Garfield's proof of the Pythagorean theorem

Garfield's proof of the Pythagorean theorem is an original proof the Pythagorean theorem invented by James A. Garfield (November 19, 1831 – September 19, 1881), the 20th president of the United States. The proof appeared in print in the New-England Journal of Education (Vol. 3, No.14, April 1, 1876).[1][2] At the time of the publication of the proof Garfield was not the President, he was only the Congressman from Ohio. He assumed the office of President on March 4, 1881, and served in that position only for a brief period up to September 19, 1881.[3] Garfield was the only President of the United States to have contributed anything original to mathematics. The proof is nontrivial and, according to the historian of mathematics, William Dunham, "Garfield's is really a very clever proof."[4] The proof appears as the 231st proof in The Pythagorean Proposition, a compendium of 370 different proofs of the Pythagorean theorem.[5]

Garfield in 1881

The proof

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Diagram to explain Garfield's proof of the Pythagorean theorem

In the figure,   is a right-angled triangle with right angle at  . The side-lengths of the triangle are  . Pythagorean theorem asserts that  .

To prove the theorem, Garfield drew a line through   perpendicular to   and on this line chose a point   such that  . Then, from   he dropped a perpendicular   upon the extended line  . From the figure, one can easily see that the triangles   and   are congruent. Since   and   are both perpendicular to  , they are parallel and so the quadrilateral   is a trapezoid. The theorem is proved by computing the area of this trapezoid in two different ways.

 .
 

From these one gets

 

which on simplification yields

 

References

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  1. ^ G., J. A. (1876). "PONS ASINORUM". New England Journal of Education. 3 (14): 161. ISSN 2578-4145. JSTOR 44764657.
  2. ^ Sid J. Kolpas. "Mathematical Treasure: James A. Garfield's Proof of the Pythagorean Theorem". maa.org. Mathematical Association of America. Retrieved 29 November 2023. (The article appeared in the peer-reviewed online journal Convergence published by the Mathematical Association of America.)
  3. ^ Alonso Del Arte (February 2019). "A future president once published a mathematical proof". medium.com. Retrieved 29 November 2023.
  4. ^ William Dunham (1994). The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities. New York: John Wiley & Sons. p. 99. ISBN 0-471-53656-3.
  5. ^ Elisha Scan Loomis (1940). The Pythagorean Proposition (2 ed.). Washington DC: National Council of Teachers of Mathematics. p. 109. ISBN 978-0-87353-036-1. Retrieved 28 November 2023. (A collection of 370 different proofs of the Pythagorean theorem.)