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The procedure of the Section of Pieces of Areas in Li Ye and Yang Hui’s works: genealogy of diagrams and equations

Published online by Cambridge University Press:  02 October 2020

Charlotte-V. Pollet Open the ORCID record for Charlotte-V. Pollet [Opens in a new window]
Charlotte-V. Pollet*
Affiliation:
Center for General Education, National Chiao-Tung University
*
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Argument

The study of algebra in China has often focused on the algebraic “procedure of the Celestial Source.” Its geometrical ancestors are less known. The Yigu yanduan, authored by Li Ye (1192-1279), presents the procedure alongside its two geometrical counterparts, the “Section of Pieces [of Areas]” and the “Old Procedure.” The three procedures are known to represent three generations of algorithms used to set up quadratic equations. A similar geometrical procedure appears in a treatise written by Yang Hui (second half of thirteenth century). Although the procedures look alike at first glance, the two treatises reveal different moments in their work on the relation between counting materials and geometrical representation. This study challenges their chronology trying to identify the meander of the geometrical roots of the Celestial Source. The construction of negative coefficients plays a pivotal role in this mutation and shows several layers of composition.

Keywords

algorithmic equationgeometrynegative coefficientChinaLi YeYang Hui
Type
Research Article
Information
Science in Context , Volume 33 , Issue 1 , March 2020 , pp. 37 - 63
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© The Author(s) 2020. Published by Cambridge University Press

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References

References

Ye, Li. 1248. 測圓海鏡 (Ceyuan haijing [Sea Mirror of Circle Measurements]).Google Scholar
Ye, Li. 1259. 益古演段 (Yigu yanduan [The Development of Pieces of Areas according to the Collection Augmenting the Ancient Knowledge]).Google Scholar
Both edited in:Google Scholar
(1) Edition of 1789, 四庫全書, 文淵閣 (Wen yuan ge Siku quanshu [Complete Library of the Four Treasuries, Wenyuan pavilion]). Original edition, National Palace Museum, Taiwan.Google Scholar
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(3) Edition of 1798, 知不足齋叢書 (Zhibuzu zhai congshu [Collected Works of the Private Library of the Never Sufficient Knowledge]). Reprint in中國科學技術典籍通彙: 數學卷 (Zhongguo kexue jishu dianji tong hui: Shuxue juan [Chinese Science and Technology Books Collectanea, Mathematics]). Guo Shuchun (ed) 郭书春. 河南教育出版社 (Henan jiaoyu chubanshe [Henan education press]), Vol. 1 1993.Google Scholar
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Chemla, Karine. 1982. “Etude du livre Reflets des mesures du cercle sur la mer de Li Ye.” Ph. D. diss., University of Paris XIII.Google Scholar
Chemla, Karine. 1993. “李冶《測圓海鏡》的結構及其對數學知識的表述 (Li ye ceyuan hai jing de jiegou ji qi dui shuxue zhishi de biaoshu [The Structure and Representation of Mathematical Knowledge of Ceyuan haijing by Li Ye]).” 數學史研究文集第五輯 (Shuxue shi yanjiu wenji di wu ji [History of mathematics study series, volume 5]), 123-142.Google Scholar
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Chemla, Karine, Shuchun, Guo. 2004. Les neuf chapitres: Le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.Google Scholar
Chemla, Karine. 2017. “Changing Mathematical Culture, Conceptual History, and the Circulation of Knowledge.” In Culture without Culturalism: The Making of Scientific Knowledge, edited by Chemla, Karine and Keller, Evelyn Fox. Durham, North Carolina: Duke University Press.Google Scholar
Dauben, Joseph. 2007. “Chinese Mathematics”. In The Mathematics of Egypt, Mesopotamia, China, India and Islam, A Source Book, edited by Katz, Victor, 187–380. Princeton and Oxford: Princeton University Press.Google Scholar
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Guo, Shuchun. 郭书春. 1991. 中国古代数学 (Zhonguo gudai shuxue [Mathematics in ancient China]), 1-174. 山东教育出版社 (Shandong jiaoyu chubanshe [Shandong educational press]).Google Scholar
Guo, Shuchun. 郭書春. 2010. 中國科學技術史. 數學卷. (Zhongguo kexue jishu shi: Shuxue juan [History of Chinese science and technology: Mathematics]). 科學出版社, 北京. (Kexue chubanshe [Beijing Science Press]).Google Scholar
Peng Yoke, Ho. 1973. “Li Chih”. In Dictionary of Scientific Biography, edited by Gillispie, Charles, vol. 8:313–320. Charles Scribner’s Son.Google Scholar
Jock, Hoe. 2008. The Jade Mirror of the Four Unknowns by Zhu Shijie. Christchurch. New Zealand: Mingming bookroom.Google Scholar
Horiuchi, Annick. 2000. “La notion de yanduan: Quelques réflexions sur les méthodes ‘algébriques’ de la résolution de problèmes en Chine aux X° et XI°.” In Oriens-Occidens 3:235–258.Google Scholar
Kong, Guoping. 孔國平. 1987. 李冶传 (Li Ye zhuan [Biography of Li Ye]). 北京: 河北教育出版社 (Beijing: Hebei jiaoyu chubanshe [Beijing: Hebei educational press]).Google Scholar
Kong, Guoping. 孔國平. 1996. 測圓海鏡導讀 (Ceyuan haijing daodu [Introduction to Ceyuan Haijing]). In 中華傳統數學名著導讀叢書 (Zhonghua chuantong shuxue mingzhu daodu congshu [Collection of masterpieces in Chinese traditional mathematics]), 湖北教育出版社 (Hubei jiaoyu chubanshe [Hubei educational press]).Google Scholar
Kong, Guoping. 孔國平. 1999. 李冶朱世杰與金元數學 (Li Ye Zhu Shijie yu jinyuan shuxue [Li Ye, Zhu Shijie and Jin-Yuan Mathematics]). In 中國數學史大系 (Zhongguo shuxueshi daxi [Collection of the history of Chinese mathematics]). 河北科學技術出版社 (Hebei kexue jishu chubanshe [Hebei science and technology press]).Google Scholar
Lam, Lay-Yong. 1977. A Critical Study of Yang Hui suanfa: A Thirteenth-Century Chinese Mathematical Text. Singapore: Singapore University Press.Google Scholar
Lam, Lay-Yong. 1984. “Li Ye and his Yi Gu Yan Duan: Old mathematics in Expanded Sections.” Archive for History of Exact Sciences 29:237–266.Google Scholar
Lam Lay-Yong, Ang Tian Se. 2004. Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China. Singapore: World Scientific.CrossRefGoogle Scholar
Li, Di. 李迪. 1997. 天元術與李冶 (Tianyuanshu yu Li Ye [The procedure of celestial source and Li Ye]). 中國數學通史. 宋元卷. 第四章 (Zhongguo shuxue tongshi. Song Yuan juan. Di si zhang. [Collection on the history of Chinese mathematics.,Song Yuan, chap. 4]). 江蘇教育出版社 (Jiangsu jiaoyu chubanshe [Jiangsu educational press]):184-239.Google Scholar
Li, Yan, Shiran, Du. 1987. Chinese Mathematics: A Concise History. Translated by Crossley, J. N. and Lun, W.C.. Oxford: Clarendon Press.Google Scholar
Martzloff, Jean-Claude. 1987. Histoire des mathématiques chinoises. Paris: Masson. English edition: A History of Chinese Mathematics. 1997. Springer.Google Scholar
Mei, Rongzhao, 梅荣照. 1966. 李冶及其數學著作 (Li ye ji qi shuxue zhuzuo [Li Ye and his mathematical works]). In 宋元術學史論文集 (Song Yuan shuxue shi lunwenji [Collection of articles on Song-Yuan history of mathematics]), edited by Qian Baocong, 104-148. 科學出版社 (Kexue chubanshe [Technology press]).Google Scholar
Morgan, Daniel Patrick. 2015. “What Good’s a Text? Textuality, Orality, and Mathematical Astronomy in early Imperial China.” Archives internationales d’histoire des sciences, Académie international d’histoire des sciences. 65(2):549–572.Google Scholar
Needham, Joseph. 1954. Science and Civilisation in China 3:42–151. Cambridge UK: Cambridge University Press.Google Scholar
Pollet, Charlotte. 2012. Comparison of Algebraic Practices in Medieval China and India. 2 vols. Ph.D diss. University of Paris. 7/National Taiwan Normal University.Google Scholar
Pollet, Charlotte. 2014. “The Influence of Qing Dynasty Editorial Work on the Modern Interpretation of Mathematical Sources: The Case of Li Rui’s Edition of Li Ye’s Mathematical Treatises.” Science in Context 27 (3):385–422.CrossRefGoogle Scholar
Pollet, Charlotte, and Jiamin, Ying. 2017. “One Quadratic Equation, Different Understandings: The 13th Century Interpretation by Li Ye and Later Commentators in the 18th and 19th Centuries.” Journal for History of Mathematics 30 (3):137–162.CrossRefGoogle Scholar
Te, Gusi. 特古斯. 1990. 劉益及其佚著 “議古根源” (Liu Yi ji qi yi zhu “Yigu gen yuan” [Liu Yi and his lost work, the ‘Yigu genyuan’]. In 数学史研究文集, (Shuxueshi yanjiu wenji [Research on history of mathematics series]). Edited by Li Di李迪, 九章出版社 (Jiu zhang chubanshe [The Nine Chapters Press]) 1:56-63.Google Scholar
Volkov, Alexei. 2001. ‘Le Bacchette.’ In ‘La scienza in Cina’ Chemla, Bray, Fu, Huang and Metaile (Eds). In ‘Storia della scienza’ (8 vols.). Edited by S. Petrucioli, et al. Instituto della Enciclopedia Italiana, section I, vol. 2:125-133.Google Scholar
Xu, Yibao. 徐义保. 1990. 對 “益古集”的復原與研究 (Dui “yi gu ji” de fuyuan yu yanjiu [Restoration and Research on the Collection Augmenting the Ancient [Knowledge]). In 數學史研究文集 (Shuxueshi yanjiu wenji [Research on History of Mathematics Series]). Edited by Li Di 李迪, 九章出版社 (Jiu zhang chubanshe [The Nine Chapters Press]) 1:149-165.Google Scholar
Yiwen, Zhu. 2016. “Different Cultures of Computation in Seventh Century China from the Viewpoint of Square Root Extraction.” Historia Mathematica 43 (1):3–25.Google Scholar
Zürcher, Erik. 2007. The Buddhist Conquest of China: The Spread and Adaptation of Buddhism in Early Medieval China, 3rd ed. Leiden: Brill.Google Scholar
Ye, Li. 1248. 測圓海鏡 (Ceyuan haijing [Sea Mirror of Circle Measurements]).Google Scholar
Ye, Li. 1259. 益古演段 (Yigu yanduan [The Development of Pieces of Areas according to the Collection Augmenting the Ancient Knowledge]).Google Scholar
Both edited in:Google Scholar
(1) Edition of 1789, 四庫全書, 文淵閣 (Wen yuan ge Siku quanshu [Complete Library of the Four Treasuries, Wenyuan pavilion]). Original edition, National Palace Museum, Taiwan.Google Scholar
(2) Edition of 1789, 四庫全書, 文津閣 (Wen jin ge Siku quanshu [Complete Library of the Four Treasuries, Wenjin Pavilion]). Vol. 799. Reprint. 2005.Google Scholar
(3) Edition of 1798, 知不足齋叢書 (Zhibuzu zhai congshu [Collected Works of the Private Library of the Never Sufficient Knowledge]). Reprint in中國科學技術典籍通彙: 數學卷 (Zhongguo kexue jishu dianji tong hui: Shuxue juan [Chinese Science and Technology Books Collectanea, Mathematics]). Guo Shuchun (ed) 郭书春. 河南教育出版社 (Henan jiaoyu chubanshe [Henan education press]), Vol. 1 1993.Google Scholar
Hui, Yang, 楊輝, 1275. 楊輝算法, Yang Hui suanfa [Yang Hui computational methods]. 2nd book: 田畝比類乘除捷法, Tian mu bilei cheng chu jie fa [Fast methods of multiplication and division related to various categories of fields and their measures]. 1433 Manuscript preserved in Beijing library based on a 1378 Ming edition, reprint in中國科學技術典籍通彙: 數學卷 (Zhongguo kexue jishu dianji tong hui: Shuxue juan [Chinese sScience and technology books collectanea, Mathematics]). Guo Shuchun (ed), 郭书春. 河南教育出版社 (Henan jiaoyu chubanshe [Hunan educational press]). 1993. Vol 1.Google Scholar
Shijie, Zhu, 朱世杰, 1303. 四元玉鉴, si yuan yu jian [The Jade Mirror of the Four Unknowns]. Translated into English by Ch’en Tsai Hsin (Chen Zaixin) 陳在新, 1925, reedited and completed by Guo Shuchun, 郭書春, and Guo Jinhai,郭金海. 大中華文庫, 遼寧教育出版社 (da zhonghua wenku [Library of the Chinese Classics], Liaoning jiaoyu chubanshe [Liaoning educational press]). 2006Google Scholar
Chemla, Karine. 1982. “Etude du livre Reflets des mesures du cercle sur la mer de Li Ye.” Ph. D. diss., University of Paris XIII.Google Scholar
Chemla, Karine. 1993. “李冶《測圓海鏡》的結構及其對數學知識的表述 (Li ye ceyuan hai jing de jiegou ji qi dui shuxue zhishi de biaoshu [The Structure and Representation of Mathematical Knowledge of Ceyuan haijing by Li Ye]).” 數學史研究文集第五輯 (Shuxue shi yanjiu wenji di wu ji [History of mathematics study series, volume 5]), 123-142.Google Scholar
Chemla, Karine. 1996. “Positions et changements en mathématiques à partir des textes chinois des dynasties Han à Song-Yuan: Quelques remarques.” Extrême-Orient, Extrême-Occident 18:115–147.Google Scholar
Chemla, Karine, Shuchun, Guo. 2004. Les neuf chapitres: Le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod.Google Scholar
Chemla, Karine. 2017. “Changing Mathematical Culture, Conceptual History, and the Circulation of Knowledge.” In Culture without Culturalism: The Making of Scientific Knowledge, edited by Chemla, Karine and Keller, Evelyn Fox. Durham, North Carolina: Duke University Press.Google Scholar
Dauben, Joseph. 2007. “Chinese Mathematics”. In The Mathematics of Egypt, Mesopotamia, China, India and Islam, A Source Book, edited by Katz, Victor, 187–380. Princeton and Oxford: Princeton University Press.Google Scholar
Guo, Xihan. 郭煕汉. 1996. 楊輝算法導讀 (Yang Hui suanfa daodu [Introduction to the Yang Hui suanfa]). In 中華傳統數學名著導讀叢書. 李迪 (Zhonghua chuantong shuxue mingzhu daodu congshu [Collection of masterpieces in Chinese traditional mathematics]), edited by湖北教育出版社. (Hubei jiaoyu chubanshe, [Hubei Educational Press]).Google Scholar
Guo, Shuchun. 郭书春. 1991. 中国古代数学 (Zhonguo gudai shuxue [Mathematics in ancient China]), 1-174. 山东教育出版社 (Shandong jiaoyu chubanshe [Shandong educational press]).Google Scholar
Guo, Shuchun. 郭書春. 2010. 中國科學技術史. 數學卷. (Zhongguo kexue jishu shi: Shuxue juan [History of Chinese science and technology: Mathematics]). 科學出版社, 北京. (Kexue chubanshe [Beijing Science Press]).Google Scholar
Peng Yoke, Ho. 1973. “Li Chih”. In Dictionary of Scientific Biography, edited by Gillispie, Charles, vol. 8:313–320. Charles Scribner’s Son.Google Scholar
Jock, Hoe. 2008. The Jade Mirror of the Four Unknowns by Zhu Shijie. Christchurch. New Zealand: Mingming bookroom.Google Scholar
Horiuchi, Annick. 2000. “La notion de yanduan: Quelques réflexions sur les méthodes ‘algébriques’ de la résolution de problèmes en Chine aux X° et XI°.” In Oriens-Occidens 3:235–258.Google Scholar
Kong, Guoping. 孔國平. 1987. 李冶传 (Li Ye zhuan [Biography of Li Ye]). 北京: 河北教育出版社 (Beijing: Hebei jiaoyu chubanshe [Beijing: Hebei educational press]).Google Scholar
Kong, Guoping. 孔國平. 1996. 測圓海鏡導讀 (Ceyuan haijing daodu [Introduction to Ceyuan Haijing]). In 中華傳統數學名著導讀叢書 (Zhonghua chuantong shuxue mingzhu daodu congshu [Collection of masterpieces in Chinese traditional mathematics]), 湖北教育出版社 (Hubei jiaoyu chubanshe [Hubei educational press]).Google Scholar
Kong, Guoping. 孔國平. 1999. 李冶朱世杰與金元數學 (Li Ye Zhu Shijie yu jinyuan shuxue [Li Ye, Zhu Shijie and Jin-Yuan Mathematics]). In 中國數學史大系 (Zhongguo shuxueshi daxi [Collection of the history of Chinese mathematics]). 河北科學技術出版社 (Hebei kexue jishu chubanshe [Hebei science and technology press]).Google Scholar
Lam, Lay-Yong. 1977. A Critical Study of Yang Hui suanfa: A Thirteenth-Century Chinese Mathematical Text. Singapore: Singapore University Press.Google Scholar
Lam, Lay-Yong. 1984. “Li Ye and his Yi Gu Yan Duan: Old mathematics in Expanded Sections.” Archive for History of Exact Sciences 29:237–266.Google Scholar
Lam Lay-Yong, Ang Tian Se. 2004. Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China. Singapore: World Scientific.CrossRefGoogle Scholar
Li, Di. 李迪. 1997. 天元術與李冶 (Tianyuanshu yu Li Ye [The procedure of celestial source and Li Ye]). 中國數學通史. 宋元卷. 第四章 (Zhongguo shuxue tongshi. Song Yuan juan. Di si zhang. [Collection on the history of Chinese mathematics.,Song Yuan, chap. 4]). 江蘇教育出版社 (Jiangsu jiaoyu chubanshe [Jiangsu educational press]):184-239.Google Scholar
Li, Yan, Shiran, Du. 1987. Chinese Mathematics: A Concise History. Translated by Crossley, J. N. and Lun, W.C.. Oxford: Clarendon Press.Google Scholar
Martzloff, Jean-Claude. 1987. Histoire des mathématiques chinoises. Paris: Masson. English edition: A History of Chinese Mathematics. 1997. Springer.Google Scholar
Mei, Rongzhao, 梅荣照. 1966. 李冶及其數學著作 (Li ye ji qi shuxue zhuzuo [Li Ye and his mathematical works]). In 宋元術學史論文集 (Song Yuan shuxue shi lunwenji [Collection of articles on Song-Yuan history of mathematics]), edited by Qian Baocong, 104-148. 科學出版社 (Kexue chubanshe [Technology press]).Google Scholar
Morgan, Daniel Patrick. 2015. “What Good’s a Text? Textuality, Orality, and Mathematical Astronomy in early Imperial China.” Archives internationales d’histoire des sciences, Académie international d’histoire des sciences. 65(2):549–572.Google Scholar
Needham, Joseph. 1954. Science and Civilisation in China 3:42–151. Cambridge UK: Cambridge University Press.Google Scholar
Pollet, Charlotte. 2012. Comparison of Algebraic Practices in Medieval China and India. 2 vols. Ph.D diss. University of Paris. 7/National Taiwan Normal University.Google Scholar
Pollet, Charlotte. 2014. “The Influence of Qing Dynasty Editorial Work on the Modern Interpretation of Mathematical Sources: The Case of Li Rui’s Edition of Li Ye’s Mathematical Treatises.” Science in Context 27 (3):385–422.CrossRefGoogle Scholar
Pollet, Charlotte, and Jiamin, Ying. 2017. “One Quadratic Equation, Different Understandings: The 13th Century Interpretation by Li Ye and Later Commentators in the 18th and 19th Centuries.” Journal for History of Mathematics 30 (3):137–162.CrossRefGoogle Scholar
Te, Gusi. 特古斯. 1990. 劉益及其佚著 “議古根源” (Liu Yi ji qi yi zhu “Yigu gen yuan” [Liu Yi and his lost work, the ‘Yigu genyuan’]. In 数学史研究文集, (Shuxueshi yanjiu wenji [Research on history of mathematics series]). Edited by Li Di李迪, 九章出版社 (Jiu zhang chubanshe [The Nine Chapters Press]) 1:56-63.Google Scholar
Volkov, Alexei. 2001. ‘Le Bacchette.’ In ‘La scienza in Cina’ Chemla, Bray, Fu, Huang and Metaile (Eds). In ‘Storia della scienza’ (8 vols.). Edited by S. Petrucioli, et al. Instituto della Enciclopedia Italiana, section I, vol. 2:125-133.Google Scholar
Xu, Yibao. 徐义保. 1990. 對 “益古集”的復原與研究 (Dui “yi gu ji” de fuyuan yu yanjiu [Restoration and Research on the Collection Augmenting the Ancient [Knowledge]). In 數學史研究文集 (Shuxueshi yanjiu wenji [Research on History of Mathematics Series]). Edited by Li Di 李迪, 九章出版社 (Jiu zhang chubanshe [The Nine Chapters Press]) 1:149-165.Google Scholar
Yiwen, Zhu. 2016. “Different Cultures of Computation in Seventh Century China from the Viewpoint of Square Root Extraction.” Historia Mathematica 43 (1):3–25.Google Scholar
Zürcher, Erik. 2007. The Buddhist Conquest of China: The Spread and Adaptation of Buddhism in Early Medieval China, 3rd ed. Leiden: Brill.Google Scholar