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 `pageok` [Eugene Volokh, July 12, 2006 at 3:32pm] Trackbacks More Presidents: Yes, but which President is credited for having come up with an original proof of the Pythagorean Theorem? James Garfield — what, you thought it was Thomas Jefferson? And he apparently did it while serving in Congress, yet. Why don't we have Congressmen like that these days? (link) anonyomousss (mail): where can we find the proof? 7.12.2006 4:50pm (link) David Matthews (mail): Glad you asked! Here's a video clip I made of it (requires DivX codec -- I'll also post up a quicktime format one....) Proof 7.12.2006 4:53pm (link) David Matthews (mail): Here's the MUUUUCH larger QuickTime Version (like 5+ megs): QuickTime Proof 7.12.2006 4:55pm (link) John Armstrong (mail): I never did like this "proof". Firstly, the trapezoid in question is half the square in a much older proof. Secondly, arguing the areas of the figures from analytic geometry (the essential source of the formulae) makes me very leery that the Pythagorean theorem was not used as justification of the methods themselves, thus begging the question. 7.12.2006 5:03pm (link) Adrienne (mail) (www): Garfield also had the ability to write in Greek with one hand while writing in Latin with the other. 7.12.2006 5:40pm (link) David Matthews (mail): "Firstly, the trapezoid in question is half the square in a much older proof." That's my problem with it. "Secondly, arguing the areas of the figures from analytic geometry (the essential source of the formulae)...." Actually, no. These areas can be found in an entirely coordinate-free (non-analytic) context, so there's no danger of a hidden use of a "distance formula." It's easy to show that the area of a triangle is 1/2 the area of a circumscribing rectangle, and a trapezoid is in general just the sum of a rectangle and at most two triangles. But I share your dislike of this proof, in spite of having animated it.... 7.12.2006 6:00pm (link) David Matthews (mail): "a trapezoid is in general just the sum of a rectangle and at most two triangles." I'm wrong about that, come to think of it. Anyway, this special trapezoid is luckily just the union of one rectangle and one triangle.... 7.12.2006 6:05pm (link) jimbino (mail): So who was the last president fluent in both English and a modern foreign language? And who was the first president to be circumcised? 7.12.2006 6:20pm (link) Walt Quist (mail): I could not get the mathworld link to work - it timed out. Not sure why Eugene uses the localhost links since they don't work on another pc. I agree that they don't make congressmen like that anymore. Some of us were pretty good at math in high school and college but never came up with a proof for anything. I never learned to write Greek or Latin with either hand. The old guys got a much better education than I did in the 40's and 50's and MUCH better than today. 7.12.2006 7:29pm (link) The Divagator (mail) (www): So who was the last president fluent in both English and a modern foreign language? If I had to guess (assuming that fluent means fluent and not the piss-ant phrase-book stuff some of them affect), I would say either FDR or Hoover. George HW Bush and Nixon are maybes, but I don't recall their fluency in a foreign language. Does anyone know the answer? 7.12.2006 9:01pm (link) AppSocRes (mail): While he was a student Napoleon discovered and proved a very significant geometric theorem, which is named after him. I first ran across Garfield's proof about ten years ago in a little book called The Most Beautiful Mathematical Formulas. I was impressed by the relative clarity and brevity of the proof then and still am now. Garfield's proof of the Pythagorean Theorem may be the most elegant I've ever seen. I majored in math and have a pretty good background in geometry: I've waded through a good bit of Euclid's Elements and Hilbert's analysis and reformulation. Is some of the prejudice against Garfield's proof just because it is partially algebraic and hence not in the spirit of Euclid or does it derive from some latent prejudice against a succesful Republican politician of the gilded age? 7.12.2006 9:29pm (link) Charlie (Colorado) (mail): Depends on what you mean by "fluent" but GWB can conversationally ad lib. I guess there's some question whether texmex Spanish is a "modern" language, as it preserves significant locutions from the 16th c. 7.12.2006 10:05pm (link) Glenn W Bowen (mail): his home is in my home town, Mentor, Ohio. Mrs. Garfield, a granddaughter-in-law, lived behind the place in a little colonial house. I remember his pistols and saber from the civil war being on display. 7.12.2006 10:52pm (link) Frank Drackmann (mail): I bet President Garfield would be pissed if he knew his major legacy was a cartoon about an annoying fat cat. 7.13.2006 9:48am (link) David Matthews (mail): "Is some of the prejudice against Garfield's proof just because it is partially algebraic and hence not in the spirit of Euclid..." I think the biggest prejudice is the one that John Armstrong mentioned, above, that it's really just a needless complication of the more famous, much older, and more elegant "three squares" proof. The algebraic part is merely a convenience. It could easily be rewritten completely in terms of Euclid's postulates, but that would make it a lot harder for us modern folks (meaning Garfield, as well) to follow. But it still seems to me to be a much more productive use of chamber time in the House of Representatives than what we currently get from our legislators. 7.13.2006 11:17am (link) Mikeyes (mail): Theodore Roosevelt was fluent in German, French and Dutch. He was able to converse in Dutch with a famous South African hunter on his post presidential hunting trip and with the Kaiser in German on his return to Europe and then the US. But that is probably not the answer. 7.13.2006 11:50am (link) jimbino (mail): Right Mikeyes, I think TR was the last American president who could converse in a living foreign language. It's embarrassing to be associated with the American flag. I don't think TR was circumcised. 7.13.2006 12:14pm (link) Traveler: EV asks: "Why don't we have Congressmen like that these days?" Perhaps if we elected more Congressmen and Presidents who are graduates of the nation's finest liberal arts college, rather than those educated in the stifling atmosphere of the Ivy League... 7.13.2006 12:29pm (link) jimbino (mail): How about requiring mastery of English for a B.A. degree, one foreign language for an M.A. and two for a PhD? There seems to be no such thing as an education, let alone a liberal arts education, in America anymore. 7.13.2006 12:50pm (link) Frank Drackmann (mail): Foreign languages are overrated. I learned German by being exposed to it as a child and living there for 6 years, it wasn't a great intellectual accomplishment. 7.13.2006 2:19pm (link) Ken B: Pretty nice proof BUT ... isn't it actually circular? Doesn't the theorem on the area of a trapezoid rely on the PT? 7.13.2006 3:48pm (link) Jack31415 (mail): Jimbino - A PhD in science used to require literacy (at least at the level of reading journal articles) in two second languages and the ability to converse in one of them. I know this because my Dad was quite proud of his (awful) German. He was a Physics PhD from the University of Colorado. Reading journal articles in any of the Romance languages isn't all that hard, given time and a dictionary. I believe that this requirement is dropped now. English fluency appears to be the de facto standard in the sciences. 7.13.2006 6:31pm (link) Andrew Hyman (mail) (www): Cool proof. Garfield rocks. 7.13.2006 8:17pm (link) Andrew Hyman (mail) (www): Incidentally, a special case of the Pythagorean Theorem is for a right isosceles triangle. The square of the hypoteneuse equals the sum of the squares of the other two sides. This special case is extremely easy to prove: just take a square, draw a big "X" inside connecting all four corners, and calculate the area of the square two different ways. I wonder how long this special case of the Pythagorean Theorem had been proved prior to the first proof of the full-blown Pythagorean Theorem. It also may be that either the special case or the general result were known empirically before they were actually proved. An interesting historical question, methinks. 7.13.2006 10:45pm (link) Frank Drackman (mail): I always liked the "FarSide" cartoon of this scientist working at a chalkboard on a massive proof. The board is filled with equations and symbols and down toward the bottom it says "And then a miracle happens" before it goes on with the equations. 7.14.2006 9:39am `pageok` `pageok`