Advertisement

Parts in Chinese Mathematical Texts. Interpreting the Chapter Form of The Nine Chapters on Mathematical Procedures

  • Karine ChemlaEmail author
  • Dahai Zou
Chapter
Part of the Why the Sciences of the Ancient World Matter book series (WSAWM, volume 1)

Abstract

This article starts with an observation. The chapters that compose the earliest extant mathematical book handed down through the written tradition in China, The Nine Chapters on Mathematical Procedures, and that are mentioned in the title of the book, are a key feature distinguishing this book from the manuscripts excavated from tombs sealed in China during, roughly speaking, the Qin and early Western Han dynasties. We argue that as early as the Eastern Han dynasty, the names of these chapters were related to an interpretation of an expression (‘the nine parts of mathematics’) occurring in a text essential for Confucian scholars, Zhou Rites. We thus suggest that these facts allow us to make sense of both the third century commentator Liu Hui’s account of the history of The Nine Chapters, and the difference between The Nine Chapters and the newly discovered manuscripts with respect to the textual parts composing them. We argue that in Liu Hui’s view, the fact that The Nine Chapters was organized into such chapters was a feature manifesting the adherence of this mathematical book to a set of texts to which the first Qin emperor had gone to great lengths to control the access. This hypothesis allows us to interpret Liu Hui’s assertion that the transmission of The Nine Chapters was badly affected by the first Qin emperor’s book burning policy, at the same time as it accounts for why the newly excavated documents are not divided into chapters of this kind. These conclusions suggest that the chapters, and their names, might have played a key role in The Nine Chapters, these textual parts constituting an essential feature distinguishing this Classic from other mathematical writings. In the second part of our argument, we discuss a mathematical interpretation of the meaning of these chapters and the chapterization.

Bibliography

Primary Sources

  1. Ban Gu 班固. 1964. Han shu 漢書 (History of the [former] Han dynasty). Beijing: Zhonghua shuju 中華書局.Google Scholar
  2. Chemla, Karine, and Guo Shuchun. 2004. Les neuf chapitres. Le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.Google Scholar
  3. Fan Ye 范曄. 1973. Hou Han shu 後漢書 (Records on the Later Han dynasty). Beijing: Zhonghua shuju 中華書局.Google Scholar
  4. Guo Qingfan 郭慶藩. ed. 1985. Zhuang Zi Jishi 莊子集釋 (Collected explanations of Master Zhuang). Beijing 北京: Zhonghua Shuju 中華書局.Google Scholar
  5. Guo Shuchun 郭書春. ed. 1990. Hui jiao Jiu zhang suan shu 匯校九章算術. Shenyang 瀋陽: Liaoning jiaoyu chubanshe 遼寧教育出版社.Google Scholar
  6. Guo Shuchun 郭書春. ed. 1993. Zhongguo kexue jishu dianji tonghui. Shuxue juan 中國科學技術典籍通彙.數學卷. Zhengzhou: Henan jiaoyu chubanshe.Google Scholar
  7. Hubei sheng Jingzhou shi Zhouliangyuqiao yizhi bowuguan 湖北省荆州市周梁玉橋遺址博物館. ed. 2001. Guanju Qin Han mu jiandu 關沮秦漢墓簡牘 (Bamboo and wooden strips from Qin and Han tombs at Guanju). Beijing: Zhonghua Shuju 中華書局.Google Scholar
  8. Jia Gongyan 賈公彦. 1987. Xu Zhou li feixing 序周禮廢興 (Preface on the rise and decline of Zhou Rites). In Zhou li zhengyi 周禮正義 (Correct meaning of Zhou Rites). In Shi san jing zhushu 十三經注疏 (Commentaries and explanations on the Thirteen Classics), ed. Ruan Yuan 阮元, 635–636. Beijing: Zhonghua Shuju 中華書局.Google Scholar
  9. Jingzhou shi bowuguan 荆州市博物館. ed. 2005. Guodian chu mu zhujian 郭店楚墓竹簡 (Bamboo strips from a Chu tomb at Guodian). Beijing: Wenwu Chubanshe 文物出版社.Google Scholar
  10. Qian Baocong 錢寶琮. ed. 1963. Suanjing shishu 算經十書 (Qian Baocong jiaodian 錢寶琮校點) (Critical punctuated edition of The Ten Classics of Mathematics). (2 vols.). Beijing 北京: Zhonghua shuju 中華書局.Google Scholar
  11. Ruan Yuan 阮元, ed. 1980. Shi san jing zhushu 十三經注疏 (Commentaries and explanations on The Thirteen Classics). Reprint. Beijing: Zhonghua shuju.Google Scholar
  12. Shanghai tushuguan, and Beijing daxue tushuguan. 1980. Songke suanjing liu zhong 宋刻算經六種 (Six mathematical Canons printed during the Song). Beijing: Wenwu chubanshe.Google Scholar
  13. Sima Qian 司馬遷. 1963. Shi ji 史記 (Historical Records). Beijing 北京: Zhonghua shuju 中華書局.Google Scholar
  14. Sun Yirang 孫詒讓. ed. 1987. Zhou li zhengyi 周禮正義 (Correct meanings of Zhou Rites. Edition and commentary). Beijing: Zhonghua shuju 中華書局.Google Scholar
  15. Yang Jialuo. 楊家駱主編. ed. 1978. Zhongguo tianwen lifa shiliao 中國天文曆法史料 (Historical Material on Chinese Astronomy and Calendar). Taipei: Dingwen shuju 鼎文書局.Google Scholar
  16. Zhangjiashan ersiqi hao hanmu zhujian zhengli xiaozu 張家山二四七號漢墓竹簡整理小組 (Group collating the bamboo strips from tomb no. 247 at Zhangjiashan). eds. 2001. Zhangjiashan hanmu zhujian [ersiqi hao mu] 張家山漢墓竹簡[二四七號墓] (Bamboo strips from the tomb of Han Dynasty excavated from Zhangjiashan (tomb no. 247)). Beijing 北京: Wenwu chubanshe 文物出版社.Google Scholar
  17. Zhu Hanmin 朱漢民 and Chen Songchang 陳松長. eds. 2011. Yuelu shuyuan cang Qinjian (er) 嶽麓書院藏秦簡 (貳) (Bamboo or wooden strips of Qin Dynasty preserved at Yuelu Academy, Vol. II). Shanghai: Shanghai cishu chubanshe 上海辭書出版社.Google Scholar

Secondary Literature

  1. Bielenstein, Hans. 1980. The Bureaucracy of Han times. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  2. Boltz, William. 1993. Chou li 周禮. In Early Chinese texts: a bibliographical guide, ed. Michael Loewe, 24–32. Berkeley, Calif.: Society for the Study of Early China: Institute of East Asian Studies University of California Berkeley.Google Scholar
  3. Chemla, Karine. 1992. Résonances entre démonstration et procédure: Remarques sur le commentaire de Liu Hui (IIIe siècle) aux Neuf chapitres sur les procédures mathématiques (1er siècle). In Regards obliques sur l’argumentation en Chine. Extrême-Orient, Extrême-Occident. Vol. XIV, ed. Karine Chemla, 91–129. Saint-Denis: Presses Universitaires de Vincennes.Google Scholar
  4. Chemla, Karine. 2008a. Antiquity in the shape of a canon. Views on antiquity from the outlook of mathematics. In Perceptions of Antiquity in Chinese Civilization, eds. Dieter Kuhn and Helga Stahl, 191–208. Heidelberg: Edition Forum.Google Scholar
  5. Chemla, Karine. 2008b. Classic and commentary: An outlook based on mathematical sources, Preprint 344/ Max-Planck-Institut für Wissenschaftsgeschicht. Berlin: Max-Planck-Institut für Wissenschaftsgeschichte.Google Scholar
  6. Chemla, Karine, and Guo Shuchun. 2004. Les neuf chapitres. Le Classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.Google Scholar
  7. Chen Songchang 陳松長. 2009. Yuelu shuyuan suo cang qinjian zongshu 嶽麓書院所藏秦簡綜述 (Summary on Qin bamboo or wooden strips preserved in Yuelu Academy). Wenwu 文物 (Cultural Relics) 2009(3): 76–88.Google Scholar
  8. Chen Zhenghong 陳正宏 and Tan Beifang 談蓓芳. 2004. Zhongguo jinshu jianshi 中國禁書簡史. Shanghai: Xuelin chubanshe 學林出版社.Google Scholar
  9. Cullen, Christopher. 2004. The Suàn shù shū 筭數書 ‘Writings on reckoning’: A translation of a Chinese mathematical collection of the second century BC, with explanatory commentary. Needham Research Institute Working Papers 1. Cambridge (UK): Needham Research Institute.Google Scholar
  10. Cullen, Christopher. 2007. The Suàn shù shū 筭數書, ‘Writings on reckoning’: Rewriting the history of early Chinese mathematics in the light of an excavated manuscript. Historia Mathematica 34: 10–44.CrossRefGoogle Scholar
  11. Fu, Zhengyuan. 1996. China’s legalists. The earliest totalitarians and their art of ruling. Armonk: M. E. Sharpe.Google Scholar
  12. Guo Shuchun 郭書春. 1984. Jiu zhang suanshu he Liu Hui zhu zhong zhi lü gainian ji qi yingyong shixi《九章算術》和劉徽注中之率概念及其應用試析 (Analysis of the concept of and its uses in The Nine Chapters on Mathematical Procedures and Liu Hui’s commentary). Kejishi Jikan 科技史集刊 (Journal for the History of Science and Technology) 11: 21–36.Google Scholar
  13. Guo Shuchun 郭書春. 1992. Gudai shijie shuxue taidou Liu Hui 古代世界數學泰斗劉徽 (Liu Hui, a great master of mathematics in the ancient world). Ji’nan濟南: Shandong kexue jishu chubanshe 山東科學技術出版社.Google Scholar
  14. Guo Shuchun 郭書春. 1997. Zhang Cang yu Jiuzhang Suanshu 張蒼與《九章算術》(Zhang Cang and The Nine Chapters on Mathematical Procedures). In Ke shi xinchuan: Qingzhu Du Shiran xiansheng congshi kexueshi yanjiu 40 zhounian xueshu lunwenji 科史薪傳:庆祝杜石然先生從事科學史硏究40周年學術論文集 (Transmission of the flame in the history of science. Collection of articles to celebrate the anniversary of Professor Du Shiran’s Forty years of research in the history of science), eds. Liu Dun 劉鈍, Han Qi 韓琦 et al., 112–121. Shenyang瀋陽: Liaoning jiaoyu chubanshe 遼宁教育出版社.Google Scholar
  15. Hu Pingsheng 胡平生. 1998. Fuyang shuanggudui Han jian shushu shu jianlun 阜陽雙古堆漢簡數術書簡論 (Simple discussion on Han Dynasty books on numbers and procedures written on bamboo strips and unearthed at Fuyang). In Chutu wenxian yanjiu 出土文献研究 (Studies on Unearthed Documents), ed. Zhongguo wenhua yichan yanjiuyuan 中國文化遺產研究院 4, 12–30. Beijing 北京: Zhonghua shuju 中華書局.Google Scholar
  16. Hucker, Charles O. 1985. A dictionary of official titles in Imperial China. Stanford, Calif.: Stanford University Press.Google Scholar
  17. Kern, Martin. 2010a. Early Chinese literature. Beginnings through Western Han. In The Cambridge History of Chinese Literature. Vol. I, eds. Kang-i Sun Chang and Stephen Owen, 1–115. Cambridge (UK): Cambridge University Press.Google Scholar
  18. Kern, Martin. 2010b. Offices of writing and reading in the Rituals of Zhou. In Statecraft and classical learning: The rituals of Zhou in East Asian history, eds. Benjamin Elman and Martin Kern, 65–93. Leiden & Boston: Brill.Google Scholar
  19. Li Di 李迪. 1997. Zhongguo shuxue tongshi. shanggu dao wudai juan 中國數學通史. 上古到五代卷 (General history of mathematics in China. Vol. I: From early ancient times to the Five Dynasties). Nanjing 南京: Jiangsu jiaoyu chubanshe 江蘇教育出版社.Google Scholar
  20. Li Di 李迪. 2000. Guanyu zhujian ‘Suanshu shu’ de ruogan wenti 關於竹簡“算數書”的若干問題 (Several questions on the bamboo strip manuscript Writings on Mathematics on bamboo strips). In 橫地清先生七十大壽紀念志 (Collected papers for the seventieth anniversary of Yokochi Kiyoshi 横地清), ed. Suzuki Masahiko (鈴木正彦), 21–24. Papers of the conference held on December 2, 2000.Google Scholar
  21. Li Jimin 李繼閔. 1982. Jiuzhang suanshu zhong de bilü lilun “九章算術” 中的比率理論 (The theory of ratios in The Nine Chapters on mathematical procedures). In Jiuzhang suanshu yu Liu Hui 九章算術與劉徽 (The Nine Chapters on Mathematical Procedures and Liu Hui), ed. Wu Wenjun, 228–245. Beijing: Beijing Shifan Daxue Chubanshe 北京師範大學出版社.Google Scholar
  22. Li Jimin 李繼閔. 1990. Dongfang shuxue dianji Jiuzhang suanshu ji qi Liu Hui zhu yanjiu 東方數學典籍——《九章算術》及其劉徽注研究 (Research on the Oriental mathematical Classic The Nine Chapters on Mathematical Procedures and on its Commentary by Liu Hui). 西安 Xi’an: 陝西人民教育出版社 Shaanxi renmin jiaoyu chubanshe.Google Scholar
  23. Li Rui 李銳. 2010. Qin fenshu kao 秦焚書考 (Critical research on the “book burning” of the Qin Dynasty). Renwen Zazhi人文雜志 (Journal of Humanities) 5: 140–146.Google Scholar
  24. Li Xueqin 李學勤. 1998. Cong chutu jianbo tandao xieshulü 從出土簡帛談到《挾書律》(Discussion of the statute for keeping books based on unearthed bamboo strips and silks). In Zhou Qin Han Tang yanjiu 周秦漢唐研究 (Studies on Zhou, Qin, Han to Tang Dynasties) 1: 1–7. Xi’an西安: Sanqin chubanshe三秦出版社.Google Scholar
  25. Li Yan 李儼. 1958. Zhongguo shuxue dagang (xiuding ben) 中國數學大綱 (修訂本) (An outline of the history of mathematics in China. Revised edition). Beijing: Kexue chubanshe 科學出版社 (Science Press).Google Scholar
  26. Li Yan 李儼. 1955. Zhongguo suanxue shi 中國算學史 (History of mathematics in China). Shanghai: Shangwu yinshuguan 商務印書館.Google Scholar
  27. Liu Qiyu 劉起釪. 1991. Zhou li zhenwei zhi zheng jiqi shuxie cheng de zhengshi yiju《周禮》真僞之爭及其書寫成的真實依據 (The debate on the authenticity of Zhou Rites and the actual evidence on how it was written). In Gushi xubian 古史續辨 (Continuation of the making of distinctions in ancient history), ed. Liu Qiyu 劉起釪, 619-653. Beijing: Zhonghua shuju 中華書局.Google Scholar
  28. Liu Shipei 劉師培. 2006. Jingxue jiaokeshu 經學教科書 (Textbook of studies on Confucian Classics). Ningwu 寧武: Ningwu nanshi jiaoyin 寧武南氏校印.Google Scholar
  29. Liu Xiaogan 劉笑敢. 1988. Zhuang Zi zhexue jiqi yanbian 莊子哲學及其演變 (The philosophy of Master Zhuang and its development). Beijing: Zhongguo shehui kexue chubanshe 中国社会科学出版社.Google Scholar
  30. Ma Feibai 馬非百. 1982. Qin ji shi 秦集史 (History of the Qin Dynasty based on a collection of documents). Beijing: Zhonghua shuju 中華書局.Google Scholar
  31. Meng Wentong 蒙文通. 1933. Jingxue jueyuan 經學抉原 (Looking for the origin of studies on Confucian Classics). Shanghai: Shangwu yinshuguan 商務印書館.Google Scholar
  32. Neininger, Ulrich. 1983. Burying the scholars alive: on the origin of a Confucian martyrs’ legend. In East Asian civilizations. New attempts at understanding traditions, Vol. II: Nation and Mythology, eds. Krzysztof Gawlikowski, Wolfram Eberhard, Carl-Albrecht Seyschab, 121–136. Munich: Simon & Magiera Publishers.Google Scholar
  33. Nylan, Michael. 1994. The ‘Chin Wen/Ku Wen’ controversy in Han times. T’oung Pao (Second Series) 80 (1–3): 83–145.Google Scholar
  34. Nylan, Michael. 2001. The five ‘Confucian’ Classics. New Haven: Yale University Press.Google Scholar
  35. Peng Hao 彭浩. 2001a. Guodian chujian Lao Zi jiaodu 郭店楚簡《老子》校讀 (Collations and explanations on Master Lao on Chu bamboo strips unearthed from Guodian). Wuhan 武漢: Hubei renmin chubanshe 湖北人民出版社.Google Scholar
  36. Peng Hao 彭浩. 2001b. Zhangjiashan hanjian Suanshu shu zhushi 張家山漢簡算數書注釋 (Commentary on Writings on Mathematics written on the Han bamboo strips excavated at Zhangjiashan). Beijing: Kexue chubanshe 科學出版社.Google Scholar
  37. Peng Hao 彭浩. 2005. Zhangjiashan hanjian Suanshu shu yu Jiuzhang suanshu 張家山漢簡《算數書》與《九章算術》(The Han manuscript on bamboo strips from Zhangjiashan Writings on Mathematics and The Nine Chapters on Mathematical Procedures). In Changsha sanguo wujian ji bainian lai jianbo faxian yu yanjiu guoji xueshu yantaohui lunwenji 長沙三國吳簡暨百年來簡帛發現與研究國際學術研討會論文集 (Proceedings of the International Conference on the Discovery and Study of Wu Strips (Three Kingdoms) Unearthed at Changsha and the Strips and Silks Discovered in the Last Century), ed. Changshashi wenwu kaogu yanjiusuo 長沙市文物考古研究所編, 330–334. Beijing: Zhonghua shuju 中華書局.Google Scholar
  38. Petersen, Jens Østergård. 1995. Which books did the First Emperor of Ch’in burn? On the meaning of Pai chia in early Chinese sources. Monumenta Serica. Journal of Oriental Studies 42: 1–52.Google Scholar
  39. Pian Yuqian 騈宇騫 and Duan Shu’an 段書安. 2006. Ershi shiji chutu jianbo zongshu 二十世紀出土簡帛綜述 (Summary of the unearthed strips and silk manuscripts in the twentieth century). Beijing: Wenwu chubanshe 文物出版社.Google Scholar
  40. Qian Baocong 錢寶琮. 1964. Zhongguo shuxueshi 中國數學史 (History of Mathematics in China). Beijing: Kexue chubanshe 科學出版社.Google Scholar
  41. Sun Wenqing 孫文青.1931. Jiuzhang suanshu yuanliu kao 九章算術源流考 (Examination of the origins and developments of The Nine Chapters on mathematical procedures). Nü shida xueshu jikan 女師大學術季刊 2(1):1-60.Google Scholar
  42. Van Ess, Hans. 1994. The old text/new text controversy. Has the 20th century got it wrong? T’oung Pao 80 (1–3): 146–170.Google Scholar
  43. Vihan, Jan. 2012. Language, likeness, and the Han phenomenon of convergence. PhD. diss. Harvard University.Google Scholar
  44. Volkov, Alexei. 2014. Mathematics education in East- and Southeast Asia. In Handbook on the history of mathematics education, eds. Alexander Karp and Gert Schubring, 55–72, 79–82. New York: Springer.Google Scholar
  45. Watson, Burton. 1968. The complete works of Chuang tzu. New York: Columbia University Press.Google Scholar
  46. Xiao Can 蕭燦. 2010. “Yuelu shuyuan cang qinjian shu yanjiu” 嶽麓書院藏秦簡《數》研究 (A study of the mathematical text Mathematics on the bamboo strips preserved at Yuelu Academy). Ph.D. diss. Changsha 長沙: Hunan University 湖南大學.Google Scholar
  47. Xiao Can 蕭燦 and Zhu Hanmin 朱漢民. 2010. Gougu xinzheng: Yuelu shuyuan cang qinjian shu de xiangguan yanjiu 勾股新證——岳麓書院藏秦簡《數》的相關研究 (New evidence for the Pythagoras theorem: related studies based on the Qin mathematical text Mathematics written on bamboo strips and preserved at Yuelu Academy. Ziran kexueshi yanjiu 自然科學史研究 (Studies in the History of Natural Sciences 29(3): 313–318.Google Scholar
  48. Zhang Qicheng 張啓成. 2000. Dui liu jing wei rujia jingdian shuo de zai sikao 對六經爲儒家經典說的再思考 (Rethinking the viewpoint that the six Classics were Confucian Classics). Guizhou daxue xuebao (shehui kexue ban) 貴州大學學報(社會科學版) (Journal of Guizhou University (Edition of Social Sciences)). 18 (5): 74–78.Google Scholar
  49. Zou Dahai 鄒大海 . 2001a. Zhongguo shuxue de xingqi yu xianqin shuxue 中國數學的興起與先秦數學 (The emergence of mathematics in China and the mathematics in pre-Qin Period). Shijiazhuang 石家莊: Hebei kexue jishu chubanshe 河北科學技術出版社.Google Scholar
  50. Zou Dahai 鄒大海. 2001b. Chutu Suanshu shu chutan 出土《算數書》初探 (A preliminary research on the unearthed Writings on Mathematics). Ziran kexueshi yanjiu 自然科學史研究 (Studies in the History of Natural Sciences) 20(3): 193–205.Google Scholar
  51. Zou Dahai 鄒大海. 2003. Cong Suanshu shu he qinjian kan shanggu liangmi de bilü 從《算數書》和秦簡看上古粮米的比率 (The ratios between different types of grains in high antiquity: A new research based on Writings on Mathematics and Qin strips). Ziran kexueshi yanjiu 自然科學史研究 (Studies in the History of Natural Sciences) 22(4): 318–328.Google Scholar
  52. Zou Dahai 鄒大海. 2004. Cong Suanshu shu yu Jiuzhang suanshu de guanxi kan suanfa shi shuxue wenxian zai shanggu shidai de liuchuan. 從《算數書》與《九章算術》的關係看算法式數學文獻在上古時代的流傳 (The transmission of mathematical documents in the form of computational methods during the early period of ancient times: A research on the basis of the study on the relationship between Writings on Mathematics and The Nine Chapters on Mathematical Procedures). Gannan shifan xueyuan xuebao 贛南師範學院學報 (Journal of Gannan Teachers’ College) 6: 6–10.Google Scholar
  53. Zou Dahai 鄒大海. 2005. Shuihudi bamboo strips of Qin Dynasty and mathematics in pre-Qin period. Kaogu 考古 (Archaeology). 2005 (6): 57–66. An English translation can be found in Frontiers of History in China, Vol.2, no.4, Oct. 2007. Higher Education Press, Springer, pp. 632-654. An abridged English version was published in Chinese archaeology 7, Beijing: China Social Sciences Press, 2007, pp. 132-136.Google Scholar
  54. Zou Dahai 鄒大海. 2007. Cong Suanshu shu ying buzu wenti kan shanggu shidai de ying bu zu fangfa 從《算數書》盈不足問題看上古時代的盈不足方法 (The method of excess and deficit in high antiquity: An investigation on the basis of the problems of excess and deficit in the Suanshu shu). Ziran kexueshi yanjiu 自然科學史研究 (Studies in the History of Natural Sciences) 26 (3): 312–323.Google Scholar
  55. Zou Dahai 鄒大海. 2008. Chutu jiandu yu zhongguo zaoqi shuxueshi 出土簡牘與中國早期數學史 (The unearthed bamboo strips and the history of mathematics in early China). Renwen yu shehui 人文與社會 (Journal of Humanism and Society) 2 (2): 71–98.Google Scholar
  56. Zou Dahai 鄒大海. 2009. Guanyu Suanshu shu, Qinlü he shanggu liangmi jiliang danwei de jige wenti 關於《算數書》、秦律和上古糧米計量單位的幾個問題 (Several questions on Writings on Mathematics, the statutes of Qin and measurement units of grains and husked grains during the early period of ancient time). Neimenggu shifan daxue xuebao (ziran kexue hanwen ban) 內蒙古師範大學學報(自然科學漢文版) (Journal of Neimenggu Normal University (Chinese Edition of Natural Sciences)) 38(5): 508–515.Google Scholar
  57. Zou Dahai 鄒大海. 2010a. Cong chutu zhujian kan zhongguo zaoqi weishu suanti jiqi shehui beijing 從出土竹簡看中國早期委輸算題及其社會背景 (The problems concerning weishu and their social context in early China: A research based on the unearthed bamboo strips). Hunan daxue xuebao (shehui kexue ban) 湖南大學學報(社會科學版) (Journal of Hunan University (Social Sciences)) 24(4): 5–10.Google Scholar
  58. Zou Dahai 鄒大海. 2010b. Cong chutu jiandu wenxian kan zhongguo zaoqi de zhengfushu gainian 從出土簡牘文獻看中國早期的正負數概念 (Concepts of positive and negative numbers in early China: an investigation based on documents written on unearthed bamboo and wooden strips). Kaogu xuebao 考古學報 (Journal of Archaeology (Original English title: Acta Archaeologica Sinica)) 2010(4): 481–504.Google Scholar
  59. Zou Dahai 鄒大海. 2010c. Cong chutu wenxian kan shanggu yishi zhidu yu zhengfushu gainian. 從出土文獻看上古醫事制度與正負數概念 (Medical institutions and concepts of positive and negative numbers in the early Ancient Period: an investigation based on unearthed documents. Zhongguo lishi wenwu 中國歷史文物 (China History and Cultural Relics (Original English title: Journal of National Museum of China)) 2010(5): 69–76. Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS and Université Paris Diderot, Laboratory SPHERE, UMR 7219Paris, Cedex 13France
  2. 2.Institute for the History of Natural SciencesChinese Academy of SciencesBeijingChina

Personalised recommendations