Systematic Treatise on Computational Methods, Volume 3

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Suppose a rectangular field has a width of 32 steps, and it is only stated that the diagonal exceeds the length by 8 steps. What is the area?

Treated as the difference between the length and the hypotenuse original: "直如股弦差" (zhí rú gǔ xián chā). This indicates the problem is modeled on a right-angled triangle where the diagonal is the hypotenuse and the length is the "height" or "gu".

Halve the result, just as in the calculation for a right-angled triangular area original: "折半如勾股積" (zhé bàn rú gōu gǔ jī). This note refers to the underlying geometric logic where a rectangle is two identical right-angled triangles.

Answer: 1,920 square steps.

Method: Set the width of 32 steps and multiply it by itself 32 × 32 to obtain 1,024 steps. Separately, take the excess of 8 steps and multiply it by itself 8 × 8 to obtain 64 steps. Subtract the smaller value from the larger to get a remainder of 960 steps, which serves as the dividend Shi (實): the number to be divided in a division operation. Double the excess of 8 steps to make 16 steps, which serves as the divisor Fa (法): the number by which the dividend is divided. Divide [the dividend by the divisor] to obtain a length of 60 steps. Multiply this by the width of 32 steps to obtain the area. This matches the query.

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Suppose a rectangular field has a length of 60 steps, and it is only stated that the diagonal exceeds the width by 36 steps. What is the area?

Treated as the difference between the width and the hypotenuse original: "直如勾弦差" (zhí rú gōu xián chā). Here, the diagonal is compared to the "base" or "width" (gou) of the triangle.

Halve the result, just as in the calculation for a right-angled triangular area

Answer: 1,920 square steps.

Method: Set the length of 60 steps and multiply it by itself 60 × 60 to obtain 3,600 steps. Separately, take the excess of 36 steps and multiply it by itself 36 × 36 to obtain 1,296 steps. Subtract the smaller value from the larger to get a remainder of 2,304 steps, which serves as the dividend. Double the excess of 36 steps to make 72 steps, which serves as the divisor. Divide [the dividend by the divisor] to obtain a width of 32 steps. Multiply this by the length of 60 steps to obtain the area. This matches the query.