SteeleJ., Observations and predictions of eclipse times by early astronomers (Dordrecht and Boston, 2000), 21–83.
2.
NeugebauerO., Astronomical cuneiform texts (3 vols, London, 1955), i, 41–277.
3.
SachsA. J. and HungerH., Astronomical Diaries and related texts from Babylon (3 vols, Vienna, 1988–95).
4.
See, especially, BrittonJ. P., “The structure and parameters of column “Φ”, in From ancient omens to statistical mechanics: Essays on the exact sciences presented to Asger Aaboe, ed. by BerggrenJ. L. and GoldsteinB. R. (Copenhagen, 1987), 23–36; and idem, “Lunar anomaly in Babylonian astronomy”, in Ancient astronomy and celestial divination, ed. by SwerdlowN. M. (Cambridge, Mass., 1999), 187–254; Brack-BernsenL., “Babylonische Mondtexte: Beobachtung und Theorie”, in Die Rolle der Astronomie in den Kulturen Mesopotamiens, ed. by GalterH. D. (Graz, 1993), 331–58; eadem, Zur Entstehung der babylonischen Mondtheorie (Stuttgart, 1997); eadem, “Goal-Year Tablets: Lunar data and predictions”, in Ancient astronomy and celestial divination, ed. by Swerdlow, 149–77.
5.
AaboeA.BrittonJ. P.HendersonJ. A.NeugebauerO., and SachsA. J., Saros cycle dates and related Babylonian astronomical texts (Transactions of the American Philosophical Society, lxxxi/6; Philadelphia, 1991)), 16.
6.
BrittonJ. P., “An early function for eclipse magnitudes in Babylonian astronomy”, Centaurus, xxxii (1989), 1–52, esp. pp. 4–13. As Britton notes (pp. 2–3), the limits of the nodal zone for lunar eclipses (within which the lunar latitude is sufficiently small for a lunar eclipse to occur) vary from 9.5° to 12.2° before or after a node and, since the nodal elongation of the sun increases by roughly 30.7° per month, lunar eclipses cannot occur in consecutive months. There are two nodal zones, one about the ascending node, and the other about the descending node. After 6 months, the nodal elongation of the sun accumulates to about 184°, and so it moves from one nodal zone to the next, i.e., its nodal elongation from the second node is about 4° greater than it was from the initial node; hence, the sun will again be in a nodal zone unless it was too close to the end of the initial nodal zone. When the sun is in the nodal zone near one of the limits (at which time the moon is 180° from the sun), the eclipse may not be noticed because of its small magnitude. See also AaboeA., “Remarks on the theoretical treatment of eclipses in Antiquity”, Journal for the history of astronomy, iii (1972), 105–18. For historical background in support of Britton's interpretation of the data, see Steele, Eclipse times (ref. 1), 78ff.
7.
Aaboe, Saros cycle (ref. 5), 14.
8.
See MoesgaardK. P., “The Full Moon Serpent: A foundation stone of ancient astronomy?”, Centaurus, xxiv (1980), 51–96. Cf.Britton, “Column Φ” (ref. 4), 34, n. 4.
9.
Aaboe, Saros cycle (ref. 5), 18–20.
10.
SmartW. M., Text-book on spherical astronomy (Cambridge, 1965), 420.
For the contrary claim, see Britton, “Lunar anomaly” (ref. 4), 192f.
14.
Rochberg-HaltonF., Aspects of Babylonian celestial divination: The lunar eclipse tablets of Enūma Anu Enlil (Vienna, 1988).
15.
NeugebauerO., A history of ancient mathematical astronomy (Berlin and New York, 1975), 484ff.
16.
Britton, “Column Φ” (ref. 4), 25f; idem, “Lunar anomaly” (ref. 4), 211f. Britton believes that the relationship, 251 synodic months = 269 anomalistic months, was known to the author(s) of System A, although it does not occur explicitly there (privately communicated).
See Brack-Bernsen, “Babylonische Mondtexte” (ref. 4), 349.
23.
GoldsteinB. R., “The obliquity of the ecliptic in ancient Greek astronomy”, Archives internationales d'histoire des sciences, xxxiii (1983), 3–14, esp. p. 8; and BowenA. C. and GoldsteinB. R., “Hipparchus' treatment of early Greek astronomy,”Proceedings of the American Philosophical Society, cxxxv (1991), 233–54, esp. pp. 237f and 248f. Although the Babylonians did not appeal to deductive demonstrations, their facility with numbers was certainly on a par with the Greeks, and this theorem was well within their grasp.
24.
It is difficult to believe that these month lengths were derived independently and, while it is tempting to consider that the value for the synodic month was derived directly from the value for the anomalistic month, this seems unlikely, for (as noted in Appendix 2): (269/251) · 27;33,16,30 = 29;31,50,11,36d rather than 29;31,50,8,20d.
25.
Britton, “Column Φ” (ref. 4), 24. Britton (“Lunar anomaly” (ref. 4), 242f) remarks that this value for the anomalistic month is the “sole month-length in the System A theory that is explicitly defined”.
26.
NeugebauerO., “Saros” and lunar velocity in Babylonian astronomy (Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser, xxxi/4; Copenhagen, 1957), 12ff; cf.Neugebauer, History (ref. 2), 501. Neugebauer considered a unit of 1H such that 6H = 1d, whereas the Babylonians used a unit, corresponding to time-degrees, called ush such that 360 ush = 1d, i.e., 1H = 60 ush. Judging from Britton's remark (see ref. 25), this value for the synodic month played no role in the theory of System A.
27.
Britton has withdrawn the suggestion he made previously (“Column Φ” (ref. 4), 24f): Privately communicated.
28.
For the step in the Babylonian computation where the error occurs, see Neugebauer, “Saros” (ref. 26), 13, n. 16.
29.
See, e.g., ToomerG. J., Ptolemy's Almagest (New York and Berlin, 1984), 176, n. 10; and AaboeA., “On the Babylonian origin of some Hipparchian parameters”, Centaurus, iv (1955), 122–5.
30.
Britton, “Lunar anomaly” (ref. 4), 211ff.
31.
If the upper bound is taken to be the value for the synodic month in the Saros text where m = 29;31,50,19,11,4,56d, then 223m ≈ 6585;20,1,18,… d and the result is slightly higher: 29;31,50,7,20d.
