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Visual Representations of Arithmetical Operations Performed with Counting Instruments in Chinese Mathematical Treatises

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Researching the History of Mathematics Education

Part of the book series: ICME-13 Monographs ((ICME13Mo))

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Abstract

The chapter is devoted to visual representations of two counting instruments, the counting rods (suanzi 算子, chousuan 籌算) and the abacus (suanpan 算盤) , found in Chinese mathematical treatises compiled prior to the early 17th century. The former instrument is generally believed to be replaced by the latter in the mid-second millennium AD. The role of the visual representations of these two instruments in didactical practices has never been given due attention by historians of mathematics education. The author provides a preliminary analysis of the images of the instruments in two mathematical textbooks, the Shendao dabian lizong suanhui 神道大編曆宗算會 compiled in 1558 by Zhou Shuxue 周述學 and Panzhu suanfa 盤珠算法 completed in 1573 by Xu Xinlu 徐心魯 and conjectures that these images were used mainly for didactical purposes.

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Notes

  1. 1.

    For brief descriptions of the instruments see, for example, Martzloff (1997), Jami (1998), and Volkov (1998, 2017).

  2. 2.

    For a description of mathematics education in China, Korea, and Japan prior to 1500 see Volkov (2014).

  3. 3.

    Only a very short description of the instrument is found in the Shu shu ji yi 數術記遺 (Procedures of “numbering” recorded to be preserved [for posterity]) compiled in the 2nd century AD, see Guo and Liu (2001, p. 448); for descriptions of the counting rods found in texts not directly related to mathematics, see Li (1929) and Volkov (2017).

  4. 4.

    See, for example, a description found in the Sun zi suan jing 孫子算經, see Guo and Liu (2001, p. 262); for translations, see Berezkina (1963, p. 23), Lam and Ang (2004, p. 193).

  5. 5.

    For a hypothesis concerning the author of the treatuse see Volkov (2012, p. 517, note a).

  6. 6.

    See Guo and Liu (2001, p. 262); for translations and interpretations, see Berezkina (1963, pp. 23–24), Lam and Ang (2004, pp. 193–195).

  7. 7.

    For the history and contents of the treatise see Berezkina (1957), Kawahara (1980 ), Chemla and Guo (2004), and Dauben et al. (2013).

  8. 8.

    See Guo and Liu (2001, pp. 121–125, 173–183). For translations, see Chemla and Guo (2004, pp. 361–368, 371–385, 617–659), Dauben et al. (2013, pp. 373–409, 415–455, 905–1033).

  9. 9.

    On this College see, for example, Volkov (2014, pp. 66–68); on the history of the College and its functioning in the first millennium AD see Volkov (2012, pp. 514–523).

  10. 10.

    See Qin ([1247] 1993). For a comprehensive study of this treatise see Libbrecht (1973 ) .

  11. 11.

    See, for example, Li (1999, pp. 483–485); for a systematic study of the treatise see Yang (2003 ).

  12. 12.

    For translations and discussions of the latter procedure see Berezkina (1963, pp. 23–24), Lam and Ang (2004, pp. 194–195).

  13. 13.

    This reconstruction has one obvious weak point: the operations necessary to conduct it appear to be more complex than those used at the following steps for the division itself.

  14. 14.

    The digit “3” in the upper row was missing in the copy of the treatise used for this translation.

  15. 15.

    Once again, the actual process used to test the digit 4 is not known.

  16. 16.

    Han (1993, p. 1141).

  17. 17.

    Kodama (1970) on p. 23 reproduces two pages of each of these two versions.

  18. 18.

    Li (1999, p. 486, n.1) mentions this copy of the treatise.

  19. 19.

    My preliminary investigation of the versions Xu ([1573a] 1970/1994) and Xu ([1573b] 1993) suggests that they were printed with the same printing blocks, that is, that they are textually identical. However, their full titles differ, which means that the title on the wooden block used for printing the first page was re-carved.

  20. 20.

    The whereabouts of the copy reproduced in Guo (1993) [that is, (Xu [1573b] 1993)] are not specified in the introduction by Han (1993) (even though Han Qi mentions that a copy of it is preserved in the Cabinet Library, the copy Xu ([1573b] 1993) does not have the seal of this library on page 1b, unlike the copy Xu ([1573a] 1970). It is known that one more copy of the treatise is preserved in the Institute for Research for Humanities, Kyoto University (Japan), but this copy remains unavailable to me.

  21. 21.

    This use of colors is repeated throughout the entire first chapter: in the diagrams placed in the upper sections of pp. 1b, 2b, 3b, 4b, 5b, 6b, 7b, 8b, 12a–b, 13a–b, 14a–b, 15a–b, 16a–b, 17a–b, 18a–b, 19a–b the “active” beads are white, while in the diagrams in the upper sections of pp. 2a, 3a, 4a, 5a, 6a, 7a, 8a, 9a the “active” beads are black.

  22. 22.

    In Tables 1, 2, 3 “(10)” means that one upper bead and five lower beads are “active”; “(5)” means that the five is represented with five active beads in the lower section, while “5” means that five is represented as one active bead in the upper section.

  23. 23.

    The number 1111111101 is not explicitly mentioned in the treatise; it can be restored as the sum of the amount set on the abacus (9876(5)4312) plus 123456789. See also Chen (2013, p. 25).

  24. 24.

    See, for example, the opening part of the Jiuzhang suanfa bilei daquan九章算法比類大全 by Wu Jing 吳敬 (fl. 1450) (Wu [1450] 1993, p. 16); for a translation and discussion see Chen (2013, pp. 54–58).

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  1. National Tsing Hua University, Hsinchu, Taiwan

    Alexei Volkov

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  1. DIMA—Dipartimento di Matematica, Genoa University, Genoa, Italy

    Fulvia Furinghetti

  2. Teachers College, Columbia University, New York, New York, USA

    Alexander Karp

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Volkov, A. (2018). Visual Representations of Arithmetical Operations Performed with Counting Instruments in Chinese Mathematical Treatises. In: Furinghetti, F., Karp, A. (eds) Researching the History of Mathematics Education. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-68294-5_14

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