560 tian miao
problems in the GGSX have the same layout in general, but also the parts of
every problem are similarly arranged in a formal way.
Concerning the fi ve parts of each problem in the main text of GGSX ,
there is not much that can be said about the fi rst three. Th e structure of
the presentation of each problem and its solution remains the same for
the whole book. Th e only changes concern the numerical values in the
problem and the answers as well as the concrete procedures. We shall focus
our analysis on the last two parts of the presentation of each problem: the
outline and the explanation.
Let us begin our analysis of the structure of these parts of the problems in
the GGSX with an inspection of the outline of the calculations. For each of
the thirty-three problems contained in the book, Li Rui gives an outline of
the calculations. And except for the fi rst three problems, they all bring into
play the tianyuan method. 16 Th e fi rst step is to set up the celestial unknown.
In addition, Li Rui follows a strict rule in choosing the unknown. Th e rule16 Tianyuan algebra is a method for solving problems. It makes use of polynomials with one
indeterminate, expressed according to a place-value system, in order to fi nd out an algebraic
equation that solves the problem. Th e equation was also written down according to a place-Figure 16.2 Li Rui’s diagram for his explanation for the fourth problem in Detailed
Outline of Mathematical Procedures for the Right-Angled Triangle.gougousquare of gugu hypotenusesubtractguhypotenuseit was called ‘ Gougu procedure’), as given by Liu Hui around the year 263. Strictly speaking,
the demonstration is not a rigorous one, and it is unknown whether it refl ects Liu Hui’s
original proof or not. For Li Rui’s demonstration of the Pythagorean theorem, see Tian Miao,
forthcoming. For Liu Hui and his proof of the Pythagoras theorem, see Wu Wenjun 1978 ; Guo
Shuchun 1992 ; Chemla 1992 ; CG2004.