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{{Infobox scientist
| name = Archimedes of Syracuse
| image = Domenico-Fetti Archimedes 1620.jpg
| caption = ''Archimedes Thoughtful'' by [[Domenico Fetti|Fetti]] (1620)
| birth_date = ''c''. 287&nbsp;BC
| birth_place = [[Syracuse, Sicily]]<br />[[Magna Graecia]]
| death_date = ''c''. 212&nbsp;BC (aged around 75)
| death_place = Syracuse
| residence = [[Syracuse, Sicily]]
| field = [[Mathematics]]<br />[[Physics]]<br />[[Engineering]]<br />[[Astronomy]]<br />[[Invention]]
| known_for = [[Archimedes' principle]]<br />[[Archimedes' screw]]<br />[[Fluid statics|hydrostatics]]<br />[[lever]]s<br />[[Archimedes' use of infinitesimals|infinitesimals]]
}}
'''Archimedes of Syracuse''' ({{lang-grc-gre|[[wikt:Ἀρχιμήδης|Ἀρχιμήδης]]}}; {{circa|287}}&nbsp;BC&nbsp;– {{circa|212}}&nbsp;BC) was a [[Greeks|Greek]] [[Greek mathematics|mathematician]], [[physicist]], [[engineer]], [[inventor]], and [[astronomer]].<ref>{{cite web|title=Archimedes (c.287 - c.212 BC)|url=http://www.bbc.co.uk/history/historic_figures/archimedes.shtml|work=BBC History|accessdate=2012-06-07}}</ref> Although few details of his life are known, he is regarded as one of the leading [[scientist]]s in [[classical antiquity]]. Among his advances in [[physics]] are the foundations of [[Fluid statics|hydrostatics]], [[statics]] and an explanation of the principle of the [[lever]]. He is credited with designing innovative [[machine]]s, including [[siege engine]]s and the [[Archimedes' screw|screw pump]] that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors.<ref name="death ray"/>
 
Archimedes is generally considered to be the greatest [[mathematician]] of antiquity and one of the greatest of all time.<ref>{{cite book |last=Calinger |first=Ronald |title=A Contextual History of Mathematics |year=1999 |publisher=Prentice-Hall |isbn=0-02-318285-7 |page=150 |quote=Shortly after Euclid, compiler of the definitive textbook, came Archimedes of Syracuse (ca. 287&nbsp;212 BC), the most original and profound mathematician of antiquity.}}</ref><ref>{{cite web |url=http://www-history.mcs.st-and.ac.uk/Biographies/Archimedes.html |title=Archimedes of Syracuse |accessdate=2008-06-09 |publisher=The MacTutor History of Mathematics archive |date=January 1999}}</ref> He used the [[method of exhaustion]] to calculate the [[area]] under the arc of a [[parabola]] with the [[Series (mathematics)|summation of an infinite series]], and gave a remarkably accurate approximation of [[pi]].<ref>{{cite web|title = A history of calculus |author=O'Connor, J.J. and Robertson, E.F.|publisher = [[University of St Andrews]]| url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html |date=February 1996|accessdate= 2007-08-07| archiveurl= http://web.archive.org/web/20070715191704/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html| archivedate= 15 July 2007 <!--DASHBot-->| deadurl= no}}</ref> He also defined the [[Archimedes spiral|spiral]] bearing his name, formulae for the [[volume]]s of [[solid of revolution|solids of revolution]], and an ingenious system for expressing very large numbers.
 
Archimedes died during the [[Siege of Syracuse (214–212 BC)|Siege of Syracuse]] when he was killed by a [[Roman Republic|Roman]] soldier despite orders that he should not be harmed. [[Cicero]] describes visiting the tomb of Archimedes, which was surmounted by a [[sphere]] [[inscribe]]d within a [[cylinder (geometry)|cylinder]]. Archimedes had proven that the sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest of his mathematical achievements.
 
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&nbsp;AD by [[Isidore of Miletus]], while commentaries on the works of Archimedes written by [[Eutocius of Ascalon|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]],<ref>{{cite web|title = Galileo, Archimedes, and Renaissance engineers |author=Bursill-Hall, Piers|publisher = sciencelive with the University of Cambridge| url = http://www.sciencelive.org/component/option,com_mediadb/task,view/idstr,CU-MMP-PiersBursillHall/Itemid,30|accessdate= 2007-08-07}}</ref> 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.<ref>{{cite web|title = Archimedes&nbsp;– The Palimpsest|publisher =[[Walters Art Museum]]|url = http://www.archimedespalimpsest.org/palimpsest_making1.html|accessdate=2007-10-14|archiveurl =http://web.archive.org/web/20070928102802/http://www.archimedespalimpsest.org/palimpsest_making1.html <!--DASHBot-->|archivedate =2007-09-28}}</ref>
 
==Biography==
Archimedes was born ''c''. 287&nbsp;BC in the seaport city of [[Syracuse, Sicily]], at that time a self-governing [[Colonies in antiquity|colony]] in [[Magna Graecia]], located along the coast of [[Southern Italy]]. The date of birth is based on a statement by the [[Byzantine Greeks|Byzantine Greek]] historian [[John Tzetzes]] that Archimedes lived for 75 years.<ref>[[T. L. Heath|Heath, T. L.]], ''Works of Archimedes'', 1897</ref> In ''[[The Sand Reckoner]]'', Archimedes gives his father's name as Phidias, an [[astronomer]] about whom nothing is known. [[Plutarch]] wrote in his ''[[Parallel Lives]]'' that Archimedes was related to King [[Hiero II of Syracuse|Hiero II]], the ruler of Syracuse.<ref>{{cite web|title = ''Parallel Lives'' Complete e-text from Gutenberg.org|author=[[Plutarch]]|publisher = [[Project Gutenberg]]| url = http://www.gutenberg.org/etext/674|accessdate=2007-07-23| archiveurl= http://web.archive.org/web/20070711045124/http://www.gutenberg.org/etext/674| archivedate= 11 July 2007 <!--DASHBot-->| deadurl= no}}</ref> A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure.<ref name="mactutor">{{cite web | author=O'Connor, J.J. and Robertson, E.F.|url = http://www-history.mcs.st-andrews.ac.uk/Biographies/Archimedes.html|title = Archimedes of Syracuse|publisher = University of St Andrews|accessdate = 2007-01-02| archiveurl= http://web.archive.org/web/20070206082010/http://www-history.mcs.st-andrews.ac.uk/Biographies/Archimedes.html| archivedate= 6 February 2007 <!--DASHBot-->| deadurl= no}}</ref> It is unknown, for instance, whether he ever married or had children. During his youth, Archimedes may have studied in [[Alexandria]], [[Ancient Egypt|Egypt]], where [[Conon of Samos]] and [[Eratosthenes|Eratosthenes of Cyrene]] were contemporaries. He referred to Conon of Samos as his friend, while two of his works (''[[Archimedes' use of infinitesimals|The Method of Mechanical Theorems]]'' and the ''[[Archimedes' cattle problem|Cattle Problem]]'') have introductions addressed to Eratosthenes.{{Ref_label|A|a|none}}
 
Archimedes died ''c''. 212&nbsp;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 {{nowrap|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 not be harmed.<ref name="death">{{cite web |first=Chris |last=Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Death/Histories.html|title = Death of Archimedes: Sources|publisher = [[Courant Institute of Mathematical Sciences]]|accessdate = 2007-01-02| archiveurl= http://web.archive.org/web/20061210060235/http://www.math.nyu.edu/~crorres/Archimedes/Death/Histories.html| archivedate= 10 December 2006 <!--DASHBot-->| deadurl= no}}</ref>
 
[[File:Cicero Discovering the Tomb of Archimedes by Benjamin West.jpeg|thumb|right|''Cicero Discovering the Tomb of Archimedes'' by [[Benjamin West]] (1805)]]
The last words attributed to Archimedes are "Do not disturb my circles" ({{lang-el|μή μου τοὺς 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.<ref name="death"/>
 
The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a [[sphere]] and a [[cylinder (geometry)|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&nbsp;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 was 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.<ref>{{cite web |first=Chris |last=Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html|title = Tomb of Archimedes: Sources|publisher = Courant Institute of Mathematical Sciences|accessdate = 2007-01-02| archiveurl= http://web.archive.org/web/20061209201723/http://www.math.nyu.edu/%7Ecrorres/Archimedes/Tomb/Cicero.html| archivedate= 9 December 2006 <!--DASHBot-->| deadurl= no}}</ref> A tomb discovered in a hotel courtyard in Syracuse in the early 1960s was claimed to be that of Archimedes, but its location today is unknown.<ref>{{cite web |first=Chris |last=Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Tomb/TombIllus.html|title = Tomb of Archimedes&nbsp;– Illustrations|publisher = Courant Institute of Mathematical Sciences|accessdate = 2011-03-15}}</ref>
 
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.<ref>{{cite web| first=Chris |last=Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Siege/Polybius.html|title = Siege of Syracuse| publisher = Courant Institute of Mathematical Sciences|accessdate = 2007-07-23| archiveurl= http://web.archive.org/web/20070609013114/http://www.math.nyu.edu/~crorres/Archimedes/Siege/Polybius.html| archivedate= 9 June 2007 <!--DASHBot-->| deadurl= no}}</ref>
 
==Discoveries and inventions==
 
===Archimedes' principle===
{{main|Archimedes' principle}}
[[File:Archimedes water balance.gif|thumb|right|180px|Archimedes may have used his principle of buoyancy to determine whether the golden crown was less [[density|dense]] than solid gold.]]
 
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, 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.<ref>{{cite web|title = ''De Architectura'', Book IX, paragraphs 9–12, text in English and Latin|author= [[Vitruvius]]| publisher = [[University of Chicago]]|url = http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Vitruvius/9*.html|accessdate=2007-08-30}}</ref> 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,<ref>{{cite web|title = Incompressibility of Water|publisher =[[Harvard University]]|url = http://www.fas.harvard.edu/~scdiroff/lds/NewtonianMechanics/IncompressibilityofWater/IncompressibilityofWater.html|accessdate=2008-02-27| archiveurl= http://web.archive.org/web/20080317130651/http://www.fas.harvard.edu/~scdiroff/lds/NewtonianMechanics/IncompressibilityofWater/IncompressibilityofWater.html| archivedate= 17 March 2008 <!--DASHBot-->| deadurl= no}}</ref> 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 (word)|Eureka]]!" ({{lang-el|"εὕρηκα}}!," meaning "I have found it!"). The test was conducted successfully, proving that silver had indeed been mixed in.<ref>{{cite web|title = Buoyancy|author= [[HyperPhysics]]| publisher =[[Georgia State University]]|url = http://hyperphysics.phy-astr.gsu.edu/Hbase/pbuoy.html|accessdate=2007-07-23| archiveurl= http://web.archive.org/web/20070714113647/http://hyperphysics.phy-astr.gsu.edu/hbase/pbuoy.html#c1| archivedate= 14 July 2007 <!--DASHBot-->| deadurl= no}}</ref>
 
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.<ref name="inaccuracy">{{cite web |first=Chris |last=Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html|title = The Golden Crown|publisher = [[Drexel University]]|accessdate = 2009-03-24| archiveurl= http://web.archive.org/web/20090311051318/http://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html| archivedate= 11 March 2009 <!--DASHBot-->| deadurl= no}}</ref> Archimedes may have instead sought a solution that applied the principle known in [[fluid statics|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 [[buoyancy|buoyant force]] equal to the weight of the fluid it displaces.<ref>{{cite web|title = ''Archimedes' Principle''|first=Bradley W |last=Carroll |publisher=[[Weber State University]]|url =http://www.physics.weber.edu/carroll/Archimedes/principle.htm|accessdate=2007-07-23| archiveurl= http://web.archive.org/web/20070808132323/http://physics.weber.edu/carroll/Archimedes/principle.htm| archivedate= 8 August 2007 <!--DASHBot-->| deadurl= no}}</ref> Using this principle, it would have been possible to compare the density of the golden crown to that of solid gold by balancing the crown on a scale with a gold reference sample, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly. [[Galileo Galilei|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."<ref name="galileo">{{cite web |first=Chris |last=Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Crown/bilancetta.html|title = The Golden Crown: Galileo's Balance|publisher = [[Drexel University]]|accessdate = 2009-03-24| archiveurl= http://web.archive.org/web/20090224221137/http://math.nyu.edu/~crorres/Archimedes/Crown/bilancetta.html| archivedate= 24 February 2009 <!--DASHBot-->| deadurl= no}}</ref> 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.<ref name="kingcrown">[http://www.jstor.org/stable/27690606 O. A. W. Dilke. Gnomon. 62. Bd., H. 8 (1990), pp. 697-699 Published by: Verlag C.H.Beck]</ref><ref>Marcel Berthelot - Sur l histoire de la balance hydrostatique et de quelques autres appareils et procédés scientifiques, Annales de Chimie et de Physique [série 6], 23 / 1891, pp. 475-485</ref> 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.<ref name="kingcrown" />
 
===Archimedes' screw===
{{main|Archimedes' screw}}
[[File:Archimedes-screw one-screw-threads with-ball 3D-view animated small.gif|thumb|left|The [[Archimedes' screw]] can 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|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.<ref>{{cite book |last=Casson|first= Lionel|authorlink= Lionel Casson|title=Ships and Seamanship in the Ancient World|year=1971 |publisher= Princeton University Press |isbn=0-691-03536-9}}</ref> According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a [[Gymnasium (ancient Greece)|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 {{nowrap|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]].<ref>{{cite web|title = ''Sennacherib, Archimedes, and the Water Screw: The Context of Invention in the Ancient World''|author=Dalley, Stephanie. [[John Peter Oleson|Oleson, John Peter]]| publisher = ''Technology and Culture'' Volume 44, Number 1, January 2003 (PDF)| url =http://muse.jhu.edu/journals/technology_and_culture/toc/tech44.1.html|accessdate=2007-07-23}}</ref><ref>{{cite web|title = Archimedes' screw&nbsp;– Optimal Design|author=Rorres, Chris| publisher =Courant Institute of Mathematical Sciences|url =http://www.cs.drexel.edu/~crorres/Archimedes/Screw/optimal/optimal.html |accessdate=2007-07-23}}</ref><ref>[[:File:Archimedes-screw one-screw-threads with-ball 3D-view animated.gif|An animation of an Archimedes' screw]]</ref> The world's first seagoing [[steamboat|steamship]] with a [[propeller|screw propeller]] was the ''[[SS Archimedes]]'', which was launched in 1839 and named in honor of Archimedes and his work on the screw.<ref>{{cite web |title = SS Archimedes|publisher= wrecksite.eu |url = http://www.wrecksite.eu/wreck.aspx?636|accessdate=2011-01-22}}</ref>
 
===Claw of Archimedes===
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.<ref>{{cite web |first=Chris |last=Rorres|title = Archimedes' Claw&nbsp;– Illustrations and Animations&nbsp;– a range of possible designs for the claw| publisher = Courant Institute of Mathematical Sciences|url = http://www.math.nyu.edu/~crorres/Archimedes/Claw/illustrations.html|accessdate=2007-07-23}}</ref><ref>{{cite web|title = Archimedes' Claw&nbsp;– watch an animation|first=Bradley W |last=Carroll|publisher = Weber State University| url = http://physics.weber.edu/carroll/Archimedes/claw.htm|accessdate=2007-08-12| archiveurl= http://web.archive.org/web/20070813202716/http://physics.weber.edu/carroll/Archimedes/claw.htm| archivedate= 13 August 2007 <!--DASHBot-->| deadurl= no}}</ref>
 
===Heat ray===
[[File:Archimedes Heat Ray conceptual diagram.svg|thumb|right|Archimedes may have used mirrors acting collectively as a [[parabolic reflector]] to burn ships attacking [[Syracuse, Sicily|Syracuse]].]]
 
The 2nd century AD author [[Lucian]] wrote that during the [[Siege of Syracuse (212 BC)|Siege of Syracuse]] (''c.'' 214–212&nbsp;BC), Archimedes destroyed enemy ships with fire. Centuries later, [[Anthemius of Tralles]] mentions [[burning-glass]]es as Archimedes' weapon.<ref>''Hippias'', 2 (cf. [[Galen]], ''On temperaments'' 3.2, who mentions ''pyreia'', "torches"); [[Anthemius of Tralles]], ''On miraculous engines'' 153 [Westerman].</ref> The device, sometimes called the "Archimedes heat ray", was used to focus sunlight onto approaching ships, causing them to catch fire.
 
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.<ref>{{cite web |author=[[John Wesley]] |url = http://wesley.nnu.edu/john_wesley/wesley_natural_philosophy/duten12.htm| title = ''A Compendium of Natural Philosophy'' (1810) Chapter XII, ''Burning Glasses''|publisher = Online text at Wesley Center for Applied Theology|accessdate = 2007-09-14 |archiveurl = http://web.archive.org/web/20071012154432/http://wesley.nnu.edu/john_wesley/wesley_natural_philosophy/duten12.htm <!--DASHBot--> |archivedate = 2007-10-12}}</ref> 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. This would have used the principle of the [[parabolic reflector]] in a manner similar to a [[solar furnace]].
 
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&nbsp;m). The mirrors were pointed at a plywood {{nowrap|mock-up}} of a Roman warship at a distance of around 160&nbsp;feet (50&nbsp;m). When the mirrors were focused accurately, the ship burst into flames within a few seconds. The plywood ship had a coating of [[bitumen|tar]] paint, which may have aided combustion.<ref>{{cite news|title = Archimedes' Weapon| publisher = [[Time (magazine)|Time Magazine]]|date = November 26, 1973| url = http://www.time.com/time/magazine/article/0,9171,908175,00.html?promoid=googlep|accessdate=2007-08-12}}</ref> A coating of tar would have been commonplace on ships in the classical era.{{Ref_label|D|d|none}}
 
In October 2005 a group of students from the [[Massachusetts Institute of Technology]] carried out an experiment with 127 one-foot (30&nbsp;cm) square mirror tiles, focused on a {{nowrap|mock-up}} wooden ship at a range of around 100&nbsp;feet (30&nbsp;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&nbsp;°F).<ref>{{cite web|title = How Wildfires Work|author= Bonsor, Kevin| publisher = [[HowStuffWorks]]| url = http://science.howstuffworks.com/wildfire.htm|accessdate=2007-07-23| archiveurl= http://web.archive.org/web/20070714174036/http://science.howstuffworks.com/wildfire.htm| archivedate= 14 July 2007 <!--DASHBot-->| deadurl= no}}</ref><ref>[http://www.engineeringtoolbox.com/fuels-ignition-temperatures-d_171.html Fuels and Chemicals&nbsp;– Auto Ignition Temperatures]</ref>
 
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.<ref name="death ray">{{cite web|title = Archimedes Death Ray: Testing with MythBusters|publisher = MIT| url = http://web.mit.edu/2.009/www//experiments/deathray/10_Mythbusters.html|accessdate=2007-07-23}}</ref>
 
In December 2010, ''MythBusters'' again looked at the heat ray story in a special edition featuring [[Barack Obama]], entitled ''President's Challenge''. Several experiments were carried out, including a large scale test with 500 schoolchildren aiming mirrors at a {{nowrap|mock-up}} of a Roman sailing ship 400&nbsp;feet (120&nbsp;m) away. In all of the experiments, the sail failed to reach the 210 °C (410&nbsp;°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.<ref name="death ray2">{{cite web|title = TV Review: MythBusters 8.27&nbsp;– President's Challenge| url = http://fandomania.com/tv-review-mythbusters-8-27-presidents-challenge/|accessdate=2010-12-18}}</ref>
 
===Other discoveries and inventions===
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]].<ref name="lever rorres">{{cite web |first=Chris |last=Rorres|url = http://www.math.nyu.edu/~crorres/Archimedes/Lever/LeverLaw.html|title = The Law of the Lever According to Archimedes|publisher = [[Courant Institute of Mathematical Sciences]]|accessdate = 2010-03-20}}</ref><ref name="lever clagett">{{cite book |first=Marshall |last=Clagett|url = http://books.google.com/?id=mweWMAlf-tEC&pg=PA72&lpg=PA72&dq=archytas+lever&q=archytas%20lever| title = Greek Science in Antiquity| publisher = Dover Publications|accessdate = 2010-03-20 |isbn=978-0-486-41973-2 |year=2001}}</ref> 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." ({{lang-el|δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω}})<ref>Quoted by [[Pappus of Alexandria]] in ''Synagoge'', Book VIII</ref> Plutarch describes how Archimedes designed [[block and tackle|block-and-tackle]] [[pulley]] systems, allowing sailors to use the principle of [[lever]]age to lift objects that would otherwise have been too heavy to move.<ref>{{cite web|author=Dougherty, F. C.; Macari, J.; Okamoto, C.|title = Pulleys|publisher=[[Society of Women Engineers]]|url = http://www.swe.org/iac/lp/pulley_03.html|accessdate=2007-07-23| archiveurl= http://web.archive.org/web/20070718031943/http://www.swe.org/iac/LP/pulley_03.html| archivedate= 18 July 2007 <!--DASHBot-->| deadurl= no}}</ref> 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.<ref>{{cite web |url = http://www.tmth.edu.gr/en/aet/5/55.html| title = Ancient Greek Scientists: Hero of Alexandria|publisher = Technology Museum of Thessaloniki|accessdate = 2007-09-14| archiveurl= http://web.archive.org/web/20070905125400/http://www.tmth.edu.gr/en/aet/5/55.html| archivedate= 5 September 2007 <!--DASHBot-->| deadurl= no}}</ref>
 
[[Cicero]] (106–43&nbsp;BC) mentions Archimedes briefly in his [[dialogue]] ''[[De re publica]]'', which portrays a fictional conversation taking place in 129&nbsp;BC. After the capture of Syracuse ''c.'' 212&nbsp;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|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:
 
{{quote|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.&nbsp;— 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.<ref>{{cite web|title = ''De re publica'' 1.xiv §21|author= [[Cicero]]| publisher =thelatinlibrary.com|url = http://www.thelatinlibrary.com/cicero/repub1.shtml#21|accessdate=2007-07-23}}</ref><ref>{{cite web|title =''De re publica'' Complete e-text in English from Gutenberg.org|author=[[Cicero]]|publisher = [[Project Gutenberg]]|url= http://www.gutenberg.org/etext/14988|accessdate=2007-09-18| archiveurl= http://web.archive.org/web/20070929122153/http://www.gutenberg.org/etext/14988| archivedate= 29 September 2007 <!--DASHBot-->| deadurl= no}}</ref>}}
 
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 {{nowrap|''[[On Sphere-Making]]''}}. Modern research in this area has been focused on the [[Antikythera mechanism]], another device built {{circa|100}}&nbsp;BC that was probably designed for the same purpose.<ref>{{cite web|title = Discovering How Greeks Computed in 100 B.C. |first=John |last=Noble Wilford|work = [[The New York Times]]|url = http://www.nytimes.com/2008/07/31/science/31computer.html?_r=0|date=July 31, 2008|accessdate=2013-12-25}}</ref> Constructing mechanisms of this kind would have required a sophisticated knowledge of [[Differential (mechanical device)|differential gearing]].<ref>{{cite web|title = The Antikythera Mechanism II|publisher = [[Stony Brook University]]|url = http://www.math.sunysb.edu/~tony/whatsnew/column/antikytheraII-0500/diff4.html|accessdate=2013-12-25}}</ref> 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.<ref>{{cite web|title = Spheres and Planetaria |first=Chris |last=Rorres|publisher = Courant Institute of Mathematical Sciences|url = http://www.math.nyu.edu/~crorres/Archimedes/Sphere/SphereIntro.html|accessdate=2007-07-23}}</ref><ref>{{cite news|title = Ancient Moon 'computer' revisited|publisher = BBC News|date = November 29, 2006| url = http://news.bbc.co.uk/1/hi/sci/tech/6191462.stm|accessdate=2007-07-23}}</ref>
 
==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."<ref>{{cite web|title = Extract from ''Parallel Lives''|author= [[Plutarch]]| publisher = fulltextarchive.com| url = http://fulltextarchive.com/pages/Plutarch-s-Lives10.php#p35|accessdate=2009-08-10}}</ref>
[[File:PiArchimede4.svg|thumb|right|Archimedes used [[Pythagoras' Theorem]] to calculate the side of the 12-gon from that of the [[hexagon]] and for each subsequent doubling of the sides of the regular polygon.]] Archimedes was able to use [[infinitesimal]]s in a way that is similar to modern [[Integral|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 3{{frac|1|7}} (approximately 3.1429) and 3{{frac|10|71}} (approximately 3.1408), consistent with its actual value of approximately 3.1416.<ref>{{cite web|title =Archimedes on measuring the circle|author= Heath, T.L.| publisher =math.ubc.ca| url =http://www.math.ubc.ca/~cass/archimedes/circle.html|accessdate=2012-10-30}}</ref> He also proved that the [[area]] of a circle was equal to π multiplied by the [[square]] of the [[radius]] of the circle (πr<sup>2</sup>). 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.<ref>{{cite web|title = Archimedean ordered fields|author= Kaye, R.W.| publisher = web.mat.bham.ac.uk| url = http://web.mat.bham.ac.uk/R.W.Kaye/seqser/archfields|accessdate=2009-11-07}}</ref>
 
In ''[[Measurement of a Circle]]'', Archimedes gives the value of the [[square root]] of 3 as lying between {{frac|265|153}} (approximately 1.7320261) and {{frac|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."<ref>Quoted in Heath, T. L. ''Works of Archimedes'', Dover Publications, ISBN 0-486-42084-1.</ref> It is possible that he used an [[iteration|iterative]] procedure to calculate these values.<ref>{{cite web|title = The Computation of Pi by Archimedes|author= McKeeman, Bill| publisher =Matlab Central| url = http://www.mathworks.com/matlabcentral/fileexchange/29504-the-computation-of-pi-by-archimedes/content/html/ComputationOfPiByArchimedes.html#37|accessdate=2012-10-30}}</ref>
 
[[File:Parabolic segment and inscribed triangle.svg|thumb|right|As proven by Archimedes, the area of the [[parabole|parabolic]] segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure.]]
In ''[[The Quadrature of the Parabola]]'', Archimedes proved that the area enclosed by a [[parabola]] and a straight line is {{frac|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 [[Series (mathematics)#History of the theory of infinite series|infinite]] [[geometric series]] with the [[Geometric series#Common ratio|common ratio]] {{frac|1|4}}:
 
:<math>\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. \;</math>
 
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 line]]s, and so on. This proof uses a variation of the series {{nowrap|[[1/4 + 1/16 + 1/64 + 1/256 + · · ·]]}} which sums to {{frac|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 of Syracuse|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 {{lang|grc|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 [[Names of large numbers|vigintillion]], or 8{{e|63}}.<ref>{{cite web|title = The Sand Reckoner |first=Bradley W |last=Carroll|publisher = Weber State University| url = http://physics.weber.edu/carroll/Archimedes/sand.htm|accessdate=2007-07-23| archiveurl= http://web.archive.org/web/20070813215029/http://physics.weber.edu/carroll/Archimedes/sand.htm| archivedate= 13 August 2007 <!--DASHBot-->| deadurl= no}}</ref>
 
==Writings==
The works of Archimedes were written in [[Doric Greek]], the dialect of ancient [[Syracuse, Sicily|Syracuse]].<ref>Encyclopedia of ancient Greece By Wilson, Nigel Guy  [http://books.google.com/books?id=-aFtPdh6-2QC&pg=PA77 p. 77] ISBN 0-7945-0225-3 (2006)</ref> 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 [[polyhedron|polyhedra]], while [[Theon of Alexandria]] quotes a remark about [[refraction]] from the {{nowrap|now-lost}} ''Catoptrica''.{{Ref_label|B|b|none}} During his lifetime, Archimedes made his work known through correspondence with the mathematicians in [[Alexandria]]. The writings of Archimedes were collected by the [[Byzantine Empire|Byzantine]] architect [[Isidore of Miletus]] (''c''. 530&nbsp;AD), while commentaries on the works of Archimedes written by [[Eutocius of Ascalon|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&nbsp;AD), and Latin by [[Gerard of Cremona]] (''c.'' 1114–1187&nbsp;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.<ref>{{cite web|title = Editions of Archimedes' Work|publisher = Brown University Library| url = http://www.brown.edu/Facilities/University_Library/exhibits/math/wholefr.html|accessdate=2007-07-23| archiveurl= http://web.archive.org/web/20070808235638/http://www.brown.edu/Facilities/University_Library/exhibits/math/wholefr.html| archivedate= 8 August 2007 <!--DASHBot-->| deadurl= no}}</ref> 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.<ref>{{cite web|title = The Galileo Project: Hydrostatic Balance|author=Van Helden, Al|publisher = [[Rice University]]| url = http://galileo.rice.edu/sci/instruments/balance.html|accessdate=2007-09-14| archiveurl= http://web.archive.org/web/20070905185039/http://galileo.rice.edu/sci/instruments/balance.html| archivedate= 5 September 2007 <!--DASHBot-->| deadurl= no}}</ref>
 
===Surviving works===
[[File:Esfera Arquímedes.jpg|thumb|right|A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases. A [[sphere]] and [[Cylinder (geometry)|cylinder]] were placed on the tomb of Archimedes at his request.]]
* ''[[On the Equilibrium of Planes]]'' (two volumes)
:The first book is in fifteen propositions with seven [[Axiom|postulates]], while the second book is in ten propositions. In this work Archimedes explains the ''[[Torque|Law of the Lever]]'', stating, "Magnitudes are in equilibrium at distances reciprocally proportional to their weights."
:Archimedes uses the principles derived to calculate the areas and [[center of mass|centers of gravity]] of various geometric figures including [[triangle]]s, [[parallelogram]]s and [[parabola]]s.<ref name="works">{{cite web |author =Heath, T.L.|url = http://www.archive.org/details/worksofarchimede029517mbp|title = The Works of Archimedes (1897). The unabridged work in PDF form (19&nbsp;MB)| publisher = [[Internet Archive|Archive.org]]|accessdate = 2007-10-14| archiveurl= http://web.archive.org/web/20071006033058/http://www.archive.org/details/worksofarchimede029517mbp| archivedate= 6 October 2007 <!--DASHBot-->| deadurl= no}}</ref>
* ''[[Measurement of a Circle|On the Measurement of a 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 gives an [[Approximations of π|approximation]] of the value of pi ({{pi}}), showing that it is greater than {{frac|223|71}} and less than {{frac|22|7}}.
* ''[[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 (mathematics)|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 coordinate system|polar coordinates]] ({{math|''r''}}, {{math|θ}}) it can be described by the equation
::<math>\, r=a+b\theta</math>
:with [[real number]]s {{math|a}} and {{math|b}}. This is an early example of a [[Curve|mechanical curve]] (a curve traced by a moving [[point (geometry)|point]]) considered by a Greek mathematician.
* ''[[On the Sphere and Cylinder|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 [[circumscribe]]d [[cylinder (geometry)|cylinder]] of the same height and [[diameter]]. The volume is {{frac|4|3}}{{pi}}{{math|''r''}}<sup>3</sup> for the sphere, and 2{{pi}}{{math|''r''}}<sup>3</sup> for the cylinder. The surface area is 4{{pi}}{{math|''r''}}<sup>2</sup> for the sphere, and 6{{pi}}{{math|''r''}}<sup>2</sup> for the cylinder (including its two bases), where {{math|''r''}} is the radius of the sphere and cylinder. The sphere has a volume {{nowrap|two-thirds}} that of the circumscribed cylinder. Similarly, the sphere has an area  {{nowrap|two-thirds}} that of the cylinder (including the bases). A sculpted sphere and cylinder were placed 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 [[cross section (geometry)|sections]] of [[Cone (geometry)|cones]], spheres, and paraboloids.
* ''[[On Floating Bodies]]'' (two volumes)
:In the first part of this treatise, Archimedes spells out the law of [[wikt:equilibrium|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 {{nowrap|self-gravitating}}, since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.
 
: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: {{quote|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]]''
:In this work of 24 propositions addressed to Dositheus, 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]] {{frac|1|4}}.
[[File:Stomachion.JPG|thumb|right|''[[Ostomachion|Stomachion]]'' is a [[dissection puzzle]] in the [[Archimedes Palimpsest]].]]
* ''[[Ostomachion|(O)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 could be assembled into the shape of a square. Dr. Netz calculates that the pieces can be made into a square 17,152 ways.<ref>{{cite news|title = In Archimedes' Puzzle, a New Eureka Moment|author= Kolata, Gina| publisher =[[The New York Times]] |date = December 14, 2003| url = http://query.nytimes.com/gst/fullpage.html?res=9D00E6DD133CF937A25751C1A9659C8B63&sec=&spon=&pagewanted=all|accessdate=2007-07-23}}</ref> The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded.<ref>{{cite web|title = The Loculus of Archimedes, Solved|author= Ed Pegg Jr.| publisher =[[Mathematical Association of America]] |date = November 17, 2003| url = http://www.maa.org/editorial/mathgames/mathgames_11_17_03.html|accessdate=2008-05-18| archiveurl= http://web.archive.org/web/20080519094951/http://www.maa.org/editorial/mathgames/mathgames_11_17_03.html| archivedate= 19 May 2008 <!--DASHBot-->| deadurl= no}}</ref> The puzzle represents an example of an early problem in [[combinatorics]].
:The origin of the puzzle's name is unclear, and it has been suggested that it is taken from the [[Ancient Greek]] word for throat or gullet, stomachos ({{lang|grc|στόμαχος}}).<ref>{{cite web |first=Chris |last=Rorres|url = http://math.nyu.edu/~crorres/Archimedes/Stomachion/intro.html|title = Archimedes' Stomachion| publisher = Courant Institute of Mathematical Sciences|accessdate = 2007-09-14| archiveurl= http://web.archive.org/web/20071026005336/http://www.math.nyu.edu/~crorres/Archimedes/Stomachion/intro.html| archivedate= 26 October 2007 <!--DASHBot-->| deadurl= no}}</ref> [[Ausonius]] refers to the puzzle as ''Ostomachion'', a Greek compound word formed from the roots of {{lang|grc|ὀστέον}} (''osteon'', bone) and {{lang|grc|μάχη}} (machē&nbsp;– fight). The puzzle is also known as the Loculus of Archimedes or Archimedes' Box.<ref>{{cite web |url = http://www.archimedes-lab.org/latin.html#archimede| title = Graeco Roman Puzzles| publisher =Gianni A. Sarcone and Marie J. Waeber|accessdate = 2008-05-09| archiveurl= http://web.archive.org/web/20080514130547/http://www.archimedes-lab.org/latin.html| archivedate= 14 May 2008 <!--DASHBot-->| deadurl= no}}</ref>
* ''[[Archimedes' cattle problem]]''
: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 in Alexandria. Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous [[Diophantine equation]]s. There is a more difficult version of the problem in which some of the answers are required to be [[square number]]s. This version of the problem was first solved by A. Amthor<ref>Krumbiegel, B. and Amthor, A. ''Das Problema Bovinum des Archimedes'', Historisch-literarische Abteilung der Zeitschrift Für Mathematik und Physik 25 (1880) pp. 121–136, 153–171.</ref> in 1880, and the answer is a very large number, approximately 7.760271{{e|206544}}.<ref>{{cite web |first=Keith G |last=Calkins|url = http://www.andrews.edu/~calkins/profess/cattle.htm|title = Archimedes' Problema Bovinum| publisher = [[Andrews University]]|accessdate = 2007-09-14|archiveurl = http://web.archive.org/web/20071012171254/http://andrews.edu/~calkins/profess/cattle.htm <!--DASHBot-->|archivedate = 2007-10-12}}</ref>
* ''[[The Sand Reckoner]]''
:In this treatise, Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the [[Heliocentrism|heliocentric]] theory of the [[Solar System|solar system]] proposed by [[Aristarchus of Samos]], as well as 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{{e|63}} 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.<ref>{{cite web|title =English translation of ''The Sand Reckoner'' |publisher = [[University of Waterloo]]| url = http://www.math.uwaterloo.ca/navigation/ideas/reckoner.shtml|accessdate=2007-07-23| archiveurl= http://web.archive.org/web/20070811235335/http://www.math.uwaterloo.ca/navigation/ideas/reckoner.shtml| archivedate= 11 August 2007 <!--DASHBot-->| deadurl= no}}</ref>
* ''[[The Method of Mechanical Theorems]]''
:This treatise was thought lost until the discovery of the [[Archimedes Palimpsest]] in 1906. In this work Archimedes uses [[Archimedes' use of infinitesimals|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 language|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.<ref>{{cite web|title = Archimedes' Book of Lemmas| publisher = [[cut-the-knot]]| url = http://www.cut-the-knot.org/Curriculum/Geometry/BookOfLemmas/index.shtml|accessdate= 2007-08-07| archiveurl= http://web.archive.org/web/20070711111858/http://www.cut-the-knot.org/Curriculum/Geometry/BookOfLemmas/index.shtml| archivedate= 11 July 2007 <!--DASHBot-->| deadurl= no}}</ref>
 
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.{{Ref_label|C|c|none}} However, the first reliable reference to the formula is given by [[Hero of Alexandria|Heron of Alexandria]] in the 1st century AD.<ref>{{cite web|title = Heron of Alexandria |author=O'Connor, J.J. and Robertson, E.F.|publisher = [[University of St Andrews]]| url = http://www-history.mcs.st-and.ac.uk/Biographies/Heron.html|date=April 1999|accessdate= 2010-02-17}}</ref>
 
==Archimedes Palimpsest==
{{main|Archimedes Palimpsest}}
[[File:Archimedes Palimpsest.jpg|thumb|upright|In 1906, The Archimedes Palimpsest revealed works by Archimedes thought to have been lost.]]
The foremost document containing the work of Archimedes is the [[Archimedes Palimpsest]]. In 1906, the Danish professor [[Johan Ludvig Heiberg (historian)|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.<ref>{{cite web|title = Reading Between the Lines|author= Miller, Mary K.| publisher= [[Smithsonian (magazine)|Smithsonian Magazine]]|date=March 2007| url= http://www.smithsonianmag.com/science-nature/archimedes.html| accessdate=2008-01-24| archiveurl= http://web.archive.org/web/20080119024939/http://www.smithsonianmag.com/science-nature/archimedes.html?| archivedate= 19 January 2008 <!--DASHBot-->| deadurl= no}}</ref> 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 City|New York]].<ref>{{cite news|title = Rare work by Archimedes sells for $2 million|publisher = [[CNN]]|date = October 29, 1998| url = http://edition.cnn.com/books/news/9810/29/archimedes/|accessdate=2008-01-15| archiveurl = http://web.archive.org/web/20080516000109/http://edition.cnn.com/books/news/9810/29/archimedes/| archivedate = May 16, 2008}}</ref> 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 [[Suda|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 {{nowrap|[[x-ray]]}} [[light]] to read the overwritten text.<ref>{{cite news|title = X-rays reveal Archimedes' secrets|publisher = BBC News|date = August 2, 2006| url = http://news.bbc.co.uk/1/hi/sci/tech/5235894.stm|accessdate=2007-07-23| archiveurl= http://web.archive.org/web/20070825091847/http://news.bbc.co.uk/1/hi/sci/tech/5235894.stm| archivedate= 25 August 2007 <!--DASHBot-->| deadurl= no}}</ref>
 
The treatises in the Archimedes Palimpsest are: ''On the Equilibrium of Planes, On Spirals, [[Measurement of a Circle]], On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems'' and ''Stomachion''.
 
==Legacy==
[[File:FieldsMedalFront.jpg|thumb|The [[Fields Medal]] carries a portrait of Archimedes.]]
* There is a [[impact crater|crater]] on the [[Moon]] named [[Archimedes (crater)|Archimedes]] (29.7° N, 4.0° W) in his honor, as well as a lunar mountain range, the [[Montes Archimedes]] (25.3° N, 4.6° W).<ref>{{cite web |title=Oblique view of Archimedes crater on the Moon |author=Friedlander, Jay and Williams, Dave |publisher=[[NASA]] |url=http://nssdc.gsfc.nasa.gov/imgcat/html/object_page/a15_m_1541.html |accessdate=2007-09-13| archiveurl= http://web.archive.org/web/20070819054033/http://nssdc.gsfc.nasa.gov/imgcat/html/object_page/a15_m_1541.html| archivedate= 19 August 2007 <!--DASHBot-->| deadurl= no}}</ref>
* The [[asteroid]] [[3600 Archimedes]] is named after him.<ref>{{cite web |title=Planetary Data System |publisher=NASA |url=http://starbrite.jpl.nasa.gov/pds-explorer/index.jsp?selection=othertarget&targname=3600%20ARCHIMEDES |accessdate=2007-09-13| archiveurl= http://web.archive.org/web/20071012171730/http://starbrite.jpl.nasa.gov/pds-explorer/index.jsp?selection=othertarget&targname=3600+ARCHIMEDES| archivedate= 12 October 2007 <!--DASHBot-->| deadurl= no}}</ref>
* The [[Fields Medal]] for outstanding achievement in mathematics carries a portrait of Archimedes, along with a carving illustrating his proof on 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).<ref>{{cite web |title=Fields Medal |publisher=[[International Mathematical Union]] |url=http://www.mathunion.org/medals/Fields/AboutPhotos.html |accessdate=2007-07-23| archiveurl = http://web.archive.org/web/20070701033751/http://www.mathunion.org/medals/Fields/AboutPhotos.html| archivedate = July 1, 2007}}</ref>
* Archimedes has appeared on postage stamps issued by [[East Germany]] (1973), [[Greece]] (1983), [[Italy]] (1983), [[Nicaragua]] (1971), [[San Marino]] (1982), and [[Spain]] (1963).<ref>{{cite web |first=Chris |last=Rorres |url=http://math.nyu.edu/~crorres/Archimedes/Stamps/stamps.html |title=Stamps of Archimedes |publisher=Courant Institute of Mathematical Sciences |accessdate=2007-08-25}}</ref>
* The exclamation of [[Eureka (word)|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]].<ref>{{cite web |title=California Symbols |publisher=California State Capitol Museum |url=http://www.capitolmuseum.ca.gov/VirtualTour.aspx?content1=1278&Content2=1374&Content3=1294 |accessdate=2007-09-14| archiveurl= http://web.archive.org/web/20071012123245/http://capitolmuseum.ca.gov/VirtualTour.aspx?content1=1278&Content2=1374&Content3=1294| archivedate= 12 October 2007 <!--DASHBot-->| deadurl= no}}</ref>
* A movement for civic engagement targeting universal access to health care in the US state of [[Oregon]] has been named the "Archimedes Movement," headed by former Oregon Governor [[John Kitzhaber]].<ref>{{cite web|url=http://www.archimedesmovement.org/|title=The Archimedes Movement}}</ref>
 
==See also==
{{col-begin|width=auto}}
{{col-break}}
* [[Arbelos]]
* [[Axiom of Archimedes|Archimedes' axiom]]
* [[Archimedes number]]
* [[Archimedes paradox]]
* [[Buoyancy|Archimedes principle]] of buoyancy
{{col-break}}
* [[Archimedes' screw]]
* [[Archimedean solid]]
* [[Archimedes' circles|Archimedes' twin circles]]
* [[Archimedes' use of infinitesimals]]
* [[Archytas]]
{{col-break}}
* [[Diocles (mathematician)|Diocles]]
* [[List of things named after Archimedes]]
* [[Methods of computing square roots]]
* [[Pseudo-Archimedes]]
{{col-break}}
* [[Salinon]]
* [[Steam cannon]]
* [[Syracusia]]
* [[Vitruvius]]
* [[Zhang Heng]]
{{col-end}}
 
==Notes==
'''a.''' {{Note_label|A|a|none}}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 {{nowrap|''c.'' 280–220 BC}}, suggesting that Archimedes may have been an older man when writing some of his works.
 
'''b.''' {{Note_label|B|b|none}}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.''' {{Note_label|C|c|none}}[[Carl Benjamin Boyer|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&nbsp;— ''k''&nbsp;=&nbsp;√(''s''(''s''&nbsp;−&nbsp;''a'')(''s''&nbsp;−&nbsp;''b'')(''s''&nbsp;−&nbsp;''c'')), where ''s'' is the semiperimeter&nbsp;— was known to Archimedes several centuries before Heron lived. Arabic scholars also attribute to Archimedes the 'theorem on the broken [[chord (geometry)|chord]]'&nbsp;... Archimedes is reported by the Arabs to have given several proofs of the theorem."
 
'''d.''' {{Note_label|D|d|none}} "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.<ref>{{cite book|last=Casson|first=Lionel|title=Ships and seamanship in the ancient world|year=1995|publisher=The Johns Hopkins University Press|location=Baltimore|isbn=978-0-8018-5130-8|pages=211–212|url=http://books.google.com/books?id=sDpMh0gK2OUC&pg=PA18&dq=why+were+homer%27s+ships+black#v=onepage&q=why%20were%20homer's%20ships%20black&f=false}}</ref>
 
==References==
{{Reflist|colwidth=30em}}
 
==Further reading==
{{Wikisource1911Enc|Archimedes}}
* {{cite book |last=[[Carl Benjamin Boyer|Boyer, Carl Benjamin]]|title=A History of Mathematics|year=1991|publisher= Wiley|location= New York|isbn=0-471-54397-7}}
* {{cite book|last=Clagett|first=Marshall|authorlink=Marshall Clagett|title=Archimedes in the Middle Ages|location=Madison, WI|publisher=University of Wisconsin Press|volume= 5 vols|year=1964-1984}}
* {{cite book |last=[[Eduard Jan Dijksterhuis|Dijksterhuis, E.J.]] |title=Archimedes|year=1987 |publisher= 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.
* {{cite book |last=Gow |first=Mary |title=Archimedes: Mathematical Genius of the Ancient World|year=2005|publisher=Enslow Publishers, Inc |isbn=0-7660-2502-0}}
* {{cite book |last=Hasan |first=Heather |title=Archimedes: The Father of Mathematics|year= 2005|publisher=Rosen Central |isbn=978-1-4042-0774-5}}
* {{cite book |author=[[T. L. Heath|Heath, T.L.]]|title=Works of Archimedes|year=1897 |publisher=Dover Publications |isbn=0-486-42084-1}} Complete works of Archimedes in English.
* {{cite book |last=Netz, Reviel and Noel, William |title=The Archimedes Codex|year=2007|publisher=Orion Publishing Group|isbn= 0-297-64547-1}}
* {{cite book |last=[[Clifford A. Pickover|Pickover, Clifford A.]]|title =Archimedes to Hawking: Laws of Science and the Great Minds Behind Them|year=2008 |publisher= [[Oxford University Press]] |isbn=978-0-19-533611-5}}
* {{cite book |last=Simms, Dennis L. |title=Archimedes the Engineer|year=1995 |publisher= Continuum International Publishing Group Ltd |isbn=0-7201-2284-8}}
* {{cite book |last=Stein, Sherman |title=Archimedes: What Did He Do Besides Cry Eureka?|year=1999 |publisher= Mathematical Association of America|isbn=0-88385-718-9}}
 
===''The Works of Archimedes'' online===
* Text in Classical Greek: [http://www.wilbourhall.org PDF scans of Heiberg's edition of the Works of Archimedes, now in the public domain]
* In English translation: [http://www.archive.org/details/worksofarchimede029517mbp ''The Works of Archimedes''], trans. T.L. Heath; supplemented by [http://books.google.com/books?id=suYGAAAAYAAJ ''The Method of Mechanical Theorems''], trans. L.G. Robinson
 
==External links==
{{Sister project links|commons=Category:Archimedes|v=Ancient Innovations|n=Particle accelerator reveals long-lost writings of Archimedes|s=Author:Archimedes|b=FHSST Physics/Forces/Definition}}
{{Spoken Wikipedia|Archimedes.ogg|2009-03-31}}
{{In Our Time|Archimedes|b00773bv|Archimedes}}
* {{InPho|thinker|2546}}
* {{PhilPapers|search|archimedes}}
* [http://www.archimedespalimpsest.org/ The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland]
* [http://mathdb.org/articles/archimedes/e_archimedes.htm The Mathematical Achievements and Methodologies of Archimedes]
* {{MathPages|id=home/kmath038/kmath038|title=Archimedes and the Square Root of 3}}
* {{MathPages|id=home/kmath343/kmath343|title=Archimedes on Spheres and Cylinders}}
* [http://www.cs.drexel.edu/~crorres/bbc_archive/mirrors_sailors_sakas.jpg Photograph of the Sakkas experiment in 1973]
* [http://web.mit.edu/2.009/www/experiments/steamCannon/ArchimedesSteamCannon.html Testing the Archimedes steam cannon]
* [http://www.stampsbook.org/subject/Archimedes.html Stamps of Archimedes]
* [http://www.dailymail.co.uk/sciencetech/article-2050631/Eureka-1-000-year-old-text-Greek-maths-genius-Archimedes-goes-display.html Eureka! 1,000-year-old text by Greek maths genius Archimedes goes on display] ''[[Daily Mail]]'', October 18, 2011.
 
{{Greek mathematics}}
{{Ancient Greece topics}}
{{featured article}}
 
{{Authority control|LCCN=n/80/104666|VIAF=29547910|GND=118503863}}
 
<!-- Metadata: see [[Wikipedia:Persondata]] -->
{{Persondata
|NAME=Archimedes
|ALTERNATIVE NAMES=
|SHORT DESCRIPTION=ancient Greek mathematician, physicist and engineer
|DATE OF BIRTH=circa 287 BC
|PLACE OF BIRTH=[[Syracuse, Sicily]], [[Magna Graecia]]
|DATE OF DEATH=circa 212 BC
|PLACE OF DEATH=[[Syracuse, Sicily]], [[Magna Graecia]]
}}
{{DEFAULTSORT:Archimedes}}
[[Category:Archimedes| ]]
[[Category:287 BC births]]
[[Category:212 BC deaths]]
[[Category:3rd-century BC Greek people]]
[[Category:3rd-century BC writers]]
[[Category:People from Syracuse, Sicily]]
[[Category:Ancient Greek engineers]]
[[Category:Ancient Greek inventors]]
[[Category:Ancient Greek mathematicians]]
[[Category:Ancient Greek physicists]]
[[Category:Hellenistic-era philosophers]]
[[Category:Doric Greek writers]]
[[Category:Sicilian Greeks]]
[[Category:Sicilian mathematicians]]
[[Category:Sicilian scientists]]
[[Category:Murdered scientists]]
[[Category:Geometers]]
[[Category:Ancient Greeks who were murdered]]
[[Category:Ancient Syracusians]]
[[Category:Fluid dynamicists]]
[[Category:Buoyancy]]
 
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