Ancient and Medieval Values for the Mean Synodic Month
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References
1. Neugebauer O., “From Assyriology to Renaissance art”, Proceedings of the American Philosophical Society, cxxxiii (1989), 391–403.
2. Goldstein B. R., “On the Babylonian discovery of the periods of lunar motion”, Journal for the history of astronomy, xxxiii (2002), 1–13, espec. pp. 10–11 (note that on p. 11, line 3, ‘4267’ is a mistake for 6247 synodic months).
3. For the mean value of column G (in uš) in System B that yields the length of the mean synodic month, see Neugebauer O., Astronomical cuneiform texts [henceforth ACT] (3 vols, London, 1955), i, 78. The standard value, 29;31,50,8,20d, occurs in this form in Text No. 210, line 6 (ibid., i, 272) but, as far as I can tell, it does not occur in any other published Babylonian text. Neugebauer also identified a Greek papyrus that contains a fragment of an ephemeris according to System B (including column G) which implied, among other things, that the System B value for M was available in the Greek world in its proper context: idem, “A Babylonian lunar ephemeris from Roman Egypt”, in A scientific humanist: Studies in memory of Abraham Sachs, ed. by Leichty E., et al. (Philadelphia, 1988), 301–4. For more recent treatment of this text, see Jones A., “A Greek papyrus containing Babylonian lunar theory”, Zeitschrift für Papyrologie und Epigraphik, cxix (1997), 167–72.
4. Geminus, xviii. 3 (Geminos, Introduction aux phénomènes, ed. and transl. by Aujac G. (Paris, 1975), 94), says that 19756d = 669m (equivalent to 6585;20d = 223m), and this relationship implies that M = 29;31,50,18,…d. Ptolemy also reports an 18-year period of 6585;20d and ascribes it to “the even more ancient [astronomers]”: Almagest, iv.2 (Toomer G. J., Ptolemy's Almagest (London, 1984), 175). See also Goldstein, “Periods” (ref. 2), 1–2.
5. Neugebauer, ACT (ref. 3), i, 272: Text No. 210, line 10.
6. Steele J. M., “A simple function for the length of the Saros in Babylonian astronomy”, in Under one sky: Astronomy and mathematics in the ancient Near East, ed. by Steele J. M., Imhausen A. (Alter Orient und Altes Testament, ccxcvii (Muenster, 2003), 405–20). I am most grateful to Dr Steele for providing me with a preprint of his paper and for allowing me to summarize part of it here. A fuller discussion of the analysis will appear in a forthcoming paper by Lis Brack-Bernsen and John Steele.
7. For early lunar eclipse observations by the Babylonians (including the eclipse of −746 Feb. 6), see Britton J. P., “Scientific astronomy in pre-Seleucid Babylon”, in Die Rolle der Astronomie in den Kulturen Mesopotamiens, ed. by Gaiter H. D. (Graz, 1993), 61–76, espec. p. 63, n. 4; cf. Steele J. M., Observations and predictions of eclipse times by early astronomers (Dordrecht and Boston, 2000), 43–45.
8. Cf. Britton J. P., “Lunar anomaly in Babylonian astronomy”, in Ancient astronomy and celestial divination, ed. by Swerdlow N. M. (Cambridge, MA, 1999), 187–254, espec. p. 207.
9. Ptolemy, Almagest, iv.2; Toomer, Almagest (ref. 4), 175–6. Toomer G. J. (“Hipparchus' empirical basis for his lunar mean motions”, Centaurus, xxiv (1981), 97–109) offers a reconstruction of Hipparchus's methods and the eclipses he might have used to confirm the Babylonian parameter.
10. Copernicus N., De revolutionibus, iv.4 (Nuremberg, 1543), 101 v; cf. Aaboe A., “On the Babylonian origin of some Hipparchian parameters”, Centaurus, iv (1955), 122–5; and Swerdlow N. M., Neugebauer O., Mathematical astronomy in Copernicus's De revolutionibus (New York and Berlin, 1984), 199.
11. For al-Ḥajjāj's translation, see Leiden, MS Or. 680, 50b:6. As noted by Mancha J. L. (“The Provençal version of Levi ben Gerson's tables for eclipses”, Archives internationales d'histoire des sciences, xlviii (1998), 269–352, espec. p. 309), this variant appears in Gerard of Cremona's translation of the Almagest, ed. 1515, 36r; and in George of Trebizond's Latin translation of the Almagest from Greek, ed. 1528, 33r. Copernicus annotated a copy of Gerard of Cremona's translation of the Almagest (ed. 1515): See Czartoryski P., “The library of Copernicus”, Studia copernicana, xvi (1978), 355–96, espec. p. 372. See now Mancha J. L., “A note on Copernicus' ‘correction’ of Ptolemy's mean synodic month”, to appear in Suhayl.
12. Isaac Israeli, Liber jesod olam seu Fundamentum mundi, ed. by Goldberg B., Rosenkranz L. (2 vols, Berlin, 1846–48), i, 49b, col. 1; al-Bīrūnī, al-Qānūn al-mascūdī (3 vols, Hyderabad, 1954–56), ii, 730. Abraham Bar Ḥiyya (Sefer ha−cibbur, ed. by Filipowski H. (London, 1851), 37) also says that Ptolemy's value for the synodic month is 29;31,50,8,9,20d.
13. For Ibn Yūs, see Leiden, MS Or. 143, p. 20; and for al-Biṭrūjī, see Goldstein B. R., Al-Biṭrūjī: On the principles of astronomy (2 vols, New Haven, 1971), i, 145; cf. Neugebauer O., Ethiopic astronomy and computus (Vienna, 1979), 18.
14. Al-Bīrūnī, Chronology of ancient nations, transl. by Sachau C. E. (London, 1879), 143, 147; cf. 408, 410.
15. Pingree D., “The fragments of the works of Yacqūb ibn Ṭāriq”, Journal of Near Eastern studies, xxvii (1968), 97–125, espec. p. 99; al-Bīrūnī, India, transl. by Sachau C. E. (2 vols, London, 1888), ii, 15–16; cf. i, 350, 368.
16. Goldstein B. R., Ibn al-Muthannā's commentary on the astronomical tables of al-Khwārizmī (New Haven, 1967), 18; Kennedy E. S., Pingree D., Haddad F. I., The book of the reasons behind astronomical tables by cAlī ibn Sulaymān al-Hāshimī (Delmar, NY, 1981), 105, 234–6.
17. Babylonian Talmud, Seder moced, Tractate Rosh ha-shanah, 25a; transl. by Epstein I. (London, 1938), 110. See also, Pirḳê de Rabbi Eliezer, transl. by Friedlander G. (London, 1916), 43: “The total of the days of the lunar month is 29 ½ days, 40 minutes, and 73 parts.” The date of this Hebrew text is disputed, but a version of it was probably composed in the eighth century.
18. Kennedy E. S., “Al-Khwārizmī on the Jewish calendar”, Scripta mathematica, xxvii (1964), 55–59; reprinted in idem, Studies in the Islamic exact sciences (Beirut, 1983), 661–5.
19. Al-Bīrūnī, Chronology (ref. 14), 143; Abraham Ibn Ezra, Sefer ha−cibbur, ed. by Halberstam S. Z. H. (Lyck, 1874), 3a; Abrahm Bar Ḥiyya, Sefer ha−cibbur (ref. 11), 45; Maimonides, Sanctification of the new moon, transl. by Gandz S., with supplementation and an introduction by Obermann J., and an astronomical commentary by Neugebauer O. (New Haven, 1956), 27, 33, 114.
20. See, e.g., Stern S., Calendar and community: A history of the Jewish calendar, 2nd century BCE–10th century CE (Oxford and New York, 2001), 201–4.
21. Neugebauer O., “Studies in ancient astronomy, VII: Magnitudes of lunar eclipses in Babylonian astronomy”, Isis, xxxvi (1945) 10–15, espec. pp. 12–13; idem, ACT (ref. 3), i, 39, 47. For an extensive recent study of Babylonian metrology, see Powell M. A., “Masse und Gewichte”, in Reallexikon der Assyrologie und Vorderasiatischen Archäologie, ed. by Edzard D. O. (9 vols, Berlin, 1932–), vii (1987–90), 457–517: On the barleycorn, see pp. 458, 478–9.
22. Reiner E., Pingree D., “A Neo-Babylonian report on seasonal hours”, Archiv für Orientforschung, xxv (1977), 50–55; Rochberg-Halton F., “Babylonian seasonal hours”, Centaurus, xxxii (1989), 146–70.
23. King D. A., “Too many cooks: A new account of the earliest Muslim geodetic measurements”, Suhayl, i (2000), 207–41, espec. pp. 223, 235: 4.
24. Cf. Stern, Calendar (ref. 20), 204. Note also that 1p = 0;0,0,8,20d, i.e., 1p is the difference between 29;31,50,8,20d and 29;31,50d (the value for M in the Muslim calendar).
25. Zonta M., “La tradizione ebraica dell'Almagesto di Tolomeo”, Henoch, xv (1993), 325–50, espec. p. 332; cf. Mancha, “Provençal version” (ref. 11), 307; Levi ben Gerson's Astronomy, chap. 64: Paris, Bibliothèque nationale de France, MS Heb. 724, 127a: 21 ff.
26. Levi ben Gerson's Astronomy, chap. 82: Paris, Bibliothèque nationale de France, MS Heb. 724, 155a; cf. Goldstein B. R., Levi ben Gerson's astronomical tables (New Haven, 1974), 106.
27. Chabás J., “The astronomical tables of Jacob ben David Bonjorn”, Archive for history of exact sciences, xlii (1991), 279–314, espec. pp. 283–4.
28. Tabulae astronomice illustrissimi Alfontij regis castelle, ed. by Ratdolt E. (Venice, 1483), d7r; cf. Swerdlow N. M., “The length of the year in the original proposal for the Gregorian calendar”, Journal for the history of astronomy, xvii (1986), 109–18, espec. pp. 115–16.
29. Toomer G. J., “A survey of the Toledan tables”, Osiris, xv (1968), 5–174, espec. p. 80.
30. Ptolemy, Upotheseôn ton planômenon [The Planetary hypotheses], ed. and transl. by Heiberg J. L., Nix L., Heiberg J. L., Claudii Ptolemaei Opera astronomica minora (Leipzig, 1907), 70–145, espec. p. 79.
31. Neugebauer O., A history of ancient mathematical astronomy (3 vols, Berlin and New York, 1975), ii, 902.
32. For the Hebrew text, see Levi ben Gerson's Astronomy (ref. 26); for the Latin text, see Mancha “Provençal version” (ref. 11), 308.
33. Goldstine H. H., New and full moons 1001 B.C. to A.D. 1651 (Philadelphia, 1973).
34. Toomer, Almagest (ref. 4), 204; Goldstein B. R., “Medieval observations of solar and lunar eclipses”, Archives internationales d'histoire des sciences, xxix (1979), 101–56, espec. p. 143.
35. In fact, according to The Times atlas of the world (2nd edn revised, Boston, 1971), the longitude of Orange is 4;48°E and the longitude of Alexandria is 29;55°E, for a difference of 25;7° ≈ 1;40h. The longitude of Montpellier is 3;53°E, and so it is not on the same meridian as Orange.
36. Goldstein, Astronomical tables (ref. 26), 229–38.
37. Cf. Mancha, “Provençal version” (ref. 11), 308.
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