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Archimedes
c. 287 BCE – c. 212 BCE • Syracuse, Sicily • Mathematician · Physicist · Inventor
Archimedes of Syracuse stands alone at the summit of ancient science. Born around 287 BCE on the island of Sicily, he spent most of his life in Syracuse — a flourishing Greek colony — and produced a body of mathematical and physical work so advanced that it would not be equaled for nearly two thousand years. He calculated pi to unprecedented accuracy, derived the formula for the surface area and volume of a sphere, laid the groundwork for integral calculus fifteen centuries before Newton and Leibniz, and invented war machines that held the might of Rome at bay. He accomplished all of this without algebra, without modern notation, and without peers anywhere near his level.
The story everyone knows is the one about the bath. King Hiero II of Syracuse had commissioned a golden crown and suspected the goldsmith of diluting it with silver. He asked Archimedes to determine whether the crown was pure gold without melting it down. The problem occupied the mathematician until, lowering himself into a bath, he noticed the water level rise and immediately understood: a body displaces its own volume of fluid, so he could measure the crown's volume precisely by submerging it. He was so overcome that he reportedly ran naked through the streets of Syracuse crying "Eureka!" — I have found it. Whether the story is literally true scarcely matters. It captures something authentic about Archimedes: a mind so consumed by intellectual fire that the rest of the world barely existed.
His mathematical achievements were staggering in their range and depth. In Measurement of a Circle, he bounded pi between 223/71 and 22/7 by inscribing and circumscribing polygons of 96 sides around a circle and computing each perimeter by hand. In On the Sphere and Cylinder, he proved that a sphere has exactly two-thirds the volume and surface area of its circumscribing cylinder — a result he valued so highly that he asked for a diagram of a sphere inside a cylinder to be engraved on his tomb. In The Method, rediscovered only in 1906 in a scraped-down Byzantine manuscript, he described an approach to finding areas and volumes that is recognizably a precursor to integral calculus: slicing shapes into infinitesimally thin strips and summing them to find the whole.
He also wrote The Sand-Reckoner, a treatise designed to prove that he could express any number, however enormous — including the number of grains of sand needed to fill the universe. To do so, he invented a notation for representing colossal quantities and, in passing, cited the heliocentric model of Aristarchus of Samos as the only cosmological framework large enough for his calculation to be interesting. The work is mathematics, philosophy, and sheer audacity in equal measure.
"Give me a place to stand, and I shall move the Earth." — Archimedes, on the power of the lever
When Rome came for Syracuse in 214 BCE, Archimedes turned his mind to war. He designed catapults and ballistae calibrated to fire at multiple ranges, so that as Roman ships approached the harbor walls they came under continuous bombardment from every distance. More terrifyingly, he engineered the "Claw of Archimedes" — a crane mounted on the walls with a grappling hook that could seize an enemy ship, hoist its prow clear of the water, and drop it to founder. Roman soldiers became so frightened that the mere sight of a rope or pole appearing above the walls would send them fleeing, certain it was another of his devices. The Roman general Marcellus reportedly said, in black humor, that Archimedes was using the sea as a writing-board for his mathematical diagrams.
Syracuse fell in 212 BCE by treachery, not by force of arms. Archimedes was killed by a Roman soldier. The most famous version of the story holds that he was working on a geometric problem drawn in the sand when the soldier arrived and ordered him to come along. Archimedes, not yet realizing the city had fallen, said: "Don't disturb my circles." The soldier killed him. Marcellus, who had given explicit orders that Archimedes was not to be harmed, was furious. He arranged a proper burial and, according to Plutarch, treated the mathematician's family with respect.
Born in Syracuse, Sicily. His father Phidias was an astronomer.
Studies in Alexandria, Egypt under followers of Euclid; befriends the mathematician Conon of Samos.
Writes On the Equilibrium of Planes — the first rigorous mathematical treatment of the lever and center of gravity.
Solves King Hiero's crown problem; discovers the principle of buoyancy — later known as Archimedes' Principle.
Completes On the Sphere and Cylinder, proving the volume relationship he considered his greatest result.
Roman siege of Syracuse begins; Archimedes deploys his war machines to hold off the legions of Marcellus.
Syracuse falls to Rome; Archimedes is killed. Cicero later finds and restores his neglected tomb in 75 BCE.
The principle of buoyancy is taught to every physics student on Earth. Archimedes' method of exhaustion underlies integral calculus. His work on levers and mechanical advantage still defines how engineers think about force multiplication. His value of pi was used by navigators and builders for centuries. The Archimedes screw — a helical device for raising water — is still manufactured today for irrigation and sewage treatment. But perhaps his greatest legacy is the demonstration that one human mind, armed only with pure reason and geometric intuition, can unlock secrets of the physical universe that appear impenetrable. He set the template for what scientific genius looks like: obsessive, joyful, and utterly transformative of everything that follows.