Method of exhaustion

The method of exhaustion proves area and volume results by trapping a quantity between figures whose measures converge toward it. Ancient Greek mathematicians used it as a rigorous precursor to integral calculus.

Eudoxus's principle: A proportionality principle that compares magnitudes without requiring them to share a unit or be commensurable. It supplies the proportional reasoning behind classical exhaustion proofs.
Eudoxus of Cnidus: A fourth-century BCE Greek mathematician and astronomer associated with the theory of proportion and the method of exhaustion. His theory of proportion grounded the method's rigorous comparison of magnitudes.
Euclid's proof of the circle area theorem: A proof that the areas of circles are proportional to the squares of their diameters. Euclid uses exhaustion to show that no proposed proportionality factor can differ from the true one.
Indivisibles: A geometric approach that treats figures as collections of infinitely many lower-dimensional elements. Unlike exhaustion, indivisibles reason directly from constituent lines or slices.
Double reductio ad absurdum: A proof strategy that rules out both a quantity being too large and being too small. Exhaustion proves equality by excluding both possible inequalities.
Euclid's Elements: A thirteen-book Greek mathematical treatise that organized geometry, number theory, and related results. Book XII uses exhaustion to prove results about circles, pyramids, cones, and spheres.
Archimedes' measurement of the circle: A short treatise proving that a circle's area equals the area of a right triangle with legs equal to its radius and circumference. The proof bounds the circle between polygonal areas to establish an exact equality.
Cavalieri's principle: A principle that compares areas or volumes by matching lengths or areas of parallel cross-sections. It reaches geometric area and volume results by a different comparison strategy.
Inscribed polygon: A polygon whose vertices lie on a curve and whose interior lies within the enclosed region. Inscribed figures provide lower bounds for the area being exhausted.
Archimedes: A Greek mathematician and engineer of the third century BCE known for work on geometry, mechanics, and hydrostatics. He refined exhaustion into some of antiquity's most powerful area and volume proofs.

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