Systematic Treatise on Arithmetic, Vol. 6 (1592) - Page 22

Cutting from the Outer Circumference of a Ring

Outer circumference: 72 paces Inner circumference: 24 paces Width: 8 paces Original: "diameter" (jing). In ring-field problems, this refers to the radial width of the ring. Area: 99 paces

Cutting from the Inner Circumference of a Ring

Outer circumference: 72 paces Middle circumference: 42 paces Inner circumference: 24 paces Width: 3 paces

Take the remaining area of 285 fifteen paces and multiply it by twelve to get 3,420 paces. Also, square the outer circumference of 72 to get 5,184 paces. Subtract the smaller from the larger to get a remainder of 1,764 paces as the shi: dividend. Use the kaipingfang: square root extraction method to find the middle circumference of 42 paces. Subtract the inner circumference of 24 paces to get a remainder of 18. Divide this by six to get the width of 3 paces, which matches the problem. The divisor six is used because the text employs the "ancient pi" ratio where the circumference is three times the diameter; thus, the difference between circumferences is six times the radial 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. One desires to cut an area of 99 paces starting from the inner circumference. We ask: what are the resulting middle circumference and the width of the cut?

Answer: The middle circumference is 42 paces; the width is 3 paces.

Method: First, add the outer and inner circumferences together and halve the sum. Multiply this by the total width to find the total area of 384 four paces. Subtract the target area of 99 paces from this total; the remainder of 285 five paces is the area of the remaining section from the outer circumference. The author simplifies the problem by calculating the total area and subtracting the inner cut to turn it into an "outer cut" problem, which was solved on the previous page.