Working on and with division in early China, 3 rd century BCE-7 th century CE

Abstract : When compared with the mathematical manuscripts from the earliest decades of the Chinese empire recently excavated, the oldest mathematical book in Chinese handed down through the written tradition, The Nine Chapters (1st century C.E.), presents interesting differences. The texts of its procedures regularly make use of theoretical terms that are not found in the manuscripts, or at least not with the same theoretical meaning. Some of these terms are read by the commentators as related to the reasons why the procedures are correct: “make communicate 通 tong," “equalize tong,” lü 率—a concept of number as defined relatively to other numbers. Others point out that a certain structure has been identified in a set of operations. This is the case primarily for one of the terms by means of which division is prescribed 除 chu, which is the basis for a whole terminological complex absent from the earliest manuscripts. I do not mean to say that theoretical terms are completely absent from the manuscripts. Writings on Mathematical Procedures, excavated from a tomb sealed ca 186 B.C.E., contains some such terms. Moreover, where they occur is interesting. However, the extent and the nature of the phenomenon cannot be compared to what we have in The Nine Chapters. Interestingly enough, a great deal of these terms occur in relation to division or related procedures, predominantly the rule of three. These clues strongly suggest that division has been an important topic of reflection in some Chinese milieus between the 3rd century B.C.E. and the first century C.E. The hypothesis is supported by many other facts. Division, for instance, with closely related procedures like the rule of three, is a major topic in the manuscripts, in which many procedures are devoted to its various types of execution. One has to bear in mind that the operands to which division is applied were usually complex measured quantities. Division was also the only operation for which operands are designated by technical terms. Further, elementary operations, such as changes of units, essential to the practice of division can be correlated to theoretical discussions developed in the commentaries on The Nine Chapters and to the emergence of mathematical concepts. The article suggests that the execution of division was carried out through mainly four basic phases. On one of them, the transformation of the operands into the smallest possible integers with respect to a single measuring unit, there seems to me a great continuity of practice between the manuscripts and The Nine Chapters, the main difference being the introduction of technical terms to designate procedures. On the next two elementary operations, through which the result is produced bit by bit, we can perceive a break between the time when the known manuscripts were written and the time of the compilation of The Nine Chapters. I argue that this break might have been correlated to a change in the number system with which integers were written down with counting rods. The Nine Chapters bears witness to the emergence of a system of practice with the instrument of computation, centered on division and the then opposed operation of multiplication, a system that remained in use for centuries. Finally, the fourth phase related to the statement of the result, using fractions. It evidences transformations and provides evidence on how the concept of fraction to which ancient Chinese sources testify might have emerged in the context of the ancient way of computing divisions.
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Karine Chemla. Working on and with division in early China, 3 rd century BCE-7 th century CE. Karine Chemla; Agathe Keller; Christine Proust. Cultures of computation and quantification in the ancient world, In press. ⟨halshs-01954353⟩



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