Eudoxus from Knidos

Eudoxos of Knidos ( Greek Εὔδοξος Eúdoxos ; * probably between 397 and 390 BC in Knidos ; † probably between 345 and 338 BC in Knidos) was a Greek mathematician , astronomer , geographer , doctor , philosopher and lawgiver of antiquity . Except for fragments, his works have been lost. Therefore, his scientific achievements are only known or can be inferred from reports by other authors. With his mathematical representation of the celestial body movements, he made a significant contribution to the geometry of astronomy. In mathematics he founded the general theory of proportions.

Life

The traditional chronological classification of the lifetime of Eudoxus is incorrect. The doxographer Diogenes Laertios quotes a communication from Apollodorus of Athens , from which the period from 407 to 404 results for the birth and the period 355 to 352 for the death, since Eudoxus is said to have died in his 53rd year of life. This can but not applicable, since Eudoxus expressly refers to the death of Plato in 348/347 in one of his works. Since he survived Plato, his birth probably falls between 397-390 and his death between 345 and 338.

Eudoxus came from a humble background. His hometown of Knidos had been around since the 5th century BC. Famous for its medical school rivaling that of Kos . Allegedly he studied mathematics with Archytas of Taranto and medicine with Philistion of Lokroi . For a student relationship with Archytas, who lived in Taranto , a longer study visit to Italy would have to be accepted, which is doubted in research; the tradition of an alleged trip to Sicily is considered implausible. At the age of about 23 he went to Athens because of the reputation of the Socratics who taught there . This first stay in Athens lasted only two months and probably led to his first meeting with Plato. According to a message communicated by Diogenes Laertios, Plato initially refused to accept him as a student, but the doxographer Sotion of Alexandria, quoted by Diogenes, reports that he was able to take part in Plato's lectures. In any case, Eudoxus soon returned to Knidos.

Around 365/364 Eudoxus traveled to Egypt accompanied by a fellow citizen, the doctor Chrysippus. A letter of recommendation from King Agesilaus II of Sparta paved the way for him to see Pharaoh Nectanebos I. The stay lasted sixteen months. He was particularly interested in the knowledge of the Egyptian priests, into whose astronomy he gained insight. After returning from Egypt he went to Kyzikos on the south coast of the Marmara Sea , where he gave lessons. From there he visited the influential Persian satrap Maussolos , who was open to cultural issues .

He later moved to Athens with a considerable number of his students. There, according to a controversial tradition, he is said to have rivaled Plato as a teacher. Whether he entered the Platonic Academy and to what extent he can be called a Platonist and in turn influenced Plato cannot be clearly determined. In any case, there was a contact to the academy that was probably close. According to a report that goes back to Aristoxenus of Taranto, Aristotle entered the academy “under Eudoxus” or “at the time of Eudoxus”. In older research, this formulation was interpreted to mean that Aristotle entered when Plato was on his second journey to Sicily, and that Eudoxus, as Plato's deputy, headed the academy. This is not plausible, however, since Eudoxos was still young at the time and could not have worked in the academy for long, if he ever actually belonged to it.

After his stay in Athens, Eudoxus began teaching again, probably again in Kyzikos. He later returned to Knidos. There he worked as a legislator for his fellow citizens, with whom he enjoyed a high reputation; it also found recognition across the region. He had an observatory built in Knidos .

The doctor Chrysippus, who accompanied him to Egypt, the mathematicians Menaichmos and Deinostratos and the astronomer Polemarchus of Kyzikos belonged to the students of Eudoxus .

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

Works

Although Eudoxus achieved his most important achievements in the field of geometry, not a single title of a relevant work has survived. His mathematical discoveries are therefore only known from the writings of other authors. Nothing has come down to us about philosophical works; possibly he did not write any, but only presented his views orally.

Eudoxus wrote several astronomical writings that are only known from mentions or reproductions of their content in later literature:

• “Phenomena” (Phainómena) , his first astronomical work. He called a revised version "mirror" ( Énoptron ; meaning: mirror of the world order). The script consisted of three books. The first contained a description of the relative positions of the stars, the second dealt with their positions in relation to the celestial sphere and its divisions, and the third contained a catalog of the stars with information about their respective rising and setting. The famous poem “Phenomena and Signs” by Aratos von Soloi has been preserved , the first part of which contains a free translation of the “Phenomena” of Eudoxus into verses.
• "About speeds" (Peri tachōn) , his main astronomical script , in which he explained the movements of the planets. Under “planets” (“ changing stars ”) one understood the moon, sun, Venus, Mercury, Mars, Jupiter and Saturn; the planets beyond Saturn were not yet known. Eudoxos proceeded from his geocentric worldview , which is based on the assumption of an immovable earth around which the spherical shells ( spheres ) rotate, which are assigned to the movable celestial bodies and the fixed stars. The speeds referred to in the title mean the different speeds of rotation of the spheres. Aristotle describes the system of Eudoxus succinctly in his metaphysics .
• "About the extinction caused by the sun" (Peri aphanismōn hēliakōn) . Here Eudoxos set out his method by which the time of the rise and set of a star is determined when sunlight makes precise observation impossible.
• "The eight-year cycle" (Oktaetērís) , in which Eudoxus explained an astronomical calendar based on an eight-year cycle. It appears to be the oldest work with this title on this subject, which was often discussed later, but it is unclear whether the traditional title comes from Eudoxus. His original work can only be partially reconstructed; Eratosthenes rightly considered the version available to him, which received much attention in antiquity, to be inauthentic.
• Astronomía , an astronomical calendar in verse that contained legends about the stars. This didactic poem was a late work.

The treatise Eudoxi ars astronomica , previously attributed to him and preserved on papyrus , does not come from Eudoxus . However, this treatise contains, among other things, material on astronomy and calendar calculation that goes back to him.

In the geographical area, Eudoxus wrote a literary “description of the earth” (Gēs períodos) , which is one of his later works. In it he also dealt with cultural-geographic topics.

What the writings attributed to Eudoxus were about, which Diogenes Laertios referred to as "dog dialogues" (Kynōn diálogoi) , is unknown. Diogenes invokes Eratosthenes and adds that, according to other authors, these dialogues were written by Egyptians in their language and Eudoxus only translated them into Greek. The latter would require appropriate knowledge of the Eudoxus language and has therefore been questioned.

philosophy

A report by Aristotle is available on two statements by Eudoxus on philosophical questions. One concerns the theory of ideas , the other the doctrine of the good . In both questions Eudoxus takes a view that fundamentally contradicts that of Plato.

Theory of ideas

Presumably Eudoxos did not develop his own doctrine of ideas, but only discussed an individual question of the doctrine of ideas on the occasion of a given occasion, whereby he was possibly only concerned with a hypothetical consideration. The problem was the question of how the participation of the individual things in the ideas comes about. Eudoxus thought that he could solve it with a doctrine of mixing; the ideas are mixed with the perceptible objects. Aristotle compares this to the admixture of a color to what she has colored. It is uncertain whether this comparison goes back to Eudoxus. How Eudoxos envisioned the mixture is unclear; Apparently he started from a natural-philosophical mixture concept common to the pre-Socratics and, in contrast to Plato, assumed a local presence of ideas in things. In contrast to Aristotle, he also wanted to adhere to Plato's doctrine of an existence of ideas separated from things. This earned him the charge of inconsistency. The traditional counter-argument is that he materializes ideas and incorporates them into the transience of the material world and that they lose their simplicity and immutability in the process. As a result, they would lose their specific ontological status and would no longer be ideas in the sense of the Platonic doctrine of ideas.

ethics

In ethics , Eudoxos took a hedonistic position in determining the good . He equated good with joy. He argued that joy is sought by all beings - rational and unreasonable; it is therefore the general good, the good par excellence. In contrast to other goods, it is not praised because it is above all commendable goods. Since it is striven for for its sake and not for the sake of another good, it has a higher rank than those goods which appear desirable for a specific purpose. Every good to which it is added becomes more desirable; therefore it is the true good, namely that which does not gain through something else, but only through itself. It is disputed whether Eudoxus also held the conviction that the pleasure principle belongs to the essence of the deity and therefore the desire for pleasure, which is in all living beings, aims be one and the same in the cosmos, ultimately on the deity in whom the highest pleasure is realized.

mathematics

Eudoxus founded the general theory of proportions. He was able to include irrational sizes for the first time , as his theory of proportions can also be applied to incommensurable sizes. His definitions of ratio ( lógos ) and proportion ( analogía ) are handed down in the fifth book of Euclid's elements .

Research has suggested that the Archimedean axiom named after Archimedes actually came from Eudoxus. Eudoxus apparently knew the initial problem, but to what extent he dealt with it is unclear.

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

Eudoxos dealt with the problem of doubling the cube, which was intensely discussed in antiquity ("Delic problem"). He found an unspecified solution for this by cutting curves; their intersections yielded the two mean proportions to the edge of the given and that of the sought cube, which are necessary to solve the problem. Furthermore, as Plutarch reports, Eudoxos also invented a mechanical device for the approximate construction of two middle proportions.

Astronomy and geography

Model with two spheres (here rings) for the representation of the movements of the sun relative to the earth

In astronomy, Eudoxos was concerned with mathematically representing the movements of the planets with standstills and reversals, which appear to be irregular within the framework of his geocentric view of the world. Allegedly, Plato had set the research task of reducing the planetary revolutions to regular circular movements. For each planet as well as for the sun and moon, Eudoxus assumed a separate system of concentric spherical shells (spheres) that rotate uniformly around mutually inclined axes at different speeds and in different directions. Because of the common center - the earth - one speaks of the theory of homocentric spheres . Each celestial body is fixed at the equator of the innermost shell assigned to it. The shells belonging to a celestial body are attached to each other at their poles. This transfers their movements to the innermost shell and thus to the star. Eudoxus assigned four spheres each to the five planets known to him, and three each to the sun and moon. The first (outermost) planetary sphere in each case causes the daily orbit of the planet from east to west, the second its annual orbit in the zodiac from west to east, the third and fourth together produce the (apparent) eight-shaped loop movement , the Eudoxos "hippopede" (horse fetter) called. One sphere was sufficient for the fixed stars. This gave Eudoxus a total of 27 spheres. The question of a connection between the individual spherical systems did not arise for him, because his conception was a purely mathematical, not physically founded hypothesis that resolved complex relationships into simple elements (regular circular movements).

In research it is partly suspected, partly denied that Eudoxos was already trying to fulfill the later widespread demand of “ saving the phenomena ” and was even the originator of it in the sense of the formulation of a research principle. The exact meaning of this expression is controversial. According to one interpretation, the original aim was to show the apparent movements of the heavenly bodies as a result of their true movements. It was only later that the principle of “saving the phenomena” was understood to mean that a theory is only required to require that the calculations carried out with it correspond to every observation result, and not that it accurately reproduces physical reality. According to another interpretation, “saving” the phenomena meant subsequently modifying or supplementing a mathematical description that is supposed to do justice to given physical or natural philosophical assumptions so that it “saves the phenomena”, i. H. certain phenomena that appear as anomalies within their framework are also taken into account.

The system of Eudoxos only enabled approximate solutions and did not take into account all the anomalies of the movements of the celestial bodies known at the time. Therefore it was later enlarged by Callippus of Kyzikos , who added further spheres.

An important weakness of the system was that it could not explain the fluctuations in brightness of the planets, which suggest changes in their distance from Earth. The Aristotle commentator Simplikios pointed this out in late antiquity .

Eudoxus also determined the distances and proportions of the earth, moon and sun. How he proceeded is unknown. The order of the moveable celestial bodies was the same for him at the time: from inside to outside moon, sun, Venus, Mercury, Mars, Jupiter, Saturn.

He is also said to have determined the circumference of the earth. Aristotle, referring to "mathematicians", whom he does not name, indicates a circumference of 400,000 stadia . This far too high number is probably an estimate; the first recorded calculation is that of Eratosthenes from the 3rd century BC. The estimate given by Aristotle is often ascribed to Eudoxus, but there is no evidence for it.

Cicero reports that Eudoxus was the first to write the constellations on a celestial globe after Thales had made such a sphere. Accordingly, Eudoxus was not the inventor of the celestial globe, but the first to map the stars.

According to a catalog of inventions handed down by Vitruvius , Eudoxus was considered the inventor of an aráchnē ("spider"). In later times this term was used to describe the movable disk of a flat astrolabe (a two-dimensional representation of the celestial sphere). This rotating disk depicted the fixed star sky and could reproduce its daily rotation. Such an astrolabe did not yet exist at the time of Eudoxus. His arachne was probably a preliminary stage, a transparent disk that formed the main component of a star clock. It carried a picture of the fixed star sky that resembled a cobweb.

As a geographer, Eudoxos was particularly concerned with the division of the earth into climatic zones . He drew a map of the Oikumene (the known, populated part of the earth's surface) in the temperate zone of the northern hemisphere . For him, the Oikumene had the shape of a rectangle that was twice as long as it was wide. The border was formed by the Iberian Peninsula in the west, India in the east, Ethiopia in the south and the Scythian region in the north . In the south, the temperate zone was followed by the “burned zone”, the northern half of which extended to the equator. In the southern hemisphere, the climatic conditions were mirror images. There he assumed in the southern temperate climatic zone, in which the Nile rises, one of the Oikumene analogous, also inhabited Gegenoikumene. He determined the geographical latitude ( énklima ) of a region according to the ratio of the longest day to the shortest or shortest night; whether he has already calculated it in latitudes is uncertain. For his place he came to a ratio of 5: 3; Another result he determined was 12: 7. It is unclear whether the two statements refer to different locations or whether one is to be understood as a correction of the other. In his measurements he used a gnomon , a vertical stick whose shadow is observed in the horizontal plane. The ratio we were looking for resulted from the ratio of the shortest and longest midday shadow observed at the two solstices .

reception

Antiquity

To what extent Plato's Dialogue Philebos was intended as an anti-Hedonist answer to Eudoxus' argument for equating the good with joy or pleasure is unclear; Opinions differ in research.

In his Nicomachean Ethics, Aristotle was critical of Eudoxus' theory of pleasure. In doing so, he encountered the difficulty that the train of thought of Eudoxus was partly very similar to arguments that he himself put forward elsewhere, but the resulting equation of the good with the pleasure was not acceptable to him, since he was the striving of the "many" Enjoyment was considered vulgar and animal. So he made do with an argument ad hominem : Eudoxus had not found favor with his arguments because of their persuasiveness, but because of his own character, which was so good that he was not suspected of defending a vulgar hedonism. In principle, however, Aristotle shared Eudoxus' conviction that the normative basic term "good" should be defined in action theory as "the goal towards which everything strives". Possibly he took over the equation of the (real or supposed) good with the striving, which he placed at the beginning of the Nicomachean Ethics , from Eudoxus. In his lost work "On the Ideas", Aristotle also put forward arguments against Eudoxus' theory of ideas as part of his criticism of the Platonic doctrine of ideas.

Euclid incorporated a number of Eudoxus findings into his elements . This concerned the theory of proportions and their geometric applications as well as volume determinations.

The planetary system of Eudoxus remained decisive in the extended version of Callippos in Greek astronomy until it was used from the end of the 3rd century BC. By the epicyclic theory . In particular, it shaped the astronomical ideas of Aristotle. Aristotle tried to give the mathematical model a physical foundation by conceiving the spheres as real material realities, increasing their number and assuming that they are all interconnected. So Eudoxus indirectly influenced the physical world view of the Middle Ages through the Aristotle reception .

The "phenomena" received attention from many ancient authors. They were used, among others, by Philip of Opus , Aratos of Soloi and Geminos of Rhodes ; Vitruvius had only indirect access via an intermediate source. Aratos wrote a constitution. In the 2nd century BC The astronomer Hipparchus of Nikaia wrote a critical commentary on the "phenomena" of Eudoxus and the constitution of Aratos.

Strabo praises the geographic competence of Eudoxus. He reports that the historian Polybius appreciated Eudoxus' account of historical geography; this included the founding stories of Greek cities as well as information about the origins of their populations and migration movements. As Strabo 25/24 BC BC or soon afterwards traveled through Egypt, he was shown rooms in Heliopolis in which Eudoxus had allegedly stayed. This legend was part of an Egyptian propaganda that traced Greek wisdom and science back to Egyptian roots. Seneca said that Eudoxus brought knowledge of planetary movements from Egypt to Greece.

Philostratus regards Eudoxus as an oratorically gifted philosopher and forerunner of the "second sophistry ". Diogenes Laertios rates his works as extremely valuable.

Modern

Eudoxos has received an extremely high level of attention in modern research. Its system of spheres in particular has been the subject of intense discussion by astronomy historians since the 19th century. From 1828–1830 the astronomer Christian Ludwig Ideler submitted an attempt at reconstruction, and in 1849 an investigation by Ernst Friedrich Apelt brought further insights. The reconstruction of the astronomer Giovanni Schiaparelli published in 1877 proved to be groundbreaking. It dominated until the late 20th century and is still considered "classic" today, but has weaknesses that have led to the emergence of alternative hypotheses.

In 1935 the lunar crater Eudoxus , in 1973 the Mars crater Eudoxus and in 1998 the asteroid (11709) Eudoxos was named after Eudoxos.

Text output

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

literature

• David R. Dicks: Early Greek Astronomy to Aristotle . Thames and Hudson, London 1970, ISBN 0-500-40013-X , pp. 151-189.
• Friedrich Heglmeier: The homocentric spheres of Eudoxus and Kallippos and the error of Aristotle . Erlangen 1988 (dissertation).
• Hans Krämer : Eudoxos from Knidos . In: Hellmut Flashar (ed.): Older Academy - Aristoteles - Peripatos (= outline of the history of philosophy . The philosophy of antiquity. Vol. 3). 2nd, reviewed and expanded edition, Schwabe, Basel 2004, ISBN 3-7965-1998-9 , pp. 56–66, 146–149.
• Jean-Pierre Schneider: Eudoxe de Cnide . In: Richard Goulet (ed.): Dictionnaire des philosophes antiques . Volume 3, CNRS Éditions, Paris 2000, ISBN 2-271-05748-5 , pp. 293-302.
• Hans-Joachim Waschkies : From Eudoxus to Aristotle. The continuing effect of the Eudoxian theory of proportions in the Aristotelian theory of the continuum . Grüner, Amsterdam 1977, ISBN 90-6032-086-7 .

Web links

• Literature by and about Eudoxus von Knidos in the catalog of the German National Library
• John J. O'Connor, Edmund F. Robertson :  Eudoxus of Cnidus. In: MacTutor History of Mathematics archive . [partially outdated state of research]
• Don Allen: Eudoxus of Cnidus (1997) [biographical information only partially correct]

Remarks

1. ↑ See on this François Lasserre (Ed.): Die Fragmente des Eudoxos von Knidos , Berlin 1966, pp. 137-139, 254 f .; Hans-Joachim Waschkies: From Eudoxus to Aristoteles , Amsterdam 1977, pp. 34–40; Friedrich Heglmeier: The homocentric spheres of Eudoxus and Kallippos and the error of Aristotle , Erlangen 1988, pp. 8-15.
2. ↑ Diogenes Laertios 8,8,86 and 8,8,89.
3. ↑ François Lasserre (Ed.): The fragments of Eudoxus von Knidos , Berlin 1966, p. 146; Hans Krämer is less skeptical: Eudoxos from Knidos . In: Hellmut Flashar (ed.): Older Academy - Aristoteles - Peripatos (= Outline of the History of Philosophy. The Philosophy of Antiquity. Vol. 3), 2nd edition, Basel 2004, pp. 56–66, here: 57. Philip Merlan : Studies in Epicurus and Aristotle , Wiesbaden 1960, p. 100 considers the trip to be historical.
4. ↑ This Chrysippus was possibly a relative of the famous doctor Chrysippus of Knidos , with whom he should not be confused. On the dating François Lasserre (ed.): The fragments of Eudoxus by Knidos , Berlin 1966, p. 139 f.
5. ↑ Diogenes Laertios 8,8,87.
6. ↑ Hans Krämer: Eudoxos from Knidos . In: Hellmut Flashar (Ed.): Older Academy - Aristoteles - Peripatos (= Outline of the History of Philosophy. The Philosophy of Antiquity. Vol. 3), 2nd edition, Basel 2004, pp. 56–66, here: 57; Jean-Pierre Schneider: Eudoxe de Cnide . In: Richard Goulet (ed.): Dictionnaire des philosophes antiques , Vol. 3, Paris 2000, pp. 293-302, here: 297.
7. ↑ This is how Kurt von Fritz thinks : The theory of ideas of Eudoxus von Knidos and its relationship to the Platonic theory of ideas . In: Kurt von Fritz: Writings on Greek Logic , Vol. 1, Stuttgart 1978, pp. 147–169, here: 167 and Philip Merlan: Studies in Epicurus and Aristotle , Wiesbaden 1960, p. 99.
8. ↑ Hans-Joachim Waschkies: From Eudoxos to Aristoteles , Amsterdam 1977, pp. 41–52; Hermann Schmitz : Aristotle's theory of ideas , vol. 2: Platon and Aristoteles , Bonn 1985, p. 159 f.
9. ↑ See on this work John Gardiner-Garden: Eudoxos, Skylax and the Syrmatai . In: Eranos 86, 1988, pp. 31-42, here: 31 f. and the detailed attempt at reconstruction by Friedrich Gisinger : The description of the earth by Eudoxus von Knidos , Berlin 1921, p. 15 ff.
10. ^ François Lasserre (ed.): The fragments of Eudoxus von Knidos , Berlin 1966, p. 269; Joachim Quack disagrees : The role of hieroglyphs in the theory of the Greek vowel alphabet . In: Wolfgang Ernst , Friedrich Kittler (ed.): The birth of the vowel alphabet from the spirit of poetry , Munich 2006, pp. 75–98, here: 88 f. and John Gwyn Griffiths: A Translation from the Egyptian by Eudoxus . In: The Classical Quarterly NS 15, 1965, pp. 75-78.
11. ↑ For Eudoxos' concept of ideas see Kurt von Fritz: Die Ideenlehre des Eudoxos von Knidos and their relationship to the Platonic doctrine of ideas . In: Kurt von Fritz: Writings on Greek Logic , Vol. 1, Stuttgart 1978, pp. 147–169; François Lasserre (ed.): The fragments of Eudoxus by Knidos , Berlin 1966, pp. 149–151; Hans Krämer: Eudoxos from Knidos . In: Hellmut Flashar (Ed.): Older Academy - Aristoteles - Peripatos (= Outline of the History of Philosophy. The Philosophy of Antiquity. Vol. 3), 2nd edition, Basel 2004, pp. 56–66, here: 59– 61; Russell M. Dancy: Two Studies in the Early Academy , Albany 1991, pp. 23-56; Hermann Schmitz: The theory of ideas of Aristoteles , Vol. 2: Platon and Aristoteles , Bonn 1985, pp. 157-161.
12. ↑ Hans Krämer: Eudoxos from Knidos . In: Hellmut Flashar (Ed.): Older Academy - Aristoteles - Peripatos (= Outline of the History of Philosophy. The Philosophy of Antiquity. Vol. 3), 2nd edition, Basel 2004, pp. 56–66, here: 64– 66; François Lasserre (ed.): The fragments of Eudoxus by Knidos , Berlin 1966, pp. 13 f., 151–156; Heinrich Karpp: Studies on the philosophy of Eudoxus von Knidos , Würzburg 1933, pp. 6-27.
13. ↑ This hypothesis is represented by Wolfgang Schadewaldt : Hellas and Hesperien , 2nd edition, vol. 1, Zurich and Stuttgart 1970, pp. 644–655. For other views see François Lasserre (Ed.): Die Fragmente des Eudoxos von Knidos , Berlin 1966, p. 156 f.
14. ↑ Hans-Joachim Waschkies: From Eudoxos to Aristoteles , Amsterdam 1977, pp. 308-318.
15. ↑ François Lasserre (ed.): The fragments of Eudoxus von Knidos , Berlin 1966, pp. 20-22, 163-166.
16. ↑ Hans Krämer: Eudoxos from Knidos . In: Hellmut Flashar (ed.): Older Academy - Aristoteles - Peripatos (= Outline of the history of philosophy. The philosophy of antiquity. Vol. 3), 2nd edition, Basel 2004, pp. 56–66, here: 63 holds this controversial news for believable; also Hermann Schmitz: Die Ideenlehre des Aristoteles , vol. 2: Platon and Aristoteles , Bonn 1985, p. 166. Jürgen Mittelstraß is one of the representatives of the opposite view : The rescue of phenomena , Berlin 1962, pp. 1–4, 150– 155.
17. ↑ For the model of the spheres see Friedrich Heglmeier: The Greek Astronomy at the Time of Aristotle. A new approach to the spherical models of Eudoxus and Callippus . In: Antike Naturwissenschaft and their reception , Vol. 6, 1996, pp. 51–71, here: 53–61; Erkka Maula: Studies in Eudoxus' Homocentric Spheres , Helsinki 1974, pp. 14 ff. On the hippopede see John D. North: The Hippopede . In: Anton von Gotstedter (Ed.): Ad radices , Stuttgart 1994, pp. 143–154.
18. ↑ This is the traditional and still prevailing doctrine in research. Larry Wright takes an opposing position: The Astronomy of Eudoxus: Geometry or Physics? In: Studies in History and Philosophy of Science , Vol. 4, 1973/74, pp. 165-172. He thinks that Eudoxos sought a description and explanation of the actual physical conditions.
19. ↑ Hans Krämer: Eudoxos from Knidos . In: Hellmut Flashar (Ed.): Older Academy - Aristoteles - Peripatos (= Outline of the History of Philosophy. The Philosophy of Antiquity. Vol. 3), 2nd edition, Basel 2004, pp. 56–66, here: 62 f .; Leonid Zhmud : "Saving the phenomena" between Eudoxus and Eudemus. In: Gereon Wolters , Martin Carrier (eds.): Homo Sapiens and Homo Faber , Berlin 2005, pp. 17–24; Martin Carrier: Saving the Phenomena. On the changes in an ancient research principle . In: Gereon Wolters, Martin Carrier (eds.): Homo Sapiens and Homo Faber , Berlin 2005, pp. 25–38.
20. ^ Fritz Krafft : The Mathematikos and the Physikos . In: Old Problems - New Approaches , Würzburg 1964, pp. 5–24. Krafft thinks that the task of "saving" phenomena (anomalies) has not yet arisen for Eudoxos.
21. ↑ See also Walter Burkert : Weisheit und Wissenschaft , Nürnberg 1962, p. 302 and note 3; Friedrich Heglmeier: The homocentric spheres of Eudoxus and Kallippos and the error of Aristotle , Erlangen 1988, p. 242.
22. ↑ François Lasserre (ed.): The fragments of Eudoxus from Knidos , Berlin 1966, pp. 17, 211.
23. ↑ Klaus Geus : Eratosthenes of Cyrene. Studies on the Hellenistic History of Culture and Science , Munich 2002, p. 226, note 86.
24. ↑ Cicero, De re publica 1,14,22.
25. ↑ François Lasserre (ed.): The fragments of Eudoxus by Knidos , Berlin 1966, pp. 158-160. For other interpretations of the arachne, see Bernard R. Goldstein, Alan C. Bowen: A New View of Early Greek Astronomy . In: Isis Vol. 74, 1983, pp. 330-340, here: 335-337; Erkka Maula: The Spider in the Sphere. Eudoxus' arachne . In: Philosophia (Athens) Vol. 5/6, 1975/76, pp. 225-258.
26. ^ François Lasserre (ed.): The fragments of Eudoxus von Knidos , Berlin 1966, p. 239 f .; Friedrich Gisinger: The description of the earth by Eudoxus von Knidos , Berlin 1921, p. 13 f.
27. ↑ Stephan Heilen : Eudoxos von Knidos and Pytheas von Massalia , in: Wolfgang Hübner (Ed.): History of Mathematics and Natural Sciences in Antiquity , Vol. 2: Geography and related Sciences , Stuttgart 2000, pp. 55–73, here : 58-60.
28. ↑ Árpád Szabó : Eudoxus and the problem of the tendon tables . In: Jürgen Wiesner (Ed.): Aristoteles. Work and Effect , Vol. 1, Berlin 1985, pp. 499-517, here: 502-510. Alan C. Bowen and Bernard R. Goldstein take a contrary position to this prevailing doctrine: Hipparchus' Treatment of Early Greek Astronomy. The Case of Eudoxus and the Length of Daytime . In: Proceedings of the American Philosophical Society Vol. 135, 1991, pp. 233-254. They doubt that the figures quoted are based on measurements or observations and consider it possible that the two ratios were arithmetically derived from theoretical considerations.
29. ↑ Hans Krämer: Eudoxos from Knidos . In: Hellmut Flashar (ed.): Older Academy - Aristoteles - Peripatos (= Outline of the History of Philosophy. The Philosophy of Antiquity. Vol. 3), 2nd edition, Basel 2004, pp. 56–66, here: 66; Justin CB Gosling, Christopher CW Taylor: The Greeks on Pleasure , Oxford 1982, pp. 157-164; Hermann Schmitz: The theory of ideas of Aristoteles , vol. 2: Platon and Aristoteles , Bonn 1985, pp. 161–166; Dorothea Frede is skeptical : Plato: Philebos. Translation and Commentary , Göttingen 1997, pp. 390–394.
30. ↑ Aristotle, Nicomachean Ethics 1172b; see. 1095b. See Roslyn Weiss: Aristotle's Criticism of Eudoxan Hedonism . In: Classical Philology Vol. 74, 1979, pp. 214-221.
31. ↑ Aristotle, Nicomachean Ethics 1094a; see Franz Dirlmeier : Aristoteles: Nikomachische Ethik , 8th edition, Berlin 1983, p. 266 f .; Otfried Höffe : Aristoteles , 2nd edition, Munich 1999, p. 202 f.
32. ↑ See Friedrich Heglmeier: The Greek Astronomy at the Time of Aristotle. A new approach to the spherical models of Eudoxus and Callippus . In: Ancient natural science and their reception , Vol. 6, 1996, pp. 51–71, here: 69 f.
33. ↑ On Strabon's Eudoxos reception see Johannes Engels : The Strabonian cultural geography in the tradition of ancient geographical writings and their significance for ancient cartography . In: Orbis Terrarum Vol. 4, 1998, pp. 63-114, here: 73-76.
34. ↑ Seneca, Naturales quaestiones 7,3,2.
35. ↑ On the history of science see Friedrich Heglmeier: The homocentric spheres of Eudoxus and Kallippos and the error of Aristotle , Erlangen 1988, pp. 69–89. For alternative solutions see Ido Yavetz: On the Homocentric Spheres of Eudoxus . In: Archives for History of Exact Sciences 52, 1998, pp. 221-278; Ido Yavetz: A New Role for the Hippopede of Eudoxus . In: Archive for History of Exact Sciences 56, 2002, pp. 69-93.
36. ↑ lunar photo of the day March 26, 2010; 11709 Eudoxos (1998 HF20) JPL Small-Body Database Browser.