It’s hard to imagine now, but the Republican Party used to have some very intelligent men. If you go back roughly a century and a half, you might find an example. Or as Lincoln might say, seven score and three years ago.
While serving his seventh term as a congressman from Ohio, James Garfield published a proof of the familiar Pythagorean theorem, which relates the longest side (the hypotenuse) of a right triangle to its shorter sides.
No one could remember having seen such a proof before. That’s what the editors said of Garfield’s proof in the April 1, 1876 issue of the New-England Journal of Education. Five years after that, Garfield was President of the United States.
When Garfield proved it, the Pythagorean theorem was a bona fide theorem, unlike the Fermat conjecture, which was still more than a century away from its first proof by Andrew Wiles in 1994.
It is often worthwhile in math to prove things that have already been proven. For example, an elementary proof of the Fermat conjecture would be much appreciated, since every page of the Wiles proof is completely over my head.
The Pythagorean theorem states that in a right triangle (a triangle with one angle of 90 degrees, or π/2 radians, or 100 grads), with the lengths of the shorter sides being a and b, and the length of the longest side being c, the following equation is true: a² + b² = c².
For example, given a triangle with sides measuring 3 units, 4 units and 5 units respectively, we see that 3² + 4² = 5² = 9 + 16 = 25.
The theorem was first proven, if not by Pythagoras himself, by one of his students.
Or, for all we know, it may have been proven even earlier by someone in Africa or Asia. It’s also possible someone in the Americas proved it long ago, though it is perhaps less likely that any evidence has survived.
In the 12th Century, Bhaskara came up with a proof that starts out a lot like the Pythagorean proof, and it also ends up with four equal triangles surrounding a square to produce a larger square.
Each side of the inner square is also the hypotenuse of a triangle, thereby proving the theorem.
Both are simple proofs which anyone who has honestly made it through a geometry class should be able to understand. What more could we want in a proof?
For one thing, it might seem inelegant to have to duplicate the initial triangle at all. If we try to find another way to prove the theorem, we might come up with a far more elegant new proof.
Or it might just be new but no more elegant than existing proofs. Or we could just wind up merely reaffirming the standard proof, but even that outcome is worth the effort, I think.
Garfield’s proof is no more elegant than Bhaskara’s proof, but it was original, and it is perhaps better for students who are more algebraically-oriented than geometrically-oriented.
Like almost everyone else who has proven the Pythagorean theorem, Garfield starts out by constructing a right triangle. It’s a logical way to begin the proof.
The only requirement is that the triangle has one right angle. The other two angles may both be 45 degrees (π/4 radians or 50 grads) and thus the two shorter sides are equal in length.
However, Garfield’s proof is more convincing, in my opinion, if the narrower angles and shorter side lengths are distinct.
Then we duplicate the triangle, and place the copy so that side a of the original aligns with side b of the copy, or side b of the original aligns with side a of the copy.
The important thing is that we have one side of the original and one side of the copy (neither side the hypotenuse) forming a continuous line segment of length a + b.
This kind of looks like a trapezoid. It just needs one line to complete the trapezoid.
So far this looks like half of Bhaskara’s proof (see Figure 0 above). But from this point forward, Garfield’s proof involves quite a bit more algebra than Bhaskara’s proof.
The area of the trapezoid is ((a + b)²)/2. The trapezoid is made of two right triangles that have c as their hypotenuse, and one isosceles triangle that is also a right triangle.
The isosceles triangle has c for both of its shorter sides, and its own hypotenuse that we’re not concerned with right now.
However, the area of the isosceles triangle is calculated easily enough: it’s half c². The area of the original triangle is ab/2, so the combined area of the original triangle and its copy is ab.
This gives us the following equation: ((a + b)²)/2 = ab + c²/2. Multiplying both sides by 2, we get (a + b)² = 2ab + c². Using FOIL (First + Outer + Inner + Last), we verify that (a + b)² = a² + 2ab + b².
So our equation now stands as a² + 2ab + b² = 2ab + c². Hopefully at this point it’s obvious where Garfield is going with this.
Subtracting 2ab from both sides, we obtain a² + b² = c², thus completing the proof.
So Garfield’s proof still requires the duplication of a triangle, and a heck of a lot more algebraic rewriting than either the Pythagorean proof or Bhaskara’s proof.
It is nevertheless an intellectual accomplishment, of the sort we should not expect from any Republican politician today.
Maybe Garfield could have accomplished a lot as president. He did make it past the 100-day mark, but on July 2, 1881, a lawyer who was upset that Garfield had denied him a diplomatic post shot the president.
The assassin declared Vice President Chester A. Arthur (R) was now president, but Arthur didn’t take over right away.
Garfield spent the next few weeks hoping someone could find the precise location of a bullet in his lower back, in order to remove it.
Alexander Graham Bell, who received a patent for the telephone the same year Garfield’s proof of the Pythagorean theorem was published, tried to find the bullet with his metal detector prototype. Unfortunately Bell failed and Garfield died on September 19, 1881.
While it is important to remember that Garfield denied his eventual assassin the post of U. S. Ambassador to France, we should also remember that he appointed Frederick Douglass to the post of Recorder of Deeds for the District of Columbia.
It’s one of those tidbits about Garfield that tends to be forgotten right along with his proof of the Pythagorean theorem.