a2 + b2 = c2. Remember that from high school math class? That's the Pythagorean theorem, which shows that in a right triangle, where the shorter legs are a and b, the sum of their squares is equal to the square of the longest leg, the hypotenuse, c.
But for husband and wife mathematics team Robert and Ellen Kaplan, there's much more to this equation. In their new book Hidden Harmonies, they write that the Pythagorean theorem is an ancient oak in the landscape of thought.
"It's the foundation of all our navigation in and beyond the world," Robert Kaplan tells NPR's Robert Siegel. "It lets us know that we live on a flat surface, relatively speaking. It allows us to send rockets to the moon and beyond them."
Ellen Kaplan says she remembers memorizing the theorem in school — but that it wasn't until much later that she began to truly appreciate its elegance. "To find that there was a beautiful logic behind it, to find that it spread to all sorts of different realms, not only in mathematics, was just a revelation in the writing of the book," she says.
Another surprising fact about the Pythagorean theorem — it wasn't really discovered by Pythagoras. "Pythagoras was a shaman," Robert Kaplan explains. In the sixth century B.C., Pythagoras set up a colony in the southern part of Italy. It was there that his followers came up with the formula, "and, of course, Pythagoras took or was given credit for it."
Euclid, the Greek mathematician, also played a central role in the formula, explains Ellen Kaplan. "He is the person who was actually trying to take all of these handymen's 3-4-5 rules and other observed relationships and not only make them clear, but to prove that they're true — which is something that no one had done before with mathematics."
Though the 3-4-5 rule is the best known, "there are many, many, infinitely many combinations of numbers such that a2 + b2 = c2," Robert Kaplan says. But it gets more complicated. Kaplan explains that a member of the Pythagorean community "showed that if you take a right triangle whose legs are 1 and 1, the hypotenuse of it by the theorem that this community had come up with would not itself be a whole number, or a ratio of whole numbers." It would be an irrational number.
According to dramatic (though unproven) legend, Kaplan says, the Pythagoreans "thanked him so much for his proof, asked him to step to the edge of the nearest cliff — and off he went."
Now, 2,500 years after Pythagoras' time, it's a lot easier to determine the length of a triangle side, or the degree of an angle. "This is one of the tragic effects of deciding that calculus is the pons asinorum," says Ellen Kaplan (that translates to "the bridge of asses" or "the bridge of fools," and it refers to one of Euclid's basic math problems — a really tough one — a test that would frustrate and stymie the inexperienced and allow only the skilled to continue into the more advanced mathematical proofs.) "People have said, 'Oh, there's no point in geometry because you can just get some artificial device that will tell you these things — lengths are the same or the angles are the same.'"
And as for proofs, why prove something that's already been proven before? "Young mathematicians are being brought up in a style of thinking which is purely memorization of a practical definition," Ellen Kaplan says.
The Kaplans would like to see more do-it-yourself discovery in math education. "Ellen and I began our math circles at Harvard in 1994 to bring young math students back to the way of being discoverers and inventors themselves," Robert Kaplan explains. "In our classes, we just pose a problem and let them go to work on it together — geometry, algebra, calculus, it's all there to be played with. And as Plato says, because we're the playthings of the gods, playing is the most serious of our activities."