A Mathematician for President

[Image courtesy of the Images of American Political History.]

In 1876, a politician made mathematical history. James Abram Garfield, the honorable Congressman from Ohio, published a brand new proof of the Pythagorean Theorem in The New England Journal of Education. He concluded, “We think it something on which the members of both houses can unite without distinction of party.”

The Politician’s Proof

To show that $c^2 = a^2 + b^2$ , Garfield used a trapezoid:

Given: Trapezoid ACED, constructed using congruent right triangles ABC and BDE.

(1) Show that ABD is a right triangle.

Alex showed this at the bottom of The Pythagorean Proof.

(2) Find the total area of the three triangles.

$Area = \frac12 a b + \frac12 a b + \frac12 c^2 = ab + \frac12 c^2$

(3) Find the area of the whole trapezoid.

$Area = \frac12 \left( height \right) \left( sum\; of\; bases \right)$

$= \frac12 \left( a + b \right) \left( a + b \right)$

$= \frac12 \left( a^2 + 2ab + b^2 \right)$

(4) The whole trapezoid is equal to the sum of its parts.

$ab + \frac12 c^2 = \frac12 \left( a^2 + 2ab + b^2 \right)$

(5) Simplify the equation.

$ab + \frac12 c^2 = \frac12 \left( a^2 + 2ab + b^2 \right)$

$ab + \frac12 c^2 = \frac12 a^2 + ab + \frac12 b^2$

$\frac12 c^2 = \frac12 a^2 + \frac12 b^2$

$c^2 = a^2 + b^2$

Q.E.D.

The Rest of the Story

Garfield went on to become the 20th president of the United States. He won his election by the narrowest margin in U.S. history — less than 10,000 votes, or 1/10 of 1%. He was the last president born in a log cabin, and he appointed Abraham Lincoln’s son to his Cabinet.

Then in 1881, just four months after taking office, President Garfield was shot.

One bullet grazed his arm, but a second one lodged somewhere inside. Sixteen doctors dug around (with unwashed hands). No one could find the bullet. Inventor Alexander Graham Bell tried a new idea: a metal detector. He said he found the bullet, much deeper than they first thought. Doctors operated, without sterilization or success. It turned out that Bell had detected a metal mattress spring.

More than three months after he was shot, President Garfield died. An autopsy showed that the bullet would not have killed him, if the doctors had left him alone.

The gunman was hanged for assassination.

The doctors sent a bill to the government — and got paid.

To Be Continued…

Read all the posts from the May/June 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.

Alexandria Jones and the Mosaic Tile Mystery

9 thoughts on “A Mathematician for President”

I loved it, … But Garfield was not JUST a politician, he was a Professor of Math in Ohio before he became a senator… I copied the intro and linked your post at my blog with some additional comments.. you have a great site…
We who teach the non-stay at home math kids, thank you for all the hard work..

Pat

I had heard that, too, or read it somewhere, but the sources I found as I revised this article from my old newsletter all said he taught “Classics” or ancient languages.

i dont understand…!

In order to understand this proof, you need to first master basic algebra and geometry. If you have passed algebra 1 and high school geometry, this should be easy to you. If you haven’t gotten that far in math, then don’t worry about understanding this proof — just keep working to learn the basics.

Beautiful pictures,
If you go here http://www.pballew.net/arithm11.html#pythagor
you will see a proof that is very visual, and very similar… Keep in mind that the white areas is not changing, so the square a^2 and b^2 added together, make up the same area as the square built on side c,with area c^2. The white area is equal to both a^2 + b^2, and also to c^2, so they must be equal.
I hope that helps… it is better with the pictures, so give it a look.
Pat

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