# First appearance of the "four triangles and a square" proof of the Pythagorean Theorem

A well-known proof of the Pythagorean Theorem is illustrated in the figure below:

This figure shows a square with side lengths $$a + b$$, dissected into four right triangles (each with area $$\frac 12 ab$$) and a square (of side $$c$$). A basic area computation shows that $$(a + b)^2 = 2ab + c^2$$, from which one can conclude that $$a^2 + b^2 = c^2$$.

It's notable that the proof corresponding to this diagram is not the same as the one found in Euclid's Elements (Book I, Prop. 47). Hence my question: What is the first known appearance of this figure in print? According to the Wikipedia article on the Pythagorean theorem,

English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements, and mentions the proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof.

This would seem to suggest that Bretschneider (d. 1878) and Hankel (d. 1873) published, or at least described, this diagram somewhere in their writings. However, this claim falls short of saying that this proof (or the diagram) originates with them. How far back can this diagram and its corresponding proof be traced?

• I have always thought that this proof was attributed to Bhaskara II in the 12th century.
– nwr
Commented Aug 29, 2023 at 2:59
• The diagram in the proof attributed to Bhaskara is not exactly the same as the one that appears above... Commented Aug 29, 2023 at 3:20
• Of course this proof is not in Euclid. Area is only done much later in Euclid. Euclid's proof is a "tour de force" using as little machinery as possible. Proposition I.47 is the "climax" of Book I. Commented Aug 29, 2023 at 16:46
• @GeraldEdgar: However, we must not forget that there is a 2nd. proof of the Pythagorean theorem in Eucl. VI-XXXI. Commented Aug 29, 2023 at 22:03

I suggest that you take a look at page 49 of the 2nd. edition of "The Pyhagorean Proposition" by Elisha S. Loomis: this is the first book at which one must take a look when trying to unravel matters related to the many different proofs of Eucl. I-XLVII.

Basically the author says that the proof in which you are interested can already be found on vol. I, p. 159 of a publication entitled "Mathematical Monthly" and that it is attributed there to a Rev. A. D. Wheeler of Brunswick, Me. It appears that the said Monthly was a short-lived journal that circulated sometime between 1858 and 1861.

Finally, one can infer from the notes in that page of Loomis's book that the very diagram in question also appears in T. Simpson's "Elements of geometry" (London, 1760, p. 27).

• Thank you for the citation to Simpson's Elements. The diagram appears on p. 27 but the use of the diagram to prove the Pythagorean Theorem is found on p. 33. Commented Aug 29, 2023 at 13:03

This will need confirmation (as there might be an even earlier version), but off the top of my head, I believe the earliest "visual proof" of the Pythagorean theorem was demonstrated in the Chinese Zhoubi Suanjing or Chou Pei Suan Ching (周髀算經), which is roughly dated around c. 100 B.C.E.- c. 100 C.E. Although others claim it's dated between c. 200 B.C.E - c. 200 C.E. In it you'll find the Gougu Theorem (AKA Chinese Pythagorean Theorem), which is similar to the visual proof that you're describing.

• If I understand correctly, the author is not asking about the oldest "visual proof" of the Pythagorean proposition but about the first appearance in print of the proof based on the diagram he shared. On the other hand, even though the diagram in the proof of the Gougu Theorem may be related to the diagram under discussion, they are a wee bit different from each other. Commented Aug 29, 2023 at 12:58
• This diagram is clearly in the same spirit, but it is somewhat over-specific, in that the triangle in question is explicitly a 3-4-5 right triangle. I am looking for an example that has the generic quality of the one in my post. Commented Aug 29, 2023 at 13:04
• does this work use the word Behold? Commented Sep 1, 2023 at 23:49
• @Bortolossi, no, "behold" was stated in the Bhaskara II version. Commented Sep 2, 2023 at 0:41