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Circle Measurements in Ancient China

  • Lam Lay-Yong
  • Ang Tian-Se

Abstract

This paper discusses the method of Liu Hui (3rd century) for evaluating the ratio of the circumference of a circle to its diameter, now known as π A translation of Liu’s method is given in the Appendix. Also examined are the values for or given by Zu Chongzhi (429–500) and unsurpassed for π a millenium. Although the method used by Zu is not extant, it is almost certain that he applied Liu’s method. With the help of an electronic computer, a table of computations adhering to Liu’s method is given to show the derivation of Zu’s results. The paper concludes with a survey of circle measurements in China. © 1986 Academic Press, Inc.

Keywords

Decimal Place Song Dynasty Mathematical Text Decimal Fraction Standard History 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ang Tian Se. 1977. Chinese computation with the counting-rods. Papers on Chinese Studies, University of Malaya 1, 97–109.Google Scholar
  2. Bai Shangshu [ap]. 1982. Cong Wang Mang Jiang qi dao Liu Xin yuan lu [er] [From Wang Mang’s measuring vessel to Liu Xin’s ratio of circumference to diameter]. Beijing daxue xuebao, No. 2, 75–79.Google Scholar
  3. Bai Shangshu 1983. Jiu zhang suanshu zhushi [es] [Annotations on Nine chapters of the mathematical art]. Beijing: Kexue chubanshe.Google Scholar
  4. Beckmann, P. 1970. A history of ir (pi). All page references are to the second edition, Boulder, Colo.: Golem, 1971.Google Scholar
  5. Gupta, R. C. 1975. Some ancient values of pi and their use in India. Mathematics Education 9, 1–5. He Shaogeng [bb]. 1983. Method for determining segment areas and evaluation of a. In Ancient Google Scholar
  6. China’s technology and science. Compiled by the Institute of the History of Natural SciencesGoogle Scholar
  7. Chinese Academy of Sciences. Beijing: Foreign Language Press.Google Scholar
  8. Heath, T. L. (ed.). 1897. The works of Archimedes. All page references are to the reissued edition, New York: Dover, 1953.Google Scholar
  9. Heath, T. L. 1921. A history of Greek mathematics. 2 vols. London/New York: Oxford Univ. Press (Clarendon).Google Scholar
  10. Lam Lay-Yong. 1969. The geometrical basis of the ancient Chinese square-root method. /.sis 61, 92101.Google Scholar
  11. Lam Lay-Yong and Shen Kangsheng. 1985. The Chinese concept of Cavalieri’s principle and its applications. Historia Mathematica 12, 219–228.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Li Di [bg]. 1962. Da kexuejia Zu Chongzhi [et] [The great scientist Zu Chongzhi]. Shanghai: Renm in chubanshe.Google Scholar
  13. Li Di [bg]. 1982. Jiu zhang suanshu zhengming wentide gai shu [eu] [A summary of the various viewsGoogle Scholar
  14. on the Jiu zhang suanshu]. In Jiu zhang suanshu yu Liu Hui [cw] [The Jin zhang suanshu and Liu Hui], Wu Wenjun [cv], ed., pp. 28–50. Beijing: Shifan daxue chubanshe.Google Scholar
  15. Mikami, Y. 1913. The development of mathematics in China and Japan. All page references are to the second edition, New York: Chelsea, 1974.Google Scholar
  16. Needham, J. 1959. Science and civilisation in China. Vol. 3. Cambridge: Cambridge Univ. Press.zbMATHGoogle Scholar
  17. Neugebauer, O. 1952. The exact sciences in antiquity. All page references are to the second edition, Providence, R.I.: Brown Univ. Press, 1957.zbMATHGoogle Scholar
  18. Qian Baocong [al]. 1923. Zhongguo suan shu zhong zhi zhoulu yanjiu [ex] IA study of ar in Chinese mathematical texts]. In Qian Baocong kexueshi lunwen xuanji [cy] [Selected essays of Qian Baocong on the history of Chinese science], pp. 50–74. All page references are to the 1983 book edition, Beijing: Kexue chubanshe 1983.Google Scholar
  19. Qian Baocong (ed.). 1963. Suanjing shi shu [cz] [Ten mathematical manuals]. Shanghai: Zhonghua shuju.Google Scholar
  20. Qudan Xida [da]. 729. Kaiyuan zhan jing [aq] [Kaiyuan treatise on astrology].Google Scholar
  21. Ruan Yuan [db]. 1799. Chou ren zhuan [dc] [Biographies of mathematicians and scientists]. In Guoxue jiben congshu. Taipei: Shangwu yinshuguan, 1968.Google Scholar
  22. Song shu [ar]. [Standard history of the Liu Song dynasty]. 1973 edition. Beijing: Zhonghua shuju. Sui shu [y] [Standard history of the Sui dynasty]. 1973 edition. Beijing: Zhonghua shuju.Google Scholar
  23. Sun Zhifu [an]. 1955. Zhongguo gudai shuxue jia guanyu yuanzhoulu yanjiu de chengjiu [dd] [Achievements in the research of it by ancient Chinese mathematicians]. Shuxue Tongbao, No. 5, 5–12.Google Scholar
  24. Wagner, D. 1978. Doubts concerning the attribution of Liu Hui’s commentary on the Chiu-chang suan-shu. Acta Orientalia 39, 199–212.Google Scholar
  25. Wang, L., and Needham, J. 1955. Homer’s method in Chinese mathematics: Its origins in the root-extraction procedures of the Han dynasty. T’oung Pao 43, 345–401.CrossRefGoogle Scholar
  26. Xu Chunfang [ao]. 1957. Zhong suan jiade jihexue yanjiu [de] [Research on geometry by Chinese mathematicians]. Hong Kong.Google Scholar
  27. Yan Dunjie [as]. I936a. Sui shu lull zhi Zu Chongzhi yuan lu ji shi shi [df] [An analysis of records for it by Zu Chongzhi as found in the harmonics and calendar chapters of Sui shu]. Xue yi zazhi [dg] (Shanghai) 15, No. 10 27–57.Google Scholar
  28. Yan Dunjie 1936b. Zhongguo suanxuejia Zu Chongzhi ji qi yuanzhou lu zhi yanjiu [dh] [Chinese mathematician Zu Chongzhi and his work on ir]. Xue yi zazhi [dg] (Shanghai) 15, No. 5, 37–50.Google Scholar
  29. Youschkevitch, A. P., and Rosenfeld, B. A. 1973. Al-Kashi (or Al-Kashani), Ghiyath al-din Jamshid Mas’ud. In Dictionary of scientific biography, Vol. 7, pp. 255–262. New York: Scribner’s.Google Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Lam Lay-Yong
    • 1
  • Ang Tian-Se
    • 2
  1. 1.Department of MathematicsNational University of SingaporeSingaporeRepublic of Singapore
  2. 2.Department of Chinese StudiesUniversity of MalayaKuala LumpurMalaysia

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