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國立臺灣師範大學數學系博士班博士論文 指導教授： 洪萬生博士 林力娜博士 Exploring the Features

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國立臺灣師範大學數學系博士班博士論文 指導教授： 洪萬生博士 林力娜博士 Exploring the Features of Algebra in Medieval China: the Case of Yigu yanduan. 研 究 生： 博佳佳 中 華 民 國 一百零一 年 七 月 1 FOREWORD The little story This project started in 2007, when Karine Chemla suggested me to use my knowledge of Sanskrit and Chinese for mathematics. I was teaching philosophy and studied Sanskrit and modern simplified Chinese at university. In order to improve my Chinese and to learn mathematics, I was sent to the department of mathematics of National Taiwan Normal University and prescribed to read the Yigu yanduan by Li Ye to teach myself mathematics in classical Chinese. We decided to build a joint thesis partnership. And this is where the adventure really started. First, the more I was reading the Yigu yanduan, the less it resembled to its secondary literature. This treatise had much to say than expected and it became the heart of my research. This dissertation for NTNU is the first part of a bigger work. The reader will find here a short version of my research and my translation, having for focus the reading of Li Ye’s treatise. In the complete version of my work for Paris VII, one will find the elements of comparison with some treatises written in Sanskrit, the complete translation of the Yigu yanduan and some translations of Sanskrit texts. During this adventure, I have been also confronted to cultural gaps. Not only there was a gap between everyday life in Taiwan and France, and also a gap of scientific practices between the two universities. There are different work cultures between the French and the Taiwanese way of “doing science”, but also different work cultures between Sanskritists and Sinologists, between historians and historians of mathematics in a same university… sometimes contradictory, and always bewildering differences. I do apologize for the reader who will have to suffer my writing. I was asked to write in English (or at least in this international language one dares calling English), while English is not my native language. A native speaker will help me the transform my Globish-Frenglish into real English in a near future. This research was done thanks to the patronage of Taiwan Ministry of Education and the Conseil regional d’ile de France, whose financial helps made everything possible. I also want to thank all those who accompanied me during these years: 劉容真教授, 英家銘, who was like a big brother for me, 黃美倫, and of course all the team of NTNU maths department! I thank also the girl’s band (Sylvia, Laura and Alex) for their patient listening. Je tiens a remercier mon mari pour son indefectible soutien, et notre Leonard pour les fous rire. 2 3 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 1 Table of Content 1. Introduction to the Yigu yanduan 06 1.1 Methodological aspects 06 1.2 General description 10 1.3 State of art 12 1.4 Source of the Yigu yanduan: the Yiguji 17 2. The Qing dynasty editors’ work 20 2.1 .Commentaries to the Yigu yanduan 20 2.2 Status of the available editions 24 2.2.1. Editorial notes and corrections to the discursive part. 25 2.2.2 Treatment of diagrams. 29 2.2.3 Treatment of polynomials. 35 3. Statements of problems: questions of interpretation. 42 3.1 order of problem (part 1) 43 3.2 diagrams and statements 48 3.3 The use of data from the statement 49 4. Description of the Art of the Celestial Source, 天元術. 56 4.1 Description 59 4.1.1 descriptive example 59 4.1.2 generic description 66 4.2 manipulations on the counting support 69 4.2.1 Writing numbers 69 4.2.2. names of positions on the support 72 4.2.3 arithmetic of polynomials 75 i- addition and subtraction 76 ii- multiplication 78 iii- division 81 4.3 from the extraction of square root to the concept of equation 84 5. Description of the procedure of Section of Pieces of Areas, 條段. 93 5.1 圖 as diagrams 97 4 5.2 diagrams and equation 100 5.3 transformation of diagrams 107 5.4 geometrical configuration and arithmetical configuration 114 5.5 negative and positive coefficients 119 5.6 order of problem (part 2): The analogy 132 Conclusion 150 Supplements. 1. Table of editorial notes and table of differences between the characters in the different editions of Yigu yanduan. 152 2. Table of equations in Yigu yanduan. 158 3. Samples of translation 163 i. Jing Zhai gu jing tu 163 ii. Yigu yanduan’s preface by Li Ye 165 iii. Problem one 167 iv. Problem two 178 v. Problem three 187 vi. Problem forteen 198 vii. Problem twenty one 209 viii. Problem thirty six 217 ix. Problem forty five 225 x. Problem sixty three 230 4. Lexicon 238 5. Bibliography 244 5 Exploring the Features of Algebra in Medieval China: The Case of Yigu yanduan 1. INTRODUCTION 1.1 Methological aspects One of the best ways to understand the mathematical reasoning of ancient treatises is to work through translations. Here I worked on a complete translation with mathematical commentaries of a mathematical treatise written in Chinese with a comparative edition of its Chinese version. The text presents a list of problems and examples on linear and quadratic equations. I translated each of the problems twice: in a literal translation and in a transcription into modern mathematical language. The use of the modern mathematical terminology solely for the purpose of comparison would lead to standardization of practice under the criteria of our contemporary reading, and would prevent the valorisation of the specificities of ancient text. What is a familiar object today recovers several different practices in the past. And we have to keep in mind that mathematical objects are cultural products elaborated by the work of different cultures which did not use the same concepts in a same way. In modern transcription, all quadratic equations look the same: ax² + bx + c = 0. And the all mathematical treatises would be reduced to sets of solutions and setting up of linear and quadratic equations while what looks at first sight the same, is in fact hiding differences. Following [Chemla Karine, 1995], one will consider what a modern mathematician would write as ax² + bx + c = 0. For us today, this object represents multiple aspects. It can be considered as an operation, but it can also be thought of as an assertion of equality. In another respect, the relation represented by this equation can be tackled in various ways so as to determine the value of the unknown quantity x. There are various kinds of solutions: those by radicals, numerical ones like the so-called “Ruffini-Horner” procedure, and geometrical solutions, among others. [Chemla Karine, 1995] shows that this combination of elements of such diverse natures is not to be found as such in ancient documents and that they did not have undergo a linear development, “whereby a first conception of equations would be progressively enriched until we attain the complexity of the situation sketched out above”. On the contrary, she finds various ancient mathematical writings wherein the 6 elements distinguished above are scattered and dissociated, and other writings which combine some of them. Therefore, it may be that the history of algebraic equations has, on one hand, to be conceived as a combination of several kinds of equations; and, on the other hand, as syntheses between some of these aspects when they happen to meet.
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