It’s a boring result but a gorgeous proof.

Because if you have mis-matched halves you also have “unequal radii” and that means you’re not a circle. So we have proved that two things are incompatible with one another: You can’t be both a circle, and have mis-matched halves. A circle is a figure that’s equally far away from the midpoint in all directions. But this means that the thing wasn’t a circle to start with. Then one radius is longer than the other. Now, draw a radius in that direction, from the midpoint of the circle to the place on the perimeter where the two halves don’t match up. So there must be some place where one of the two pieces is sticking out beyond the other. The pieces were not equal, we assumed, so when you flip one on top of the other they don’t match up. Take one of the pieces and flip it onto the other. And we suppose that those two pieces are not the same. Very well, so we have a line going through the midpoint of a circle, and it’s cut into two pieces. Suppose the diameter does not divide the circle into two equal halves. This is going to be a proof by contradiction. It’s not about the theorem, it’s about the proof. How can you fall in love with geometry by proving something so trivial and obvious?īut don’t despair. Pretty disappointing, isn’t it? What a lame theorem. That love-at-first-sight moment, that theorem that opened our eyes to the power of mathematical proof, was: That a diameter cuts a circle in half. So, here we go: What was the first theorem ever proved? What was the spark that started the wildfire of axiomatic-deductive mathematics? The best guess, based on historical evidence, goes like this. Let’s analyse that question, the credibility question, in a bit more depth later, but first let’s take the stories at face value and see how we can relive the creation of deductive geometry as it is conveyed in these Greek histories.

History is perhaps mixed with legend in those kinds of accounts, but key aspects are likely to be quite reliable. But we still sort of know what Thales proved, more or less. Hundreds of years before we have any direct historical sources for Greek geometry. The Greek tradition tells us who had this lightbulb moment: Thales. Why would anyone sit down and say to themselves “I’m gonna prove some theorems today” when nobody had ever done such a thing before? How could that idea enter someone’s mind out of the blue like that? How did proofs begin? It’s like a chicken-or-the-egg conundrum. Or subscribe with your favorite app by using the address below

Geometry proofs