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Encyclopedia > Archimedes
Classical Greek philosophy
Ancient philosophy
Archimedes Thoughtful by Fetti (1620)

Name Archimedes was a celebrated mathematician and engineer of ancient Greece. ... Image File history File linksMetadata Size of this preview: 450 Ã— 600 pixel Image in higher resolution (1364 Ã— 1818 pixel, file size: 548 KB, MIME type: image/jpeg) Archimedes Thoughtful by Domenico Fetti, 1620 Alte Meister Museum, Dresden (Germany/Deutschland) Source/Quelle: http://archimedes2. ... Domenico Fetti (c1589-1624) was an Italian painter born in Rome. ...

Archimedes of Syracuse (Greek: Άρχιμήδης)

Birth

c. 287 BC (Syracuse, Sicily, Magna Graecia) Syracuse (Italian Siracusa, Sicilian Sarausa, Greek , Latin Syracusae) is an Italian city on the eastern coast of Sicily and the capital of the province of Syracuse. ... Magna Graecia around 280 b. ...

Death

c. 212 BC (Syracuse)

Euclid of Alexandria
Natural philosophy Euclid Euclid of Alexandria (Greek: ) (ca. ... Natural philosophy or the philosophy of nature, known in Latin as philosophia naturalis, is a term applied to the objective study of nature and the physical universe that was regnant before the development of modern science. ...

Main interests

mathematics, physics, engineering, astronomy For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Engineering is the discipline of acquiring and applying knowledge of design, analysis, and/or construction of works for practical purposes. ... For other uses, see Astronomy (disambiguation). ...

Notable ideas

Hydrostatics, Levers,
Infinitesimals Hydrostatics, also known as fluid statics, is the study of fluids at rest. ... For the Portuguese town and parish, see Lever, Portugal. ... The ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse was the first mathematician to make explicit use of infinitesimals. ...

Archimedes died during the Siege of Syracuse, when he was killed by a Roman soldier despite orders that he should not be harmed. At his request, his tomb carried a carving of his favorite mathematical proof. Modern experiments have tested claims that he built a "death ray" capable of setting ships on fire at a distance, and that he constructed a device that could sink ships by lifting them out of the water. [3] The discovery of previously unknown works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.[4] The Siege of Syracuse was fought from 214 BC to 212 BC between the rebellious city of Syracuse, and a Roman army under Marcellus sent to put down the citys rebellion. ... The Archimedes Palimpsest is a palimpsest on parchment in the form of a codex which originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors. ...

Carl Friedrich Gauss is said to have remarked that Archimedes was one of the three epoch-making mathematicians, with the others being Sir Isaac Newton and Ferdinand Eisenstein.[5] Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Sir Isaac Newton in Knellers portrait of 1689. ... Ferdinand Gotthold Max Eisenstein (April 16, 1823 - October 11, 1852) was a German mathematician. ...

## Biography

This bronze statue of Archimedes is at the Archenhold Observatory in Berlin. It was sculpted by Gerhard Thieme and unveiled in 1972.

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 had ordered him not to be harmed.[9] Combatants Roman Republic Carthage Commanders Publius Cornelius Scipioâ€ , Tiberius Sempronius Longus Publius Cornelius Scipio Africanus, Gaius Flaminiusâ€ , Fabius Maximus, Claudius Marcellusâ€ , Lucius Aemilius Paullusâ€ , Gaius Terentius Varro, Marcus Livius Salinator, Gaius Claudius Nero, Gnaeus Cornelius Scipio Calvusâ€ , Masinissa, Minuciusâ€ , Servilius Geminusâ€  Hannibal Barca, Hasdrubal Barcaâ€ , Mago Barcaâ€ , Hasdrubal Giscoâ€ , Syphax... Marcus Claudius Marcellus (ca. ... A siege is a military blockade of a city or fortress with the intent of conquering by force or attrition, often accompanied by an assault. ...

The last words attributed to Archimedes are "Do not disturb my circles" (Greek: μή μου τούς κύκλους τάραττε), 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.[9] For other uses, see Latin (disambiguation). ...

Μη μου τους κύκλους τάραττε – Do not disturb my circles Image File history File links Archimedes_circles. ...

Listen to the last words attributed to Archimedes.

Problems listening to the file? See media help.

The sphere has 2/3 the surface area and volume of the circumscribing cylinder. A sphere and cylinder were carved on the tomb of Archimedes at his request.

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 Universal History 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.[11] Polybius (c. ... A portrait of Titus Livius made long after his death. ...

## Discoveries and inventions

The story about the golden crown does not appear in the known works of Archimedes, but in his treatise On Floating Bodies he gives the principle known in hydrostatics as Archimedes' Principle. This states that a body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid.[14] Hydrostatics, also known as fluid statics, is the study of fluids at rest. ... In physics, buoyancy is an upward force on an object immersed in a fluid (i. ...

While Archimedes did not invent the lever, he wrote the earliest known rigorous explanation of the principle involved. According to Pappus of Alexandria, his work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." (Greek: "δος μοι πα στω και ταν γαν κινάσω")[15] 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.[16] For the Portuguese town and parish, see Lever, Portugal. ... Pappus of Alexandria is one of the most important mathematicians of ancient Greek time, known for his work Synagoge or Collection (c. ... This block and tackle on a davit of the Mercator is used to help lower a boat. ... For the band, see Pulley (band). ... Leverage is related to torque; leverage is a factor by which lever multiplies a force. ...

The Archimedes' screw was operated by hand and could raise water efficiently.

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 Hieron 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.[17] 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. The screw was a machine 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. Versions of the Archimedes' screw are still in use today in developing countries. 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.[18][19][20] Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Archimedes screw. ... Athenaeus (ca. ... The Greek ship Syracusia had a length of 55 meters (180. ... The gymnasium functioned as a training facility for competitors in public games. ... The Birth of Venus, (detail) by Sandro Botticelli, 1485 For other uses, see Aphrodite (disambiguation). ... Archimedes screw. ... Hanging Gardens redirects here. ...

The Claw of Archimedes is another 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 on to 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.[21][22] The Claw of Archimedes was an ancient weapon devised by Archimedes to defend the seaward portion of Syracuses city wall against amphibious assault. ...

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.[23] Replica catapult at ChÃ¢teau des Baux, France For the handheld Y-shaped weapon, see slingshot. ... A modern non-digital odometer A Smiths speedometer from the 1920s showing odometer and trip meter An odometer is a device used for indicating distance traveled by an automobile or other vehicle. ... Osama was here and he doesnt enjoy this site???? the red sox won and i am one happy camper. ...

Cicero (106 BC–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 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: For other uses, see Cicero (disambiguation). ... For other uses, see Dialogue (disambiguation). ... De re publica is a work by Cicero, written in six books 54-51 BC, in the format of a Socratic dialogue, that is to say: Scipio Africanus Minor (who had died a few decades before Cicero was born) takes the role of wise old man, that is an obligatory... Marcus Claudius Marcellus (ca. ... For the Defense and Security Company, see Thales Group. ... Another article concerns Eudoxus of Cyzicus. ... Gaius Sulpicius Gallus, Roman general, statesman and orator. ... Lucius Furius Philus was a consul of ancient Rome in 136 BC. He was a member of the Scipionic circle, and particularly close to Scipio Aemilianus. ...

 “ 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. – When Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that position which was its shadow on the Earth, when the Sun was in line.[24][25] ”

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 from classical antiquity that was probably designed for the same purpose. Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing. 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.[26][27] For the song by Ai Otsuka, see Planetarium (song) // A planetarium is a theatre built primarily for presenting educational and entertaining shows about astronomy and the night sky, or for training in celestial navigation. ... A small orrery showing earth and the inner planets An orrery is a mechanical device that illustrates the relative positions and motions of the planets and moons in the solar system in heliocentric model. ... Pappus of Alexandria is one of the most important mathematicians of ancient Greek time, known for his work Synagoge or Collection (c. ... On Sphere-Making is the title of a lost work by Archimedes, mentioned by Pappus of Alexandria. ... The Antikythera mechanism (main fragment). ... In an automobile and other four-wheeled vehicles, a differential is a device, usually consisting of gears, that allows each of the driving wheels to rotate at different speeds, while supplying equal torque to each of them. ...

### "Death ray"

Archimedes may have used mirrors acting as a parabolic reflector to burn ships attacking Syracuse

Lucian wrote that during the Siege of Syracuse (c. 214–212 BC), Archimedes repelled an attack by Roman forces with a burning glass.[28] The device was used to focus sunlight on to the approaching ships, causing them to catch fire. This claim, sometimes called the "Archimedes death ray", 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.[29] 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 on to a ship. This would have used the principle of the parabolic reflector in a manner similar to a solar furnace. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... A parabolic reflector (also known as a parabolic dish or a parabolic mirror) is a reflective device formed in the shape of a paraboloid of revolution. ... Syracuse (Italian Siracusa, Sicilian Sarausa, Greek , Latin Syracusae) is an Italian city on the eastern coast of Sicily and the capital of the province of Syracuse. ... Lucian. ... The Siege of Syracuse was fought from 214 BC to 212 BC between the rebellious city of Syracuse, and a Roman army under Marcellus sent to put down the citys rebellion. ... A burning-glass is a large convex lens that can concentrate the suns rays onto a small area, heating up the area and thus resulting in ignition of the surface exposed. ... Descartes redirects here. ... This article is about the metal alloy. ... For other uses, see Copper (disambiguation). ... A parabolic reflector (also known as a parabolic dish or a parabolic mirror) is a reflective device formed in the shape of a paraboloid of revolution. ... solar oven A solar oven or solar furnace is a way of harnessing the suns power to cook food. ...

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 mocked-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 weapon was a feasible device 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 flash point, which is around 300 degrees Celsius (570 °F), and this is hotter than the maximum temperature produced by a domestic oven.[30] When Mythbusters broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (or "failed") due to of the length of time and ideal weather conditions required for combustion to occur. [31] â€œMITâ€ redirects here. ... MythBusters is an American popular science television program on the Discovery Channel starring special effects experts Adam Savage and Jamie Hyneman, who use basic elements of the scientific method to test the validity of various rumors and urban legends in popular culture. ... This page is a candidate for speedy deletion. ... For other uses, see Flash point (disambiguation). ...

A similar test of the "Archimedes death 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 is flammable and may have aided combustion.[32] Skaramagas, Skaramanga, Skaramaga, Skaramangas or Skaramagkas (Greek: Î£ÎºÎ±ÏÎ±Î¼Î±Î³ÎºÎ¬ or Î£ÎºÎ±ÏÎ±Î¼Î±Î³ÎºÎ¬Ï‚) older forms Skaramangas and Scaramanga is a small town in the western part of Athens, Greece. ... This article is about the capital of Greece. ... Ewer from Iran, dated 1180-1210CE. Composed of brass worked in repoussÃ© and inlaid with silver and bitumen. ...

## Mathematics

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.”[33] Mestrius Plutarchus (Greek: Î Î»Î¿ÏÏ„Î±ÏÏ‡Î¿Ï‚; 46 - 127), better known in English as Plutarch, was a Greek historian, biographer, essayist, and Middle Platonist. ...

Archimedes used the method of exhaustion to approximate the value of π.

In The Measurement of a Circle, Archimedes gives the value of the square root of 3 as being more than 265/153 (approximately 1.732) and less than 1351/780 (approximately 1.7320512). The actual value is around 1.7320508076, making this a very accurate estimate. He introduced this result without offering any explanation of the method used to obtain 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."[34] In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ...

In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He expressed the solution to the problem as a geometric series that summed to infinity with the ratio 1/4: Image File history File links No higher resolution available. ... A parabolic segment. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... For other uses, see Infinity (disambiguation). ... This article is about the mathematical concept. ...

$sum_{n=0}^infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + cdots = {4over 3}. ;$

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 is a variation of the infinite series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3. A secant line of a curve is a line that intersects two or more points on the curve. ... In mathematics, a series is a sum of a sequence of terms. ... Archimedes figure with a = 3/4 In mathematics, the infinite series 1/4 + 1/16 + 1/64 + 1/256 + Â· Â· Â· is an example of one of the first infinite series to be summed in the history of mathematics; it was used by Archimedes circa 250-200 BC.[1] Its sum is...

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 around the myriad. The word is based on the Greek for uncountable, murious, and was also used to denote the number 10,000. He proposed a number system using powers of myriad myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8×1063 in modern notation.[35] The Sand Reckoner (Greek: Î¨Î±Î¼Î¼Î¯Ï„Î·Ï‚, Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. ... Hiero II, tyrant of Syracuse from 270 to 215 BC, was the illegitimate son of a Syracusan noble, Hierocles, who claimed descent from Gelo. ... For other uses, see Myriad (disambiguation). ...

## Writings

Archimedes is said to have remarked about the lever: "Give me a place to stand on, and I will move the Earth."
• On the Equilibrium of Planes (two volumes)
The first book is in fifteen propositions with seven postulates, while the second book is in ten propositions. In this work Archimedes explains the Law of the Lever, stating:
 “ Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance. ”
Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, paraboloids, and hemispheres. [36]
• On the Measurement of the Circle
This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of Conon of Samos. In Proposition II, Archimedes shows that the value of π (Pi) is greater than 223/71 and less than 22/7. The latter figure was used as an approximation of π throughout the Middle Ages and is still used today when a rough figure is required.
• On Spirals
This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation
with real numbers a and b. This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.
• On the Sphere and the Cylinder (two volumes)
In this treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter. The volume is 4/3πr³ for the sphere, and 2πr³ for the cylinder; the surface area is 4πr² for the sphere, and 6πr² for the cylinder, where r is the radius. The sphere will have two thirds of the volume and surface area of the cylinder. A carving of this proof was used on the tomb of Archimedes at his request.
• On Conoids and Spheroids
This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.
• On Floating Bodies (two volumes)
In the first part of this treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not self-gravitating, since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.
Archimedes is commemorated on a Greek postage stamp from 1983.
In the second part, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, similar to the way that icebergs float. Archimedes' principle of buoyancy is given in the work, stated as follows:
 “ 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. ”
• The Quadrature of the Parabola
A work of 24 propositions addressed to Dositheus. In this treatise Archimedes proves by two methods that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He achieves this by calculating the value of a geometric series that sums to infinity with the ratio 1/4.
• Stomachion
This is a dissection puzzle similar to a Tangram, and the treatise describing it was found in more complete form in the Archimedes Palimpsest. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Research published by Dr. Reviel Netz of Stanford University in 2003 argued that Archimedes was attempting to determine how many ways the pieces of paper could be assembled into the shape of a square. The figure given by Dr. Netz is that the pieces can be made into a square in 17,152 ways. The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded. The Stomachion represents an example of an early problem in combinatorics. Stomachion is the Greek word for stomach, στομάχιον; the reason for the name is unknown.[37][38]
This work was discovered by Gotthold Ephraim Lessing in a Greek manuscript consisting of a poem of 44 lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773. It is addressed to Eratosthenes and the mathematicians at the University of Alexandria. Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations. There is a more difficult version of the problem in which some of the answers are required to be square numbers. This version of the problem was first solved by a computer in 1965, and the answer is a very large number, approximately 7.760271×10206544.[39]
In this treatise, Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos (concluding that "this is impossible"), contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8×1063 in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner or Psammites is the only surviving work in which Archimedes discusses his views on astronomy.[40]
• The Method of Mechanical Theorems
This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.

### Apocryphal works

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.[41] Arabic can mean: From or related to Arabia From or related to the Arabs The Arabic language; see also Arabic grammar The Arabic alphabet, used for expressing the languages of Arabic, Persian, Malay ( Jawi), Kurdish, Panjabi, Pashto, Sindhi and Urdu, among others. ... Sir Thomas Little Heath (October 5, 1861 - March 16, 1940) was a mathematician, classical scholar, historian of ancient Greek mathematics, and translator. ... Marshall Clagett (January 23, 1916 - October 21, 2005) was an American scholar who specialized in the history of science. ...

It has also been claimed by the Arab scholar Abu'l Raihan Muhammed al-Biruni 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.[42] A triangle with sides a, b, and c. ... Hero (or Heron) of Alexandria (Greek: Î‰ÏÏ‰Î½ Î¿ Î‘Î»ÎµÎ¾Î±Î½Î´ÏÎµÏÏ‚) (c. ...

## Archimedes Palimpsest

Main article: Archimedes Palimpsest

The treatises in the Archimedes Palimpsest are: On the Equilibrium of Planes, On Spirals, The Measurement of the Circle, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems and Stomachion.

## Legacy

The Fields Medal carries a portrait of Archimedes.

There is a crater on the Moon named Archimedes (29.7° N, 4.0° W) in his honor, and a lunar mountain range, the Montes Archimedes (25.3° N, 4.6° W).[46] The asteroid 3600 Archimedes is named after him.[47] Image File history File links Metadata Size of this preview: 624 Ã— 600 pixelsFull resolution (800 Ã— 769 pixel, file size: 115 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links Metadata Size of this preview: 624 Ã— 600 pixelsFull resolution (800 Ã— 769 pixel, file size: 115 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... The obverse of the Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ... Tycho crater on Earths moon. ... This article is about Earths moon. ... Archimedes is a large lunar impact crater on the eastern edges of the Mare Imbrium. ... Montes Archimedes is a mountain range on the Moon. ... For other uses, see Asteroid (disambiguation). ... 3600 Archimedes is a small main belt asteroid. ...

The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with his proof concerning the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to him which reads in Latin: "Transire suum pectus mundoque potiri" (Rise above oneself and grasp the world).[48] The obverse of the Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ...

Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982) and Spain (1963).[49] This article is about the state which existed from 1949 to 1990. ...

The exclamation of Eureka! attributed to Archimedes is the state motto of California. In this instance the word refers to the discovery of gold near Sutter's Mill in 1848 which sparked the California gold rush.[50] Eureka (Eureka!, or Heureka; Greek (later ); IPA: (modern Greek), (ancient Greek, both former and later forms), Anglicised as ) is a famous exclamation attributed to Archimedes. ... This article is about the U.S. state. ... Sutters Mill in 1850. ... The California Gold Rush (1848â€“1855) began shortly after January 24, 1848 (when gold was discovered at Sutters Mill in Coloma). ...

This article or section contains information that has not been verified and thus might not be reliable. ... An Archimedes number, named after the ancient Greek scientist Archimedes, to determine the motion of fluids due to density differences, is a dimensionless number in the form where: g = gravitational acceleration (9. ... The Archimedes Paradox states that an object can float in water that has less volume than the object itself, if its average density is less than that of water. ... In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... The ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse was the first mathematician to make explicit use of infinitesimals. ... This article presents and explains several methods which can be used to calculate square roots. ... For other uses, see Zhang Heng (disambiguation). ...

## Notes and references

### Notes

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. Conon of Samos (circa 280 BC - circa: 220 BC) was a Greek mathematician and astronomer. ...

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. On Sphere-Making is the title of a lost work by Archimedes, mentioned by Pappus of Alexandria. ... Theon (c. ... The Sand Reckoner (Greek: Î¨Î±Î¼Î¼Î¯Ï„Î·Ï‚, Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. ... Sir Thomas Little Heath (October 5, 1861 - March 16, 1940) was a mathematician, classical scholar, historian of ancient Greek mathematics, and translator. ...

c. ^ Boyer, Carl Benjamin A History of Mathematics (1991) ISBN 0471543977 "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=sqrt(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." Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... A chord of a curve is a geometric line segment whose endpoints both lie on the curve. ...

### References

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6. ^ T. L. Heath, Works of Archimedes, 1897
7. ^ Plutarch. Parallel Lives Complete e-text from Gutenberg.org. Project Gutenberg. Retrieved on 2007-07-23.
8. ^ O'Connor, J.J. and Robertson, E.F.. Archimedes of Syracuse. University of St Andrews. Retrieved on 2007-01-02.
9. ^ a b Rorres, Chris. Death of Archimedes: Sources. Courant Institute of Mathematical Sciences. Retrieved on 2007-01-02.
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11. ^ Rorres, Chris. Siege of Syracuse. Courant Institute of Mathematical Sciences. Retrieved on 2007-07-23.
12. ^ Vitruvius. De Architectura, Book IX, paragraphs 9–12, text in English and Latin. University of Chicago. Retrieved on 2007-08-30.
13. ^ HyperPhysics. Buoyancy. Georgia State University. Retrieved on 2007-07-23.
14. ^ Carroll, Bradley W. Archimedes' Principle. Weber State University. Retrieved on 2007-07-23.
15. ^ Quoted by Pappus of Alexandria in Synagoge, Book VIII
16. ^ Pulleys. Society of Women Engineers. Retrieved on 2007-07-23.
17. ^ Casson, Lionel (1971). Ships and Seamanship in the Ancient World. Princeton University Press. ISBN 0691035369.
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19. ^ Rorres, Chris. Archimedes Screw - Optimal Design. Courant Institute of Mathematical Sciences. Retrieved on 2007-07-23.
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28. ^ Hippias, C.2.
29. ^ John Wesley. A Compendium of Natural Philosophy (1810) Chapter XII, Burning Glasses. Online text at Wesley Center for Applied Theology. Retrieved on 2007-09-14.
30. ^ Bonsor, Kevin. How Wildfires Work. HowStuffWorks. Retrieved on 2007-07-23.
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• Boyer, Carl Benjamin (1991). A History of Mathematics. New York: Wiley. ISBN 0-471-54397-7.
• Dijksterhuis, E.J. (1987). Archimedes. Princeton University Press, Princeton. ISBN 0-691-08421-1.  Republished translation of the 1938 study of Archimedes and his works by an historian of science.
• Gow, Mary (2005). Archimedes: Mathematical Genius of the Ancient World. Enslow Publishers, Inc. ISBN 0-7660-2502-0.
• Hasan, Heather (2005). Archimedes: The Father of Mathematics. Rosen Central. ISBN 978-1404207745.
• Heath, T.L. (1897). Works of Archimedes. Dover Publications. ISBN 0-486-42084-1.  Complete works of Archimedes in English.
• Netz, Reviel and Noel, William (2007). The Archimedes Codex. Orion Publishing Group. ISBN 0-297-64547-1.
• Simms, Dennis L. (1995). Archimedes the Engineer. Continuum International Publishing Group Ltd. ISBN 0-720-12284-8.
• Stein, Sherman (1999). Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. ISBN 0-88385-718-9.

Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Eduard Jan Dijksterhuis (1892-1965), Historian of Science from the Netherlands. ... Thomas Little Heath (October 5, 1861 - March 16, 1940) was a mathematician, classical scholar, historian of ancient Greek mathematics, and translator. ...

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