How did Archimedes use the method of exhaustion without assuming an actual infinity?

Archimedes utilized the method of exhaustion to calculate areas and volumes by avoiding the concept of actual infinity, which was rejected by the ancient Greeks. Instead, he employed a rigorous logical procedure based on finite approximations.123

To measure a figure, Archimedes would "exhaust" its area or volume by inscribing or circumscribing it with simpler shapes, such as polygons or prisms, whose dimensions were known. He proved his results by demonstrating that any value other than the one he proposed could be "exhausted" or undercut by these finite approximations.4

For example, when calculating the area of a parabolic segment, he used an infinite geometric series but avoided direct contact with infinity by showing that the sum of any finite number of terms could be made to exceed any value less than the total area. By proving that the area could not be anything other than his calculated result, he established a rigorous bound without requiring the existence of an actual infinite quantity. This approach effectively treated the desired value as a limit, though it relied on the theory of proportions rather than modern limit notation.4513