PRESIDENT GARFIELD, AND THE PYTHAGOREAN THEOREM

From THE MATHEMATICAL UNIVERSE, by William Dunham, copyright 1994, John Wiley & Sons, Inc.

Garfield, Dunham reveals, was the author of a unique proof of the Pythagorean Theorem.

This, obviously, was back in the days when U.S. political leaders still admired a classical education. How far are we now from the oratory of an Abraham Lincoln, or from the inspired crusade for classical education in free, public schools by a Thaddeus Stevens! And how unusual for a U.S. president and Congressman to be involved in geometrical constructions! But perhaps this story further fills in the picture of why such a political leadership was considered a threat to the British aristocracy who had just seen their support for the Confederacy go up in flames.

Dunham's book, which is arranged to present different discoveries and personalities in the history of mathematics "from A to Z", includes under the letter "H", the chapter "Hypotenuse", in which he discusses 3 different proofs of the Pythagorean Theorem. These include the "Chinese Proof" embodied in the hsuan- thu diagram of a square tilted in another square, dated from somewhere between 1000BC - 1 AD; the proof by John Wallis in the 17th century, based on the proportionality of similar triangles; and finally, the proof developed by then Congressman James Garfield in 1876, based on the construction of a trapezoid containing a right triangle.

Dunham discusses on pages 95-96 the mathematical interests of presidents from Washington to Grant as follows:

"U.S. presidents, whatever abilities they exhibit in other spheres, are seldom known for their mathematical powers. ... Historically, however, some chief executives have possessed mathematical talent. One was George Washington, an accomplished surveyor, who endorsed mathematics with the following words: 'The investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings .... From the high ground of mathematical and philosophical demonstration, we are insensibly led to far nobler speculations and sublime meditations.'[fn: Edmund Ingalls, "George Washington and Mathematics Education," Mathematics Teacher, Vol 47, 1954, p. 409.]"

Dunham continues:

"Abraham Lincoln was also a strong advocate of mathematics. As a young adult studying law, Abe recognized the need to sharpen his reasoning skills, to learn what it meant to prove a point by means of a sound logical argument. As he later recalled in an autobiographical sketch: 'I said, "Lincoln, you can never make a lawyer if you do not understand what demonstrate means"; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what "demonstrate" means, and went back to my law studies.' [fn: James Mellon, ed., The Face of Lincoln, Viking, New York, 1979, p. 67.]"

"Radical in his politics and fiercely patriotic, he left academe to join the Union Army when war erupted in 1861. Interestingly the math teacher turned out to be a fine soldier. Garfield quickly rose in the ranks until he was appointed chief of staff for Union General John Rosecrans. In 1863, Garfield shifted from the U.S. Army to the U.S. House of Representatives, where he spent the next 17 years as a Radical Republican intent upon reforming, if not punishing the South. ... In 1880 James A. Garfield earned the Republican nomination for the presidency and narrowly defeated another Civil War hero, the Democrat Winfield Scott Hancock, in that fall's election. At his inauguration in March of 1881, our mathematical president promised to improve the educational opportunities of all Americans, because, 'It is the high privilege and sacred duty of those now living to educate their successors and fit them, by intelligence and virtue, for the inheritance which awaits them.'[fn: The Inaugural Addresses of the American Presidents, annotated by Davis Newton Lott, Holt, Rinehart and Winston, NY, 1961, p. 146.][1]

"But promise was about all that came from the Garfield administration, for on July 2, 1881, having served less than four months, he was shot by a disgruntled office seeker while boarding a train in Washington. ... he lingered until mid-September before death overtook him. ... Politically, his was a life with its greatest dreams left unfulfilled. But he left a mark upon mathematics."

The construction itself proceeds by dropping two parallel lines from the legs of a right triangle, and then connecting those lines with a base line in such a way that the right triangle is now embedded in a trapezoid -- and the trapezoid itself is cut into 3 right triangles. The demonstration then compares the area of the trapezoid with the areas of the 3 right triangles and concludes that the sum of the squares of the legs of the right triangle is equal to the square of the hypotenuse of the right triangle.

Dunham comments that "Garfield's is really a very clever proof", and shows later that the demonstration is a mirror image of the hsuan-thu diagram of the "Chinese proof". The author then concludes:

"Here again we see the benefits of looking at the trapezoid's area from two different viewpoints. As the author of the New England Journal article wryly observed, 'we think it something on which the members of both houses can unite without distinction of party.'"

Footnote:

"His zeal to extend educational opportunities to the masses, early showed itself and continued with uniform intensity throughout life. It was a necessary corollary to his ambition for human equality in the matter of liberties, rights and punishments. To him, education was the best means afforded to human kind to obtain equal opportunity in life, and he never veered from his staunch conviction that public schools were a fundamental requisite for the maintenance of our form of Government. p. 145...

He kept up his propaganda, however, and at public gatherings where toasts were offered, his would be "Education -- May the film be removed from the eyes of Pennsylvania and she learn to dread ignorance more than taxation." P. 146

[from Thaddeus Stevens, by Thomas Frederick Woodley, The Telegraph Press, Harrisburg, PA, 1934.]