summary.
Exploring Features of Algebra in Medieval China: The Case of Yigu yanduan.

The Yigu yanduan, 益古演段, was written in 1259 by Li Ye, 李冶, and published later in 1282. The Yigu yanduan presents itself as a list of 64 problems in three rolls. All the problems are related to the same topic which at first sight looks very pragmatic and simple: that is calculating the diameter or side of a field inside of which there is a pond. But the central topic of the Yigu yanduan is in fact the construction and formulation of quadratic equations derived from problems on squares, rectangles and circles.
The statute of this text was interpreted by historians as being an introduction to the Ceyuan haijing, 測圓海鏡, the other mathematical masterpiece written by Li Ye in 1248, and published as the same time as the Yigu yanduan. The Yigu yanduan has long been regarded as a kind of text for popular purpose and remained in the shadow of the Ceyuan haijing. The book is still considered as a list of simplified examples in the procedure of the Celestial Source. The purpose of this study is to confront this point of view, to explain why there was such a misunderstanding and to put into light a peculiar field in Chinese mathematics. I show that this book is in fact masterpiece treatise whose practices can be related to the famous Han dynasty classic, the Nine Chapters in Mathematical Art, 九章算術 and its commentary by Liu Hui (3rd century). The focus must be redirected on another procedure: the section of areas.
This study was done through careful comparison of all remaining available Qing dynasty editions of the Yigu yanduan, collection and reproduction of all the diagrams, and translation of the 64 problems. This study first shows how the Qing dynasty editors work with ancient sources and how their editorial choices mislead our modern interpretation. The systematic study of diagrams shows that one of the most important features of the Yigu yanduan is in fact a practice of manipulation of figures performed by the reader. The heart of the book relies on a non discursive practice: drawing and visualizing manipulations of figures.
Key word: history of algebra, diagrams, transformation, tabular settings, analogy

summary.
Exploring Features of Algebra in Medieval China: The Case of Yigu yanduan.

The Yigu yanduan, 益古演段, was written in 1259 by Li Ye, 李冶, and published later in 1282. The Yigu yanduan presents itself as a list of 64 problems in three rolls. All the problems are related to the same topic which at first sight looks very pragmatic and simple: that is calculating the diameter or side of a field inside of which there is a pond. But the central topic of the Yigu yanduan is in fact the construction and formulation of quadratic equations derived from problems on squares, rectangles and circles.
The statute of this text was interpreted by historians as being an introduction to the Ceyuan haijing, 測圓海鏡, the other mathematical masterpiece written by Li Ye in 1248, and published as the same time as the Yigu yanduan. The Yigu yanduan has long been regarded as a kind of text for popular purpose and remained in the shadow of the Ceyuan haijing. The book is still considered as a list of simplified examples in the procedure of the Celestial Source. The purpose of this study is to confront this point of view, to explain why there was such a misunderstanding and to put into light a peculiar field in Chinese mathematics. I show that this book is in fact masterpiece treatise whose practices can be related to the famous Han dynasty classic, the Nine Chapters in Mathematical Art, 九章算術 and its commentary by Liu Hui (3rd century). The focus must be redirected on another procedure: the section of areas.
This study was done through careful comparison of all remaining available Qing dynasty editions of the Yigu yanduan, collection and reproduction of all the diagrams, and translation of the 64 problems. This study first shows how the Qing dynasty editors work with ancient sources and how their editorial choices mislead our modern interpretation. The systematic study of diagrams shows that one of the most important features of the Yigu yanduan is in fact a practice of manipulation of figures performed by the reader. The heart of the book relies on a non discursive practice: drawing and visualizing manipulations of figures.
Key word: history of algebra, diagrams, transformation, tabular settings, analogy

Primary sources:
李冶, Li Ye:
• 益古演段, Yigu yanduan.1259
- 四庫全書,文淵閣, Wenyange siku quanshu, original edition from National Palace Museum, Taiwan.
- 四庫全書,文津閣Wenjinge siku quanshu, vol. 799. Reprint. 2005.
- 知不足齋叢書, reprint in中國科學技術典籍通彙。郭书春. 河南教育出版社. 1993. Vol 1.
• 敬齋古今黈. 12??. 學術筆記叢刊, 北京中華書局出版社, 劉德權 點校. 1995
• 測圓海鏡, Ceyuan haijing.1248.
-知不足齋叢書, reprint in中國科學技術典籍通彙。郭书春. 河南教育出版社. 1993. Vol 1.
- 四庫全書,文津閣Wenjinge siku quanshu, vol. 799. Reprint. 2005.

朱世杰, Zhu Shijie, 四元玉鉴, si yuan yu jian, 1303. translated into English by Ch’en Tsai Hsin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. 2006
秦九韶, Qin Jiushao, 數學九章, shuxue jiu zhang, 1247, in 四庫全書,文津閣Wenjinge siku quanshu, Reprint. 2005. Vol. 798.

阮元 Ruan Yuan, 疇人傳 Chou ren chuan, 1799. 中華漢語工具書書庫. 案徽教育出版社.Vol.82.

元史,Yuan shi, Official history of the Yuan, 1370. 四庫全書,文津閣Wenjinge siku quanshu,. Reprint. 2005. vol. 290. Ch.160. pp.11-13

Nārāyana, Bījaganitāvatamsa. Xerox copies from Benares Sanskrit College, Sarasvati Bhavana,
Manuscripts B1: No. 35579, B2: No. 98699.

Secondary sources:
[Abhyankar S.K. 1980]. Bhaskaracharya’s Bijaganita and its English Translation. Bhaskaracharya Pratishthana. Poona.
[Adler Joseph, 1999]. "Zhou Dunyi: The Metaphysics and Practice of Sagehood", in Sources of Chinese Tradition, William Theodore De Bary and Irene Bloo, (Eds). 2nd ed., 2 vols. Columbia University Press.
[Allar André, 1978]. “A propos d’un algorisme latin de Frankenthal: une méthode de recherche”, Janus, vol.65, pp.119-41.
[Ang tian se, 1978]. “Chinese interest in right angle triangles”, Historia Mathematics, 5, pp. 253-266
[Bag Amulya Kumar, 1979]. Mathematics in Ancient and Medieval India. Chaukhambha Orientalia. Varanasi, Delhi.
[Breard Andrea, 1999]. “Re-Kreationeines mathematischen Konzeptes im chinesischen Diskurs. “Reihen” vom 1.bis zum 19. Jahrhundert”. Stuttgart: Boethius, Franz Steiner Verlag: 42
[Breard Andrea, 2000]. “La recomposition des mathematiques chez Zhu Shijie: la constitution d’un domaine specifique autour du nombre “quatre””.Oriens-Occidens, Sciences, mathématiques et philosophie de l’antiquité à l’âge classique. 3. p. .259-278
[Breard Andrea, 2000]. “La recomposition des mathematiques chez Zhu Shi-jie: la constitution d’un domaine specifique autour du nombre “quatre”. ”. Oriens-Occidens 3 :259-277
[Bottermans Jack, 2008]. The Book of Games. Strategy, tactics and history. Ed. Sterling. New York-London.
[Bronkhorst Johannes, 2001]. “Pāṇini and Euclid: Reflections on Indian Geometry”. Journal of Indian Philosophy, kluwer academic publishers. 29: 43-80.
[Colebrooke Henry Thomas, 1817]. Algebra with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhascara. London. John Murray. Reed. S.R. Sarma (Eds). Sharada Publishing House. Delhi. 2005.
[Chen Ellen, 1989]. The Tao Te Ching: A New Translation and Commentary. Paragon House.
[Chemla Karine, 1982]. Etude du livre “reflets des mesures du cercle sur la mer” de Li Ye. Thèse de doctorat de l’université Paris XIII. Not published.
[Chemla Karine, 1993]. “李冶, 測圓海鏡,的結構及其對數學知識的表述”, 數學史研究文集第五輯. 123-142.
[Chemla Karine, 1994a]. “Different concepts of equations in The Nine Chapters on Mathematical Procedures 九章算術and in the Commentary on it by Liu Hui (3rd century)”. Historia Scientiarum. Vol 4-2.
[Chemla Karine, 1994b]. “Essais sur la signification mathématique des marqueurs de couleur chez Liu Hui (3eme sciècle)”. In A. Peyraube, I. Tamba, A. Luca (eds), Mélanges en hommage a Alexis Rygaloff, Cahiers de linguisitique. Asie Orientale, 23. N.1:61-76. http://www.persee.fr
[Chemla Karine, 1994c]. “Similarities Between Chinese and Arabic Mathematical Writings: (I) Root Extraction”. Arabic Sciences and Philosophy, Cambrige Univserity Press. vol 4:207-266
[Chemla Karine, 1994]. “Nombres, opérations et équations en divers fonctionnements: quelques méthodes de comparaison entre des procédures élaborées dans trois mondes différent”, in I.Ang and P.E Will (eds). Nombres, astres, plantes et viscères. Sept essais sur l’histoire des sciences et des techniques en Asie orientale, Paris. Collège de France, Institut des Hautes Etudes Chinoises (Mémoires de l’Institut des Hautes Etudes Chinoises, XXXV) :1-36.
[Chemla Karine, 1995]. “Algebraic Equations East and West Until the Middle Ages”. East Asian Science:Tradition and Beyond., Kansai University Press, Osaka, Japan. 83-89.
[Chemla Karine, 1996]. «Positions et changements en mathématiques à partir des textes chinois des dynasties Han a Song-Yuan. Quelques remarques ». Extrême-Orient, Extrême –Occident 18 :115-147
[Chemla Karine, 1997]. “Fractions and Irrational Between Algorithm and Proof in Ancient China”. Studies in History of Medecine and Science. Hakeem Abdul Hameed (Eds). Jamia Hamdard. New Delhi. Vol. Xv, No.1-2 : 31-53
[Chemla Karine, 2000]. “Les problèmes comme champs d’interprétation des Algorithmes dans les Neuf Chapitres sur les procédures mathématiques et leurs commentaires. De la résolution des systèmes d’équations linéaires”. Oriens-Occidens. Sciences, mathématiques et philosophie de l’antiquité à l’âge classique. 3, 189-234
[Chemla Karine, 2001]. « Variété des modes d’utilisation des Tu dans les textes mathématiques des Song et Yuan ». Pre-print. Conference “From Image to Action: The function of Tu-Representation in East Asian Intellectual Culture”, Paris, 2001. http://hal.ccsd.cnrs.fr, section Philosophie, sub-section “histoire de la logique et des mathématiques”.
[Chemla Karine, Agathe Keller, 2002]. “The Sanskrit karaṇīs and the Chinese mian (side). Computation with Quadratic Irrationals in Ancient China and India”. In Y. Dold-Samplonius.J.W. Dauben, M. Flokerts, B. Van Dalen (Eds) From China to Paris : 2000 Years Transmission of Mathematical Ideas. Franz Steiner Verlag. Boethius. 46 :87-132
[Chemla Karine, Guo Shuchun, 2004]. Les neuf chapitres. Le classique mathématique de la Chine ancienne et ses commentaires. Paris. Dunod.
[Chemla Karine, 2004]. “What is the content of this book? A Plea for Developing History of Sciences and History of Text Conjointly” Chemla Karine (ed). History of Science, History of Text, Dordrecht-Boston-London: Kluwer Academic Publishers, 238: 201- 230
[Chemla Karine, 2005]. “Geometrical Figures and Generality in Ancient China and Beyond: Liu Hui and Zhao Shuang, Plato and Thabit ibn Qurra”. Science in Context, Cambridge University Press. 18:123-166
[Chemla Karine, 2006a]. “Artificial Language in the Mathematics of Ancient China”. Journal of Indian Philosophy, Springer, 34 :31-56.
[Chemla Karine, 2006b]. “La généralité, valeur épistémologique fondamentale des mathématiques de la Chine ancienne”. In Hommage rendu a Jean Filliozat. Institut de France, Académie des inscriptions et belles-lettres. Séance du 17 novembre 2006. Paris. MMVI : 113-146
[Chemla Karine, 2009]. “Apprendre à lire: la démonstration comme élément de pratique mathématique”. Figures de la preuve. Communications de l’Ecole des Hautes Etudes en Science Sociales. Centre Edgar Morin. Seuil: 84-101
[Chemla Karine 2010a]. “Changes and Continuities in the Use of Diagrams Tu in Chinese Mathematical Writings (Third Century to Fourtheen Century) [1]”. East Asian Science, Technology and Society, 4: 303-326.
[Chemla Karine, 2010b]. 從古代中國數學的觀點探對知識論文化. 祝平一主編, 中國史新論科技史分冊: 科技與中國社會, 台北: 聯徑出版社:181-270
[Chia Lucille, 2002]. Printing for Profit: the Commercial Publishers of Jianyang, Fujian (11th -17th centuries). Harvard-Yenching Institute Monograph Series 56.
[Clark Walter, 1930]. The Aryabhatiya of Aryabhata. An ancient Indian Work on Mathemaics and Astronomy. The university of Chicago Press, Chicago, Illinois.
[Cullen Christopher, 1996], Astronomy and mathematics in ancient China: the Zhou bu suan jing, Needham Research Institute Studies, 1, Cambridge, GB, Cambridge University Press.
[Datta Bibhutibhusan, 1933]. “The algebra of Nārāyana”, Isis 19, 472-85.
[Datta Bibhutibhusan, Singh Avadhesh Narayan, 1935]. History of Hindu Mathematics, A Source Book. Part I and II. Asia Publishing House. Bombay, India.
[Datta Bibhutibhusan, 1993]. Ancient Hindu Geometry. The science of the Sulba. Cosmo Publications. India, New Delhi.
[Dauben Joseph, 2000]. “Medieval Mathematics”, in The History of Mathematics from Antiquity to the present: A selective Annotated Bibliography. Eds. Dauben Joseph. Revised edition on CD-RO, Ed. Albert C. Lewis. American Mathematical Society.
[Dauben Joseph, 2007]. “Chinese Mathematics”. Victor Katz (Ed). The Mathematics of Egypt, Mesopotamia, China, India, and Islam, A Source Book. Princeton University Press. Princeton and Oxford. 187-380
[Filliozat Pierre-Sylvain, 2004]. “Ancient Sanskrit Mathematics: an Oral Tradition and a Written Literature”. Chemla Karine (Ed). History of Science, History of Text, Dordrecht-Boston-London: Kluwer Academic Publishers, 238: 137-157.
[Gi Meisheng, 1999]. Le chemin du souffle. Culture et Science Chinoise, Paris.
[Guo Xihan.郭煕汉 1996]. 楊輝算法導讀. 中華傳統數學名著導讀叢書. 李迪(eds). 湖北教育出版社
[Guo Shuchun, 郭書春, 1982]. 九章算術 中的整數句股形研究; 科技史文集, 第8輯:54-66.
[Guo Shuchun, 郭書春, 1991]. 中國古代數學. 山東教育出版社, pp. 1-174
[Guo Shuchun, 郭書春, 2010]. 中國科學技術史. 數學卷. 科學出版社, 北京 .
[Hayashi Takao, 1995]. The Bhakhshālī Manuscript, An ancient Indian mathematical Treatise. Groningen: Egbert Forsten.
[Hayashi Takao, Kusuba Takanori, 1998]. “Twenty-One Algebraic Normal Forms from Citrabhanu”. Historia Mathematica 25:1-21.
[Hayashi Takao, 2004]. “Two Benares Manuscripts of Nārāyana Pandita’s Bījaganitāvatamsa”, Burnett Charles, Hogendijk Jan, Plofker Kim, Yano Michio, (eds). Studies in the History of the Exact Sciences in Honour of David Pingree. Leiden-Boston: Brill. 386-496.
[Hayashi Takao, 2009]. “Bījagaṇita of Bhāskara”. Michio Yano and Ken Saito (Eds). Sciamus, Source and Commentary in Exact Science. Kyoto. Vol.10:1-303
[He Yunpo,何云波, 2001]. 圍棋與中國文化. 北京.人民出版社.
[Ho Peng Yoke, 1973]. “Li Chih”, in Charles Gillispie (Ed). Dictionary of Scientific Biography. Ed. Charles Scribner’s son. Vol. 8. pp.313-320.
[Hoe John, 2008]. The Jade Mirror of the Four Unknowns by Zhu Shijie. Mingming bookroom. Christchurch. New Zealand.
[Horiuchi Annick. 2000]. “La notion de yanduan: quelques réflexions sur les méthodes “algébriques”. de la résolution de problèmes en Chine aux X° et XI°, Oriens-Occidens 3 : 235-258.
[Horng Wann-Sheng, 1993a]. “Chinese Mathematics at the Turn of the 19th Century : Jiao Xun, Wang Lai and Li Rui”. Cheng-hung Lin and Daiwei Fu (Eds), Philosophy and Conceptual History of Science in Taiwan. Kluwer Academic Publishers:167-208.
[Horng Wann-Sheng, 洪萬生, 1993b]. 談天三友. 台北. 明文書局. 297.
[Høyrup Jens, 2004]. “Mahāvīra’s geometrical problems: traces of unknown links between Jaina and Mediterranean mathematics in the classical ages”. History of the mathematical sciences. Hindustan Book Agency, New Delhi. P. 83-95
[Høyrup Jens, 2006]. “Pre-modern “algebra”: A concise survey of that which was shaped into the technique and discipline we know”. The way through science and philosophy. http://rucforsk.ruc.dk
[Keller Agathe, 2000]. Un commentaire indien du VIIeme siècle. Bhāskara et le gaṇita-pāda de l’Aryabhaṭīya. Thèse de doctorat. Université Paris VII.
[Keller Agathe, 2005]. “Making Diagrams speak, in Bhāskara I’s commentary on the Aryabhațiya”. Historia Mathematica 32: 257-302
[Keller Agathe, 2006a]. Expounding the Mathematical Seed. Vol I and II. Basel-Boston-Berlin: Biskhauser Verlag.
[Keller Agathe, 2006b]. “Comment on a écrit les nombres dans le sous-continent indien, histoire et enjeux”. In Hommage rendu a Jean Filliozat. Institut de France, Académie des inscriptions et belles-lettres. Séance du 17 novembre 2006. Paris. MMVI : 65-81
[Keller Agathe, 2007]. “Qu’est ce que les mathématiques ? Les réponses taxinomiques de Bhaskara, un commentateur, mathématicien et astronome Indien du VII° siècle”. http://halshs.archives-ouvertes.fr/.
[Keller Agathe, 2009]. “On Sanskrit Commentaries Dealing with Mathematics (VIIth-XIIth century) ”. http://halshs.archives-ouvertes.fr/
[Keller Agathe, 2011]. “Dispelling Mathematical Doubts. Assessing Mathematical Correctness of Algorithms in Bhāskara’s Commentary on the Mathematical Chapter of the Aryabhatiya”. Karine Chemla (Eds). On the History and Historiography of Mathematical Proof. Cambridge University Press. http://halshs.archives-ouvertes.fr/
[Keller Agathe, 2011]. George Peacock’s Arithmetic in the changing landscape of the history of mathematics in India. Indian Journal of History of Science. 46.2 :205-233
[Kong Guoping,孔國平,1987]. 李冶传. 北京 : 河北教育出版社.
[Kong Guoping,孔國平,1996]. 測圓海鏡導讀. 中華傳統數學名著導讀叢書. 李迪(eds). 湖北教育出版社
[Kong Guoping,孔國平,1999].李冶朱世杰與金元數學,中國數學史大系, 河北科學技術出版社.
[Kripa Nath Sinha, 1985]. “Śrīpati: An Eleventh-Century Indian Mathematician”. Historia Mathematica. 12: 25-44
[Kripa Nath Sinha, 1986]. “Algebra of Śrīpati: An Eleventh Century Indian Mathematician”, Ganita Bhāratī 8, 27-34.
[Kusuba Takanori, 1994]. “Combinatoric and Magic Square in India: A Study of Nārāyana Pandita’s Ganitakaumudi”, Ann Arbor. Chapter 13-14.
[Lacrosse Joachim, 2005]. Philosophie comparée Grèce, Inde, Chine. Librairie philosophique Vrin. Paris.
[Lam Lay-Yong, 1977]. A critical study of Yang Hui suanfa, a Thirteen Century Chinese Mathematical Text, Singapore, Singapore University Press.
[Lam Lay-Yong, 1979]. “Chu Shih-chieh’s Suan-hsüeh ch’i-meng (Introduction to Mathematical Studies)”. Archive for History of Exact Sciences. Springer-Verlag. Vol.21:1-28
[Lam Lay-Yong, 1984]. “Li Ye and his Yi Gu Yan Duan (old mathematics in Expanded Sections)”. Archive for history of exact sciences. Singapore. 29: 237-266
[Lam Lay-Yong, Ang Tian Se, 2004]. Fleeting Footsteps, Tracing the Conception of Arithemtic and Algebra in Ancient China. World Scientific, Singapore.
[Le Blanc Charles, 1985]. Huai-nan Tzu: Philosophical Synthesis in Early Han Thought: The Idea of Resonance (Kan-Ying) With a Translation and Analysis of Chapter Six. Hong Kong University Press.
[Li Di, 李迪, 1997]. 天元術與李冶. 中國數學通史. 宋元卷. 第四章. 江蘇教育出版社.pp. 184-239.
[Li Jimin, 李繼閔, 1982], 劉徽對整句股數的研究; 科技史文集; 第8輯:51-53
[Li Jimin, 李繼閔1990], 東方數學典籍 -“九章算術” 及其劉徽注研究; 陜西人民教育出版社: 1-492.
[Li Yan, 李儼, 1926], 重差術源流及其新注 “學藝”, 第8卷 第8 期:1-15.
[Li Yan, 李儼, 1955]. 中國史論叢, 第四章, 科學出版社. pp. 1-365.
[Li Yan, Du Shiran, 1987]. Chinese Mathematics, A Concise History. Translated by Crossley, J.N and Lun W.C. Clarendon Press. Oxford.
[Libbrecht Ulrich, 1973]. Chinese mathematics in the thirteen century. The Shu-shu chiu-chang of Ch’in Chiu-shao. The MIT Press, Cambridge.
[Lin Longfu,林隆夫, 2008]. 莉拉沃蒂 ,婆什迦罗. 北京科學出版社.
[Liu Shancheng, 刘善承, 2007]. 中国围棋史. 成都时代出版社.
[Lloyd Geoffrey, Nathan Sivin, 2002]. The way and the word. Science and medicine in Early China and Greece. Yale University Press. New Haven and London.
[Martzloff Jean-Claude, 1988]. Histoire des mathématiques chinoises. Masson. Paris. For English edition: A History of Chinese Mathematics, Springer, 1997.
[Mei Rongzhao,梅荣照, 1966]. 李冶及其數學著作, 錢宝琮(ed.), 宋元術學史論文集, 科學出版社 :104-148.
[Mei Rongzhao, 梅荣照, 1984], 劉徽的方程理論; 劉徽的句股理論 ;“科學史集刊” 第11集: 63-76 ; 77-95.
[Mikami Yoshio, 1913]. The Development of Mathematics in China and Japan. Leibzig, New-York, BG Teubner. Ch.12. Li Yeh. 79-84.
[Narashima Roddam, 2007]. “Epistemology and Language in Indian Astronomy and Mathematics”, Journal of Indian Philosophy, 35: 521-541.
[Needham Joseph, 1954], Science and Civilisation in China. Cambridge. Vol.III: 42-151.
[Needham Joseph, Colin A. Ronan, 1978]. The Shorter Science and Civilisation in China. Cambridge University Press.
[Netz Reviel, 1999]. The Shaping of Deduction in Greek Mathematics. A Study in Cognitive History. Cambridge University Press. Ideas in Context 51.
[Patte François, 2004]. L’œuvre mathématique et astronomique de Bhāskarācārya, le Siddhāntaśiromaṇi I.II. Ecole pratique des hautes études. Sciences historiques et philologiques. Hautes études orientales 38. Extrême orient 4. Droz. Vol. I. II.
[Patte François, 2006]. L’algèbre en Inde au XIIe siècle. In Hommage rendu a Jean Filliozat. Institut de France, Académie des inscriptions et belles-lettres. Séance du 17 novembre 2006. Paris. MMVI : 83-101
[Pingree David, 1970/95]. Census of the Exact Sciences in Sanskrit, Series A, 5 vols American Philosophical Society.
[Plofker Kim, 2007]. “Mathematics in India”. Victor Katz (Ed). The Mathematics of Egypt, Mesopotamia, China, India, and Islam, A Source Book. Princeton University Press. Princeton and Oxford. 385-512
[Plofker Kim, 2009]. Mathematics in India. Princeton University Press.

[Pollock Sheldon, 2002]. Introduction: Working Papers on Sanskrit Knowledge Systems on the Eve of Colonialism. Journal of Indian Philosophy, 30 (5), 431-9.

[Qian Baocong, 錢寶琮, 1937]. 中國數學中之整數句股形研究 “數理雜志”. 第1卷 第3 期: 94-112.
[Qian Baocong, 錢寶琮, 1963]. 算徑十書. 北京, 中華書局.

[Qian Baocong, 錢寶琮, 1964]. 中國數學史. 科學出版社.

[Qu Anjing, 1997]. “On Hypothenuse Diagrams in Ancient China”. Centaurus. Vol. 39: 193-210.

[Raina Dhruv, 1999]. Nationalism, Institutional Science and the Politics of Knowledge: Ancient Indian Astronomy and Mathematics in the Landscape of French Enlightenment Historiography. Doctoral Dissertation. Göteborgs University. Rapport Nr. 201.

[Rashed Roshdi, 1974] The development of Arabic Mathematics : between arithmetic and algebra. Translation A. F. W. Armstrong. Kluwe Academic Publisher, vol.156

[Rashed Roshdi, 1978]. “L’extraction de la racine et l’invention des fractions décimales XIe-XIIe siècles”, Archive for History of Exact Sciences, vol.18 (1978), pp.191-243.

[Rashed Roshdi, 1983]. « L’idée de l’algèbre selon Al-Khwarizmi », Fundamenta Scientia, vol.4, pp. 87-100.

[Rashed Roshdi, 1997]. « L’algèbre », in Roshdi Rashed (Eds), Histoire des sciences arabes, 3 vol., Paris, Seuil.
[Robinet Isabelle, 1990]. "The Place and Meaning of the Notion of Taiji in Taoist Sources Prior to the Ming Dynasty," History of Religions 23.4: 373-411.
[Robinet Isabelle, 2008]. "Wuji and Taiji 無極 • 太極Ultimateless and Great Ultimate", in The Encyclopedia of Taoism, ed. Fabrizio Pregadio, Routledge, pp. 1057–9.
[Sarasvati Amma, 1979]. Geometry in Ancient and Medieval India. Delhi: Motilal Banarsidass.
[Sarma, Sreeramula Rajeswara, 2002]. “Rule of Three and its variation in India”. In Y. Dold-Samplonius.J.W. Dauben, M. Flokerts, B. Van Dalen (Eds). From China to Paris: 2000 Years Transmission of Mathematical Ideas. Franz Steiner Verlag. Boethius. 46 :133-156.
[Sarton George, 1927]. Introduction to the history of science. Vol. II. Part. II:627-628.

[Singh Parmanand 1998/2002]. “The gaṇita Kaumudī of Nārāyaṇa Paṇḍita”, Gaṇita Bhāratī 20, 1998, 25-82 (chaps. I-III); 21, 1999, 10-73 (Chap.IV); 22, 2000, 19-85 (Chaps. V-XII); 23, 2001, 18-82 (Chap.XIII); 24, 2002, 35-98 (Chap. XIV).
[Siu Man-Keung, Volkov Alexei, 1999]. “Official Curriculum in Traditional Chinese Mathematics: How did Candidates Pass the Examinations?”, Historia Scientiarum, vol. 9-1, pp.85-99
[Shukla Kripa Shankar, 1970]. “Nārāyana paṇdita’s Bījaganitāvatamsa, part I”, Akhila Bharatiya Sanskrit Parishad. Lucknow.
[Srinivasiengar, C.N. 1967]. The History of Ancient Indian Mathematics. The World Press Private LTD. Calcutta.
[Staal Fritz, 1999]. “Greek and Vedic Geometry”. Journal of Indian Philosophy. Kluwer Academic Publishers. 27:105-127.
[Te Gusi, 特古斯, 1990]. 劉益及其佚著 “議古根源”. 李迪 (ed), 数学史研究文集, 九章出版社1: 56-63.
[Tian Miao, 1999]. “Jiegenfang, Tianyuan, and Daishu: Algebra in Qing China”. Historia Scientarum, vol.9-1. Pp. 101-119
[Volkov Alexei, 1991a]. “Structure d'un traité mathématique: l'exemple du Hai dao suan jing”. In: K.Chemla, A.Volkov, V.Lichtmann (eds.), Modèles et structures des textes chinois anciens (Extrême-Orient Extrême-Occident, no. 13), Paris: PUV, pp. 93-100.
[Volkov Alexei, 1991b]. “La structure des textes chinois anciens: quelques remarques”. In: K.Chemla, A.Volkov, V.Lichtmann (eds.), Modèles et structures des textes chinois anciens (Extrême-Orient Extrême-Occident, no. 13), Paris: PUV, pp. 155-161.
[Volkov Alexei, 1992]. “Analogical Reasoning in Ancient China. Some Examples”. In Chemla (Eds), regards obliques sur l’argumentation en Chine. Extreme-Orient, Extreme-Occident, 14: 15-48
[Volkov Alexei, 1994]. “Transformations of geometrical objects in Chinese mathematics and their evolution”. Alleton.V and Volkov.A (Eds). Notions et perceptions du changement en Chine, Paris, Collège de France, Institut des Hautes Etudes Chinoises (Mémoires de l’Institut des Hautes Etudes Chinoises, vol. XXXVI), pp.133-148.
[Volkov Alexei, 1996]. “Lam Lay-Yong, Ang Tian-Se. Fleeting Footsteps.” Archives Internationales d’Histoire des Sciences, vol. 46, no. 136, pp. 155-159.
[Volkov Alexei, 1996-1997]. “Science and Taoism: An Introduction”. Taiwanese Journal for Philosphy and History of Science. N 8. Vol 5: 1-58
[Volkov Alexei, 2001]. “Le Bacchette”, in Chemla, Bray, Fu, Huang and Metaile (Eds). “La scienza in Cina, Storia della scienza”, 8 vol. Enciclopedia Italiana, section I, vol. II.
[Volkov Alexei, 2006]. “Le raisonnement par analogie dans les mathématiques chinoises du premier millénaire de notre ère”. Durand-Richard M-J. L’analogie dans la démarche scientifique. L’Harmattan. Collection Histoire des Sciences, série études.
[Volkov Alexei, 2007]. “Geometrical Diagrams in Traditional Chinese Mathematics”. Francesca Bray, Vera Dorofeeva-Lichtmann, Georges Metailie (Eds), Graphics and text in the Production of Technical Knowledge in China. Sinica Leidensia vol.79. Brill. Leiden-Boston. 425-460.
[Wilhelm Richard, Baynes Cary, 1967]. The I Ching or Book of Changes, With foreword by Carl Jung. 3rd. ed., Bollingen Series XIX. Princeton NJ: Princeton University Press (1st ed. 1950)
[Wu Wenjun, 吴文俊, 1985]. 益古演段. 李冶的數學成就. 中國數學史大系. 第二編. 第三章:104-130.
[Wu Wenjun, 吴文俊, 1987]. 秦九韶 與 “數書九章”. 中國數學史研究叢書之二. 北京師範大學出版社.
[Wylie Alexander, 1852]. “Jotting on the Science of the Chinese: Arithmetic.” North China Herald, Aug-Nov. 108. Repr. Copernicus, 1882, 2, 169, 183.

[Xu Yibao,徐义保 1990] 对”益古集”的复原与研究, (research and restoration of the yiguji), 李迪 (ed), 数学史研究文集,九章出版社1: 149-165.
[Yano Michio, 1984]. “Kushyar ibn Labban’s book on Astrology”, The Bulletin of the international Institute for Linguistic Science, vol. V, pp. 67-89.
[Ying Jia-Ming, 2010]. “The Kujang sulhae 九章術解: Nam Pyong-Gil’s reinterpretation of the mathematical methods of the Jiuzhang suanshu”. Historia Mathematica, Elsevier, 38: 1-27
[Yushkevich Adolf Pavlovich, 1955] “O dostizenijax kitajskix ucenyx v oblasti mathematiki,” (On the achievements of the Chinese scholars in the field of mathematics). In Iz Istorii Nauki I Texniki Kitaja (Essays in the History of Science and technology in China), Moscow.
[Zhang Dainian, Edmund Ryden,2002]. Key Concepts in Chinese Philosophy. Yale University Press
[Zhou Hanguang, 周瀚光, 1987]. 論李冶的科學思想. 中國數學史論文集 (三). 吳文後(eds). 山東教育出版社:73-80.