Eudoxus of Cnidus

All of his original works are lost, though some fragments are preserved in Hipparchus' Commentaries on the Phenomena of Aratus and Eudoxus.

Eudoxus, son of Aeschines, was born and died in Cnidus (also transliterated Knidos), a city on the southwest coast of Anatolia.

At the age of 23, he traveled with the physician Theomedon—who was his patron and possibly his lover[6]—to Athens to study with the followers of Socrates.

He spent two months there—living in Piraeus and walking 7 miles (11 km) each way every day to attend the Sophists' lectures—then returned home to Cnidus.

His friends then paid to send him to Heliopolis, Egypt for 16 months, to pursue his study of astronomy and mathematics.

From Egypt, he then traveled north to Cyzicus, located on the south shore of the Sea of Marmara, the Propontis.

According to some sources,[citation needed] c. 367 he assumed headship (scholarch) of the Academy during Plato's period in Syracuse, and taught Aristotle.

[citation needed] He 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.

In mathematical astronomy, his fame is due to the introduction of the concentric spheres, and his early contributions to understanding the movement of the planets.

Eudoxus is considered by some to be the greatest of classical Greek mathematicians, and in all Antiquity second only to Archimedes.

[10] He rigorously developed Antiphon's method of exhaustion, a precursor to the integral calculus which was also used in a masterly way by Archimedes in the following century.

[11] Eudoxus introduced the idea of non-quantified mathematical magnitude to describe and work with continuous geometrical entities such as lines, angles, areas and volumes, thereby avoiding the use of irrational numbers.

In doing so, he reversed a Pythagorean emphasis on number and arithmetic, focusing instead on geometrical concepts as the basis of rigorous mathematics.

Some Pythagoreans, such as Eudoxus's teacher Archytas, had believed that only arithmetic could provide a basis for proofs.

Induced by the need to understand and operate with incommensurable quantities, Eudoxus established what may have been the first deductive organization of mathematics on the basis of explicit axioms.

The change in focus by Eudoxus stimulated a divide in mathematics which lasted two thousand years.

In combination with a Greek intellectual attitude unconcerned with practical problems, there followed a significant retreat from the development of techniques in arithmetic and algebra.

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.

Ancient Greek mathematicians calculated not with quantities and equations as we do today; instead, a proportionality expressed a relationship between geometric magnitudes.

Eudoxus is credited with defining equality between two ratios, the subject of Book V of the Elements.

The complexity of the definition reflects the deep conceptual and methodological innovation involved.

The Archimedean property, definition 4 of Elements Book V, was credited to Eudoxus by Archimedes.

[12] In ancient Greece, astronomy was a branch of mathematics; astronomers sought to create geometrical models that could imitate the appearances of celestial motions.

Identifying the astronomical work of Eudoxus as a separate category is therefore a modern convenience.

The inclusion of a third sphere implies that Eudoxus mistakenly believed that the Sun had motion in latitude.

Aristotle was concerned about the physical nature of the system; without unrollers, the outer motions would be transferred to the inner planets.

A major flaw in the Eudoxian system is its inability to explain changes in the brightness of planets as seen from Earth.

Astronomers responded by introducing the deferent and epicycle, which caused a planet to vary its distance.

Aristotle, in the Nicomachean Ethics,[14] attributes to Eudoxus an argument in favor of hedonism—that is, that pleasure is the ultimate good that activity strives for.