Archimedes — Key Ideas to Explore

The Mechanical Method in Practice

Heath devotes considerable space to Archimedes’ mechanical method, which treats geometric figures as if they were physical bodies on a lever. The excerpts describe how Archimedes would “produce the diameter or axis in the direction away from the figure” and imagine it as a lever. He then concentrated elements of an unknown figure X at one point on the lever, while elements of a known figure B acted at their actual positions. By balancing moments about a fulcrum, he derived the area or volume of X.

Heath lists the results Archimedes obtained this way: the area of a parabolic segment, volumes of a sphere and spheroid, volumes of segments of these solids, and centers of gravity for hemispheres, spherical segments, and conicoid segments. The method was not a final proof but a discovery tool; Archimedes always followed it with a rigorous geometrical demonstration by exhaustion.

Two Solid Figures from the Lost Treatise

A specific treatise, described in the excerpts, presents two solids: one is a cylinder inscribed in a rectangular parallelepiped on a square base, cut by a plane through a side of one square face and the parallel diameter of the opposite base. The resulting solid resembles a “horse’s hoof,” and Archimedes proved its volume is one-sixth of the parallelepiped. The second solid arises from two cylinders inscribed in a cube, each cylinder’s base a circle in opposite square faces; their intersection produces a rounded solid whose volume is two-thirds of the cube.

Heath notes that Archimedes first proved these volumes by the mechanical method, then gave rigorous geometrical proofs by exhaustion. The manuscript of this treatise is unfortunately incomplete, but Heath’s account preserves the essence of Archimedes’ reasoning.

Biographical Framing and Source Use

Heath’s biography is structured around Archimedes’ works, not his life story. The first chapter sketches the known facts: his birth about 287 BCE, his father Phidias an astronomer, his study at Alexandria, his friendship with Conon and Eratosthenes, and his return to Syracuse where he “lived a life entirely devoted to mathematical research.” Heath emphasizes that Archimedes’ mechanical inventions were, in Plutarch’s phrase, “the diversions of geometry at play,” and that Archimedes himself attached no importance to them.

The excerpts reveal Heath’s reliance on Polybius and Plutarch for the siege of Syracuse and Archimedes’ death in 212 BCE. He does not invent details; where sources are silent, he says little. The biography thus functions as an introduction to the mathematics, not a full life narrative.

Heath’s Archimedes is best approached as a companion to reading the original works. The mechanical method and the two solid figures are presented with enough clarity to follow the reasoning, but the excerpts are incomplete—the treatise on the hoof-shaped solid and the intersecting cylinders is only partly preserved. Readers interested in the full mathematical details should consult Heath’s translations of Archimedes’ works, where the rigorous proofs are given. This biography provides the historical and conceptual framework.

Reading about Archimedes’ quiet proofs, I recalled sitting with my father, tracing circles in the dust. The weight of a perfect argument felt like a held breath. Another afternoon, the same stillness came from a small book, though. The Second Story of Meno A Continuation of Socrates' Dialogue with Meno in Which the Boy Proves Root 2 is Irrational — Context and Discussion somehow brought that same ache of discovering a shape that refuses to be contained.