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Scholarly Mathematics in the Rēš Temple

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Scholars and Scholarship in Late Babylonian Uruk

Part of the book series: Why the Sciences of the Ancient World Matter ((WSAWM,volume 2))

Abstract

The Rēš temple in Uruk was home to a community of scholars who pursued various mathematical topics. In this contribution, the evidence for scholarly mathematics in the Rēš is compiled and investigated. The archival and institutional contexts of scholarly mathematics in the Rēš are briefly sketched. A new analysis of three mathematical tablets is presented. Possible connections between the mathematical tablets from the Rēš and those from earlier libraries in Uruk are explored.

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Notes

  1. 1.

    Perhaps a consequence of measures taken by Xerxes in the wake of two uprisings against his reign (Waerzeggers 2003/4). For the history of the Eanna archive see also van Driel (1998); Pedersén (1998: 205–206); Jursa (2005).

  2. 2.

    For this library see Hoh (1979), Farber (1987), Frahm (2002), Robson (2008: 237–240), Ossendrijver (forthcoming). On one astrological tablet, SpTU 1 94, Iqîšâ identifies himself as an ‘enterer of the temple of Anu and Antu’ (ēreb bīt Anu u Antu); for this priestly title see Funk (1984: 106–168), Waerzeggers (2010: 46).

  3. 3.

    For instance, W 22656/02 (SpTU, 4 178), a building ritual, see Clancier (2009: 404). The tablet remains unedited.

  4. 4.

    For these tablets see the contribution by Proust in the present volume.

  5. 5.

    W 22260a (SpTU 1, 101) and W 22309a + b (SpTU 1, 102); see the comments in Steele (2015).

  6. 6.

    W 23016 (SpTU 5, 316); for an edition see Friberg (2007: 453).

  7. 7.

    W 22661/3a(+)b (SpTU 5, 317).

  8. 8.

    See Clancier (2009: 404).

  9. 9.

    Frahm (2002).

  10. 10.

    For the tablets with mathematical astronomy see Neugebauer (1955) and Ossendrijver (2012); for the microzodiac texts and calendar texts see Weidner (1967).

  11. 11.

    For the horoscopes see Rochberg (1998); for the planetary compilation see Steele (2000).

  12. 12.

    See the group of tablets labeled W 186 in Kose (1998: 469) and Lindström (2003: 211). Most of these are kept in Istanbul (siglum: U), some in Berlin (VAT). As far as known they include 79 tablets with mathematical astronomy .

  13. 13.

    According to information from the archives of the Vorderasiatisches Museum provided by J. Marzahn (3 November 2016) these tablets were acquired in Baghdad from the dealer Chajjat (also spelled Khayat) at an unknown date. Further unprovenanced scholarly tablets from the Rēš that were acquired through the antiquities market are kept in Chicago (siglum: A) and Yale (siglum: MLC); they include 29 tablets with mathematical astronomy (Neugebauer 1955; Ossendrijver 2012).

  14. 14.

    Clancier (2009: 86–90).

  15. 15.

    For this library see Pedersén (1998: 209–210).

  16. 16.

    U 109 + 114 + VAT 9154 (Lindström 2003: 217), a template table of lunar system B (Neugebauer 1955: No. 171). It is not clear which of the three fragments were found in court VII.

  17. 17.

    The cuneiform sign that indicates vanishing digits (0) was only introduced in the fifth century BCE.

  18. 18.

    Ossendrijver (2011a, b).

  19. 19.

    Thureau-Dangin (1922: No. 31).

  20. 20.

    See Allotte de la Füye (1932), Cazalas (1932), Neugebauer (1932, 1935: 14–22).

  21. 21.

    See Bruins (1970), Knuth (1972), Friberg (1986), Proust (2000), Friberg (2005: 294–298), Friberg and al-Rawi (2016: 43–49).

  22. 22.

    This differs from the usual arrangement of snaking numerical columns on astronomical tables, which proceed from left to right on the obverse, but from right to left on the reverse.

  23. 23.

    The original version of the numeral 9 analogously consists of three rows of three units, but by ca. 450 BCE a distinct sign for the 9 consisting of three diagonally arranged small wedges was introduced.

  24. 24.

    Two exceptions occur in obv. ii.5 (20.16) and obv. ii.32 (20.12).

  25. 25.

    Neugebauer (1932); see also Cazalas (1932), Neugebauer (1934), Friberg (2005: 292–293), Friberg and al-Rawi (2016: 47).

  26. 26.

    This is because any change from (p, q, r) to (p + 2 k, q + k, r + k), where k is an integer, amounts to a multiplication by 60k. Hence there are infinitely many combinations of p, q and r that result in the same grid point, corresponding to the same sequence of sexagesimal digits in the floating notation .

  27. 27.

    This final entry must probably be viewed as the incipit of a following tablet.

  28. 28.

    Friberg (1986: 84).

  29. 29.

    Neugebauer (1932: 199–200).

  30. 30.

    There are only five instances where an open circle is located in between two filled circles on a straight line. They interrupt four linear sequences of filled circles oriented along the p axis. All other filled circles are part of uninterrupted sequences. Even the number in the lower right corner, (0, −22, 0), ends a complete sequence of eight numbers extending all the way to (0, 19, 0) in the upper left corner. The fact that the interruptions occur along the p direction suggests that the numbers to the right and left of them were not computed by doubling and halving, respectively.

  31. 31.

    Depending on the factors used, the stepsize along the path may comprise several units of p, q or r.

  32. 32.

    See Proust (2000), Friberg and al-Rawi (2016: 44–45).

  33. 33.

    The digits 6, 7 and 8 can be misread as 3, 4 and 5, respectively, or vice versa, if one row of vertical wedges is overlooked or spuriously added.

  34. 34.

    The errors in obv. ii 32, rev. ii 1, 2 were first reported by de la Füye (1932), then also by Neugebauer (1935: 23).

  35. 35.

    See also Friberg and al-Rawi (2016: 44–45).

  36. 36.

    I owe this suggestion to an oral presentation by Christopher Woods at the Rencontre Assyriologique Internationale, Philadelphia 2016.

  37. 37.

    Note that the 0 that is missing from n (rev. ii 1) occurs at about the same position as the identical sign : that separates n from \( \bar{n} \) in the surrounding rows. Furthermore, the immediately following digit 29 coincides with the initial digit of \( \bar{n} \) in this part of the table. The resulting confusion may explain the omission of the 0.

  38. 38.

    In fact, the version of n written in obv. ii 32 is irregular, so it does not have a reciprocal in the Babylonian sense.

  39. 39.

    See Knuth (1972: 676).

  40. 40.

    If the 34 in n (obv. ii 32) is not corrected to 44 before multiplying by 20 then digits 14–15 of the product would equal 51.28 instead of 54.48. In that case the presence of the correct sequence 54.48 would be difficult to explain.

  41. 41.

    Bruins (1970).

  42. 42.

    For a reconstruction of the complete Standard Table, or ‘First Tablet’, see Friberg (2005: 294–305), and Friberg (2005: 461–462). A very similar reconstruction was presented by Britton (1991–3: 76–77), differing only in that it lacks entry No. 1 from Friberg’s table. New textual finds have confirmed Friberg’s version (Ossendrijver 2014: 156–160).

  43. 43.

    See Friberg and al-Rawi (2016: 26–37) for an edition of SpTU 4 174 and Friberg and al-Rawi (2016: 28–42) for a table of reciprocals from the Neo Babylonian library of the Ebabbar in Sippar in which the initial digit of n ranges from 1 to 3. The only other published Late Babylonian table of reciprocals of numbers n with an initial digit beyond 1 is BM 41101 (Aaboe 1965), probably from Babylon . Since it only preserves numbers n having initial digit 4 it is of no concern here.

  44. 44.

    The lacking entries are (17, 18, 0), (8, 10, 0), (14, 10, 0), (7, 7, 0), and (19, 0, 10). All but one of them concern numbers n or \( \bar{n} \) having at least six digits, but otherwise no pattern could be identified among them.

  45. 45.

    This tablet was owned by Nidinti-Anu of the Ekur-zākir clan. For an edition see Koch (2005: 297–312).

  46. 46.

    For an edition see Brack-Bernsen and Hunger (2002).

  47. 47.

    See Ossendrijver (2012: 25).

  48. 48.

    Ossendrijver (2012: No. 67).

  49. 49.

    E.g. A 3405 (Steele 2000), A 3406 + U 147 + 160 (ACT 186), A 3409 (ACT 800a), and A 3415 (ACT 400).

  50. 50.

    Neugebauer and Sachs (1945: Text Y).

  51. 51.

    Friberg (1997: 302–304, 324–325).

  52. 52.

    Ossendrijver (2012: 600).

  53. 53.

    AO 6477 (Ossendrijver 2012: No. 42) and A 3413 (Ossendrijver 2012: No. 93).

  54. 54.

    Friberg (1997: 302, 324).

  55. 55.

    Neugebauer and Sachs (1945: 141, fn. 328a).

  56. 56.

    Friberg (1997: 302–304; 324–325).

  57. 57.

    The absolute interpretation of 45 as 45,0 and of all other numbers in this problem is implied by steps 1–6.

  58. 58.

    For this algorithm, see Friberg (1997: 316–334).

  59. 59.

    Friberg (1997: 285–286).

  60. 60.

    Powell (1987–1990: 483–484); Friberg (1997: 277, 292–3, 301).

  61. 61.

    Neugebauer and Sachs (1945: 143–144).

  62. 62.

    For the Late Babylonian mathematical terminology see Ossendrijver (2012: 26–27).

  63. 63.

    Aaboe (1964: 26–28); Friberg (1997: 302–304).

  64. 64.

    Robson (1999: 93–110).

  65. 65.

    Powell (1987–1987: 428–429).

  66. 66.

    Neugebauer (1935: 103).

  67. 67.

    For the archaeological context of U 91 see Lindström (2003: 211, 220). The findspot of the fragment W 169 (Warka X 45) appears to be unknown, because it is neither listed in Lindström (2003) nor in Kose (1998).

  68. 68.

    Aaboe (1968/69).

  69. 69.

    Aaboe (1999).

  70. 70.

    Ossendrijver (2014: Text I).

  71. 71.

    Some of these principle numbers are only preserved on the duplicate from Babylon (Aaboe 1999).

  72. 72.

    See Ossendrijver (2012: 69, 71, 82, 103).

  73. 73.

    While 3.30 only occurs at the end of regular numbers , the sequences 3.31–3.39 do occur in the interior of such numbers.

  74. 74.

    For the final digits of regular numbers see Bruins (1970: 99–104), Friberg (1999: 148–155).

  75. 75.

    Thureau-Dangin (1922: No. 33); Neugebauer (1935: 96–107), Høyrup (2002: 390–391); Friberg (2007: 323, 351, 444, 458); Robson (2007: 175–177).

  76. 76.

    Huber (1957: 279–281).

  77. 77.

    For these problems see also Friberg (2007: 458).

  78. 78.

    George (1992: 109–119, 414–434, pls. 24–25, No. 13).

  79. 79.

    Robson (2007: 177).

  80. 80.

    Kose (1998: 469, No. 653); Lindström (2003: 212).

  81. 81.

    Van Dijk (1980: No. 96); George (1992: 199).

  82. 82.

    Lindström (2003: 214).

  83. 83.

    Van Dijk (1980: No. 87). An edition by E. Robson is available on the Oracc website (http://oracc.org/cams/gkab/P363351).

  84. 84.

    Ossendrijver (2011a, b).

  85. 85.

    Ossendrijver (2012: 19–27).

  86. 86.

    Steele (2016); Ossendrijver (Forthcoming).

  87. 87.

    Ossendrijver (2014: 149–150).

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Acknowledgements

The research leading to these results has received funding from the Excellence Cluster TOPOI (DFG Grant EXC264) and the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 269804.

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  1. Institute of Philosophy, Humboldt University of Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Mathieu Ossendrijver

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  1. Laboratoire SPHERE UMR 7219, Université Paris Diderot—CNRS, Paris, France

    Christine Proust

  2. Department of Egyptology and Assyriology, Brown University, Providence, RI, USA

    John Steele

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Ossendrijver, M. (2019). Scholarly Mathematics in the Rēš Temple. In: Proust, C., Steele, J. (eds) Scholars and Scholarship in Late Babylonian Uruk. Why the Sciences of the Ancient World Matter, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-04176-2_6

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