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How did Archimedes use thin slices to compare the volumes of solids?

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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 utilized a heuristic technique, detailed in his work Method Concerning Mechanical Theorems, to discover formulas for areas and volumes by conceptualizing solids as being composed of an infinite number of infinitesimally thin strips or sections.12

To compare the volumes of two solids, he divided each figure into a corresponding family of these thin, parallel cross sections. He then "balanced" these pairs of sections against each other on a notional lever, applying the law of the lever to determine the ratio of the two original figures based on the weights of their corresponding parts.12

While this "mechanical" method allowed him to arrive at key discoveries—such as the volume of a sphere—Archimedes explicitly noted that it was a tool for discovery rather than a rigorous proof. For his formal mathematical proofs, he relied on the method of exhaustion, a rigorous technique of successive finite approximation established by Eudoxus of Cnidus.123