Welcome to Numberholic! I’m your host, Sejari, and you’re watching the third episode of Proven Yet Hard to Believe. In this series, we uncover fascinating stories behind some of the most significant proofs and constructions in the history of mathematics. Today, we’re diving into a cornerstone of ancient Greek mathematics: the Pythagorean Theorem. While it’s a household name, there’s much more to its story than meets the eye. Are you ready? Let’s get started!
When you think of ancient Greek mathematics, the most iconic proof is undoubtedly the Pythagorean Theorem. It’s so well-known that we might be tempted to skip it altogether. But wait—there’s a common misunderstanding we need to clear up!
Did you know that there are over 400 different proofs of the Pythagorean Theorem? Back in the early 20th century, Elisha S. Loomis published a book compiling 344 of these proofs. Since then, even more have been discovered. Isn’t that incredible? Today, we’ll take a look at just a few of the most intriguing ones.
First, let’s explore the diagram traditionally credited to Pythagoras himself—an elegant construction that supposedly marked his first successful proof.
Next, we have the approach by Bhaskara, the Indian mathematician, whose brilliant method we’ve introduced in a previous episode.
Did you know even Leonardo da Vinci contributed his own proof to the theorem? What a fascinating connection!
But the story doesn’t stop there. The Pythagorean Theorem wasn’t limited to Greece and India. In ancient Chinese mathematics, it appears under the name Gougu Theorem. One notable example is found in the Zhoubi Suanjing, an ancient text that likely originated as early as the 11th century BCE during the Zhou dynasty, though it was expanded and compiled into its current form by the 2nd century CE. Isn’t it amazing to think about the cultural exchange and parallel developments?
Interestingly, the Zhoubi Suanjing and Euclid’s Elements were written around the same period, and both serve as collections of mathematical knowledge up to their respective times. However, the Gougu Theorem might predate Pythagoras himself by 500 years! That raises an important question: can we call these illustrations true “proofs”?
If we were to accept these as proofs, there is evidence suggesting that the Babylonians knew the Pythagorean Theorem much earlier.
Take a look at this simple visual transformation. While the diagram might seem to “prove” a theorem at a glance, there’s an issue. Can you spot where this extra square suddenly came from? Shifting the shapes leaves unexplained gaps, which undermines the logical rigor we demand from a proof.
And that’s why intuitive, picture-based demonstrations alone aren’t enough. Pythagoras’s true achievement wasn’t just creating diagrams but establishing the logical underpinnings of the theorem. He formalized definitions and propositions such as:
- A right angle is half of a straight angle.
- The sum of the two acute angles in a right triangle equals a right angle.
These concepts, combined with earlier results attributed to Thales—like the property that angles in a semicircle are right angles and the base angles of an isosceles triangle are equal—allowed Pythagoras to deliver a rigorous proof. Isn’t it fascinating to see how these pieces come together?
Fast-forward two centuries to Euclid, who extended these ideas further. He generalized the interior angle sum property to any triangle, proving in Elements Book One, Proposition 32, that the sum of the interior angles of a triangle equals two right angles. It may seem trivial now, but this leap required mastering the abstract concept of “parallelism,” which wasn’t solidified until Euclid’s time.
Euclid’s method, celebrated for its elegance, eventually traveled through the Arab world, where this particular proof earned the nickname “Bride’s Chair.” As we conclude today’s episode, let’s take a moment to appreciate this stunning method and its enduring legacy.
If you enjoyed today’s exploration, don’t forget to like the video and subscribe to Numberholic! I’m Sejari, and I’ll see you next time on Proven Yet Hard to Believe, where we continue to uncover the mathematical wonders that shaped history. See you soon!