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Suan Shu Shu A Book on Numbers and Computations

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An Erratum to this article was published on 29 March 2008

Abstract

In December and January of 1983–1984, archaeologists excavating the tomb of an ancient Chinese provincial bureaucrat at a Western Han Dynasty site near Zhangjiashan, in Jiangling county, Hubei Province, discovered a number of books on bamboo strips, including inter alia works on legal statutes, military practice, and medicine. Among these was a previously unknown mathematical work on some 200 bamboo strips, the Suan shu shu, or Book of Numbers and Computations. Based upon other works found in the tomb, especially a copy of the Er nian lü ling (Laws and Decrees of the Second Year (of the reign of empress Lü, i.e. Lü Hou)), archaeologists have dated the tomb to ca. 186 BCE (Lü Hou’s regency lasted from 188 to 180 BCE). The Suan shu shu, as the earliest yet discovered work devoted specifically to mathematics from ancient China, has stirred considerable interest among Chinese historians of science. The translation and commentary offered here draw extensively on the works cited in Sect. 3 below. Several appendixes devoted to specific issues related to translating the Suan shu shu, including its title and the problem of determining English equivalents for various commodities that arise in the text, may be found in Appendix II.

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References

  • Bai 1983: Bai Shangshu Jiuzhang suanshu zhushi (Commentaries and Explanations for the Nine Chapters on Mathematical Procedures), Beijing: Kexue chubanshe, 1983.

  • Berezkina 1957: Berezkina, Elvira Ivanovna, “Matematika v devyati knigakh” (Mathematics in Nine Books, a Russian translation of the Nine Chapters on Mathematical Procedures), Istoriko-matematicheskie issledovaniya (Studies on the History of Mathematics), 10 (1957), pp. 427–584.

  • Bodde 1986: Bodde, Derk, “The State and Empire of Ch’in,” in [Twichett and Loewe 1986, vol. 1, pp. 20–102].

  • Boltz 1999: Boltz, William G., “Language and Writing,” Chapter 2 in [Loewe and Shaughnessy 1999, pp. 74–123].

  • Bray 1984: Bray, Francesca, Agriculture, Vol. 6, Part II of Biology and Biological Technology, in Science and Civilisation in China, Joseph Needham, ed., Cambridge, England: Cambridge University Press, 1984.

  • Chang 1983: Chang Te-Tzu, “The Origins and Early Cultures of the Cereal Grains and Food Legumes,” in The Origins of Chinese Civilization, David N. Keightley, ed., Berkeley, CA: University of California Press, 1983, pp. 65–77.

  • Chemla 1997: Chemla, Karine, “What is at Stake in Mathematical Proofs from Third-Century China?” Science in Context, 10(2)(1997), pp. 227–251.

  • Chemla 1997–1998: “Fractions and Irrationals between Algorithm and Proof in Ancient China,” Studies in History of Medicine and Science, 15(1–2)(1997–1998), pp. 31–54.

  • Chemla 2003: “Generality Above Abstraction. The General Expressed in Terms of the Paradigmatic in Ancient China’s Mathematics,” Science in Context, 16(3)(2003), pp. 413–458.

  • Chemla and Guo 2004: Chemla, Karine, and Guo Shuchun, Les Neuf chapitres. Le Classique mathématique de la Chine ancienne et ses commentaires, Paris: Dunod, 2004.

  • Chen and Chen 1987: Chen Yuejun and Chen Yanping Suan shu shu yu Jiu zhang suan shu (The Suan shu shu and the Nine Chapters on Mathematical Procedures), in Hubei sheng kaigu xuehui lunwenji (I) (Collected Papers of the Archaeological Society of Hubei Province), 1 (1987), pp. 220–222.

  • Chuan and Kraus 1975: Chuan Han-Sheng and Richard A. Kraus, Mid-Ch’ing Rice Markets and Trade: An Essay in Price History, Cambridge, MA: Harvard East Asian Monographs 54, Harvard University Press, 1975.

  • Cullen 1996: Cullen, Christopher, Astronomy and Mathematics in Ancient China: the Zhou bi suan jing, Cambridge, England: Cambridge University Press, 1996.

  • Cullen 2004: The Suan shu shu Writings on Reckoning,’ Cambridge, England: Needham Research Institute Working Papers, No. 1, 2004. see: http://Wenwuw.nri.org.uk/suanshushu.html

  • Dauben 2004: Dauben, Joseph W., The Suan shu shu (A Book on Numbers and Computations). Preliminary Investigation,” in Form, Zahl, Ordnung. Studien zur Wissenschafts- und Technikgeschichte. Ivo Schneider zum 65. Geburtstag, Rudi Seising, Menso Folkerts, and Ulf Hashagen, eds., Stuttgart: Fritz Steiner Verlag, 2004, pp. 151–168.

  • Dauben 2005: “Three Multi-Tasking Problems in the Suan shu shu, the Oldest Yet-Known Mathematical Work from Ancient China,” Physica et Historia—Festschrift für Andreas Kleinert zum 65. Geburtstag, Susan Splinter, Sybille Gerstengarbe, Horst Remane, and Benno Parthier, eds., Acta Historica Leopoldina, 45 (2005), pp. 25–34.

  • Dauben 2006: “ Suan Shu Shu (A Book on Numbers and Computation): Two Problems in Collating, Interpreting and Translating the Suan Shu Shu,” Actas del IX Congreso de la Sociedad Expañola de Historia de las Ciencias y de las Técnicas, Cádiz: Sociedad Expañola de Historia de las Ciencias y de las Técnicas, 2006, vol. 1, pp. 199–218.

  • Du 1988: Du Shiran Jiangling Zhangjiashang zhujian Suan shu shu chutan (A Preliminary Study on the Bamboo Strips Suan shu shu Unearthed from Zhangjiashang, Jianglin), Ziran kexueshi yanjiu (Studies in the History of Natural Sciences), 7(3)(1988), pp. 201–204.

  • Duan and Hai 2004: Duan Yaohui and Hai Hong Xian Qin shuxueshi yanjiu de xin jinzhan—pinglun Zou Dahai zhu Zhongguo shuxue de xingqi yu xian Qin shuxue (Recent Progress of Historical Research of Pre-Qin Mathematics—Review of Zou Dahai’s Chinese Mathematics in the Pre-Qin Period), Ziran kexue shi yanjiu (Studies in the History of Natural Science), 23(2)(2004), pp. 168–172.

  • Duan and Zou 2003: Duan Yaoyong and Zou Dahai Suan shu shu zhong “Yi yuan cai fang” “Yi fang cai yuan” liang wen jiaozheng (New Collations of the Two Problems “Yi yuan cai fang” and “Yi fang cai yuan” in the Unearthed Mathematical Book Suan shu shu), Ziran kexueshi yanjiu (Studies in the History of Natural Science), 22(2)(2003), pp. 168–172.

  • Gao 1974: Gao Shufan Zhengzhong xingyinyi zonghe da zidian (Zhengzhong Comprehensive Dictionary of the Form, Pronunciation, and Meaning (of Chinese Characters)), Taipei: Zhengzhong shuju, 1974.

  • Guo (GSR) 2004: Guo Shirong Suan shu shu kanwu (Corrections for the Suan shu shu) Neimenggu shifan daxue xuebao. Ziran kexue hanwenban (Journal of Inner Mongolia Normal University, Natural Science Edition), 30(3)(2001), pp. 276–285.

  • Guo (GSC) 1990: Guo Shuchun Huijiao Jiu zhang suan shu (A Critical Textual Study of the Nine Chapters on Mathematical Procedures), Shenyang: Liaoning jiaoyu chubanshe, 1990.

  • Guo (GSC) 2000: “Theoretical Research Tendency of Mathematics in Pre-Qin Times Viewed from Suan shu shu,” Proceedings of the HPM 2000 Conference: History of Mathematics Education, Challenges for a New Millennium. A Satellite Meeting of ICME-9, Horng Wann-Sheng and Lin Fou-Lai, eds., Taipei: Department of Mathematics, National Taiwan Normal University, 2000, Vol. I, pp. 35–40.

  • Guo (GSC) 2001: Suan shu shu jiaokan (Collation of the Suan shu shu) Zhongguo kexu shiliao (China Historical Materials of Science and Technology), 22(3)(2001), pp. 202–219.

  • Guo (GSC) 2002: Shilun Suan shu shu de lilun gongxian yu bianzua (Some Views on Theoretical Aspects of the Suan shu shu and its Compilation), Faguo hanxue (Sinologie Française), vol. 6. Beijing: Zhonghua shuju, 2002, pp. 505–537.

  • Guo (GSC) 2003: Suan shu shu chutan (A Preliminary Study on the Suan shu shu), Guoxue yanjiu (Studies on National Heritages), 11(2003), pp. 307–349.

  • Guo (GSC) 2004: Huijiao Jiu zhang suan shu (A Critical Textual Study of the Nine Chapters on Mathematical Procedures), revised edition, 2 vols., Shenyang: Liaoning jiaoyu chubanshe; also printed in Taiwan as Huijiao Jiu zhang suan shu (A Critical Textual Study of the Nine Chapters on Mathematical Procedures), Taipei: Jiuzhang chubanshe.

  • Guo and Liu 1998: Guo Shuchun and Liu Dun Jiaodian Suanjing shishu (Collation of the Ten Mathematical Classics), Shenyang: Liaoning jiaoyu chubanshe, 1998 (2 vols.); also printed in a single volume in Taiwan as Guo Shuchun and Liu Dun Jiaodian Suanjing shishu (Collation of the Ten Mathematical Classics), Taipei: Jiuzhang chubanshe, 2001.

  • Guo and Lan 2003: Guo Zhengzhong and Lan Keli (Christian Lamouroux), eds., Shuxue dianji suoyin Qin Han zhi Song shehui jingli shiliao (Index of Mathematical Books from the Qin-Han to the Song. Materials for socio-economic history), Shenyang: Liaoning jiaoyu chubanshe, 2003.

  • Harper 1998: Harper, Donald J., Early Chinese Medical Literature: The Mawangdui Medical Manuscripts, London: Royal Asiatic Society, Sir Henry Wellcome Asian Series, 1998.

  • Horng 2000a: Horng Wann-Sheng Suan shu shu chutan (On the Suan shu shu: A Preliminary Study), Shida xuebao: Kexue jiaoyu lei (Journal of National Taiwan Normal University: Science and Education Edition), 45(2)(2000), pp. 77–91.

  • Horng 2000b: Suan shu shu de jize lunzheng (Several Examples of Logical Reasoning in the Suan shu shu), Taiwan lishi xuehui Tongxun (Newsletter of the Taiwan Historical Society), 11 (2000), pp. 44–52.

  • Horng 2002: Guangyu Suan shu shu lunzheng de yige beizhu (A Note on Several Examples of Logical Reasoning in the Suan shu shu), HPM HPM Tongxun (HPM Newsletter), 5(10)(2002), pp. 1–8.

  • Horng and Lin 2002: Horng Wann-Sheng and Lin Cang-yi Suan shu shu bufen timing de zai jiaokan (Further Collation on The Names of Certain Sections in the Suan shu shu), HPM HPM Tongxun (HPM Newsletter), 5(2,3)(2002), pp. 6–24.

  • Horng et al. 2006: Horng Wann-Sheng Lin Cang-yi Su Huiyu and Su Junhong Shuzhi qiyuan—Zhongguo shuxueshi kaizhang Suan shu shu (Origins of Number—Suan Shu Shu: The First Chapter in the History of Chinese Mathematics), Taipei: Taiwan Commercial Press, 2006.

  • Hsu 1980: Hsu Cho-yun, Han Agriculture. The Formation of Early Chinese Agrarian Economy (206 B.C.-A.D. 220), Seattle, WA: University of Washington Press, 1980.

  • Hsu 1999: “The Spring and Autumn Period,” in [Loewe and Shaughnessy 1999, pp. 545–586].

  • Hulsewé 1955: Hulsewé, A.F.P., Remnants of Han Law. Introductory Studies and an Annotated Translation of Chapters 22 and 23 of the History of the Former Han Dynasty, Leiden: E.J. Brill, 1955.

  • Hulsewé 1981: “Weights and Measures in Ch’in Law,” in State and Law in East Asia. Festschrift Karl Bünger, Dieter Eikemeier and Herbert Franke, eds., Wiesbaden: Otto Harrassowitz, 1981, pp. 25–39.

  • Hulsewé 1986: “Ch’in and Han Law,” in [Twitchett and Loewe 1986, vol. 1, pp. 520–544].

  • HYDCD 1987: Hanyu da cidian (Comprehensive Dictionary of the Chinese Language), see [Luo Zhufeng 1987].

  • Ji 2004: Ji Zhigang Zantan yu yihan: du Peng Hao zhu Zhangjiashan hanjian Suan shu shu zhushi (Praise and Regrets: Some Thoughts on Peng Hao’s Commentaries on the Suan shu shu), Ziran kexueshi yanjiu (Studies in the History of Natural Sciences), 23(1)(2004), pp. 91–96.

  • Ji 2005: Suan shu shu “shao guang” “da guang” erwen de shidou yu jiaokan (Annotations and Collations of two problems, “shao guang” and “da guang,” in the Suan shu shu), Ziran kexueshi yanjiu (Studies in the History of Natural Sciences), 24(3)(2005), pp. 229–235.

  • Jiang 1996: Jiang Renjie , ed., Shuowen jiezi jizhu (Selected Commentaries on the Dictionary of Chinese Characters (by Xu Shen, 2nd century, CE)), Shanghai: Shanghai guji chubanshe, 1996, in 3 vols.

  • Jochi 2001: Jochi Shigeru Sansûsho nihongo yaku (Japanese Translation of the Suan Shu Shu), Wasan kenkyûjo kiyô (Bulletin of the Wasan Institute), 4 (2001), pp. 19–46 (in Japanese).

  • Lam 1980: Lam Lay Yong, “The Chinese Connection Between the Pascal Triangle and the Solution of Numerical Equations of Any Degree,” Historia Mathematica, 7 (1980), pp. 407–424.

  • Lam 1994: “Jiu Zhang Suanshu (Nine Chapters on the Mathematical Art): An Overview,” Archive for History of Exact Sciences, 47(1)(1994), pp. 1–51.

  • Lam 1997: “Zhang Qiujian Suanjing (the mathematical classic of Zhang Qiujian): An Overview,” Archive for History of Exact Sciences, 50(3–4)(1997), pp. 201–240.

  • Lam and Ang 1992: Lam Lay Yong and Ang Tian Se, Fleeting Footsteps. Tracing the Conception of Arithmetic and Algebra in Ancient China, Singapore: World Scientific, 1992; rev. ed. 2004.

  • Li 1997: Li Di Zhongguo shuxue tongshi, shanggu dao wudai juan (A General History of Chinese Mathematics, from antiquity to the fifth century volume), Nanjing: Jiangsu jiaoyu chubanshe, 1997.

  • Li 2004: Zhongguo shuxue tongshi, Ming-Qing juan (A General History of Chinese Mathematics, Ming-Qing volume), Nanjing: Jiangsu jiaoyu chubanshe, 2004.

  • Li 1985: Li Xueqin Zhongguo shuxue shi shang de zhongda faxian (A Very Important Discovery for the History of Chinese Mathematics), Wenwu tiandi (Journal of Cultural Relics), 1(1985), p. 47.

  • Li and Du 1987: Li Yan and Du Shiran A Concise History of Chinese Mathematics, John N. Crossley and Anthony W.-C. Lun, trans., Oxford: Clarendon Press: 1987.

  • Loewe 1961: Loewe, Michael, “The Measurement of Grain During the Han Period,” T’oung Pao, 49 (1961), pp. 64–95.

  • Loewe 1967: Records of Han Administration, vol. 2: Documents, Cambridge, England: Cambridge University Press, 1967.

  • Loewe 1974: Crisis and Conflict in Han China, 104 BC to AD 9, London: George Allen & Unwin, 1974.

  • Loewe and Shaughnessy 1999: Loewe, Michael, and Edward L. Shaughnessy, eds., The Cambridge Modern History of Ancient China, Cambridge, England: Cambridge University Press, 1999.

  • Luo 1987: Luo Zhufeng ed., Hanyu da cidian (Comprehensive Dictionary of the Chinese Language), Shanghai: Shanghai cishu chubanshe, 1987, in 12 volumes.

  • Martzloff 1987: Martzloff, Jean-Claude, Histoire des mathématiques chinoises, Paris: Masson, 1987.

  • Martzloff 1997: A History of Chinese Mathematics, Stephen S. Wilson, trans., New York: Springer-Verlag, 1997.

  • Needham and Wang 1959: Needham, Joseph, and Wang Ling, Science and Civilisation in China. Vol. 3: Mathematics and the Sciences of the Heavens and the Earth, Cambridge, England: Cambridge University Press, 1959.

  • Ohkawa et al., 2004: Ohkawa Toshitaka Harikae Toshio and Tamura Makoto Sansûsho chu ni okeru yottsu no sandai ni tsuite (Concerning Four Computational Problems in the Suan shu shu), paper presented at a Conference on the Suan shu shuand Pre-Qin Mathematics, Institute for History of Natural Science, Chinese Academy of Sciences, Beijing, August 12–14, 2004, preprint. See: http://pal.las.osaka-sandai.ac.jp/~suanshu/j/publications.html

  • Ohkawa and Tamura 2004: Ohkawa Toshitaka and Tamura Makoto Chokasan kankan Sansusho inshitsu kai (Explanation of the Yinqi section of the Suan shu shu on Han bamboo slips from Zhangjiashan), (Journal of Osaka Sangyo University, Humanities), 114 (2004.10), pp. 1–21. See: http://pal.las.osaka-sandai.ac.jp/~suanshu/j/publications.html

  • Pan 1983: Pan Jixing “Lacquer and Lacquer Technique,” in Ancient China’s Technology and Science, Beijing: Foreign Languages Press, 1983, pp. 202–212.

  • Peng 2000: Peng Hao Zhongguo zuigu de shuxue zhuzuo Suan shu shu (The Earliest Arithmetic Book in Chinese, Suan shu shu), Wenwu (Cultural Relics), 9(2000), pp. 85–90.

  • Peng 2001: Zhangjiashan hanjian Suan shu shu zhushi (Commentary on the Suan shu shu Han Bamboo Slips from Zhangjiashan), Beijing: Kexue chubanshe, 2001.

  • Qian 1963: Qian Baocong ed., Suanjing shishu (Ten Mathematical Classics), Beijing: Zhonghua shuju, 1963.

  • Qian 1966: Song Yuan shuxueshi lunwenji (Collected Papers on the History of Mathematics During the Song and Yuan Dynasties), Beijing: Kexue chubanshe, 1966.

  • Sadao 1986: Sadao, Nishijima, “The Economic and Social History of Former Han,” in [Twichett and Loewe 1986, vol. 1, pp. 595–601].

  • Schrimpf 1963: Schrimpf, Robert, “La Collection mathématique ‘Souan king che chou.’ Contribution à l’histoire des mathématiques chinoises des origins au VIIe siècle de notre ère,” unpublished doctoral dissertation, Rennes, France, 1963.

  • Shaughnessy 1987: Shaughnessy, Edward L., Before Confucius: Studies in the Creation of the Chinese Classics, Albany, NY: State University of New York Press, 1987.

  • Shen, Crossley, and Lun 1999: Shen Kangshen , John N. Crossley, and Anthony W.-C. Lun, The Nine Chapters on the Mathematical Art: Companion and Commentary, Oxford: Oxford University Press, 1999.

  • Shuihudi 1977: Shuihudi Qinmu zhujian (Bamboo Texts from a Qin Tomb at Shuihudi), Beijing: Wenwu chubanshe, 1977, in 7 vols.

  • Shuowen 1996: Shuowen jiezi jizhu (Selected Commentaries on the Dictionary of Chinese Characters), by Xu Shen (2nd century, CE), Jiang Renjie, ed., Shanghai: Shanghai guji chubanshe, 1996, in 3 vols.

  • Swann 1950: Swann, Nancy Lee, Food and Money in Ancient China, Princeton, NJ: Princeton University Press, 1950; repr. New York: Octagon Books, 1974.

  • Swetz and Kao 1977: Swetz, Frank J., and T.I. Kao, Was Pythagoras Chinese? An Examination of Right Triangle Theory in Ancient China, University Park, PA: The Pennsylvania State University Press, 1977.

  • Tongxun 2000: Su Yiwen Su Junhong Su Huiyu Chen Fengzhu Lin Cangyi Huang Qingyang and Ye Jihai Suan shu shu jiaokan (Collation of the Suan shu shu), HPM HPM Tongxun (HPM Newsletter), 3(11)(2000), pp. 2–20.

  • Twitchett and Loewe 1986: Twitchett, Denis, and Michael Loewe, eds., The Cambridge History of China, Cambridge, England: Cambridge University Press, 1986.

  • Vogel 1968: Vogel, Kurt, tr., Chiu chang suan shu: Neun Bücher arithmetischer Technik, Ein Chinesiches Rechenbuch für den praktischen Gebrauch aus den frühen Hanzeit, Braunschweig: Friedrich Vieweg Verlag (Ostwalds Klassiker), 1968.

  • Wang 1956: Wang Ling, “The Chiu chang suan shu and the History of Mathematics during the Han Dynasty,” Ph.D. dissertation, Trinity College, Cambridge University, 1956. Ph.D. dissertation no. 2917–2918.

  • Wenwu 1976: Yunmeng Qin suan shiwen (A Transcription of Ch’in Dynasty Inscribed Bamboo Slips Unearthed at Yunmeng, Hubei Province), Wenwu (Cultural Relics), 7 (1976), pp. 1–11.

  • Wenwu 2000: Jiangling Zhangjiashan Hanjian zhengli xiaozu (Study Group for the Collation and Arrangement of the Han Dynasty Bamboo Slips from Zhangjiashan, Jiangling County), ed., Jiangling Zhangjiashan Hanjian Suan shu shu shiwen (Transcription of the Bamboo Suan shu shu or A Book of Arithmetic from Jiangling), Wenwu (Cultural Relics), 9 (2000), pp. 78–84.

  • Yu 1994: Yu Zhongxin Hanjian kaoli (Studying Calendars from Han Dynasty Wood and Bamboo slips), Taipei: Wenjin chubanshe, 1994.

  • Zhangjiashan 2001: Zhangjiashan ersiqi hao Hanmu zhujian zhengli xiaozu (Study Group for the Collation and Arrangement of the Bamboo Slips from Zhangjiashan Han Dynasty Tomb 247), ed., Zhangjiashan Hanmu zhujian (Bamboo Slips from Han Tombs at Zhangjiashan), Beijing: Wenwu chubanshe, 2001, pp. 83–96; 249–272.

  • Zou 2001: Zou Dahai Chutu Suan shu shu chutan (Preliminary Research on the Unearthed Mathematics Book Suan shu shu), Ziran kexueshi yanjiu (Studies in the History of Natural Sciences), 20(3)(2001), pp. 193–205.

  • Zou 2003: Cong Suan shu shu he Qin jiankan shanggu liangmi de bilu (Some Exchange Rates of Rice and Grain as Recorded in the Suan shu shu and Some Other Strips of Qin Dynasty), Ziran kexueshi yanjiu (Studies in the History of Natural Sciences), 22(4)(2003), pp. 318–328.

  • Zou 2004: Chu tu Suan shu shu jiaoshi yi ze (New Collation and Annotation on a Section of the Unearthed Suan shu shu), Dongnan wenhua (Southeast Culture), (2004)(2), pp. 83–85.

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Correspondence to Joseph W. Dauben.

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Communicated by J.-C. Martzloff

An erratum to this article can be found at http://dx.doi.org/10.1007/s00407-008-0023-0

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Dauben, J.W. Suan Shu Shu A Book on Numbers and Computations. Arch. Hist. Exact Sci. 62, 91–178 (2008). https://doi.org/10.1007/s00407-007-0124-1

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