Systematic Treatise on Arithmetic, Vol. 6

Sectioning the Slanted Field

Identical to a trapezoid original: 梯形 (tixin). A four-sided shape with two parallel sides of different lengths.

If the middle width is 6 paces, what is the length of the section at the southern The "south" usually refers to the wider, bottom base in traditional Chinese surveying diagrams. end?
Answer: The length of the southern section is 8 paces.

The method states: Set the middle width of the cut 6 paces and subtract the upper width 4 paces, leaving a remainder of 2 paces. Multiply this by the length 30 paces to get 60 paces as the dividend (shi: the value to be divided). Then, subtract the northern (upper) and southern (lower) widths from each other to get a remainder of 8 paces, which serves as the divisor (fa: the measure used to divide). Divide the dividend by this to obtain the result. Note: Mathematically, 60 divided by 8 is 7.5. The answer "8" and the subsequent calculations suggest the original author may have used a total length of 32 paces in their calculations, despite the text stating 30.

〇 To find the length of the lower section: Subtract the middle width from the lower width to get a remainder of 6 paces. Multiply this by the original length to get 192 paces as the dividend. Subtract the upper and lower widths to get a remainder of 8 paces as the divisor. Divide to get the length of the lower section, which is 24 paces, fitting the problem. Calculation: 192 ÷ 8 = 24. Here, the text implies an original length of 32 (since 6 × 32 = 192).

Now, if the length of the lower section is 24 paces, what is the middle width?
Answer: 6 paces.

The method states: Subtract the upper width 4 paces from the lower width to get a remainder of 8 paces as the dividend. Use the original length 30 paces as the divisor and divide. For every 1 pace of length, the width difference kuo chai: the rate at which the field narrows or widens is 0.25 original: 2 fen 5 li. Use this as a constant and multiply it by the length of the lower section 24 paces.