# Archimedes

**Archimedes of Syracuse** (;^{[2]} Ancient Greek: Ἀρχιμήδης, romanized: *Arkhimḗdēs*; Doric Greek: [ar.kʰi.mɛː.dɛ̂ːs]; c. 287 – c. 212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer.^{[3]} Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Generally considered the greatest mathematician of antiquity and one of the greatest of all time,^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]} Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola.^{[10]}

Other mathematical achievements include deriving an accurate approximation of pi, defining and investigating the spiral bearing his name, and creating a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics, including an explanation of the principle of the lever. He is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.

Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, which Archimedes had requested be placed on his tomb to represent his mathematical discoveries.

Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus in Byzantine Constantinople, while commentaries on the works of Archimedes written by Eutocius in the sixth century AD opened them to wider readership for the first time. The relatively few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance, while the discovery in 1906 of previously unknown works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.^{[11]}^{[12]}^{[13]}

Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years.^{[14]} In *The Sand Reckoner*, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known. Plutarch wrote in his *Parallel Lives* that Archimedes was related to King Hiero II, the ruler of Syracuse.^{[15]} A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure.^{[16]} It is unknown, for instance, whether he ever married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene were contemporaries. He referred to Conon of Samos as his friend, while two of his works (*The Method of Mechanical Theorems* and the *Cattle Problem*) have introductions addressed to Eratosthenes.^{[a]}

Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he must not be harmed.^{[18]} Marcellus called Archimedes "a geometrical Briareus".^{[19]}

The last words attributed to Archimedes are "Do not disturb my circles", a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is often given in Latin as "*Noli turbare circulos meos*," but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch. Valerius Maximus, writing in *Memorable Doings and Sayings* in the 1st century AD, gives the phrase as "" – "... but protecting the dust with his hands, said 'I beg of you, do not disturb this.'" The phrase is also given in Katharevousa Greek as "μὴ μου τοὺς κύκλους τάραττε!" (*Mē mou tous kuklous taratte!*).^{[18]}

*...sed protecto manibus puluere 'noli' inquit, 'obsecro, istum disturbare'*

The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. In 75 BC, 137 years after his death, the Roman orator Cicero was serving as quaestor in Sicily. He had heard stories about the tomb of Archimedes, but none of the locals were able to give him the location. Eventually he found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription.^{[20]} A tomb discovered in the courtyard of the Hotel Panorama in Syracuse in the early 1960s was claimed to be that of Archimedes, but there was no compelling evidence for this and the location of his tomb today is unknown.^{[21]}

The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege of Syracuse given by Polybius in his *The Histories* was written around seventy years after Archimedes' death, and was used subsequently as a source by Plutarch and Livy. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city.^{[22]}

The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. According to Vitruvius, a votive crown for a temple had been made for King Hiero II of Syracuse, who had supplied the pure gold to be used, and Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith.^{[23]} Archimedes had to solve the problem without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density.
While taking a bath, he noticed that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the volume of the crown. For practical purposes water is incompressible,^{[24]} so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "Eureka!" (Greek: "εὕρηκα, *heúrēka*!", meaning "I have found [it]!").^{[23]} The test was conducted successfully, proving that silver had indeed been mixed in.^{[25]}

The story of the golden crown does not appear in the known works of Archimedes. Moreover, the practicality of the method it describes has been called into question, due to the extreme accuracy with which one would have to measure the water displacement.^{[26]} Archimedes may have instead sought a solution that applied the principle known in hydrostatics as Archimedes' principle, which he describes in his treatise *On Floating Bodies*. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.^{[27]} Using this principle, it would have been possible to compare the density of the crown to that of pure gold by balancing the crown on a scale with a pure gold reference sample of the same weight, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly. Galileo considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself."^{[28]} In a 12th-century text titled *Mappae clavicula* there are instructions on how to perform the weighings in the water in order to calculate the percentage of silver used, and thus solve the problem.^{[29]}^{[30]} The Latin poem *Carmen de ponderibus et mensuris* of the 4th or 5th century describes the use of a hydrostatic balance to solve the problem of the crown, and attributes the method to Archimedes.^{[29]}

A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of Naucratis described how King Hiero II commissioned Archimedes to design a huge ship, the *Syracusia*, which could be used for luxury travel, carrying supplies, and as a naval warship. The *Syracusia* is said to have been the largest ship built in classical antiquity.^{[31]} According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes' screw was purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a low-lying body of water into irrigation canals. The Archimedes' screw is still in use today for pumping liquids and granulated solids such as coal and grain. The Archimedes' screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon.^{[32]}^{[33]}^{[34]} The world's first seagoing steamship with a screw propeller was the SS *Archimedes*, which was launched in 1839 and named in honor of Archimedes and his work on the screw.^{[35]}

The Claw of Archimedes is a weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker", the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped onto an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it. There have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitled *Superweapons of the Ancient World* built a version of the claw and concluded that it was a workable device.^{[36]}^{[37]}

Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse.
The 2nd century AD author Lucian wrote that during the Siege of Syracuse (c. 214–212 BC), Archimedes destroyed enemy ships with fire. Centuries later, Anthemius of Tralles mentions burning-glasses as Archimedes' weapon.^{[38]} The device, sometimes called the "Archimedes heat ray", was used to focus sunlight onto approaching ships, causing them to catch fire. In the modern era, similar devices have been constructed and may be referred to as a heliostat or solar furnace.^{[39]}

This purported weapon has been the subject of ongoing debate about its credibility since the Renaissance. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes.^{[40]} It has been suggested that a large array of highly polished bronze or copper shields acting as mirrors could have been employed to focus sunlight onto a ship.

A test of the Archimedes heat ray was carried out in 1973 by the Greek scientist Ioannis Sakkas. The experiment took place at the Skaramagas naval base outside Athens. On this occasion 70 mirrors were used, each with a copper coating and a size of around five by three feet (1.5 by 1 m). The mirrors were pointed at a plywood mock-up of a Roman warship at a distance of around 160 feet (50 m). When the mirrors were focused accurately, the ship burst into flames within a few seconds. The plywood ship had a coating of tar paint, which may have aided combustion.^{[41]} A coating of tar would have been commonplace on ships in the classical era.^{[d]}

In October 2005 a group of students from the Massachusetts Institute of Technology carried out an experiment with 127 one-foot (30 cm) square mirror tiles, focused on a mock-up wooden ship at a range of around 100 feet (30 m). Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. It was concluded that the device was a feasible weapon under these conditions. The MIT group repeated the experiment for the television show *MythBusters*, using a wooden fishing boat in San Francisco as the target. Again some charring occurred, along with a small amount of flame. In order to catch fire, wood needs to reach its autoignition temperature, which is around 300 °C (570 °F).^{[42]}^{[43]}

When *MythBusters* broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (or failed) because of the length of time and the ideal weather conditions required for combustion to occur. It was also pointed out that since Syracuse faces the sea towards the east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors. *MythBusters* also pointed out that conventional weaponry, such as flaming arrows or bolts from a catapult, would have been a far easier way of setting a ship on fire at short distances.^{[44]}

In December 2010, *MythBusters* again looked at the heat ray story in a special edition entitled "President's Challenge". Several experiments were carried out, including a large scale test with 500 schoolchildren aiming mirrors at a mock-up of a Roman sailing ship 400 feet (120 m) away. In all of the experiments, the sail failed to reach the 210 °C (410 °F) required to catch fire, and the verdict was again "busted". The show concluded that a more likely effect of the mirrors would have been blinding, dazzling, or distracting the crew of the ship.^{[45]}

While Archimedes did not invent the lever, he gave an explanation of the principle involved in his work *On the Equilibrium of Planes*. Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas.^{[46]}^{[47]} According to Pappus of Alexandria, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." (Greek: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω)^{[48]} Plutarch describes how Archimedes designed block-and-tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move.^{[49]} Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.^{[50]}

Cicero (106–43 BC) mentions Archimedes briefly in his dialogue *De re publica*, which portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse c. 212 BC, General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms, constructed by Archimedes and used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Cicero mentions similar mechanisms designed by Thales of Miletus and Eudoxus of Cnidus. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius Philus, who described it thus:

*Hanc sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in aere illo quot diebus in ipso caelo succederet, ex quo et in caelo sphaera solis fieret eadem illa defectio, et incideret luna tum in eam metam quae esset umbra terrae, cum sol e regione.*

This is a description of a planetarium or orrery. Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled *On Sphere-Making*. Modern research in this area has been focused on the Antikythera mechanism, another device built c. 100 BC that was probably designed for the same purpose.^{[53]} Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing.^{[54]} This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.^{[55]}^{[56]}

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote: "He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life."^{[57]}
Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. Through proof by contradiction (reductio ad absurdum), he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π. In *Measurement of a Circle* he did this by drawing a larger regular hexagon outside a circle and a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step. As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of π lay between 31/7 (approximately 3.1429) and 310/71 (approximately 3.1408), consistent with its actual value of approximately 3.1416.^{[58]} He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle (πr^{2}). In *On the Sphere and Cylinder*, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. This is the Archimedean property of real numbers.^{[59]}

In *Measurement of a Circle*, Archimedes gives the value of the square root of 3 as lying between 265/153 (approximately 1.7320261) and 1351/780 (approximately 1.7320512). The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this result without offering any explanation of how he had obtained it. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results."^{[60]} It is possible that he used an iterative procedure to calculate these values.^{[61]}

In *The Quadrature of the Parabola*, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio 1/4:

If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and so on. This proof uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.

In *The Sand Reckoner*, Archimedes set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote: "There are some, King Gelo (Gelo II, son of Hiero II), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited." To solve the problem, Archimedes devised a system of counting based on the myriad. The word is from the Greek μυριάς *murias*, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion, or 8×10^{63}.^{[62]}

The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse.^{[63]} The written work of Archimedes has not survived as well as that of Euclid, and seven of his treatises are known to have existed only through references made to them by other authors. Pappus of Alexandria mentions *On Sphere-Making* and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the now-lost *Catoptrica*.^{[b]} During his lifetime, Archimedes made his work known through correspondence with the mathematicians in Alexandria. The writings of Archimedes were first collected by the Byzantine Greek architect Isidore of Miletus (c. 530 AD), while commentaries on the works of Archimedes written by Eutocius in the sixth century AD helped to bring his work a wider audience. Archimedes' work was translated into Arabic by Thābit ibn Qurra (836–901 AD), and Latin by Gerard of Cremona (c. 1114–1187 AD). During the Renaissance, the *Editio Princeps* (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin.^{[64]} Around the year 1586 Galileo Galilei invented a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes.^{[65]}

Any body wholly or partially immersed in a fluid experiences an upthrust equal to, but opposite in sense to, the weight of the fluid displaced.

Archimedes' *Book of Lemmas* or *Liber Assumptorum* is a treatise with fifteen propositions on the nature of circles. The earliest known copy of the text is in Arabic. The scholars T.L. Heath and Marshall Clagett argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. The *Lemmas* may be based on an earlier work by Archimedes that is now lost.^{[74]}

It has also been claimed that Heron's formula for calculating the area of a triangle from the length of its sides was known to Archimedes.^{[c]} However, the first reliable reference to the formula is given by Heron of Alexandria in the 1st century AD.^{[75]}

The foremost document containing the work of Archimedes is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople and examined a 174-page goatskin parchment of prayers written in the 13th century AD. He discovered that it was a palimpsest, a document with text that had been written over an erased older work. Palimpsests were created by scraping the ink from existing works and reusing them, which was a common practice in the Middle Ages as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th century AD copies of previously unknown treatises by Archimedes.^{[76]} The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. On October 29, 1998 it was sold at auction to an anonymous buyer for $2 million at Christie's in New York.^{[77]} The palimpsest holds seven treatises, including the only surviving copy of *On Floating Bodies* in the original Greek. It is the only known source of *The Method of Mechanical Theorems*, referred to by Suidas and thought to have been lost forever. *Stomachion* was also discovered in the palimpsest, with a more complete analysis of the puzzle than had been found in previous texts. The palimpsest is now stored at the Walters Art Museum in Baltimore, Maryland, where it has been subjected to a range of modern tests including the use of ultraviolet and x-ray light to read the overwritten text.^{[78]}

**a.** ** ^{^}** In the preface to

*On Spirals*addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works.

**b.** ** ^{^}** The treatises by Archimedes known to exist only through references in the works of other authors are:

*On Sphere-Making*and a work on polyhedra mentioned by Pappus of Alexandria;

*Catoptrica*, a work on optics mentioned by Theon of Alexandria;

*Principles*, addressed to Zeuxippus and explaining the number system used in

*The Sand Reckoner*;

*On Balances and Levers*;

*On Centers of Gravity*;

*On the Calendar*. Of the surviving works by Archimedes, T.L. Heath offers the following suggestion as to the order in which they were written:

*On the Equilibrium of Planes I*,

*The Quadrature of the Parabola*,

*On the Equilibrium of Planes II*,

*On the Sphere and the Cylinder I, II*,

*On Spirals*,

*On Conoids and Spheroids*,

*On Floating Bodies I, II*,

*On the Measurement of a Circle*,

*The Sand Reckoner*.

**c.** ** ^{^}** Boyer, Carl Benjamin

*A History of Mathematics*(1991) ISBN 0-471-54397-7 "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula —

*k*= √

*s*(

*s*−

*a*)(

*s*−

*b*)(

*s*−

*c*), where

*s*is the semiperimeter — was known to Archimedes several centuries before Heron lived. Arabic scholars also attribute to Archimedes the 'theorem on the broken chord' ... Archimedes is reported by the Arabs to have given several proofs of the theorem."

**d.** ** ^{^}** "It was usual to smear the seams or even the whole hull with pitch or with pitch and wax". In Νεκρικοὶ Διάλογοι (

*Dialogues of the Dead*), Lucian refers to coating the seams of a skiff with wax, a reference to pitch (tar) or wax.

^{[85]}