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Archimedes developed exact methods for geometric measurement. In Measurement of a Circle by Archimedes, he proves the relation between a circle's area, radius and circumference and obtains rational bounds on its circumference-to-diameter ratio.^[ref]

The bounds $223/71<\pi <22/7$ come from comparing inscribed and circumscribed polygons. Increasing the number of sides traps the circular quantity between computable geometric quantities. The argument illustrates the power of a rigorous bound: one need not express a number exactly to establish an accurate approximation with a guaranteed error.^[ref]