US President Garfield's Proof of the Pythagorean Theorem

The 20th United States President, James Garfield, came up with a proof of the Pythagorean theorem, while a congressman, thus "proving" at least one exception to Mark Twain's wonderful quote: "First you have your Idiots. Then there's Congressmen. But I repeat myself."

I just found this out toDAY. I found it out by reading Robert Crease's "The Great Equations.", a book I give my strongest recommendation to, especially to my dear and dearest Philosopher friends, Phil Warnell and George Hagel, both of Canada as fate would have it, as Crease is the Department Chair of Philosophy at Stony Brook University.

As George Musser Jr. and Lee Smolin have said, if Physics needs anyone, it's more Philosophers. The reason we need them in Science in General and Physics (greatest of sciences because it is the most fundamental) in particlular, is because Philosophers are THE experts in challenging assumptions. Too much of Science has gone wildcat crazy speculative, and we need more people to reign the wildest ones in, partly because of how whorish Science Writers (other than the best) are; they'll publish any damned fool idea, these days. I mean, it's embarrassing.

The problem is that Philosophers have to raise their Math and Science Knowledge Quotients. Math isn't hard, it just seems this way. Crease knows that, he writes about it. Read, study, and learn. We'll all be better off in the long run.

So anyway, The Pythagorean Theorem's roots are lost in antiquity, but we know the ancient Indians and Chinese knew of it, and the Babylonians knew of it first (so far) as it's on the Plimpton 322 cuneiform tablet, currently on display at Columbia University in NYC.

The COOLEST thing about The Pythagorean Theorem  is that is can be proved in oh so many ways; there are whole books with as many different proofs as the authors can collect, such that Pythagorean proofs, proofs that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the two other sides, are a small Mathematical cottage industry in their own right.

So, here is Garfield's. From this webpage by Angie Head, if you wish to see more. There are many more.

Begin:

The twentieth president of the United States gave the following proof to the Pythagorean Theorem. He discovered this proog five years before he become President. He hit upon this proof in 1876 during a mathematics discussion with some of the members of Congress. It was later published in the New England Journal of Education.. The proof depends on calcultaing the area of a right trapezoid two different ways. The first way is by using the area formula of a trapezoid and the second is by suming up the areas of the three right triangles that can be constructed in the trapezoid. He used the following trapezoid in developing his proof.

First, we need to find the area of the trapezoid by using the area formula of the trapezoid.
A=(1/2)h(b1+b2) area of a trapeziod

In the above diagram, h=a+b, b1=a, and b2=b.

A=(1/2)(a+b)(a+b)
=(1/2)(a^2+2ab+b^2).

Now, let's find the area of the trapezoid by summing the area of the three right triangles.
The area of the yellow triangle is
A=1/2(ba).

The area of the red triangle is
A=1/2(c^2).

The area of the blue triangle is
A= 1/2(ab).

The sum of the area of the triangles is
1/2(ba) + 1/2(c^2) + 1/2(ab) = 1/2(ba + c^2 + ab) = 1/2(2ab + c^2).

Since, this area is equal to the area of the trapezoid we have the following relation:
(1/2)(a^2 + 2ab + b^2) = (1/2)(2ab + c^2).

Multiplying both sides by 2 and subtracting 2ab from both sides we get




concluding the proof.

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