The Young Avestan and Babylonian Calendars and the Antecedents of Precession

1. Almagest (Des Claudius Ptolemäus Handbuch der Astronomie), translated by Manitius K. i (Leipzig, 1912), 144f.

2. Manitius's German translation reads mindestens, “at least”, which is evidently wrong, Incidentally, after completing the present paper I found that R. Mercier, in his important article on “Studies in the medieval conception of precession”, Part II, Archives internationales d'histoire des sciences, xxvii (1977), 33–71, deals with this passage at p. 50. As will be seen in what follows, I do not share his opinion that the Hipparchian year “is designed [my emphasis] to fit the 19 year cycle”. Hipparchus merely states that his year does fit the 19-year cycle extremely closely.

3. From the equation 126007^d1^h = 345 sidereal rotations of the Sun —7;30°, one obtains for the sidereal year 365 ¼ 1/100^d, and for precession 1° in 77 Egyptian years. See Neugebauer O., A history of ancient mathematical astronomy (Berlin, 1975; hereafter cited as HAMA), 297f.

4. HAMA, 298.

5. To my knowledge, Aaboe A., “On the Babylonian origin of some Hipparchian parameters”, Centaurus, iv (1955–56), 122–5, was the first to point out that the division 126007^d1^h/4267 yields, not 29;31,50,8,20 (hereafter called m), but 29;31,50,8,9, … (correctly m' = 29;31,50,8,9,18 = 29·53059331; the error of the Babylonian value, m—m', sums up to 16·8^s in 235, and to 5^m4^s in 4267, months) and that Copernicus (De revolutionibus IV, 4) noticed this discrepancy and silently replaced Ptolemy's value by 29;31,50,8,9,20 (cf. also HAMA, 310). Aaboe is undoubtedly right in assuming that Hipparchus, contrary to Ptolemy's account (Almagest IV, 2), did not compute his parameter, m, by carrying out the above division, but just took it over from Babylonian sources. We thus get 4267 m' = 126007^d1^h0^m2^s, and 4267 m = 126007^d1^h5^m6^s, i.e. practically identical results. Now there are two Babylonian partial eclipses recorded in Almagest IV, 9 (Manitius, i, 239 and 241) which Hipparchus may have used to test his result: (1) —501 Nov. 19/20 (cf. Aaboe, l.c.), maximum according to Almagest at 23^h36^m M.T. Babylon. 126007^d later, on —156 Nov. 14/15, there occurred an eclipse which (according to Goldstine, New and Full Moons 1001 B.C. to A.D. 1651 (Philadelphia, 1973)) reached its maximum at 0^h20^m M.T. Bab. This implies an excess over 126007^d of 44^m, which, however, should be reduced by 10^m because the Babylonian eclipse actually had its maximum at 23^h46^m (Goldstine; Oppolzer, no longer reliable for that period, has 0^h24^m). (2) —490 Apr. 25/26, max. according to Almagest at 23^h30^m, to be compared with —145 Apr. 22/23, max. at 0^h14^m (Goldstine; Oppolzer has 0^h23^m). Here again we have the same excess of 44^m, and this Hipparchus might have taken as a satisfactory confirmation of his round figure “1^h”. In reality, however, the eclipse of —490 reached its maximum at 22^h28^m (Goldstine; Oppolzer has 22^h55^m), whence the excess actually was 1^h46^m. Inevitably there will always be the possibility of an error in the determination of the maximum phase of an eclipse, but an uncertainty of one hour, as in this case, is impossible since the half-duration of the eclipse (magnitude according to Oppolzer 1·1 inches) was only 35^m. The most plausible explanation is that the Babylonian text had “1 ½ hour before midnight” and that Ptolemy's “1/2 hour” is a mistake. Errors of 10–15^m in the determination of the maximum phase of a partial eclipse seem unavoidable. A still greater uncertainty has to be taken into account in the case of total eclipses such as those of —156 (magn. 17·9, half-duration of totality 47^m) and of —145 (magn. 20·7, half-duration of totality 51^m). It thus seems out of the question that Hipparchus could obtain his estimate of “1 hour” from observation. Note: For the period 126007^d1^h see also my review of van der Waerden, Die Anfänge der Astronomie (Groningen, 1966), in Gnomon, xliv (1972), 529–37 (see p. 534f).

5a. Or, expressed in months: 1 year = 12;22,6,18,57 = 12·36842105 months (see p. 622).

6. According to Geminus's elaborate report (Gemini elementa astronomiae, ed. and transl. Manitius C. (Leipzig, 1898), 110ff), the astronomers of the school of Euctemon, Philippus and Callippus, after having stated the faultiness of the octaëteris, “brought forth as another period the 19-year cycle. They had found from observation (), namely, that in 19 years, 6940 days or 235 months are contained, including the intercalary ones, and that there are 7 intercalary in these 19 years.” It is not clear what those astronomers, in order to test the equation “19 years = 235 months”, had actually observed and found equal to a round number of days, 6940: The length of 19 years or that of 235 months. As for the term ‘year’, it goes without saying that it is always the year defined by the recurring seasons which is intended. Since, however, this ‘tropical’ year is only 20 minutes shorter than the sidereal, it requires very careful observations to recognize that the two are not identical. In order to assess the relation between tropical years and days, considering that a number of days without fraction was found, the Greek astronomers must have selected equinoxes or solstices 19 (or n × 19) years apart. This is very improbable. At any rate the year length resulting, 365 5/19 = 365·2632^d, is even higher than that of the sidereal (365·2564^d) and comes close to the Babylonian sidereal year of System A: P_A = 365·2606 (see below). This might indicate that actually not tropical but sidereal phenomena (heliacal risings, etc.) had been observed and that observations of the cardinal points (as Euctemon's of the summer solstice of —431) served only to establish an approximate connection between the seasons and the sidereal calendar, such as attested by the parapegmata ascribed to Euctemon, Eudoxus and Callippus. If, however, they were to have observed the following two pairs of lunar eclipses immediately preceding Euctemon's solstice observation of —431: —454 Nov. 9, 19;20^h M.T. Athens —435 Nov. 9, 18;45 M.T. Athens Δ = 6940^d—35m —450 Mar. 4, 21;00 M.T. Athens —431 Mar. 4, 21;20 M.T. Athens Δ = 6940^d+20m they would have felt entitled to vindicate a strict identity between the numbers of years, days and months as mentioned by Geminus since the excess of —35^m and +20^m over 6940^d is not observable. Indeed there is an overwhelming number of such pairs (or rather triplets since there are always three, apparently never more, in one series), but not all of them of course yield the same result. Thus that of —448 Aug. 6, 7;40^h (maximum phase not observable in Athens) was preceded by another on —467 Aug. 6, 19;00^h, the difference being only c. 6939·5^d, from which a year length of 365·235^d (≈ 365 4/17) would have resulted. In the same way the Babylonians could find the named period relation roughly confirmed by a similar triplet of lunar eclipses, the last of which occurred on —502 Jan. 10, c. 75^d before the introduction of the 19-year cycle: —540 Jan. 10, 4;51^h M.T. Babylon —521 Jan. 10, 4;32 M.T. Babylon Δ = 6940^d—19m —502 Jan. 10, 4;56 M.T. Babylon Δ = 6940^d+24^m Contrary to the Greek procedure, however, their mathematical analysis was based exclusively on the exact length of the month, which saved them the detours characterizing the Greek approach. On the other hand, from 235 months = 6940^d there follows the faulty month length, not mentioned explicitly, of 29·5319^d, while Geminus at the outset (pp. 100–1) gives the much better value 29 ½₃ = 29·5303^d, and thereafter (par. 43; the passage may be a later insertion by a scholiast) the exact Babylonian value, 29;31,50,8,20^d = 29·530594^d. Only then, after having told us about the efforts made to construct a lunisolar calendar based on a system of intercalations and omissions of days fitting these inaccurate values, Geminus states that it was found at discord with the “time of the year” (i.e. the year determined by the seasons), which, as established by “observations stretching over a number of years ()”, comprises 365 ¼ days. This figure, suspicious on account of the simple fraction 1/4, was eventually accepted by the school of Callippus as the correct length of the (tropical) year. Nineteen such years equal 6939 ¾ days; by division by 235 (not carried out by Geminus) a new (indeed better, though nowhere else attested) value for the length of the month: 29·53085^d, was obtained. The Callippic 76-year cycle as a final result of these repeated modifications of the underlying parameters worked well enough, but the way which led to it seems to evidence an undeniable lack of understanding of the actual problems involved. By contrast, Hipparchus's approach, as demonstrated in the above, is strictly different. He first derives a value for the tropical year, excellent for his time, from his observations of equinoxes and solstices and of the change in longitude of stars near the ecliptic. Only then does he state that, using the best available value (the Babylonian) for the length of the month, the 19-year cycle furnishes the same tropical year length as that resulting from his observations.

7. Parker R. A. and Dubberstein W. H., Babylonian chronology 626 B.C.–A.D. 75 (Providence, 1956).

8. After Parker-Dubberstein (see ref. 7), pp. 30 and 31.

9. The introduction of the new intercalation cycle probably caused the calendar makers to assume that the sequence of intercalary months must be changed. Only after the omitted intercalation in the 3rd year did they understand that the 19-year cycle had to be composed of 2 octaëterides and the 2 first years of a 3rd octaëteris followed by a normal year of 12 months, as illustrated here (figures in italics indicate intercalary years): 19-year cycle: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 … Octaëterides: 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 1 1 ….

10. For details, see Ginzel F. K., Handbuch der mathematischen und technischen Chronologie, i (Leipzig, 1906), 266. The Turkish māliye years are crude solar years serving tax purposes. They carry the same numbers as the corresponding Hegira years. In order to keep this going (33 Hegira years = 32 solar years), every 33 years one year number has to be jumped over or “deleted” (sivis).

11. For details see my chapter on “Old-Iranian calendars”, written for and destined to appear (if God wills) in Cambridge history of Iran, vol. ii.

12. See Cameron G. G., “Persepolis treasury tablets”, Oriental Institute publications 65 (Chicago, 1948), and Hallock R. T., “Persepolis fortification tablets”, ibid. 92 (1969).

13. Instead of the complicated old-Iranian, I use the modern Persian names, see Table 2a. In Table 2b the correspondences between Egyptian and Iranian months are given as valid in the 1st, 2nd and 8th periods.

14. A continuation of this intercalation procedure beyond 1 (rel.) = XII (civil) is not possible without disturbing the parallelism between the two calendars; as a consequence of the jump over one year of 365 days, namely, a phase difference occurs after the 13th intercalation (13 × 30 = 390 = 365+25), which causes the civil year to run 5 days ahead of the religious. See below (Section 2(d)), and Table 3.

15. See the concluding paragraph of the Introduction.

16. Kholjī Shāh, Kūshyār Abu'l-Hasan, al-Bīrūnī Quṭb, al-Shīrāzī al-Dīn; see Hyde, Historia religionis veterum Persarum (1700), ch. 17, pp. 203f, and Ginzel F. K. (see ref. 10), i, pp. 290ff.

17. The chronology of ancient nations (London, 1879), 12f.

18. Unfortunately the explanation is lost. It probably was contained in the lacuna on p. 45 of the Arabic text (p. 55 of Sachau's translation).

19. Computed from the Sun's mean motion in one synodic month, μ = 29;6,19,20° = 29·10537037°; 360°:μ = P_B'. A computation from Δ = M—m = 1;51,19,20 and d = 0;18 entails the value generally accepted for the length of the year according to System B: P_B = 12;22,8,53,20 months (see HAMA, 533). P_B thus becomes even higher than P_A. I consider P_B' more trustworthy than P_B because the latter is derived from the equation P_B = 2Δ/d, where the constant difference d = 0;18 (without further sexagesimals) stands as a denominator < 1. P_B thus is extremely sensitive even to minute modifications of d. The value of P_B would correspond to d' = 0·3000069036 = 0;18,0,1,29, and the difference d'—d would sum up to 1” in 40 months, or 6” in one 19-year cycle.

20. The equation, 1 “world-year of the Persians” = 360,000 years = 131,493,240 days, yields a sidereal year length of 365;15,32,24^d = 365·25900^d (see Kennedy E. S. and van der Waerden B. L., “The world-year of the Persians”, Journal of the American Oriental Society, lxxxiii (1963), pp. 318 and 325), still used by Abū Ma^cshar. As is seen, it comes very close to L as well as to P_B'.

21. 1508/1507 × 365 = 365·24220.

22. According to the Julian calendar, the limits resulting of course are a.d. 1004 and 1007, both 15 March.

23. Note that his Chronology was finished 6 years before, in a.d. 1000.

24. The fact that this passage, found in ch. 2, which offers a general survey of the nature of months and years, is at variance with all the other passages dealing with the particulars of the Persian calendar, obviously demonstrates that it was written later, after Bīrūnī had obtained reliable information from some of the surviving experts.

25. Here, as in any similar case, the question whether a value found identical in different contexts was borrowed or derived independently, can be decided upon only if it does not agree with the theoretical, modern, value; see above, Section 2(d).

26. See ref. 19.

27. HAMA, 368f.

28. Amazingly enough, Seleucid astronomers sovereignly disregard the foundations on which their calendar rests by dividing the year schematically into 4 equal parts 3;5,31,35 months long (i.e. one-fourth of a tropical year). Comparing, for the period s.e. 132–50 (—179 to —161) the dates for the solstices and the equinoxes derived from the “Uruk scheme “(see HAMA, 357ff) with modern computation, we find that summer solstice falls throughout c. 3 days too late, winter solstice 4–5 days, and vernal equinox even 4–6 days. Only in the case of autumn equinox do we have a better agreement: 1–2 days late, which, despite Neugebauer's warning (see HAMA, 360ff, and p. 206 of my HAMA review in the last issue of JHA) might be interpreted to the effect that, at least on one occasion or the other, autumn equinox has actually been observed and then been taken as a starting point. The question remains open what purpose such utterly inaccurate dates can have served. Another question is, how it could happen that the Babylonians of the Seleucid period were no longer aware that their calendar furnished them an ideal point of departure, in view of the fact that each Ululu₂ year begins on, or very near to, vernal equinox.

29. Ibid., 528, II B 8: “Solar mean motion and length of year”.

30. Neugebauer O., Astronomical cuneiform texts, i (London, [1955]), 271ff.

31. See above, Introduction.

32. The corresponding difference, A_T—A_mod. is 6;37,26^m. The error committed by using the 19-year cycle thus sums up to 1^d in 217 years, whence the last two cycles recorded in Parker-Dubberstein (see ref. 7), a.d. 49 and 68, begin 4 to 5 days after the equinox: March 24 jul. = 26 greg.

33. Applying as the only practicable means the method of corresponding altitudes.

34. See above, Introduction.