In the third century BC, Archimedes proved the area formula for a circle. His method harnessed the power of infinity in a way that would set the stage for calculus to emerge nearly 2,000 years later, but it also revealed a paradox at the heart of math that wasn't addressed rigorously until the 1800s. Archimedes was the greatest mathematical thinker of his time, and among the most important mathematicians of all time. In addition to writing a number of books that expanded the ancient Greek understanding of geometry, he was a powerful inventor, creating the first compound pulleys as well as a type of water pump that is still in use today. Archimedes was so passionate about math that he would often forget to bathe and eat. He died at the tip of a sword when he refused in invading Roman soldiers' orders because he was too engrossed in a math problem. But more on that later. So what was so special about circles anyway? To answer that, let's start with rectangles. It's pretty easy to see how we can divide this rectangle into little squares on a grid and multiply length by width to find the area. But when we try to do the same thing with a circle, the diagram is a complete mess. The circle doesn't line up with the grid, no matter what we do. Back in Archimedes' time, mathematicians and philosophers were already wrestling with this problem. They were fine with areas of shapes the head straight lines, like rectangles and triangles, but working with circles and curves was not as simple. There's some disagreement here about who knew what when, but it seems like Udoxus of Nidus had shown that the area of a circle was proportional to the square of the radius, over 100 years before Archimedes, and Euclid knew that all circles are similar regardless of size. But Archimedes is credited with identifying pi as we know it today and with proving the area formula for a circle. To understand the thinking behind Archimedes' proof of the area formula for a circle, let's start by dividing a circle into four pieces like we're slicing at pizza. Now notice that the circumference has also been split in four. So if we rearrange these into an interlocking shape, we can see that there are two slices pointing up and two slices pointing down, so the distance across both the top and the bottom parts of this shape is half the total circumference. Using modern notation, we'd say that this is equal to pi times the radius. Meanwhile, the other sides are equal to the length of the radius, and we can make those vertical by cutting one of the slices in half and moving it to the other side like that. Now this shape doesn't really look like anything familiar yet though, so let's try cutting the circle into smaller pieces and seeing what happens. If I do eight pieces instead of four, you can already see that the top and bottom sides are the same length that they were before, but now they're starting to smooth out and get a bit straighter. If we keep going, I'll cut the circle into 16 pieces, then 32, then 64, and we get the idea of what's happening. To smaller the slices, the closer this thing comes to looking like a perfect rectangle, and we know how to find the area of that. Now it's actually still not a rectangle though, because if we zoom in closely enough, we can see that the sides are bumpy instead of smooth. But we could keep making these slices smaller, and here's the kicker. According to this logic, if we cut the circle into an infinite number of slices, then based on what we saw, the circle will form a perfect rectangle with an area equal to the radius times half of this circumference. But this is where the problems begin. Archimedes and other mathematicians of his time knew that cutting a circle into an infinite number of infinitely small pieces actually doesn't make any sense. If something is infinitely small, doesn't that mean its size has to be equal to zero? And if you add up an infinite number of things that all have a size of zero, how could the sum be anything other than zero? But if being infinitely small means something other than being equal to zero, wouldn't adding up an infinite number of them just equal infinity? And even stranger, how could it be that some collections of infinitely small things could add up to a different amount than other collections of infinitely small things? This was all a big problem at the time, so Archimedes needed a way to use infinity without actually using infinity. He decided to base his proof on a method from Udoxus called exhaustion, where he showed that the area of the circle was neither bigger than the rectangle nor smaller than the rectangle and therefore had to be exactly the same size. Now Archimedes actually formed the sectors of his circle into a right triangle rather than a rectangle, but I showed you the rectangle explanation in this video because it's a bit easier to visualize. Archimedes got away with not really talking about infinity, but it was clear that this issue wasn't going anywhere. Mathematicians that wanted to study curved shapes were eventually going to have to reckon directly with the infinitely small. But for thousands of years, this concept was basically untouchable in Europe. During the Middle Ages, the works of Archimedes and Euclid seemed to have been lost entirely. The only copies that survived the early part of the era had been passed down over the centuries by scribes and Constantinople who often didn't understand what they were writing. By the 8th and 9th centuries CE, thinkers in India and the Islamic world were wrestling with their own versions of these concepts as well and even independently made some of the same discoveries as Archimedes. Back in Europe, by the 16th and 17th centuries, scientists and philosophers like Firmat, Galileo and Kepler started to engage with the idea of the infinitely small ones more. But there were still prominent detractors such as René Descartes and Thomas Hobbes. In 1632, the Roman Jesuits launched a war on this idea and its proponents, attempting to ban the concept from ever being taught or even mentioned. But by the late 1600s, a sickly and ill-tempered young mathematician named Isaac Newton, who had studied Archimedes' work closely, was about to turn the world on its head. When Newton was still in his early 20s, he developed his theories of calculus by exploding what could be done with this idea of the infinitely small. But for decades, he avoided sharing his theories with the world. When he finally did reveal his work, he wrote that the thinking was useful for solving problems, but it wasn't rigorous enough for proofs. Even Isaac Newton himself hadn't yet found a way to talk clearly about infinity. It wasn't until the 1800s when Augustine Louis Koshy finally found a rigorous method for describing infinitesimally small values with his epsilon delta proofs. In some ways, the basic strategy behind these proofs was actually remarkably similar to the method of exhaustion shown in Archimedes' circle proof 2100 years earlier. Koshy's method allowed things to be infinitely close together without actually reaching. His notation survives to this day, alongside the notation created by Leibniz, who's credited with inventing calculus independently from Newton. Armed with newfound legitimacy, calculus became a driving force behind many of modern civilizations' greatest achievements through the 19th and 20th centuries. The idea of describing motion and change by dividing things into infinitely small pieces allowed for innovations like radio, computers, GPS, ultrasounds, mapping the human genome, and much more. It's not an exaggeration to say that this one idea changed the world. For Archimedes, circles were far from his only achievement. His accomplishments included mathematically proving the laws of levers, discovering the method known as the Archimedes' principle for measuring the volume of objects using water, improving that the volume and surface area of the sphere are two-thirds of the volume and surface area of an enclosing cylinder. This last one was his favorite proof and is printed on his tombstone. It is said that in the year 1500, all of Europe knew less about math than Archimedes did on the day that he died 1700 years earlier. Now about that day, Archimedes spent the final year of his life defending his home of Syracuse Sicily from the invading Roman army led by Marcus Claudius Marcellus. Archimedes contributed to the war effort by creating weapons to keep the Romans at bay, including a device for capsizing boats known as the Claw. The Romans knew that Archimedes was an important military and intellectual asset, and General Marcellus ordered him to be captured unharmed. But when that Roman soldier found him, legend has it that Archimedes was deep in contemplation and refused to step away. It is said that his final words were do not disturb by circles.