Nasr al-Din al-Tusi's Proof of the Pythagorean
Theorem. Nasr al-Din al-Tusi (d. 1274 AD) was a renowned
Khorasani Muslim Mathematician who reexamined Euclidean geometry
[Khorasan is today's Iran/Afghanistan]. In this plate, one can read, in
Arabic, Nasr al-Din al-Tusi's version of Euclid's proof of the Pythagorean
Theorem. (see the discussion on p. 16 of  for a generalization and
an analogy). It is claimed that the oldest proof goes back to the Chinese from about 3000 years ago (circa 1800 BC; what you
will find in this hyperlinked slide is their version; it is the easiest
proof). The oldest records of 'Pythagorean' numbers are found in clay
tables dating back to the 1600-1800's BC found in Babylon, Iraq; see .
There are many proofs of this joyful fact online like
here and here. In this document from about 900 years ago, we
explain some of the features of "Arab" mathematics and offer a translation
as well. Al-Tusi was not Arab. However, Arabic was the lingua franca of
science in his time. This proof is already available in English
online. It is said that Pythagoras (whose father was `Lebanese' and mother
was `Greek', but spent most of his life and died as a `Sicilian' in
Syracuse) learned his mathematics from the Babylonians. The statement of his theorem was found in
their ancient texts.
It's all Greek to me...
Vatican Copy of Euclid's "Elements" (Pythagora's theorem appears in the middle of the text)
BG, RA, KC [and TA as to make ABGR and CATK into squares]. CAR is a single
line since BAR and BAC are right angles [qaimayn]. This is also the case
[of the line] BAT. From point A, we drop the line AL, parallel to BD. The
angle DBA is obtus. Thus [the angle] BAL is smaller than [the angle] BAC.
The line [AL] cuts the segment BC at the point M. The square [sat.h] BH is
cut into the rectangles BL and LC. We draw the segments GC and AD. In CBG
and DBA, we have two equal [mutasawi] triangles [having two equal sides
and equal angles at B].
Now the square
[with base] BC is equal to the [sum] of the squares [with bases] BA and
AC, and this is what we wanted to show. QED.
Al-Nairizi's Proof of the Pythagorean Theorem Using Tiling (900 AD)
Musamma [Definition]. Consider the trapezoid ABCD. Under such a construction, the segment BD is said to be "of the same base AB of the parallels AD, BC."
Lemma. If BD is of the same base AB of the parallels AD, BC, then
Proof. We calculate
Area (BCD) = Area of Trapezoid - Area (ABD) = (a b) + (a c)/2 - a (b+c)/2 = (a b)/2 = 1/2 Area Rectangle (ABCE).
 See for example the beautiful exposition in the book "Patterns of Plausible Reasoning", Volume I: Induction and Analogy in Mathematics, by G. Pólya, Reprint of the 1954 original. Princeton University Press, Princeton, NJ, 1990, pp. 15-17. In The Random Walks of George Pólya (published by Mathematical Association of America, Washington, DC, 2000), Gerald L. Alexanderson recounts many wonderful anecdotes about Pólya. Being very proud of his library, it is mentioned that he once showed two translations of "Patterns of Plausible Reasoning", one in Arabic and the other in Hebrew. Said Prof. Pólya, "This is, how you make peace." Pólya was of Hungarian Jewish background whose family converted to Catholicism at the turn of the previous century ("Random Walks" alludes to a question of acceptance in Europe for Jews once they converted to Christianity--in `those days', the world was divided into believers and infidels). Pólya had a beautiful mind and a kind demeanor.  Key websites on Pythagoras many incarnations... The Chinese and the Babylonian connections date to somewhere in the 1600-1800 BC, though a key Chinese record in the form of the Zhou-Bi Suàn-Jing (the hypothenus diagram) dates from the 3rd Century AD