Systematic Treatise on Arithmetic, Vol. 8

4 meaning 400 acts as the dividend: the number to be divided. Extract the square root to obtain 20 inches. Doubling this result gives 2 feet In this traditional system, 10 inches equals 1 foot; thus 20 inches doubled would be 40 inches or 4 feet, though the text concludes here at 2 feet, suggesting a halving step or specific geometric context from the preceding page, which matches the question.

Suppose there is a 股 (Gu): the vertical side or height of a right triangle 36 paces long. It is only stated that the 弦 (Xian): the hypotenuse is 18 paces longer than the 勾 (Gou): the horizontal base. The question asks: what are the lengths of the base and the hypotenuse?

The answer is: The base is 27 paces, and the hypotenuse is 45 paces.

The method says: Take the height of 36 paces and square it (multiply it by itself) to get 1,296 paces. Separately, take the 18-pace This refers to the difference between the base and the hypotenuse difference between the hypotenuse and the base as the "base difference" and square it to get 324 paces. Subtract these two figures to get a remainder of 972 paces, which acts as the dividend. Double the difference of 18 to get 36 paces as the divisor. Divide the dividend by this to find the base of 27 paces. Add the difference of 18 paces to this to find a hypotenuse length of 45 paces. This matches the question.

〇 Another method, called "Finding the Hypotenuse from the Difference," says: Square the height to get 1,296 paces as the dividend. Use the hypotenuse-base difference of 18 as the divisor. Dividing these gives the sum of the base and the hypotenuse as 72 paces. Add this sum to the difference of 18 paces to get 90. Halve this result to get a hypotenuse of 45 paces. Subtract the difference of 18 paces from the hypotenuse, and the remaining 27 paces is the value of the base.