Bust of Pythagoras

If you remember nothing else from high school geometry, you probably remember the Pythagorean Theorem: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. Or to put it in a formula:

(Credit: pedica018/Shutterstock)

The formula is useful in construction, surveying and navigation, among other practical pursuits such as determining the size of a monitor or television for buying. The theorem also led to mathematical breakthroughs such as calculus.

But who was Pythagoras, the man behind the famous and useful theorem?

On mathematics and mysticism

Other than the fact that he was born on the Greek island of Samos around 569 BC and died around 475 BC, not much is known about him. Pythagoras left no writings, but founded a sect (or what some would consider it a cult): The Divine Brotherhood of Pythagoras. His followers are often simply called the Pythagoreans, a secret society dedicated to a combination of scientific and mystical precepts.

They were an eccentric bunch and the legends surrounding Pythagoras and his followers are delightful. It is said that Pythagoras refused to eat beans and did not allow his followers to do so because he believed that beans had a soul. It is also said that the Pythagoreans were very suspicious of sex and according to Jordan Ellenberg in his book How not to make mistakes: The power of mathematical thinkingthey may have believed that Earth had a twin planet on the other side of the sun.

It is said that some of Pythagoras’ followers believed that he was a god. Yet it is said that they threw him into the sea and left him to die because the Pythagorean theorem was too, as we might say today, destructive.

Although the last myth is as unlikely as the others – and the poor sod thrown overboard is usually defined as Hippasus of Metapontus, another Pythagorean, not Pythagoras himself—is certainly appropriate given the impact of the theorem. It was dangerous, at least for the Pythagorean worldview. In developing these specific properties of the triangle, Pythagoras stumbled upon the existence of irrational numbers. And that was a big problem.

“Everything’s a Number”

The basic principle of Pythagoreanism is that numbers are the essence of everything, as stated in their motto: “Everything is number.” They had some good reasons for looking at things that way. They were the ones who established that musical intervals correspond to the lengths of the strings of a stringed instrument and derived Golden meanor golden ratio, from studying patterns in nature, such as nautilus shells and flower petals.

As David Foster Wallace explains in Everything and More: A Compact History of Infinity, “the Pythagoreans’ attempts to articulate the connections between mathematical reality and the physical world were part of the larger project of pre-Socratic philosophy, which was primarily to provide a rational, non-mythopoetic explanation of what is real and where it comes from.” They wanted to understand how things worked—absent thunder gods—and everywhere they looked, they found numbers.

Before the discovery of the Pythagorean theorem, the Greeks believed that all numbers could be expressed either as a whole number or as a fraction – the ratio of two whole numbers. But the Pythagorean theorem blew a big hole in that notion. Hiding the theorem in plain sight was a very disconcerting thing. If, for example, you look at a right triangle with two sides of 1 inch (or foot or something), the hypotenuse is a number whose square is 2.

What is the square root of 2? Something the Pythagoreans couldn’t handle: an irrational number; that is, a number that cannot be written as a fraction (or ratio). For us, it just means that the math is a little more challenging (okay, maybe a lot more challenging). For the Pythagoreans, this was a challenge to their entire worldview, which was built on the supremacy, even divinity, of numbers, which did not do such strange things.

Of course, now we have all kinds of weird numbers: imaginary numbers, transcendental numbers, the truly destructive zero plus quantum mechanics. And we mess around more or less well – we even manage to make airplanes fly and invent computers.

An older version of the theorem

The writings of other members of society were often attributed to Pythagoras himself rather than the actual writer, possibly out of respect (or perhaps as a form of self-defense). So it’s really not fair to attribute all the weirdness or all the credit to Pythagoras.

Much of what we attribute to Pythagoras may have been the work of another Pythagorean or several. But real or made up, Pythagoras was not the first to understand the famous ratio. Clay tablets found in present-day central Iraq show that Babylonian mathematicians were aware of the basics of the famous theorem at least 1,000 years before it drove the Pythagoreans crazy. We don’t know how the Babylonians reacted, but we can hope that they resisted pushing anyone off the boat.

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