The method of exhaustion

Archimedes finds the area of a parabolic segment by exhausting it with triangles, anticipating key ideas of the calculus by two millennia. We shall explore exhaustion and some paradoxical failures.

Archimedes and quadrature of the parabola

Archimedes (3rd century BC) set himself the task of calculating the area of a parabolic segment, the region between a parabola and a linear chord crossing it, as shown in the figure here.

His treatise, the Quadrature of the parabola, develops the geometric theory of parabolas and contains two proofs of his soluti…