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Text Box: Another method was to take a string with twelve cubits marked out on it, and stake it out in a triangle with three cubits on one side, four on another, and five on the other. Of course the unit of measurement could be anything...a cubit, a foot, a meter, an inch, a yard...it was the relative lengths of 3 by 4 by 5 that resulted in a right triangle. But there is a property of this 3-4-5 proportion that makes it even more curious. Take the two smaller sides and square their lengths: 3x 3 = 9, and 4 x 4 = 16. 9 + 16 = 25, or 5 x 5. Another way to say this is that if you make a square out of each side, and add the areas of the two smaller squares, you get the area of the larger square.
Pythagoras found that this held, not just for the 3 by 4 by 5 triangle, but for any right triangle. He started with a what was just a useful tool and discovered a fundamental rule of nature. What the Pythagorean Theorem, also called the 47th Proposition of Euclid, says, is that for any right triangle, that is, any triangle containing a 90-degree angle, the square of the "hypotenuse," the longer side, equals the sum of the squares of the two shorter sides. Today over a hundred ways have been found to prove this proposition. To explain any of them requires drawing diagrams, which I can't do in this setting, but all these proofs arrive at that moment of epiphany when the pieces come together like a jigsaw puzzle, and I can't help thinking of the day 2500 years ago when the puzzle was first solved. This morning Pythagoras woke up in a world of chaos, variety and inexactness, but now the universe has changed, and Pythagoras has caught nature red-handed in the act of displaying order and following rules. Imagine a caveman looking at the Astrodome, imagining it is just a big hill, then going inside and suddenly understanding the architecture that holds it up. Pythagoras had peeked under the veneer of the universe, and found that space had a kind of architecture, and that architecture was made of numbers. To us, looking at this from the vantage point of a couple of thousand years later, The 47th problem might seem a little less dramatic. It is, after all, just another one of the laws of nature. We have to remember that to the Pythagoreans, it was a new and wonderful thing to find that there were any laws of nature. Even now, we can't explain why space fits together this way, we're just so used to seeing it that we tend to overlook the implications of a world ruled by numbers. Now, it was possible to use Geometry to make predictions, not just on paper, but in the field. You could indirectly tell the length of something it was impossible to measure directly. If you knew the lengths of two sides of a right triangle, you could predict the length of the third, and always be right. The world obeyed numbers, not at random times, but always. Armed with this insight, Pythagoras taught that numbers were even more real than the world they described. He uncovered the basis of music theory when he found that you could pluck a string to make one note, then divide the string exactly by two, and pluck it to make the note one octave higher. By dividing the string length exactly by three, four and five, notes were produced which harmonized with the first. To the Pythagoreans, they were discovering a divine language of pure mathematics. To us, they were discovering that the universe could be described, predicted, and understood. Pythagoras supposedly was so inspired by the discovery of what we now call the 47th Problem of Euclid that he sacrificed a hecatomb, a hundred oxen, to the Muses in gratitude. If so, I would think he got off cheap. This single discovery has echoed through history. The entire science of trigonometry is based on it. Mapmaking, astronomy, architecture, even space travel would be impossible without it. The English political philosopher Thomas Hobbes, whose Leviathan was one of the most important books of the seventeenth century, probably would never have achieved the fame he did if he had not, at the age of 42, glanced at a copy of Euclid's Elements in a friend's study, opened to the 47th proposition. Hobbes was supposedly so shocked by the implications of the theorem that he exclaimed, "By God, this is impossible!" This single revelation apparently motivated Hobbes to a fevered lifelong study of geometry, and later physics, philosophy, and political science. It has been speculated that any civilized race will have at some point in its history discovered the Pythagorean Theorem. As I was writing this talk I suddenly remembered years ago reading a book by Pierre Boulle, who also wrote The Bridge on the River Kwai. The book I remembered was called Monkey Planet, and was made into a movie called "Planet of the Apes." In the book, an astronaut travels to a far-away planet ruled by a race of intelligent chimpanzees. To complicate matters, there are humans on the planet, but they are brute animals, like apes are here. The chimpanzees, speaking their own ape language, are unable to make sense of our hero's French, and consider him also a brute until he draws for them a diagram of the 47th proposition of Euclid, whereupon they realize he is a civilized, intelligent being, like themselves. As it has been passed down through the ages, this theorem has grown from a useful geometric principle into a symbol of the harmony of the universe. The Greek historian Plutarch, who lived in the first century AD said that the 3-4-5 triangle had become a symbol for the Egyptian gods, Osiris, Isis, and Horus, reminding us of the manner in which the Christian Trinity is sometimes represented by an equilateral triangle. We are told in the Monitor that the