Fangcheng Procedure in the Nine Chapters on the Mathematical Arts Revisited

Written by Kat Jivkova. Some historiography has often made a division between Western and non-Western mathematics. Yet, Suanshu's Nine Chapters suggest so-called "Western" mathematics may owe a greater debt to ideas developed in China.

The “Chinese origins of Western science” theory (Xixue Zhongyuan) has challenged historiography that has divided Western and non-Western mathematics. New analyses regarding the fangcheng procedure in the Nine Chapters on the Mathematical Arts (Jiuzhang Suanshu) has refuted this historiography. The previous consensus of historians of Chinese mathematics, exemplified by Jean-Claude Martzloff, suggests that the Nine Chapters showed little development beyond the initial practice of fangcheng, however a closer examination of the extant source outlines the comparable developments in several mathematical practices from its antecedent, the Book of Computation (Suan shu shu). Looking at the reconstructed forms of the Nine Chapters, the uniquely visual methods of ancient Chinese mathematicians challenge the misconception that the West was responsible for the discovery of linear algebra in the sixteenth and seventeenth century – on the contrary: fundamental theories of mathematics, specifically this notion of linear algebra, were already known to the Chinese one thousand years prior to this.

The Nine Chapters, arguably one of the oldest Chinese books on mathematics, demonstrates complexities of early Chinese mathematics. There is still discussion as to when the book was compiled, with some historians identifying it as a first century CE text, and others preferring to date it to its oldest extant source in the form of Liu Hui’s commentary in 263CE. In the preface of his commentary, Liu Hui, an eminent mathematician of imperial China, offers an insight into the history of the book. He gives explanation for its interrupted transmission, “Formerly, the cruel Qin burnt the books”, which is a reference to the book’s burning order of the first Emperor, Shi Huang Di, in 213BCE. This suggests that the Nine Chapters was a product of earlier mathematical materials which existed prior to the Qin Dynasty. The book may also be traced back to the Eastern Han Dynasty in the works of Zhang Cang and Geng Shouchang, in combination with later reconstructions after its damage. These various editions of the book, though creating issues of inconsistency, produce opportunity for illustrating different interpretations of the “method” (shu) to solve mathematical problems. Identifying the features of the original sources from which later reconstructions were based on reveals the foundations of early Chinese mathematics in its rawest form. The earliest source, the Bao edition of 1213CE, was completed in the Northern Song Dynasty and includes reprints of the commentaries of Liu Hui, alongside other aspiring literati. The incorporation of ‘shu’ is evident in this edition in the sub-commentary of Li Chunfeng, an imperial court diviner, who commented on how to use factorisation on the area of a circle (kai yuan) in order to find its circumference. This edition, along with that of the Great Encyclopaedia of Yongle Reign and further explanations from the Southern Song, can be pieced together to pinpoint early Chinese understanding of mathematics. 

Of course, to state that all early Chinese mathematicians had a flawless understanding of the mathematics of the Nine Chapters would be a simplification. Liu Hui did not fully understand fangcheng, further highlighting its complexity: “Those who are clumsy in the essential principles vainly follow this original procedure”. Nonetheless, Liu Hui was a revolutionary mathematician of his time and acknowledged the function of mathematics as a universal one, capable of measuring both the immediate and the distant. His arcane level of investigation was incredibly rare. A lack of mathematical understanding is further evident in Li Chunfeng’s failure to offer commentary in the most difficult chapters of the Nine Chapters: “Excess and Deficit” and “Fangcheng”, both of which relate to linear algebra. Li’s sub-commentary of Liu Hui’s calculation of the area of a circle gives a negligible amount of mathematical analysis and uses a less precise estimate of the number  than Liu Hui’s calculations. Donald Wagner observed:

Liu Hui was a very important original thinker and mathematician, while Li Chunfeng was a mediocre mathematician who often misunderstood Liu Hui.

Mistakes and misunderstandings such as these have caused historians to believe in the regression of Chinese mathematics from the Qing to the Tang Dynasty. However, the book showcases developments in fangcheng procedure, defined as a “rectangle of measures” by Liu Hui. This procedure was similar to the modern methods of Gaussian elimination or “row reduction”, concerned with linear equation problems, and likewise used counting rods. These counting rods were placed in columns representing different conditions, with red rods used for positive numbers, and black for negative. Problem 1, the singular passage which explains fangcheng procedure, uses a method for elimination. The problem begins with an initial placement of the rods, which in modern terms can be solved by rewriting this as an augmented matrix. After a series of multiplications and subtractions of the various columns of the matrix, the solution can be found through back substitution, which avoids fractions until the final step of the method. This back substitution is the reason Liu Hui struggled with understanding fangcheng procedure.

Though fangcheng in itself emphasised a development in mathematical knowledge from the Qing Dynasty onwards, the complexities of the Nine Chapters can truly be seen from the commentaries of Liu Hui and Chunfeng, which transformed the book into a distinct and challenging mathematical text. The first translations of the book into western languages, completed by Èl’vira Ivanova Berezkina into Russian and Kurt Vogel, only translated its fundamental text, lessening the grandeur of Chinese mathematical thought. The commentaries contain explanations of the proofs and ‘shu’ within the book, which highlight the advanced reasoning of Chinese mathematicians. Overlooking the significance of these commentaries is undoubtedly the reason why previous Western scholars failed to recognise any developments in the Nine Chapters regarding fangcheng practice. Alexei Volkov suggested that earlier dismissals of Liu Hui’s commentary were partly due to a compliance towards traditional approaches of studying classical texts which stated, “the status of the classics was always higher than that of the commentaries.” Volkov argued that the context of Liu Hui’s commentary “changed its status”, inspired by the works of A.P. Yushkevich who in 1961 reasoned that it may enhance scholarly understanding of Chinese mathematics and the reasoning behind it. The differences between the Book of Computation and the Nine Chapters further shows the developments of Chinese mathematics from what existed before the Qin Dynasty, the main change being the emergence of the practice of fangcheng in Chapter 8. Although the Book of Computation included methods for determining roots for squares and cubes, this was done with the “ying bu zu method of excess deficiency” in contrast to the more elaborate algorithms of fangcheng in the Nine Chapters.

The Nine Chapters is one source that suggests the Chinese origins of Western mathematical theory. The Chinese origins of Western science theory can therefore very well be argued, using the Nine Chapters as a key source of evidence. Through a further analysis of the book, the fangcheng procedure is shown to be sophisticated, emphasised by the misunderstandings of Li Chunfeng, and an important origin of linear algebra. There are notably limitations to the text, mainly hermeneutic issues, in the interpretation of the basic meanings of mathematical terms. The translating of texts generally raises problems, as Volkov notes:

One still can wonder whether the conventional format of translation can satisfactorily render Chinese mathematical texts.

 A more specific example lies in the definition of the word ‘mian’ in Liu Hui’s commentary, which has been given an ambiguous meaning of either “irrational number” or “the side” of a square root. However, future archaeological research may solve these interpretative problems and perhaps give more details about the identity of Liu Hui, or a more accurate estimation of when the Nine Chapters were compiled. What is currently known is that the Nine Chapters introduced into China an impressive mathematical standard which remained the basis for knowledge up to the “renaissance” of the Song Dynasty.

Written by Kat Jivkova


Bréard, Andrea. Nine Chapters on Mathematical Modernity. Cham: Springer International Publishing AG, 2019.

Chemla, Karine, and Zou, Dahai. “Parts in Chinese Mathematical Texts. Interpreting the Chapter Form of The Nine Chapters on Mathematical Procedures.” In Pieces and Parts in Scientific Texts, 91-133. Cham: Springer International Publishing, 2018.

Dauben, Joseph W. “Jiu Zhang Suan Shu (Nine Chapters on the Art of Mathematics) ― An Appraisal of the Text, Its Editions, and Translations.” Sudhoffs Archiv 97, no. 2 (2013): 199-235.

Hart, Roger. The Chinese Roots of Linear Algebra Baltimore: Johns Hopkins University Press, 2011.

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