Archimedes Doc

Archimedes (287-212 BCE) I am going to be scaling the intelligence of Archimedes, like showcasing his feats, based on some pretty well explained videos by Ben Syversen and also other sources, like from Wikipedia articles. How Archimedes Almost Broke Math with Circles In the 3rd century BC, Archimedes proved the area formula of the circle. This was a very big challenge back then because a circle is curved and this was 2000 years before stuff like calculus was invented. Back then, they could only find areas of shapes with straight lines, sure some breakthroughs regarding circles had been made by people before him, but none found out how to derive its area. He also identified pi as we know it today. Archimedes did this by dividing a circle into pieces and realized that the more slices he cut the circle into, the more and more it would resemble a rectangle with lengths of half a circumference and the radius. From what he saw, the circle would be a perfect rectangle with side lengths C/2 and R. So ofc the formula would be A = R*(C/2) aka A = R*Pi*R = Pi*R^2, the formula we know today. However, this did cause a paradox because well wouldn't an infinitesimal slice of something be 0? How can it be anything other than 0 and if they weren't each 0, wouldn't adding an infinite amount of them give infinity? That is what the ancients had a problem with. Another strange thing was how a collection of infinite things added up to a different number compared to a collection of other …