Archimedes of Syracuse (pronounced AR-kih-MEE-deez; around 287 to around 212 BC) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the city of Syracuse in Sicily. Although not much is known about his life, his surviving work shows he was one of the most important scientists of classical antiquity and one of the greatest mathematicians in history. Archimedes used ideas about very small numbers and a method called exhaustion to carefully prove many geometry rules, such as the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a paraboloid, the volume of a hyperboloid, and the area of a spiral.
Archimedes also calculated an approximation of pi (π), studied a special kind of spiral now called the Archimedean spiral, and created a system using exponents to write very large numbers. He was one of the first to use math to explain physical events, such as how objects balance (statics) and how they float (hydrostatics). His work in this area included proving the law of the lever, using the idea of a center of gravity, and describing the law of buoyancy, which is now called Archimedes’ principle. In astronomy, he measured the apparent size of the Sun and the size of the universe. He may have built a device called a planetarium to show the movements of celestial objects, possibly an early version of the Antikythera mechanism. He also designed tools like the screw pump, compound pulleys, and war machines to help protect Syracuse from attacks.
Archimedes died during the siege of Syracuse when a Roman soldier killed him, even though he was told not to be harmed. Cicero later visited his tomb, which had a sphere and a cylinder, as Archimedes had requested to honor his most important math discovery.
Archimedes’ math writings were not widely known in ancient times. Some scholars in Alexandria read his work, but the first full collection of his writings was made around 530 AD by Isidore of Miletus in Byzantine Constantinople. In the same century, Eutocius wrote explanations of Archimedes’ work, making it more accessible. During the Middle Ages, his writings were translated into Arabic in the 9th century and later into Latin in the 12th century. These translations helped scientists during the Renaissance and Scientific Revolution. In 1906, new writings by Archimedes were found in the Archimedes Palimpsest, offering fresh insights into his methods.
Biography
The details of Archimedes's life are not well known. A biography of Archimedes mentioned by Eutocius was supposedly written by his friend Heraclides Lembus, but this work is lost, and modern scholars doubt that Heraclides wrote it.
Based on a statement by the Byzantine Greek scholar John Tzetzes, who said Archimedes lived for 75 years before dying in 212 BC, Archimedes is estimated to have been born around 287 BC in the seaport city of Syracuse, Sicily. At that time, Syracuse was a self-governing colony in Magna Graecia. In The Sand-Reckoner, Archimedes mentions his father’s name as Phidias, an astronomer about whom nothing else is known. Plutarch wrote in Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse, but Cicero and Silius Italicus suggest he was of humble origin. It is unknown whether Archimedes married, had children, or visited Alexandria, Egypt, during his youth. However, his surviving works, addressed to Dositheus of Pelusium and Eratosthenes of Cyrene, suggest he worked with scholars in Alexandria. In the preface to On Spirals, Archimedes wrote that "many years have elapsed since Conon's death." Conon of Samos lived around 280–220 BC, which suggests Archimedes may have been older when writing some of his works.
Another story credits Archimedes with solving a problem for King Hiero II, known as the "wreath problem." According to Vitruvius, who wrote about this two centuries after Archimedes’ death, King Hiero II commissioned a golden wreath for a temple and provided pure gold to the goldsmith. However, the king suspected the goldsmith had replaced some of the gold with silver. Unable to make the goldsmith confess, the king asked Archimedes to investigate. Later, while stepping into a bath, Archimedes noticed that the water level rose more as he sank deeper. He realized this could be used to measure the volume of the wreath. Excited, he ran through the streets naked, shouting "Eureka!" (meaning "I have found it!"). Vitruvius wrote that Archimedes then compared the wreath to equal weights of gold and silver, placing them in water. The wreath displaced more water than the gold and less than the silver, proving it was made of a mix of gold and silver.
A different account appears in the Carmen de Ponderibus, a 5th-century Latin poem once attributed to Priscian. This account describes placing gold and silver on a balance scale and then submerging the scale in water. The difference in density between the metals caused the scale to tip, demonstrating the principle now known as Archimedes’ principle. This principle, described in Archimedes’ treatise On Floating Bodies, states that a body immersed in a fluid experiences an upward force equal to the weight of the fluid it displaces. Galileo Galilei, who invented a hydrostatic balance in 1586 inspired by Archimedes’ work, believed this method likely matched what Archimedes used.
Much of Archimedes’ engineering work likely came from meeting the needs of his home city, Syracuse. Athenaeus of Naucratis wrote in Deipnosophistae that King Hiero II commissioned Archimedes to design a massive ship, the Syracusia, the largest ship of classical antiquity. According to Athenaeus, Archimedes oversaw its launch. Plutarch tells a slightly different story, saying Archimedes boasted he could move any large weight. Hiero then challenged him to move a ship. These accounts include details that are historically unlikely, and the stories differ on how Archimedes accomplished the task. Plutarch claims Archimedes used a block-and-tackle pulley system, while Hero of Alexandria attributed the feat to Archimedes’ invention of the baroulkos, a type of windlass. Pappus of Alexandria credited Archimedes with using mechanical advantage, the principle of leverage, and quoted him as saying, "Give me a place to stand on, and I will move the Earth."
Athenaeus, possibly misquoting Hero’s account of the baroulkos, also mentions Archimedes using a "screw" to remove water from the Syracusia. This device, sometimes called Archimedes’ screw, likely existed before Archimedes. No contemporaries who described its use (Philo of Byzantium, Strabo, or Vitruvius) credited Archimedes with inventing it.
Archimedes’ greatest reputation in antiquity came from his role in defending Syracuse during the Roman siege. Plutarch wrote that Archimedes designed war machines for King Hiero II, but they were never used during Hiero’s lifetime. In 214 BC, during the Second Punic War, when Syracuse switched sides from Rome to Carthage, the Roman army under Marcus Claudius Marcellus attacked the city. Archimedes allegedly oversaw the use of war machines, delaying the Romans for a long time. Plutarch, Livy, and Polybius described these machines, including improved catapults, cranes that dropped heavy lead weights onto Roman ships, and devices that used an iron claw to lift ships out of the water and drop them, causing them to sink.
A less credible account, not mentioned by Plutarch, Polybius, or Livy, claims Archimedes used "burning mirrors" to focus sunlight on Roman ships, setting them on fire. The earliest account of ships being set on fire, by the 2nd-century satirist Lucian of Samosata, does not mention mirrors, only stating ships were set on fire by artificial means, which may have involved burning projectiles. Galen, writing later in the same century, was the first to mention mirrors. Nearly 400 years after Lucian and Galen, Anthemius attempted to reconstruct Archimedes’ hypothetical reflector geometry. The device, sometimes called "Archimedes’ heat ray," has been debated since the Renaissance. René Descartes rejected it as false, while modern researchers have tested its feasibility with mixed results.
There are conflicting accounts of Archimedes’ death during the Roman siege of Syracuse. The oldest account, from Livy, says Archimedes was killed by a Roman soldier while drawing figures in the dust, unaware of who he was. Plutarch wrote that a soldier demanded Archimedes follow him, but Archimedes refused,
Mathematics
Archimedes is best known for designing mechanical devices, but he also made important contributions to mathematics. He used methods developed by earlier mathematicians to find new results and created new techniques for solving problems.
In Quadrature of the Parabola, Archimedes explained a rule from Euclid’s Elements that shows the area of a circle is proportional to its diameter. This rule, now called the Archimedean property, states that the larger of two unequal areas can exceed the smaller one enough to surpass any given area if the difference is added to itself repeatedly. Before Archimedes, mathematicians like Eudoxus used this rule, known as the "method of exhaustion," to calculate the volume of shapes such as tetrahedrons, cylinders, cones, and spheres. These calculations are detailed in Book XII of Euclid’s Elements.
In Measurement of a Circle, Archimedes used the method of exhaustion to prove that the area of a circle equals the area of a right triangle with a base equal to the circle’s circumference and a height equal to its radius. To estimate the value of π, he compared the circle to regular polygons with increasing numbers of sides. He started with hexagons inside and outside the circle, then doubled the number of sides repeatedly. After four steps, when the polygons had 96 sides each, he determined that π was between 3 1/7 (about 3.1429) and 3 10/71 (about 3.1408), which is close to π’s actual value of approximately 3.1416. In the same work, he also showed that the square root of 3 lies between 265/153 (about 1.7320261) and 1351/780 (about 1.7320512).
In Quadrature of the Parabola, Archimedes used the method of exhaustion to prove that the area between a parabola and a straight line is 4/3 times the area of a triangle inscribed within that region. He expressed this result as an infinite geometric series with a common ratio of 1/4. The first term is the area of the triangle, the second term is the combined area of two smaller triangles, and so on. This series, 1/4 + 1/16 + 1/64 + 1/256 + …, adds up to 1/3.
Archimedes also used this method to calculate the surface areas of spheres and cones, the area of ellipses, and the area inside an Archimedean spiral.
In The Method of Mechanical Theorems, Archimedes introduced a new approach using the law of the lever to measure areas and volumes. He first outlined this method in Quadrature of the Parabola alongside a geometric proof but explained it more fully in The Method of Mechanical Theorems. He claimed to have used this mechanical method to find solutions to mathematical problems before proving them with the method of exhaustion.
Archimedes also developed ways to represent very large numbers. In The Sand Reckoner, he created a counting system based on the Greek word myriad, meaning 10,000. He used powers of myriad of myriads (100 million) to calculate a number larger than the grains of sand needed to fill the universe. He concluded that the number of grains required would be 8 vigintillion, or 8 × 10^63. This showed that mathematics can describe extremely large numbers.
In The Cattle Problem, Archimedes posed a challenge to mathematicians in Alexandria to count the number of cattle in the Herd of the Sun. This problem required solving several simultaneous Diophantine equations. A more complex version of the problem demands answers that are square numbers, resulting in a number approximately equal to 7.760271 × 10^206.
In a lost work described by Pappus of Alexandria, Archimedes proved that there are exactly thirteen semiregular polyhedra.
Writings
Archimedes shared his discoveries through letters to mathematicians in Alexandria. These letters were originally written in Doric Greek, the language spoken in ancient Syracuse.
The following works are listed in order based on new rules for organizing historical information developed by Knorr (1978) and Sato (1986).
One short work includes three statements. It is written as a letter to Dositheus of Pelusium, a student of Conon of Samos. In the second statement, Archimedes estimates the value of pi (π), showing that it is more than 223/71 (3.1408…) and less than 22/7 (3.1428…).
In a treatise also called Psammites, Archimedes finds a number larger than the total grains of sand needed to fill the universe. This work mentions the heliocentric theory of the Solar System proposed by Aristarchus of Samos, as well as ideas about Earth’s size and distances between celestial bodies. It also describes how to measure the Sun’s apparent diameter. Using a number system based on powers of the myriad, Archimedes concludes that the number of grains of sand needed to fill the universe is 8 × 10 in modern notation. The introduction notes that Archimedes’ father was an astronomer named Phidias. The Sand Reckoner is the only surviving work where Archimedes discusses astronomy.
In The Sand Reckoner, Archimedes discusses measurements of Earth, the Sun, and the Moon, as well as Aristarchus’ heliocentric model of the universe. Without using trigonometry or a table of chords, Archimedes determines the Sun’s apparent diameter by first describing the tool used for observations (a straight rod with pegs or grooves), applying corrections to the measurements, and then providing an upper and lower bound for the result to account for errors.
Ptolemy, quoting Hipparchus, mentions Archimedes’ observations of solstices in the Almagest. This would make Archimedes the first known Greek to record multiple solstice dates and times over several years.
The work On the Equilibrium of Planes has two books. The first includes seven postulates and fifteen statements, while the second has ten statements. In the first book, Archimedes proves the law of the lever, which states that:
Earlier descriptions of the lever principle appear in a work by Euclid and in Mechanical Problems, a text associated with the Peripatetic school, which followed Aristotle. Some scholars believe Mechanical Problems was written by Archytas.
Archimedes uses these principles to calculate the areas and centers of gravity of shapes such as triangles, parallelograms, and parabolas.
In a work of 24 statements addressed to Dositheus, Archimedes proves that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with the same base and height. He uses two methods: applying the law of the lever and calculating the sum of an infinite geometric series with a ratio of 1/4.
In a two-volume treatise to Dositheus, Archimedes proves a result he considered his greatest achievement: the relationship between a sphere and a cylinder that surrounds it, sharing the same height and diameter. The sphere’s volume is (4/3)πr, and the cylinder’s volume is 2πr. The sphere’s surface area is 4πr, and the cylinder’s surface area (including its two bases) is 6πr, where r is the radius of the sphere and cylinder.
A work of 28 statements to Dositheus defines what is now called the Archimedean spiral. It is the path traced by a point moving away from a fixed point at a constant speed along a line that rotates at a constant angular speed. In modern polar coordinates (r, θ), the spiral is described by the equation r = a + bθ, where a and b are real numbers.
This is an early example of a mechanical curve studied by a Greek mathematician.
A work of 32 statements to Dositheus includes calculations of the areas and volumes of cone sections, spheres, and paraboloids.
On Floating Bodies has two books. In the first, Archimedes explains the law of fluid equilibrium and proves that water forms a spherical shape around a center of gravity. Archimedes’ principle of buoyancy states that:
In the second part, Archimedes calculates the equilibrium positions of paraboloid sections, likely representing idealized ship hull shapes. Some sections float with their base underwater and their top above water, similar to icebergs.
Also called the Loculus of Archimedes or Archimedes’ Box, this is a dissection puzzle similar to a Tangram. A treatise describing it was found in the Archimedes Palimpsest. Archimedes calculates the areas of the 14 pieces, which can be arranged to form a square. Reviel Netz of Stanford University suggested in 2003 that Archimedes explored how many ways the pieces could be arranged into a square. Netz calculated that there are 17,152 possible arrangements, or 536 when excluding those that are identical through rotation or reflection. The puzzle is an early example of a combinatorics problem.
The origin of the puzzle’s name is unclear. It may come from the Greek word stomachos, meaning “throat” or “gullet.” Ausonius called the puzzle Ostomachion, a Greek compound word from osteon (“bone”) and machē (“fight”).
In a work addressed to Eratosthenes and mathematicians in Alexandria, Archimedes challenges them to count the number of cattle in the Herd of the Sun, a problem involving solving several Diophantine equations. Gotthold Ephraim Lessing discovered this work in a 44-line Greek poem in the Herzog August Library in Wolfen
Legacy
Archimedes is often called one of the most important mathematicians and scientists in history. Most historians agree that he was the greatest mathematician of ancient times.
Archimedes was known for his work with machines and inventions during ancient times. Athenaeus wrote about how Archimedes helped build the largest ship of his time, called the Syracusia. Apuleius mentioned Archimedes' studies of light reflections. Plutarch said Archimedes did not care much for machines and focused only on pure math, but modern scholars believe this was not true. Plutarch likely said this to support his own beliefs. Archimedes' math work was not widely known in ancient times except among mathematicians in Alexandria. The first complete collection of his work was made around 530 AD by Isidore of Miletus in Constantinople. Eutocius later added comments that helped more people read and understand Archimedes' writings.
Archimedes' work was translated into Arabic by Thābit ibn Qurra (836–901 AD) and later into Latin through Arabic by Gerard of Cremona (c. 1114–1187). Later, his work was translated directly from Greek to Latin by William of Moerbeke (c. 1215–1286) and Iacobus Cremonensis (c. 1400–1453).
During the Renaissance, the first printed edition of Archimedes' work, called the Editio princeps, was published in Basel in 1544 by Johann Herwagen. This edition included his writings in Greek and Latin and influenced many scientists during the Renaissance and the 17th century.
Leonardo da Vinci admired Archimedes and credited him with inventing the Architonnerre. Galileo Galilei called Archimedes "superhuman" and "my master." Christiaan Huygens said, "I think Archimedes is comparable to no one," and tried to copy his work. Gottfried Wilhelm Leibniz said, "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times."
In Sicily, a bronze coin was found with Archimedes' portrait on one side and a cylinder and sphere with the letters "ARMD" on the other. The coin is now lost, but some believe it was made in Rome for Marcellus, who brought two spheres of Archimedes to Rome.
Carl Friedrich Gauss considered Archimedes and Isaac Newton as his greatest influences. Moritz Cantor, who studied under Gauss, said Gauss once claimed only three mathematicians made major contributions: Archimedes, Newton, and Eisenstein. Alfred North Whitehead said, "In the year 1500, Europe knew less than Archimedes, who died in 212 BC." Reviel Netz said, "Western science is but a series of footnotes to Archimedes," calling him "the most important scientist who ever lived." Eric Temple Bell wrote that any list of the greatest mathematicians would include Archimedes, Newton, and Gauss.
In 1906, lost works of Archimedes were discovered in the Archimedes Palimpsest, giving new insights into his methods.
The Fields Medal, an award for top achievements in mathematics, features a portrait of Archimedes and a carving showing his proof about the sphere and cylinder. The medal includes a quote from a 1st-century AD poet: "Rise above oneself and grasp the world."
The first seagoing steamship with a screw propeller, the SS Archimedes, was launched in 1839 and named after him.
Archimedes has appeared on postage stamps from East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).
The word "Eureka!" is California's state motto. It is linked to Archimedes' discovery but actually refers to the finding of gold in 1848, which started the California gold rush.
A crater on the Moon and a mountain range, the Montes Archimedes, are named after him in his honor.