exploring the features of algebra in medieval China:the case of yigu yanduan

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(1)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.

(2) 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.

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(4) 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.

(5) Table of Content. 1. Introduction to the Yigu yanduan 1.1 Methodological aspects 1.2 General description 1.3 State of art 1.4 Source of the Yigu yanduan: the Yiguji. 06 06 10 12 17. 2. The Qing dynasty editors’ work 2.1 .Commentaries to the Yigu yanduan 2.2 Status of the available editions 2.2.1. Editorial notes and corrections to the discursive part. 2.2.2 Treatment of diagrams. 2.2.3 Treatment of polynomials.. 20 20 24 25 29 35. 3. Statements of problems: questions of interpretation. 3.1 order of problem (part 1) 3.2 diagrams and statements 3.3 The use of data from the statement. 42 43 48 49. 4. Description of the Art of the Celestial Source, 天元術. 4.1 Description 4.1.1 descriptive example 4.1.2 generic description 4.2 manipulations on the counting support 4.2.1 Writing numbers 4.2.2. names of positions on the support 4.2.3 arithmetic of polynomials i- addition and subtraction ii- multiplication iii- division 4.3 from the extraction of square root to the concept of equation. 56 59 59 66 69 69 72 75 76 78 81 84. 5. Description of the procedure of Section of Pieces of Areas, 條段. 5.1 圖 as diagrams. 93 97 4.

(6) 5.2 diagrams and equation 5.3 transformation of diagrams 5.4 geometrical configuration and arithmetical configuration 5.5 negative and positive coefficients 5.6 order of problem (part 2): The analogy Conclusion. 100 107 114 119 132 150. 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.

(7) 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.

(8) 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. In this study, I will try to identify some various aspects of equations that were elaborated separately. Secondly, I will attempt to understand what synthesis between some of them resulted in and what kind of transformation did various elements undergo. Consequently I wonder if our concept of “equation” refers to the same object in ancient traditions; if the idea of “unknown” recovers the same reality. To compare results that look the same is not sufficient. Nothing can substitute the analysis of a mathematical text first as a text. A text is not always a narration, a description or a presentation; it is a testimony on results and concepts and contains traces of activities linked to their interpretation. This is the reason why I chose the focus on literal translation first and only secondly on the mathematical transcription, the latter being indispensable but not sufficient. Literal translation implies to take into account the manifestation of mathematical objects inside the text and the relations induced by the way of “talking about” these objects. Some differences are perceptible in the ways of expressing, of shaping, structuring the discourse. I want to show that what can be recognised as the same object occurs with different status in various sources, and that the history of mathematics should not only describe the evolution of procedures to solve equations, but also the evolution of the nature of equation. There were different concepts of equation available in the world, and it has been retrospectively that they have been identified as the same object, because our analysis of ancient sources uses contemporary concepts that were designed through their synthesis. I have therefore to identify some of these elements: The ways in which what we would recognize as an equation manifests itself in the sources are different. In various traditions, equations are identified as mathematical objects of different kinds, hence, they are worked out in different ways, and the ways of solving equations have developed along different lines. My research starts with the study and the translation of a Chinese text, the Yigu yanduan (益古演段), written by Li Ye (李冶) in 1259, which is a collection of 64 problems. Li Ye is one of the famous scholars of the Song-Yuan time period1. His literary name was Renqing (仁卿), and his appellation was Jingzhai (敬齋). He was born in a bureaucratic family in 1192. He was originally known as Li Zhi (李治) but when he discovered that his name was the same as the Tang emperor whose dynastic title was Tang Gaozong (唐高宗), 1. Biographies of Li Ye can be found in English in [Mikami Yoshio, 1913], p.80; [Ho Peng Yoke, 1973] p.313-320; [Lam Lay Yong, 1984], p. 237-9; [Li Yan, Du Shi-ran, 1987], p.114; in Chinese in [Mei Rongzhao, 1966], p. 107. His life is the object of several notices since the Yuan dynasty, 1370 in 元史, Yuan shi, ch.160, for the first one, and in 1799 in the inventory of biographies of scientists, 疇人傳, chourenchuan, by 阮元, Ruan Yuan, for the last one. I will not treat this material in my present work.. 7.

(9) he changed it to Li Ye. In 1230 he passed the civil service examination and was offered a post in the government. However, his service was shortened as the district to which he was assigned fell to the Mongols in 1232. He took refuge in the north, and finally gave up all hope of an official career when the Mongols conquered the Jurchen kingdom in 1234. In such impoverished situation he devoted himself to studies. The first outcome in mathematics was the Ceyuan haijing (測圓海鏡) written in 1248. He continued to live a scholarly recluse life in the mounts Fenglong in Hebei (河北), having people coming to him for instruction. In this environment he produced the Yigu yanduan in 1259. In 1260, Kubilai Khan had on several occasions approached Li Ye for advice on state affairs and astrological interpretations. When Kubilai ascended the throne, Li Ye was offered an official post, which he declined twice. He died in the mounts Fenglong in 1279. Li Ye is also the author of the following works: Fan shuo (泛說，supernumerary talks), Jing Zhai gu jing tu (敬齋古今黈, commentary of Jing Zhai on things old and new), Jing Zhai wen ji (敬齋文集，collection of works by Jing Zhai), Bi shu cong xue (壁書叢削，amendments of books on the wall shelves) among them only the Jing Zhai gu jin tu still exists containing some quotations of Fan shuo. We do not know what happened to these books and why they disappeared 2. According to the biography of Li Ye written in Official history of the Yuan, 元史, Li Ye asked his son to burn all his works except the Ceyuan haiijng, because he felt that it alone would be of use to future generations. We do not know to what extent his wishes were carried out. But the Yigu yanduan and the Jing Zhai gu jing tu3 survived the fire4. The Ceyuan haijing was the object of many studies in history of mathematics and is systematically quoted in studies concerning algebra in China, but on the contrary, the Yigu yanduan has not really been systematically studied5. Only three studies give an analysis of the content through several examples: Mei Rongzhao in 1966, Kong Guoping in 1987 and 1999 and Lam Lay-yong in 19846. The latest available edition of the Chinese text was done by Li Rui in 1789. A reprint of it has been published by Guo Shuchun in 19937. Compared to the attention given to the Ceyuan haijing, the Yigu yanduan remained in the shadow. Nevertheless, this treatise is interesting for our purpose because of the diversity of methods that are proposed to set up quadratic equations for each of the problems and also by the 2. See [Ho Peng Yoke, 1973] See sample of translation in supplement. 4 The wileness to preserve only his mathematical masterpiece could be interpreted as an attitude of Li Ye in favour of what we call mathematics. But it could also be that Li Ye wanted to respect the Taoist philosophy, and consider that philosophy is not something to be said, nor written. The very first sentence of the Taoist canon, Dao De Jing, “道可道非常道, 名可名非常名”, “the way that can be said is not the eternal way, the name that can be said is not the eternal name”, was often interpreted as a negation of all possibilities of language to express philosophy. Therefore, it could be that the will of Li Ye is, paradoxically, also in favour of philosophy. Although only one of the philosophical books by Li Ye is still extant, we should not forget that the majority of his works were philosophical, and may be the reading of the Jing Zhai gu jing tu deserves more attention. This reading could contribute to reflexion on what we categorize as “mathematics” and “philosophy”. 5 See Part I. Introduction, state of art. 6 More details are given about the content and conclusion of these studies in the introduction of the part I.A. 7 For the history of the edition of the Chinese text, see introduction of part I.A. 3. 8.

(10) fact that it was considered as accessible for any reader8. Each of the 64 problems is provided two or three different procedures of setting up equations. The first method is qualified as “algebraic” is named the Tian Yuan,天元 , procedure of Celestial Source; the second one is qualified as “geometrical” and named Tiao Duan, 條段, Section of Pieces [of Areas] and the third one is namely the Jiu shu, 舊術, the old procedure. Li Ye states that these methods, or some of them, are issued from older treatises and, according to historians, the purpose of his treatise is to spread those out for any lambda reader. This makes us think that the Yigu yanduan can be considered as an outcome and a standardization of more ancient mathematical practices than as a revision of mathematics in Chinese. This makes the Yigu yanduan seem very representative of the diversity of algebraic practices of that time and deserves more attention. From the translation of this treatise, I want to identify what culture of algebra we are going to discover, what were the different procedures for the setting up and the resolution of quadratic equations and under which aspect is presented what we usually identify as equation. I will first start with the description of some algebraic procedures, and show how texts reassemble them. The way of gathering procedures manifests some practices. I will describe the mode of writing of what we call polynomials and equations. I will see how these objects are represented and inserted in the text and what kind of distinction one can make between what we identify as polynomial and equation. Most of the time, these are presented in tabular settings, but I will show that these settings have different status. As treatise present like a list, I will try to understand what justifies the order of problems or examples. To understand how the structure is organised will provide us clues on the intentions of authors and on how to read and interpret these texts. The Yigu yanduan contains many diagrams; I am going to investigate what are the different ways of reading and the different functions of geometrical figures. My purpose is to characterize this peculiar mathematical practice and understand the variety of natures of equation.. 8. See Part I. Introduction.. 9.

(11) 1.2 General description.. The Yigu yanduan, Development of Pieces [of areas according to] the Improvement of Ancient [Collection]9, 益古演段, was written in 1259 by Li Ye, 李冶, and published later in 128210. 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: that is calculating the diameter or side of a field inside of which there is a pond. Each problem follows the same pattern and the treatise seems very repetitive. 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 peculiarity in this text is that it introduces and differentiates two distinct methods for setting up quadratic equations. The first method, is named tian yuan,天元, and will be referred here as the procedure of Celestial Source, and the second, is named tiao duan, 條段 , the procedure of Section of Pieces [of Areas]. The first one is qualified as algebraic while the secod is said geometrical by historians, as we will see later. A third procedure, which seems geometrical too is added to twenty three of the problems, and is title “old procedure”, jiu shu, 舊術. The status of this text was interpreted by historians11 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 question this point of view, to explain why there was such a misunderstanding and to put into light a peculiar field in Chinese mathematics. I will show that this book is in fact treatise dealing with a mathematical object which was new at the Song-Yuan period, and whose mathematical practices can be related to the famous Han dynasty classic, the Nine Chapters on Mathematical Procedures12, 九章算術. The focus must be redirected on the other procedure: the Sections of Pieces [of Areas]. I will show that this procedure concerns practices of geometrical diagrams and that these practices were not new by the time of Li Ye. Available mathematical books anterior to the Song dynasty are deprived from their geometrical illustrations. In this context, the Yigu yanduan becomes a. 9. Development (演) of Pieces (段) [of areas according to] the Improvement (益) of Ancient (古) [Collection]. The character 益, yi, can also be translated by “to increase”, “to augment” or “benefit” and “adventage”. In order to stick to the idea of amelioration and enrichment, I chose to translate it by “improvement”. 10 [Kong Guoping,孔國平,1987]. p.166. 11 See Introduction, part B, State of art. 12 I follow Karine Chemla’s translation of the title, but will later refer to the classic as “the Nine Chapters”.. 10.

(12) precious source. It testifies of continuity and changes inside a tradition of practice around diagrams. Here follows the general description of problems. First the statement of a problem, introduced by 今有, jin you, gives the area of field less the area of the pond and one or several distances, usually side, diameter or diagonal. Then other distances that were not given in the statement are asked (問, wen) and the answers (答, da) immediately follow. The statement is systematically followed by a diagram representing the field and the pond, inside of which one or several of distances given in statement or in the answer are drawn. Some of the dimensions are also written down as a caption. The problem is thereafter solved according to the first procedure, tian yuan, starting with choosing the unknown and ending by establishing an equation that the unknown satisfies. The procedure describes how to find the coefficients of the different terms of the equation and gives a list of operations and manipulations on a counting support that lead to these coefficients. These coefficients are presented using tabular settings of two or three rows. On each of the rows, one sets a term of the equation. The rows are ordered by degree, the top row being the constant term, and the third one being x². The procedure ends with the statement of the equation. Li Ye does not describe how to solve the equation, the reader is supposed to know how to extract its root. There are several possible roots to the equation, but Li Ye gives only one them. We do not know if Li Ye considers the other roots. The given root of the equation is the quantity corresponding to one of the value of the unknown used to solve the problem. Li Ye ends this part with indicating how to find the different other final answers of the problem once one knows the root of the equation. Then, follows the solution by a second procedure: the solution by Sections of Areas, tiao duan. The general characteristic of this second procedure is to derive the terms of equation from geometry. This part contains first a description of each coefficients of the equation introduced by the sentence “依條段之求”. Li Ye indicates the operations that lead to transform the data of the statement into the coefficients of the equation. Each coefficient is coupled with fixed positions on a counting support, namely, the “dividend”, 實, shi; the “joint”, 從, cong; and the “constant divisor”, 常法, chang fa. The translation of these terms results from a choice made historians13 who saw a strong analogy between the procedure of division and the procedure of the root extraction. But for the moment, one will consider the shortcut associating them to, respectively, the constant term, the term in x and the term in x² of an equation. Right after this first sentence, follows a small portion of text composed of one, or sometimes two, diagrams and of an explanation which titled 義, yi, which I translate by “meaning”. The “meaning” has the shape of a small commentary whose object is the diagram. It mostly states how to identify the terms of the equation from the. 13. [Chemla Karine, Guo Shuchun, 2004], [Lam Lay-Yong, Ang Tian Se, 2004], [Li Yan, Du Shiran, 1987], among others.. 11.

(13) diagram. It is difficult to give a general description of this part, because each of the “meanings” points out the specificity of the case which is treated. Twenty three of the problems are presented with another third method, which is called “old procedure”, jiu shu, 舊術. Only one of these twenty three problems is given with a diagram. Although diagrams are mostly absent, from the vocabulary, which is almost the same as the one used in the Section of Pieces [of Areas], one can deduce that the procedure was geometrical too. The old procedure is usually very briefly stated and has the same structure as the first sentence of the Sections of Pieces [of Areas]: only the operations constructing the coefficients are stated with the same references to the three positions of the counting support. Some of the problems are also presented with variations of procedures. These variations are introduced by the expression 又法, you fa, “another method”, in pb.3; 40; 44; 56. They are placed at the end of the procedure of Section of Pieces [of Areas]. The problem 6 presents three different methods with for each a new diagram. We also notices the two following peculiarity for other problems. The problem 11 is composed of two problems with two different statements. And the problems 44 ; 59 and 60 are presented without any procedure of Section of Pieces [of Areas]. But they are not deprived of geometrical procedure, because they are accompanied with the “old procedure”. We keep in mind these notifications for later. The difficulty in reading the Yigu yanduan is to clarify the relations between the three procedures. Several interpretations were proposed to understand why Li Ye assembled these different procedures together. I will now introduce them.. 1.3 State of art.. [Lam Lay Yong, 1984] noticed that two hypothesis are possible, either “the tian yuan was new and Li Ye has taken the opportunity to justify its algebraic reasoning by falling back upon the traditional equivalent geometrical meaning” or “as an equation derived by the old method through the tiao duan concept was not easy to understand, Li Ye used the tian yuan method to elucidate the origin of the tiao duan method and explain it by means of clear geometrical figures14”. That is to say, in the first case, the procedure of Celestial Source was new and needed to be demonstrated by a well known ancient procedure, the Section of pieces [of Areas]. And in the second case, the “old procedure” and its derived form, the Section of Pieces of [Areas], were confuse, or may beforgotten, and needed to be re14. [Lam Lay-yong, 1984]. p. 264.. 12.

(14) explained by the mean of a famous well known procedure, the Celestial Source. And Li Ye added diagrams illustrating the procedure of Section of Pieces [of Areas]. For both hypotheses the procedure of Section of Pieces [of Areas] is older than the procedure of the Celestial Source. But in the first case, the Section of Pieces [of Areas] was well known, while in the second hypothesis, the Celestial Source was the well known one. The second hypothesis was the one proposed by [Mei Rongzhao, 1966]15 , and Lam Lay Yong did not choose any of the hypotheses. Both authors agree that there is difference between the old procedure and the Section of Pieces [of Areas]: in their view, the part of the text containing the Section of Pieces [of Areas] is created by Li Ye, while the old procedure is copied from an older source. And thus twenty of three of the solutions, names “old procedure” are borrowed from an ancient source. Another studies by [Kong Guoping, 1999]16 also suggested the Celestial Source was derived from the ancient geometrical procedure. A study of the procedure Section of Pieces [of Areas] by [Annick Horiuchi, 2000]17 confirms that the procedure (Celestial Source) was new and takes its source and inspiration from an old geometrical method, which is the Section of Pieces [of Areas]. The originality of Kong Guoping is that he not only claims that the Celestial Source was a new and was the method chosen by Li Ye to clarify an ancient procedure, but also that the whole geometrical procedure is a method borrowed from a predecessor18. The geometrical procedure is in fact presented with two different names, the Section of Pieces of [Areas] and the old procedure. That is: the Section of Pieces [of Areas] and the old procedure are not so different from each other. This hypothesis is different from the one presented by Mei Rongzhao. According to Mei Rongzhao, only the “old procedure” is borrowed from a predecessor. The present study will confirm the hypothesis by Kong Guoping and try to identify other items which can be attributed to more ancient sources. The various translations of the four characters of the title 益古演段, Yigu yanduan, also testify of the multiple interpretations of the status of the book. Should one consider it as a text book, a theoretical treatise or as pragmatic text? The oldest occurrence of translation, “Exercises and applications improving the ancient methods”, was proposed by [Sarton George, 1927]19 . This first translation shows that the Yigu yanduan was considered as a kind of miscellany whose object is practical (field measurements). Translations agree on the purpose to improve an ancient method, insisting on differentiating the “old” from the “new”. For example, it was later translated by “New Step in Computation” by [Libbrecht Ulrich, 1973]20 and by J-.N. Crossley in [Li Yan, Du Shiran. 1987]21. [Lam Lay Yong, 1984]22 also proposed her own translation: “Old mathematics in Expanded Sections”. But, in all 15. [Mei Rongzhao, 1966]. p. 143. [Kong Guo-ping, 1999]. p. 173; 197. 17 [Annick Horiuchi. 2000]. p.253. 18 See Introduction, Sources of the Yigu yanduan. 19 [Sarton George, 1927]. p.627. 20 [Libbrecht Ulrich, 1973]. p.19. 21 [Li Yan, Du Shiran. 1987]. p.114. 22 [Lam Lay Yong, 1984]. p. 237. 16. 13.

(15) these translations, it is difficult to understand which of the characters are translated by “computation”, “application” or “mathematics”. There are also two more literal translations into French: “Le yan duan (development of pieces of area) du Yiguji” by [Horiuchi Annick, 2000] 23 and “Le déploiement des pièces d’aires pour la [collection] augmentant les [connaissances] anciennes” by [Chemla Karine, 2001]24. For the first time, these two translations both take into account that Li Ye, in his preface first and title also, refers to an older book, the Yiguji or “collection augmentant les [connaissances] anciennes” (益古集), and that the Yigu yanduan is not only improving but also, to some extent, presenting the ancient method. And one notices that the expression “yan duan” names a type of procedure and is the main object of the title25. I will come back to the question of the translation of the title and justify my choice in conclusion. Whatever the translation of the title or the statue of the different procedures, the Yigu yanduan has been considered as “a revision of a work for beginners in the « celestial element » method”26 , as a book “devoted to the method of tian yuan shu27”, or as “an “introduction” to the Sea Mirror of Circle Measurement”28. It is generally thought that Li Ye “took the opportunity to explain the tian yuan shu method in a less complicated manner after finding his first book (the Ceyuan haijing) too difficult for people to understand 29”. And while the focus still remains on the procedure of the Celestial Source, the procedure of Section of Pieces [of Areas] is neglected, or sometime published without its diagrams30, or sometime not even mentioned31. There are several reasons for that. The first reason is that the procedure of the Celestial Source was set forth with a higher level of difficulty in the other major mathematical work by Li Ye, the Sea Mirror of the Circle Measurements, Ceyuan haijing, 測圓海鏡. This treatise was completed in 1248 and published in 1282, alike the Yigu yanduan. The Ceyuan haijing knew a successful posterity and the Yigu yanduan remained in the shadow of the Ceyuan haijing32. The Ceyuan haijing is. 23. [Horiuchi Annick, 2000]. p.238. “The yan duan (development of pieces of areas) of the Yiguji”. [Chemla Karine, 2001]. p.12-13. “the deployment of pieces of areas for the [collection] augmenting the ancient [knowledge]”. 25 These two publications in French are not strictly dedicated to the yigu yanduan. The first one is dedicated to the understanding of the procedure of section of pieces of areas based on the reading of one of the parts of the Yang Hui suanfa, 楊輝算法, the tian mu bilei chengchu jiefa, 田畝比類乘除捷法 written by Yang Hui, 楊輝, in 1275. The second one is dedicated to the change of use and meaning of the character tu, 圖, “diagram” from the Han and during the Song dynasty. 26 [Li Yan, Du Shiran, 1987]. p 114. 27 Guo Shuchun’s introduction to Ch’en Tsai Hsin translation of Zhu Shijie’s Jade Mirror of the Four Unknown. 2006. I. p.46. 28 [Dauben Joseph, 2007]. p. 327. 29 [Ho Peng Yoke, 1978]. p. 319. 30 [Dauben Joseph, 2007]. p. 329. [Guo Shuchun, 2010], p. 370-73. 31 [Ho Peng Yoke, 1978]. pp. 313-320, [Li yan, Du Riran, 1987]. 32 [Chemla Karine, 1982];[Chemla Karine, 1993]; [Kong Guoping,1996] are strictly dedicated to the Ce yuan haijing. [Mei Rongzhao, 1966], [Li Yan, 1954],[Kong Guoping, 1988]; [Kong Guoping, 1999], [Guo Shuchun, 2010] among other devoted one chapter to the reading of Ceyuan haijing and few pages for the Yigu yanduan. 24. 14.

(16) said to be the crystallized thought of Li Ye’s studies on the Art of the Celestial Source, while the Yigu yanduan would represent his effort in popularizing the method33. Altough there are some mentions of the procedure of the Celestial Source in other Chinese mathematical works34, Li Ye gives the earliest testimony of its practice. Therefore it is impossible to deduce from other materials if the procedure was common or not. The absence of other sources has for other consequence that the works of Li Ye is presented as “the first truly algebraic works in China35”. The assimilation between Li Ye’s mathematics and the disciplin of algebra seems to be continuous in history of sciences. In 1978, Ho Peng Yoke quoted George Sarton’s 1927 book in the following way: “Li Ye was indeed, as George Sarton says, essentially an algebraist36”. In more recent works, the procedure of the Celestial Source is still directly assimilated to “algebraic procedure37”: it is “the Chinese algebraic process of logically setting up algebraic expressions and finding a relation between these expressions to derive an equation38” or “the “technique of celestial element” is roughly similar to the method used in present-day textbooks in algebra39”. Despite the loss of sources anterior to the 13th century China concerning algebra and the fact that the texts preserved seems to ignore each other, one supposes that reflections having for object what we identify as equation were hold during the Song dynasty40. This led historians to think that the 13th century is the acme of algebra in China41. This is the case of Jean-Claude Martzloff: “pour certains mathématiciens chinois du 13e siècle, Li Zhi, Zhu Shi-jie et leurs émules –l’algèbre, c’est le tian yuan shu, c’est-à-dire l’art de la primordialité céleste 42“. If historians recurrently refer themselves to the notion of “algebra” concerning 13th century China, few are those trying to precise the content or relevance of this notion. The question was already raised by [Horiuchi Annick, 2000], [Breard Andrea, 2000] and [Chemla Karine, 2000]. However, no one would question that one faces a field of research far different from the topic we call algebra at the present day. If one considers the procedure of Celestial Source is algebraic, the present study intends to show that “algebra” can also take a different aspect. I will focus on algebraic practices involved in the other procedure, the Section of Pieces [of Areas]. The second reason why the interpretation focuses on the Celestial Source is due to the reading of the preface to each book written by Li Ye. In his preface to the Ceyuan haijing, 33. [Mei Rongzhao, 1966]. p. 147. [Lam Lay Yong, 1984]. p.247. [Kong Guoping, 1999], p. 173. See part III. Description of the Procedure of Celestial Source, 35 [Dauben Joseph, 2007], p. 324. 36 [Ho Peng Yoke, 1978], p.320. See [Sarton George, 1927], p. 627. 37 [Dauben Joseph, 2007], p. 323. 38 [Lam Lay Yong, 1983], p.243. 39 [Li Yan, Du Shiran, 1987], p.138. 40 [Horiuchi Annick, 2000], pp. 183-187. 41 [Li Yan, Du Shiran, 1987], Ch.5 42 [Martzloff Jean-Claude,1988], p. 242. English edition, p. 258: « Finally, for certain 13th century Chinese mathematicians, Li Zhi, Zhu Shijie and their emulators, algebra was the tian yuan technique”. We notice that the English edition does not propose any translation of the expression “tian yuan” and the last part of the French sentence “c’est-à-dire l’art de la primordialité céleste” is not translated into English. 34. 15.

(17) Li Ye complained about the government’s apparent Philistine attitude to mathematics of his time. When he wrote the preface to Yigu yanduan, he shifted his former complaint to the mathematicians themselves. He blamed them for being narrow-minded and unwilling to impart their knowledge magnanimously. They wrote in such an abstruse and guarded manner that the true mathematical knowledge was not revealed43. From this it was inferred that Li Ye writes the Yigu yanduan to correct the prevailing trend and to show how a useful mathematical technique such as the tian yuan could be learned and mastered even by beginners. And this interpretation of the preface would be confirmed by the fact that a large proportion of the treatise is occupied by repetitive scripts dedicated to the procedure of the Celestial Source and the remaining part is mostly “only” filled with diagrams and small discursive parts concerning these diagrams, and that three of the problems are given without procedure of Section of Pieces [of Areas]. But the two characters 天元, tian yuan, never appear in the preface. On the contrary, only the Section of Pieces [of Areas] is mentioned. Li Ye writes that he modified it by providing diagrams. One can indeed understand that the book is meant to be accessible (my punctuation and translation): 近世有某者，以方圓移補成編，號「益古集」，真可與劉李相頡頏。余猶恨其悶匿而不盡發，遂再為移補條段細繙圖式，使粗知十百者，便得入室啗其文，顧不快哉？.“[For instance], a book entitled Collection Improving the Ancient [Knowledge] (益古集) was compiled recently with reshaped (移補) [solutions to geometric problems of] rectangles and circles. It is indeed an equivalent of Liu Hui and Li Chunfeng. However, I detest its reserved style, and hence added detailed diagrams (細繙圖式)44 of how to reshape the Sections of Pieces [of Areas] (條段). Isn't it a great joy that the book will thus be accessible (入室) with basic knowledge (粗知) now?” There are two expressions evoking “popularization”. The first one is 入室, ru wu, which literally means “to enter the room”. It reminds of the teaching of martial arts, where students are allowed to enter the apartments of the master to receive the true teaching once they mastered by themselves the basics45. The expression is metaphoric and quite ambiguous. The second expression is more obvious more a modern reader. That is 粗知, cu zhi, translated also by “ordinary”, “vulgar” or even “coarse” knowledge. If Li Ye is doing a work of popularization, we have to wonder how he is 43. My punctuation and translation: 今之為算者，未必有劉、李之工，而褊心跼見，不肎曉然示人，惟務隱互錯糅，故為溟涬黯黮，惟恐學者得窺其彷彿也。不然，則又以淺近觕俗，無足觀者，致使軒轅隸首之術，三五錯綜之妙，盡墮於市井沾沾之見，及夫荒邨下里，蚩蚩之民，殊可憫悼。“On the other hand, contemporary mathematicians (算者), who do not necessarily study as comprehensively as Liu Hui or Li Chunfeng, are narrow-minded and short-sighted. Instead of making it clear, they prefer rendering it as implicit and intricate as possible in order to make the mathematics appear opaque and obscure. They prevent even a glimpse of its simulation being caught by others. Otherwise, some of them opt to deal with merely the basic and well-known part that does not worth looking into. Consequently, the methods (術) of the ancients Xuan Yuan (軒轅) and Li Shou (隸首) along with the sophisticated art of numbers (三五錯綜之妙) become something with which everyone in the town can be self-satisfied. It is such a pity that they actually know just as much as ignorant villagers”. 44 I do not know if the expression 圖式, tu shi, names “diagrams and [tabular] configuration” or “diagrams” only. 45 [Gu Meisheng, 1999] Preface.. 16.

(18) doing it. Is it a work of promotion, simplification, generalization or just dissemination? All these concepts are not synonyms. For example, a work of transmission does not always imply a simplification. What is “basic” or “ordinary” according to Li Ye, is not obvious for a modern reader. There is another consequent question: what was the content of the book he wants to transmit? There is almost no information on the context of the writing of the Yigu yanduan. We just know that Li Ye lived as a recluse when he wrote it. We guess that he probably had disciples, but we do not know if this book was dedicated to them and if they were trained in mathematics. We do not know who the supposed readers of the Yigu yanduan are. For the moment, none of the elements confirm the link with the Ceyuan haijing and a popularization of the procedure of the Celestial Source. We do not know if the simplification of the procedure of Celestial Source is the main topic of the Yigu yanduan.. 1.4 The source of the Yigu yanduan.. As we saw previously, in the preface of Yigu yanduan, Li Ye justifies his motivation in writing this book. Li Ye was so impressed by an older book, named Yiguji, 益古集, Collection Improving the Ancient [knowledge]46, that he compared its content to the works of the two famous commentators of the Nine Chapters of the third and seventh centuries, respectively, Liu Hui (劉徽) and Li Chunfeng (李淳風). But he found the presentation obscure and incomplete, he, therefore, decided to revise it and to add diagrams to make it clearer. We do not know how far the text was revised and which part exactly remain in the Yigu yanduan. Actually, there are no traces of the Yiguji, and the author and the date of the text are still disputable. According to Mei Rongzhao47, it is probable that the book can be attributed to a certain Jiang Zhou, 蔣周, originated from Ping Yang, 平陽, in Shan Xi, 山西. Mei Rongzhao’s argument is based on two references to a book whose first two characters in the title are Yi gu, 益古. In his preface to 朱世杰, Zhu Shijie’s Precious Mirror of Four Elements, Si yuan yu jian, 四元玉鉴, Zu Yi, 祖頤, in 1303, gives a list of works to which later readers are indebted for the knowledge of the art of Celestial Source. The first book on the list titled Yi gu by Jiang Zhou is mentioned as one the treatises contributing to the elaboration of the. 46. The title is translated by “Continuation of the ancients” by [Hoe John, 2008]. p.v and “collection of old mathematics” by [Lam Lay Yong, 1984]. p.239. 47 [Mei Rongzhao, 梅荣照 1966]. p.139.. 17.

(19) art of the Celestial Source48. At the end of the sixteenth century, the Ming mathematician Cheng Dawei, 程大位, compiled a list of mathematical texts produced between 1078 and 1224, among them a book titled Yigu suanfa, 益古算法, computing method improving the ancient [knowledge]. Those could be the book titled Yiguji by Li Ye. Xu Yibao49 found another mention of the same person from the Song dynasty. The chapter fourteen of Zhizhai shu lu jieti, 直齋書錄解題 (1244) by Chen Zhensun, 陳振孙, mentioned the Yingyong suanfa, 應用算法, Computing method for application, written before 1080 by a certain Jiang Shunyuan, 蔣舜元, from Ping Yang. Xu Yibao argues that Jiang Zhou and Jiang Shunyuan are the two names of a same person. Another book from the Yuan dynasty presents some solutions of problems using the procedure of section of area. One section of the Yang Hui suanfa, 楊輝算法, Yang Hui’s Methods of Computation50, named the Tian mu bilei chengchu jiefa, 田畝比類乘除捷法,51, Fast methods of multiplication and division related to [various] categories of fields and [their] measures, written by Yang Hui, 楊輝, in 1275, presents some extracts of an older, and lost, work: the Yigu genyuan, 議古根源, Discussion on the origin of ancient methods52, written by Liu Yi, 劉益. According to Te Gusi53, Liu Yi lived in Zhongshan, 中山, in Hebei, 河北, at the end of 10th century or beginning of 11th century. After comparing the problems studied in the Yang Hui suanfa and the ones presented in the Yigu yanduan, Xu Yibao54 deduced that the Yiguji was probably composed in the middle of the 11th century and was itself based on the Yigu genyuan. Both books were dedicated to the procedure of Sections of Pieces [of Areas], and the procedure of the Celestial Source was after derived from the latter. That is why the Yiguji was considered as a book that is a source. It is often accepted that the solution to 23 of the 64 mathematical problems named “old method” are directly borrowed from Yiguji, as it is implied by the name55. According to the preface written by the editor of the Yigu yanduan in 1282, the Yiguji contained 70 problems. But in the 18th century, the commentators of the Yigu yanduan wrote in their preface that it was less, probably 64 problems. One also noticed that three of the problems of the Yigu yanduan are not provided with the procedure named “section of area”, while they are given a procedure of Celestial Source and of the old method. But the question of the relation between the twenty three problems with old procedure and the sixty one 48. Preface to Zhu Shijie by Zu Yi 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, in 2006. 49 [Xu Yibao, 徐义保, 1990]. p. 67. 50 I use the translation of titles by [Li Yan, Du Shiran, 1987]. For the present purpose, I will not discuss these translations. 51 Translation of the title by [Volkov Alexei, 2007], p. 445. Practical Rules of Arithmetic for Surveying is the translation of the title by [Lam Lay Yong, 1977] 52 [Horiuchi Annick, 2000]. p. 238. “Réflexion sur les fondements des méthodes anciennes ». 53 [Te Gusi, 特古斯, 1990] . p. 56. 54 [Xu Yibao, 徐义保 1990]. p.72. 55 [Lam lay yong, 1984]. p.241.. p.64. [Mei Rongzhao, 梅荣照 1966]. p. 140. [Kong Guoping, 孔國平,1999]. p. 174. [Ho Peng Yoke, 1978], p. 319.. 18.

(20) problems with section of area was never elucidated. The present study will renew the question by showing we can identify more than only twenty three solutions with “old procedure” are borrowed from the Yiguji.. 19.

(21) 2. The Qing dynasty editors’ work56.. The relation between the different procedures was also already the object of commentaries of the Yigu yanduan at the end of the 18th Century. And those commentaries put the light on the procedure of the Celestial Source, and they might be the source of the interpretation of the Yigu yanduan as a collection of problems for beginners in the art of the Celestial Source. I will try to identify the source materials they were using and show how the correction made by the Qing dynasty editors to the sources for publication lead to this interpretation.. 2.1 Commentaries to the Yigu yanduan The oldest available edition of the Yigu yanduan is dated from the 18th century, and this edition contains commentaries. A first commentary, introduced by the character 案, an, was added in 1789 by one of the editors of the Imperial Encyclopaedia, the Complete Library of the Four Branches of Literature, Siku quanshu, 四庫全書57. This commentary is either inserted in two small columns inside the text written by Li Ye, or in one column at the end of one of the procedure. The author of this commentary remains anonymous. One could attribute the commentary to Dai Zhen, 戴震, who was in charge of the edition of the section on mathematics and other scientific subject with Li Huang, 李潢58, and who, according to Li Yan59, commented the Ceyuan haijing. A part of the commentary aims to explain and interpret the procedure of the Celestial Source in the light of the procedure of “borrowing the root”, jie gen fang,借根方 –the later being an algebraic method for establishing and solving equations of higher degree introduced by the Jesuits in the late 17 th century60. Added to this commentary, each of the mathematical expressions from the problem 1 to the problem 4 are translated into other mathematical terms issued from this method.. 56. Qing dynasty: 1644-1911 The Siku Quanshu, variously translated as the Imperial Collection of Four; Emperor's Four Treasuries ; Complete Library in Four Branches of Literature;or Complete Library of the Four Treasuries, is the largest collection of books in Chinese history. During the height of the Qing dynasty in the 18th century CE, the Emperor Qianlong commissioned the Siku Quanshu, to demonstrate that the Qing could surpass the Ming dynasty of 1403, which was the world's largest encyclopedia at the time. 58 [Lam Lay Yong, 1984]. p.240. 59 [Li Yan, 李儼, 1955] 60 [Tian Miao, 1999]; [Horng Wannsheng, 1993] 57. 20.

(22) For example: 以自增乘得. 案:此即一千六百步,八十池徑,一平方併 ○. 1600 Augmenting this by self-multiplying yields. tai. 80 1. Commentary: That is the sum (併) of one thousand six hundred bu and eighty diameters of the pond and one square (平方)61.. Other parts of the commentary add some supplements in order to clarify procedures, like in pb.24 and 43 where procedures for extracting the root are given. The last part of the commentary aims to propose alternative solutions to the procedure of the Section of Pieces [of Areas] of three problems (pb.38, 54, 56) when the editor finds that the procedure is not clear enough. And this part also proposes some corrections to diagrams (pb.53, 61). There are only two editorial notes concerning corrections of characters, one in the preface, another one in the problem 63. Nine years later, in 1798, Li Rui, 李銳, (1773-1817) added his own commentary while editing the mathematical part of the Collected Works of the Private Library of the never sufficient knowledge, Zhibuzu zhai congshu，知不足齋叢書, under the direction of Ruan Yuan, 阮元. He totally disagrees with the mathematical interpretation of the commentary made in the edition of the Siku quanshu and shows how the procedure of the Celestial Source and the procedure of Borrowing the Root are different62 (pb. 1, 11, 40). He explains the way of writing and reading mathematical expressions (pb. 1, 11). He also adds corrections to characters in editorial notes (see table 1), and proposes corrections to some diagrams in the Section of Pieces [of Areas] (pb. 38, 61, 62, 64). But his discourse is mainly dedicated to the dissociation of the two procedures, Borrowing the Root and Celestial Source. This is precisely the part of the Yigu yanduan which will attract the attention of later mathematicians. The Korean mathematician, Nam Pyong-Gil (1820-1869) wrote the Muihae, 無異解 (Solutions of No Difference [between “Tian Yuan Shu” and “Jie Gen Fang”]) in 185563. He copied four of the problems (pb.1, 7, 11b, 40) of the Yigu yanduan but removed all the diagrams and the Sections of Pieces [of Areas], kept the commentary by the editor of the 61. The expression 平方, ping fang, for square never appear in Li Ye’s vocabulary. See [Horng Wann-Sheng, 1993a] and [Horng Wann-Sheng, 1993b]. I will not discuss the argument concerning the two procedures of the art of Celestial Source and Borrowing the root in this work and keep the reading of this part of the commentaries for further studies. 63 [Ying Jiaming. 2010]. p.9. Thank to Ying Jiaming for providing me with a copy of Muihae. 62. 21.

(23) Siku quanshu and Li Rui and added his own commentary to the problem 7. By focusing on the art of Celestial Source only, he reinterpreted the treatise with his contemporary knowledge estimating that Li Rui was wrong. Those commentaries all focused on the procedure of the Celestial Source due to the intensive debate concerning the achievements of Western and Chinese mathematics. At the beginning the 18th century, the attribution of a Chinese origin to the Western mathematics is precisely based on the identification of the procedure of the Celestial Source with the method “borrowing the root”. The procedure of the Section of Pieces [of areas] was already covered by shadows. I will also show what modification the Qing dynasty editors made to the sources. But before reaching this point, one has to question another part of the discourse which also has the shape of a commentary, and which will help us later to understand the intention of Li Ye. A third hand can be distinguished among these commentaries. A part of the text is presented in two small columns inserted between main sentences without any introductive character. Although it has the presentation of commentary, it was never been considered as such. None of the readers of Yigu yanduan wondered why the main discourse is punctuated by small texts in two columns. As this commentary has never been considered by any historians, I deduce that they implied that Li Ye himself is the author of this part. The paternity of this commentary has never been put into question. This commentary details some algorithms, justifies some results or clarifies some quantities. Li Ye does not use so often this process in his other mathematical work, the Ce yuan hai jing, and, in the Yigu yanduan, this type of commentary appears mainly in the procedure of the Celestial Source, and three times in the procedure of section of area (pb.14, 15, 18). We thus wonder why Li Ye produced a commentary to his own text or if one has to consider this discourse in small column as a commentary. I will try to answer to this question in the chapter devoted to the order of problems. Another part of the discourse, explicitly from Li Ye, seems at first sight also to play the same role as a commentary. A short paragraph written in one column is introduced by the character yi, 義, “meaning”, twice in the procedure of the Celestial Source for justifying an algorithm (pb. 43, 55). And the temptation is to read this part as a commentary. This same character introduces the discourse placed at the end of the procedures of the section of Pieces [of Areas] of each problem. The purpose of this part of the discourse is to explain the link between the sentence describing the coefficients of the equation which begins the procedure of the section of Pieces [of Areas] and the diagrams which immediately follow. It shows how to identify the terms of the equation from that diagram. And this part of the Section of Pieces [of Areas] is not a commentary. The interpretation of the role of the “meaning” in interaction with the diagram is in the heart of the question. Karine Chemla, in her lexicon to the translation of the famous classic of the Han dynasty64, shows that the 64. [Chemla Karine, Guo Shuchun, 2004], p. 1023.. 22.

(24) “meaning” is correlated with the commentators who want to address the correctness of algorithm. The “meaning” consists in explications of more general or fundamental operations and the commentary shows that the algorithm commented is an instanciation. According to her, to clarify the “meaning” and, to identify a small number of operations which are sufficient to express it, seems to be one of the more important features of the demonstration in ancient China. This direction of interpretation provides new keys to understand the real purpose of the Yigu yanduan. One can thus wonder why such sentences are introduced by Yi in the procedure of Celestial Source and what is the purpose of Yi in the Section of Pieces [of Areas]. I will tackle this question in the chapter dedicated to the order of problems.. 23.

(25) 2.2 Status of the available editions.. There are several available complete editions of the Yigu yanduan: the one collated in the Siku quanshu, 1789, and in the Zhibu zu zhai congshu, 1798. All later editions of Yigu yanduan are copied from the critical edition made by Li Rui in 1798. The first edition of Yigu yanduan and/or the manuscript written from Li Ye’s hands are lost; we have to rely on the Qing dynasty editions to figurate the content of the “original” treatise. As a consequence of this study I will also question what the term “original” means. The version of the text which is inserted in the Siku Quanshu is based on the version of the Yongle Dadian, 永樂大典, compiled between 1403 and 1408, which is now lost65. As Li Rui responds to the Siku quanshu commentary, it is widely accepted66 that the Zhibuzu zhai congshu edition is based only on the edition of the Siku quanshu. We want to put this point into question and try to identify the source materials. From the actually available editions of Yigu yanduan, no one can imagine how the book was before its edition in the Yongle Dadian. There is a gap of five centuries between the first edition of the Yigu yanduan in 1282 -date testified by its preface- and the Qing dynasty editions. But comparing precisely the available editions of the Yigu yanduan, we notice that although the editions look at first sight the same, they in fact hide many differences; and these differences are significant. These differences can reveal how the text was before its insertion in the collection of the siku quanshu and how the editor modified or followed the documents they used for their edition. One will see in a later chapter that it is difficult to separate the history of the transmission of a text from the history of its interpretation. Li Rui wrote several editorial notes which are precious clues to learn about the materials he is using. But in order to compare these notes with the the Siku quanshu, I have first to wonder which of the editions of the Siku quanshu was used by Li Rui. There were originally seven copies of the Siku quanshu. These copies are named according to the places where they were originally stored in: the wen yan ge,文淵閣, which was stored in the palace of the same name in the forbidden city of Beijing, 北京故宮; the wen su ge, 文溯閣, which was stored in Shenyang palace,瀋陽故宮 in the northern province of Liaoning, 遼寧; the wen yuan ge,文源閣 in the ancient summer palace of Beijing,北京圓明園; the wen jin ge,文津閣, in the summer palace of Cheng-de, 承德避暑山莊; the wen zong ge, 文宗閣, in a temple of Zhenjiang, 鎮江金山寺, in the southern province of Jiang-su, 江蘇; the wen hui 65. [Kong Guoping, 孔國平 1987]. pp.166-169 Only [Lam Lay Yong] and [Kong GuoPing] wrote a paragraph on the history of the editions of the Yigu yanduan. There are no chapters dedicated to this subject in other publications. 66. 24.

(26) ge,文匯閣 in 揚州大觀堂 in Jiang-su; the wen lan ge,文瀾閣, which was in Hangzhou, 杭州. Only four copies among the seven original ones are still (or partly) available: the wen yan ge, 文淵閣, the wen jin ge, 文津閣, the wen su ge, 文溯閣, and the wen lan ge, 文瀾閣. Only the wen yan ge and the wen jing ge were reprinted in 20th century. Li Rui used to lived in the province of Jiangsu, so he might have access either to the Wen zong ge, the Wen hui ge, or even the wen lan ge edition. But the two first ones disappeared in the fire of 1853, the last is a patial reconstruction made in 1880 after the original edition was burnt in 1861. Therefore, I could not access these original sources. From the history of the other work by Li Ye, the Ceyuan haijing, we know that the edition by Li Rui of the Ceyuan haijing is based on a copy made by Ruan Yuan of the wen lan ge67. The later could also be the source of the edition of 1789 of the Yigu yanduan. As one cannot rely on the actual edition of the wen lan ge, I proceeded to a comparison of the wen yan ge and the wen jin ge editions of the Yigu yanduan to have an idea of the gap between the different editions of the Siku quanshu. I transformed the reprint into a searchable database of 45000 caracters. Then I proceed to a word by word comparison with the edition made by Li Rui. The comparisons are reported in Table I and II. The editions that are compared here are: the Zhibuzu zhai congshu commented by Li Rui as it was reproduced by Guo Shuchun (Eds.) in 中國科學技術典籍通彙, Zhongguo kexue jishu dianji tonghui, 河南教育出版社Henan jiaoyu chubanshe, 1993, vol. 1. And two editions of the Siku quanshu; the wen yan ge, which original edition is available at Taiwan National Palace Museum, gugong bowuyuan, 故宮博物館, and a reprint of edition of wen jin ge. I will refer to the first one as LR and to the second ones as respectively WYG and WJG.. 2.2.1 Editorial notes and correction to the discursive part. Li Rui added 16 editorial notes in his edition pointing out what corrections were made to the material he was using for his edition. I noticed that for some of the sentences, he signals modifications he made to the text, while the same text remains identical in both editions of the Siku quanshu. If Li Rui says that he corrected some characters in one sentence, I would expect to find the sentence without any correction in the Siku quanshu. For example, when Li Rui writes that he suppressed the character “to have”, you, 有; this character is in both editions of the siku quanshu and was suppressed from the edition by Li Rui. There are differences between the editions and these differences are signalled by an editorial note. That is what happened for most of the cases, but concerning three cases, I could not see any differences between all the three editions. For example, in pb 3 and 39, Li Rui writes that two characters are missing in the diagram, and the characters are not 67. [Mei Rongzhao, 梅荣照, 1966]. p. 111.. 25.

(27) missing in any of the editions. In pb 38, Li Rui said that the charcter wei, 為, is wrong and that he suppressed it. There are no character wei in any of the Siku quanshu I consulted [see Table 1]. The editorial notes written by Li Rui do not always match with the text copied in the different editions of the Siku quanshu. Another observation is interesting concerning editorial notes. Another editorial note, this time written by the editor of Siku quanshu, is also significant. In the pb 53, the editor of Siku quanshu, mentions that he replaced the square in the diagram by a rectangle to make the reading easier, and the diagram in Siku quanshu edition contains a rectangle (part containing 甲 in Figure 1,). In Li Rui edition, although we can read the commentary of the Siku quanshu, the diagram contains a square (idem in figure 2).. Figure 1. Figure 2. After consulting the different editorial notes, we will now turn to examine other differences which are not signalled in editorial notes. Thoses are presented in Table 1 and 2 in the supplement of this study. I could also notice a total of 95 other differences between the edition by Li Rui and the two Siku quanshu. [See table 2]. In 73 cases, WYG and WJG are the same. In only 3 cases, the three editions show three different versions (pb. 4; 6; 58). For 13 cases, WYG and LR are the same, and in the 6 other cases WJG and LR are the same.. 26.

(28) The copy in the WYG is more reliable than the WJG. The WJG contains in fact several mistakes which seems be due to dictation: like 圓, “circle”, instead of 元, “original” (pb.43); or 十, “ten” instead of 實, “dividend”, (pb. 48). Some are also graphical mistakes, like: 十, “ten”, instead of 千, “thousand” (pb.10), or 去, “to go”, instead of 云, “to say” (pb.23), differing of only one stroke. For my purpose, I take into account only the other 73 cases mentioned above, that is when the WJG and WYG are identical and present differences with the edition by Li Rui. Among the 73 cases, 35 cases are due to synonymy, for example: 原 instead 元, which both are pronounced yuan and means “original”68; 以 instead of 依, both pronounced yi and translated as “according to”. Or they are due to syntax divergences, like 一十八 in Siku quanshu instead of 十八 in LR, which in both cases means “eighteen”. We cannot consider that, in these 35 cases, one of the editions is mistaken. But among the 73 cases, the other 38 cases are indeed due to syntaxical or vocabulary mistakes. In 8 cases the Siku quanshu is correct and LR is wrong. Therefore, there are 30 cases for which LR is correct and the Siku quanshu is wrong, for example, in pb.25, one reads 167 in Siku quanshu instead of 176 in LR (pb.25), or 古徑, “diameter according the ancient lü” in LR instead of 方徑, “side and diameter” in the Siku quanshu (pb.43), the later being incorrect in the two copies of the Siku quanshu. We also notice that a sentence in pb. 40 is in LR and is missing the two Siku quanshu, while the same kind of disparition in pb.34 is signalled by an editorial note. Among these mistakes, 16 of them are corrected by an editorial note by Li Rui as we saw. So the question is why Li Rui did not write editorial notes about the 14 other mistakes? From these evidences, it seems that Li Rui is doing two things. Li Rui first responds to the commentary of the editor of the siku quanshu concerning the difference between the Celestial Source and “Borrowing the root”, and secondly, he is doing a critical work of the materials (on discourse and diagram) he is using. It is impossible to answer definitvely to the question of the identification of the materials. He could be using a copy of the Wen lan ge which discourse contains differences with its original source in the Siku quanshu. He could have access to the “original” edition of the Wen lan ge, or may be to another edition of the Siku quanshu. It could be also that he consulted the Yongle dadian or a copy of it, whether we cannot say if he had access to it. Or may be Li Rui copied and corrected the oldest edition of the Yigu yanduan he could find and inserted his commentaries and the ones of the siku quanshu inside, or at least he collated at least two editions available to him. But it seems that Li Rui might have in hand two different editions of Yigu yanduan when he prepares the Zhibu zuzhai congshu edition. At this step of the study, speaking of the Siku quanshu as a uniform and unique book vanishes. Now we will see that it becomes difficult to apply the qualificative of “original” to any of the source.. 68. Here, we did not compare the commentaries in the Siku quanshu with their copy in Li Rui’s edition. The character yuan in question here does not concern the case presented in a later paragraph. This later paragraph is dealing with the use of this character yuan in commentaries.. 27.

(29) A second detail is worthy to be mentioned. In his editorial notes, when Li Rui signals corrections, he allways refers to the previous edition of the text written by Li Ye by systematically naming it yuan ben, 元本. The commentary of Siku quanshu uses another character to refer to text which is copied. The character, yuan, 原, is used by the editor of Siku quanshu in pb. 53, 54, 61, 63. Both characters, 元 and 原, can be translated by “original”. After checking others commentaries of different others mathematical texts that are compiled in Guo Shuchun collectanea, it seems that the character yuan 原, is normally used to refer to text which is copied indinstinctivly of it provenance69. Here “original” means the material used as sources. In other texts whose commentaries are attributed to Li Rui70, like the 疇人傳71, Chourenzhuan, the editorial notes use the following characters: yuan ben, 元本, for yuan dynasty editions, ming ben, 明本, for ming dynasty editions, and yuan ben, 原本, for the source editions which copied72. Li Rui in his commentary of Ceyuan haijing73 also systematically uses the character yuan, 元, for mentioning the same kind of corrections, and one knows74 that Li Rui’s edition of Ceyuan haijing is not only based on the siku quanshu but also on a manuscript referred as Yuanchaoben, 元抄本75 which belonged to a certain Ding Jie, 丁杰. This manuscript contains the print of the seal of Ruan Yuan who provided documents to Li Rui76. Mei Rongzhao estimates this manuscript to be written between 1310 (Yuan dynasty) and 1381 (Ming dynasty) and this manuscript is actually still available in Beijing National Library. Would it be impossible that the edition of Yigu yanduan followed a similar path and that Li Rui is consulting an edition dated from the Yuan dynasty or at least that he is attributing to the Yuan dynasty? I am aware that such arguments are not enough to show that Li Rui is not reading the Yongle dadian or a copy of the Wen lan ge, but these arguments are sufficient to question the origin of the sources of the available editions. 49052 characters compose the Yigu yanduan, the discrepancies between editions concern only 95 characters. At least, I can deduce from the meticulousness of Li Rui’s editorial notes that his edition might be quite faithful to an older edition of Yigu yanduan, whatever this previous edition is. Nevertheless we can admire the precision and the quality of the work of the editors of the Qing dynasty.. 69. See re-print of commentaries of 海島算經, 九章算術, 五曹算經, 負侯陽算經 in Guo Shuchun (Eds.), 1993. [Li Yan, Du Shiran, 1987]. p.232 71 阮元, 疇人傳.Vol.82. 72 Idem. p. 197, 198, 247, 274 73 See reprint in Guo Shuchun (Eds), 1993, T 1. p.771. 70. 74. [Mei Rongzhao, 梅荣照, 1966], p.110-1 [Kong Guoping, 孔國平, 1987], p.159 76 Idem. [Mei Rongzhao, 梅荣照, 1966]. p.110-1. [Kong Guoping, 孔國平, 1987]. p.159. 75. 28.

(30) 2.2.2 Treatment of diagrams. This quality of work can be also admired in respect to the edition of diagrams, although this work is not visible at first sight. As the present study relies on a collection of diagrams, I wanted to check how reliable the diagrams provided by the current editions are. I compared carefully the two editions of the Zhibuzu zhai cong shu and the wenyange Sikuquanshu, and it appears that there are very few differences concerning the shape diagrams. Making exception of the diagram of pb.3, when there are differences, those are explicitly mentioned by commentators, like in the example above. I observed that in fact the two editors kept exactly the same proportion in reproducing diagrams. The page format of Wenyange edition of the Siku quanshu is bigger than the Zhibuzu zhai cong shu. The first one is a manuscript, while the second one is printed with woodblocks. A woodblock contains the script of a whole page. After measuring the dimensions of each diagram reproduced in the statement of problems in the original edition of the Wenyange siku quanshu preserved in the National Palace Museum in Taiwan, I noticed that those diagrams were constructed in order to be proportional to the data presented in the statement of problems. Let’s examine a sample [table I] of figures issued from the same category of problems: a square pond inside a circular field. On this sample, one can see that the dimensions of the square are changing from one problem to the other, while the circle keeps the same dimensions.. 29.

(31) Table I. samples of measurements and data from diagrams in the statement, wenyange edition of Siku quanshu. problem 15. 16. 17. 20. Diagram presented in Wenyange siku Data in the Measurement in quanshu problem millimeters Diameter: 120 bu Diameter: 52 mm. Side: 52 bu. Side: 22 mm. Diameter: 72 bu. Diameter: 52 mm. Side: 18 bu. Side: 13 mm. Diameter: 54 bu. Diameter: 52 mm. Side:24 bu. Side: 23 mm. Diameter: 60 bu. Diameter:52. Side: 15 bu. Side:13. A cross product for each case shows the measurements are proportional to the data given in the statement. The same operation was done on every diagram reproduced in the statement in the edition of Wenyange siku quanshu: all diagrams are proportional to data. I compared these measurements with other ones taken in others editions of the Siku 30.

(32) quanshu and of the Zhibuzu zhai congshu. The same observation can be made, although there is a loss of accuracy in the Zhibuzu zhai congshu. May be one can attribute the latter to technique of carving used in blockprinting. One notices that while the editions of the siku quanshu and Li Rui edition have different sizes of pages, the shapes of the diagrams are the same and they keep the same proportions: Li Rui’s edition looks like a reduction of the siku quanshu and diverse corrections that were added to diagrams show that the editors paid lot of attention to the dimensions. If there is work of the editors on the proportion of the diagrams in the statement of problem, the process in less clear concerning diagrams in the Section of Pieces [of Areas] as one will see in a later example (pb.38). The difference between the Wenyange Siku quanshu and Li Rui’s edition concerning the diagram in Section of Pieces [of Areas] of the problem 3 is on this point significant. One notices that the side of the expanded square and the diameter of the expanded circle are exact representations of the dimensions of the diagram in the statement multiplied by 1.4, the latter being proportional to data, in the Siku quanshu [Figure.3]. In both [figure.3] and [figure.4], the expanded circle is represented by dotted lines. But in Li Rui edition, the central circle is smaller. Actually, the dimension of this circle is the dimension of the circle given in the statement reduced exactly by 1.4 [Figure 4]. The consequence is that instead of having squares marked by dotted lines, one has rectangles (part containing the character 從). This mistake is not the result of a problem of carving or technical drawing. It is due to computation. This mistake in the edition by Li Rui shows that there is practice of measurement for publication of mathematical figures.. Figure 3. Figure 4. 31.