九章算術·句股（節選）
The Nine Chapters on the Mathematical Art: Base and Upright (excerpt)
Anonymous (Han dynasty compilers) 佚名 (compiled c. 1st century BCE – 1st century CE)
Qin–Han · Compiled by the 1st century CE

Cold Library · Open Stacks · S-002045 · https://coldlibrary.com/s/002045
Rights: Han dynasty mathematical text; public domain everywhere. Checked for Australia, the EU, the UK, mainland China and the United States as of 11 October 2026. Guidance, not legal advice.
Source: Wikisource, https://zh.wikisource.org/wiki/九章算術 (retrieved 2026-10-11)
English translation: Claude (an AI), 2026 · CC0 1.0
Generated 2026-10-14 from https://coldlibrary.com/en/open-stacks/nine-chapters-base-and-upright

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今有句三尺，股四尺，問：為弦幾何？

荅曰：五尺。

今有弦五尺，句三尺，問：為股幾何？

荅曰：四尺。

今有股四尺，弦五尺，問：為句幾何？

荅曰：三尺。

句股術曰：句股各自乘，并，而開方除之，即弦。

又股自乘，以減弦自乘，其餘開方除之，即句。

又句自乘，以減弦自乘，其餘開方除之，即股。

今有圓材徑二尺五寸，欲為方版，令厚七寸。問：廣幾何？

荅曰：二尺四寸。

術曰：令徑二尺五寸自乘，以七寸自乘減之，其餘開方除之，即廣。

今有木長二丈，圍之三尺。葛生其下，纏木七周，上與木齊。問：葛長幾何？

荅曰：二丈九尺。

術曰：以七周乘三尺為股，木長為句，為之求弦。弦者，葛之長。

今有池方一丈，葭生其中央，出水一尺。引葭赴岸，適與岸齊。問：水深、葭長各幾何？

荅曰：水深一丈二尺；葭長一丈三尺。

術曰：半池方自乘，以出水一尺自乘，減之，餘，倍出水除之，即得水深。加出水數，得葭長。

今有立木，繫索其末，委地三尺。引索卻行，去本八尺而索盡。問：索長幾何？

荅曰：一丈二尺、六分尺之一。

術曰：以去本自乘，令如委數而一，所得，加委地數而半之，即索長。

今有竹高一丈，末折抵地，去本三尺。問：折者高幾何？

荅曰：四尺、二十分尺之十一。

術曰：以去本自乘，令如高而一，所得，以減竹高而半其餘，即折者之高也。

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今有句三尺，股四尺，问：为弦几何？

荅曰：五尺。

今有弦五尺，句三尺，问：为股几何？

荅曰：四尺。

今有股四尺，弦五尺，问：为句几何？

荅曰：三尺。

句股术曰：句股各自乘，并，而开方除之，即弦。

又股自乘，以减弦自乘，其余开方除之，即句。

又句自乘，以减弦自乘，其余开方除之，即股。

今有圆材径二尺五寸，欲为方版，令厚七寸。问：广几何？

荅曰：二尺四寸。

术曰：令径二尺五寸自乘，以七寸自乘减之，其余开方除之，即广。

今有木长二丈，围之三尺。葛生其下，缠木七周，上与木齐。问：葛长几何？

荅曰：二丈九尺。

术曰：以七周乘三尺为股，木长为句，为之求弦。弦者，葛之长。

今有池方一丈，葭生其中央，出水一尺。引葭赴岸，适与岸齐。问：水深、葭长各几何？

荅曰：水深一丈二尺；葭长一丈三尺。

术曰：半池方自乘，以出水一尺自乘，减之，余，倍出水除之，即得水深。加出水数，得葭长。

今有立木，系索其末，委地三尺。引索却行，去本八尺而索尽。问：索长几何？

荅曰：一丈二尺、六分尺之一。

术曰：以去本自乘，令如委数而一，所得，加委地数而半之，即索长。

今有竹高一丈，末折抵地，去本三尺。问：折者高几何？

荅曰：四尺、二十分尺之十一。

术曰：以去本自乘，令如高而一，所得，以减竹高而半其余，即折者之高也。

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English translation by Claude (an AI)

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.