The Empty and the Full: Li Ye and the Way of Mathematics

The following sections are included:

• Acknowledgements
• Conventions
• Preface
• Contents

One of the best ways to understand the mathematical reasoning of Chinese treatises is to work through translations, not because translations transpose a discourse from one language to another, but because they fail to do so. However, I am deeply indebted to many wonderful works translating Chinese mathematical texts. Without them, this present study would have not been possible. But the peculiarity of the Chinese treatise I present here is that it reaches the limits of translation and of language. More concretely, as a translator, I have tried to be faithful to what Humboldt calls das Fremde in his preface to the translation of Aeschylus’ Agamemnon. There is transformation in translation. Meaning is always movement. The strangeness of language can be seen strikingly in the work of translation here. Humbold uses the term fremdheit (translated as “foreignness”, or “strangeness”) to describe the sense of reader has when the translator seems to have chosen what sounds strange or odd, as though a mistake has been made in translation. Das Fremde (translated as the “strange”, or “unfamiliar”) describes the reader’s sense of reading something that is recognisable, that has been translated appropriately, but that gives the feeling of reading that word for the first time [Eco (2004)]. Humboldt showed that a proper understanding of language in its original form is not transparent or unproblematic. Das Fremde always disturbs the reader. The present study aims to translate some written excerpts of a mathematical treatise, but also to translate the invisible part of the same treatise…

Despite their scarcity, diagrams play an important role in Chinese mathematics. In China, diagrams are indissociable from the history of elaboration of algebraic objects. To understand how diagrams and equations are related, recent eras offer a better point of departure than the Song–Yuan period. This chapter starts with a description of the algebraic procedure to set up quadratic equation, which is at least in part diagrammatic in origin. After the description of the procedure presented in Problem 1 of the Development of Pieces [of Areas], and its general history has been established, the difference between the concept of equation as found in the procedure and modern concepts can be discussed. The origin of these differences lies in the practice of executing divisions and extracting square roots with counting rods and/or diagrams, as has been show by experts in history of Chinese mathematics. Then, a description of the geometrical procedure is introduced. This description clarifies that (1) diagrams play an argumentative role and (2) they are, ontologically, literal expressions of the mathematical object (such as the quadratic equation). An example shows how to visualise diagrams as the movement of areas and not as static lines. The chapter contains two problems, Problem 1 exemplifies the basis and Problem 21 provides a clear picture of transformations. All coloured diagrams are modern and thus artificial representation. These diagrams attempt to show, as a cartoonist might, the different cinematic moments of a movement. Their weakness lies in their modernity. Modern objects always hide contemporary concerns and purposes, probably different from Li Ye or Yang Hui’s intentions. The reconstructed diagrams remain just a contemporary attempt to reconstruct ancient objects. While intended to represent movement by picturing it at separate instances, they have paradoxically stopped, like still-frames of a motion picture. However, the stop-motion animations are the best didactic tool to guide a reader through a lost ‘way of seeing’ mathematical object.

Before tackling the question of diagrams, a detour through the famous Celestial Source procedure is necessary. This procedure stands as a lighthouse in the history of mathematics written in Chinese. More familiar than the geometrical procedure, the Celestial Source procedure is a good starting point for understanding didactic elements of the Chinese version of the quadratic equation.

The following sections are included:

• Diagram and Text
  • Meaning of Yanduan
  • Different Types of Diagram
• Diagram as Equation
  • Translation
  • Mathematical Transcription
  • Interpretation

Diagrams are not only objects to be published in mathematical treatises. The published materials are relics of set of practices. The diagrams are indeed mathematical object, but in the present case, they are also evidence for specific culture of discourse which relates to their philosophical background. In the Development of Pieces [of Areas], diagrams were objects to be drawn, visualised as transitory. They are silent textual objects, part of speechless discourse…

From the examples of problems 1 and 21, it is possible to order the steps in the practice of using diagrams and texts. First, the data of the problem were read; then, the practitioner manipulated the rods and drew a diagram. The last step was to transcribe the terms of the equation into sentences such as those of [1.8] or [21.8], where the results of computation were given in fine. At this step, the counting board was set up to directly extract the square root. Diagrams were drawn by readers before the text was written. The diagrams we see now in the edition of Development of Pieces [of Areas] were added by Li Ye at the very end of the process, as an editorial practice…

Movement is a key to understanding the diagrams of the Development of Pieces [of Areas]. Several different possible transformations can be observed in the diagrams of the Development of Pieces [of Areas]. Not only areas are moved virtually and cut into pieces but they are also filled and emptied. The relation between plain and empty areas lies at the core negative and positive coefficients. Indeed, the question is how to represent a negative coefficient by mean of areas. Through a comparison of Li Ye and Yang Hui’s works, it is possible to reconstruct some steps in the development of the elaboration of quadratic equation. It is also possible to distinguish several layers in the composition of the text transmitted by Li Ye. This section draws upon Problems 18 and 5 of the Development of Pieces [of Areas] and Problem 49 of Yang Hui’s Methods of Computation. Problems 18 and 5 offer cases from which the Old Procedure can be reconstructed and enable the interpretation of the Section of Pieces [of Areas] as containing several procedures depending on the different reading and cutting of the diagrams. In light of Problem 49 of Yang Hui’s Method of Computation, a clear evolution of concept of negative coefficient emerges. To understand the paths of these transmission and interpretation, it is first necessary to visualize the different operations of transformations as performed on diagram (illustrated by Problem 2).

The reader of the Development of Pieces [of Areas] is supposed to visualise several transformations of the diagrams in order to set up equations. The number of transformations is not infinite, and these transformations are carefully chosen. It is possible to inventory each type of transformation and describe their mathematical meaning. In fact, the first chapter of the Development of Pieces [of Areas] uses only six types of transformations for the Section of Pieces [of Areas]. These transformations work like fundamental operations of arithmetic. They are the six models of bricks with which it is possible to construct every geometrical algorithm. In the other two chapters of the treatise, the operations increase in sophistication and operate on objects that are more complex but essentially remain the same. These operations appear after the reader has gained experience in reproducing problems at the point when the pattern of vocabulary and drawing become repetitive. This chapter establishes elements necessary to Chapter 7 wherein the combination of operations becomes meaningful…

How may the equations in the Section of Pieces [of Areas] procedure be understood? How may an equation be rendered into modern notation when the statement of equality, the designation of the unknown variable, and the sign of numbers are unwritten? The Development of Pieces [of Areas] does not explain how negative and positive quantities are differentiated. There are no traces of a symbolic notation. The characters fu 負, ‘negative’ and zheng 正, ‘positive’, usually name negative or positive quantities but they do not appear in the text concerning the Section of Pieces [of Areas]. These characters fu and zheng appear only in the commentary on Problem 14 by Li Ye, and not in the procedure itself: ‘[Whenever there is] a negative adjunct (fu cong) with a positive corner (zheng yu) or a positive adjunct (zheng cong) with a negative corner (fu yu), the dividend (shi) remains always the same. That is why the edge and the adjunct are used to differentiate’. Conversely, the term ‘empty’ (xu 虛) and ‘augmented’ (yi 益) are associated with the expressions transcribed as negative quantities in modern mathematical terms. In the Celestial Source procedure, the character xu appears twice in Problems 1 and 2. In these cases, the character names an area expressed according to a polynomial composed of two terms of different powers. This expression of the area will be cancelled by its expression by a constant term to make the equation. How can a character used to name a negative coefficient in one procedure later be used to name entire polynomials in another procedure of the same treatise? Are these characters synonyms for fu, ‘negative’? Their semantic range must be considered as well as their transcription into modern notation. This transcription depends on knowledge of the sign of a given term. On this point, it is interesting to look at how historians recorded negative terms…

The following sections are included:

• Diagram of the Old Procedure
• Computing Coefficients with the Old Procedure
  • Multiplication by Three
  • Counting Support Row and Signs of Coefficients

The study of the transformation of figures calls into question the concept of negative and positive coefficients. Through the imaginary movement of areas, the movement of emptying or filling areas, and the movement of cutting or assembling rectangles into ribbon, a progression of expressing equations emerges. By Liu Yi’s time, distinct expressions referred to negative and positive quantities. However, because equations are based on the construction of diagrams and negative areas are not intuitively obvious, the geometric elaboration of negative coefficients presented interesting issues. The positive and negative quantities are treated independently on the counting surface; the equation represents a perceived correspondence between tabular exposition and diagrammatic representation. However, the distinctive elements are not clearly stated: negative coefficients, equalities, and unknowns are all absent from the discourse. The evolution of computational practices with diagrams, especially the geometric representation of a mathematical object called ‘adjunct’ which was filled or emptied, ensured that the negative quantities became associated with negative coefficients. At the time of Li Ye and Yang Hui, negative coefficients were clearly constructed and quadratic equations were free from geometrical representations. In their treatises, Li Ye and Yang Hui present several vignettes in the elaboration of quadratic equations with negative coefficients…

The order in which the problems of Development of Pieces [of Areas] are presented also contains clues to the development of mathematical concepts. The sixty-four problems are not randomly ordered, nor do they reflect increasing complexity. The construction of the diagram helps to reveal the logic behind the order in which the problems are presented…

The key to understanding the content of Development of Pieces [of Areas] is not its narrative thread. The structure of the treatise reveals its most important concern. The problems of the Development of Pieces [of Areas] are not randomly ordered. There is a system of classification based on a field survey of problem types which is a pretext for abstract thought. In some of the problems, the numerical data are transformed into other types of data. The purpose is to simplify the problem and thereby link the problem with previously solved problem. This method consists of reducing the data of a sophisticated problem to a previously worked simple example. This method reveals a primary classification of problems according to the data given in their statements. This chapter describes the process of reduction…

The process of reducing data to derive a solution via a previously solved problem expresses an analogical order. Besides the order of data based on areas and distances given in the statement, problems are also ordered by analogy. Several elements of the geometric procedure presented inside the diagram-equation direct this order. The order of problem is tacitly implied by the diagram, as seen here in Problems 36 and 45. The way of grouping problems implies a specific use of memorisation. Reductions, as seen in previous chapter, ease memorisation and reduce the amount of mental labour. The practitioner is saved the labour of repeating similar treatment in situations that are established as related through the visual quotations in diagrams. However, the role of memory is not only bound to retain procedures and geometrical patterns. Problem 36 below highlights another role for memorisation and another system of organization of problems.

Problems in the Development of Pieces [of Areas] are ordered according to the possible combinations of the operations presented in Chapter 4. These combinations are applied to various figures and data. It seems that the Section of Pieces [of Areas] procedure explores the entanglements of several sets of combinations:

• A finite set of combinations of types of figures (circles, squares…). (Chapter 7).
• A finite set of combinations of types of data (Area and distance given in statement). (Chapter 7).
• A finite set of combinations of transformations on geometrical areas. (Chapter 4).
• A finite set of mathematical objects (fractions, half-areas, powers, full or empty areas…)

An investigation of the structure leads to a deeper understanding of the problems and procedures and to a different mathematical meaning, which remains to be unveiled. Chemla’s experience with the Nine Chapters led her to believe that that classic is also subject to the same phenomenon of reference between problems [Chemla (1997a)]. Some studies have already underlined the importance of analogical reasoning in China, such as [Volkov (1992)], which demonstrated its role in the commentary by Liu Hui to the Nine Chapters. Liu Hui prescribes ‘mending the void with the excess’ (yi ying bu xu 以盈補虛) as a procedure to solve problems in the first (Problems 25–26) of the Nine Chapters. This process is used for the computation of the area of an isosceles triangle, which is part of a figure called gui tian 圭田 . Half a triangle, ABC, is applied as a ‘missing’ part ADE to complete the rectangle ADEC…

Li Ye sought to make the content of an ancient book, the Collection Augmenting the Ancient [knowledge] (eleventh century) more accessible to readers by illustrating it with diagrams. A systematic study of the diagrams shows that one of the most important features of the Development of Pieces [of Areas] is the practice of transformation of figures. The heart of the book rests on non-discursive practices: drawing and visualising the transformation of figures and manipulating counting rods on a support. The topic is reflected in the title, Yigu yanduan, now translated as ‘Development of Pieces [of Areas] [according to] [the collection] Augmenting the Ancient [Knowledge]’. That is, the procedure of development by the Section of Pieces of Areas presented in the treatise titled the Collection Augmenting the Ancient [knowledge] (augmented with diagrams)…

The following sections are included:

• Appendix A Supplements
  • Tables
  • Classic of Computation of Master Sun, Division
  • Classic of Computation of Master Sun, the Extraction of a Square Root
  • Web of Problems
• Appendix B Chinese Glossary
• Appendix C Chinese Names

The following sections are included:

• Bibliography
• Index