The Gougu Theorem: the Pythagorean Theorem in Ancient China

The gougu theorem (勾股定理, gōugǔ dìnglǐ) is the Chinese name for the Pythagorean theorem. Gou and gu are the two legs of a right triangle and xian is the hypotenuse, so the rule reads: gou squared plus gu squared equals xian squared. It appears in the Zhoubi Suanjing and the Nine Chapters on the Mathematical Art, both compiled during the Han dynasty (206 BCE – 220 CE).

The Chinese tradition, independent of Greece as far as any evidence shows, has its own style: problems first, then procedures, then justifications by cutting and moving areas. For the famous "hypotenuse diagram" that proves the theorem, see the Chinese proof of the Pythagorean theorem.

Gou, gu and xian: the vocabulary

The names make most sense when you picture a gnomon, an upright post used to measure the sun's shadow.

勾 gōu (also written 句) — "hook"

The shorter leg. In shadow measurement it is the shadow lying flat on the ground. In our notation it is a.

股 gǔ — "thigh"

The longer leg, standing upright like a leg. The Zhoubi says plainly: "the gnomon is the gu; the true shadow is the gou." In our notation it is b.

弦 xián — "bowstring"

The hypotenuse, stretched across like a string between the ends of a bow. This is the standard term in the Nine Chapters; the older Zhoubi passages say 徑隅 jìng yú ("the diagonal between the corners") or 邪 xié ("slant"). In our notation it is c.

The Zhoubi Suanjing: Shang Gao and the Duke of Zhou

The Zhoubi Suanjing (周髀算經), "The Mathematical Classic of the Zhou Gnomon", is a book of astronomy and the mathematics needed for it. Bi literally means "thighbone" but here refers to the gnomon, and the text links the two leg-words directly: the gnomon is the gu, the "thigh" of the right triangle. The title's Zhou can mean the Zhou dynasty or "circuit", so it has also been translated as "the gnomon and the circular paths of heaven".

The book opens with a conversation. The Duke of Zhou asks the official Shang Gao how the ancients could measure heaven, which cannot be climbed by steps, and the earth, which cannot be measured with a rule: where do numbers come from? Shang Gao answers that the art of number comes from the circle and the square, the square from the carpenter's square (矩, jǔ), and then gives the example that has made the passage famous:

故折矩，以為句廣三，股修四，徑隅五。
"So bend the carpenter's square: let the gou be 3 wide and the gu 4 long; then the diagonal from corner to corner is 5." (our translation)

The next few lines describe forming a square on the outside, halving it and arranging carpenter's squares around it, ending with the remark that two carpenter's squares together come to 25: "this is called accumulating carpenter's squares." They are terse, and scholars translate and interpret them differently. In the third century CE the commentator Zhao Shuang added a diagram and a general argument, the "hypotenuse diagram" explained on the Chinese proof page.

The general rule: Chen Zi measures the sun

Shang Gao's passage only gives the 3-4-5 case. The general rule appears later in the same book, in a dialogue in which a student, Rong Fang, learns from a teacher named Chen Zi how to work out the sun's distance from gnomon shadows:

若求邪至日者，以日下為句，日高為股，句股各自乘，并而開方除之，得邪至日。
"To find the slant distance to the sun, take [the distance to the point] beneath the sun as gou and the height of the sun as gu. Multiply gou and gu each by itself, add them, and extract the square root: this gives the slant distance to the sun." (our translation)

That is the theorem in words, c = √a² + b², stated as a procedure. Chen Zi's numbers are 60,000 li along the ground and 80,000 li up, giving 100,000 li:

How old is the Zhoubi?

Tradition placed the dialogue in the lifetime of the Duke of Zhou, around 1000 BCE, and you will still see claims that China knew the theorem three thousand years ago. Modern historians do not accept that. The book as we have it is a Han-dynasty compilation; estimates for its present form range from about the first century BCE to the first century CE. Christopher Cullen, its English translator, argues that it is best understood as a product of Han astronomy. Some passages, possibly including the Shang Gao dialogue, may preserve older material, but the Duke of Zhou is a literary frame, not a witness. In China the theorem is sometimes also called 商高定理, the "Shang Gao theorem".

The Nine Chapters: the gougu rule

The Nine Chapters on the Mathematical Art (九章算術, Jiǔzhāng Suànshù) is the great problem book of early Chinese mathematics. It was assembled from earlier material over a long period; most historians place its final form between roughly 100 BCE and 100 CE. Its ninth and last chapter, Gougu, contains 24 problems on right triangles. The opening problems apply the rule directly; later ones combine it with algebra and similar triangles.

The rule is given as a procedure, almost word for word the same as Chen Zi's: multiply gou and gu each by itself, add, and extract the square root to get the xian. The chapter also gives the reverse procedures — subtract the square of one leg from the square of the hypotenuse to find the other leg — which is exactly how you find a missing side today.

Lengths are in zhang, chi and cun: 1 zhang = 10 chi and 1 chi = 10 cun. A Han-period chi was roughly 23 cm, but the arithmetic does not depend on that.

Classic gougu problems, solved

Each of these problems hides a right triangle in an everyday scene. The statements are paraphrased; the answers are the Nine Chapters' own, checked here in modern notation. For more practice of the same kind, see the Pythagorean theorem word problems.

The Nine Chapters solves this without algebra symbols. Its procedure: square half the pond's width, subtract the square of the height above water, and divide by twice that height. Here (25 − 1) ÷ 2 = 12. That recipe is exactly the line 2x = 24 above, written as a rule.

Try two more

Both of these are also in chapter 9. Solve them before opening the answers.

Liu Hui and the out-in principle (263 CE)

Liu Hui (c. 220 – c. 280) lived in the kingdom of Wei during the Three Kingdoms period. His commentary on the Nine Chapters, traditionally dated to 263 CE, does what the original text does not: it explains why the procedures work.

For the gougu rule he uses what historians call the out-in complementary principle (出入相補, chūrù xiāngbǔ): cut a figure into pieces, move the pieces without overlapping, and the area stays the same. He colors the square on the gou red and the square on the gu blue-green, then moves pieces of them "out" and "in", each to a matching place, until together they fill the square on the xian. His original diagram is lost, so modern reconstructions of the exact cuts differ; one is animated on the Chinese proof page. The idea is the same one behind the rearrangement proof.

Liu Hui also put the theorem to work in two famous directions:

• Calculating π. Starting from a hexagon inscribed in a circle of radius 1, he used the gougu rule twice per step to find the side of a polygon with twice as many sides. For the 12-gon: half the hexagon's side is 1 over 2, the distance from the center to that side is √1 − 1 over 4 ≈ 0.8660, the gap to the circle is about 0.1340, and the new side is √0.5² + 0.1340² ≈ 0.5176. Repeating this up to 96 and 192 sides gives an area of about 3.1410, which justified the value π ≈ 3.14.
• Measuring the unreachable. His Sea Island Mathematical Manual (海島算經, Haidao Suanjing), nine problems on finding heights and distances from sightings with poles, was originally attached to his commentary on chapter 9. It relies on gougu reasoning and similar triangles, the same tools surveyors use today (see the theorem in navigation).

Independent discovery and the question of "first"

Was the gougu theorem discovered before Pythagoras? On the evidence, the surviving Chinese texts are later than him: they took their present form in the Han dynasty, four or more centuries after Pythagoras (c. 570–c. 495 BCE) and two or more after Euclid (c. 300 BCE). They are also later than the Old Babylonian tablets such as Plimpton 322 and probably later than the Indian Śulba Sūtras.

That does not make the Chinese work derivative. There is no sign of Greek influence: the vocabulary, the gnomon setting, the problem-and-procedure style and the dissection justifications are all distinctively Chinese. Frank Swetz and T. I. Kao examine this tradition in detail in Was Pythagoras Chinese? (1977) and treat it as an independent development. The honest answer to "who discovered it?" is that several cultures did — see the history of the Pythagorean theorem for the full timeline.