Behold!

[Colour picture which says it all]

The above picture is my favourite proof of Pythagoras' theorem. Filling in the details is left as an exercise to the reader.

Disclaimer: I have learned quite a bit about this and other proofs of the Pythagoras theorem since last time I edited this page. I now know that much of what you read below is wrong or misguided. Until I can find the time to improve the page, you should read this with a skeptical eye. (Always good advice anyhow.) It's not all wrong, of course. But to give just one example of the wrongness, the Chou pei suan ching and Zhoubi suanjing are one and the same: They are just transliterations of the Chinese phrase 周髀算經 in, respectively, the Wade–Giles and the pinyin systems of transcription (the pinyin version should be zhōubì suànjīng, really).
(Disclaimer added 2005-07-21; it may still take weeks for me to get around to a major overhaul of this page.)

Is this the oldest proof?

This proof is sometimes referred to as the Chinese square proof, or just the Chinese proof. The righthand picture above appears in the Chou pei suan ching (ca. 1100 B.C.E.), for the special (3,4,5) pythagorean triple. See also Development of Mathematics in Ancient China.

According to David E. Joyce's A brief outline of the history of Chinese mathematics, however, the earliest known proof of Pythagoras is given by Zhoubi suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) (c. 100 B.C.E.-c. 100 C.E.)

In the The MacTutor History of Mathematics archive there is a section devoted to Chinese mathematics. The overview section at that section also mentions the Zhoubi suanjing and its proof – and a bit of controversy over whether it really is a proof as well.

I have been told that this proof, with the exclamation `Behold!', is due to the Indian mathematician Bhaskara II (approx. 1114-1185). A web page at the Aurora University's Mathematics department attributes a slightly different proof, together with the "Behold!" exclamation, to Bhaskara, and refers to the present proof simply as the proof without words.

The controversy over who had the first proof will probably last forever. Part of the reason is that the notion of what is considered proof changes with time. Hence, rather than obsessing over who was first, let us instead throw away our prejudices and marvel at the ingenuity and analytic abilities of our distant ancestors.

Note added 2002-11-15: Today I attended a very interesting lecture by Jöran Friberg (retired, formerly at Chalmers University of Technology in Göteborg, Sweden).

Friberg, a leading authority on Babylonian mathematics, presented convincing evidence that the old Babylonians were aware of the Pythagoras theorem around 1800 B.C.E. In the clay tablets from the time one finds many examples of calculations of a geometric nature which depend heavily on Pythagoras. Moreover, they had a geometric proof of the algebraic identity (a+b)2=a2+b2+2ab which is essentially obtained by contemplating the left picture above. (Of course, they did not write it algebraically as I did here, but thought of the squares as real geometric objects, and also 2ab as two a×b rectangles.) They were also very adept at generalizing from known results and computing areas by moving bits around to arrive at better known areas, so there is little doubt that they could have found the above proof. Friberg is convinced that they did, though there is no firm evidence of this. It should be recognized that the Babylonians had no concept of axiomatization and abstract proof as we know them from Euklid. Instead, they were absolute masters at all kinds of practical calculations.

Friberg also presented evidence of Babylonian influence on Greek mathematics (indirectly, via the Egyptians). Among other things, Pappus has a simplified version of Euklid's proof of the Pythagoras theorem which seems influenced by Babylonian methods - although he, like Euklid, uses shear transforms to distort rectangles to parallellograms of equal area and back, which is very un-Babylonian.

Disclaimer: The above is just my interpretation of what Friberg told us in the lecture. I may well have misunderstood or misrepresented some points, so don't blame him if I wrote something blatantly wrong. (Later, Friberg told me that I have indeed not represented him accurately in all respects, but he did not elaborate. So take the above with quite a large grain of salt.)

There is a whole web page entitled Pythagoras's theorem in Babylonian mathematics at the The MacTutor History of Mathematics archive.

Note added 2002-12-03: I finally got around to reading the August 2002 issue of the AMS Notices, which has an interesting article (PDF format) Learning from Liu Hui? A Different Way to Do Mathematics by Christopher Cullen. One interesting reference therein: A proof of the Pythagorean Theorem by Liu Hui (third century AD) by Donald B. Wagner.

Other pointers

Dead links

The following documents, which I used to refer to, have moved and not reappeared where I have been able to find them.


There is a PostScript version of the picture too.

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Harald Hanche-Olsen Last update: 2005-07-20 22:10 UTC