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.
The Jiuzhang suanshu (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, 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.
The third-century commentator Liu Hui [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 which 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.
Notes
[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,
'seven ingenious pieces') see e.g. [Gardner 1974].
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
science3, 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 mathematica6,
164-188.
Wu Wenjun
(ed.) 1982.
'Jiuzhang suanshu' yu Liu Hui [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 maintained
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:
(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:
, and
the longer leg is 4 chi, what is the hypotenuse? 
[2]
gives the following explanation for the first algorithm above:
,
'seven ingenious pieces') see e.g. [Gardner 1974].
(ed.) 1963.
[Ten mathematical classics]. 2 vols. Beijing:
Zhonghua shuju.
[The history of mathematics in China]. Beijing:
Kexue Chubanshe; reprinted 1981.
(ed.) 1982.
[The Jiuzhang
suanshu and Liu Hui]. Beijing: Beijing Shifan Daxue
Chubanshe. English summary, pp. 334-335.
