Archimedes's Integrals

One of my staff (thanks Fergus) told me ages ago about Archimedes' proof that the volume of a sphere is 4/3 π R³ (where R is the radius of the sphere). It is a very very cool proof and it's high time I shared it! One of the reasons it is so cool is that it uses the concept that a volume can be produced by stacking up a whole lot of thin slices. This is the idea behind integration, and Archimedes used this idea thousands of years before Newton or Leibniz.

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