The History of Mathematical Proof in Ancient Traditions

568 tian miao

methods and leaving the reader with no doubt about mathematical knowl-

edge were the essence of Western mathematics. Two years aft er the publi-

cation of the Jihe yuanben , in 1609, Xu Guangqi composed Gougu yi ( Th e

Principle of Gougu ). 28 To interpret the word yi , we have to briefl y mention

what Xu Guangqi said in the preface of another book, Celiang fayi (1607).

Peter Engelfriet off ers the following analysis:

He [Xu Guangqi] makes a distinction that proves very important in his conception

of Western mathematics: a distinction between methods and yi. Th e word yi can

take on a wide range of meanings, but it is obvious that in this context it must refer

to the proofs and explanations given in Western mathematics. For Xu Guangqi

states explicitly that only aft er the Jihe yuanben had been translated was it possible

to transmit the yi of the methods. Moreover, the Western methods of surveying are

not essentially diff erent from the methods transmitted in the Zhoubi suanjing and

the Jiuzhang suanshu. What makes Western mathematics more valuable is that it

supplies explanations which show why the methods are correct. 29

In his Gougu yi , Xu Guangqi sums up the main topics Chinese mathema-

ticians addressed with respect to gougu problems. He stresses that these

problems could only be solved on the basis of the Pythagorean theorem, 30

to be found as Proposition 47 of Book i in the Elements. He argues:

In the old Nine Chapters , there are also [methods] of fi nding the gou and gu from

each other, [fi nding] the inscribed square and circle, and [fi nding] the sums and the

diff erences from each other. 31 But it is only capable of stating the methods, and it is

not capable of discussing its principles ( yi ). Th e methods established [in it] are in

disorder and shallow, and do not bear reading. 32

What is signifi cant for us here is that both Xu Guangqi and Li Rui use

the word yi. While Xu argues that Th e Nine Chapters did not talk about

28 See Engelfriet 1998 : 297–8.

29 Engelfriet 1998 : 297. Engelfriet discusses the meaning of yi and the origin of this term in Xu

Guangqi ’s book in more detail in Engelfriet 1993.

30 Th e example Xu Guangqi quotes earlier in his text is from the Zhoubi suanjing (Th e

Mathematical Canon of the Zhoubi , dating from the beginning of the fi rst century bce ). Th is

text contains a general statement of the Pythagorean theorem, including a paragraph which

could be regarded as a general proof of it (see Ch’en Liang-ts’so 1982 , Li Jimin 1993 ). In the

third century, Zhao Shuang and Liu Hui present clearer proofs in their commentaries to the

Zhoubi Suanjing and Th e Nine Chapters , respectively. See Qian Baocong 1982 ; Guo Shuchun

1985 ; CG2004: 704–45.

31 Th e Nine Chapters of Mathematical Procedures (dated from the fi rst century bce to the fi rst

century ce ). Th is is one of the most important mathematical classics of ancient China. Th e

ninth chapter of this book is devoted to gougu methods. See Guo Shuchun 1985.

32 Xu Guangqi, preface.