The Pythagorean Theorem is without doubt the most famous theorem in mathematics. Just about every high school student knows it by heart: A squared plus B squared equals C squared. That is, given a right triangle (a triangle two of whose sides meet at a right angle), the sum of the squares on the legs is equal to the square on the third side (the hypotenuse).
Although no one knows who first discovered this theorem, it’s named after Pythagoras because it was his school that made it known to the world. We learned about Pythagoras and his followers in my July 7 column, “Math, Music, and Mysticism.”
How the theorem was first proved isn’t known, either. But an easy proof, which may well be the Pythagorean proof, is provided in the illustration. In the diagram on the lower left, four equal right triangles and the squares on their legs add up to the large square; in the diagram on the lower right, the same four right triangles and the square on their hypotenuse add up to the same large square. So the squares on the legs must equal the square on the hypotenuse.
Another proof is provided in Euclid’s Elements, written in about 300 BC. This work has been called the most successful textbook ever written. It formed the basis of the math curriculum until the Nineteenth Century – for over 2,000 years, in other words – and the number of its editions since the invention of the printing press is second only to the Bible.
The climax of Book I of the Elements is the proof of the Pythagorean Theorem, which amounts to cutting the square on the hypotenuse into triangular and quadrilateral shapes and rearranging them on the two smaller squares, a bit like a jigsaw puzzle.
The Pythagorean Theorem has been independently discovered by cultures around the globe. There are literally hundreds of proofs. In his book The Pythagorean Proposition, Elisha Loomis collects and classifies about 350 different proofs.
Numerous historical figures have tried their hand at this noble pursuit, among them James Garfield, the only U.S. president known to have proved a theorem. His explanation, which uses a trapezoid, appeared in 1876. Albert Einstein devised a new proof as well.
The Pythagorean Theorem prompts the idea that directions at right angles are more or less independent of each other. There is a three-dimensional version of the theorem, and from there it was only natural to extend it to four dimensions, to five dimensions, and so on. It’s hard to exaggerate the far-reaching consequences of this idea.
For one thing, it allows us to turn vibrations into a kind of geometry, in which the basic vibrational modes are viewed as being at “right angles” to one another. These independent modes are added to one another in vibrating strings and drumheads, producing a variety of sounds. Using the same idea, electronic synthesizers combine simple sound waves to imitate any type of musical instrument.
The Pythagorean idea also appears in the theory of atoms. There are many similarities between the possible arrangements of electrons around the nucleus of a hydrogen atom and the simple harmonics of a vibrating string.
Pythagoras built his philosophy on the idea that nature is a kind of harmony. So it’s only fitting that the theorem named after him pervades the modern theories of music and the atom. The great esteem in which this theorem has been held throughout history is enshrined in the legend that he sacrificed a hecatomb of oxen – one hundred animals – to the gods in celebration of his discovery.
But the English mathematician Charles Dodgson, better known by his pen name Lewis Carroll, under which he wrote Alice’s Adventures in Wonderland, humorously objected to such extravagance: “One can imagine oneself, even in these degenerate days, marking the epoch of some brilliant scientific discovery by inviting a convivial friend or two, to join one in a beefsteak and a bottle of wine. But a hecatomb of oxen! It would produce a quite inconvenient supply of beef.”
Michael Ortiz is an associate professor of mathematics at Sul Ross State University-Rio Grande College in Uvalde. He contributes a monthly math/science-related column to the Uvalde Leader-News.