Эвдокс Книдский: Древнегреческий астроном и математик

Greek astronomer and mathematician (c.390–c.340 BC)

Eudoxus of Cnidus ('/juː//d//ə//k//s//ə//s/; Εὔδοξος ὁ Κνίδιος, Eúdoxos ho Knídios; ) was an ancient Greek astronomer, mathematician, doctor, and lawmaker. He was a student of Archytas and Plato. All of his original works are lost, though some fragments are preserved in Hipparchus' Commentaries on the Phenomena of Aratus and Eudoxus. Spherics by Theodosius of Bithynia may be based on a work by Eudoxus.

Eudoxus, son of Aeschines, was born and died in Cnidus (also transliterated Knidos), a city on the southwest coast of Anatolia. The years of Eudoxus' birth and death are not fully known but Diogenes Laërtius gave several biographical details, mentioned that Apollodorus said he reached his acme in the 103rd Olympiad (368–365 BC), and claimed he died in his 53rd year. From this 19th century mathematical historians reconstructed dates of 408–355 BC, but 20th century scholars found their choices contradictory and prefer a birth year of His name Eudoxus means "honored" or "of good repute" (εὔδοξος, from eu "good" and doxa "opinion, belief, fame", analogous to the Latin Benedictus). According to Diogenes Laërtius, crediting Callimachus' Pinakes, Eudoxus studied mathematics with Archytas (of Tarentum, Magna Graecia) and studied medicine with Philiston the Sicilian. At the age of 23, he traveled with the physician Theomedon—who was his patron and possibly his lover—to Athens to study with the followers of Socrates. He spent two months there—living in Piraeus and walking 7 mile (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. He traveled south to the court of Mausolus. During his travels he gathered many students of his own. Around 368 BC, Eudoxus returned to Athens with his students. According to some sources, c. 367 he assumed headship (scholarch) of the Academy during Plato's period in Syracuse, and taught Aristotle. 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. He had one son, Aristagoras, and three daughters, Actis, Philtis, and Delphis. 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. His work on proportions shows insight into irrational numbers and the linear continuum: it allows rigorous treatment of continuous quantities and not just whole numberss 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, and inspired Richard Dedekind's work on the real numbers. Craters on Mars and the Moon are named in his honor. An algebraic curve (the Kampyle of Eudoxus) is also named after him.

Eudoxus is considered by some to be the greatest of classical Greek mathematicians, and in all Antiquity second only to Archimedes. Eudoxus was probably the source for most of book V of Euclid's Elements. 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. In applying the method, Eudoxus proved such mathematical statements as: areas of circles are to one another as the squares of their radii, volumes of spheres are to one another as the cubes of their radii, the volume of a pyramid is one third the volume of a prism with the same base and altitude, and the volume of a cone is one third that of the corresponding cylinder. 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.

Callippus, a Greek astronomer of the 4th century, added seven spheres to Eudoxus's original 27 (in addition to the planetary spheres, Eudoxus included a sphere for the fixed stars). Aristotle described both systems, but insisted on adding "unrolling" spheres between each set of spheres to cancel the motions of the outer set. 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. Because the spheres are concentric, planets will always remain at the same distance from Earth. This problem was pointed out in Antiquity by Autolycus of Pitane. Astronomers responded by introducing the deferent and epicycle, which caused a planet to vary its distance. However, Eudoxus's importance to astronomy and in particular to Greek astronomy is considerable.

Aristotle, in the Nicomachean Ethics, attributes to Eudoxus an argument in favor of hedonism—that is, that pleasure is the ultimate good that activity strives for. According to Aristotle, Eudoxus put forward the following arguments for this position:
All things, rational and irrational, aim at pleasure; things aim at what they believe to be good; a good indication of what the chief good is would be the thing that most things aim at. Similarly, pleasure's opposite—pain—is universally avoided, which provides additional support for the idea that pleasure is universally considered good. People don't seek pleasure as a means to something else, but as an end in its own right. Any other good that you can think of would be better if pleasure were added to it, and it is only by good that good can be increased. Of all of the things that are good, happiness is peculiar for not being praised, which may show that it is the crowning good.