Abraham Lincoln - A President Trained by Euclid

In studying law, Abraham Lincoln (16th United States of America President) came across the word demonstrate frequently. He searched for further meaning. Consulting Websters, he found “certain proof” and “proof beyond the possibility of doubt.” 

So, what did Lincoln do? He studied the works of Euclid – six volumes of geometric proofs and he mastered all of them. Proofs where a proposition is demonstrated beyond doubt based on a foundation of facts and logic. We don’t know Lincoln as a mathematician, but we do know him as a famous debater and orator.

Geometry in high school is the point at which students either really start to understand the mathematical process or really despise math.

It’s in geometry where we learn how to set-up a proof. We start with given axioms and step, by step, we establish a proof. At the end of the proof, it is the practice to write “Q.E.D.” Which is the Latin acronym for Q.E.D., Quod erat demonstrandum -

– That which is to be demonstrated.

 Some of this chain of logic is seen in the 1^st paragraph of the Gettysburg Address (note the term proposition):

             "Four score and seven years ago our fathers brought forth upon this continent, a new nation, conceived in Liberty, and dedicated to the proposition that all men are created equal."

Another example is from AL’s private notes in 1854 (perhaps formulating an opposing viewpoint to those who tried to justify slavery):

“If A can prove, however conclusively, that he may, of right, enslave B – why may not B snatch the same argument, and prove equally, that he may enslave A.” Source: Abraham Lincoln: A Life by Michael Burlingame, Johns Hopkins University Press, 2013.

Also, in the ending paragraph of AL 1^st inaugural address (more of a music analogy but linked by chords and harmony to math):

The mystic chords which, proceeding from so many battlefields and so many patriot graves, pass through all the hearts and all hearths in this broad continent of ours, will yet again harmonize in their ancient music when breathed upon by the guardian angel of the nation.

So, Lincoln learned from the discipline of geometry how to make a solid, logical argument and how make a proof.

1.   Carpenter, F.B. The Inner Life of Abraham Lincoln: Six Months at the White House Hurd & Houghton, New York, NY (1874).

Article: From Euclid to Abraham Lincoln, Logical Minds Think Alike (nautil.us)

Pythagorean Presidential Smarts

So this blog isn't wasn't you are expecting if you "hate" or "love" our president. 144 years ago, James Garfield developed a proof to the Pythagorean Theorem. A few years later, he was elected the 20th President of the United States of America in 1880. 

There are many proofs to the Pythagorean Theorem. Garfield's proof, while similar to some prior proofs, is very concise and easy to understand.

Take any right-angle triangle (ABC) and make a duplicate triangle (A'AC') that is rotated and adjacent to it that you arrange as shown:

Add a line connecting points B and A'.

Now, name the sides of each triangle for the point at its opposite angle. Note, the the length A' = A and C' = C:

Garfield calculated the area of the trapezoid two ways. First, by addition of parts - adding the area of the three triangles is C²/2 + AB/2 + (AB)/2 =  C²/2 + AB

The second way to calculated the area of the trapezoid is the using the formula, base times 1/2 the sum of the two sides, or: (A + B) x 1/2 x (A + B).= A²/2 + B²/2 + AB

Since both calculated areas represent the same, C²/2 + AB = A²/2 + B²/2 + AB

Subtract AB from both sides of the equation and multiply both sides by 2, the result is:

C² = A² + B²