Archimedes of Syracuse (c. 287 BC): Mathematician and Engineer

Archimedes of Syracuse (c. 287–212 BC) was the greatest mathematician of the ancient world and one of its most inventive engineers. Working in the Greek city of Syracuse in Sicily, he calculated areas and volumes of curved shapes, set out the laws of the lever and of floating bodies, and designed machines that held off a Roman army for two years. He was killed when Rome captured Syracuse in 212 BC, but his writings shaped mathematics and physics from the Renaissance to the age of Newton.

Background: Syracuse in the 3rd century BC

Syracuse, founded by Corinthian colonists in 733 BC, was the largest and richest Greek city in Sicily. In Archimedes’ lifetime it was ruled by Hiero II, a general who took power around 270 BC and later called himself king. Hiero ruled for more than 50 years and, after a brief early war with Rome, became a loyal Roman ally during the First Punic War.

Sicily lay between two great powers, Rome and Carthage. Their rivalry shaped Syracuse’s fate and, in the end, Archimedes’ own.

Early life and education

The usual birth date of about 287 BC comes from the Byzantine writer John Tzetzes, who says Archimedes died at the age of 75. In his work The Sand Reckoner, Archimedes mentions his father Phidias, an astronomer. Plutarch says he was a relative and friend of King Hiero, although this may be an exaggeration.

Archimedes probably spent time in Alexandria in Egypt, the leading centre of Greek science. He corresponded with scholars there, including the astronomer Conon of Samos, Dositheus and Eratosthenes, head of the Library. Many of his treatises begin as letters to these men, sending results for them to check.

Mathematical discoveries

Archimedes is best known in mathematics for finding areas and volumes of curved figures, a problem that later led to integral calculus. He used the “method of exhaustion”, developed earlier by Eudoxus, which traps a shape between ever closer inner and outer approximations.

His main results include:

• On the Sphere and Cylinder: the volume and surface area of a sphere are two-thirds of those of the cylinder that encloses it. Archimedes regarded this as his finest discovery and, according to Plutarch, asked for a sphere and cylinder to be carved on his tomb.
• Measurement of a Circle: using polygons with 96 sides, he showed that pi lies between 3 10/71 and 3 1/7 (about 3.1408 and 3.1429).
• Quadrature of the Parabola: the area of a segment of a parabola is four-thirds of a certain inscribed triangle, proved by summing an infinite series.
• On Spirals and On Conoids and Spheroids: studies of curves and solids that went far beyond earlier Greek geometry.
• The Sand Reckoner: a system for naming very large numbers, used to estimate how many grains of sand would fill the universe (about 10 to the power of 63 in modern notation). The work also preserves the only direct report of Aristarchus’ heliocentric theory.

In The Method of Mechanical Theorems, addressed to Eratosthenes, Archimedes explained how he discovered results by imagining figures balanced on a lever, then proved them rigorously by geometry. This rare glimpse of a mathematician’s working method was unknown to modern scholars until 1906.

Mechanics and hydrostatics

In On the Equilibrium of Planes Archimedes gave a mathematical proof of the law of the lever: weights balance at distances inversely proportional to their size. He also showed how to find the centres of gravity of triangles, parallelograms and parabolic segments.

In On Floating Bodies he founded hydrostatics, the science of fluids at rest. He stated what is now called Archimedes’ principle: a body in a fluid is buoyed up by a force equal to the weight of fluid it displaces. He used it to analyse the stability of floating shapes, a problem relevant to ships.

Several famous sayings and devices are linked with his work on mechanics. Pappus of Alexandria, writing about five centuries later, reports him boasting that, given a place to stand, he could move the Earth. Plutarch tells that he moved a fully loaded ship single-handed using a system of pulleys to show Hiero what mechanics could do. The water-raising screw that bears his name is traditionally credited to him, though some historians think similar devices existed earlier.

The Eureka story

The best-known story about Archimedes comes from the Roman architect Vitruvius in the 1st century BC. Hiero suspected that a goldsmith had mixed silver into a gold crown. Archimedes is said to have noticed, while stepping into a bath, that his body displaced water, and to have run home shouting “Eureka” (“I have found it”).

Vitruvius says he compared the water displaced by the crown with that displaced by equal weights of gold and silver. Many historians doubt the story or suggest that Archimedes, if he tested the crown at all, used a more precise method based on his principle of buoyancy, such as weighing the crown in water against a balance.

The siege of Syracuse

After Hiero II died in 215 BC, his grandson and successor Hieronymus abandoned the Roman alliance for Carthage during the Second Punic War. Hieronymus was murdered soon after, but Syracuse stayed on the Carthaginian side. In 214 or 213 BC a Roman army and fleet under Marcus Claudius Marcellus attacked the city.

Polybius, Livy and Plutarch describe how Archimedes’ machines defended the walls. Catapults of different ranges struck the Romans at every distance. Cranes dropped heavy stones on ships, and iron “claws” lifted the bows of ships out of the water and let them fall, capsizing or smashing them. Plutarch says the Romans grew so nervous that they fled whenever a rope or beam appeared over the wall.

The direct assault failed, and Marcellus settled into a blockade. In 212 BC, during a festival of Artemis, Roman troops climbed a poorly guarded section of wall and took the outer city. The rest of Syracuse fell soon after and was plundered.

Death of Archimedes

Ancient writers agree that Archimedes was killed by a Roman soldier during the capture of the city and that Marcellus regretted it, having ordered him spared. Plutarch gives several versions. In the best known, Archimedes was absorbed in a geometry problem and refused to go with a soldier until he had finished, and the soldier killed him. In others, he was carrying instruments the soldiers mistook for valuables.

The words often attributed to him, “Do not disturb my circles”, appear in different forms in later writers and are probably legendary. Marcellus reportedly took two astronomical globes from Syracuse to Rome, one of them a mechanical model of the heavens made by Archimedes.

Key people

Hiero II (ruled c. 270–215 BC) was Archimedes’ patron. Ancient stories link Archimedes with royal projects, including the giant ship Syracusia.

Marcus Claudius Marcellus (c. 270–208 BC) was the Roman general who besieged and captured Syracuse. He was five times consul and was killed in an ambush by Hannibal’s forces in 208 BC.

Eratosthenes of Cyrene was head of the Library of Alexandria and one of Archimedes’ most important correspondents.

Myths and facts

Burning mirrors. A popular story says Archimedes set Roman ships on fire with mirrors focusing sunlight. It does not appear in Polybius, Livy or Plutarch and is first found in much later writers. Modern tests, including student experiments at MIT in 2005, suggest it would have been very difficult to do against moving ships.

The tomb. This story has a good source. In 75 BC Cicero, serving as a Roman official in Sicily, found Archimedes’ neglected tomb overgrown with brambles outside Syracuse, recognising it by the carved sphere and cylinder.

Legacy and the Archimedes Palimpsest

Archimedes’ works survived through Byzantine copies and Arabic translations. William of Moerbeke translated most of them into Latin in 1269, and printed Greek and Latin editions appeared in 1544. Galileo, Kepler and others studied him closely, and his methods anticipated parts of calculus developed by Newton and Leibniz.

In 1906 the Danish scholar Johan Ludvig Heiberg identified a 13th-century prayer book in Constantinople as a palimpsest, a manuscript whose earlier text had been scraped off and overwritten. Underneath was a 10th-century copy of Archimedes, including the only Greek text of The Method and the puzzle known as the Stomachion. The manuscript was sold at auction in New York in 1998 for 2 million US dollars and was studied with multispectral imaging at the Walters Art Museum in Baltimore from 1999.

Archimedes’ portrait appears on the Fields Medal, the leading international prize in mathematics.

Timeline

• c. 287 BC: Archimedes is born in Syracuse.
• c. 270 BC: Hiero II takes power in Syracuse.
• 215 BC: Hiero II dies; Syracuse soon joins Carthage.
• 214 or 213 BC: Marcellus begins the siege of Syracuse.
• 212 BC: Syracuse falls; Archimedes is killed.
• 75 BC: Cicero rediscovers Archimedes’ tomb.
• 1269: William of Moerbeke translates Archimedes into Latin.
• 1906: Heiberg identifies the Archimedes Palimpsest.

Why it matters

Archimedes combined rigorous proof with practical mechanics in a way no one matched until the Scientific Revolution. His results on areas, volumes, levers and buoyancy are still taught, and his methods pointed the way towards calculus and mathematical physics.