Squaring the circle. Three different perspectives


All students know the specific formula of the circle’s area. They have learned this thing in school, without a previous proofs neither justification. Possibly because the reasoning developed by Archimedes, to find this algorithm, is too complex, and it can not be understood by children who are not yet trained in the art of the logic and the reasoning by reductio ad absurdum. The most ancient cultures, as those Egyptian and Babylonian, devised a formula to calculate these surfaces, though not as perfect as that discovered by Archimedes. Before Alexandrian geometer, Antiphon, Hippocrates and Euclid they tried to determine the circle’s quadrature by approximating polygons to the curve. Centuries later, Evangelista Torricelli put together both methods coming from Archimedes and Bonaventura Cavalieri, by completing the circle and the right triangle with indivisible magnitudes. In this article we will review all these approaches, by watching how these constructions discussed the issue from different perspectives: Babylonian tessellations, exhaustion method, and indivisibles. Moreover, we can leave aside the more intricate deductions in order to show our students the underlying heuristic to the conceptualization of the circle’s measure. nike sb

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