 So, both trisecting the angle and squaring the circle can be solved using the curve known as the quadratrix, which was described by Hippias of Ellis around 430 BC. Ellis is a part of Greece that's on the western side of the Peloponnesian Peninsula. Hippias belonged to a school of philosophy known as the Sophists. Generally speaking, the Sophists had a bad reputation because they taught other people for money. And it's worth pointing out that the objection was not based on the idea that knowledge should be free, but more based on the idea that unless you were independently wealthy, you had no business teaching other people how to think. Now, Hippias invented a curve, and the curve is known by its Latin name, the quadratrix. Unfortunately, none of Hippias' writings have survived, and so we only know what he did based on others writing about him several centuries after his death. But evidently, he described the quadratrix as follows. Suppose you have a square and a quadrant of a circle. Let the top side fall towards the bottom, and at the same time, let a radius of the circle also turn towards the bottom. And here's the trick. Let's synchronize their movements so the radius and the side reach the bottom simultaneously. The intersection of the turning radius and the falling side form the quadratrix. So the quadratrix can be used to trisect the angle. So to trisect an angle, let's place the vertex at the center of the circle and set one leg along the base. We'll identify where the angle meets the quadratrix, and the point on the quadratrix, where one third of the height will be a point on the trisected angle. What's worth noting here is this isn't just limited to the trisection of an angle. We can replace one third with any fractional about that we want. The quadratrix allows us to take an angle and divide it into any number of pieces. Dynastratus, who lived around 350 BC and was the brother of Monochmas, is credited with squaring the circle using the quadratrix. His construction relies on finding the terminal point, and there's just one problem. That terminal point doesn't exist. So the quadratrix is formed by the intersection of the falling line and the turning radius, and at the bottom the line and the radius coincide. There's no intersection point. Of course today we know how we'd handle this, we'd define that terminal point as a limit. And so this may actually be the first appearance of a true limit in mathematics. Now if we could locate the terminal point h, Dynastratus proved that the length of the arc AC is to the radius AD, as the radius AD is to the length DH. We'll ignore one potential problem. Isn't that a ratio between a straight line and a curved line? Now since AD and DH are line segments, we can find the line PQ, where PQ is AD as AD is to DH. Because this means that PQ is equal in length to arc AC. And so this gives us a line that's equal in length to one quarter of the circumference of a circle, and then we can use this to square the circle. We'll leave those details to the viewer. And so by the fourth century, all three of these classical problems were solved. You could duplicate the cube, you could trisect the angle, you could square the circle. So why do we say they're impossible? Because you're playing a game of chess. Here's one way to win. Your opponent, player 1, starts off pawn to king 4. And your move, I'll give you $5,000 to lose. And your opponent, concedes. And this is an illustration of an important idea. Winning isn't everything, it's how you win that matters. Around 400 BC an ideal emerged in geometric problems. Problems in geometry should be solved using only a collapsible compass and a straight edge. Using any other device is considered cheating. And so while we can duplicate the cube using conic sections, trisect the angle and square the circle using the quadratrix, these are not compass and straight edge constructions, and so they do not count as solutions. And in fact, the impossibility of being able to solve these three problems is exactly like the impossibility of winning a game of chess if the only piece you have left is the king.