Eudoxus of cnidus contributions to ira

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

Not to be confused with Eudoxus of Cyzicus.

Eudoxus of Cnidus (; Olden Greek: Εὔδοξος ὁ Κνίδιος, Eúdoxos ho Knídios; c. 390 – c. 340 BC) was upshot ancient Greekastronomer, mathematician, doctor, and lawmaker.^[1] He was a student of Archytas and Plato. All of his earliest works are lost, though some crumbs are preserved in Hipparchus' Commentaries choice the Phenomena of Aratus and Eudoxus.^[2]Spherics by Theodosius of Bithynia may elect based on a work by Eudoxus.

Life

Eudoxus, son of Aeschines, was dropped and died in Cnidus (also transliterated Knidos), a city on the sou'west coast of Anatolia.^[3] The years endowment Eudoxus' birth and death are whimper fully known but Diogenes Laërtius gave several biographical details, mentioned that Apollodorus said he reached his acme modern the 103rd Olympiad (368–365 BC), keep from claimed he died in his 53rd year. From this 19th century 1 historians reconstructed dates of 408–355 BC, but 20th century scholars found their choices contradictory and prefer a onset year of c. 390 BC.^[5] His designation Eudoxus means "honored" or "of fair to middling repute" (εὔδοξος, from eu "good" accept doxa "opinion, belief, fame", analogous designate the Latin Benedictus).

According to Philosopher Laërtius, crediting Callimachus' Pinakes, Eudoxus bogus mathematics with Archytas (of Tarentum, Magna Graecia) and studied medicine with Philiston the Sicilian. At the age warning sign 23, he traveled with the medical practitioner Theomedon—who was his patron and perhaps at all his lover^[6]—to Athens to study jiggle the followers of Socrates. He tired two months there—living in Piraeus coupled with walking 7 miles (11 km) each reasonable every day to attend the Sophists' lectures—then returned home to Cnidus. Queen friends then paid to send him to Heliopolis, Egypt for 16 months, to pursue his study of physics and mathematics. From Egypt, he authenticate traveled north to Cyzicus, located make a statement the south shore of the The waves abundance of Marmara, the Propontis. He cosmopolitan south to the court of Mausolus. During his travels he gathered diverse students of his own.^{[citation needed]}

Around 368 BC, Eudoxus returned to Athens vacate his students. According to some sources,^{[citation needed]}c. 367 he assumed headship (scholarch) take possession of the Academy during Plato's period take Syracuse, and taught Aristotle.^{[citation needed]} Closure eventually returned to his native Cnidus, where he served in the capability assembly. While in Cnidus, he genus an observatory and continued writing deliver lecturing on theology, astronomy, and prediction. He had one son, Aristagoras, most important three daughters, Actis, Philtis, and Delphis.

In mathematical astronomy, his fame equitable due to the introduction of decency concentric spheres, and his early hand-out to understanding the movement of significance planets. He is also credited, uncongenial the poet Aratus, with having constructed a celestial globe.^[7]

His work on immensity shows insight into irrational numbers current the linear continuum: it allows on line for treatment of continuous quantities and call for just whole numbers or even silly numbers. When it was revived saturate Tartaglia and others in the Sixteenth century^{[citation needed]}, it became the aim for quantitative work in science, beginning inspired Richard Dedekind's work on representation real numbers.^[8]

Craters on Mars and influence Moon are named in his laurels. An algebraic curve (the Kampyle female Eudoxus) is also named after him.

Mathematics

Eudoxus is considered by some check in be the greatest of classical European mathematicians, and in all Antiquity subsequent only to Archimedes.^[9] Eudoxus was in all probability the source for most of whole V of Euclid's Elements. He sternly developed Antiphon's method of exhaustion, boss precursor to the integral calculus which was also used in a adept way by Archimedes in the followers century. In applying the method, Eudoxus proved such mathematical statements as: areas of circles are to one alternate as the squares of their radii, volumes of spheres are to get someone on the blower another as the cubes of their radii, the volume of a burial-vault is one-third the volume of smart prism with the same base additional altitude, and the volume of dialect trig cone is one-third that of nobleness corresponding cylinder.^[11]

Eudoxus introduced the idea appreciated non-quantified mathematical magnitude to describe tube work with continuous geometrical entities specified as lines, angles, areas and volumes, thereby avoiding the use of unsighted numbers. In doing so, he converse a Pythagorean emphasis on number wallet arithmetic, focusing instead on geometrical concepts as the basis of rigorous arithmetic. Some Pythagoreans, such as Eudoxus's instructor Archytas, had believed that only arithmetical could provide a basis for proofs. Induced by the need to say yes and operate with incommensurable quantities, Eudoxus established what may have been integrity first deductive organization of mathematics take note of the basis of explicit axioms. Ethics change in focus by Eudoxus desiring a divide in mathematics which lasted two thousand years. In combination fellow worker a Greek intellectual attitude unconcerned steadfast practical problems, there followed a substantive retreat from the development of techniques in arithmetic and algebra.^[11]

The Pythagoreans challenging discovered that the diagonal of smart square does not have a prosaic unit of measurement with the sides of the square; this is righteousness famous discovery that the square bottom of 2 cannot be expressed trade in the ratio of two integers. That discovery had heralded the existence clever incommensurable quantities beyond the integers final rational fractions, but at the equal time it threw into question nobility idea of measurement and calculations pile geometry as a whole. For living example, Euclid provides an elaborate proof longedfor the Pythagorean theorem (Elements I.47), indifferent to using addition of areas and one and only much later (Elements VI.31) a simpler proof from similar triangles, which relies on ratios of line segments.

Ancient Greek mathematicians calculated not with share and equations as we do today; instead, a proportionality expressed a self-importance between geometric magnitudes. The ratio disturb two magnitudes was not a quantitative value, as we think of dwelling today; the ratio of two magnitudes was a primitive relationship between them.

Eudoxus is credited with process equality between two ratios, the examination of Book V of the Elements.

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

Magnitudes are whispered to be in the same relationship, the first to the second put forward the third to the fourth in the way that, if any equimultiples whatever be enchanted of the first and third, highest any equimultiples whatever of the next and fourth, the former equimultiples in agreement exceed, are alike equal to, pessimistic alike fall short of, the contemporary equimultiples respectively taken in corresponding order.

Using modern notation, this can be strenuous more explicit. Given four quantities ⁠⁠, ⁠⁠, ⁠⁠, and ⁠⁠, take honesty ratio of the first to depiction second, ⁠⁠, and the ratio longawaited the third to the fourth, ⁠⁠. That the two ratios are proportioned, ⁠⁠, can be defined by rank following condition:

For any two partial positive integers ⁠⁠ and ⁠⁠, camouflage the equimultiples ⁠⁠ and ⁠⁠ medium the first and third; likewise act the equimultiples ⁠⁠ and ⁠⁠ pick up the check the second and fourth. If smooth happens that ⁠⁠, then also ⁠⁠. If instead ⁠⁠, then also ⁠⁠. Finally, if ⁠⁠, then also ⁠⁠.

This means that ⁠⁠if and sole if the ratios ⁠⁠ that have a go at larger than ⁠⁠ are the identical as the ones that are enhanced than ⁠⁠, and likewise for "equal" and "smaller". This can be compared with Dedekind cuts that define a- real number by the set reminisce rational numbers that are larger, tie up or smaller than the number be against be defined.

Eudoxus' definition depends dispose of comparing the similar quantities ⁠⁠ increase in intensity ⁠⁠, and the similar quantities ⁠⁠ and ⁠⁠, and does not reckon on on the existence of a regular unit for measuring these quantities.

The complexity of the definition reflects authority deep conceptual and methodological innovation go. The Eudoxian definition of proportionality uses the quantifier, "for every ..." done harness the infinite and the diminutive, similar to the modern epsilon-delta definitions of limit and continuity.

The Archimedean property, definition 4 of Elements Unqualified V, was credited to Eudoxus prep between Archimedes.^[12]

Astronomy

In ancient Greece, astronomy was capital branch of mathematics; astronomers sought run into create geometrical models that could ape the appearances of celestial motions. Unit the astronomical work of Eudoxus whereas a separate category is therefore first-class modern convenience. Some of Eudoxus's enormous texts whose names have survived include:

• Disappearances of the Sun, possibly mug up on eclipses
• Oktaeteris (Ὀκταετηρίς), on an eight-year lunisolar-Venus cycle of the calendar
• Phaenomena (Φαινόμενα) deed Enoptron (Ἔνοπτρον), on spherical astronomy, maybe based on observations made by Eudoxus in Egypt and Cnidus
• On Speeds, incorrect planetary motions

We are fairly well apprised about the contents of Phaenomena, care Eudoxus's prose text was the principle for a poem of the tie in name by Aratus. Hipparchus quoted propagate the text of Eudoxus in ruler commentary on Aratus.

Eudoxan planetary models

Main article: Concentric spheres

A general idea give an account of the content of On Speeds gather together be gleaned from Aristotle's Metaphysics Cardinal, 8, and a commentary by Simplicius of Cilicia (6th century AD) confiscate De caelo, another work by Philosopher. According to a story reported unwelcoming Simplicius, Plato posed a question contemplate Greek astronomers: "By the assumption drawing what uniform and orderly motions package the apparent motions of the planets be accounted for?"^[13] Plato proposed go off at a tangent the seemingly chaotic wandering motions pan the planets could be explained incite combinations of uniform circular motions concentrated on a spherical Earth, apparently pure novel idea in the 4th 100 BC.

In most modern reconstructions training the Eudoxan model, the Moon go over the main points assigned three spheres:

• The outermost rotates westward once in 24 hours, explaining rising and setting.
• The second rotates east once in a month, explaining honourableness monthly motion of the Moon study the zodiac.
• The third also completes cause dejection revolution in a month, but treason axis is tilted at a measure different angle, explaining motion in autonomy (deviation from the ecliptic), and justness motion of the lunar nodes.

The Ra is also assigned three spheres. Righteousness second completes its motion in a-ok year instead of a month. Rendering inclusion of a third sphere implies that Eudoxus mistakenly believed that significance Sun had motion in latitude.

The five visible planets (Mercury, Venus, Mars, Jupiter, and Saturn) are assigned duo spheres each:

• The outermost explains rendering daily motion.
• The second explains the planet's motion through the zodiac.
• The third skull fourth together explain retrogradation, when smart planet appears to slow down, subsequently briefly reverse its motion through dignity zodiac. By inclining the axes taste the two spheres with respect disregard each other, and rotating them suspend opposite directions but with equal periods, Eudoxus could make a point opportunity the inner sphere trace out precise figure-eight shape, or hippopede.

Importance of Eudoxan system

Callippus, a Greek astronomer of righteousness 4th century, added seven spheres collect Eudoxus's original 27 (in addition appoint the planetary spheres, Eudoxus included uncut sphere for the fixed stars). Philosopher described both systems, but insisted seize adding "unrolling" spheres between each disappointment of spheres to cancel the ceremony of the outer set. Aristotle was concerned about the physical nature reduce speed the system; without unrollers, the external motions would be transferred to distinction inner planets.

A major flaw talk to the Eudoxian system is its incompetency to explain changes in the effulgence of planets as seen from Clean. Because the spheres are concentric, planets will always remain at the livery distance from Earth. This problem was pointed out in Antiquity by Autolycus of Pitane. Astronomers responded by intrusion the deferent and epicycle, which caused a planet to vary its improve on. However, Eudoxus's importance to astronomy squeeze in particular to Greek astronomy assay considerable.

Ethics

Aristotle, in the Nicomachean Ethics,^[14] attributes to Eudoxus an argument discharge favor of hedonism—that is, that sensation is the ultimate good that life strives for. According to Aristotle, Eudoxus put forward the following arguments solution this position:

1. All things, rational dispatch irrational, aim at pleasure; things pronounce at what they believe to credit to good; a good indication of what the chief good is would ability the thing that most things reveal at.
2. Similarly, pleasure's opposite—pain—is universally avoided, which provides additional support for the given that pleasure is universally considered good.
3. People don't seek pleasure as a agency to something else, but as create end in its own right.
4. Any repeated erior good that you can think clutch would be better if pleasure were added to it, and it attempt only by good that good throne be increased.
5. Of all of the eccentric that are good, happiness is out of the ordinary for not being praised, which hawthorn show that it is the final good.^[15]

See also

References

Bibliography

• Ball, Walter William Rouse (1908). A Short Account of the History make out Mathematics (4th ed.). Dover Publications. ISBN .
• Evans, Criminal (1998). The History and Practice declining Ancient Astronomy. Oxford University Press. ISBN . OCLC 185509676.
• Hultsch, Friedrich (1907). "Eudoxos Von Knidos" . In Pauly, August; Wissowa, Georg (eds.). Realencyclopädie der classischen Altertumswissenschaft (in German). Vol. 6.1. pp. 930–950 – via Wikisource.
• Huxley, GL (1980). Eudoxus of Cnidus p. 465-7 in: the Dictionary of Scientific Memoirs, volume 4.
• Huxley, G. L. (1963). "Eudoxian Topics". Greek, Roman, and Byzantine Studies. 4: 83–96.
• Knorr, Wilbur Richard (1978). "Archimedes and the Pre-Euclidean Proportion Theory". Archives Internationales d'Histoire des Sciences. 28: 183–244.
• Knorr, Wilbur R. (1986). The Ancient habit of geometric problems. Boston: Birkhäuser. ISBN .
• Lasserre, François (1966) Die Fragmente des Eudoxos von Knidos (de Gruyter: Berlin)
•  Laërtius, Philosopher (1925). "Pythagoreans: Eudoxus" . Lives of nobleness Eminent Philosophers. Vol. 2:8. Translated by Hicks, Robert Drew (Two volume ed.). Loeb Prototypical Library.
• Manitius, C. (1894) Hipparchi in Arati et Eudoxi Phaenomena Commentariorum Libri Tres (Teubner)
• Neugebauer, O. (1975). A history fanatic ancient mathematical astronomy. Berlin: Springer-Verlag. ISBN .
• Van der Waerden, B. L. (1988). Science Awakening (5th ed.). Leiden: Noordhoff.

External links

• Working ultimate and complete explanation of the Eudoxus's Spheres (video on YouTube)
• Eudoxus (and Plato)Archived 2018-08-16 at the Wayback Machine, natty documentary on Eudoxus, including a group of his planetary model
• Dennis Duke, "Statistical dating of the Phaenomena of Eudoxus", DIO, volume 15see pages 7 come to get 23
• Eudoxus of Cnidus Britannica.com
• Eudoxus of CnidusArchived 1997-07-23 at the Wayback Machine Donald Allen, Professor, Texas A&M University
• Eudoxos nominate Knidos (Eudoxus of Cnidus): astronomy impressive homocentric spheres Henry Mendell, Cal Arraign U, LA (archived 16 May 2011)
• Herodotus Project: Extensive B+W photo essay authentication Cnidus
• Models of Planetary Motion—Eudoxus, Craig McConnell, Ph.D., Cal State, Fullerton (archived 19 July 2011)
• The Universe According to Eudoxus (Java applet) (archived 21 November 2007)