A Presidential Pythagorean Proof

James Garfield, circa 1870-1880. This image is in the public domain in the U.S. because its copyright has expired.

James Abram Garfield was born on this day, November 19, in 1831. Had an unstable, delusional stalker?s bullets and nineteenth-century medical ?care? not cut short his life just six months into his presidency, he would be 181 today (more on that later). Garfield was an intelligent man who studied some math in college, but contemporary documents tend to highlight his skills and interests in preaching, debate and English rather than mathematics. It is not clear how he became involved with one of the most famous theorems in geometry.

The Pythagorean theorem describes the relationship between the side lengths of a right triangle. The square of the hypotenuse, the longest side, is equal to the sum of the squares on the other two sides, or more familiarly, a²+b²=c². As with many true things, there are multiple ways to prove this theorem. In a March 7, 1876 diary entry, Garfield, then a congressman from Ohio, mentioned showing a new proof to a mathematics professor at Dartmouth University. Later that year, the proof was published in the New-England Journal of Education (now known simply as the Journal of Education).

The article, which only takes the bottom third of one column, begins, ?In a personal interview with Gen. James A. Garfield, Member of Congress from Ohio, we were shown the following demonstration of the pons asinorum, which he had hit upon in some mathematical amusements and discussions with other M. C.?s [members of Congress]. We do not remember to have seen it before, and we think it something on which the members of both houses can unite without distinction of party.?

Oddly, pons asinorum (Latin for ?bridge of asses?) was usually used to refer to the isosceles triangle theorem in Euclid?s Elements, which holds that in an isosceles triangle (a triangle with two sides of the same length), the angles opposite the congruent sides are themselves congruent. The proof of this theorem was sometimes considered the first challenging problem in the Euclid?s classical geometry text. In this case, however, it is clear that the pons asinorum is the Pythagorean theorem.

The diagram accompanying Garfield's proof of the Pythagorean Theorem in the New-England Journal of Education, 1876. Image: Evelyn Lamb

This helpful diagram illustrates Garfield?s proof. In this trapezoid, the two smaller right triangles are congruent to each other, and the large triangle is an isosceles right triangle. Its non-hypotenuse sides are the hypotenuses of the smaller triangles. The proof relies on finding the area of the trapezoid in two different ways: by using the area formula for the trapezoid and by adding up the areas of the three triangles. Because the area is the same no matter how we dissect the trapezoid, we end up with equations relating the side lengths of the smaller right triangles. (For full details and some more proofs of the Pythagorean theorem, check out the Wolfram MathWorld page on the Pythagorean theorem.)

Originality is sometimes in the eye of the beholder: Garfield?s exact argument does not appear anywhere else prior to the 1876 journal article, but it is extremely similar to a proof from the classical Chinese astronomy and mathematics text Zhou Bing Suan Jing, which was probably compiled during the first century BCE.

The diagram from the classical Chinese proof of the Pythagorean theorem. This image is in the public domain in the U.S. because its copyright has expired.

Garfield?s trapezoid is equivalent to this diagram cut along a diagonal of the tilted, thick-edged square. It is unlikely that Garfield ever saw the proof from this text, as it was not translated into English until 1996, but it is possible that the same proof appeared in another text that he did know.

Had Garfield lived to age 181, he would surely have been glad to know that 181 is part of two different Pythagorean triples, sets of integers that satisfy the equation a²+b²=c² and therefore make up the side lengths of a right triangle. Of course, when we allow square roots to get involved, any integer can be a side length of a right triangle; what is special about Pythagorean triples is that all three side lengths are whole numbers. 3-4-5 is probably the most famous Pythagorean triple. It works because 3²+4²=9+16=25=5². The Pythagorean triples containing 181 are 19-180-181 and 181-16380-16381, so 181 is both the longest and shortest side of a right triangle with integer side lengths.

If Garfield did a little more mathematical investigation, he might be dismayed to realize that his age is not special: all positive integers (except for 1 and 2) are part of at least one Pythagorean triple.

The odd numbers are easiest to see. Suppose a triangle has one side of length n and one side of length n+1. The difference of the squares of these sides will be an odd integer. In fact, (n+1)²-n²=2n+1. So if 2n+1 is itself a square (for example, if n=4, then n+1=5 and 2n+1=9, which is 3²), then the third side of the triangle is an integer, written ? 2n+1 . If k is any odd integer greater than one, its square can be written k²=2n+1 for some whole number n. In that case, the numbers k, n and n+1 form a Pythagorean triple. So every odd integer k is in some Pythagorean triple.

For even numbers, we know that the number 4 appears in the Pythagorean triple 3-4-5. We also know that if we double all the side lengths of a triangle, we get a triangle with the same ratio of side lengths. For example, 3-4-5 ?inflates? to 6-8-10, another Pythagorean triple. The interested reader can use these facts to prove that every even integer greater than 2 appears as a side length of a Pythagorean triple.

Unfortunately, President Garfield is not able to be here to enjoy his doubly Pythagorean 181^st birthday. But perhaps his Pythagorean proof is still ?something on which the members of both houses can unite without distinction of party.?

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