Proofs of Pythagoras // 47

Mnesarchus, was a merchant from Tyre; his mother, Pythais, was from Samos. They may have met when Mnesarchus brought corn to Samos during a famine, and was publicly thanked by being made a citizen.

Pythagoras studied philosophy under Pherekydes. He prob- ably visited another philosopher, Thales of Miletus. He attended lectures given by Anaximander, a pupil of Thales, and absorbed many of his ideas on cosmology and geometry. He visited Egypt, was captured by Cambyses II, the King of Persia, and taken to Babylon as a prisoner. There he learned Babylonian mathematics and musical theory. Later he founded the school of Pythagoreans in the Italian city of Croton (now Crotone), and it is for this that he is best remembered. The Pythagoreans were a mystical cult. They believed that the universe is mathematical, and that various symbols and numbers have a deep spiritual meaning.

Various ancient writers attributed various mathematical theorems to the Pythagoreans, and by extension to Pythagoras ­ notably his famous theorem about right-angled triangles. But we have no idea what mathematics Pythagoras himself originated. We don't know whether the Pythagoreans could prove the theorem, or just believed it to be true. And there is evidence from the inscribed clay tablet known as Plimpton 322 that the ancient Babylonians may have understood the theorem 1200 years earlier ­ though they probably didn't possess a proof, because

........................................... Babylonians didn't go much for proofs anyway.

Proofs of Pythagoras Euclid's method for proving Pythagoras's Theorem is fairly complicated, involving a diagram known to Victorian school- boys as `Pythagoras's pants' because it looked like underwear hung on a washing line. This particular proof fitted into Euclid's development of geometry, which is why he chose it. But there are many other proofs, some of which make the theorem much more obvious. 48 // Proofs of Pythagoras

Pythagoras's

pants.

One of the simplest proofs is a kind of mathematician's jigsaw puzzle. Take any right-angled triangle, make four copies, and assemble them inside a carefully chosen square. In one arrangement we see the square on the hypotenuse; in the other, we see the squares on the other two sides. Clearly, the areas concerned are equal, since they are the difference between the area of the surrounding square and the areas of the four copies of the triangle.

(Left) The square on the hypotenuse (plus four triangles). (Right)

The sum of the squares on the other two sides (plus four

triangles). Take away the triangles . . . and Pythagoras's

Theorem is proved.

Then there's a cunning tiling pattern. Here the slanting grid is formed by copies of the square on the hypotenuse, and the other grid involves both of the smaller squares. If you look at how one slanting square overlaps the other two, you can see how to cut the big square into pieces that can be reassembled to make the two smaller squares.

two page view?

all devices