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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]

Thomas Heath was a historian of mathematics whose translations and commentaries helped make ancient Greek mathematical texts accessible to modern readers. Alongside a career in the civil service, he published work on Diophantus, Apollonius of Perga, Archimedes and Euclid, including a three-volume account of Euclid's Elements.^[ref] His scientific importance lies in connecting the surviving text with its mathematical argument: translating the words alone does not explain how an ancient construction or proof fits into its original setting. His commentary remains useful when comparing historical geometric reasoning with modern algebraic notation.