Eudoxus of Cnidus

Eudoxus of Cnidus (Greek Εὔδοξος Eudoxus; * probably 397-390 BC in Cnidus, † probably 345-338 BC in Cnidus ) was a Greek mathematician, astronomer, geographer, physician, philosopher and legislator of antiquity. His works are lost except for fragments. Therefore, his scientific achievements are only known from reports of other authors or open up. With its mathematical representation of the celestial bodies movements he made ​​a significant contribution to the geometrization of astronomy. In mathematics, he founded the general theory of proportions.

• 6.1 antiquity
• 6.2 Modern

Life

The traditional chronology of the life of Eudoxus is incorrect. The Diogenes Laertius Doxograph cites a message of Apollodorus of Athens, from the results in the birth of the period 407-404 and for the death, as Eudoxus to have died in his 53rd year of life, the period from 355 to 352 This can but not accurate, as Eudoxus in one of his works specifically occurred at a time indication on the 348/347 death of Plato refers. Since he survived Plato, his birth, his death probably about falls in the period 397-390, roughly in the period 345-338.

Eudoxus was born in modest circumstances. His home city of Knidos was in the 5th century BC for its medical school, which rivaled that of Kos, famous. Supposedly he studied with Archytas mathematics and Philistion of Locri medicine. For a student relative to Archytas who lived in Tarentum, but a longer period of study in Italy would have to be assumed, which is doubted in research; the tradition to an alleged trip to Sicily is considered implausible. At the age of about 23, he went to Athens, as the reputation of the teaching there Socratic attracted him. This first stay in Athens lasted only two months and resulted in probably his first encounter with Plato. According to a message communicated by Diogenes Laertius Plato initially refused to accept him as a student, but the reports quoted by Diogenes Doxograph Sotion of Alexandria, he was able to attend lectures of Plato. Anyway Eudoxus soon returned to Cnidus.

To 365/364 Eudoxus traveled in the company of a fellow citizen, physician Chrysippus, to Egypt. A recommendation letter from the king Agesilaus II of Sparta paved the way for him to Pharaoh Nectanebo I. The stay lasted sixteen months. His particular interest was the knowledge of the Egyptian priests, in their astronomy he gained insight. After returning from Egypt he went to Cyzicus on the southern coast of the Marmara Sea, where he gave lessons. From there he attended the influential, open-minded for cultural concerns Persian satrap mouse solos.

Later he moved with a considerable number of his students to Athens. There he is said to have a controversial according to tradition rivals as a teacher with Plato. Whether he joined the Platonic Academy and to what extent it can be called a Platonist, and in turn influenced Plato, is not clearly determined. Anyway, there was a contact to the Academy, which was probably closely. According to a report, which goes back to Aristoxenus of Tarentum, Aristotle came " under Eudoxus " or " at the time of Eudoxus " in the academy. This formulation has been interpreted in the earlier research so that the entrance of Aristotle was, as Plato was on his second journey to Sicily, and that Eudoxus served as deputy Plato's Academy. However, this is not plausible, as Eudoxus was then still young and may have worked in the Academy not long, if he ever actually belonged to her.

After the stay in Athens Eudoxus began again an independent teaching, probably again in Cyzicus. Later he returned to Cnidus. There he worked as a legislator for his fellow citizens, with whom he enjoyed high reputation; well as nationally he found recognition. He built an observatory in Cnidus.

Among the pupils of Eudoxus included the physician Chrysippus, who accompanied him to Egypt, the mathematician and astronomer Menaechmus and Dinostratus Polemarchos of Cyzicus.

Diogenes Laertius mentions Eudoxus had three daughters and a son named Aristagoras.

Works

Although Eudoxus accomplished his most important achievements in the field of geometry, not a single title of the relevant work has survived. His mathematical discoveries are therefore known only from the writings of other authors. From philosophical works, nothing is handed down; he may not have written, but his views stated orally.

Eudoxus wrote several astronomical writings, which are known in later literature only mentions or reproductions of their contents:

• " Phenomena " ( phainomena ), his first astronomical work. A revised version he called " mirror " ( Énoptron; meant: Mirror of the World Order ). The writing consisted of three books. The first contained a description of the relative positions of the stars, the second treated their positions relative to the celestial sphere and their organizations, in the third, a catalog of stars was with details of the current rising and setting. Get the famous poem " phenomena and character" of Aratus of Soli, the free transmission contains " phenomena " of Eudoxus in verses in the first part.
• " Over-speeding " ( Peri TACHON ), his main astronomical work in which he explained the movements of the five known planets and the sun and the moon. He went out of his geocentric world view that is based on the assumption of a stationary earth to rotate the spherical shells ( spheres ) that are associated with the moving celestial bodies and the fixed stars. With the speeds mentioned in the title of the different rotational speeds of the spheres are meant. Aristotle describes the system of Eudoxus scarce in his metaphysics.
• "On the extinction caused by the sun " ( aphanismōn Peri Heliakon ). Here Eudoxus explained his method, with which the time of the ascent and demise of a star is determined when the sunlight makes an accurate observation impossible.
• " The eight-year cycle " ( Oktaetērís ) in which Eudoxus explained an astronomical calendar, which is based on an eight year cycle. It appears to be the oldest so -titled work on this later often discussed topic to act, but it is unclear whether the traditional title of Eudoxus comes. His original work is only partially reconstructed; Eratosthenes gave a him present version, which was a lot of attention in the ancient world, rightly for not authentic.
• Astronomía, an astronomical calendar in verse, the acting of the stars contained the word. This didactic poem was a late work.

Not by Eudoxus comes of him earlier treatise ascribed Eudoxi ars astronomica, which is available on papyrus. However, this paper contains, among other material for astronomy and calendar calculations, which goes back to him.

In geographical area Eudoxus wrote a fictional figure " geography " (Ges periodos ), one of his late works. In it, he also dealt with cultural geography topics.

What had writings for a content, called the Diogenes Laertius as " dogs Dialogues" ( Kynon diálogoi ), is unknown. Diogenes refers to Eratosthenes and supplemented by other authors it were originally to dialogues in Egyptian language that Eudoxus have translated only into Greek. The latter would require knowledge of the languages ​​of Eudoxus and is therefore hardly credible.

Philosophy

About two opinions of Eudoxus to philosophical questions is a report of Aristotle. One concerns the theory of ideas, the other instruction of the good. In both questions Eudoxus represents a view that contradicts that of Plato's fundamental.

Theory of Ideas

Probably Eudoxus has not developed its own theory of ideas, but discussed only in special circumstances a single question of the theory of ideas, which you might wish merely hypothetical whereas him. The problem was the question of how the participation of individual things comes to ideas about. Eudoxus thought he could solve it with a mixture of teaching; the ideas are added to the perceptible objects. Aristotle compares this with the admixture of a color to her stained. It is uncertain whether this comparison is due to Eudoxus. How Eudoxus has presented the mixture is unclear; apparently he went from a natural philosophy, common in the pre-Socratics mix concept and took unlike Plato at a local presence of the ideas in things. In contrast to Aristotle, he also wanted to hold on to Plato's doctrine of a separate existence from the things of ideas. This earned him the accusation of inconsistency. The traditional counter- argument is that he materialize the ideas and inclusive in the transience of the material world and that they thereby lose their simplicity and immutability. Thus they would lose their specific ontological status, so would be no ideas in the sense of the Platonic doctrine of ideas more.

Ethics

In ethics Eudoxus was of a hedonistic position in the determination of good. He put the good with the same joy. He argued that the joy might of all beings - rational and irrational - sought; it is therefore the common good, the good per se. You will not be commended to other goods, in contrast, as they stand above all laudable goods. Since they 'll strives for her sake and not the sake of another good, you get a higher rank than to the goods that appear to be desirable for a particular purpose. Each product, which they will annexed, will thus desirable; Therefore, it is the true good, namely, that which does not win by something else, but only by itself is disputed whether Eudoxus was of the conviction, the pleasure principle inherent in the nature of the Godhead and therefore targets the striving for pleasure, which in all living things in the cosmos are one and the same is, ultimately, on the deity in the highest pleasure is realized.

Mathematics

Eudoxus established the general theory of proportions. He was able for the first time include the irrational sizes, as his theory of proportion is also applicable to incommensurable magnitudes. His definitions of ratio ( lógos ) and proportion ( analogia ) have been handed down in the fifth book of Euclid's Elements.

In research, it has been suggested that one named after Archimedes axiom Archmedische actually comes from Eudoxus. The initial problem has Eudoxus knew obviously, but to what extent he has dealt with it is unclear.

He examined the volume ratios of bodies and showed that the volume of a pyramid of the respective prism, and a third that of a cone corresponding to one third of the corresponding cylinder. He used an infinitesimal calculation method, the method of exhaustion for his proof. Using this method he was able to determine the radius and the ratio of the circular area and the ball volume.

Eudoxus dealt with in antiquity intensively discussed problem of the doubling cube ( " Delos problem "). He found it an unspecified solution known by the intersection of curves; the points of intersection were necessary for solving the problem is proportional to the two middle edge of the given and the desired cube. Further, Eudoxus invented, such as Plutarch, also a mechanical device for approximate construction of two mean proportionals.

Astronomy and geography

In astronomy, it was Eudoxus matter of presenting in the context of his geocentric irregularly appearing movements of the planets with shutdowns and return lines mathematically. Supposedly Plato had made ​​the research task, due to the Planetenumläufe regular circular motions. For each of the planets and sun and moon Eudoxus took our own system of concentric spherical shells ( spheres ) which rotate at different speeds and in different directions likely to mutually inclined axes. Because of the common center - the earth - it is called the theory of homocentric spheres. Each celestial body is fixed at the equator of the innermost shell associated with it. Belonging to a celestial body shells are attached to their poles together. This will transfer their movements to the innermost shell, and thus on the star. The five planets known to him had Eudoxus four spheres to the sun and the moon three. Each of the first (outermost ) planet sphere causes the daily rotation of the planet from east to west, the second its annual path in the zodiac from west to east, the third and fourth together generate the (apparent) eight -shaped loop movement that Eudoxus " Hippopede " (horses ankle) called it. For the fixed stars was enough for a sphere. This Eudoxus came to a total of 27 spheres. The question of a relationship between the individual spheres systems with each other turned on him not, because his conception was a purely mathematical, not physical well-founded hypothesis, the complex relations into simple elements (regular circular motions ) dissolved.

The research is partly assumed and partly denied that Eudoxus already tried to fulfill " saving the phenomena " which later spread requirement of, and even the author was in accordance with the formulation of a research principle. The exact meaning of this expression is controversial. According to one interpretation, it was originally a matter of having the apparent motions of the heavenly bodies as a result of their true movements. Only later do we have understood by " saving the phenomena " the principle that is to require a theory only, that are consistent with their calculations carried out with each observation result, and not that it reflects physical reality truthfully. According to another interpretation was the " Rescue " of the phenomena meant a mathematical description, the predetermined physical or natural philosophical assumptions to be met, as subsequently modify or supplement that they " save the phenomena ", ie certain phenomena that appear as anomalies in their context, also be considered.

The Eudoxus of the system allowed only approximate solutions and was not aware of all the known anomalies of the celestial body movements. Therefore, it was later expanded by Callippus of Cyzicus, who added more spheres.

An important weakness of the system was that it could not explain the variations in brightness of the planets, which suggest changes to their distance from Earth. Then pointed the Aristotle commentator Simplicius in Late Antiquity.

Eudoxus also determined the distances and proportions of earth, moon and sun. As he was going on here, is unknown. The order of the moving celestial body was with him at that time common: from the inside out Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn.

He also should have determined the earth's circumference. Aristotle is appealing to " mathematician ", which he calls not by name, a circumference of the earth of 400,000 stadiums. In this way too high number is probably an estimate; the first recorded calculation is that of Eratosthenes in the 3rd century BC The notified by Aristotle estimate is often attributed to Eudoxus, but there is no evidence for it.

Cicero reported Eudoxus had the first registered the constellations on a celestial globe, having already Thales had made such a ball. Thus, Eudoxus was not the inventor of the celestial globe, but the first, who told the stars charted.

A traditional Vitruvius catalog of inventions, according to Eudoxus was the inventor of Arachne ( "spider" ). This term referred to in later times the movable plate of a flat astrolabe ( a two-dimensional representation of the celestial sphere ). This rotating disk formed from the fixed stars and could reflect its daily rotation. However, such astrolabe did not exist at the time of Eudoxus. His arachne was probably a precursor, a transparent disk, which formed the main component of a star clock. She wore a picture of the fixed stars, which resembled a spider's web.

As a geographer to Eudoxus was particularly concerned with the division of the earth into climates. He drew a map of lying in the temperate zone of the northern hemisphere Oikumene ( the known inhabited part of the earth's surface ). For him, the Oikumene had the shape of a rectangle that was twice as long as wide. The boundary formed in the west, the Iberian Peninsula, to the east India, Ethiopia in the south and in the north the territory of the Scythians. In the south joined the temperate zone, the " burnt zone " whose northern half handed to the equator. In the southern hemisphere the climatic conditions were mirror images. There he took in the southern temperate zone, in which springs from the Nile, one of the Oikumene analog, also inhabited Gegenoikumene. He determined the latitude ( énklima ) a region on the ratio of the longest to the shortest day or the shortest night; whether he has already calculated in latitude, is uncertain. For his place he came to a ratio of 5: 3; Discovered another result of it was 12: 7 It is unclear whether the two statements refer to different places or is regarded as an adjustment to the other. He used a gnomon, a vertical rod whose shadow is observed in the horizontal plane for his measurements. From the ratio of the shortest and the longest noon shadow, which were observed at the two solstices, the ratio was looking for.

Reception

Antiquity

To what extent was Plato's dialogue Philebus intended as antihedonistische answer to the reasoning of Eudoxus for the equating of good with the joy or pleasure, is unclear; in research, opinions about it apart.

Aristotle in his Nicomachean Ethics was a critically on Eudoxus ' like teaching. He came across the difficulty that the reasoning of Eudoxus partially arguments that he put forward elsewhere, were very similar, the resulting inferred equating of good with pleasure, however, was not acceptable to him, as he the pursuit of the "many " after enjoyment considered vulgar and bestial. Therefore, he made ​​do with an argument ad hominem: Eudoxus had with his arguments not proved popular because of their persuasion, but because of his own character, who was so good that you did not suspect him to defend a vulgar hedonism. Basically told Aristotle but the conviction of Eudoxus that the basic normative concept of " good" theory of action is to determine as "the destination to which everything aims ". Maybe he assumed the equation of ( real or supposed ) good with striven, he asked at the beginning of the Nicomachean Ethics, by Eudoxus. In his lost work "On the idea " did Aristotle in the context of his critique of Plato's theory of ideas and arguments against the theory of ideas of Eudoxus ago.

Euclid took a number of decisions of the Eudoxus in his elements. This was the theory of proportions and their geometric applications, and volume determinations.

The planetary system of Eudoxus was instrumental in the extended version of Callippus in Greek astronomy until it was ousted from the late 3rd century BC by the epicycle. In particular, it influenced the astronomical ideas of Aristotle. Aristotle tried to substantiate the mathematical model physically by the spheres conceived as a real material circumstances, their number still increased and assumed that they were all connected. So Eudoxus influence on the reception of Aristotle led indirectly to the physical world of the Middle Ages.

The " phenomena " received attention in many ancient authors. They were used, among others, Philip of Opus, Aratus of Soli and Geminus of Rhodes; Vitruvius had only indirect access through an intermediate source. Aratus wrote a Versfassung. In the 2nd century BC the astronomer Hipparchus of Nicaea wrote a critical commentary on the " phenomena " of Eudoxus and the Versfassung of Aratus.

Strabo praises the geographical competence of Eudoxus. He says the historian Polybius had estimated Eudoxus ' account of the historical geography; these included founding stories of Greek cities, as well as information about the lineage relationships of their populations and migration. As Strabo 25/24 BC or shortly thereafter traveled to Egypt, he was shown in Heliopolis rooms in which Eudoxus had allegedly stopped. This legend was part of an Egyptian propaganda, which led back Greek wisdom and knowledge on Egyptian roots. Seneca said that Eudoxus had brought the knowledge of the planetary movements from Egypt to Greece.

Philostratus considered Eudoxus as rhetorically gifted philosopher and forerunner of the " Second Sophistic ". Diogenes Laertius rated his works to be extremely valuable.

Modern

In modern research Eudoxus has exceptionally strong attention. In particular, his spheres system is discussed intensively since the 19th century astronomy historians. 1828-1830 put the astronomer Christian Ludwig Ideler before a reconstruction attempt in 1849 brought an investigation by Ernst Friedrich Apelt further insights. The 1877 reconstruction of the astronomer Giovanni Schiaparelli published proved to be groundbreaking. They dominated until the late 20th century and is still considered a " classic ", but has weaknesses that have led to the emergence of alternative hypotheses. 1935, the lunar crater Eudoxus, 1973, the Mars crater Eudoxus and 1998, the asteroid ( 11709 ) Eudoxus was named after Eudoxus.

Text output

• François Lasserre (eds.): The fragments of Eudoxus of Cnidus. De Gruyter, Berlin, 1966 ( critical edition with commentary; see the very critical review by Gerald J. Toomer in Gnomon Vol 40, 1968, pp. 334-337. ).