Square of the Hypotenuse

There’s a delightful mathematical moment in the movie Merry Andrew, when Danny Kaye, playing schoolmaster Andrew Larabee, breaks into song to teach the Pythagorean theorem.

I was reminded of this scene by a sentence in an article about the Pythagorean theorem in the October issue of Mathematics Magazine. The Pythagorean theorem “is probably the only nontrivial theorem in mathematics that most people know by heart,” comments Darko Veljan of the University of Zagreb in Croatia.

Indeed, if there is one thing that someone might remember from grade school mathematics, it’s the fact that “the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides.”

According to one legend, Pythagoras (c.580 B.C.-c.500 B.C.) discovered the theorem while waiting to see Polycrates, the tyrannical ruler of the city of Samos. Cooling his heels in a palace hall, Pythagoras spent the time pondering the floor’s square tiling. He imagined how a diagonal line cutting across a square would divide the square into two right triangles. He noted that the area of a square erected over the diagonal is double the area of the square erected on an adjacent side. In other words, the square on the hypotenuse is equal to the sum of the squares on the triangle’s two legs. Pythagoras came to believe that the same relationship would hold when the legs have unequal lengths.

The theorem’s history, however, is more complex than this legend would suggest. The use of the 3-4-5 triangle for constructing a right angle, for instance, goes back to much earlier times in Egypt, Babylon, and China. In his textbook The History of Mathematics, Roger Cooke of the University of Vermont describes how the Babylonians might have discovered the Pythagorean theorem more than 1,000 years before Pythagoras.

Basing his account on a passage in Plato’s dialogue Meno, Cooke suggests that the discovery arose when someone, either for a practical purpose or perhaps just for fun, found it necessary to construct a square twice as large as a given square. Simply doubling a square’s side actually quadruples the square’s area. If you contemplate the quadrupled square for a while, you might think to join the midpoints of adjacent sides–in effect, drawing the diagonals of the four copies of the original square.

“Since these diagonals cut the four squares in half, they will enclose a square twice as big as the original one,” Cooke notes. Someone ‘playing’ with the figure might then consider the effect of joining points on adjacent sides when they are no longer the midpoints but at a given distance from the corners of a square.

“Doing so creates a square in the center of the larger square surrounded by four copies of a right triangle whose hypotenuse equals the side of the center square; it also creates the two squares on the legs of that right triangle and two rectangles that together are equal in area to four copies of the triangle,” Cooke writes. This construction adds up to the Pythagorean theorem.

“The Pythagorean theorem was an early example of an important fact rediscovered independently and often,” Veljan remarks. Moreover, more than 400 different proofs of the theorem are known today, he adds.

Veljan offers the following modern-day visualization of the Pythagorean theorem’s area relationship: A pizza shop makes three sizes of pizzas; their diameters are the sides of a right triangle. The big pizza must then be equal to the sum of the two smaller pizzas.

The famous Babylonian clay tablet known as Plimpton 322 (see http://www.nsm.iup.edu/ma/gsstoudt/history/images/plimpton.html) goes a step further. Dating from the period between 1900 B.C. and 1600 B.C., the tablet has columns of numbers that apparently represent what are now called Pythagorean triples.

The whole numbers a, b, and c are a Pythagorean triple if a and b are the lengths of two sides of a right triangle with hypotenuse c, so a² + b² = c².

In general, for any number k, the corresponding Pythagorean triple is a = 2k + 1, b = 2k(k + 1), and c = b + 1. For example, when k = 1, a = 3, b = 4, and c = 5. When k = 2, a = 5, b = 12, and c = 13.

The Babylonians used a sexagesimal, or base 60, number system. The Plimpton tablet has several columns of numbers, written in cuneiform script. The following table shows the numbers in two of the columns written in decimal notation. One apparent error is corrected (4825 replaces 11521 in the second row).

119	169
3367	4825
4601	6649
12709	18541
65	97
319	481
2291	3541
799	1249

Mathematics historian Howard Eves has conjectured that each pair of numbers represents two of the three members of a Pythagorean triple, corresponding to one side and the hypotenuse of a right triangle.

The numbers also fit the following formula for finding Pythagorean triples: a = 2uv, b = u² – v², and c = u² + v², where u and v are relatively prime, one number is odd while the other is even, and u is greater than v. For example, when u = 12 and v = 5, b = 119 and c = 169 (as given in the first row of the table) and a must be 120.

It’s straightforward to extend the Pythagorean formula to right triangles in three and higher dimensions. For example, for a rectangular box that is a units long, b units wide, and c units high, the diagonal d obeys the following relationship: d² = a² + b² + c². Moreover, you can look for analogous relationships for triangles on the surface of a sphere, on the hyperbolic plane, and in other spaces.

Generalizing the Pythagorean equation for triangles with integer sides to powers greater than 2 leads to Fermat’s last theorem and the so-called ABC conjecture (see The Amazing ABC Conjecture, December 6, 1997).

Though more than 2,500 years old, Veljan concludes, “this ‘folklore’ theorem remains eternally youthful, as many people continue to find new interpretations, generalizations, analogues, proofs, and applications.”