16 A formal system of the Gougu method: a study
on Li Rui’s Detailed Outline of Mathematical
Procedures for the Right-Angled Triangle
T i a n M i a o
In contrast to the deductive structure developed in Euclid’s Elements , which
is always taken as the model for ancient Greek mathematical reasoning, the
structure of most ancient Chinese mathematical books could be described
as that of a collection of problems and procedures. Moreover, these pro-
cedures were mostly described within the context of numerical problems.
As historians have argued, in some ancient Chinese mathematical texts
there are proofs establishing the correctness of the algorithms included.
However, these proofs were mostly written by subsequent mathematicians,
and were contained in commentaries attached to related procedures. 1
Th erefore, as these proofs were specifi cally brought to bear on procedures
that were taken from texts that already existed, they could seldom form
a system by themselves, and hence the reasoning model in them looks
fl exible. Th is raises two related questions: when did Chinese mathemati-
cians think of developing a formal system of mathematics in their books?
Moreover, could the mathematical results developed in ancient China be
presented systematically and formally?
In this chapter, I shall rely on a Chinese mathematical book, the
Gougu Suanshu Xicao (hereaft er abbreviated as GGSX, Detailed Outline of
Mathematical Procedures for the Right-Angled Triangle , 1806), to investigate
these questions. Furthermore, I hope that the discussion will shed some
light upon questions such as why and in which context a formal system of
mathematics emerged in China.
(^1) Th e best-known examples of proofs in ancient Chinese mathematical texts are those Liu Hui
provided in his commentary to Jiuzhang suanshu. For greater detail, see Guo Shuchun 1987 ;
552 Chemla 1992 ; CG2004: 3–70; Wu Wenjun 1978.