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The study of algebra in China has often focused on the algebraic “procedure of the Celestial Source.” Its geometrical ancestors are less known. The Yigu yanduan, authored by Li Ye (1192-1279), presents the procedure alongside its two geometrical counterparts, the “Section of Pieces [of Areas]” and the “Old Procedure.” The three procedures are known to represent three generations of algorithms used to set up quadratic equations. A similar geometrical procedure appears in a treatise written by Yang Hui (second half of thirteenth century). Although the procedures look alike at first glance, the two treatises reveal different moments in their work on the relation between counting materials and geometrical representation. This study challenges their chronology trying to identify the meander of the geometrical roots of the Celestial Source. The construction of negative coefficients plays a pivotal role in this mutation and shows several layers of composition.
Recent studies in Sinology have shown that Qing dynasty editors acted as
philologists. This paper argues that the identification of their philological
methods and editorial choices suggests that their choices were not totally
neutral and may have significantly shaped the way modern historians interpreted
specific works edited by mathematicians of that dynasty. A case study of the
re-edition in 1798 of a Song dynasty treatise, the Yigu yanduan
(1259), by a Qing dynasty mathematician will illustrate this point. At the end
of the eighteenth century, Li Rui (1773–1817) was asked to prepare an
edition of the mathematical works written by Li Ye (1192–1279) for a
private collection. Li Rui was a talented mathematician, but he was also a
meticulous editor and trained philologist. He adopted his editorial model from
the preparation of the imperial encyclopaedia, the Siku
quanshu, but Li Rui also made some corrections to the text in an effort
to restore an older version of Li Ye's treatises that had been lost.
Convinced of the Chinese origin of algebra, Li Rui used philological techniques
to recover the lost materials and to restore the roots of “Chinese
mathematics.” The Yigu yanduan contains two
algebraic procedures to set up quadratic equations, one from the procedure of
Celestial Source (tian yuan shu) and the
other from the Section of Pieces [of Areas] (tiao
duan). Curiously, the second procedure has not yet attracted the
attention of scholars so far, although Li Rui's edition is the one
typically used by twentieth-century historians of mathematics. Today, the
Celestial Source characterizes “Chinese
algebra.” However, the specific concerns of Li Rui about the
procedure of Celestial Source, combined with his editorial
methods, contributed to this perspective.
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