In one of Kepler’s lesser-known works, Volume-Measurement of Barrels, he does this in three dimensions, slicing barrels into planes and summing the planes together. Kepler, at least, wasn’t afraid of a glaring problem: as Δx goes to zero, the sum becomes equivalent to adding an infinite number of zeros together—a result that makes no sense. Kepler ignored the problem; though adding infinite zeros together was gibberish from a logical point of view, the answer it yielded was the right one.
