MELISSA BLOCK, host:
This is ALL THINGS CONSIDERED from NPR News. I'm Melissa Block.
ROBERT SIEGEL, host:
And I'm Robert Siegel.
A squared plus B squared equals C squared is one of those landmarks' life-changing truths one absorbs in high school mathematics. The numbers three, four and five never looked the same once you've taken on board this wisdom. That in a right triangle whose shorter legs are A and B, the sum of their squares is equal to the square of the largest leg, the hypotenuse, C.
Well, for Robert and Ellen Kaplan, husband and wife, this formula is not just a watershed in the history of mathematics, it is such rich material, they've written an entire book about it, "Hidden Harmonies: The Lives and Times of the Pythagorean Theorem." And the Kaplans join us from Amherst, Massachusetts.
Welcome to the program.
Mr. ROBERT KAPLAN (Mathematician): Thank you very much. What a pleasure to be on.
Ms. ELLEN KAPLAN (Mathematician): Delighted to be here.
SIEGEL: And you write the Pythagorean theorem is an ancient oak in the landscape of thought. How important a milestone, first, Robert Kaplan, is the Pythagorean theorem?
Mr. KAPLAN: It's the foundation of all our navigation in and beyond the world. 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.
(Soundbite of laughter)
Mr. KAPLAN: It's the essence of our navigating.
SIEGEL: Wow. Ellen Kaplan, do you remember learning the Pythagorean theorem yourself?
Ms. KAPLAN: Yes. As people used to in schools of that era, you're just told A squared plus B squared equals C squared. Shut up. And 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.
SIEGEL: Now, first subject, authorship of the theorem. And we should point out first, who does not have any claim to authorship of the Pythagorean theorem?
Ms. KAPLAN: Well, Pythagoras.
SIEGEL: That would be Pythagoras.
(Soundbite of laughter)
Ms. KAPLAN: That's Pythagoras. Well, it's very hard to tell about Pythagoras. He could have been down a hole and doing it and just not telling anybody.
Mr. KAPLAN: There was a - or I should say is a contemporary statistician named Robert Stigler, who in 1961 formulated Stigler's law of eponymy. No theorem of any importance is named after its actual founder, but after someone else. He calls it Stigler's theorem because, as he points out, it's not his.
(Soundbite of laughter)
SIEGEL: It's like all Cretans are liars that you're saying.
Mr. KAPLAN: Exactly. Pythagoras was a shaman. That's a sixth century B.C. in the southern part of Italy. He'd come from the island of Samos and founded a colony where his followers broke up into two groups, the hearers, the akousmatikoi; and the knowers, the mathematikoi. And it was among those that the theorem arose, and, of course, Pythagoras took or was given credit for it.
SIEGEL: Ultimately, we have to read - if we're going to read about the Pythagorean theorem, we must read about Euclid. I assume his role is somehow central here.
Ms. KAPLAN: Not only is his role central, I think his role is creative because he is the person who was actually trying to take all of these handymen's three-four-five 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.
SIEGEL: Now, I should note here, we've made mention of the three-four-five Pythagorean triplet. It was my - this was Siegel's theorem. When I was...
(Soundbite of laughter)
SIEGEL: ...a math tutor in high school, that the college board on the SAT only cares about the three-four-five, the five-12-13 and the isosceles right triangle. If you got those down pat, you're OK. That's worth two or three questions right there.
(Soundbite of laughter)
Mr. KAPLAN: Those are the prominent ones. Of course, it turns out there are many, many, infinitely many combinations of numbers such that A squared plus B squared equals C squared - whole numbers. Or in that isosceles case that you've mentioned, one squared plus one squared, if the two sides are one and one, gives you, for the hypotenuse, not a whole number. It's going to be the square root of two, which is a real thorn in the side of mathematicians of that day, because the square root of two wasn't a rational number, a ratio of two whole numbers.
Ms. KAPLAN: And there's just wonderful confusion between ratio and rational. And so if you have something which you think is rational thought, but it's not a rational number, that's very difficult.
SIEGEL: This actually introduces one of my favorite moments in your book, the proof by the ancient skeptics that the square root of two is not some fraction that we could figure out if we just carried it out to enough decimal places or whatever. It is an irrational number.
Mr. KAPLAN: The dramatic story, perhaps it's not true, but it seems as if there was one member of the Pythagorean community named Hypsous(ph). Hypsous showed that if you take a right triangle whose legs are one and one, 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 roughly 14/10, but exactly, well, exactly it will never be a ratio of whole numbers. It will be irrational. The Pythagoreans, so the story goes, thanked him so much for his proof, asked him to step to the edge of the nearest cliff and off he went.
(Soundbite of laughter)
SIEGEL: So much for irrationality in...
Mr. KAPLAN: Yes.
SIEGEL: ...the view of the Pythagoreans. By the way, I was reminded in reading your book that when I went to high school, I spent a year doing plane geometry. It was, I think, my favorite year, and that the course just seemed to be pure problem solving and pure reason, if you will. Do kids today still do a year of doing plane geometry or - I thought my kids passed over it much more quickly than that.
Ms. KAPLAN: This is one of the tragic effects of deciding that calculus is the pons asinorum. People have said, oh, there's no point in geometry because you can just get some artificial device that will tell you that these things - lengths are the same or the angles are the same.
And as for proving things, there's really no need to prove something that's been proved once before. So young mathematicians are being brought up in a style of thinking, which is purely memorization of a practical definition.
Mr. KAPLAN: Which is why 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. We start with 4-year-olds. We go on through 70-year-olds.
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.
SIEGEL: Well, Robert and Ellen Kaplan, thanks so much for talking with us today.
Mr. KAPLAN: Thank you so much, Robert.
Ms. KAPLAN: Thank you very much.
Mr. KAPLAN: It's been a real pleasure.
Ms. KAPLAN: A delightful conversation.
SIEGEL: The Kaplans are the authors of "Hidden Harmonies: The Lives and Times of the Pythagorean Theorem."
(Soundbite of song, "The Square of the Hypotenuse")
Unidentified Group: (Singing) The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides.
SIEGEL: And here's a note for all those of us who are Latin-impaired. Ellen Kaplan referred to calculus as having become a pons asinorum. That translates to the bridge of asses or the bridge of fools, and it refers to one of Euclid's basic math problems - his first, really tough one - a pons asinorum is a test that will frustrate and stymie the inexperienced and allow only the skilled to continue, in this case, into the more advanced mathematical proofs.
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