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Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic pe ...
astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...
,
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
, scholar, and student of Archytas and
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
. All of his original works are lost, though some fragments are preserved in
Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the e ...
' commentary on Aratus's poem on
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
. '' Sphaerics'' by Theodosius of Bithynia may be based on a work by Eudoxus.


Life

Eudoxus was born and died in
Cnidus Knidos or Cnidus (; grc-gre, Κνίδος, , , Knídos) was a Greek city in ancient Caria and part of the Dorian Hexapolis, in south-western Asia Minor, modern-day Turkey. It was situated on the Datça peninsula, which forms the southern sid ...
(also spelled Knidos), which was a city on the southwest coast of
Asia Minor Anatolia, tr, Anadolu Yarımadası), and the Anatolian plateau, also known as Asia Minor, is a large peninsula in Western Asia and the westernmost protrusion of the Asian continent. It constitutes the major part of modern-day Turkey. The re ...
. The years of Eudoxus' birth and death are not fully known but the range may have been , or . His name Eudoxus means "honored" or "of good repute" (, from ''eu'' "good" and ''doxa'' "opinion, belief, fame"). It is analogous to the Latin name Benedictus. Eudoxus's father, Aeschines of
Cnidus Knidos or Cnidus (; grc-gre, Κνίδος, , , Knídos) was a Greek city in ancient Caria and part of the Dorian Hexapolis, in south-western Asia Minor, modern-day Turkey. It was situated on the Datça peninsula, which forms the southern sid ...
, loved to watch stars at night. Eudoxus first traveled to
Tarentum Tarentum may refer to: * Taranto, Apulia, Italy, on the site of the ancient Roman city of Tarentum (formerly the Greek colony of Taras) **See also History of Taranto * Tarentum (Campus Martius), also Terentum, an area in or on the edge of the Camp ...
to study with Archytas, from whom he learned
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. While in
Italy Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the homonymous geographical ...
, Eudoxus visited
Sicily (man) it, Siciliana (woman) , population_note = , population_blank1_title = , population_blank1 = , demographics_type1 = Ethnicity , demographics1_footnotes = , demographi ...
, where he studied medicine with Philiston. At the age of 23, he traveled with the physician Theomedon—who (according to
Diogenes Laërtius Diogenes Laërtius ( ; grc-gre, Διογένης Λαέρτιος, ; ) was a biographer of the Greek philosophers. Nothing is definitively known about his life, but his surviving ''Lives and Opinions of Eminent Philosophers'' is a principal sour ...
) some believed was his lover—to
Athens Athens ( ; el, Αθήνα, Athína ; grc, Ἀθῆναι, Athênai (pl.) ) is both the capital and largest city of Greece. With a population close to four million, it is also the seventh largest city in the European Union. Athens dominates a ...
to study with the followers of
Socrates Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no t ...
. He eventually attended lectures of
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
and other philosophers for several months, but due to a disagreement they had a falling-out. Eudoxus was quite poor and could only afford an apartment at
Piraeus Piraeus ( ; el, Πειραιάς ; grc, Πειραιεύς ) is a port city within the Athens urban area ("Greater Athens"), in the Attica region of Greece. It is located southwest of Athens' city centre, along the east coast of the Saro ...
. To attend Plato's lectures, he walked the in each direction each day. Due to his poverty, his friends raised funds sufficient to send him to Heliopolis,
Egypt Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a List of transcontinental countries, transcontinental country spanning the North Africa, northeast corner of Africa and Western Asia, southwest corner of Asia via a land bridg ...
, to pursue his study of astronomy and mathematics. He lived there for 16 months. 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 Mausolus ( grc, Μαύσωλος or , xcr, ���𐊠���𐊸𐊫𐊦 ''Mauśoλ'') was a ruler of Caria (377–353 BCE) and a satrap of the Achaemenid Empire. He enjoyed the status of king or dynast by virtue of the powerful position created by ...
. During his travels he gathered many students of his own. Around 368 BC, Eudoxus returned to Athens with his students. According to some sources, around 367 he assumed headship ( scholarch) of the Academy during Plato's period in Syracuse, and taught
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
. 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 Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing th ...
, astronomy, and
meteorology Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...
. 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 The cosmological model of concentric (or homocentric) spheres, developed by Eudoxus, Callippus, and Aristotle, employed celestial spheres all centered on the Earth. In this respect, it differed from the epicyclic and eccentric models with multip ...
, and his early contributions to understanding the movement of the
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
s. His work on proportions shows insight into
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s; it allows rigorous treatment of continuous quantities and not just whole numbers or even
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s. When it was revived by
Tartaglia Tartaglia may refer to: *Tartaglia (commedia dell'arte), Commedia dell'arte stock character *Angelo Tartaglia (1350 or 1370–1421), Italian condottiero * Niccolò Fontana Tartaglia (1499/1500–1557), Venetian mathematician and engineer *Ivo Tarta ...
and others in the 16th century, it became the basis for quantitative work in science, and inspired the work of Richard Dedekind. Craters on
Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin at ...
and the
Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
are named in his honor. An algebraic curve (the Kampyle of Eudoxus) is also named after him.


Mathematics

Eudoxus is considered by some to be the greatest of
classical Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic pe ...
mathematicians, and in all
Antiquity Antiquity or Antiquities may refer to: Historical objects or periods Artifacts *Antiquities, objects or artifacts surviving from ancient cultures Eras Any period before the European Middle Ages (5th to 15th centuries) but still within the histo ...
second only to
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
. Eudoxus was probably the source for most of book V of Euclid's . He rigorously developed Antiphon's
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
, 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.Morris Kline, ''Mathematical Thought from Ancient to Modern Times'' Oxford University Press, 1972 pp. 48–50 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. The Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with 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 Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
provides an elaborate proof of 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, but instead they used proportionalities to express the relationship between quantities. Thus the ratio of two similar quantities was not just a numerical value, as we think of it today; the ratio of two similar quantities was a primitive relationship between them. Eudoxus was able to restore confidence in the use of proportionalities by providing an astounding definition for the meaning of the equality between two ratios. This definition of proportion forms the subject of Euclid's Book V. In Definition 5 of Euclid's Book V we read: By using modern-day notation, this is clarified as follows. If we take four quantities: ''a'', ''b'', ''c'', and ''d'', then the first and second have a ratio a/b; similarly the third and fourth have a ratio c/d. Now to say that a/b = c/d we do the following: For any two arbitrary integers, ''m'' and ''n'', form the equimultiples ''m''·''a'' and ''m''·''c'' of the first and third; likewise form the equimultiples ''n''·''b'' and ''n''·''d'' of the second and fourth. If it happens that ''m''·''a'' > ''n''·''b'', then we must also have ''m''·''c'' > ''n''·''d''. If it happens that ''m''·''a'' = ''n''·''b'', then we must also have ''m''·''c'' = ''n''·''d''. Finally, if it happens that ''m''·''a'' < ''n''·''b'', then we must also have ''m''·''c'' < ''n''·''d''. Notice that the definition depends on comparing the similar quantities ''m''·''a'' and ''n''·''b'', and the similar quantities ''m''·''c'' and ''n''·''d'', and does not depend on the existence of a common unit of measuring these quantities. The complexity of the definition reflects the deep conceptual and methodological innovation involved. It brings to mind the famous fifth postulate of Euclid concerning parallels, which is more extensive and complicated in its wording than the other postulates. The Eudoxian definition of proportionality uses the quantifier, "for every ..." to harness the infinite and the infinitesimal, just as do the modern
epsilon-delta definition Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
s of limit and continuity. Additionally, the Archimedean property stated as definition 4 of Euclid's book V is originally due not to Archimedes but to Eudoxus.


Astronomy

In
ancient Greece Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cu ...
, 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: * ''Disappearances of the Sun'', possibly on eclipses * ''Oktaeteris'' (Ὀκταετηρίς), on an eight-year lunisolar-Venus cycle of the calendar * ''Phaenomena'' (Φαινόμενα) and ''Enoptron'' (Ἔνοπτρον), on
spherical astronomy Spherical astronomy, or positional astronomy, is a branch of observational astronomy used to locate astronomical objects on the celestial sphere, as seen at a particular date, time, and location on Earth. It relies on the mathematical methods of ...
, probably based on observations made by Eudoxus in Egypt and Cnidus * ''On Speeds'', on planetary motions We are fairly well informed about the contents of ''Phaenomena'', for Eudoxus's prose text was the basis for a poem of the same name by Aratus.
Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the e ...
quoted from the text of Eudoxus in his commentary on Aratus.


Eudoxan planetary models

A general idea of the content of ''On Speeds'' can be gleaned from
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
's ''Metaphysics'' XII, 8, and a commentary by Simplicius of Cilicia (6th century AD) on ''De caelo'', another work by Aristotle. According to 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 the Eudoxan model, the Moon is assigned three spheres: * The outermost rotates westward once in 24 hours, explaining rising and setting. * The second rotates eastward once in a month, explaining the monthly motion of the Moon through the
zodiac The zodiac is a belt-shaped region of the sky that extends approximately 8° north or south (as measured in celestial latitude) of the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. The pa ...
. * The third also completes its revolution in a month, but its axis is tilted at a slightly different angle, explaining motion in latitude (deviation from the ecliptic), and the motion of the lunar nodes. 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 Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin at ...
,
Jupiter Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousand ...
, and Saturn) are assigned four spheres each: * The outermost explains the daily motion. * The second explains the planet's motion through the zodiac. * The third and fourth together explain retrogradation, when a planet appears to slow down, then briefly reverse its motion through the zodiac. By inclining the axes of the two spheres with respect to each other, and rotating them in opposite directions but with equal periods, Eudoxus could make a point on the inner sphere trace out a figure-eight shape, or
hippopede In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotatio ...
.


Importance of Eudoxan system

Callippus Callippus (; grc, Κάλλιππος; c. 370 BC – c. 300 BC) was a Greek astronomer and mathematician. Biography Callippus was born at Cyzicus, and studied under Eudoxus of Cnidus at the Academy of Plato. He also worked with Aristotle at ...
, 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 Greek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the Ancient Greek, Hellenistic, Greco-Roman, and Late Antiquity eras. It is not limited geographically to Greece or to ...
is considerable.


Ethics

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
, in the '' Nicomachean Ethics'', attributes to Eudoxus an argument in favor of
hedonism Hedonism refers to a family of theories, all of which have in common that pleasure plays a central role in them. ''Psychological'' or ''motivational hedonism'' claims that human behavior is determined by desires to increase pleasure and to decr ...
—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.This particular argument is referenced in Book One.


See also

*
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
* Euclid's * Eudoxus reals (a fairly recently discovered construction of the real numbers, named in his honor) * Delian problem * Incommensurable magnitudes *
Speusippus Speusippus (; grc-gre, Σπεύσιππος; c. 408 – 339/8 BC) was an ancient Greek philosopher. Speusippus was Plato's nephew by his sister Potone. After Plato's death, c. 348 BC, Speusippus inherited the Academy, near age 60, and remaine ...


References


Bibliography

* * * * * * * * Lasserre, François'' (''1966)'' Die Fragmente des Eudoxos von Knidos'' (de Gruyter: Berlin) * * Manitius, C. (1894) ''Hipparchi in Arati et Eudoxi Phaenomena Commentariorum Libri Tres ''(Teubner) * *


External links


Working model and complete explanation of the Eudoxus's Spheres

Eudoxus (and Plato)
a documentary on Eudoxus, including a description of his planetary model
Dennis Duke, "Statistical dating of the Phaenomena of Eudoxus", ''DIO'', volume 15
''see pages 7 to 23''
Eudoxus of Cnidus
Britannica.com

Donald Allen, Professor, Texas A&M University

Henry Mendell, Cal State U, LA



Craig McConnell, Ph.D., Cal State, Fullerton *

(
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
applet) {{DEFAULTSORT:Eudoxus Of Cnidus 5th-century BC births 4th-century BC deaths 4th-century BC Greek physicians 4th-century BC philosophers Academic philosophers Ancient Greek astronomers Ancient Greek mathematicians Ancient Greek ethicists Ancient Cnidians Students of Plato 4th-century BC mathematicians People from Muğla Province