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Mathematics Education in Oriental Antiquity and Middle Ages

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  • Mathematical Treatise
  • Song Dynasty
  • Tang Dynasty
  • Twelfth Century
  • Mathematical Text

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Fig. 4.1

Notes

  1. 1.

    In this chapter, the pinyin transliteration system adopted in P.R. of China and European sinology is used for the ancient and medieval Chinese personal names, terms, and titles of treatises. However, the reader should be aware that the reading of Chinese characters went through considerable changes during the period from the late first millennium BC to ca. AD 1500, which render the pinyin transliteration a pure convention used solely to facilitate referring to names and titles and irrelevant to their actual historical pronunciation. The names of Chinese authors from Hong Kong and Taiwan are transliterated according to the transliteration systems adopted by the authors themselves. In all cases, I used traditional Chinese characters for personal names and titles of publications and provided their simplified versions currently used in the P.R. of China only when they were originally used by the authors.

  2. 2.

    The term jiu shu could be understood as “nine numbers,” yet its comparison with the other “arts” such as, for example, wu she 五射 (five [styles of] archery) and liu shu 六書 (six [styles of] writing), suggests that the word shu 數 here most likely meant “operations with numbers” or “numerical procedures”; see below on alternative interpretations.

  3. 3.

    The term jiu zhang 九章 that appears in the titles of a number of Chinese mathematical treatises, including that of the extant editions of the Jiu zhang suan shu, is usually rendered in the Western historiography as “nine chapters,” resulting in such translations as The Nine Chapters on the Mathematical Art. For a discussion of the meaning of this term, see Volkov 2010, p. 281, n. 1.

  4. 4.

    The text literally reads “six years,” yet traditional Chinese age count starts from the moment of conception.

  5. 5.

    Li ji zhu shu 禮記注疏 (Book of Rites, with Commentaries and Explanations), Siku quanshu 四庫全書 edition, juan 28, p. 27b.

  6. 6.

    Cullen (2007, p. 34) strongly doubted the degree of Liu Hui’s awareness of the actual early textual history of the treatise.

  7. 7.

    In this chapter I use the character suan 筭 instead of suan 算 in the titles of a treatise or the names of institutions if in at least one source the former character was used instead of the latter. The original meanings of these two characters were not the same: according to the etymological dictionary Shuo wen jie zi 說文解字 (Explaining [Written] Signs and Analyzing [Compound] Characters) by Xu Shen 許慎 (AD 55?–149?), the character suan 筭 meant “counting rods,” while suan 算 meant “operations performed with the counting rods,” that is, “computations.”

  8. 8.

    The treatise was published for the first time in 2000 (SSS 2000) and later reproduced photographically (SSS 2001, 2006); it was translated into Japanese (Jochi 2001; Ohkawa et al. 2006), modern Chinese (Horng et al. 2006), and English (Cullen 2004; Dauben 2008). For references to publications devoted to the Suan shu shu, see Ohkawa et al. 2006, pp. 168–169; Dauben 2008, pp. 172–177; and Zou 2008, pp. 95–98.

  9. 9.

    A short excerpt from the Suan shu 算術 was published in 2008 (SS 2008) and studied by Karine Chemla and Ma Biao (2011).

  10. 10.

    The original Chinese text reads “nine nine eighty one eight nine seventy two…”; the symbols of multiplication and equality are added for the convenience of the modern reader.

  11. 11.

    Sun 2000, p. 138; see also Lee 2000, p. 515, n. 230. Even though Lee did not find any evidence supporting this statement of Sun, he agrees that mathematics was systematically taught in state-run institutions prior to the Northern Wei.

  12. 12.

    For biographical data of Zhen Luan, see Volkov (1994).

  13. 13.

    JTS 1975, p. 2036. This treatise in one volume, commented by Zhen Luan, was apparently different from the treatise bearing the same title but subdivided into two volumes and listed on the same page as compiled by Li Chunfeng李淳風 (602–670).

  14. 14.

    JTS 1975, p. 2039. Zhen Luan is mentioned as the compiler of the treatise.

  15. 15.

    JTS 1975, p. 2039. Zhen Luan is mentioned as the compiler of both texts. Some authors believe that the catalog contains a scribal error and Zhen Luan was not the compiler of the three-volume treatise; see JTS 1975, p. 2083, n. 6. One cannot rule out the possibility that this entry resulted from an error made by a copyist when writing the title of another treatise presumably authored by Zhen Luan, the Wu jing suan shu 五經筭術.

  16. 16.

    In JTS 1975 this treatise is mentioned as compiled by Liu Hui.

  17. 17.

    Different sources provide different figures; see Li 1933 [1977], p. 255. The title boshi literally means “serviceman (shi) of broad [knowledge] (bo),” hence “erudite,” as Hucker (1988, p. 389, no. 4746) suggests.

  18. 18.

    Li 1933 [1977]. This number looks problematic, especially when compared with the number of students enrolled in this program during the Tang dynasty. One cannot rule out the possibility that the number was miswritten by a copyist (who, e.g., wrote 八十, “80”, instead of 十八, “18”) or that 80 was no more than a projected number of students, while the actual number was smaller.

  19. 19.

    Li 1933 [1977], p. 261. Des Rotours (1932) also states that mathematics examinations must have been conducted during the period 627–649 (p. 28) and conjectures that they must have started around 629–632 (p. 129, n. 1). Hucker (1988, p. 461, no. 5856), who claimed that “Tang did not duplicate the Sui school until 657,” apparently did not take into consideration the works of Li Yan and des Rotours.

  20. 20.

    Mentioned as “14 to 19” in Chinese sources, see des Rotours 1932, p. 136.

  21. 21.

    Modern authors use various renderings of the name of this institution: Hucker (1988, p. 299) suggests “Directorate of Education” while Lee (2000, passim) prefers “Directorate of National Youth.”

  22. 22.

    Martzloff (1997) on p. 123 erroneously claims that editorial work was done during the period 618–627; on p. 125 he contradicts himself when saying, equally erroneously, that this work was carried out from 644 to 648.

  23. 23.

    With perhaps only one exception: a copy of the treatise Shu shu ji yi was found by Bao Huanzhi 鮑澣之 (fl. ca. 1200) in a Daoist monastery; its thirteenth-century printed edition, unlike other extant treatises, does not contain an opening part with the names of the Tang dynasty editors and therefore may have been based on either an incomplete copy of the Tang edition or even on a version based on a pre-Tang edition (Volkov 1994).

  24. 24.

    The lifetime of Li Ji is unknown; Guo Shuchun argues that Li must have been active after 712 and no later than the early ninth century; see Guo 1989, pp. 198–199.

  25. 25.

    The descriptions are found in the Tang liu dian 唐六典 (Six Codes of the Tang [Dynasty]), completed in 738, in the Jiu Tang shu, and in the Xin Tang shu; see TLD 1983, Chap. 21, p. 10b; JTS 1975, vol. 6, p. 1892; XTS 1956, Chap. 44, p. 2a.

  26. 26.

    The Jiu Tang shu provides only the titles of the textbooks but not the duration of their study. For a translation of the description found in the Xin Tang shu, see des Rotours 1932, pp. 139–142, 154–155; see also Siu 1995, p. 226, Siu and Volkov 1999.

  27. 27.

    Hucker (1988, p. 461, no. 5856) is apparently wrong when claiming that in 657 the “prescribed student enrollment was set at only 10.” He probably was misled by the quota of students, 10, adopted in the early ninth century; see below.

  28. 28.

    No specific names of the programs are provided in the original documents; the adjectives “regular,” “advanced,” and “compulsory” are added on the basis of the contents of the programs (Siu and Volkov 1999 Volkov 2012a).

  29. 29.

    See Lee 2000, p. 138, for a slightly different rendering.

  30. 30.

    See des Rotours 1932, p. 128, n. 1 for a description of the origin of the candidates.

  31. 31.

    One cannot help but notice that the curriculum of the school corresponded to the doctoral exams, formally open for even those candidates who did not graduate from the school, such as “provincial candidates.” It remains unknown whether instruction in the provinces was conducted on the basis of the same textbooks or whether the degree examination was designed in this way to give advantage to the graduates of the school.

  32. 32.

    Some restrictions applied in the advanced program; see des Rotours 1932, p. 155, n. 2.

  33. 33.

    This hypothesis was advanced by Siu and Volkov (1999) and amply illustrated in Siu (1999, 2004, pp. 174–177).

  34. 34.

    A piece of evidence supporting this hypothesis was found in a Vietnamese mathematical treatise; see Volkov (2012a).

  35. 35.

    A brief introduction to nine mathematical manuscripts from Dunhuang reproduced in Guo (1993, pp. 407–420) containing references to works on them published in Chinese is found in Wang 1993. Several mathematical manuscripts from Dunhuang were earlier published by Li Yan (1955, pp. 22–39); for their description and analysis, see Libbrecht 1982.

  36. 36.

    Wong (1979, p. 95) argues that the School did not have instructors of mathematics until 717.

  37. 37.

    Samguk sagi 三國史記 (Historical Records of Three [Korean] Kingdoms), juan 38, as quoted in Feng and Li (2000, p. 89). The original document presents the ages in traditional “Chinese” style, that is, actual age plus one year. It is not impossible that the Korean record contains a scribal error, and the limit age, 30 (i.e., 三十), was a miswritten 20 (二十) – that is, the actual limit age of the students was 19 (in Western style), which in this case would be close enough to the limit age in the contemporaneous Chinese school (18).

  38. 38.

    For Korean names and terms, the Revised Romanization transliteration (adopted in South Korea in 2000) is used; in cases when a name may be known to the Western reader in the McCune-Reischauer transliteration, it will be provided in parentheses.

  39. 39.

    The fact that mathematical subjects were taught by a Buddhist monk appears particularly interesting in the context of the connections between Buddhist networks and the transmission of mathematical knowledge, as mentioned above.

  40. 40.

    For Japanese terms, the Hepburn romanization system is used.

  41. 41.

    At that time a Korean scholar from the Kingdom of Paekche named Gwisil Jipsa (Kwisil Chipsa) 鬼室集斯 (Japanese reading Kishitsu Shushi) served as the head of the newly established National University; see Wong 1979, p. 86. On the role played in educational activities in Japan by instructors and students from the kingdom of Paekche defeated in 660–663 by the allied armies of Silla 新羅 (57 BC–935AD) and the Chinese Tang Empire, see Wong 1979, pp. 88–90.

  42. 42.

    Wong 1979, p. 95, provides an interesting comment on the number of students (30): “The class of mathematics […] was equal in size to its counterpart in China. However, in terms of the percentage in the whole student body, it was much larger.”

  43. 43.

    San kai and Chong cha were interpreted by Li Yan as titles of two different treatises; the total number of textbooks in this case should have been equal to 10. Feng and Li 2000 argue that the four characters “san kai chong cha” referred to one treatise, and the total number of the textbooks therefore was nine; the same hypothesis was advanced much earlier by Fujiwara (1940). This interpretation appears plausible since it fits into the description of the examination procedure found in the same source (see below).

  44. 44.

    The title of this treatise (lit. “Nine Governors” or “Nine Powers”) may suggest that its contents were somehow related to the Indian astronomical/astrological system Navagraha (Nine Celestial Bodies, i.e., five naked-eye planets, the Sun, the Moon, and two “imaginary” planets, Rahu and Ketu).

  45. 45.

    Bian ming shu li 辯明術理 (in Mandarin transliteration), see RGG 1985, vol. 2, p. 456.

  46. 46.

    The Japanese text uses the cyclic characters 甲 (Chinese jia) and otsu 乙 (Chinese yi) meaning “first” and “second,” respectively. The score of those who answered seven or eight questions is not specified.

  47. 47.

    RGG 1985, vol. 2, pp. 456–457. This description suggests that the Roku shō was considered a relatively difficult treatise.

  48. 48.

    Li 1933 [1977], p. 267; the only treatise left unmentioned is the Shū hi.

  49. 49.

    Some sources mention 210 as the total number of students, which is not consistent with the quotas of the three colleges listed above; see Yang 2003, vol. 2, p. 128.

  50. 50.

    As Qian Baocong 錢寶琮 suggested in his introduction to the treatise (Guo 1993, vol. 1, p. 309), the treatise was printed during the Song dynasty; however, Qian does not specify whether the book was printed in 1084 or later. All its printed copies were lost, according to Qian, during the Qing 清 dynasty (1644–1911).

  51. 51.

    According to Cheng Dawei 程大位 (1533–1606), ten mathematical texts were printed in 1084 and reprinted in the early thirteenth century; they included Huangdi jiu zhang 黃帝九章, Zhou bi suan jing, Wu jing suan fa 五經筭法, Hai dao suan fa 海島筭法, Sunzi suan fa 孫子筭法, Zhang Qiujian suan fa 張丘建筭法, Wu cao suan fa 五曹筭法, Qi gu suan fa 緝古筭法, Xiahou Yang suan fa 夏侯陽筭法, and Suan shu qia yi 筭術恰遺 (Li 1933 [1977], p. 280). Some of the titles listed by Cheng slightly differ from those found in the curricula of the School of Computation of the Tang and Song dynasties.

  52. 52.

    The “three [astrological] schemes” or “three cosmic boards” were the divinatory systems Tai yi 太乙, Qimen dunjia 奇門遁甲, and Liu ren 六壬 (Ho 2003, pp. 36–40, 83–84, 113–119).

  53. 53.

    To compare with the abovementioned record of the age of students in Korea.

  54. 54.

    It has been suggested that this was the Chinese treatise Xie Chawei suan fa 謝察微算法 (Computational Methods of Xie Chawei) compiled in ca. 1050; see Li et al. 1999, p. 74 and Jun 2006, p. 478.

  55. 55.

    Note that Brahmagupta counts more operations than Bhāskarācarya: he adds the Rules of Three, Five, Nine, and Eleven; the Inverse Rule of Three; Five rules to reduce fractions; barter and exchange; and rules to sell living beings.

  56. 56.

    If xi is the fineness (or “touch”) of the ith piece of gold, and yi its purity, x and y their respective value for the melted mass of gold, then xy = ∑xiyi. Therefore, x = ∑/y, and y = ∑/x.

  57. 57.

    13*10 + 12*4 + 11*2 + 10*4 = 240 = xy. Therefore in the first case, x = 240/20 = 12. In the second case, y = 240/16 = 15. In the third case, x = 240/15 = 16.

  58. 58.

    A translation into Sanskrit was made from al-Tūsī’s Persian version but obviously Euclid’s text was not understood by the pundits who undertook this work.

  59. 59.

    In this paper the references to page numbers are given according to the Taiwanese reprint of 1977.

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Keller, A., Volkov, A. (2014). Mathematics Education in Oriental Antiquity and Middle Ages. In: Karp, A., Schubring, G. (eds) Handbook on the History of Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9155-2_4

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