回首頁

回第08期

由考古發現看中國古代數學的演化（下）
從《數》、《筭數書》到《九章算術》

全文閱覽

作        者　道本周 Joseph W. Dauben

作者簡介　道本周（Joseph W. Dauben）哈佛大學博士，曾訪問普林斯頓高等研究院、劍橋大學，現在是紐約市立大學勒曼學院的科學史傑出教授。研究興趣廣泛包括科學史、數學史、科學社會學、中國科學史。著有兩本知名數學家傳記：Georg Cantor（《康托》）與Abraham Robinson（《羅賓森》）。2012 年獲美國數學學會懷特曼數學史獎。

譯        者　林倉億

譯者簡介　林倉億畢業於師大數學所，現為臺南一中教師、清華歷史所博士班學生，著有《數之起源》（合著），譯有《爺爺的證明題》（合譯）與《溫柔數學史》（合譯），曾任《HPM 通訊》副主編。

本文出處    2012 年，作者擔任交通大學人文與社會科學研究中心的訪問學者。作者將該年在清華大學演講的材料整理後，除受邀於同年9 月20日在哈佛大學費正清中國研究中心演講下列主題：The Evolution of Mathematics in Ancient China: From the Newly Discovered 數Shu and 算數書 Suan shu shu Bamboo Texts to 九章算術 the Nine Chapters on the Art of Mathematics .“，並將演講內容發表於Notices of the ICCM 2（2014）no.2。作者感謝交大邀訪，與當時郭書春、洪萬生、徐光台、琅元（Alexei Volkov）、鄒大海提供的寶貴意見。本文（上）篇已刊登於上期。

延伸閱讀 

• Netz, Reviel & Noel, William The Archimedes Codex：Revealing The Secrets of The World’s Greatest Palimpsest（2007），中文譯本：曹亮吉譯《阿基米德寶典─失落的羊皮書》（2007），天下文化。
• 阿基米德羊皮書的普及網站，可以找到各種資料，以及更仔細的成書說明：http://archimedespalimpsest.org/ 其中底下網頁提供歷史說明與影片：http://archimedespalimpsest.org/about/history/index.php
  另外，底下是阿基米德羊皮書本身頁面的專業網站 http://www.archimedespalimpsest.net
• 李宗元 〈祖沖之、球體公式及其他〉（1977），《數學傳播》第1 卷第４期。也可見《數學知識》網頁http://episte.math.ntu.edu.tw/articles/mm/mm_01_4_01/index.html
  

參考資料
References　　（●基本上同上期，但加下列一段）

《方法》書名之討論：

至於我們有興趣的這份手抄本的常見標題──The Method──雖可追溯到海伯格的版本，卻是有問題的翻譯。克諾布洛（Eberhard Knobloch）指出：「代表『程序』（procedure）這個概念的希臘文是ephodos。阿基米德在寫給埃拉托斯特尼 （Eratosthenes）的著名信件中，用的也是完全相同的概念。在這兩個例子中，ephodos一定不能譯作『方法』（method）。當阿基米德 提到方法，他指的是今日我們說的『力學方法』（mechanical method），也就是他在這封信中用的字tropos。」（[Knobloch]）泰斯巴克（C. M. Taisbak）則指出ephodos「並非今日『方法』（Method）的意義，而是『進路』（approach）的意思。當它出自阿基米德的序言時， 用『由後門進入力學問題』（●[斜體]Entering Mechanical Problems by the Back Door●）的標題比較能代表他的態度，畢竟他知道這是啟發法（heuristics）而非演繹法（deduction）。」（[Taisbak] ）

然而，我自己對 Ephodos 的解讀，則是將它譯作「攻擊」（attack）在古代與現代希臘語中的意義──不是任意的攻擊，而是有系統、有方法、小心謹慎的攻擊。就像阿基米德這類數學家對某一個特別困難、具挑戰性的問題可能會發動的攻擊。

參考資料：

一、中文部分
[白尚恕1982]白尚恕〈《九章算術》與劉徽的幾何理論〉, 吳文俊主編 《《九章算術》與劉徽》(1982)北京師範大學出版社: 137–161.
[白尚恕1983]白尚恕《《九章算術》注釋》(1983, 北京科學出版社.
[杜石然]杜石然〈祖暅之公理〉, 《數學通報》3 (1954): 9–11.
[李儼1937]李儼《中國算學史》(1937), 上海商務印書館, 1954年重印.
[李儼1954]李儼《中國古代數學史料》(1954), 上海科學技術出版社, 1963年再版.
[洪萬生] 洪萬生〈古代中國的幾何學〉,《科學月刊》(1981) 12: 22–30.
[郭世榮]郭世榮〈《算數書》勘誤〉,《內蒙古大學學報（自然科學版）》30 (2001) no. 3: 276–285.
[郭書春1990]郭書春《匯校《九章算術》》(1990), 遼寧教育出版社.
[郭書春1993]郭書春主編《中國科學技術典籍通匯: 數學卷》五冊(1993), 河南教育出版社.
[郭書春2001]郭書春〈《算數書》校勘〉,《中國科技史料》22 (2001) no.3: 202–219.
[通訊] 蘇意雯, 蘇俊鴻, 蘇惠玉, 陳鳳珠, 林倉億, 黃清陽, 葉吉海〈《算數書》校勘〉, 《HPM通訊》3 (2000) no.11: 2–20.
[彭浩2000]彭浩〈中國最早的數學著作《筭數書》〉,《文物》(2000) no. 9: 85–90.
[彭浩2001]彭浩《張家山漢簡〈算數書〉注釋》(2001), 北京科學出版社.
[楊輝]楊輝《詳解九章算法》(1261). 重印於[郭書春1993]第一冊.
[鄒大海2001]鄒大海 〈出土《算數書》初探〉,《自然科學史研究》20 (2001) no. 3: 193–205.
[錢寶琮1963]錢寶琮校點《算經十書》(1963), 北京中華書局.
[錢寶琮1964]錢寶琮《中國數學史》(1964), 北京科學出版社.
[蕭燦]蕭燦《嶽麓書院藏秦簡《數》研究》(2010), 湖南大學博士論文.
[WW2000]江陵張家山漢簡整理小組編〈江陵張家山漢簡《算數書》釋文〉, 《文物》9 (2000): 78–84.
[ZJS]張家山二四七號漢墓竹簡整理小組編《張家山漢墓竹簡（二四七號墓）》-(2001), 北京文物出版社.

二、英文部分
[Berlin]Berlin, J. “World’s Oldest Decimal Times Table Found in China.” National Geographic (April 5, 2014): http://news.nationalgeographic.com/news/2014/04/140405-chineseoldest-multiplication-table-decimal/.
[B & D]Black, M. and Davidson, R. Constantin von Tischendorf and the Greek New Testament. Glasgow: University of Glasgow Press, 1981.
[Bloom]Bloom, A. The Linguistic Shaping of Thought: A Study in the Impact of Language on Thinking in China and the West. Hillsdale, N.J.: Erlbaum Associates, 1981.
[Burnet]Burnet, J. The Fragments of Heraclitus. London, 1920.
[Chemla1]Chemla, K. “What is at Stake in Mathematical Proofs from Third-Century China?” Science in Context 10 (2) (1997): 227–251.
[Chemla2]Chemla, K. “Fractions and Irrationals between Algorithm and Proof in Ancient China.” Studies in History of Medicine and Science 15 (1–2) (1997–1998): 31–54.
[Cullen1993]Cullen, C. “Chiu chang suan shu 九章算術.” In Early Chinese Texts: a Bibliographical Guide, Michael Loewe, ed. Early China Special Monograph Series 2. Berkeley: Society for the Study of Early China, Institute of East Asian Studies, University of California, 1993.
[Cullen2002]Cullen, C. “Learning from Liu Hui? A Different Way to Do Mathematics.” Notices of the American Mathematical Society 49 (7) (2002): 783–790.
[Cullen2004]Cullen, C. “The Suan shu shu 筭數書‘Writings on reckoning’: A translation of a Chinese mathematical collectionof the second century BC, with explanatory commentary.” Needham Research Institute Working Papers 1. Cambridge: Needham Institute, 2004.
[Dauben2008]Dauben, J. W. “算數書Suan Shu Shu (A Book on Numbers and Computation). English Translation, Notes and Critical Commentary.” Archive for History of Exact Sciences 62 (2) (2008): 91–178.
[DGX] Dauben, J. W., Guo, S. and Xu, Y. 九章筭術 Nine Chapters on the Art of Mathematics. A Critical Edition and English Translation based upon a New Collation of the Ancient Text and Modern Chinese Translation. Shenyang: Liaoning Education Press, 2013.
[Elman]Elman, B. A. Review of (Bloom 1981). In The Journal of Asian Studies 42 (3) (1983): 611–614.
[馮立昇&徐義保]Feng, L. (馮立昇) and Xu, Y. (徐義保) “The Tsinghua Multiplication Table.” Mathematical Knowledge at Work in Ancient China, Special Symposium S115-A of the Twenty-Fourth International Congress of History of Science, Technology and Medicine, The University of Manchester, UK, July 23, 2013. http://www.ichstm2013.com/programme/guide/i/10953.html.
[Heiberg]Heiberg, J. L. Geometrical Solutions Derived from Mechanics. A Treatise of Archimedes. Chicago: Open Court, 1909.
[Hulswé]Hulswé, A. F. P. “Weights and Measures in Ch’in Law.” In State and Law in East Asia. Festschrift Earl Bünger, Dieter Eikemeier and Herbert Franke, eds. Wiesbaden: Otto Harrassowitz, 1981: 25–39.
[Kiang]Kiang, T. “An old Chinese way of finding the volume of a sphere.” The Mathematical Gazette 56 (1972): 88–91.
[Kirk]Kirk, G. S. The Cosmic Fragments/Heraclitus. Cambridge, UK: Cambridge University Press, 1954.
[Knobloch]Knobloch, E. “Commentary on ‘Cleomedes and the Measurement of the Earth: A Question of Procedures’ by Alan C. Bowen, Centaurus 2003, 45, pp. 59–68.” Centaurus 50 (1–2) (2008): 205.
[藍麗蓉&洪天賜]Lam Lay Yong（藍麗蓉）, Ang Tian Se（洪天賜）. Fleeting Footsteps, Tracing the Concept of Arithmetic and Algebra in Ancient China (1992). World Scientific.
[藍麗蓉]Lam, Lay-Yong (藍麗蓉). “Jiu Zhang Suanshu 九章算術 (Nine Chapters on the Mathematical Art): An Overview.” Archive for History of Exact Sciences 47 (1) (1994): 1–51.
[藍麗蓉&沈康身]Lam, L.-Y. (藍麗蓉)and Shen, K. (沈康身) “The Chinese Concept of Cavalieri’s Principle and its Applications.” Historia Mathematica 12 (1985): 219–228.
[李零&Cook]Li, L.(李零) and Cook, C. A. “Translation of the Chu Silk Manuscript.” In Defining Chu: Image and Reality in Ancient China, Constance A. Cook and John S. Major, eds. Honolulu: Hawaii University Press, 1999: 171–176.
[李儼&杜石然]Li, Yan (李儼) and Du, Shiran (杜石然). A Concise History of Chinese Mathematics. John N. Crossley and Anthony W.-C. Lun, trans. Oxford: Clarendon Press: 1987.
[Lloyd]Lloyd, G. E. R. Adversaries and Authorities. Investigations into Ancient Greek and Chinese Science. Cambridge, England: Cambridge University Press, 1996.
[Lloyd&Sivin]Lloyd, G. E. R. and Sivin, N. The Way and the Word (道 dao and λóγoς logos). Science and Medicine in Early China and Greece. New Haven: Yale University Press, 2002.
[Loewe]Loewe, M. Records of Han Administration, vol 2: Documents. Cambridge, England: Cambridge University Press, 1967.
[Martzloff1987]Martzloff, J.-C. Histoire des mathématiques chinoises. Paris: Masson, 1987.
[Martzloff1997]Martzloff, J.-C. A History of Chinese Mathematics. Stephen S. Wilson, trans. Berlin: Springer, 1997; rev. ed. 2006.
[三上義夫]Mikami, Yoshio (三上義夫). The Development of Mathematics in China and Japan (Abhandlugen zur Geschichte der mathematischen Wissenschaften 30 (1913)); repr. New York: Chelsea, 1961.
[Needham&Wang]Needham, J. and Wang, Ling (王鈴). Science and Civilisation in China. Vol. 3. Mathematics and the Sciences of the Heavens and the Earth. Cambridge: Cambridge University Press, 1959.
[Netz&Noel]Netz, R. and Noel, W. The Archimedes Codex. How a medieval prayer book is revealing the true genius of antiquity’s greatest scientist. Cambridge, MA: Da Capo Press, 2007.
[NST]Netz, R., Saito, K., and Tchernetska, N. “A New Reading of Method Proposition 14: Preliminary Evidence from the Archimedes Palimpsest (Part 1).” Sciamvs 2 (2001): 9–29; and Reviel Netz, Ken Saito, and Natalie Tchernetska. “A New Reading of Method Proposition 14: Preliminary Evidence from the Archimedes Palimpsest (Part 2).” Sciamvs 3 (2002): 109–125.
[邱瑾]Qiu, J. (邱瑾) “Ancient Times Table Hidden in Chinese Bamboo Strips.” Nature (January 7, 2014): http://www.nature.com/news/ancient-times-table-hidden-in-chinese-bamboostrips-1.14482.
[Sato]Sato, T. “A Reconstruction of The Method Proposition 17, and the Development of Archimedes’ Thought on Quadrature—Why did Archimedes not notice the internal connection in the problems dealt with in many of his works?” Historia Scientiarum 31 (1986): 61–86.
[Seidenberg]Seidenberg, A. “On the Volume of a Sphere.” Archive for History of Exact Sciences 39 (1988/89): 97–119.
[沈康身]Shen, Kangshen (沈康身). The Nine Chapters on the Mathematical Art: Companion and Commentary. John N. Crossley and Anthony W.-C. Lun, trans. Oxford: Oxford University Press, 1999.
[Swetz&Kao]Swetz, F. J. and Kao, T. I. Was Pythagoras Chinese? An Examination of Right Triangle Theory in Ancient China. University Park, PA: The Pennsylvania State University Press, 1977.
Taisbak, C. M. Historia Mathematica. Mailing List Archive: 23 Jun 1999 19:32:38 +0200 [http://sunsite.utk.edu/math_archives/.http/hypermail/historia/jun99/0149.html].
[vdW]van der Waerden, B. L. Geometry and Algebra in Ancient Civilizations. Berlin: Springer-Verlag, 1983.
[Volkov]Volkov, A. “Geometrical Diagrams in Traditional Chinese Mathematics.” In Graphics and Text in the Production of Technical Knowledge in China. The Warp and the Weft, Francesca Bray, Vera Dorofeeva-Lichtmann, and Georges Metailie, eds. Leiden: Brill, 2007: 425–460.
[Wagner1978a]Wagner, D. “Liu Hui and Zu Gengzhi on the Volume of a Sphere.” Chinese Science 3 (1978): 59–79.
[Wagner1978b]Wagner, D. “Doubts concerning the attribution of Liu Hui’s commentary on the Chiu-chang suan-shu.” Acta Orientalia (Copenhagen), 39 (1978): 199–212.
[Wagner1979]Wagner, D. “An early Chinese derivation of the volume of a pyramid: Liu Hui, third century A.D.” Historia Mathematica 6 (1979): 164–188.
[吳文俊]Wu, Wenjun (Wentsun) (吳文俊). “The out-in complementary principle.” In Ancient China’s Technology and Science, Institute for the History of Natural Sciences, Chinese Academy of Sciences, ed. Beijing: Foreign Languages Press, 1983: 66–89.
[Wylie1852]Wylie, A. “Jottings on the Science of Chinese Arithmetic.” In North China Herald (August–November, 1852); reprinted in (Wylie 1897: 159–194).
[Wylie1897]Wylie, A. Chinese Researches. Shanghai: n.p., 1897; repr. Taipei: Ch’eng-Wen Publishing, 1966. http://babel.hathitrust.org/cgi/pt?id=mdp.39015058016513#page/n8/mode/1up (accessed January, 2013).
[鄒大海2007a]Zou, Dahai (鄒大海). “Shuihudi bamboo strips of the Qin Dynasty and mathematics in Pre-Qin Period.” Frontiers of History in China 2 (4) (2007a): 632–654.
[鄒大海2007b]Zou, Dahai (鄒大海). “Shuihudi’s Bamboo Strips of Qin Dynasty and Mathematics in Pre-Qin Period.” Chinese Archaeology 7 (2007b): 132–136.