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Eudoxus of Cnidus (410 or 408 BC – 355 or 347 BC) was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy. Theodosius of Bithynia's Sphaerics may be based on a work of Eudoxus.


His name "Eudoxus" means "good opinion" or of "good fame" (in Greek Εὔδοξος), from eu = good, doxa = opinion or belief or fame). It is analogous to the latin name "Benedictus" (Benedict, Benedetto).

Eudoxus's father Aeschines of Cnidus loved to watch stars at night. Eudoxus first travelled to Tarentummarker to study with Archytas, from whom he learned mathematics. While in Italy, Eudoxus visited Sicily, where he studied medicine with Philiston.

Around 387 BC, at the age of 23, he traveled with the physician Theomedon, who according to Diogenes Laertius some believed was his lover, to Athens to study with the followers of Socrates. He eventually became the pupil of Plato, with whom he studied for several months, but due to a disagreement they had a falling out. Eudoxus was quite poor and could only afford an apartment at the Piraeus. To attend Plato's lectures, he walked the seven miles (11 km) each direction, each day. Due to his poverty, his friends raised funds sufficient to send him to Heliopolis, Egypt to pursue his study of astronomy and mathematics. He lived there 16 months. From Egypt, he then traveled north to Cyzicus, located on the south shore of the Sea of Marmara, and the Propontismarker. He traveled south to the court of Mausolus. During his travels he gathered many students of his own.

Around 368 BC, he returned to Athens with his students. Eudoxus eventually returned to his native Cnidus, where he served in the city assembly. While in Cnidus, he built an observatory and continued writing and lecturing on theology, astronomy and meteorology. He had one son, Aristagoras, and three daughters, Actis, Philtis and Delphis.


His work on proportions shows tremendous insight into numbers; it allows rigorous treatment of continuous quantities and not just whole numbers or even rational numbers. When it was revived by Tartaglia and others in the 16th century, it became the basis for quantitative work in science for a century, until it was replaced by the algebraic methods of Descartes.

Eudoxus rigorously developed Antiphon's method of exhaustion, which was used in a masterly way by Archimedes. This method is a precursor to the integral calculus.


a2x4 = b4(x2 + y2).

Also, craters on Mars and the Moonmarker are named in his honor.


The Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with the sides of the square; this is the famous discovery that the square root of 2 cannot be expressed as the ratio of two integers. This discovery had heralded the existence of incommensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole. For example, Euclid provides an elaborate proof of the Pythagorean theorem, by using addition of areas instead of the much simpler proof from similar triangles, which relies on ratios of line segments.



In Definition 5 of Euclid's Book V we read:

Let us clarify it by using modern-day notation. If we take four quantities: a, b, c, and d, then the first and second have a ratio a/b; similarly the third and fourth have a ratio c/d.

Now to say that a/b = c/d we do the following:For any two arbitrary integers, m and n, form the equimultiplesm·a and m·c of the first and third; likewise form the equimultiples n·b and n·d of the second and fourth.

If it happens that m·a > n·b, then we must also have m·c > n·d.If it happens that m·a = n·b, then we must also have m·c = n·d. Finally, if it happens that m·a n·b, then we must also have m·c n·d.







  • Disappearances of the Sun, possibly on eclipses
  • On Speeds, on planetary motions


Eudoxan planetary models






  • The outermost explains the daily motion.
  • The second explains the planet's motion through the zodiac.

Importance of Eudoxan system





See also



  1. Diogenes Laertius; VIII.87
  2. largely in book ten
  3. this particular argument is referenced in book one

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