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What did Archimedes’ mechanical method use to discover geometric results?

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Portrait of Archimedes by Giuseppe Nogari shows an elderly man with white hair and beard, holding a compass, gazing downward, set against a dark background. — Archimedes Archimedes, oil on canvas by Giuseppe Nogari, 18th century; in the Pushkin Fine Arts Museum, Moscow. Archimedes was an ancient Greek mathematician and inventor.

Archimedes’ mechanical method, as described in his treatise Method Concerning Mechanical Theorems, was a heuristic technique used to discover geometric results such as the area of a parabolic segment and the volume of a sphere.1

The process involved the following steps:
* Division: Archimedes divided two figures into an infinite number of infinitesimally thin strips.1
* Weighing: He then "weighed" each corresponding pair of these strips against each other on a notional balance. By applying the law of the lever, he could determine the ratio of the two original figures.12

Archimedes emphasized that while this mechanical approach was an effective tool for discovery, it did not constitute a rigorous mathematical proof. For his formal proofs, he relied on the method of exhaustion, a rigorous procedure involving successive finite approximations.13