Volume 3

Page 11

Suppose there is a single irregular four-sided field Si bu deng tian (四不等田): literally a "four-not-equal field," referring to a quadrilateral where the sides and angles are unequal. To measure it, cut it into three sections:
One section is a rectangular field, 40 steps long and 28 steps wide.
On the southern side, there is one right-angled triangular section; its long leg is 31 steps and its short leg is 10 steps.
On the eastern side, there is one right-angled triangular section; its long leg is 40 steps and its short leg is 4 steps.
Question: What is the total area?

Irregular Four-Sided Shapes

This follows the standard instructions without the slightest error original: "此乃准教毫忽無差" (chǐ nǎi zhǔn jiào háo hū wú chà). In traditional Chinese measurement, a "hao" and "hu" are extremely small units, similar to "hair's breadths," signifying high precision.

Method: First, take the rectangular field’s length of 40 steps and multiply it by the width of 28 steps to obtain a rectangular area Zhi ji (直積): the area of a rectangle of 1,120 square steps.

○ Next, take the southern right-angled triangular section; multiply its long leg Gu (股): the longer leg of a right triangle of 31 steps by its short leg Gou (勾): the shorter leg of a right triangle of 10 steps. Halve the result to obtain an area of 155 square steps.

○ Then, take the eastern right-angled triangular section; multiply its long leg of 40 steps by its short leg of 4 steps. Halve the result to obtain an area of 80 square steps.

○ The combined area of the three sections is 1,360 square steps.

Note: There is a slight mathematical discrepancy in the original woodblock printing. The sum of the calculated parts (1120 + 155 + 80) is actually 1,355, but the text states the total is 1,360.

**Technical