Archimedes

Archimedes of Syracuse was a Greek mathematician, physicist and engineer of the third century BCE, generally regarded as the greatest mathematician of the ancient world. He calculated areas and volumes of curved figures with methods that anticipated integral calculus, gave a close estimate of the value of pi, founded the sciences of statics and hydrostatics with the law of the lever and the principle of buoyancy, and became famous for the war machines with which he defended his city against Rome.

Archimedes lived in Syracuse, the largest Greek city of Sicily, during the Hellenistic period. Almost everything known about his life comes from Roman and Greek writers who lived long after him, such as Polybius, Livy, Plutarch and Vitruvius, and much of it is anecdote. He is said to have been about seventy-five when he died in 212 BCE, which gives a birth date of around 287 BCE, but this rests on a single late source. In one of his own works he names his father as Phidias, an astronomer. Plutarch says he was a relative or friend of Hiero II, who ruled Syracuse from about 270 to 215 BCE, and many stories link him with the king. He may have studied in Alexandria, the scientific capital of the Greek world under the Ptolemaic Kingdom, where the tradition of Euclid was strong. Whether or not he went there, he corresponded with mathematicians in Alexandria, including Conon of Samos and Eratosthenes, and sent them his results, sometimes with deliberately false statements included to catch out anyone who claimed to have proved them first.

Archimedes' main achievement was in geometry, where he went beyond the straight-line figures of earlier mathematicians to find exact results for curves and curved solids. To do this he used the method of exhaustion: he trapped a curved figure between shapes whose areas he could calculate, such as polygons inside and outside a circle, and showed that any other answer than his own would lead to a contradiction.

In Measurement of a Circle he applied this to polygons with 96 sides and showed that pi lies between 3 10/71 and 3 1/7, that is, between about 3.1408 and 3.1429. In Quadrature of the Parabola he proved that the area enclosed by a parabola and a straight line is four-thirds of the area of a particular triangle inside it, summing what is in effect an infinite series. On Spirals studies the curve now called the Archimedean spiral, and On Conoids and Spheroids finds the volumes of solids made by rotating curves. The result he valued most, in On the Sphere and Cylinder, is that a sphere has two-thirds of the volume and, counting its ends, two-thirds of the surface area of the smallest cylinder that contains it. At his request, his tomb was marked with a sphere inside a cylinder.

In On the Equilibrium of Planes Archimedes gave a mathematical proof of the law of the lever: two weights balance when their distances from the pivot are in inverse proportion to their weights. He also showed how to find the centres of gravity of triangles, parallelograms and other shapes. A famous saying attributed to him, give me a place to stand and I will move the earth, expresses the power of the lever, though it reaches us only from later writers.

On Floating Bodies founded hydrostatics. It contains what is now called Archimedes' principle: a body immersed in a fluid is pushed upwards by a force equal to the weight of the fluid it displaces, so that an object floats when it displaces its own weight of water. He went on to work out the conditions under which floating shapes such as sections of a paraboloid stay upright or tip over, a problem of real importance for ship design. The well-known story that he discovered how to test whether a goldsmith had cheated Hiero by mixing silver into a gold crown while stepping into his bath, and ran through the streets shouting eureka, I have found it, comes from Vitruvius two centuries later and is probably a legend, though the problem itself may be genuine.

In The Sand Reckoner Archimedes set out to count the grains of sand that would fill the universe, and in doing so created a system for naming very large numbers, far beyond the Greek numerals of his time; his answer is equivalent to about 10 to the power of 63. The work also preserves the main ancient account of the theory of Aristarchus of Samos that the Earth moves around the Sun. The Cattle Problem, addressed to Eratosthenes, is a puzzle in verse whose full solution involves numbers with over two hundred thousand digits. The Method of Mechanical Theorems, which explains how he used imaginary balances and thin slices of figures to discover results before proving them rigorously, was lost for centuries. It was found in 1906 by the Danish scholar Johan Ludvig Heiberg in a palimpsest in Constantinople: a tenth-century Byzantine copy of Archimedes' works whose pages had been scraped and reused for a prayer book in the thirteenth century. After the manuscript was sold at auction in 1998, modern imaging recovered much more of the hidden text, including parts of the Stomachion, a work on a dissection puzzle.

Ancient writers credited Archimedes with many machines, including compound pulleys with which he is said to have drawn a fully loaded ship onto land single-handed, and a bronze sphere showing the motions of the Sun, Moon and planets. The water-lifting screw that bears his name may be older than him, and its attribution is uncertain.

During the Second of the Punic Wars, after Hannibal's victories in Italy, Syracuse broke its alliance with the Roman Republic and sided with Carthage. A Roman army and fleet under Marcus Claudius Marcellus besieged the city from 214 BCE. Polybius, Livy and Plutarch all describe how Archimedes' catapults and cranes held off the attackers; one device, the claw, is said to have lifted Roman ships out of the water and dropped them. The story that he set ships on fire with mirrors focusing sunlight appears only centuries later and is doubted by most historians. Syracuse finally fell in 212 BCE, and Archimedes was killed by a Roman soldier, apparently against Marcellus's orders. Plutarch gives several versions of how it happened, one of them the famous story that he was absorbed in a diagram in the sand and told the soldier not to disturb his circles.

In 75 BCE the Roman orator Cicero, serving in Sicily, found Archimedes' neglected tomb near one of the city gates, recognising it by the sphere and cylinder. His works were preserved and studied in Byzantium and translated into Arabic during the Islamic Golden Age, and then into Latin. Printed editions of the sixteenth century made them widely known during the Renaissance, and Galileo Galilei, who called him superhuman, was among the many scientists who built on them. His methods for areas and volumes came closer to calculus than any other ancient work, and mathematicians still place him among the greatest who have ever lived. He remains the central figure of science in ancient Greece, and his portrait appears on the Fields Medal, the highest award in mathematics.