The early Greek mathematicians knew the formulas for the circumference and area of a circle, and they knew the formulas for the surface area and the volume of a sphere, but no one noticed that the change in formula from circumference to area is the same as the change in formula from surface area to volume, as we show below. These two changes are similar because they both go from the formula for outer envelope to the formula for what is contained: the circumference is like a fence around a field, which is the area, the thing contained. The surface area is like a balloon around the volume of air it contains.
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