Ancient and Medieval Values for the Mean Synodic Month

Neugebauer O. , “From Assyriology to Renaissance art”, Proceedings of the American Philosophical Society , cxxxiii (1989), 391–403.

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).

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. (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.

Geminus , xviii. 3 ( Geminos , Introduction aux phénomènes , ed. and transl. by Aujac G. (Paris, 1975), 94), says that 19756^d = 669^m (equivalent to 6585;20^d = 223^m), and this relationship implies that M = 29;31,50,18,…^d. Ptolemy also reports an 18-year period of 6585;20^d 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.

Neugebauer , ACT (ref. 3), i, 272: Text No. 210, line 10.

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.

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.

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.

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.

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.

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.

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-mas^cūdī (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.

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.

Al-Bīrūnī , Chronology of ancient nations , transl. by Sachau C. E. (London, 1879), 143, 147; cf. 408, 410.

Pingree D. , “The fragments of the works of Ya^cqūb 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.

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.

Babylonian Talmud , Seder mo^ced, 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.

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.

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.

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.

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.

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.

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.

Cf. Stern , Calendar (ref. 20), 204. Note also that 1^p = 0;0,0,8,20^d, i.e., 1^p is the difference between 29;31,50,8,20^d and 29;31,50^d (the value for M in the Muslim calendar).

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.

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.

Chabás J. , “The astronomical tables of Jacob ben David Bonjorn”, Archive for history of exact sciences , xlii (1991), 279–314, espec. pp. 283–4.

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.

Toomer G. J. , “A survey of the Toledan tables”, Osiris , xv (1968), 5–174, espec. p. 80.

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.

Neugebauer O. , A history of ancient mathematical astronomy (3 vols, Berlin and New York, 1975), ii, 902.

For the Hebrew text, see Levi ben Gerson's Astronomy (ref. 26); for the Latin text, see Mancha “Provençal version” (ref. 11), 308.

Goldstine H. H. , New and full moons 1001 B.C. to A.D. 1651 (Philadelphia, 1973).

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.

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.

Goldstein , Astronomical tables (ref. 26), 229–38.

Cf. Mancha , “Provençal version” (ref. 11), 308.