九章算術·句股（節選）九章算术·勾股（节选）

Suppose the base is three feet and the upright is four feet. What is the hypotenuse?

Answer: five feet.

Suppose the hypotenuse is five feet and the base is three feet. What is the upright?

Answer: four feet.

Suppose the upright is four feet and the hypotenuse is five feet. What is the base?

Answer: three feet.

The rule of base and upright: multiply the base by itself and the upright by itself, add them, and take the square root: this is the hypotenuse.

Or multiply the upright by itself and subtract it from the hypotenuse multiplied by itself; take the square root of the remainder: this is the base.

Or multiply the base by itself and subtract it from the hypotenuse multiplied by itself; take the square root of the remainder: this is the upright.

Suppose there is a round log two feet five inches across, and you want to cut from it a square-edged board seven inches thick. How wide is the board?

Answer: two feet four inches.

Rule: multiply the diameter, two feet five inches, by itself; subtract seven inches multiplied by itself; take the square root of the remainder: this is the width.

Suppose there is a tree twenty feet tall and three feet around. A vine grows at its foot and winds seven times around the tree, reaching exactly to the top. How long is the vine?

Answer: twenty-nine feet.

Rule: multiply seven turns by three feet to make the upright, take the height of the tree as the base, and find the hypotenuse. The hypotenuse is the length of the vine.

Suppose there is a square pond ten feet on a side, with a reed growing in the middle that stands one foot above the water. If the reed is pulled towards the bank, it just reaches the edge. How deep is the water, and how long is the reed?

Answer: the water is twelve feet deep; the reed is thirteen feet long.

Rule: multiply half the side of the pond by itself; subtract one foot, the height above the water, multiplied by itself; divide the remainder by twice the height above the water: this gives the depth of the water. Add the height above the water to get the length of the reed.

Suppose a rope is tied to the top of an upright post, and three feet of it lie on the ground. If you pull the rope and walk back, it runs out when you are eight feet from the post. How long is the rope?

Answer: twelve and one sixth feet.

Rule: multiply the distance from the post by itself, divide by the length lying on the ground, add the length lying on the ground to the result, and halve it: this is the length of the rope.

Suppose a bamboo ten feet tall is broken, and its tip touches the ground three feet from the foot of the stem. How high is the break?

Answer: four and eleven twentieths feet.

Rule: multiply the distance from the foot by itself, divide by the height, subtract the result from the height of the bamboo, and halve the remainder: this is the height of the break.