Archimedes (c. 287–212 BCE) was a mathematician, inventor, and investigator of physical phenomena from Syracuse, a Greek city in Sicily. A major figure in ancient Greek mathematics, he developed rigorous methods for determining areas and volumes, investigated mechanical equilibrium and floating bodies, and became associated with practical machines and military devices. His surviving writings provide firmer evidence of his achievements than the anecdotes preserved by later authors. (mathshistory.st-andrews.ac.uk)
Life and historical setting
Archimedes lived during the Hellenistic period, when Greek scholarship connected communities across the Mediterranean. His birth date is approximate. In The Sand Reckoner, he identifies his father as the astronomer Phidias. He corresponded with scholars associated with Alexandria, including Conon, Dositheus, and Eratosthenes; a period of study in Alexandria is plausible but not securely documented. His career was closely associated with Syracuse and its ruler Hieron II. (mathshistory.st-andrews.ac.uk)
During the Roman siege of Syracuse, ancient accounts credit Archimedes with designing defensive machinery. He died when forces of the Roman Republic captured the city in 212 BCE. Later narratives describe a Roman soldier killing him while he was occupied with a mathematical problem, but their versions differ. The circumstances of his death and his supposed final words cannot be established with the same confidence as the date and setting. (mathshistory.st-andrews.ac.uk)
Geometry and mathematical methods
Archimedes extended geometry through carefully structured demonstrations. His use of the method of exhaustion established exact results by comparing curved figures with rectilinear approximations and ruling out alternative values. These arguments supplied rigorous mathematical proofs without the symbolic machinery of modern analysis. Several results can now be expressed through integration, although his procedures remained geometrical. (mathshistory.st-andrews.ac.uk)
In Measurement of a Circle, he proved that a circle’s area equals that of a right triangle whose perpendicular sides equal the circle’s radius and circumference. He also bounded the ratio now denoted by π:
The calculation compared inscribed and circumscribed regular polygons, reaching polygons with 96 sides. Crucially, the result was a pair of demonstrated bounds, not a claim that π equals . (media.bloomsbury.com)
On the Sphere and Cylinder established that a sphere’s surface area is four times the area of its greatest circular section. Its volume is two-thirds that of the cylinder enclosing it with the same radius and a height equal to the sphere’s diameter. In modern notation, these results give and . (mathshistory.st-andrews.ac.uk)
In Quadrature of the Parabola, Archimedes showed that the area enclosed by a parabola and a chord is four-thirds the area of a particular inscribed triangle. The work combines mechanical reasoning with geometrical demonstration and belongs to his broader investigation of conic sections. His preface explicitly distinguishes discovering a result through mechanics from establishing it geometrically. (mathshistory.st-andrews.ac.uk)
Mechanics and floating bodies
On the Equilibrium of Planes develops a mathematical account of the lever and investigates centers of gravity of plane figures. For two weights on opposite sides of a fulcrum, equilibrium occurs when their distances are inversely proportional to their weights. Expressed in modern notation, . Archimedes organized such results around assumptions about balance and geometrical symmetry. (mathshistory.st-andrews.ac.uk)
On Floating Bodies is a foundational work of hydrostatics. Its propositions explain the behavior of solids relative to the fluid in which they are immersed. The result called Archimedes’ principle states, in modern formulation, that the buoyant force equals the weight of displaced fluid. The treatise also investigates the equilibrium positions of floating paraboloids, connecting buoyancy with shape and stability. (mathshistory.st-andrews.ac.uk)
Machines and anecdotes
The Archimedes screw, traditionally attributed to him, raises water through rotation of a helical structure. Ancient accounts connect it with his visit to Egypt; the device became associated with irrigation and drainage. Accounts of the siege describe artillery and mechanisms for attacking ships, although they do not permit every design to be reconstructed confidently. (mathshistory.st-andrews.ac.uk)
The celebrated crown story appears in Vitruvius, writing substantially later. It relates that Archimedes recognized how water displacement could reveal whether a supposedly gold crown contained silver, then ran outside shouting “Eureka,” meaning “I have found it.” This is a historical anecdote, not a documented experimental report. Likewise, stories of burning ships with mirrors should be distinguished from securely established mathematical achievements. (mathshistory.st-andrews.ac.uk)
Writings and transmission
The Sand Reckoner develops a scheme for expressing extremely large numbers and considers how many grains of sand could fill the universe. It also preserves evidence about ancient astronomy, including Aristarchus’s proposal that Earth moves around the Sun. The Method of Mechanical Theorems, addressed to Eratosthenes, explains how imagined balances and sections of figures could guide mathematical discovery. (mathshistory.st-andrews.ac.uk)
His writings survived through copying, commentary, and Latin translations, including manuscripts from the Byzantine Empire. An especially important witness is the Archimedes Palimpsest: a tenth-century mathematical manuscript whose parchment was later erased and reused for a prayer book. Johan Ludvig Heiberg examined it in 1906, recovering previously unavailable material, particularly The Method. Later conservation and imaging improved readings of its texts and diagrams. (archimedespalimpsest.org)
Most surviving Archimedean works were already known before this discovery. Their transmission helped make them available to Renaissance scholars and later investigators. His determinations of curved areas and volumes were important antecedents of calculus, but should not be equated with the general symbolic methods developed in the seventeenth century. (archimedespalimpsest.org)