Divide the doubled area 450 by the width The width of 30 was found via square root on the previous page. to obtain a cut width of 15 paces. Separately, take the original length 45 paces and subtract the cut length of 35 paces; the remainder is a lower length of 10 paces, which matches the question.

Systematic Treatise on Arithmetic, Volume 7

Page 8

Suppose there is a triangular field with a length of 75 paces and a northern width of 30 paces. Now, a section with an area of 720 square paces is to be cut starting from the northern width. The question asks: what are the length and width of this cut section?

The answer states: The cut length is 30 paces. The width at the point of the cut is 18 paces.

Another Method: Take the cut area 720 square paces and double it to get 1,440 paces. Multiply this by the original width 30 paces to get 43,200 paces as the shi: dividend. Divide this by the original length 75 paces, which acts as the fa: divisor, to obtain 576 paces.

Next, square the northern width 30 paces 30 times 30 to get 900 paces. Subtract the 576 paces from this; the remainder is 324 paces, which serves as the shi: dividend. Use the kaipingfang: square root method A traditional Chinese algorithm for extracting square roots similar to long division. to find the cut width of 18 paces.

Add this to the northern width of 30 paces for a total of 48 paces. Halve this to get 24 paces, which serves as the fa: divisor. Divide the cut area of 720 square paces by this divisor to obtain a cut length of 30 paces. This matches the question.