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The ancient Greeks certainly used calculus. Archimedes used regular polygons inscribed and circumscribing a circle to estimate the value of pi. The perimeter of the circle (the circumference) lay between the perimeters of the two sets of polygons. The limit, as the number of sides of the polygon were increased, gave pi. Unfortunately, calculations of trigonometric ratios were not sufficiently refined for Archimedes to get very accurate results.

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