Quadrature of the Parabola

Quadrature of the Parabola (Greek: Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regarding parabolas, culminating in two proofs showing that the area of a parabolic segment (the region enclosed by a parabola and a line) is that of a certain inscribed triangle.

It is one of the best-known works of Archimedes, in particular for its ingenious use of the method of exhaustion and in the second part of a geometric series. Archimedes dissects the area into infinitely many triangles whose areas form a geometric progression. He then computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment. While this was an impressive feat of real-analytical reasoning for Ancient Greek mathematics, it remained ad hoc. It wasn't until the 17th century that Cavalieri's quadrature formula anticipated the deeper and more general methods of integral calculus.

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