A formal system of the Gougu method 555
Here is a translation of the fi rst part of the table of contents, 6 in which gou is
rendered as a , gu as b and the hypotenuse as c. I designate the diff erence in
layout by two marks that I place at the beginning of each item.
- a , b (being given), fi nd c
- a , c (being given), fi nd b
- b , c (being given), fi nd a
ә a, a + b (being given), subtract a from the sum, the remainder is b , enter
into this problem by the procedure of a and b
ә a , b − a (being given), add a to the diff erence, the sum is b , enter into this
by the procedure of a and b
ә a, a + c (being given), subtract a from the sum, the remainder is c , enter
into this by the procedure of a and c
ә a, c − a (being given), add a to the diff erence, enter into this by the pro-
cedure of a and c - a , b + c (being given), fi nd b and c
- a , c − b (being given), fi nd b and c
ә b , a + b (being given), subtract the diff erence from b , the remainder is a ,
enter into this by the procedure of a and b - b, a + c (being given), fi nd a and c
- b , c − a (being given), fi nd a and c
- b , c + b (being given), subtract b from the sum, the remainder is c , enter
into this by the procedure of b and c
Table 16.1 Th e thirteen items of the ‘ Gougu Procedure’
a (gou)
b (gu)
c (xian)
b + a (gougu he )
b − a (gougu jiao)
c + a (gouxian he)
c − a (gouxian jiao)
b + c (guxian he)
c – b (guxian jiao)
a + b + c (gouhe he)
b + c – a (gouhe jiao)
a + c – b (goujiao he)
a – c + b (goujiao jiao)
(^6) For the complete table of contents, see Appendix 1. In the original text, there is no mark at the
beginning of each problem in the table of contents. In order to clarify the structure of the table
and the book, I attach a mark, a circle or a square, to each problem (see below, p. 570).