A proof of the Pythagorean Theorem by Liu Hui
(third century AD)

Historia mathematica, 1985, 12, pp. 71-3.

In this Web version I have included Chinese characters, which were not in the published version. Some statements here are no longer up to date, but I have not made any major changes.

Donald B. Wagner

The Jiuzhang suanshu 九章算術 (Arithmetic in nine chapters) is a Chinese mathematical book, probably of the first century A.D. Chapter 9, on right triangles, consists of 24 problems together with algorithms for their solution, with no explanation.[1] The Pythagorean Theorem is introduced by the first three problems:

1. If [the length of] the shorter leg [of a right triangle] is 3 chi 尺, and the longer leg is 4 chi, what is the hypotenuse?
Answer: 5 chi.

2. If the hypotenuse is 5 chi, and the shorter leg is 3 chi, what is the longer leg?
Answer: 4 chi.

3. If the longer leg is 4 chi, and the hypotenuse is 5 chi, what is the shorter leg?
Answer: 3 chi.

The method of right triangles:

Multiply the shorter leg and the longer leg each by itself, add, and extract the square root. This is the hypotenuse.

Or: Multiply the longer leg by itself; subtract this from the product of the hypotenuse by itself; extract the square root of the difference. This is the shorter leg.

Or: Multiply the shorter leg by itself; subtract this from the product of the hypotenuse by itself; extract the square root of the difference. This is the longer leg.

[Qian Baocong 1963, 241-242; cf. Vogel 1968, 90-91]

The third-century commentator Liu Hui 劉徽 [2] gives the following explanation for the first algorithm above:

The shorter leg multiplied by itself is the red square, and the longer leg multiplied by itself is the blue square. Let them be moved about so as to patch each other, each according to its type. Because the differences are completed, there is no instability. They form together the area of the square on the hypotenuse; extracting the square root gives the hypotenuse.

[Qian Baocong 1963, 241]

The diagram that Liu Hui refers to here is no longer extant, but it is clear that it must have shown a way of cutting up the squares on the two legs of the right triangle and fitting the pieces together to form the square on the hypotenuse. This method of cutting up and rearranging areas is used very often in Liu Hui's commentary; it is reminiscent of the Chinese game of 'Tangrams'.[3]

Figure 1 shows a possible reconstruction of Liu Hui's diagram. A reader who was familiar with some elementary facts concerning congruency of triangles would, after some study of this diagram, have been able to see that the theorem is correct. Undoubtedly other reconstructions are possible, but I am not aware of any other proposed reconstruction which satisfies the condition that the two smaller squares are cut up and directly fitted together to form the larger square.


[1]. The best edition of the Jiuzhang suanshu [available in 1985] is that of Qian Baocong [1963, 91-272]. Vogel [1968] gives a translation of the main text, without the later commentaries. On the date of the jiuzhang suanshu see [Qian Baocong 1964, 32-33]. Van der Waerden [1983, 36-56] and Vogel [1968, 124-138] discuss its mathematical content.

[2]. On Liu Hui and his mathematics see [van der Waerden 1983, 192-207; Wu Wenjun 1982; Wagner 1978a, b, 1979].

[3]. On Tangrams (qi qiao ban 七巧板, 'seven ingenious pieces') see e.g. [Gardner 1974].


Gardner, M. 1974. Mathematical games. Scientific American 231, No. 2 (August), 98-103; No. 3 (September), 187-191.

Qian Baocong 錢寶琮 (ed.) 1963. Suanjing shi shu 算經十書 [Ten mathematical classics]. 2 vols. Beijing: Zhonghua shuju.

Qian Baocong 1964. Zhongguo shuxue shi 中國數學史 [The history of mathematics in China]. Beijing: Kexue Chubanshe; reprinted 1981.

Vogel, K. (tr.) 1968. Neun Bücher arithmetischer Technik. Braunschweig: Vieweg.

van der Waerden, B. L. 1983. Geometry and algebra in ancient civilizations. Berlin/Heidelberg/New York: Springer.

Wagner, D. B. 1978a. Liu Hui and Tsu Keng-chih on the volume of a sphere. Chinese science 3, 59-79.
Web version.

--------- 1978b. Doubts concerning the attribution of Liu Hui's commentary on the Chiu-chang suan-shu. Acta Orientalia (Copenhagen) 39, 199-212.

--------- 1979. An early Chinese derivation of the volume of a pyramid: Liu Hui, third century A.D. Historia mathematica 6, 164-188.

Wu Wenjun吳文俊 (ed.) 1982. 'Jiuzhang suanshu' yu Liu Hui 《九章算術與劉徽》 [The Jiuzhang suanshu and Liu Hui]. Beijing: Beijing Shifan Daxue Chubanshe. English summary, pp. 334-335.

Figure 1

Professor Jöran Friberg, of the Department of Mathematics, Chalmers University of Technology, Gothenburg, Sweden (friberg@math.chalmers.se), suggests another possible diagram:

A page originated by Alexander Bogomolny gives a large number of proofs of the theorem, and several of these yield possible reconstructions of Liu Hui's diagram.

A correspondent has informed me that my construction is not original - it was discovered by Benjir von Gutheil, who died in 1914, as can be seen on this page from W. Lietzmann, Geometrische Aufgabensammlung, Ausgabe B, Leipzig & Berlin: Teubner, 1916. Click on the figure to see it enlarged: