Who was Archimedes?

One of the most repeated stories about Archimedes, recorded by the Roman architect Vitruvius roughly two centuries after his death, concerns a golden wreath commissioned by King Hiero II for a temple. Suspecting the goldsmith had secretly replaced some of the gold with silver, Hiero asked Archimedes to find a way to test the wreath without damaging it. According to the tale, Archimedes realized while lowering himself into a bath that the water rose in proportion to the volume of his body, and understood this effect could reveal the wreath's composition. He is said to have run into the streets shouting "Eureka" in his excitement, having forgotten to get dressed.

Vitruvius describes Archimedes testing the wreath by comparing how much water it displaced against equal weights of pure gold and pure silver. A separate, anonymous Latin poem called the Carmen de Ponderibus gives a different version, in which gold, silver, and the wreath were weighed against each other on a balance submerged in water, allowing differences in density to tip the scale. This second method matches the buoyancy principle Archimedes actually describes in his surviving work On Floating Bodies, where an object immersed in fluid experiences an upward force equal to the weight of fluid it displaces. The astronomer Galileo Galilei, who built his own hydrostatic balance centuries later, considered this balance-based version more likely to reflect what Archimedes actually did.

Much of Archimedes' engineering work seems to have served the practical needs of Syracuse. The writer Athenaeus describes an enormous ship called the Syracusia, commissioned by Hiero II and said to be the largest vessel built in antiquity, which Archimedes was credited with helping launch. Plutarch tells a related story in which Archimedes boasted he could move any weight given a place to stand, prompting Hiero to challenge him to move a fully loaded ship single-handedly. Writers disagree on how this was done, crediting either a pulley system or a hoisting device called the baroulkos, while Pappus of Alexandria attributed the feat to Archimedes' understanding of leverage and mechanical advantage.

When Rome besieged Syracuse in 214 BC during the Second Punic War, after the city switched its allegiance to Carthage, Archimedes reportedly built defensive war machines for the city, directing their use personally according to Plutarch. Separate accounts from Plutarch, Livy, and Polybius describe improved catapults along with cranes that dropped heavy weights onto Roman ships or used iron claws to lift and capsize them, and these machines significantly delayed the Roman advance. A later and far less certain legend claims Archimedes used mirrors to set Roman ships ablaze with concentrated sunlight; this detail is absent from the earliest accounts and only appears in writings from centuries afterward, with its plausibility still debated among historians today.

Archimedes extended a technique called the method of exhaustion, used earlier by mathematicians such as Eudoxus of Cnidus, to measure curved shapes with a rigor that anticipated modern calculus. In his work Measurement of a Circle, he approximated the value of pi by inscribing and circumscribing regular polygons around a circle and repeatedly doubling the number of sides, eventually reaching 96-sided polygons to show that pi lies between roughly 3.1408 and 3.1429.

In Quadrature of the Parabola, he proved that the area enclosed by a parabola and a straight line equals four-thirds the area of a related triangle, demonstrating this using an infinite series of shrinking triangles whose areas sum neatly to that ratio. He applied similar exhaustion-based reasoning to calculate the surface area of spheres and cones, the area of ellipses, and the area enclosed by the spiral curve that now bears his name.

Archimedes also described a separate, more intuitive method in a treatise called The Method of Mechanical Theorems, in which he used the law of the lever to balance geometric figures against each other and estimate volumes and areas before proving them formally. By his own account, he used this physical, almost experimental approach to discover results first, only afterward constructing the rigorous exhaustion-based proofs for which he became known.

Beyond geometry, Archimedes tackled the problem of expressing enormous quantities. In The Sand-Reckoner, he built a numbering system based on powers of a myriad, or ten thousand, in order to calculate a number larger than the grains of sand that would fill the universe, arriving at a figure of roughly eight vigintillion. In the separate Cattle Problem, addressed to scholars at Alexandria, he posed a system of equations whose full solution produces a number with over two hundred thousand digits.

When Syracuse finally fell to the Romans, Archimedes was killed despite an order from the Roman general Marcellus that he be spared. The earliest account, by Livy, simply states he was killed by a soldier who did not recognize him while he was drawing geometric figures in the dust. Plutarch adds that Archimedes refused to stop working on his problem when ordered to follow the soldier, prompting the fatal attack, while the writer Valerius Maximus records his final words as a plea not to disturb his drawings, a line later popularized as "do not disturb my circles," though that exact phrase appears nowhere in ancient sources.

Marcellus was said to be angered by the death, since he valued Archimedes highly and had reportedly brought two mechanical planetariums built by Archimedes back to Rome, one of which he kept for himself. When Cicero later served as a quaestor in Sicily, he located Archimedes' neglected tomb near a gate of Syracuse and had it restored, noting that it was marked with a carved sphere and cylinder representing Archimedes' proof that a sphere's volume and surface area are two-thirds those of its surrounding cylinder.

Archimedes' writings were not widely read in antiquity itself, circulating mainly among scholars in Alexandria. It was not until around 530 AD that a comprehensive compilation of his works was assembled in Constantinople, after which translations into Arabic in the ninth century and Latin in the twelfth century carried his ideas into the Islamic world and medieval Europe, where they later influenced scientists of the Renaissance and Scientific Revolution. A major turning point came in 1906, when previously lost treatises, including The Method of Mechanical Theorems, were rediscovered hidden beneath later religious text in a reused parchment known as the Archimedes Palimpsest, offering fresh insight into how he actually arrived at his results.