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The Method of Exhaustion

Trap a curve between polygons, squeeze until the gap vanishes beneath any magnitude you name — and prove the area exactly, with no infinity in sight.

How do you measure the area of a circle or the volume of a cone, shapes whose edges curve and never sit flat? The Greeks answered by squeezing: fit polygons inside and outside the figure, and keep doubling their sides so the leftover gap shrinks below any size you care to specify. By showing that any competing answer would eventually be contradicted, they proved areas and volumes exactly — a rigorous forerunner of the integral calculus, achieved two thousand years before limits were formalized.