The influence of pythagorean philosophy on the development of mathematical astronomy.

12. Maria THE INFLUENCE OF PYTHAGORE AN PHILOSOPHY ON THE DEVELOPMENT OF MATHEMATICAL AST RON OMY 117_{ 25° sas € PRE SATHAS ASS mu MA Pythagorean philosophy / ed. by Konstantin os |. Boudouris. Athens : International Center for Greek Philosophy and Culture, 1992. 257 p. ill. index). (Studies in Greek philosophy ; 7). - papers read at the third international conference on Greek philosophy, Samos, August 1991.

PETER LAU TNER 16.1b. 189.15ff., cf. 134.36-135.7. point of zed, from a different hS 17. The role of xoıvn alo«So0nomeisPlathasonisbeet nReascrudintini gs of Aristotle», in PCP ns. view, by H. J. Blumenthal, 27(1981), 1-16. MARIA K. PAPATHANASSIOU 18.2v600ev 119.6ff., oixo@ev 119.10. 19. Iè.188.35. by Priscianus to the we perceive is assignedwayalso 20. The ability to perceiveis that r. 22.1-23. but in a rather differenoft CAG, cf.XIMetSimpaphlici influence of sensus commun us. This hor venience I called theier aut For the sake of conn cont Steel, «Priscianus Lydus761en-82de2, d by F. Boss and C. voo attribution has bee udo este 34 (1972), Simplicius», in Tijdthissch.workrtoFiloPriss.cian In De anima van Psey on (2)us. However, the 2, who assign with French summar r821«noe-82tic» Changing n noticed also by C. Steel,andThePris discrepancy between theiLater Neoplahastonbee cianus. : Iamblichus, Damascius cianus foll Self. A Study on Soul in n resolved byismsayi ows r. Pris aph Met in that ng bee has and , 1978 sel Brus in DA he confronts his own views with those of Jamblichus Tamblichus while in the (p. 149-155). DR PETER LAUTNER UNIVERSITY OF BUDAPEST THE INFLUENCE OF PYTHAGOREAN PHILOSOPHY ON THE DEVELOPMENT OF MATHEMATICAL ASTRONOMY When referring to the ancient mathematical astronomy we mean the notions of spherical astronomy as well as various mathematical theories invented by Greek astronomers for the description of the motion of the planets. For this reason I do not intend to deal here with ideas concerning the physical state of the celestial bodies. It is true that the astronomical knowledge of the Pythagoreans and their general contribution to the development of astronomy are considered as a constantly controversial subject of the history of this science because of the lack of authentic early evidence'. But we cannot dispute the importance attributed by the Pythagoreans to the àdelpai émotuar of astronomy and harmonics, as follows from Socrate’s words in Plato’s Republic”. For this reason a new look at the cosmological system of Philolaus of Kroton and the theories of Archytas of Taras in relation to the evolution of the astronomical knowledge of Greeks may possibly help to the deduction of some basic conclusions regarding the cosmological model of the late Pythagoreans. In the Philolaic system, in the center of the universe there is a fire called «Hearth of the world»; the divine bodies dance around it, the following according to their order from the center: the first is the counter-earth, the second is the earth which always moves and revolves 2& évavtiag to the counter-earth, the third is the moon, the fourth is the sun, then come the five planets, and the last one is the sphere of the fixed stars’. The general characteristics of the cosmological system of the Pythagoreans can be summarized as follows:

1) The spherical universe. 2) The circular motions of the celestial bodies around a common center. 3) The idea that the common center is a fire not identified with the sun. 4) The distinction between planets and other stars. 5) The circular shape of the earth. 6) The displacement of the earth from the center of the universe and its motion around it.‘ 7) The hypothesis of the counter-earth. 8) The existence of other celestial bodies invisible to us. 9) The divinity of the celestial bodies. 10) The harmony of the spheres. The notions of the sphericity of the universe and the periodic circular motions of the celestial bodies already exist in Heraclitus, Parmenides, Empedocles, and Anaxagoras, who differ from the Pythagoreans by their assumption that the earth is motionless in the middle of the world.’ For this reason it is generally accepted that the genuinity of the Philolaic system lies mainly in the displacement of the earth from the middle of the universe and in the hypothesis of the counter-earth, which possibly aimed at raising the total number of the spheres of the celestial bodies to ten the sacred number of the Pythagoreans. From the remaining fragments of Philolaus’s work «On Nature» we can understand the importance of the role of number in the universe: nature and all in the world are composed from «unlimited and limited» elements.’ All things that can be known contain number; without this, nothing could be thought or known;® moreover all things are related to one another by the ratios of their numbers.’ Special importance is attributed to the power of the decade in the divine, the celestial, and the human life; without it all would be «infinite and invisible and unseen».!° These ideas applied to the case of astronomy lead to more substantial results apart from that of the hypothesis of the counterearth as the tenth body of the system of the world: they introduce quantitative relations between the members of the system, which are not only static, as order and relative distances of its members between them, but also dynamic, as ratios of their periods of revolution. In this point we can distinguish between mythical or simple cosmological ideas and mathematical cosmological models. Because all efforts of later Greek astronomers pointed to the PYTHAGOREAN PHILOSOPHY & MATHEMATICAL ASTRONOMY 119 principle of «saving the phenomena», i.e. to the introduction of a new or to the improvement of an already existing mathematical model describing the structure of the universe, the motion of the sphere of the fixed stars, and the motions of the planetary spheres, so that the arithmetical data from observations agree with those calculated on the basis of the theories. There is a problem: what do we know about the application of these mathematical relations in the case of Pythagorean astronomy? In the remaining fragments of Archytas’s work «Harmonic», there is a passage referring to the mathematicians who know well the dde/pea uadnuara of geometry, arithmetic, spherics, and harmonics. The reason why these sciences are so called is evident, as they treat numbers or magnitudes and their relations to one another, i.e. their ratios and proportions. But I would like to comment further on the same passage referring to «the mathematicians who gave us exact means of distinguishing the speed of stars and their heliacal risings and settings.»!! It is noteworthy that the passage mentions only one speed of the stars in contrast to the plural used for their heliacal risings and settings. The terms &rrıroAn (=heliacal rising) and dvoig (=setting) for the stars are used in astronomy to denote the various phenomena of the first appearance (or disappearance) of a fixed star a little before sunrise (or after sunset), according to the season, while the corresponding phenomena for the planets are their conjunctions or oppositions to the sun. Consequently, the reference to the «one» speed of stars implies that the passage speaks about the fixed stars, which participate in the revolution of the celestial sphere without changing their relative angular distances. Such knowledge relating the aspects of the celestial sphere to the seasons is already found in Hesiod,” and in 5th cent. B.C. the observations of Meton and Euctemon are registered on the famous «parapegmata»,! the material of which was enriched by more accurate observations of later astronomers until Ptolemy (included).'* The problem is, what are the speeds of the planets according to the Pythagoreans, as no one source mentions their order accurately; it is simply said that after the Sun come the five planets. I think the reason for the silence about the exact order of the five planets is perhaps that it was not needed, and I will immediately explain why. In antiquity the speeds of the planets refer to the time needed in order that they describe a whole circle in the zodiac, i.e. to their sidereal periods. In the case of the Moon, Mars, Jupiter and Saturn

the differences between these periods are so great that no problem has ever appeared in the estimation of their order and distances from the earth. The problem appears in the case of Mercury, Venus and the Sun, because the first two do not seem moving far from the sun and they describe a whole circle in about a year. Consequently, when sources mention that the five planets come after the sun, even without giving their sidereal periods, the order of the planetary sequence is unclear only as far as Mercury and Venus are concerned, i.e.: central fire, Counter-earth, Earth, Moon, Sun, Mercury - Venus (or inversely), Mars, Jupiter, Saturn, sphere of fixed stars. Generally, this vagueness regarding the order of the Sun, Mercury and Venus 2Ln0mRaeEkLeI2 PYTHAGOREAN PHILOSOPHY & MATHEMATICAL ASTRONOMY 121 both Platonic cosmological systems place the earth in the middle of the world. In the system exposed by Plato in the Republic, the outermost whorl of Ananke’s (=Necessity’s) spindle is the sphere of fixed stars, while the inner whorls, which correspond to the planets, the sun and the moon, are concentric and also have the same axis with the outermost whorl; moreover they move with different speeds on the same plane with it, but in opposite direction in relation to it.” On the contrary, the cosmological system in Timaeus takes into account the circle of the Same, which corresponds to the celestial equator and refers to the diurnal rotation of the celestial sphere from east to west, as well as the circle of the Different, i.e. the ecliptic, also continues in later Greek astronomy." where the planets seem moving from west to east when they are in Another question is, whether the Philolaic system took into account the obliquity of the ecliptic, i.e. that the zodiac belt, where the Sun, the Moon and the planets seem moving, is inclined in respect to the circle of the revolution of the celestial sphere, i.e. the celestial equator. Aétius referring to the motion of the earth says: direct motion, and from east to west when in retrograde motion. Moreover it is clearly said that the circles of the Same and the Different do not concide, but they intersect and are like the letter x 4 We do not know the chronological difference between the two «Some say that the earth is at rest; but Philolaus the Pythagorean Platonic models; but we can at least ask, where Plato was influenced says that it is carried in a circle round the fire on a slanting circle in a similar fashion to the sun and moon».'? As Aristotle and Simplicius do not mention inclined orbits,'” the general opinion is that all about the «inclined circle», namely the ecliptic, is Aétius’s interpolation. But the discovery of the obliquity of the ecliptic is attributed to Oinopides of Chios,'* who flourished c. 450 B. C., while Philolaus flourished c. 430 B.C. and he could have known about Oinopides’s discovery. In the literary sources there also is the opposite opinion, that Oinopides appropriated this Pythagorean discovery, or that he from. Although both models, the one in the Republic and the one in Timaeus, display great differences from the Philolaic model, they also share some common elements with it. The first is the principle of the circular motion as the only one which can be attributed to the divine celestial bodies (fixed stars and planets), and which takes place around a common center. The second is the order of the planets in the two Platonic models. According to the descriptions contained in the texts this order is as follows: Earth, Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn, sphere of the fixed stars. learned it from the Egyptians.'? Namely, the sun comes immediately after the moon”, after it Venus To two other Pythagoreans, Hicetas” and Ecphantus of Suracuse, is attributed the idea of the rotation of the earth around an axis passing through its center, which explains the apparent diurnal motion of the celestial sphere: We do not know when they exactly lived, but they are usually considered as Philolaus’s contemporaries (middle of 5th cent. B.C.). Although according to Ecphantus the earth is in the middle of the world and moves without involving change of place but by revolution,” it is very important that one of these two kinds of true motion is attributed to the earth by Philolaus and Mercury —although the latter two have the same speed as the and the other kind of motion by Ecphantus and Hicetas. According to another tradition, the cosmological system as exposed in Timaeus is Plato’s plagiarism, who bought and studied Philolaus’s works.? Let us have a closer look at this subject. Firstly, sun,* while the remaining planets display different speeds in relation to each other as well as to the former three planets.?” Plato does not give any numerical value for these speeds, as he does not write a mathematical treatise, but he says that only a few men know them, evidently the mathematicians - astronomers.*% In the Philolaic model we also have the five planets after the sun. It is true that Aétius does not mention any word on planetary speeds in the Philolaic model; in spite of it, I very much doubt that there were not some basic correspondences —even inaccurate— between planets and speeds, because of the Pythagorean «Harmony of spheres», i.e. their Harmonics applied to the spheres of the celestial bodies, and which Plato took from them.

Consequently, although Plato did not plagiarize Philolaus’s system, nevertheless he adopted some of its basic principles regarding the motion of the celestial bodies, which actually seem to describe circles on heaven. This is well understood by us, because we are familiar with basic geometrical knowledge. But in antiquity it was the Pythagorean philosophy and especially its principles regarding numbers and geometrical figures, and their relation to the universe, which initiated the use of mathematics for the description of the structure of the universe and of the motions of its parts. From Plato on, great mathematicians - astronomers improve the two models of the geocentric system, i.e. the eccentric and the epicyclic models, aiming to an explanation of the apparently irregular motions of the planets by introducing one or more cyclic paths.* Their persistence in circular orbits leads to a more and more sophisticated structure of the models by the continuous addition of new cycles and epicycles.*! Aristarchus of Samos, another mathematician - astronomer, is the main introducer of the heliocentric system. According to Archimedes, «the hypotheses of Aristarchus of Samos are that the fixed stars and the sun remain motionless, and that the earth revolves in the circumference of a circle about the sun, which lies in the middle of the orbit».” Although the passage refers to the earth, we may suggest that also the planets revolve around the sun, as the fixed stars and the sun are mentioned as motionless. Aristarchus’s system adopts the Pythagorean circular planetary orbits around a common center which is the sun. The circular orbits are in the zodiac and the path of the earth coincides with the ecliptic. If Aristarchus ascribed one circular orbit around the sun for every planet, then the rejection of its system by the astronomers may not be due to religious prejudices but to scientific reasons, because it could not «save the phenomena». Namely, it is sure that in his system there was a difference between observed and calculated planetary positions, as we now know that the planets move in elliptical orbits. This view is further supported by a case of silent deviation from the circular orbits in the geocentric Ptolemaic system. From some erroneous observations Ptolemy concluded that Mercury’s orbit had two points at which the planet was nearest to the earth, i.e. that it had two perigees.” This implied that the locus of the center of the epicycle was not a circle, but an ovoid curve with the earth situated not in the middle of its main axis, but in the side of the two perigees, namely in the narrow part of the ovoid.** As Ptolemy was a PYTHAGOREAN PHILOSOPHY & MATHEMATICAL ASTRONOMY 123 w that he violated the tradition famous geometer, it is sure he kne he did it, but only in regarding circular orbits; nevertheless to his Mercury’s case, in order to «save the phenomena» according observations. he plagiarized For the same reason Copernicus, iedalthitough with respect to the appl Aristarchus’s heliocentric idea, he ycles, great number of cycles and epic geocentric system, by using aeren calc between observed and ulatede in order to minimize the diff ces ular orbits dominated mor planetary positions. The principle of circnnin g of the 17th century than a millennium. Only in the ebegi of planetary orbits, one of Johannes Kepler formulated the threaroulaws the sun in elliptical orbits, which says that the planets moveby thendsun. one focus of which is occupied harmony of the spheres The ideas of the Pythagoreans about the move , as well as those of the planets producing sounds asandthey their correspondence to the regarding the five regular solids d, infl uenced very much Plato’s four primary elements and the worl later cosmological systems. Kepler cosmology in Timaeus, and otherthem is shown by two of his also was deeply influenced by up to, asnow,it as us scientists find works.” The influence continues and moleculafamo r structure of matter again the regular solids in atomic osophy, which strongly linked and they refer to the Pythagorean phil the numbers with the universe both in micro and macro scales.* NOTES 1970, 62. 1. D. R. Dicks, Early Greek Astronomy to Aristotle, Cornell U. Pr. here iv 2. Plat. Repub. 530d, in: Platonis opera, i-v, ed. I. Burnet, Oxford U. Pr., (1902 / repr. 1978). Stob. Ecl. I 21, 8 = 3. Aëtius I 7, 7 & M 11, 3 = Philol. Frg. A 16 & eAder17. Vorso kratiker (=DK), Philol. Fre. B 7, in: H. Diels - W. Kranz, Die Fragment Weidmann, Berlin 1952, !°1961. 4. Aristot. De caelo, B 13, 293a18, ed. D. J. Allan, Loeb, Oxford U. Pr. repr. 1973. 5. D. R. Dicks, op. cit., 44-45, 48-49, 51-55, 57.

6. Aristot., op. cit., B 13, 293a23. 7. Diog. VIII 85 = Philol. Frg. B 1. Stob. Ec/., I 21, 7a = Philol. Frg. B 2, in: PYTHAGOREAN PHILOSOPHY & MATHEMATICAL ASTRONOMY 125 23. Plat. Repub. X 616c - 617b. 24. Plat. Tim. 36b-d. DK. 25. Ibid. 38d, 39b. 8. Stob. Ecl. I 21, 7b = Philol. Frg. B 4, in: DK . 26. Plat. Repub. 617a, Tim. 39d. 9. Stob. Ecl. I prooem. cor. 3 = Philol. Frg. B 11, in: DK. 27. Plat. Repub. 617c, Tim. 36d. 10. See note 9. 28. Plat. Tim. 39c. 11. Iambl. In Nicom. 9, 1 = Archyt. Frg. B 1, in: DK. 12. Hesiod, Works and days, 383-7, 414-9, 479-80, 564-7, 571-2, 597-8, 609-11, 615-21, 663-5. 29. Aristot., op. cit., B 9, 290b12. 30. Olaf Pedersen & Mogens Pahl, Early Physics and Astronomy, MacDonald & Janes, London 1974, 74-87. 13. D. R. Dicks, op. cit., 107. 31. O. Pedersen & M. Pahl, op. cit., 90-99. 14. Claude Prolemée, Apparitions des fixes et annonces, éd. Halma, Paris 1819- 20. 32. Archimed. Arenar., I 4, in: Greek mathematics, i-ii, trans. by Ivor Thomas, Loeb, Harvard U. Pr. 1941 / repr. 1980, here ii 2. 15. O. Neugebauer, A History of Ancient Mathematical Astronomy, i-iii, Springer, 1975, here i 148. 33. Claude Ptolemée, Composition mathématique, IX 8, éd. Halma, Paris i (1813), ii (1816), here: ii 172. 16. Aëtius III 13. 1. 2 = Philol. Frg. A 21, in: DK . 34. O. Neugebauer, op. cit., here: i 163-4, iii 1253 fig. 148. O. Pedersen & M. 17. Aristot. op.cit., B 13, 293a18. Simplic. In Aristot. De caelo 511, 26, in: G. Pahl, op. cit., 96 fig. 7. 19. S. Kirk - J. E. Raven, The Presokratic Philosophers, Cambridge U. Pr. 1957 / repr. 1973, 260. 35. O. Pedersen & M. Pahl, op. cit., 308-314. 18. Theon. Smyrn., p. 198, 14. Macrob. Sat. I 17, 31 = Oinop. Frg. 7, in: DK . 36. J. Kepler, Astronomia Nova (1609).It is noteworthy that ellipse is one of the conic sections, which had been fully studied by Apollonius of Perga in his 19. Aëtius II 12, 2. Diod. I 98, 2 = Oinop. Frg. 7, in: DK. famous work on them, written about the middle of third century B.C. 20. Cic. Akad. Pr. II 39, 123 = Hiketas Frg. 1, in: DK. 37. Mysterium cosmographicum (1596), Harmonice mundi (1619). 21. Hippol. Refut. I 15 = Ekphant. Frg. 1. Aétius III 13, 3 = Ekphant. Frg. 5, in: 38. Lecture of Prof. Roald Hoffmann (Nobel Prize for Chemistry 1981) at the 22. Diog. VIII 84. 85 = Philol. Frg. A 1, in: DK . University of Athens (20 May 1991).