"The History of the Pythagorean Theorem"

Hello everyone. We're back, and this time we're talking about the history of the Pythagorean Theorem. I hope you like it. When you first hear the words Pythagorean Theorem, you might think back to junior high. You might think of the theorem itself. Or you might even remember the last time you used it in real life. I told you I'd never use it, Ms. Smith. Nowadays, it's common knowledge that the square of two smaller sides of a right triangle will be equal to the square of the hypotenuse. We've all heard of a squared plus b squared equals c squared, but for which right triangles does this apply? Well, all of them, actually. The Pythagorean Theorem applies to all right triangles and the known and unknown universe. Hitchhikers guide to the Pythagorean Theorem, coming soon. Well, contrary to its name, Pythagoras was not the first person to discover this theory. For that, we have to go back. Way back to ancient Egypt. Six thousand years ago, the ancient Egyptians learned how to make a right triangle from a Pythagorean triple. Three whole numbers that make up a right triangle. They would make 12 equal sections of rope in a chain, put three on one side, four on the other, and connect them with the remaining five, ensuring that the rope was taut. The ability to accurately make 90-degree angles was crucial to constructing the pyramids. Next, we head to ancient Babylon, the Fertile Crescent, over 4,000 years ago. Stone and clay tablets from this time express extremely accurate knowledge of a one-to-one right triangle's proportions. It is believed that Egyptian traders carried mathematics to the Northwest, and Babylonians adapted these ideas to create their own version of them. Continuing, the next time we see knowledge of the Pythagorean Theorem is just over 1,200 years later, in India, 800 BC. In the Baudiyana's book, the Subhasatra, he demonstrates that, quote, a rope stretched along the length of a diagonal of a right triangle produces an area which the vertical and horizontal sides make together, end quote. Or in other words, a squared plus b squared equals c squared. Now, finally, we reach the famed Pythagoras. Pythagoras had a school set up in Crotone, Italy, around 220 BC. He taught his students his famous principle in a manner not known to the world. Seriously, we don't know Pythagoras' original proof for his own theorem. But, even after he had proven it to the world, it was not yet common knowledge. This history is far from done. The next person to discover the Pythagorean Theorem was, surprisingly enough, the Greek mathematician, Euclid. In preposition 47 of Euclid's first book of Elements, it is stated in two largely different proofs that the squares of the two smaller sides of a right triangle will add to the equal of a third. Around the same time in China, there was a book written called Nine Chapters and the Mathematical Procedures. In it, many problems are presented, and one which demonstrates the solution for the Pythagorean Theorem. Past this, the Pythagorean Theorem has not changed in hundreds of years, and it remains under the name of Pythagoras to the current day. Many famous people have created their own proofs for the Pythagorean Theorem, including Leonardo da Vinci, Albert Einstein, and former President James Garfield. Here, I will show you a proof by President James Garfield. As you can see here, we have a trapezoid. And the area of a trapezoid is equal to its base times its height, divided by 2. I like to think of this as we're multiplying the height by the average length of the bases. So in this case, the bases are a and b, so we'll add those up and divide by 2, and we'll multiply that by a plus b, which is the height. Simplifying this, we get a plus b squared, all over 2. And, because the area equals a times b, plus c squared over 2, that's the area of the entire trapezoid as broken up into smaller right triangles, because each right triangle that's formed by sides a and b, its area is a times b over 2, and we have two of those. So multiplying out, we just get a, b, and the middle triangle is c squared divided by 2. Then, we're going to substitute in the a plus b squared over 2, in where the a is, in the second equation. Then we'll multiply everything by 2 to get rid of that pesky denominator, and we're left with a plus b squared is equal to 2ab plus c squared. We then are going to multiply out the binomial of a plus b, so we get a squared plus 2ab plus b squared, equals 2ab plus c squared. Subtracting 2ab from both sides, we are left with a squared plus b squared equals c squared, the Pythagorean Theorem. Throughout history, it seems that this simple rule has been discovered, lost, and rediscovered many times over, each person believing they're the first. And yet, Pythagoras, one of the more recent, is the one who took the name. Maybe this is an allegory for human condition, that every time you believe you're original, people have done it in the past, and people will continue to do it in the future. You can never be truly original, yet we always strive for originality. Or maybe it's just an allegory for Reddit.