564 tian miao
Th is procedure may be represented in modern algebraic terms by the
following equation:
x^2 + (a + b)x − [(a + b)^2 − c^2 ]/2 = 0
whose solution is x = a.
Li Rui’s explanation may be translated as follows:
Explanation: in the square of the sum, there are four pieces of the product of gou
( a ) and gu (b), one piece of the square of the diff erence between gou and gu. In
the square of the hypotenuse, there are twice the product of gou and gu , and one
piece of the square of the diff erence (between gou and gu ). Subtracting one from
the other, the remainder is twice the product of gou and gu. Halving it, one gets one
piece of the product, which is also the product of gou and the sum of gou and gu
minus the square of gou. Th erefore, take the sum as the negative coeffi cient of the
fi rst degree of the unknown. 21
Now, let us inquire into the process of explanation (see Figure 16.3 ). In the
fi rst step, Li Rui decomposes the two ‘squares’ mentioned at the beginning of
21 Li Rui 1806 : 12a.Figure 16.3 Li Rui’s diagram for his explanation for the eighth problem in Detailed
Outline of Mathematical Procedures for the Right-Angled Triangle.gougougugougugugougousubtractsubtractsubtractdifferencedifferencesubtract
hypotenuse
hypotenusehypotenuse