Suppose there is a circular field with a diameter of 13 paces. A segment is cut from the edge with an area of 32 square paces. We ask: what are the lengths of the chord and the sagitta In geometry, the "chord" is the straight line connecting the two points of an arc, and the "sagitta" (Latin for "arrow") is the perpendicular distance from the center of the chord to the arc. In Chinese mathematics, these are known as "xian" and "shi." of the cut?

Calculation of a Circular Segment

The method says: Double the area to get 64 paces. Square this result to get 4,096 paces, which serves as the dividend shi: the constant term in an equation to be solved. Multiply the area 32 by four to get 128 paces, which is the upper coefficient shanglian: the second-order coefficient. Also, multiply the diameter 13 paces by four to get 52 paces, which is the lower coefficient xialian: the first-order coefficient. Use 5 as the leading coefficient fuyu: the coefficient of the highest power. Apply the method for extracting the fourth-degree root Original: "kai san cheng fang." This refers to a method for solving a polynomial equation of the fourth degree, which was a sophisticated technique in traditional Chinese algebra used to approximate the dimensions of a circle's segment. to solve it.

Set the trial quotient 4 paces on the top left as the divisor. Multiply the upper coefficient by it to get 520 Note: Based on the numbers provided, 4 times 128 is 512; 520 suggests a slight variation in the recorded intermediate steps or a transcription error in the original woodblock.. Then multiply the leading coefficient 5 by the quotient 4 to get 20. Subtract this from the lower coefficient 52 paces to get a remainder of 32. Separately, square the quotient 4 to get 16, and multiply it by the remaining lower coefficient 32 to get 512. Add this to the upper coefficient 520 to get a total of 1,032. Now, use 1,032 as the divisor for the dividend to find the sagitta, which is 4.

Separately, take the area and double it to get 64. Divide this by the sagitta to get 16. Subtract the sagitta of 4, and the remainder is the chord of 12, which matches the problem.