THE GOLDEN RATIO: The Story of Phi, The
World's Most Astonishing Number, by Mario Livio, (Broadway Books, New
York: 2002), p.28
“Thus, the square on the hypotenuse
is clearly equal in area to the sum of the two smaller squares. In
his 1940 book The Pythagorean Proposition, mathematician Elisha Scott
Loomis presented 367 proofs of the Pythagorean theorem, including
proofs by Leonardo da Vinci and by the 20th president of
the United States, James Garfield.
“Even though the Pythagorean theorem
was not yet known as a “truth” characterizing all right angle
triangles, Pythagorean triples actually had been recognized long
before Pythagoras. A Babylonian clay tablet from the Old Babylonian
period (ca. 1600) contains fifteen such triples.
“The Babylonians discovered that
Pythagorean triples can be constructed using the following simple
procedure, or “algorithm.” Choose any two whole numbers p and q
such that p is larger than q. You can now form the Pythagorean
triples p2-q2; 2pq; p2+q2.
For example, suppose q is 1 and p is 4. Then p2-q2=42-12=
16-1=15; 2pq=2x4x1=8; p2+q2=42+12=16+1=17.
The set of numbers 15, 8, 17 is a Pythagorean triple because
152+82=172 (225+64=289). You can
easily show that this will work for any whole number p and q.
...Therefore, there exists an infinite number of Pythagorean triples
(a fact proven by Euclid of Alexandria).
