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Britton’s theory of the creation of Column \(\varPhi \) in Babylonian System A lunar theory

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Abstract

The following article has two parts. The first part recounts the history of a series of discoveries by Otto Neugebauer, Bartel van der Waerden, and Asger Aaboe which step by step uncovered the meaning of Column \(\varPhi \), the mysterious leading column in Babylonian System A lunar tables. Their research revealed that Column \(\varPhi \) gives the length in days of the 223-month Saros eclipse cycle and explained the remarkable algebraic relations connecting Column \(\varPhi \) to other columns of the lunar tables describing the duration of 1, 6, or 12 synodic months. Part two presents John Britton’s theory of the genesis of Column \(\varPhi \) and the System A lunar theory starting from a fundamental equation relating the columns discovered by Asger Aaboe. This article is intended to explain and, hopefully, to clarify Britton’s original articles which many readers found difficult to follow.

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Notes

  1. Neugebauer (1955).

  2. Ossendrijver (2012).

  3. Neugebauer (1955).

  4. Neugebauer (1975, p. 375).

  5. Ibid., pp. 492–496.

  6. Ibid., pp. 488–490.

  7. Neugebauer (1975, pp. 485–487), Ossendrijver (2012, pp. 145–150).

  8. Kugler (1900).

  9. Neugebauer (1955, p. 44; 1975, p. 484).

  10. Ibid.

  11. Neugebauer (1957).

  12. We will use the term increment for the difference in value, positive or negative, over a tabular interval.

  13. van der Waerden (1966).

  14. Aaboe (1968).

  15. Britton (1990, 1999, 2007, 2009).

  16. Britton (1990, pp. 65–66).

  17. Neugebauer (1957, p. 10, sections 12, 13).

  18. Neugebauer (1975, p. 487).

  19. van der Waerden (1966, p. 146–148).

  20. Ibid., p. 19.

  21. Britton (1999, p. 208).

  22. Aaboe (1968, p. 19).

  23. Ibid., p. 19, 22.

  24. Equation (2), an equivalent form of the interval rule, follows from the elementary algebraic equivalence: \(a - b = c - d <==> a + d = b + c\). Equations (1) and (2) express a relation between the actual values of the lengths of the months before reduction by a whole number of days. The definition of the entries in a given column of the ephemerides involves subtracting a constant number of days, k, (\(k=6585\) for Column \(\varPhi , k=29\) for Column G, \(k=354\) for Column \(\varLambda \)). If the same constant is subtracted from each of the summands in both sums on each side of Eq. (1), then all these individual subtractions cancel when one sum is subtracted from the other.

  25. Ibid., p. 25.

  26. Aaboe (1971).

  27. Ibid., Tables 1 and 2, pp. 8–9.

  28. Britton (2009, Table 3.2, p. 378).

  29. Steele (2002).

  30. Ibid., p. 409.

  31. Ibid., the three quotes in this paragraph are from pages 414, 417, 406, respectively.

  32. Brack-Bernsen (1980, p. 46).

  33. Ibid., p. 45.

  34. Brack-Bernsen (1990, p. 41).

  35. The lunar four, ME, ŠU, GE\(_{6}\), NA, are time intervals measured just before and just after the full moon. ME is the time from moonrise to sunset on the last evening before the full moon, ŠU is the time from moonset to sunrise on the last day before the full moon, GE\(_{6}\) is the time from sunset to moonrise on the first evening after the full moon, and NA is the time from sunrise to moonset on the first day after the full moon. All four are measured in uš. ME and GE\(_{6}\) are measured on the eastern horizon, and ŠU and NA are measured on the western horizon. The combination of observational data just before and just after the full moon and observed at both east and west horizons has the effect of canceling the influence of all factors except the lunar anomaly.

  36. Ibid., p. 44, 46.

  37. Britton (2009).

  38. Britton (2007).

  39. Britton (2009, p. 386).

  40. Ibid. p. 386.

  41. Britton (2007, p. 125).

  42. Britton (1990, pp. 64–65).

  43. For example, determining if two numbers are congruent modulo 2\(\varDelta \).

  44. Britton (2009, p. 395).

  45. Personal conversation.

  46. Britton (2007, p. 139, Table C.2 235 months, row 7, columns 19,20 and row 14, column 11).

  47. Ibid., row 20, column 14.

  48. Ibid., row 27, columns 5 and 8.

  49. Ibid., Table 4.1, p. 109.

  50. Brak-Bernsen (1980, p. 46).

  51. Britton (2009, pp. 404–405).

  52. Ibid., p. 429.

  53. Ibid., p. 429.

  54. Ibid., p. 429.

  55. See footnote 24 regarding the reduction by 6585 days or 354 days.

  56. “mod X” means “up to integral multiples of the number X” even when X is not itself an integer.

References

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Correspondence to Steven Shnider.

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Communicated by : Alexander Jones.

In memory of John Phillips Britton, (1939–2010).

Appendix

Appendix

  1. 1.

    The calculation explaining \(223d_\varPhi = - {3/28} d_\varPhi = -{3/28} \times {2{;}45{,}55{,}33{,}20} = - {0{;}17{,}46{,}40}\).

    As shown in the text, the true period for Column \(\varPhi \) is the anomalistic month as a fraction of the synodic month, p \(=\) 0.933085. The Babylonians used the value 6247/6695. The tabular period is

    $$\begin{aligned} P=p/(1-p) = {6247/448}={13{;}56{,}39{,}6{,}26}. \end{aligned}$$

    The basic relation between amplitude, tabular difference and tabular period is \({P}_\varPhi = 2\varDelta _\varPhi /d_\varPhi \). Therefore,Footnote 55

    $$\begin{aligned} ({6247}/{448}) d_\varPhi= & {} 0 (mod \,{2}\varDelta _\varPhi ),\nonumber \\ (({223} \times {28}+ {3})/{448}) d_\varPhi= & {} (({223} + {3}/{28}) / {16}) d_\varPhi = {0}\, (mod\, {2} \varDelta _\varPhi ). \end{aligned}$$

    Multiplying by the whole number 16 preserves congruence modulo integer multiples of \({2}\varDelta _\varPhi \),

    $$\begin{aligned} ({223} + {{3/28}}) d_\varPhi= & {} 0 (mod\, {2}\varDelta _\varPhi ),\quad \hbox {and}\\ 223d_\varPhi= & {} - {{3/28}}d_\varPhi = -{3/28} \times {2{;}45{,}55{,}33{,}20} \\= & {} - {0{;}17{,}46{,}40} (mod\, 2\varDelta _\varPhi ) . \end{aligned}$$
  2. 2.

    The calculation explaining the meaning of Aaboe’s equation (3.3).

    The monthly increment in Column \(\varPhi \) is \(\hbox {d}_{\varPhi }=\pm 2{;}45{,}55{,}33{,}20\) in units of uš, or time degrees, and the amplitude is \(\varDelta _{\varPhi }= 19{;}16{,}51{,}6{,}40\) uš. In rows 20 to 51 of BM 36311, where both \(\varPhi \) values are in the ascending branch on the ephemerides tables, that is, the monthly increment is positive, \(d_\varPhi \) \(=\) \(+\)2;45,55,33,20:

    $$\begin{aligned} II(n) - I(n)= & {} \varPhi _{n}- \varPhi _{n-12} = {12} d_{\varPhi }- 2\varDelta _{\varPhi } \\= & {} {{33{;}11{,}6{,}40{,}0}}- {{38{;}33{,}42{,}13{,}20}} = -{{5{;}22{,}35{,}33{,}20}}. \end{aligned}$$

    In rows 75 to 105, the decreasing branch where \(d_{\varPhi } = - 2{;}45{,}55{,}33{,}20\):

    $$\begin{aligned} II(n) - I(n)= & {} \varPhi _{n}-\varPhi _{n-12} =12 d_{\varPhi } + 2 \varDelta _{\varPhi }\\= & {} -{33{;}11{,}6{,}40{,}0 + 38{;}33{,}42{,}13{,}20= 5{;}22{,}35{,}33{,}20}. \end{aligned}$$
  3. 3.

    The derivation of Aaboe’s equation (3.1) from the interval rule.

Define an index \({j=m + 223(n-1)}\), where the tabular index, denoted n, of BM 36311 is in units of one Saros, but the initial month of the table is month number \(m+1\) which may be any month of the Saros cycle. Thus, j is the serial number of the month corresponding to row n. The columns are described as follows:

$$\begin{aligned} I(n) = (223M)_{j -12,}\quad II(n) = (223M)_{j}, \quad {III(n) = (12M)}_{j-12.} \end{aligned}$$

The entry I(n) in Column I represents the length of 223 months in excess of 6585 days, beginning at month \(m-11\) of Saros cycle n, II(n) represents the length of 223 months in excess of 6585 days, beginning at month \(m+1\) of Saros cycle n,  and III(n) represents the length in excess of 354 days of 12 months ending with month m in Saros cycle n, all giving the component dependent on the lunar anomaly alone. Note that II(n) + III(n) is the length in excess of 6939 days of 235 months (i.e., a Metonic period) beginning at month \(m -11\) in Saros n.

Substituting for the expressions on the two sides of equation (3.1)Footnote 56

$$\begin{aligned} II(n) - I(n)= & {} (223M)_{j }- (223M)_{j-12 }=d_{12}(223M)_{j-12,}\\ III(n+1) - III(n)= & {} (12M)_{j+211 }- (12M)_{j-12 }=d_{223}(12M) _{j-12}. \end{aligned}$$

Their equality is the particular case of the interval rule, \({k=12, n=223}\),

$$\begin{aligned} d_{{12}}({223}M)_{j-{12}}=d_{{223}}({12}M) _{j{{-12}}}. \end{aligned}$$

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Shnider, S. Britton’s theory of the creation of Column \(\varPhi \) in Babylonian System A lunar theory. Arch. Hist. Exact Sci. 71, 279–318 (2017). https://doi.org/10.1007/s00407-017-0189-4

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