General Source of Computational Methods: Volume Seven

Mnemonic for Seeking the Middle Width of a Triangular Field

To find the middle width of a triangle, one must reflect:

Take the lower width and multiply it by the length of the tip.

Add the main vertical length to that same tip length;

Use this sum as the divisor to find the middle width accurately.

The method states: Set the lower width at 12 inches. Multiply this by the length of the triangular tip, which is 1 foot 5 inches, to obtain 18 inches as the shi: dividend (the total to be divided). Separately, take the main vertical length of 10 feet and add the tip length of 1 foot 5 inches, totaling 11 feet 5 inches to serve as the fa: divisor (the number by which you divide). Dividing [the dividend by the divisor] yields a middle width of 1 foot 5? inches 6? fen 5? li The fen and li are decimal subdivisions of the inch; this result is approximately 1.565 inches.. This matches the inquiry.

Division of a Triangular Field

Suppose there is a triangular field where each of the three sides measures 14 paces.
It is now to be divided into three sections, and each section must be four-sided.
The question asks: What are the lengths and widths of each section?

The answer states: The total area is 84 square paces original: "84 bu". In traditional Chinese surveying, area is often expressed in square paces.; each of the three sections contains 28 square paces.
For each section, the length is 8 paces and the width is 7 paces.

Historical Note: The area of an equilateral triangle with a side of 14 is approximately 84.87. The text uses 84, a common practical approximation in period manuals. The division of a triangle into three equal quadrilaterals was a standard challenge for land surveyors to ensure equitable tax distribution among three parties.