# ddrummer realtime

## Wednesday, January 11, 2012

### .In 1876, a politician made mathematical history.

[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 newproof of the Pythagorean Theoremin 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.
$ab + \frac12 c^2 = \frac12 \left( a^2 + 2ab + b^2 \right)$
$ab + \frac12 c^2 = \frac12 a^2 + ab + \frac12 b^2$
$\frac12 c^2 = \frac12 a^2 + \frac12 b^2$
$c^2 = a^2 + b^2$

## 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.