The Pythagorean theorem

Today, like with Plato's allegory of the cave, I want to discuss one of the major developments in thinking contributed to the ancient Hellenes. When I was in high school, I was terrible at math. Especially geometry was a huge problem for me. I had nightmares about the Pythagorean theorem even years after high school--I understood the overall theory, but then my teacher dragged in a lighthouse or tent pole and I was lost. It took my brain a good couple of years to understand that those lighthouses and tent poles didn't have anything to do with math and were just an overlay intended to make math easier for people whose brain was wired differently than mine. For me, it made me hate anything connected to mathematics. In a roundabout way, I was eventually reintroduced to math in a way that made a lot more sense to me, and looking back now, the Pythagorean theorem is really not that difficult anymore. If I have readers who are struggling with this in school, trust me, it gets better.

Back to Pythagoras. Pythagoras of Samos (Πυθαγόρας ὁ Σάμιος) was an Ionian Hellenic philosopher, mathematician, and founder of the religious movement called Pythagoreanism. He was an influential voice in philosophy, religion, mysticism and science in the late 6th century BC, and is best known for the Pythagorean theorem which bears his name. Because almost everything we now know about Pythagoras was written down centuries later, it may very well be that the theorem as well as anything else attributed to Pythagoras was discovered by his colleagues, students, successors, or even his mother. That said, I'm just going to assume that Pythagoras was brilliant and came up with one of the most basic fundaments of geometry, that in any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). Gods, that sounds complicated when posed like this. The Pythagorean theorem is better known as the formula used to solve it: a² + b² = c².

The easiest form of the Pythagorean theorem

We often call the ancient Hellenes the founders of modern thinking, and although the theorem was very influential even then, it had been discovered before. What Pythagoras did--and which man other Hellenic scholars did in their time--was proof why the theorem works. Instead of simply going with it, they struggled to find proofs of the theories. To prove the Pythagorean theorem, we must realize that it does not actually concern the triangle depicted above; it concerns the squares that you can connect to them. In the example above, there will be a square with a value of 3 on all sides to the left of the shortest side (a), a larger square with a value of 4 on all sides tagged to the bottom line (b), and an even larger square with a value of (in this example) 5 on all sides tagged to the hypotenuse (c). You can see this in the image to the left, below. Pythagoras then realized that this theory works because the area encompassed by the outer square never changes, and the area of the four triangles is the same at the beginning and the end, so the black square areas must be equal, therefore a² + b² = c². Actually, we don't know if this is the proof Pythagoras used, but it sure is one that works.

 

Pythagoras's theorem has been at the core of mathematics and geometry ever since its discovery, and even though the discoveries by Pythagoras and his contemporaries at his school were supposed to be kept secret within the brotherhood that surrounded those who studied, lived, and worshipped with Pythagoras, many of them were leaked to the public eventually. This allowed those who were not part of Pythagoreanism to understand and develop the foundations Pythagoras had laid out. His influence could be seen in the work of the greats--like Plato--but the influence of his brotherhood can be found even as far down the line as Freemasonry and Rosicrucianism.

 

Pythagoras made many more brilliant discoveries in his time, many of which w will talk about at a later date. For now, I want to share with you this lengthy, but hugely interesting lecture by N.J. Wildberger at the University of New South Wales for the college course 'History of Mathematics'. In this first lecture (with two parts) a very rough outline of world history from a mathematical point of view is given, with a focus on the position the work of the ancient Greeks as following from Egyptian and Babylonian influences has in that timeline. It also introduces the Pythagoras' theorem in a very elaborate and understandable way. The disposition about Pythagoras starts at about 13 minutes in. Part b can be found here, and is equally interesting. Enjoy!

 



 

Image source: squares, Pythagorean proof