Systematic Treatise on Arithmetic
Volume Two
Page Twelve

Diagram for Finding the Height using the Height-Difference

The 3 feet emerging from the water is the difference 較 (Jiao): the difference between the height and the hypotenuse Bank Reed Pulling the reed to the bank forms the hypotenuse 弦 (Xian) The water depth of 12 feet is the height 股 (Gu): the vertical side The distance of 9 feet to the bank is the base 勾 (Gou): the horizontal side Water depth is treated as the height. The reed pulled to the bank is treated as the hypotenuse.

The method says: Set the distance of 9 feet to the bank as the base 勾. Multiply it by itself to obtain 81. Take the 3 feet emerging from the water as the height-difference 股較; multiply it by itself to obtain 9 feet. Subtract this 9 from 81, leaving 72 as the dividend 實 (Shi): the number to be divided. Double the difference of 3 feet to make 6 feet, which serves as the divisor 法 (Fa). Divide the dividend by this to find a water depth the height of 12 feet 1丈2尺. This matches the requirements of the problem.

Suppose there is a base of 9 feet, and the hypotenuse exceeds the height by 3 feet. What are the lengths of the hypotenuse and the height?

Answer: The hypotenuse is 15 feet; the height is 12 feet.

The method says: Multiply the base of 9 feet by itself to obtain 81 as the dividend. Use the excess of 3 feet as the divisor. Divide 81 divided by 3 to obtain 27 feet. Subtract the excess of 3 feet from this to get a remainder of 24 feet. Halve this to obtain a height of 12 feet. Add the excess of 3 feet to this to find the hypotenuse...