The Method of Mechanical Theorems (also referred to as The Method) is a work by Archimedes which contains the first attested explicit use of indivisibles (sometimes referred to as infinitesimals). The work was originally thought to be lost, but was rediscovered in the celebrated Archimedes Palimpsest. The palimpsest includes Archimedes' account of the "mechanical method", so called because it relies on the law of the lever, which was first demonstrated by Archimedes, and of the center of gravity, which he had found for many special cases.
Archimedes did not admit infinitesimals as part of rigorous mathematics, and therefore did not publish his method in the formal treatises that contain the results. In these treatises, he proves the same theorems by exhaustion, finding rigorous upper and lower bounds which both converge to the answer required. Nevertheless, the mechanical method was what he used to discover the relations for which he later gave rigorous proofs.
Read more about The Method Of Mechanical Theorems: Area of A Parabola, First Proposition in The Palimpsest, Volume of A Sphere, Surface Area of A Sphere, Curvilinear Shapes With Rational Volumes, Other Propositions in The Palimpsest
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