Systematic Treatise on Arithmetic, Vol. 8 (1592) - Page 17

Finding the Height and Hypotenuse using the Height-Difference

Obtaining a hypotenuse of 17 feet matches the requirements of the problem.

Suppose there is a standing pole of unknown height and a rope of unknown length. When the rope hangs straight down from the top of the pole, two feet of it lie coiled on the ground. If you pull the rope away from the base of the pole by eight feet, the tip of the rope just touches the ground. What is the height of the pole and the length of the rope?

Answer: The height of the pole is 15 feet; the length of the rope is 17 feet.

original: "1 zhang 5 chi" and "1 zhang 7 chi". In traditional Chinese units, 1 zhang equals 10 chi (feet).

This follows the same logic as the method for finding the hypotenuse using the hypotenuse-difference. Diagram of a right-angled triangle. The vertical side represents the pole, the horizontal side represents the distance the rope was pulled, and the diagonal represents the rope itself. The height is 2 feet less than the hypotenuse. The base is 9 feet less than the hypotenuse.

Diagonal side (Rope): Rope length is 17 feet 弦 (Xian): the hypotenuse
Vertical side (Pole): Pole height is 15 feet 股 (Gu): the vertical side
Base side (Base): The 8-foot distance from the pole is the base 勾 (Gou): the horizontal side

The method says: Set the 8-foot distance from the pole 8 feet as the base. Multiply it by itself to obtain 64. Take the 2 feet of rope that lay on the ground as the height-difference original: 股較 (Gu Jiao); the difference between the vertical pole and the diagonal rope; multiply it by itself to obtain 4. Subtract 4 from 64, leaving 60 as the dividend 實 (Shi): the value to be divided. Double the difference of 2 feet to make 4 feet, which serves as the divisor 法 (Fa). Divide the dividend by this to find a pole height of 15 feet the height/Gu. If you add the difference of 2 feet to the height, you obtain a rope length of 17 feet the hypotenuse/Xian. The result matches the problem.

If using the "finding the hypotenuse using the hypotenuse-difference" method: set the 8-foot distance from the pole 8 feet as the base. Multiply it by itself to obtain 64 as the dividend. Use...