Phainomena (Euclid)

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Phainomena (Greek Φαινόμενα ) is a work of theoretical and computational astronomy , which the mathematician Euclid of Alexandria around 300 BC. Has written. The main topic is the calculation of the duration of daylight on a given date at a given degree of latitude by means of spherical geometry , whereby the observation of the rising and setting of stars and celestial circles (e.g. ecliptic ) forms the basis.

Structure and authenticity

The work consists mainly of 18 propositions about the rising and setting of stars and celestial circles in a spherically imagined cosmos. It is not certain that Euclid is the author, but it is widely assumed, especially because ancient authors, such as the Greek doctor Galenus, pass this on. There are also writings on this subject from other Greek authors, in particular from Autolycus of Pitane (around the same time as Euclid) and Theodosius of Bithynia (1st century BC).

The main part is preceded by a brief introduction. It is a mixture of descriptive definitions, astronomical concepts, and introductory assumptions that are believed not to come from Euclid but represent a later Scholion to aid readers and students.

content

introduction

The following astronomical definitions and claims (some with evidence) are made:

  • The rising and setting of the fixed stars are observed at the same point, with a fixed distance between them.
  • Therefore they are on circular paths with the observer in the center. These circular orbits are parallel and have an unmoving star as a pole in the constellation Great Bear (Greek bear = arktoi). However, such a visible star did not exist in Euclid's time
  • The stars between the pole and the arctic circle are always visible, stars south of it have a rising and setting, so they are partly above and partly below the earth. The circle on which the stars are equally long above and below the earth is called the equator . The Milky Way and the ecliptic ( zodiac , zodiac ), which are at an acute angle to the fixed star circles, are wider circles .
  • From the foregoing it is concluded that the cosmos can be assumed to be spherical.
  • Further circles on the cosmos sphere (= sphere ) are the horizon , the meridian (through the poles and perpendicular to the horizon), and the tropics . The horizon, meridian, ecliptic and equator are great circles (circles on the sphere whose center is identical to that of the sphere).

The writing does not develop a new worldview, but remains in the ideas that were represented at the time of Euclid by some of the ancient scientists, e.g. the almost simultaneous Autolycus of Pitane and the somewhat earlier Eudoxos of Knidos .

The propositions

(The text below follows the paraphrases by John Lennart Berggren)

The following propositions are particularly interesting:

  • Proposition 1: The earth is the center of the cosmos.
  • Proposition 5: Of the stars that rise and set, the more northerly stars rise earlier and set later.
  • Proposition 9: Between the equator and the arctic circle the signs of the zodiac have different times of rise, those following the Cancer the longest, those following the Capricorn the shortest.

These rising and setting times of different lengths - just like the different times of daylight - are caused by the inclined position of the earth's axis to the plane of the earth's orbit around the sun. The proposition is thus a theorem for determining the duration of daylight; Euclid does not address this connection in the surviving part of the scriptures.

  • Proposition 14: Of 2 equal arcs of the ecliptic, the one that is closer to the tropic of the sun in summer remains visible longer.

The proof clearly shows Euclid's geometrical approach. He regards the ecliptic as a great circle on the sphere that intersects the great circles equator and horizon, but also the tropics at an acute angle. About the length of the distances between the intersection points he postulates sentences that are close to spherical geometry. All in all, the explanations do not come close to the level of Greek mathematics at the same time, nor do the elements of Euclid; the "evidence" is often nothing more than a slightly modified formulation of the assumption.

Babylonian astronomy already calculated the duration of daylight by adding up the rising times of the signs of the zodiac. Euclid took a geometrical approach. But only the trigonometric consideration by Hipparchus and especially Ptolemy ( Almagest , II, 8,9) brought a more convincing solution.

Tradition and survival

For a long time Euclid's work was part of the teaching texts for students of astronomy and mathematics - not for beginners, but for "higher semesters" - there are comments by the Greek mathematician and astronomer Pappos from the 4th century . However, the text has largely been replaced by more advanced scripts, e.g. B. through the Sphaerica of Theodosius of Bithynia.

David Gregory edited and translated the text in 1703. Heinrich Quantity created an edition and a translation into Latin in 1916, John Lennart Berggren in 1996 a richly commented translation into English.

Text editions and translations

  • Heinrich Quantity: Phaenomena in EUCLIDIS Phaenomena et Scripta Musica , Leipzig 1916.
  • John Lennart Berggren, RSD Thomas: Euclids Phenomena - a translation and study of a hellenistic work in spherical astronomy , Garland Publishing 1996

literature

  • John Lennart Berggren, RSD Thomas: Euclids Phenomena - a translation and study of a hellenistic work in spherical astronomy , Preface, Garland Publishing 1996
  • Otto Neugebauer : A History of Ancient Mathematical Astronomy , New York-Heidelberg-Berlin 1975

Individual evidence

  1. ^ Johan Ludvig Heiberg : History of Mathematics and Natural Sciences in Antiquity , Munich 1925, p. 35
  2. ↑ John Lennart Berggren: Euclids Phenomena , Preface, p. 1
  3. ↑ John Lennart Berggren: Euclids Phenomena , Preface, p. 8f
  4. ^ Otto Neugebauer: A History of Ancient Mathematical Astronomy , p. 748ff
  5. ^ Otto Neugebauer: A History of Ancient Mathematical Astronomy , p. 756
  6. ↑ John Lennart Berggren: Euclids Phenomena , p. 48
  7. ^ Otto Neugebauer: A History of Ancient Mathematical Astronomy , p. 748f, 756
  8. ^ Otto Neugebauer: A History of Ancient Mathematical Astronomy , p. 764
  9. ^ Otto Neugebauer: A History of Ancient Mathematical Astronomy , p. 764
  10. ^ Otto Neugebauer: A History of Ancient Mathematical Astronomy , p. 755, 749
  11. ↑ John Lennart Berggren: Euclids Phenomena , Preface, p. 2
  12. ↑ BL Van der Waerden: Awakening of Science , Basel / Stuttgart 1966, p. 321
  13. ↑ John Lennart Berggren: Euclids Phenomena , Preface, p. 15th
  14. ↑ Peter Schreiber : Euklid , Leipzig 1987, p. 71