Volume Seven

Mnemonic: If Asking for the Lower Width of a Trapezoid Below a Pointed Field

Do Not Dwell on the Combined Length of the Triangle and Trapezoid

Multiply the Upper Width by the Combined Length to Form the Dividend

Divide by the Height of the Tip to Find the Lower Width

The method states: Set the combined length of the triangular tip and the trapezoid at 12 feet. Multiply this by the upper width of 1 foot 5 inches to obtain 18 feet as the shi: dividend (the total to be divided). Use the tip length of 10 feet as the fa: divisor (the number to divide by). Dividing yields a lower width of 1 foot 8 inches, which matches the question.

---

Mnemonic for Finding the Height of a Trapezoid Added to a Pointed Field

To Find the Height of a Trapezoid Extending from a Pointed Field

Subtract the Upper Width from the Lower Width

Multiply the Remainder by the Tip's Height to Form the Dividend

Dividing by the Upper Width Gives the Height of the Trapezoid

The method states: Take the lower width of 1 foot 8 inches and subtract the upper width of 1 foot 6 inches. The remainder is 2 inches. Multiply this by the height of the triangular tip The text uses "gui chang" (圭長), referring to the vertical height of the triangular section., which is 10 feet, to obtain 2 feet as the dividend. Divide this by the upper width of 1 foot 6 inches. This yields a vertical height for the trapezoid of 1 foot 2.5 inches, which matches the question.

These calculations rely on the principles of similar triangles. In traditional Chinese mathematics, land was often divided into these "gui" (pointed) and "ti" (ladder/trapezoid) shapes. By understanding the ratio between the width and the height of the triangular tip, a surveyor could determine the dimensions of any subsequent section added to the base.