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47-584: (Redirected from Eudoxos ) Eudoxus may refer to: Eudoxus of Cnidus (c. 395–390 BC – c. 342–337 BC), Greek astronomer and mathematician, student of Plato Eudoxus of Cyzicus ( fl. c. 130 BC), Greek navigator who explored the Arabian Sea for Ptolemy VIII of Egypt Eudoxus (lunar crater) Eudoxus (Martian crater) See also [ edit ] 11709 Eudoxos , asteroid Eudoxia (name) Charaxes eudoxus , an African butterfly Topics referred to by

94-510: A / b = c / d {\displaystyle a/b=c/d} ⁠ if and only if the ratios ⁠ n / m {\displaystyle n/m} ⁠ that are larger than ⁠ a / b {\displaystyle a/b} ⁠ are the same as the ones that are larger than ⁠ c / d {\displaystyle c/d} ⁠ , and likewise for "equal" and "smaller". This can be compared with Dedekind cuts that define

141-533: A = n ⋅ b {\displaystyle m\cdot a=n\cdot b} ⁠ , then also ⁠ m ⋅ c = n ⋅ d {\displaystyle m\cdot c=n\cdot d} ⁠ . Finally, if ⁠ m ⋅ a < n ⋅ b {\displaystyle m\cdot a<n\cdot b} ⁠ , then also ⁠ m ⋅ c < n ⋅ d {\displaystyle m\cdot c<n\cdot d} ⁠ . This means that ⁠

188-425: 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

235-530: A catalogue of bishops beginning with the 1st century; Michel Le Quien mentions fifty-nine. A more complete list is found in Nicodemos, in the Greek "Office of St. Emilian" (Constantinople, 1876), 34–36, which has eighty-five names. Of particular importance are the famous Arian theologian Eunomius of Cyzicus ; Saint Dalmatius ; bishops Proclus and Germanus , who became Patriarchs of Constantinople; and Saint Emilian,

282-576: A martyr in the 8th century. Another saint who came from Cyzicus, Saint Tryphaena of Cyzicus , is the patron saint of the city. Gelasius , a historian of Arianism , who wrote about 475, was born at Cyzicus. Cyzicus remained a metropolitan see of the Greek Orthodox Church until the 1923 Convention Concerning the Exchange of Greek and Turkish Populations emptied it of Greek Orthodox faithful, whether they spoke Greek or Turkish. The last bishop of

329-465: A numerical value, as we think of it today; the ratio of two magnitudes was a primitive relationship between them. Eudoxus is credited with defining equality between two ratios, the subject of Book V of the Elements . In Definition 5 of Euclid's Book V we read: Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of

376-508: A point between 1370 and 1372 until 1387, the metropolitan was empty; Speros Vryonis speculates this was due to financial difficulties. Later in the 14th century, the sees of Chalcedon and certain patriarchal possessions in Bithynia and Hellespont were bestowed on the metropolitan of Cyzicus. In the Ottoman era , it was part of the kaza of Erdek in the province of Brusa . Cyzicus, as capital of

423-583: A real number by the set of rational numbers that are larger, equal or smaller than the number to be defined. Eudoxus' definition depends on comparing the similar quantities ⁠ m ⋅ a {\displaystyle m\cdot a} ⁠ and ⁠ n ⋅ b {\displaystyle n\cdot b} ⁠ , and the similar quantities ⁠ m ⋅ c {\displaystyle m\cdot c} ⁠ and ⁠ n ⋅ d {\displaystyle n\cdot d} ⁠ , and does not depend on

470-635: A series of earthquakes beginning in 443, with the last in 1063. Although its population was transferred to Artake before the 13th century when the peninsula was occupied by the Crusaders , in 1324 the metropolitan of Cyzicus was one of three sees in Anatolia which was able to contribute a temporary annual subsidy to the Patriarchate of Constantinople. Following its conquest by the Ottomans it underwent hard times. From

517-444: A story reported by Simplicius, Plato posed a question for Greek astronomers: "By the assumption of what uniform and orderly motions can the apparent motions of the planets be accounted for?" Plato proposed that the seemingly chaotic wandering motions of the planets could be explained by combinations of uniform circular motions centered on a spherical Earth, apparently a novel idea in the 4th century BC. In most modern reconstructions of

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564-553: 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: Cyzicus Cyzicus ( / ˈ s ɪ z ɪ k ə s / SIZ -ik-əs ; Ancient Greek : Κύζικος , romanized :  Kúzikos ; Ottoman Turkish : آیدینجق , romanized :  Aydıncıḳ )

611-485: Is different from Wikidata All article disambiguation pages All disambiguation pages Eudoxus of Cnidus 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

658-571: The Erdek and Bandırma roads, is protected by Turkey 's Ministry of Culture . The city was said to have been founded by Pelasgians from Thessaly , according to tradition at the coming of the Argonauts ; later it received many colonies from Miletus , allegedly in 756 BC, but its importance began near the end of the Peloponnesian War when the conflict centered on the sea routes connecting Greece to

705-735: The Roman province of Hellespontus , was its ecclesiastical metropolitan see . In the Notitiae Episcopatuum of Pseudo-Epiphanius, composed in about 640, Cyzicus had 12 suffragan sees ; Abydus , Baris in Hellesponto (between Sariköy and Biga), Dardanus , Germa in Hellesponto (ruins of Germaslu, Kirmasti, Girmas), Hadrianotherae (Uzuncia yayla), Ilium , Lampsacus , Miletopolis , Oca , Pionia (Avcılar), Poemanenum (Eskimanias), Troas . The province also included two autocephalous archiepiscopal sees: Parium and Proconnesus . Cyzicus had

752-692: 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

799-402: 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 c.  390 BC . His name Eudoxus means "honored" or "of good repute" ( εὔδοξος , from eu "good" and doxa "opinion, belief, fame", analogous to

846-419: 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

893-553: 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

940-775: The Black Sea. At this time, the cities of Athens and Miletus diminished in importance while Cyzicus began to prosper. Commander of the Athenian fleet Alcibiades defeated the Spartan fleet in a major naval engagement near Cyzicus known as the Battle of Cyzicus in 410 BC. Famed ancient philosopher Eudoxus of Cnidus established a school at Cyzicus and went with his pupils to Athens, visiting Plato . Later he returned to Anatolia to his hometown of Cnidus , and died circa 350 BC. The era of Olympiads in Cyzicus

987-561: The Eudoxan model, the Moon is assigned three spheres: The Sun is also assigned three spheres. The second completes its motion in a year instead of a month. The inclusion of a third sphere implies that Eudoxus mistakenly believed that the Sun had motion in latitude. The five visible planets ( Mercury , Venus , Mars , Jupiter , and Saturn ) are assigned four spheres each: Callippus , a Greek astronomer of

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1034-654: 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 miles (11 km) each way every day to attend

1081-453: The Pythagorean theorem ( Elements I.47), by using addition of areas 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 quantities and equations as we do today; instead, a proportionality expressed a relationship between geometric magnitudes. The ratio of two magnitudes was not

1128-429: 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. The Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with

1175-402: The contents of Phaenomena , for Eudoxus's prose text was the basis for a poem of the same name by Aratus . Hipparchus quoted from the text of Eudoxus in his commentary on Aratus. A general idea of the content of On Speeds can be gleaned from Aristotle 's Metaphysics XII, 8, and a commentary by Simplicius of Cilicia (6th century AD) on De caelo , another work by Aristotle. According to

1222-444: 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

1269-434: The existence of a common unit for measuring these quantities. The complexity of the definition reflects the deep conceptual and methodological innovation involved. The Eudoxian definition of proportionality uses the quantifier, "for every ..." to harness the infinite and the infinitesimal, similar to the modern epsilon-delta definitions of limit and continuity. The Archimedean property , definition 4 of Elements Book V,

1316-569: The first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. Using modern notation , this can be made more explicit. Given four quantities ⁠ a {\displaystyle a} ⁠ , ⁠ b {\displaystyle b} ⁠ , ⁠ c {\displaystyle c} ⁠ , and ⁠ d {\displaystyle d} ⁠ , take

1363-591: The first and third; likewise form the equimultiples ⁠ n ⋅ b {\displaystyle n\cdot b} ⁠ and ⁠ n ⋅ d {\displaystyle n\cdot d} ⁠ of the second and fourth. If it happens that ⁠ m ⋅ a > n ⋅ b {\displaystyle m\cdot a>n\cdot b} ⁠ , then also ⁠ m ⋅ c > n ⋅ d {\displaystyle m\cdot c>n\cdot d} ⁠ . If instead ⁠ m ⋅

1410-399: The following condition: For any two arbitrary positive integers ⁠ m {\displaystyle m} ⁠ and ⁠ n {\displaystyle n} ⁠ , form the equimultiples ⁠ m ⋅ a {\displaystyle m\cdot a} ⁠ and ⁠ m ⋅ c {\displaystyle m\cdot c} ⁠ of

1457-599: 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

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1504-425: The introduction of the concentric spheres , and his early contributions to understanding the movement of the planets . He is also credited, by the poet Aratus , with having constructed a celestial globe . His work on proportions shows insight into irrational numbers and the linear continuum : it allows rigorous treatment of continuous quantities and not just whole numbers or even rational numbers . When it

1551-556: The naval Battle of Cyzicus in 410 during the Peloponnesian War , an Athenian fleet routed and completely destroyed a Spartan fleet. At the peace of Antalcidas (387 BC), like the other Greek cities in Asia, it was made over to Persia . Alexander the Great later captured it from the Persians in 334 BC and was later claimed to be responsible for connecting the island to the mainland. The history of

1598-400: The ratio of the first to the second, ⁠ a / b {\displaystyle a/b} ⁠ , and the ratio of the third to the fourth, ⁠ c / d {\displaystyle c/d} ⁠ . That the two ratios are proportional, ⁠ a / b = c / d {\displaystyle a/b=c/d} ⁠ , can be defined by

1645-449: The same term [REDACTED] This disambiguation page lists articles associated with the title Eudoxus . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Eudoxus&oldid=896324695 " Categories : Disambiguation pages Human name disambiguation pages Hidden categories: Short description

1692-535: The see died in 1932. Today it is a titular metropolis of the Ecumenical Patriarchate of Constantinople . Since 1885, the Catholic Church lists Cyzicus as a titular see of the highest (Metropolitan) rank, but vacant since 1974. Titular metropolitans were: The site amid the marshes of Balkiz Serai is known as Bal-Kiz and entirely uninhabited, though under cultivation. The principal extant ruins are

1739-409: 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

1786-602: The temple dedicated to the Emperor Hadrian are still visible: the columns were 21.35 metres high (about 70 feet), while the highest known elsewhere, those at Baalbek in Lebanon are only 19.35 metres (about 63 feet). The structure was the largest Greco-Roman temple ever built. The monuments of Cyzicus were used by the Byzantine emperor Justinian as a quarry for the building of his Saint Sophia cathedral, and were still exploited by

1833-730: The town in Hellenistic times is closely connected with that of the Attalids of Pergamon , with whose extinction it came into direct relations with Rome . Cyzicus was held for the Romans against King Mithridates VI of Pontus who besieged it with 300,000 men in 74 BC, the Siege of Cyzicus , but it withstood him stoutly, and the siege was raised by Lucullus : the loyalty of the city was rewarded by an extension of territory and other privileges. The Romans favored it and recognized its municipal independence. Cyzicus

1880-401: The walls, dating from the fourth century, which are traceable for nearly their whole extent, and the substructures of the temple of Hadrian , the ruins of a Roman aqueduct and a theatre. The picturesque amphitheatre , intersected by a stream, was one of the largest in the world. Construction for the amphitheatre began in the middle of the first century until the end of the third. Its diameter

1927-562: Was an ancient Greek town in Mysia in Anatolia in the current Balıkesir Province of Turkey . It was located on the shoreward side of the present Kapıdağ Peninsula (the classical Arctonnesus ), a tombolo which is said to have originally been an island in the Sea of Marmara only to be connected to the mainland in historic times either by artificial means or an earthquake. The site of Cyzicus, located on

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1974-417: Was credited to Eudoxus by Archimedes. 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. Some of Eudoxus's astronomical texts whose names have survived include: We are fairly well informed about

2021-501: Was nearly 500 feet (150 m) and it is located at these coordinates 40°23′54″N 27°53′5″E  /  40.39833°N 27.88472°E  / 40.39833; 27.88472  ( Cyzicus amphitheatre ) , north of the main part of Cyzicus. Of this magnificent building, sometimes ranked among the seven wonders of the ancient world, thirty-one immense columns still stood erect in 1444. These have since been carried away piecemeal for building purposes. The colossal foundations of

2068-432: 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

2115-555: Was reckoned from 135 or 139. Owing to its advantageous position it speedily acquired commercial importance, and the gold staters of Cyzicus were a staple currency in the ancient world till they were superseded by those of Philip of Macedon . Its unique and characteristic coin, the cyzicenus , was worth 28 drachmae. During the Peloponnesian War (431–404 BC) Cyzicus was subject to the Athenians and Lacedaemonians alternately. In

2162-555: 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

2209-612: Was the leading city of Northern Mysia as far as Troas . Under Tiberius , it was incorporated into the Roman Empire but remained the capital of Mysia (afterwards, Hellespontus ) and became one of the great cities of the ancient world. There was a women's cult at Cyzicus worshiping the goddess Artemis , which was called Dolon (Δόλων). Cyzicus was captured temporarily by the Arabs led by Muawiyah I in AD 675. It appears to have been ruined by

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