The great insight of Eudoxus and Archimedes was that it doesn’t matter whether it’s a circle or a polygon with very many very short sides. The two areas will be close enough for any purpose you might have in mind. The area of the little fringe between the circle and the polygon has been “exhausted” by our relentless iteration. The circle has a curve to it, that’s true. But every tiny little piece of it can be well approximated by a perfectly straight line, just as the tiny little patch of the earth’s surface we stand on is well approximated by a perfectly flat plane.* The slogan to keep in
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