Well, that title is quite strange. Who would have thought that our 20th president and Pythagoras could be in the same sentence? Getting to that later, who even is James Garfield? Some know him as our 20th president or a mathematician with the former being more probable.
Early Life
Born in Moreland Hills, Ohio in 1831; Garfield was fatherless for most of his life and earned money for education through canoe boat teams. When he attended Geagua Academy in his teen years he supported himself with a teaching position at a district school and also worked as a school janitor on the side. In 1854 at 23 years old, he attended Williams College in Massachusetts and was one of the oldest.
Though Garfield was a very intellectual and serious student, he had many hobbies such as hunting or billiard balls. He also met his wife while he was studying at Williams. She was Lucretia Rudolph, one of the very few people whose appetite for knowledge matched his. They married in 1858 leading a happy married life.
Garfield was also quite the renaissance man, to say the least; he was knowledgeable in many subjects such as English, history, geology, and mathematics. He even passed the bar exam in 1861 shortly after he got married!
In 1859 he threw himself into politics and became part of the Ohio legislature, becoming the youngest member.
Government Life
In 1856, Garfield campaigned for John C Fremont one of the presential candidates for the newly formed Republican party. He later campaigned for Abraham Lincoln in 1860. Although, he never really liked Lincoln and referred to him as a “second-rate Illinois lawyer”. This was all around the time before the Civil War. One year later, Garfield joined the army and was thrown into the midst of the battle against the Confederacy where he managed Ohio infantry and various brigades. Through his many impressive victories in battles, he won the title of major general, the youngest to hold this title. In 1863, Garfield resigned from the Army to become part of the House of Representatives where he was elected without any campaigning.
He was known as one of the most radical republicans in Congress. Garfield repeatedly won reelection in 1862 for 18 years, and become the leading Republican in Congress. Finally, in 1881, he became president of the United States.
Sadly, around 7 months later he was assassinated. He was fatally wounded by a deranged gunman as he was prepared to board a train in Washington D.C.
Mathematician?
Well, as said above, math was one of the many subjects this renaissance man studied! It was said that mathematics was among his favorites. His notable contribution was that he created an original proof for the Pythagoras Theorem. The theorem follows as where a and b are the legs of the right triangle and c is the hypotenuse. His interpretation uses trapezoids and areas!
Summary of Proof
{maa.org, visual representation of Garfield’s proof}
He first constructed a trapezoid with two right triangles and an isosceles triangle. The isosceles triangle has two lengths of c and another unknown. We notice that this isosceles triangle is actually a right triangle. This is because in the trapezoid, Z + (90 – Z)+ ? = 180. The missing angle has to be 90. He basically equated the area of a trapezoid to the sum of the areas of the triangles.
This gives us 1/2(a+b)^2 = ab/2 + ab/2 + c^2/2
This simplifies to (a+b)^2 = 2ab + c^2 or a^2 + 2ab + b^2 = c^2
Subtracting 2ab from each side gives us a^2 + b^2 = c^2. It’s a pretty elegant proof.
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