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Cracking bones and numbers: solving the enigma of numerical sequences on ancient Chinese artifacts

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Abstract

Numerous recent discoveries in China of ancient tombs have greatly increased our knowledge of ritual and religious practices. These discoveries include excavated oracle bones, bronze, jade, stone and pottery objects, and bamboo manuscripts dating from the twelfth to fourth century BCE. Inscribed upon these artifacts are a large number of records of numerical sequences, for which no explanation has been found of how they were produced. Structural links to the Book of Changes, a divination manual that entered the Confucian canon, are evident; yet, the algorithm described therein dates to the slightly later second to first century BCE. By combining archeological and statistical evidence, we propose a new methodology that enables us to reconstruct and test cleromantic techniques which can explain how these numerical sequences were generated. Dice and divination stalk use, either in combination or separately, appear in fact to have been underlying the rather stable numerical patterns in ancient China all the way back to the late Shang dynasty (1300–1046 BCE). Bringing to light such a long-standing technique, which awaits further confirmation from the ever-growing database of newly discovered numerical and textual records, can change drastically our understanding of early Chinese history and of the historical development of sophisticated arithmetical practices and the rationalization of chance.

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Fig. 1
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Adapted from Xiao (1989, 67, 69, figs. 1 and 2)

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Notes

  1. Cleromantic divination refers to the casting of lots or die in order to predict events. Although the outcomes of such techniques can be considered mathematically as a purely random events, they implied some sort of spiritual agency in various cultural traditions.

  2. In the early history of Chinese divination, stalk sortilege, i.e., divination by dividing milfoil stalks (shi 筮), was one of the two main cleromantic techniques. It had gradually replaced osteomancy, a pyromantic technique reading the cracks in oracle bones (bu 卜).

  3. The section on the Da Yan 大衍 stalk sortilege method was codified, and in our eyes normalized, by Han scholars for producing the Yijing hexagrams in the transmitted “Xici” 繫辭 commentary to the Yijing. It does not appear in the second century BCE version of the “Xici,” our earliest preserved version, but a shorter “Xici” was found together with a version of the Yijing in the Mawangdui burial dated to 168 BCE; see Shaughnessy (1996), 21. For other versions, see Peterson (1982). Unfortunately, the latter does not note the history of the appearance of the section on the Da Yan method. He suspects that the text was intact by the first century BCE (77).

  4. Wang (2013, 21–24).

  5. For recent examples, see Zhang (2010) and Chen and Jinmin (2012). A useful historical review of the debates over whether or not these numerical sets can be read against the Zhouyi (and whether or not there are various systems of reading them) can be found in Wang and Zhou (2015, 2–19). See especially Zhang (1980–1981), Rao (1989), Cao (2002b), Pu (2011, 187–197) and Jia (2014a).

  6. The character for a 7 was graphically very close to the 1 .

  7. For a discussion on the divination method described in the Jingjue manuscript and its interpretation as a manual, i.e., a text that provides all possible results of the technique, see Bréard and Cook (2020).

  8. Wang and Zhou (2015). A survey of scholarship concerning the debate over different systems and of the available evidence is also found in Bian (2016).

  9. See, for example, the curious late Shang “whetstone” found in 1982 in a tomb near Anyang in Miaopu 苗圃. It has six different sets of numerical gua inscribed on different sides and edges: 766667, 768767, 665768, 811166, 811116, 667668. See Zhongguo shehui kexueyuan kaogu yanjiusuo Anyang gongzuodui (1986, 117) and Zheng (1986, 49, figs. 2, 3, 4).

  10. See item M80:7 from the northern area of the site Miaopu in Yinxu. Adapted from, shown in Zhongguo shehui kexueyuan kaogu yanjiusuo Anyang gongzuodui (1986, 118, fig. 11).

  11. The plastron, i.e., the nearly flat part of the shell structure of a turtle, is divided into several plates, with the hypoplastron designating the four biggest plates.

  12. Xiao (1989).

  13. Allan (1991).

  14. See Cao (1989, 638, fig. 1).

  15. A gui in ancient China could indicate the presence of a human or other demonic spirit. A gua was also itself a spirit.

  16. Cao (2002b, 65).

  17. Cao (2003). See also (Bréard and Cook 2020).

  18. See for example Zhongguo shehui kexueyuan kaogu yanjiusuo (2007, 22, fig. 158).

  19. A complication of this view is why both, 6s and 8s, presumably representing the same broken or even number value, would appear in the same gua. One explanation could lie in their graphic proximity.

  20. See Cao (2002a) and Liao (2003, 13).

  21. Liu (2016, 66, figs. 1 and 2). See the discussion in Dotson (2019). For a discussion of the earliest Khoten dice, see Nishi (2008).

  22. Tomb 2 in Ba Dong Kongbao 巴東孔包, from Sichuan wenwu 6 (2003, 6, fig. 6), reproduced in Liu (2016, 66, fig. 2).

  23. In his coffin were placed legal and divinatory texts, including Qin statutes, a daybook, and a hexagram text now known as the Guicang. A Liu Bo board was also found. The tomb dates to after 287 BCE but before 221 BCE. For the now lost twenty-three wooden dice, see Jingzhou (1995, 42, fig. 14).

  24. The term “hemerology” designates “calendar-based texts that record lucky and unlucky times and places for specified activities” Harper and Kalinowski (2017, 2).

  25. “Plastromancy” designates divination based upon patterns on a prepared and burned turtle plastron that was used as an oracle bone.

  26. For the Luoyang clay tureen decorated with numerical gua and a hunting or battle scene, see Li (2017).

  27. The jar was discovered about 3 meters underground during roadwork. A bit of broken sward was found nearby. See fig. 2 in Yao (1990).

  28. Yao (1990, 56, fig. 3).

  29. Mentioned in the Zuozhuan Zhao 12 and described in the Liji, Shiji, and Hou Hanshu. The Shiji “Huaji liezhuan 滑稽列傳” 4 notes the playing of Liu Bo and Tou Hu together.

  30. Sun (1989, juan 56, Chapter 40, 1383–1397). The alternation between two drum sounds, visually recorded as a dyadic system adds another possible dynamic to divination calculation, that is a relationship to music. For more details, see Bréard and Cook (2020).

  31. For early China, see Li (2010, 15–20). For a general overview, see Bréard (2020).

  32. Our database was constructed on the basis of Bian (2016) with minor emendations. For details on the kind of changes we introduced, see next page.

  33. See Li (2006, 213).

  34. See Bian (2016, 16).

  35. See Cao (2002b, 154–155).

  36. On the Great Commentary (Dazhuan 大傳) see Peterson (1982). The original statement is 大衍之數五十,其用四十有九。分而為二以象兩,掛一以象三,揲之以四以象四時,歸奇於扐以象閏。五歲再閏,故再扐而後掛. This statement follows a line explaining that numbers 1, 3, 5, 7 and 9 represent Heaven and 2, 4, 6, 8 and 10 represent Earth and that changes in their combinations represent the movement of spirits and ghosts.

  37. See Smith (2008, 26–27).

  38. All sixteen possible outcomes are listed in the manuscript. See Bréard and Cook (2020).

  39. See for example Ma (2014, 64).

  40. For an introduction, edition and translation to this text, see Cook and Zhao (2017).

  41. See Röllicke (1999, 24–27) and Lewis (2002).

  42. See Shandong (2014, 27).

  43. Since the polyhedra were irregular, the observed frequencies of course misrepresented slightly the probabilities of the numbers that arise when three or two dice are used.

  44. For an introduction and a full translation of the Guicang, see Shaughnessy (2014, 141–187).

  45. Liu (2014, 26–28) for example suggests that model A5 with either 58, 57 or 56 stalks underlies the numbers recorded in the Shifa manuscript. See Table 12, calculated with precise probabilities and thus slightly different from the table in Liu (2014, 28), for the obvious misfit of these models.

  46. For the mathematical model of the stalk procedure, it is actually not necessary to consider all 110 possibilities of division, since the situation is symmetric with respect to removing one stalk from either the right or the left bunch.

  47. Mathematically, we know that generally, a procedure An+1 with \( Y \) stalks, where \( Y \equiv 3 \left( {mod \, 4} \right) \), leads to the same numerical results with the same probabilities as a procedure An with \( Y - 5 \) stalks.

  48. See Chen (1998, 42–44). Assuming 55 as the number of stalks correlates with the total of (odd) heavenly numbers (1 + 3 + 5 + 7 + 9 = 25) and (even) earthly numbers (2 + 4 + 6 + 8 = 20).

  49. See Bray and Curtis (1957, 327–332) for the original discussion of this statistic.

  50. See Liu (2014, 28), Jia (2014b, 200–201) and Cheng (2014, 63).

  51. See Table 12 for the misfits of these models. For the model of Jia (2014b), see the previous footnote. Cheng (2014) proposes a model equivalent to A5 with 56 stalks Liu (2014) considers A5 with 56, 57 and 58 stalks. Both calculate only with the approximate probabilities (25%, 50%, 75%) for each iteration instead of the precise probabilities as shown in Table 7. The cumulative difference is not insignificant. For example, they obtain the probabilities 4.68%, 20.31%, 34.37%, 28.12%, 10.93% and 1.56% for obtaining the numbers 4, 5, 6, 7, 8 and 9 in case of A5 with 56 stalks.

  52. See Bréard and Cook (2020).

  53. See Cook (2006, 101, 105–107) and Kalinowski (2009, 371).

  54. See footnote 7.

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Acknowledgements

The authors are grateful to Professor Michael Lackner and his colleagues at the IKGF (International Consortium for Research in the Humanities, Universität Erlangen-Nürnberg, Germany) who provided us with a fellowship and their insight and expertise that greatly assisted our collaborative research. Cook also thanks the Institute for Advanced Study, Princeton, for support.

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  1. Faculté des Sciences d’Orsay, Université Paris-Sud, Bâtiment 407 – Rue du Doyen Georges Poitou, 91400, Orsay, France

    Andrea Bréard

  2. Department of Modern Languages and Literatures, Lehigh University, 31 Williams Hall, Bethlehem, PA, 18015, USA

    Constance A. Cook

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Bréard, A., Cook, C.A. Cracking bones and numbers: solving the enigma of numerical sequences on ancient Chinese artifacts. Arch. Hist. Exact Sci. 74, 313–343 (2020). https://doi.org/10.1007/s00407-019-00245-9

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