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Mathematics Education in Oriental Antiquity and Middle Ages


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

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Fig. 4.1


  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.

Primary Sources (for Section 1)

  • Cheng Dawei 程大位. 1990. Suan fa tong zong jiaoshi 算法統宗校釋 (An annotated edition of the Summarized fundamentals of computational methods). Hefei: Anhui jiaoyu Publishers.

    Google Scholar 

  • CM 1969 – Tr`ân VănVi 陳文為 et al. (eds.). Khâm Định Việt Sử Thông Giám Cương Mục 欽定越史通鑑綱目 (Imperially commissioned itemized summaries of the comprehensive mirror of Việt history). Taipei: Guoli zhongyang tushuguan.

    Google Scholar 

  • Guo 1993 – Guo Shuchun 郭書春 (ed.), Shu xue juan 數學卷 (Mathematical section). In Zhongguo kexue jishu dianji tonghui 中國科學技術典籍通彙 (Comprehensive collection of primary materials on science and technology in China), ed. Ren Jiyu 任繼愈. Zhengzhou: Henan Education Publishers.

    Google Scholar 

  • JTS 1975 – Liu Xu 劉昫 (ed). Jiu Tang shu 舊唐書 (Old history of the Tang [Dynasty]). Beijing: Zhonghua shuju.

    Google Scholar 

  • NHS 1985 – Nihon shoki 日本書紀 (Chronicles of Japan). Tokyo: Yoshikava Kōbunkan.

    Google Scholar 

  • RGG 1985 – Kiyohara Natsuno 清原夏野 et al., Ryō no gige 令集解 (Explanations of the codes [of the eras Taihō and Yōrō]). Tokyo: Yoshikava Kōbunkan.

    Google Scholar 

  • SS 2008 Hubei sheng wenwu kaogu yanjiusuo 湖北省文物考古研究所 and Yunmeng xian bowuguan 云梦县博物馆, Hubei Yunmeng Shuihudi M77 fajue jianbao 湖北云梦睡虎地M77发掘简报 (Short report on the excavations of the tomb M77 at Shuihudi in Yunmeng, Hubei province), Jiang Han kaogu 江汉考古, 109: 31–37, plates 11–16.

    Google Scholar 

  • SSS 2000 – Jiangling Zhangjiashan Hanjian zhengli xiao zu 江陵張家山漢簡整理小組 (Restoration team for the Han [dynasty] bamboo strips [found at] Zhangjiashan in Jiangling). “Jiangling Zhangjiashan Hanjian ‘Suan shu shu’ shiwen”《江陵張家山漢簡〈算數書〉釋文》(Transcription of the Suan shu shu of the Han [dynasty written on] bamboo strips [found at] Zhangjiashan in Jiangling). Wenwu 文物, no. 9: 78–84.

    Google Scholar 

  • SSS 2001 – Zhangjiashan Han mu zhujian (erbai sishi qi hao mu) 張家山漢墓竹簡〔二四七號墓〕 (Bamboo strips from the Han [dynasty] tomb number 247 in Zhangjiashan). Beijing: Wenwu Chubanshe.

    Google Scholar 

  • SSS 2006 – Ohkawa [= Ōgava] Toshitaka 大川俊隆 et al. Kan kan “San sū sho”. Shashinban 漢簡『算数書』. 写真版 (Photographic reproduction of the bamboo strips of the Suan shu shu on the Han [dynasty]). Kyoto: Hōyu shoten 朋友書店 [appendix to Ohkawa et al. 2006].

    Google Scholar 

  • TLD 1983 – Zhang Jiuling 張九齡 et al., Tang liu dian 唐六典 (Six codes of the Tang [Dynasty]). In Siku quanshu, Wenyuange edition 文淵閣四庫全書, vol. 595. Taibei: Taiwan Shangwu Publishers.

    Google Scholar 

  • XTS 1956 – Ouyang Xiu 歐陽脩 and Song Qi 宋祁. Xin Tang shu 新唐書 (New history of the Tang [Dynasty]). Taibei: Editorial Board for Publication of 25 Dynastic Histories.

    Google Scholar 

  • Yang 2003 – Yang Xuewei 杨学为 (ed.). Zhongguo kaoshi shi wenxian jicheng 中国考试史文献集成 (Collected materials on the history of examinations in China). Beijing: Gaodeng jiaoyu Publishers.

    Google Scholar 

Secondary Sources (for Section 1)

  • Publications in Asian Languages

    Google Scholar 

  • Cheng Fangping 程方平. 1993. Liao, Jin, Yuan jiaoyu shi 辽金元教育史 (History of education of the Liao, Jin, and Yuan [dynasties]). Chongqing: Chongqing Publishers.

    Google Scholar 

  • Feng Lisheng 冯立升, Li Di 李迪. 2000. Liu zhang, San kai xin tan《六章》、《三開》新探 (New exploration of the [Korean mathematical treatises] Liu zhang and San kai). Xibei daxue xuebao 西北大学学報, 30(1): 89–92.

    Google Scholar 

  • Fujiwara Matsusaburō 藤原松三郎. 1940. Shina sūgakushi no kenkyū III 支那數學の研究 III (Studies in the history of Chinese mathematics, part III). Tohoku sugaku zasshi 東北數學雜誌 (The Tohoku Mathematical Journal), 47: 309–321.

    Google Scholar 

  • Guo Shirong 郭世荣. 1991. Lun Zhongguo gudai de guojia tiansuan jiaoyu 论中国古代的国家天算教育 (On the state astronomical and mathematical education in ancient China). In Shuxue shi yanjiu wenji 数学史研究文集 (Collected works on the history of Chinese mathematics), ed. Li Di 李迪, issue 2: 27–30. Huhehaote: Nei Menggu daxue and Taipei: Jiuzhang Publishers.

    Google Scholar 

  • Guo Shuchun 郭书春. 1989. A preliminary inquiry into Li Ji’s Jiuzhang Suanshu Yinyi (李籍《九章算术音义》初探), Ziran kexue shi yanjiu自然科学史研究 (Studies in history of natural sciences), 8(3): 197–204.

    Google Scholar 

  • Horng Wann-Sheng 洪萬生, Lin Cang-Yi 林倉億, Su Hui-Yu 蘇惠玉, Su Jun-Hong 蘇俊鴻. 2006. Shu zhi qiyuan: Zhongguo shuxue shi kaizhang ‘Suan shu shu’ 數之起源中國數學史開章 “筭數書” (The origin of the numbers: Suan shu shu, the opening chapter of the history of mathematics in China). Taibei: Taibei shangwu yinshuguan.

    Google Scholar 

  • Jochi Shigeru 城地茂 (tr.). 2001. San sū sho Nihongo yaku 算數書日本語譯 (Japanese translation of the Suan shu shu). In Wasan kenkyuso kiyo 和算研究所紀要, 4: 19–46.

    Google Scholar 

  • Kim Yong-Woon 金容雲, Kim Yong-Guk 金容局. 1978. Kankoku sūgaku shi 韓國數學史 (A history of Korean mathematics). Tokyo: Maki shoten.

    Google Scholar 

  • Liu Caonan 劉操南. 1944. Zhou li ‘jiu shu’ jie 周禮九數解 (Explanation of [the term] ‘jiu shu’ in the Zhou li). Yishi bao (wenshi fukan) 益世報 (文史副刊), no. 19. [Not seen; quoted from Ou et al. 1994, pp. 234, 240, no. 12].

    Google Scholar 

  • Li Yan 李儼. 1933 [1977]. Tang Song Yuan Ming shuxue jiaoyu zhidu 唐宋元明數學教育制度 (The system of mathematics education [in China] of the Tang, Song, Yuan, and Ming [dynasties]). Originally published in Kexue zazhi 科學雜誌 (Science Magazine), vol. 17 (1933), no. 10, pp. 1545–1565; reproduced in Li Yan 李儼, Zhong suan shi luncong 中算史論叢 (Collected works on the history of Chinese mathematics). Beijing: Zhongguo Kexueyuan, 1954–1955, vol. 4, pp. 238–280, and in Li Ziyan 李子嚴 (= Li Yan 李儼), Zhong suan shi luncong 中算史論叢, Taibei: Taiwan shangwu, 1977, vol. 4, pt. 1, pp. 253–285.Footnote

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

    Google Scholar 

  • Li Yan 李儼. 1937. Zhongguo suanxue shi 中國算學史 (History of Chinese mathematics). Shanghai: Shangwu yinshuguan.

    Google Scholar 

  • Li Yan 李儼. 1955. Zhongguo gudai shuxue shiliao 中國古代數學史料 (Materials [for study] of ancient Chinese mathematics). 2nd ed. Shanghai: Xinhua shudian

    Google Scholar 

  • Ohkawa [= Ōgava] Toshitaka 大川俊隆 et al. 2006. [Chō ka san Kan kan “San sū sho” kenkyukai 張家山漢簡『算数書』研究会 (Society for study of the Suan shu shu of the Han [dynasty written on bamboo] strips from Zhanjiashan).] Kan kan “San sū sho”: Chūgoku saiko-no sūgakusho 漢簡『算数書』: 中国最古の数学書 (The Suan shu shu of the Han [dynasty written on bamboo] strips from Zhanjiashan: the most ancient Chinese mathematical book). Kyoto: Hōyu shoten 朋友書店.

    Google Scholar 

  • Ou Yan 瓯燕, Wen Benheng 文本亨, and Yang Yaolin 杨耀林. 1994. Cong Shenzhen chutu chengfa koujue lun woguo gudai ‘jiu jiu zhi shu’ 从深圳出土乘法口诀论我国古代九九之术 (Discussing the ancient Chinese “method of nine nines” on the basis of the mnemonic multiplication rule [written on a brick] excavated in Shenzhen). In Shenzhen kaogu faxian yu yanjiu 深圳考古发现与研究 (Shenzhen: Archeological discoveries and research), ed. Wu Zengde 吴曾德, 232–241. Beijing: Wenwu Publishers. (Originally published in Wenwu, 1991, no. 9: 78–85).

    Google Scholar 

  • Sun Hongan 孙宏安. 1996. Zhongguo gudai kexue jiaoyu shilüe 中国古代科学教育略 (Brief history of science education in ancient China). Shenyang: Liaoning jiaoyu Publishers.

    Google Scholar 

  • Sun Peiqing 孙培青 (ed.). 2000. Zhongguo jiaoyu shi 中国教育史 (History of education in China). Shanghai: Huadong Shifan Daxue Publishers.

    Google Scholar 

  • Tong Jianhua 佟健华, Yang Chunhong 杨春宏, Cui Jianqin 崔建勤 (eds.). 2007. Zhongguo gudai shuxue jiaoyu shi 中国古代数学教育史 (History of mathematics education in ancient China). Beijing: Kexue Publishers.

    Google Scholar 

  • Wang Yusheng 王渝生. 1993. Dunhuang suanshu tiyao 敦煌算書提要 (Summary of mathematical books from Dunhuang). In Guo 1993, vol. 1, 401–405.

    Google Scholar 

  • Yan Dunjie 严敦杰. 2000. Zu Chongzhi kexue zhuzuo jiaoshi 祖沖之科学著作校释/Annotated collation of Zu Chongzhi’s scientific works. Shenyang: Liaoning jiaoyu.

    Google Scholar 

  • Zhou Dongming 周东明. 1990. ‘Xi suan gang mu’ yu Yang Hui de shuxue jiaoyu sixiang《习算纲目》与杨辉的数学教育思想 (The ‘Syllabus for studying mathematics’ and Yang Hui’s philosophy of mathematics education). 华中师范大学学报 (自然科学版). Journal of Central Normal University, Natural Sciences series, 24(3): 396–399.

    Google Scholar 

  • Zou Dahai 鄒大海. 2008. Chutu jiandu yu Zhongguo zaoqi shuxueshi 出土簡牘與中國早期數學史 (Unearthed texts on bamboo strips and history of ancient Chinese mathematics). Renwen yu shehui xuebao 人文與社會學報 2(2): 71–98.

    Google Scholar 

  • Zou Dahai 邹大海. 2011. Guanyu Qin jian shuxue zhuzuo Shu de guoji huiyi zai Yuelu shuyuan juxing – ‘Yuelu shuyuan cang Qin jian (di er juan) guoji yanjiuhui’ jianjie 关于秦简数学著作《数》的国际会议在岳麓书院举行 –《岳麓书院藏秦简》(第二卷) 国际研读会简介 (On the international conference in Yuelu Academy devoted to the mathematical work Shu on [bamboo] strips [dated of] the Qin [dyansty] – a brief introduction of the Second Volume of Proceedings of the International Workshop on Qin dynasty bamboo strips held in Yuelu Academy). Ziran kexue shi yanjiu 自然科学史研究 (Studies in History of Natural Sciences), no. 1: 131–132.

    Google Scholar 

  • Publications in Western Languages

    Google Scholar 

  • Berezkina, El’vira I. 1975. Van Syao-Tun. Matematicheskii traktat o prodolzhenii drevnikh (metodov) (Wang Xiaotong. Mathematical treatise on the continuation of ancient (methods), [an annotated Russian translation]). In Istoriko-matematicheskie issledovaniya (Studies in the history of mathematics) 20: 329–371.

    Google Scholar 

  • Boltz, William G. 1993. Chou li [= Zhou li] 周禮. In Early Chinese texts. A bibliographical guide, ed. M. Loewe. Berkeley: The Society for the Study of Early China/The Institute of East Asian Studies, University of California (Berkeley): 24–32.

    Google Scholar 

  • Chemla, Karine, and Guo Shuchun. 2004. 九章算術. Les Neuf Chapitres: le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.

    Google Scholar 

  • Chemla, Karine, Ma Biao 馬彪. 2011. Interpreting a newly discovered mathematical document written at the beginning of the Han dynasty in China (before 157 B.C.E.) and excavated from tomb M77 at Shuihudi (睡虎地). Sciamvs 12: 159–191.

    Google Scholar 

  • Cullen, Christopher. 1993. Chiu chang suan shu [= Jiu zhang suan shu]. 九章算術. In Early Chinese texts. A bibliographical guide, ed. M. Loewe, 16–23. Berkeley: The Society for the Study of Early China/The Institute of East Asian Studies, University of California (Berkeley).

    Google Scholar 

  • Cullen, Christopher. 2004. The Suan shu shu ‘Writing on reckoning’: A translation of a Chinese mathematical collection of the second century BC, with explanatory commentary. Needham Research Institute working papers 1. Cambridge: Needham Research Institute.

    Google Scholar 

  • Cullen, Christopher. 2007. The Suàn shù shū 算數書, ‘Writings on reckoning’: Rewriting the history of early Chinese mathematics in the light of an excavated manuscript. Historia Mathematica 34(1): 10–44.

    Google Scholar 

  • Dauben, Joseph W. 2008. 算數書 Suan Shu Shu: A book on numbers and computations. Archive for History of Exact Sciences 62: 91–178.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Des Rotours, Robert. 1932. Le traité des examens, traduit de la Nouvelle histoire des T’ang (chap. 44–45). Paris: Librairie Ernest Leroux.

    Google Scholar 

  • Friedsam, Manfred. 2003. L’enseignement des mathématiques sous les Song et Yuan. In Éducation et instruction en Chine, vol. 2 (Les formations spécialisées), ed. C. Despeux and C. Nguyen Tri, 49–68. Paris – Louvain: Editions Peeters.

    Google Scholar 

  • Ho Peng Yoke. 2003. Chinese mathematical astrology: Reaching out to the stars. London/New York: Routledge Curzon.

    Google Scholar 

  • Hucker, Charles O. 1988. A dictionary of official titles in Imperial China. Taibei: Southern Materials Center (originally printed by Stanford University Press, 1985).

    Google Scholar 

  • Jun Yong Hoon. 2006. Mathematics in context: A case in early nineteenth-century Korea. Science in Context 19(4): 475–512.

    CrossRef  MATH  MathSciNet  Google Scholar 

  • Lam Lay Yong. 1977. A critical study of the Yang Hui suan fa. Singapore: NUS Press.

    Google Scholar 

  • Lee, Thomas H.C. 1985. Government education and examinations in Sung China. Hong Kong/New York: The Chinese University Press/St. Martin’s Press.

    Google Scholar 

  • Lee, Thomas H.C. 2000. Education in traditional China: A history. Brill: Leiden.

    Google Scholar 

  • Li Wenlin, Xu Zelin, and Feng Lisheng. 1999. Mathematical exchanges between China and Korea. Historia Scientiarum 67: 73–83.

    Google Scholar 

  • Libbrecht, Ulrich. 1982. Mathematical manuscripts from the Tunhuang [= Dunhuang] caves. In Explorations in the history of science and technology in China, ed. Li Guohao et al., 203–229. Shanghai: Chinese Classics Publishing House.

    Google Scholar 

  • Martzloff, Jean-Claude. 1997. A history of Chinese mathematics. Berlin: Springer.

    CrossRef  MATH  Google Scholar 

  • Riegel, Jeffrey K. 1993. Li chi [= Li ji] 禮記. In Early Chinese texts. A bibliographical guide, ed. M. Loewe, 293–297. Berkeley: The Society for the Study of Early China/The Institute of East Asian Studies, University of California (Berkeley).

    Google Scholar 

  • Siu Man Keung. 1993. Proof and pedagogy in ancient China: Examples from Liu Hui’s commentary on Jiu Zhang Suan Shu. Educational Studies in Mathematics 24: 345–357.

    CrossRef  Google Scholar 

  • Siu Man Keung. 1995. Mathematics education in ancient China: What lesson do we learn from it? Historia Scientiarum 4(3): 223–232.

    MATH  MathSciNet  Google Scholar 

  • Siu Man Keung. 1999. How did candidates pass the state examinations in mathematics in the Tang dynasty (618–917)? – Myth of the ‘Confucian-Heritage-Culture’ classroom. Actes du Colloque “Histoire et épistemologie dans l’éducation mathématique”, 321–334. Louvain: Université catholique de Louvain.

    Google Scholar 

  • Siu Man Keung. 2004. Official curriculum in mathematics in ancient China: How did candidates study for the examination? In How Chinese learn mathematics: Perspective from insiders, ed. Fan Lianghuo, Wong Ngai-Ying, Cai Jinfa, and Li Shiqi, 157–185. Singapore/River Edge: World Scientific.

    Google Scholar 

  • Siu Man Keung. 2009. Mathematics education in East Asia from antiquity to modern times. In Dig where you stand: Proceedings of a conference on on-going research in the history of mathematics education, ed. Kristin Bjarnadottir, Fulvia Furinghetti, and Gert Schubring, 197–208. Gardabaer, 20–24 June 2009.

    Google Scholar 

  • Siu Man Keung, and Alexei Volkov. 1999. Official curriculum in traditional Chinese mathematics: How did candidates pass the examinations? Historia Scientiarum 9(1): 85–99.

    MATH  MathSciNet  Google Scholar 

  • Sivin, Nathan. 2009. Granting the seasons. New York: Springer Science + Business Media LLC.

    MATH  Google Scholar 

  • Volkov, Alexei. 1994. Large numbers and counting rods. In Sous les nombres, le monde, Extrême-Orient Extrême-Occident, vol. 16, ed. A. Volkov, 71–92. Paris: PUV.

    Google Scholar 

  • Volkov, Alexei. 2007. Geometrical diagrams in Liu Hui’s commentary on the Jiuzhang suanshu. In Graphics and text in the production of technical knowledge in China, ed. F. Bray, V. Dorofeeva-Lichtmann, and G. Métailié, 425–459. Leiden: Brill.

    CrossRef  Google Scholar 

  • Volkov, Alexei. 2010. Commentaries upon commentaries: The translation of the Jiu zhang suan shu 九章算術 by Karine Chemla and Guo Shuchun (essay review). Historia Mathematica 37: 281–301.

    Google Scholar 

  • Volkov, Alexei. 2012a. Argumentation for state examinations: Demonstration in traditional Chinese and Vietnamese mathematics. In The history of mathematical proof in ancient traditions, ed. Karine Chemla, 509–551. Cambridge: Cambridge University Press.

    Google Scholar 

  • Volkov, Alexei. 2012b. Recent publications on the early history of Chinese mathematics (essay review). Educação Matemática Pesquisa 14(3): 348–362.

    Google Scholar 

  • Volkov, Alexei. 2013. Didactical dimensions of mathematical problems: ‘weighted distribution’ in a Vietnamese mathematical treatise. In Scientific sources and teaching contexts: Problems and perspectives, ed. C. Proust and A. Bernard. Boston Studies in Philosophy of Science. New York: Springer (forthcoming).

    Google Scholar 

  • Wong, Joseph. 1979. The government schools in T’ang China and Nara and Heian Japan: A comparative study. (An unpublished thesis submitted for the degree of Master of Arts). Canberra: Australian National University.

    Google Scholar 

References (for Section 2)

  • Chattopadhyaya, Debiprasad (ed.). 1986. History of science and technology in ancient India, the beginnings. New-Dehli: Firma Klm.

    Google Scholar 

  • Colebrooke, Henry T. 1817. Algebra, with arithmetic and mensuration. London: J. Murray.

    Google Scholar 

  • Hayashi, Takao. 2001. Mathematics (gaṇita). In Storia della Scienza, vol. II.II.X.1–7, 772–790. Rome: Istituto della Enciclopedia Italiana.

    Google Scholar 

  • Keller, Agathe. 2006. Expounding the mathematical seed, Bshāskara and the mathematical chapter of the Āryabhaṭīya. Basel: Birkhauser.

    Google Scholar 

  • Keller, Agathe. 2007. Qu’est ce que les mathématiques? Les réponses taxinomiques de Bhāskara un commentateur, mathématicien et astronome du VIIème siècle, Sciences et Frontières: 29–61. Bruxelles: Kimé.

    Google Scholar 

  • Plofker, Kim. 2007. Mathematics in India. In Mathematics in Egypt, Mesopotamia, China, India and Islam, ed. Victor Katz, 385–514. Princeton: Princeton University Press.

    Google Scholar 

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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.

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