How Archimedes almost broke maths with circles

[Kyng]

There's no doubt that Archimedes was one of the greatest mathematicians of Ancient Greece. In particular, he's credited with discovering the following formula for the area of a circle:

Area = Circumference * Radius / 2 = πr²

However, in doing so, he revealed a paradox at the heart of mathematics, which wasn't fully resolved until the 1800s. So, what was the problem?

• Basically, to prove his formula, he took a circle; chopped it up into pieces (like pizza slices), and re-arranged these slices into something that looked a bit like a rectangle. The more slices he cut the circle into, the more 'rectangular' his shape became. And he argued that, if he could cut the circle into infinitely many slices, then his shape would be a perfect rectangle, with side lengths r and πr. Then, the area of this shape (and thus, the circle) would simply be r x πr, or πr².
  
  
• However, the problem is, "adding together infinitely many infinitely small slices" doesn't really make sense . If the infinitely small slices have area 0, wouldn't the sum of those slices also have area 0? Or if they had area greater than 0, wouldn't adding them all together just give infinity? And how is it that different sums of infinitely small slices produce different totals? One solution is to modify the proof to show that the area is neither greater than πr², nor smaller than πr² (and thus must be exactly πr²) - so, you're using infinity without actually using infinity .
  
  
• However, in the 1600s, mathematicians and physicists like Fermat, Galileo, and Kepler once again had to wrestle with the infinite. Fortunately, Isaac Newton came to the rescue with the theory of calculus - but he remarked that, while his theory was useful for talking about infinity, it wasn't rigorous enough to constitute actual 'proof'. It wasn't until the 1800s that this theory was placed on solid foundations - when French mathematician Augustin-Louis Cauchy developed "epsilon-delta" arguments, that allowed things to get arbitrarily close to one another without actually touching. But that's a matter for another topic in and of itself .

Hopefully you found this interesting - and let me know if you have any questions, or anything to add !