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In 1906, Dutch historian Johan Ludvig Heiberg discovered a church document that turned out to be a copy of one of Archimedes' lost treatises written over. He was able to reconstruct much of the original writing and found Archimedes' first approach for deducing the volume of a sphere. It is mechanical and is based on the Law of the Lever. (See my earlier video on this.) I describe this mechanical approach here.