Archimedes

Life and legacy

Archimedes was born around 287 BCE in Syracuse, the most powerful Greek city in Sicily. His father, Phidias, was an astronomer, and Archimedes may have studied in Alexandria - the intellectual capital of the Hellenistic world - before returning to Syracuse, where he spent most of his life under the patronage of King Hiero II. He was the rare ancient thinker who combined supreme mathematical ability with practical engineering genius, though ancient sources suggest he regarded his theoretical work as far more important than his inventions.

His mathematical achievements were extraordinary. In On the Measurement of the Circle, he proved that pi lies between 3 10/71 and 3 1/7 (between 3.1408 and 3.1429), using a method of inscribed and circumscribed 96-sided polygons - a result of remarkable precision. In On the Sphere and Cylinder, he proved that the surface area of a sphere is four times the area of its greatest circle, and that the volume of a sphere is two-thirds that of its circumscribing cylinder - a result he was so proud of that he asked for a sphere inscribed in a cylinder to be carved on his tombstone. In The Method, rediscovered only in 1906 on a palimpsest, he revealed his technique of using mechanical principles to discover mathematical theorems before proving them rigorously, a procedure that strikingly anticipates modern integral calculus.

In physics, Archimedes established the principles that bear his name. His work On Floating Bodies founded the science of hydrostatics, proving that a body immersed in fluid is buoyed up by a force equal to the weight of the displaced fluid (Archimedes' principle). The famous story of his discovery - leaping from his bath shouting "Eureka!" after realizing how to test the purity of Hiero's crown - is told by Vitruvius and, while it may be embellished, reflects a real physical insight. His work On the Equilibrium of Planes established the mathematical theory of the lever, leading to his famous boast: "Give me a place to stand, and I will move the earth."

During the Roman siege of Syracuse (214-212 BCE), Archimedes designed war machines - including catapults, cranes that could lift and drop enemy ships, and possibly focusing mirrors (the "heat ray," though this last claim is disputed) - that repeatedly frustrated the Roman forces under Marcellus. The Roman general reportedly declared that Archimedes was like a mythological hundred-handed giant. When Syracuse finally fell in 212 BCE, Marcellus gave orders that Archimedes should be taken alive, but a Roman soldier killed him - according to the most common version of the story, while Archimedes was absorbed in a geometric diagram and told the soldier, "Do not disturb my circles."

Archimedes' influence on mathematics and physics was immense but delayed. His works were studied by Byzantine scholars and translated into Arabic, influencing Islamic mathematicians. In Europe, the full recovery of his texts during the Renaissance profoundly influenced Galileo, who called him "superhuman," and later Newton and Leibniz, the inventors of calculus. The Archimedes Palimpsest, a 10th-century Byzantine copy rediscovered in 1906 and fully analyzed with modern imaging technology in the early 2000s, revealed previously unknown works and confirmed his status as the most brilliant mathematical mind before the modern era.

Key accomplishments

• Calculated pi to unprecedented accuracy, bounding it between 3 10/71 and 3 1/7 using 96-sided polygons
• Discovered the principle of buoyancy (Archimedes' principle), founding the science of hydrostatics
• Formulated the mathematical theory of the lever and the concept of mechanical advantage
• Anticipated integral calculus in The Method, using infinitesimal techniques to calculate areas and volumes
• Designed war machines that held off the Roman siege of Syracuse for two years (214-212 BCE)
• Proved fundamental theorems on the sphere, cylinder, parabola, and spiral that remained unsurpassed for nearly two millennia

Quotes

Give me a place to stand, and I will move the earth.

Reported by Pappus of Alexandria, Collection, Book 8; on the principle of the lever

Eureka! (I have found it!)

Reported by Vitruvius, De Architectura, Book 9; on discovering how to test the purity of gold using water displacement

Related figures