Thomas Heath opens his biography of Archimedes by acknowledging that popular memory reduces the mathematician to a few anecdotes: the bath-tub cry of εὕρηκα, the boast about moving the earth, and his death during the sack of Syracuse. Heath immediately counters this by asserting that Archimedes was “the greatest mathematical genius that the world has ever seen,” a claim he supports not with hyperbole but with a detailed exposition of the mathematician’s original methods.

The excerpts show Heath drawing on Polybius and Plutarch for biographical facts—Archimedes was born about 287 BCE, studied at Alexandria with successors of Euclid, and corresponded with Conon and Eratosthenes. Yet the core of the book is not narrative but technical: Heath explains Archimedes’ mechanical method for finding areas and volumes, using the lever principle and centers of gravity, then his rigorous geometrical proofs by exhaustion.

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.