparent motions of the heavenly bodies: that of the ‘homocentric spheres’. Simplicius' account of this (*in Cael.* 492. 31 ff.), which gives its title as Περὶ ταχῶν (‘On Speeds’), reveals both the high level of mathematics and the low level of observational astronomy of the time: Eudoxus combined uniform motions of concentric spheres about different axes with great ingenuity to produce, for instance, a qualitatively correct representation of the retrogradations of some planets; but the underlying observational data are few and crude, and the discrepancies of the results with the actual phenomena often gross (for later corrections see callippus and astronomy). Its adoption in a modified form by Aristotle was responsible for its resurrection in later ages. More practical (and very influential) was Eudoxus' description of the constellations, with calendaric notices of risings and settings, which appeared in two versions, named Ἔνοπτρον and Φαινόμενα. The latter is known through its adaptation by Aratus (1) in his immensely popular poem of the same name; the commentary of Hipparchus (3) on both Eudoxus and Aratus is extant (see the edn. of Manitius (Teubner, 1894), p. 376 for refs. to Eudoxus). Another calendaric work was the Ὀκταετηρίς (‘*Eight-year* [luni-solar] *Cycle*’). The papyrus treatise named Εὐδόξου τέχνη, though composed much later, contains some elementary calendaric and astronomical information which may derive from Eudoxus. There is some evidence for Babylonian influence in Eudoxus' astronomical work.

The Γῆς περίοδος (‘Circuit of the Earth’), in several books, was a work of mathematical and descriptive geography.

## Bibliography

*Realencyclopädie der Classischen Altertumswissenschaft*, s.v. “Eudoxus 8.”Find this resource:

## Fragments and testimonia

F. Lasserre, *Die Fragmente des Eudoxos von Knidos* (1966).Find this resource:

## Mathematics

T. L. Heath, *Hist. of Greek Mathematics* 1. 320 ff..Find this resource:

O. Becker, ‘Eudoxosstudien’ 1–5, in *Quellen u. Studien z. Gesch. d. Math*. B2 and B3 (1933–6).Find this resource:

## Homocentric spheres

The classic reconstruction is G. Schiaparelli, ‘*Le sfere omocentriche di Eudosso, di Callippo e di Aristotele*’ (1875), repr. in his *Scritti sulla storia della astronomia antica* 2, Ger. trans. in *Abh. zur Gesch. d. Math*. 1 (1877), 101 ff.;

*A History of Ancient Mathematical Astronomy* 2. 674–83; but the assessment of the testimonia is in dispute: see A. C. Bowen, *Perspectives on Science* 10 (2002), 155–67 and H. Mendell, in *Ancient and Medieval Traditions in the Exact Sciences* (2000), 59–138.

## Calendar

Geminus, ed. M. Manitius (1898), 108 ff., 210 ff..Find this resource:

A. Böckh, *Über die vierjährigen Sonnenkreise der Alten* (1863).Find this resource:

Εὐδόξου τέχνη. Editio princeps, with the interesting illustrations, ed. Letronne and Brunet de Presle, *Notices et Extraits des Manuscrits* 18. 2 (1865).Find this resource:

F. Blass, *Eudoxi Ars Astronomica* (Kiel Festschrift, 1887).Find this resource:

## Geography

Fragments in Lasserre, 96ff; see further F. Gisinger, *Die Erdbeschreibung des Eudoxos von Knidos* (1921).Find this resource: