○ Song of Cutting Area from a Ring-shaped Field

To cut an area from the outer edge of a ring-shaped field,
Double the target area and multiply it by the difference between the two circumferences.
Divide the resulting number by the original width The text uses "diameter" (jing) to refer to the width or thickness of the ring segment..
Separately, square the outer circumference.
Subtract the smaller value from the larger; let the remainder be the shi: dividend.
Perform kaipingfang: square root extraction to find the inner circumference of the cut.
Subtract this new inner circumference from the outer circumference;
Divide the remainder by six to make the width of the cut section clear The divisor six is used because the author is using the ancient approximation of pi as 3. In this system, the circumference is three times the diameter, so the difference between two circumferences is 2 pi width, or 6 * width..

Suppose there is a ring-shaped field with an outer circumference of 72 paces, an inner circumference of 24 paces, and a width of 8 paces. Now, an area of 285 square paces is to be cut starting from the outer circumference. We ask: what are the inner circumference and the width of this cut section?

Answer: The inner circumference Literally "middle circumference," as it lies between the original outer and inner boundaries. is 42 paces, and the width of the cut is 5 paces.

Method: Take the area to be cut, 285 square paces, and double it to get 570 square paces. Then, subtract the original inner circumference from the outer circumference; the remainder is 48 paces, which serves as the "difference in paces." Multiply the doubled area 570 by this difference to get 27,360 square paces.

The calculation continues on the following page, where this value will be divided by the original width (8 paces) and subtracted from the square of the outer circumference (5184) before taking the square root.