Archimedes of Syracuse (pronounced AR-kih-MEE-deez; about 287–212 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 personal life, his surviving work shows he was one of the most important scientists of classical times 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. These include 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 part of a paraboloid, the volume of a part of a hyperboloid, and the area of a spiral.
Archimedes also found a way to estimate the number 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 among the first to use math to explain physical events, such as how things balance or float. His work in this area includes proving the law of the lever, using the idea of a center of gravity, and describing the law of buoyancy, now known as Archimedes’ principle. In astronomy, he measured the apparent size of the Sun and estimated the size of the universe. He may have built a device called a planetarium to show the movements of stars and planets, which might have been 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, a Roman writer, later visited Archimedes’ tomb, which had a sphere and a cylinder on top, as Archimedes had requested to honor his most important math discovery.
Archimedes’ math work was not widely known in ancient times. Some mathematicians in Alexandria read his writings, but the first full collection of his works was made around 530 AD by Isidore of Miletus in Constantinople. Eutocius’ notes on Archimedes’ work in the same century helped more people learn about his ideas. During the Middle Ages, his writings were translated into Arabic and later into Latin, influencing scientists during the Renaissance and Scientific Revolution. In 1906, new writings by Archimedes were found in the Archimedes Palimpsest, offering fresh insights into how he solved math problems.
Biography
The details of Archimedes’s life are not clear. A biography of Archimedes mentioned by Eutocius was supposedly written by his friend Heraclides Lembus, but this work has been lost. Modern scholars doubt that Heraclides actually 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 was Phidias, an astronomer about whom nothing else is known. Plutarch wrote that Archimedes was related to King Hiero II, the ruler of Syracuse, but Cicero and Silius Italicus suggest he came from a humble background. It is unknown if Archimedes ever married, had children, or visited Alexandria, Egypt, during his youth. His surviving works, addressed to Dositheus of Pelusium and Eratosthenes of Cyrene, suggest he had connections with scholars in Alexandria. In the preface to On Spirals, Archimedes writes 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.
One famous story involves Archimedes solving a problem for King Hiero II, known as the "wreath problem." According to Vitruvius, the king had a golden wreath made for a temple but suspected the goldsmith had replaced some of the gold with silver. He asked Archimedes to investigate. While taking a bath, Archimedes noticed that water levels rose when he submerged himself, realizing this could help measure the wreath’s volume. Excited by his discovery, he ran through the streets naked, shouting "Eureka!" (meaning "I have found it!"). Vitruvius describes how Archimedes compared the wreath to equal weights of gold and silver, finding the wreath displaced more water than the gold and less than the silver, proving it was not pure gold.
A different account comes from the Carmen de Ponderibus, a 5th-century Latin poem. It describes placing gold and silver on a balance scale and then submerging the scale in water. The difference in density caused the scale to tip, demonstrating the principle now known as Archimedes’ principle. This principle, which states that a body immersed in fluid experiences a buoyant force equal to the weight of the fluid it displaces, is detailed in Archimedes’ treatise On Floating Bodies. Galileo Galilei, who invented a hydrostatic balance in 1586 inspired by Archimedes’ work, believed this method was likely the one Archimedes used.
Much of Archimedes’ engineering work likely served his home city, Syracuse. Athenaeus of Naucratis wrote that King Hiero II commissioned Archimedes to design a massive ship, the Syracusia, which was said to be the largest ship of classical antiquity. Plutarch describes how Archimedes claimed he could move any large weight, leading Hiero to challenge him to move a ship. Different sources describe how this was done: Plutarch mentions a block-and-tackle pulley system, while Hero of Alexandria attributed it to a device called the baroulkos, a type of windlass. Pappus of Alexandria credited Archimedes’ use of mechanical advantage, or the principle of leverage, and quoted him saying, "Give me a place to stand on, and I will move the Earth."
Athenaeus also mentions Archimedes using a "screw" to remove water from the Syracusia. This device, sometimes called Archimedes’ screw, likely existed before his time. None of his contemporaries who wrote about it credited him with its invention.
Archimedes gained the most fame in antiquity for defending Syracuse during the Roman siege. Plutarch wrote that Archimedes built war machines for King Hiero II, but they were not used during Hiero’s lifetime. In 214 BC, during the Second Punic War, when Syracuse switched sides to support Carthage, the Romans attacked. Archimedes allegedly oversaw the use of war machines, delaying the Romans for a long time. Three historians—Plutarch, Livy, and Polybius—describe these machines, including improved catapults, cranes that dropped heavy lead weights on Roman ships, and devices that lifted ships out of the water to sink them.
A less likely account claims Archimedes used "burning mirrors" to set Roman ships on fire. This idea first appeared in the 2nd century CE, written by the satirist Lucian of Samosata, who only mentioned ships being set on fire by artificial means. The first mention of mirrors came from Galen, writing in the same century. Later, Anthemius tried to recreate Archimedes’ hypothetical mirror system, but its credibility has been debated since the Renaissance. René Descartes rejected the idea, while modern researchers have tested it with mixed results.
There are conflicting accounts of Archimedes’ death during the Roman siege of Syracuse. According to Livy, Archimedes was killed by a Roman soldier who did not recognize him while he was drawing figures in the sand. Plutarch wrote that a soldier demanded Archimedes come with him, but Archimedes refused, saying he needed to finish his work, and was then killed. Another version from Plutarch describes Archimedes carrying mathematical tools when he was killed.
Mathematics
Archimedes is often seen as a designer of mechanical devices, but he also made important contributions to mathematics. He used methods from earlier mathematicians to find new results and created new ways to solve problems.
In Quadrature of the Parabola, Archimedes wrote that a rule from Euclid’s Elements—which showed that the area of a circle is proportional to its diameter—was proven using a principle now called the Archimedean property. This principle states that the difference between two unequal areas, when added to itself repeatedly, can eventually exceed any given area. Before Archimedes, mathematicians like Eudoxus used this principle, called 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 show 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 hexagons drawn inside and outside it, then doubled the number of sides of each polygon step by step. As the polygons had more sides, their areas became closer to the circle’s area. After four steps, with 96-sided polygons, he found that π was between 3 1/7 (about 3.1429) and 3 10/71 (about 3.1408), which matches the actual value of π (approximately 3.1416). In the same work, he also estimated the square root of 3 to be between 265/153 (about 1.7320261) and 1351/780 (about 1.7320512), possibly using a similar method.
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 showed this using an infinite geometric series with a common ratio of 1/4. The first term of the series is the area of the triangle, the second term is the sum 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 an ellipse, and the area inside an Archimedean spiral.
In The Method of Mechanical Theorems, Archimedes described a new technique for measuring areas and volumes using the law of the lever. He first outlined this method in Quadrature of the Parabola alongside his geometric proof but explained it more fully in The Method of Mechanical Theorems. He claimed he used this physical method to find answers first and then used the method of exhaustion to confirm them.
Archimedes also created ways to represent very large numbers. In The Sand Reckoner, he designed a counting system based on the Greek word myriad (10,000) to calculate a number larger than the grains of sand needed to fill the universe. He used powers of 100 million (10,000 × 10,000) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion, or 8 × 10^63. This showed that mathematics can describe extremely large numbers.
In the Cattle Problem, Archimedes asked mathematicians at the Library of Alexandria to solve a puzzle about counting the number of cattle in the Herd of the Sun. This problem required solving several Diophantine equations. A more complex version of the problem involved finding answers that are square numbers, resulting in a very large number, approximately 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 work with mathematicians in Alexandria through letters. These letters were originally written in Doric Greek, the language spoken in ancient Syracuse.
The following items are listed in order based on new rules and ideas introduced by Knorr in 1978 and Sato in 1986.
This is a short work with three statements. It is written as a letter to Dositheus of Pelusium, who studied under Conon of Samos. In the second statement, Archimedes estimates the value of pi (π), showing it is more than 223/71 (3.1408…) and less than 22/7 (3.1428…).
In this work, also called Psammites, Archimedes calculates a number larger than the total grains of sand needed to fill the universe. The text mentions the heliocentric model of the Solar System proposed by Aristarchus of Samos, as well as ideas about Earth's size and distances between celestial objects. It also attempts to measure the Sun's apparent size. Using a number system based on powers of the myriad, Archimedes concludes the number of grains of sand required is 8 × 10 in modern terms. The introduction mentions Archimedes' father, 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. Without using trigonometry or a table of chords, he determines the Sun’s apparent diameter by describing the tool used (a straight rod with pegs or grooves), adjusting measurements, and giving a range 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.
There are two books in On the Equilibrium of Planes. The first has 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, which belongs to the Peripatetic school of Aristotle’s followers. Some scholars believe this work was written by Archytas.
Archimedes uses these principles to calculate areas and centers of gravity for shapes like triangles, parallelograms, and parabolas.
In a work with 24 statements addressed to Dositheus, Archimedes proves that the area between a parabola and a straight line is 4/3 the area of a triangle with the same base and height. He uses two methods: one by applying the law of the lever, and another by calculating an infinite geometric series with a ratio of 1/4.
In a two-volume work addressed to Dositheus, Archimedes discovers the relationship between a sphere and a cylinder that surrounds it, sharing the same height and diameter. The sphere’s volume is 4/3 π r, and the cylinder’s is 2 π r. The sphere’s surface area is 4 π r, and the cylinder’s is 6 π r (including its two bases), where r is the radius.
This work of 28 statements, also addressed to Dositheus, introduces the Archimedean spiral. It describes the path of a point moving away from a fixed point at a constant speed along a line that rotates at a constant rate. In modern terms, this is expressed as r = a + bθ, where a and b are real numbers.
This is an early example of a mechanical curve studied by a Greek mathematician.
This work of 32 statements, addressed to Dositheus, calculates areas and volumes of cone, sphere, and paraboloid sections.
There are two books in On Floating Bodies. The first explains the law of fluid equilibrium and proves that water forms a sphere around a center of gravity.
Archimedes’ principle of buoyancy states that:
In the second part, he calculates how sections of paraboloids float, likely modeling ship hulls. Some sections float with their base underwater and their top above water, like icebergs.
Also known as Loculus of Archimedes or Archimedes’ Box, this is a puzzle similar to a Tangram. A treatise describing it was found in the Archimedes Palimpsest. Archimedes calculates the areas of the 14 pieces, which form a square. In 2003, Reviel Netz of Stanford University suggested Archimedes studied how many ways the pieces could be arranged into a square. Netz calculated 17,152 possible arrangements, or 536 when excluding rotations and reflections. The puzzle is an early example of a combinatorics problem.
The puzzle’s name is unclear, but it may come from the Greek word stomachos, meaning “throat” or “gullet.” Ausonius called it Ostomachion, combining 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, which requires solving several Diophantine equations. Gotthold Ephraim Lessing discovered this work in a 44-line Greek poem in 1773. A harder version of the problem requires some answers to be square numbers. A. Amthor solved this version in 1880, and the answer is approximately 7.760271 × 10.
Like The Cattle Problem, The Method of Mechanical Theorems was written as a letter to Eratosthenes.
In this work, Archimedes uses a new method, an early form of
Legacy
Archimedes is often called the father of mathematics and mathematical physics. Most historians of science and mathematics agree that he was the greatest mathematician of ancient times.
Archimedes was well known for his mechanical inventions during ancient times. Athenaeus wrote in his book Deipnosophistae that Archimedes helped build the largest ship of his time, the Syracusia. Apuleius mentioned Archimedes’ work in catoptrics, which is the study of how light reflects off surfaces. Plutarch claimed that Archimedes did not care much for mechanics and focused only on pure geometry. However, modern scholars believe this was incorrect and likely a misunderstanding meant to support Plutarch’s own views. Archimedes’ mathematical writings were not widely known in ancient times except among mathematicians in Alexandria. The first full collection of his works was made around 530 AD by Isidore of Miletus in Constantinople. Eutocius, who lived in the same century, wrote commentaries on Archimedes’ works, making them more widely available for the first time.
Archimedes’ writings were translated into Arabic by Thābit ibn Qurra between 836 and 901 AD. Later, they were translated into Latin from Arabic by Gerard of Cremona, who lived from about 1114 to 1187. Direct translations from Greek to Latin were later done by William of Moerbeke, who lived around 1215 to 1286, and Iacobus Cremonensis, who lived around 1400 to 1453.
During the Renaissance, the first printed edition of Archimedes’ works, called the Editio princeps, was published in Basel in 1544 by Johann Herwagen. This edition included Greek and Latin texts and influenced many scientists during the Renaissance and again in the 17th century.
Leonardo da Vinci admired Archimedes and credited him with inventing the Architonnerre, a type of weapon. Galileo Galilei called Archimedes “superhuman” and “my master.” Christiaan Huygens said, “I think Archimedes is comparable to no one,” and he tried to copy Archimedes’ work in his early years. Gottfried Wilhelm Leibniz said, “He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.”
In the 16th century, Filippo Paruta and Leonardo Agostini reported finding a bronze coin in Sicily with Archimedes’ portrait on one side and a cylinder and sphere with the letters “ARMD” on the other. The coin is now lost, but it may have been made in Rome for Marcellus, who brought two spheres made by Archimedes to Rome.
Carl Friedrich Gauss admired Archimedes and Isaac Newton. Moritz Cantor, who studied under Gauss, said Gauss once claimed that only three mathematicians had 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, a historian of mathematics, 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, along with Newton and Gauss.
In 1906, previously lost works by Archimedes were discovered in the Archimedes Palimpsest, a medieval manuscript. These works provided new information about how Archimedes solved mathematical problems.
The Fields Medal, an award for outstanding achievements in mathematics, features a portrait of Archimedes and a carving showing his proof about the relationship between a sphere and a cylinder. The Latin quote around Archimedes’ head, “Transire suum pectus mundoque potiri” (“Rise above oneself and grasp the world”), is attributed to a 1st-century AD poet named Manilius.
The first seagoing steamship with a screw propeller was the SS Archimedes, launched in 1839. It was named after Archimedes and his work on the screw.
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 often linked to Archimedes and is the state motto of California. It refers to the discovery of gold near Sutter’s Mill in 1848, which started the California gold rush.
There is a crater on the Moon named Archimedes (29°42′N 4°00′W) and a lunar mountain range called the Montes Archimedes (25°18′N 4°36′W), both named in his honor.