42313 Reading proofs in Chinese commentaries:
algebraic proofs in an algorithmic context
Karine ChemlaTh e earliest Chinese text devoted to mathematics that has been handed
down through the written tradition, Th e Nine Chapters on Mathematical
Procedures ( Jiuzhang suanshu ), was probably compiled on the basis of older
documents and completed in the form in which we have it today in the fi rst
century ce. 1 Until recently, there was no evidence indicating the nature of
the documents that may have been used in composing Th e Nine Chapters.
However, in 1984, in a tomb that had been sealed c. 186 bce at Zhangjiashan
(today in the Hubei Province), archaeologists found a text entitled the Book
of Mathematical Procedures ( Suanshushu ) which may have been used for
this purpose. 2 Like this book that was brought to light thanks to archaeo-
logical excavations but did not survive through written transmission, Th e
Nine Chapters is mainly composed of particular problems and algorithms
for solving them, without displaying any apparent interest in establishing
1 In what follows, the title is abbreviated as Th e Nine Chapters. Th e full title would be more
accurately translated as ‘Mathematical procedures in nine chapters/patterns’. However, to
avoid confusion with titles of other Chinese mathematical books, the English translation of
which is quite close to that of Th e Nine Chapters , I give a translation that does not diverge
from the usual English title given to the book. In this volume, A. Volkov (see Chapter 15 ,
Appendix 2 ) chooses to translate the title as Computational Procedures of Nine Categories.
In the earliest document that was handed down and that outlines the history of Th e Nine
Chapters as a book, i.e. the third-century commentator Liu Hui’s preface, the process
of compilation is sketched and mentioned as having lasted more than a century. In the
introduction to Chapter 6 in CG2004, I gather the evidence on the basis of which I consider
the book to have been completed in the fi rst century ce. In this chapter, unless otherwise
stated, I follow the critical edition of Th e Nine Chapters given in CG2004. Th e reader can fi nd
in this book a complete French translation of the Classic and its traditional commentaries; see
below. Other translations of the same texts have appeared in recent years: some into modern
Chinese (Shen Kangshen 1997 ; Guo Shuchun 1998 ; Li Jimin 1998 ), one other into English
(Shen, Crossley and Lun 1999 , based on Shen 1997 ). It is impossible, within the framework of
this chapter, to comment on all the diff erences between the translation given here and these
other translations. Th e interested reader can compare the various interpretations.
2 Th e fi rst critical edition of this text can be found in Peng Hao 2001. Two translations into
English have already appeared (Cullen 2004 ; Dauben 2008 ). Th e Nine Chapters and the Book of
Mathematical Procedures have a number of similarities. For example, they deal with the same
concept of fractions, conceived of as composed of a numerator and a denominator. Moreover,
they contain similar algorithms to compute with fractions. In addition to testifying to the fact
that these elements of mathematical knowledge existed in China before 186 bce , the Book of
Mathematical Procedures provides additional information that will prove useful for us below.