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© Jeff Matthews entry June 2010

No Wonder a^{2}+b^{2}=c^{2}
was Greek to me !

—In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

(the Pythagorean theorem)

**My
plan is to go to Calabria**, the town of Crotone
(the map, below, shows the province of Crotone. Find the
'e' at the end and walk down to the town. It's at
water's edge). Specifically, I want the high school, the
*Liceo Ginnasio Classico Pitagora* at piazza
Umberto I, 15. I intend to offer to teach their English
classes for a whole week just so I can teach the kids
one of the world's most complicated pun-stories, one
that bears directly on their own town, indeed one that
connects their town and themselves to every kid in the
world who has ever studied geometry. And they won't have
to pay me. I can drive it from Naples in a few hours,
and I 'll cover my own expenses because I've got this
great idea for a t-shirt.

**T**his is the story. It
was my high-school geometry teacher's favorite. We were
grade-bound to laugh long and loud (no groaning, please)
every time he told it. We wound up with pretty good
grades and really sore diaphragms. For the politically
correct, I realize that the use of the S-word is
potentially offensive. I do know that the word *squaw*
derives from the eastern Algonquian languages and
appears in various spellings to mean 'woman' and has
often been generalized incorrectly and unfairly. I
remind you that this is just a joke, for Pete's sake.
For the zoologically correct, I do know that there are
no hippopotamuseseses in North America. I remind you
that this is just a joke, for Pete's sake. For the
Ethical-Treatment-of Animally-correct —please get a
life; I got the hides from Teddy Roosevelt. This is just
a joke, for Pete's sake.

Once upon a time, there were three pregnant Indian squaws. The first squaw was sitting on a buffalo hide, the second squaw on an antelope hide, and the third squaw on a hippopotamus hide. The first squaw gave birth to a baby boy, as did the second squaw; but the third squaw, the one on the hippo hide, had twin boys. So, what's the moral of this story?The squaw... on the hippopotamus... is equal to the... sons of the squaws... on the other two hides!

(He always spoke the last line slowly, dots and all, just to make sure we got it. He also leaned to the side a bit to show the italics.)

Crotone is a great little townthat you should visit even if you hate math; it has great beaches, a harbor and a current population of about 60,000. As long as you're there, however, know that it had a glorious presence as part of Magna Grecia. They had great athletes, famous philosophers, fine doctors and, best of all, this is where Pythagoras (image, above right) had his academy, founded in 510 BC. It was a monastic-like, ascetic order, dedicated to numbers—especially the relationshipsbetweennumbers. It all led eventually to the science of mathematics, the powerful tool for describing and predicting natural phenomena. They were so intent on serious study that they got steamed at nearby Sybaris, the citizens of which were notoriously luxuriant, flabby and given to good times. Pythagoreans sent Crotone's great wrestler, Milo, over with an army of goons and wiped out the weak and meek Sybarites.

We can forgive them. The

-naissancepart ofrenaissancewas Greek: Thales, Anaximandros, Heraclitos and a legion of others were worthy of the name "scientist," as we understand that word today: one who knows from observation and investigation. "Science" is related to the word "scissors"; a scientist therefore cuts and divides. The Greeks were the first who tried to cut and divide, figure out and label the universe on a natural rather than supernatural basis. Pythagoras was right in the middle of them, and of all the figures of early Ionian science, none had more influence on future generations than did he.

The Pythagoreans were fascinated, for example, by the mathematical relationship between the frequencies of musical tones. The interval of an octave (from a C to the next C above it=1:2) and that of a fifth (C to the next G=2:3) were considered the most harmonious and pleasing. The fact that a cube has 6 faces, 8 corners and 12 edges, numbers which are in the same relationship as those musical intervals, conferred "geometrical harmony" on the cube and made it special. They loved "perfect" numbers: those which were equal to their separate factors added together. For example, 6 (because 6=1+2+3). 28 is another (1+2+4+7+14=28). (Quick! What is the next such number? If you can figure that one out, see if you are still in time to catch the night train to Crotone; some of the old gang might still be around.) Then, they found "amicable" numbers: two numbers, each of which is equal to the factors of the other. 220 and 284, for example, (because the factors of 284 are 1, 2, 4, 71 and 142, which add up to 220, the factors of which —1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110— equal 284!) Pythagoras, himself, is said to have discovered those two numbers. (If you have others of your own, by all means go first-class. But be careful; as might be expected, they play a mean game of Blackjack down there. No matter what Pythagoras does, don't

youtake a hit on 17!)

Pythagoras, of course, is most famous for the geometry theorem which bears his name and which was the foundation of the groaner at the beginning of this brief discourse. Of all the mathematical knowledge attributed to the Pythagoreans, the most important was the realization that not every quantity can be expressed in whole numbers. True, if the shorter sides of a right triangle are 3 and 4, then the hypotenuse is 5 (3+4^{2}=5^{2}; that is, 9+16=25), a fact that delighted the Pythagoreans. But, if the shorter sides are, say, 4 and 5, then the hypotenuse is not a whole number, but winds up as 6.4031242... , an unwhole number if ever there was one. This bothered the Pythagoreans and later generations of mathematicians no end. It doesn't bother me. Their love of beauty and symmetry and their love of numbers led members of the Pythagorean school to some important views about the nature of the universe. One disciple of Pythagoras, Philolaos, even suggested that the earth was a planet and, like all other planets, was in orbit around a "central fire" in the middle of the universe.^{2}

A thousand years were to pass before the fires of knowledge kindled by the Greeks would be reignited, this time by Arab astronomers. My big regret is that the Arabs, the inventors of

al(the)jabr(reunion of broken parts) did not leave a good story, so I might have done better in high-schoolalgebra. Maybe I can get some tips from the kids in Crotone. So, about these t-shirts...

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