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45-404: He settled in a miserable village near Helicon, Ascra, vile in winter, painful in summer, never good. The 4th century BCE astronomer and general Eudoxus thought even less of Ascra's climate. However, other writers speak of Ascra as abounding in corn, Corinthian hunchbacks, and wine. By the time Eudoxus wrote, the town had been all but destroyed (by Thespiae sometime between 700 and 650 BCE),

90-510: 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

135-533: 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 ⁠

180-425: 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

225-572: A satrapy . The Persians came to dominate Egypt, but Egypt remained independent until it was made a Persian province in 485 B.C., after a revolt. The Twenty-seventh Dynasty of Egypt consists of the Persian emperors - including Cambyses, Xerxes I , and Darius the Great - who ruled Egypt as Pharaohs and governed through their satraps, as well as the Egyptian Petubastis III (522–520 BC) (and possibly

270-485: A brother of the emperor Xerxes I, and Arsames (c.454–c.406 BC). The Twenty-Eighth Dynasty consisted of a single king, Amyrtaeus , prince of Sais , who successfully rebelled against the Persians, inaugurating Egypt's last significant phase of independence under native sovereigns. He left no monuments with his name. This dynasty reigned for six years, from 404 BC–398 BC. The Twenty-Ninth Dynasty ruled from Mendes , for

315-556: A location in ancient Boeotia is a stub . You can help Misplaced Pages by expanding it . Eudoxus of Cnidus Eudoxus of Cnidus ( / ˈ juː d ə k s ə s / ; Ancient Greek : Εὔδοξος ὁ Κνίδιος , Eúdoxos ho Knídios ; c.  390  – c.  340 BC ) 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

360-684: A loss commemorated by a similarly lost Hellenistic poem, which opened: "Of Ascra there isn't even a trace anymore" ( Ἄσκρης μὲν οὐκέτ' ἐστὶν οὐδ' ἴχνος ). This apparently was a hyperbole, for in the 2nd century CE, Pausanias could report that a single tower, though not much else, still stood at the site. [REDACTED]  This article incorporates text from a publication now in the public domain :  Smith, William , ed. (1854–1857). "Ascra". Dictionary of Greek and Roman Geography . London: John Murray. 38°19′37″N 23°04′27″E  /  38.327032°N 23.074249°E  / 38.327032; 23.074249 This article about

405-465: 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

450-583: 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

495-444: 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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540-536: A wide range of archaeological finds from throughout the Levant shows an Egyptian occupation and control in the late decades of the 7th century BC. These include various Egyptian objects from several sites, ostraca and documents showing a tribute/tax system, and evidence from the fortress of Mezad Hashavyahu. Egyptian influence reached to the Euphrates area in places such as Kimuhu and Quramati . Later they were pushed back by

585-476: 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: Late Period of ancient Egypt The Late Period of ancient Egypt refers to the last flowering of native Egyptian rulers after

630-692: 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

675-568: The Third Intermediate Period in the 26th Saite Dynasty founded by Psamtik I , but includes the time of Achaemenid Persian rule over Egypt after the conquest by Cambyses II in 525 BC as well. The Late Period existed from 664 BC until 332 BC, following a period of foreign rule by the Nubian 25th Dynasty and beginning with a short period of Neo-Assyrian suzerainty , with Psamtik I initially ruling as their vassal. The period ended with

720-402: 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

765-419: 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

810-553: 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

855-561: 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

900-654: 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

945-496: The Near East. The expedition was beginning to meet with some success and made its way to Phoenicia without particular problems. unfortunately for Teos, his brother Tjahapimu was plotting against him. Tjahapimu convinced his son Nectanebo II to rebel against Teos and to make himself pharaoh. The plan was successful and the betrayed Teos had no alternative but to flee and the expedition disintegrated. The final ruler of this dynasty, and

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990-527: 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

1035-453: 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

1080-429: 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

1125-607: The city of Sais , reigned from 672 to 525 BC, and consisted of six pharaohs. It started with the unification of Egypt under Psamtik I c. 656 BC, itself a direct consequence of the Sack of Thebes by the Assyrians in 663 BC. Canal construction from the Nile to the Red Sea began. Egypt seems to have expanded into the Near East early in this period. They conquered the city of Ashdod around 655 BC, and

1170-694: The conquests of the Persian Empire by Alexander the Great and establishment of the Ptolemaic dynasty by his general Ptolemy I Soter , one of the Hellenistic diadochi from Macedon in northern Greece . With the Macedonian Greek conquest in the latter half of the 4th century BC, the age of Hellenistic Egypt began. The Twenty-Sixth Dynasty , also known as the Saite Dynasty after its seat of power

1215-402: 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

1260-444: 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

1305-448: The defeat at Carcemish , although Egyptian intervention in the Near East seems to have continued after this battle. Amasis II followed a new policy and directed his interests toward the Greek world. He annexed Cyprus during his reign. To the south, Psamtik II led a great military expedition that reached deep into upper Nubia and inflicted a heavy defeat on them. A demotic papyrus from

1350-576: The disputed Psammetichus IV ), who rebelled in defiance of the Persian authorities. The unsuccessful revolt of Inaros II (460–454), aided by the Athenians as part of the Wars of the Delian League , aspired to the same object. The Persian satraps were Aryandes (525–522 BC; 518–c.496 BC) - whose rule was interrupted by the rebel Pharaoh Petubastis III, Pherendates (c.496–c.486 BC), Achaemenes (c.486–459 BC) -

1395-434: 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,

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1440-581: The final native ruler of Egypt, was Nectanebo II who was defeated in battle leading to the re-annexation by the Achaemenid Empire . The Second Achaemenid Period saw the re-inclusion of Egypt as a satrapy of the Persian Empire under the rule of the Thirty-First Dynasty, (343–332 BC) which consisted of three Persian emperors who ruled as Pharaoh— Artaxerxes III (343–338 BC), Artaxerxes IV (338–336 BC), and Darius III (336–332 BC)—interrupted by

1485-569: 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

1530-591: 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 ⋅

1575-399: 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

1620-599: 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

1665-425: 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

1710-542: The period from 398 to 380 BC. King Hakor of this dynasty was able to defeat a Persian invasion during his reign. The Thirtieth Dynasty took their art style from the Twenty-Sixth Dynasty . A series of three pharaohs ruled from 380 to 343 BC. The first king of the dynasty, Nectanebo I , defeated a Persian invasion in 373 BC. His successor Teos subsequently led an expedition against the Achaemenid Empire in

1755-400: 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

1800-627: The reign of Ahmose II describes a small expedition into Nubia, the character of which is unclear. There is archaeological evidence of an Egyptian garrison at Dorginarti in lower Nubia during the Saite period. One major contribution from the Late Period of ancient Egypt was the Brooklyn Papyrus . This was a medical papyrus with a collection of medical and magical remedies for victims of snakebites based on snake type or symptoms. Artwork during this time

1845-409: 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

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1890-417: 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

1935-432: 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

1980-506: Was representative of animal cults and animal mummies. This image shows the god Pataikos wearing a scarab beetle on his head, supporting two human-headed birds on his shoulders, holding a snake in each hand, and standing atop crocodiles. The First Achaemenid Period (525–404 BC) began with the Battle of Pelusium , which saw Egypt ( Old Persian : 𐎸𐎭𐎼𐎠𐎹 Mudrāya ) conquered by the expansive Achaemenid Empire under Cambyses , and Egypt become

2025-555: 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

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