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{{Short description|Greek mathematician and physicist (c. 287 – 212 BC)}}
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{{Infobox scientist
| name              = Archimedes of Syracuse
| native_name       = Ἀρχιμήδης
| native_name_lang  = grc
| image             = Domenico-Fetti Archimedes 1620.jpg
| alt               = A painting of an older man puzzling over geometric problems
| caption           = ''Archimedes Thoughtful'' by [[Domenico Fetti|Fetti]] (1620)
| birth_date        = {{circa|287{{nbsp}}BC}}
| birth_place       = [[Syracuse, Sicily|Syracuse]], [[Sicily]]
| death_date        = {{circa|212{{nbsp}}BC|lk=no}} (aged approximately 75)
| death_place       = Syracuse, Sicily
| field             = [[Mathematics]]<br />[[Physics]]<br />[[Astronomy]]<br />[[Mechanics]]<br />[[Engineering]]
| known_for         = {{collapsible list|[[Archimedes' principle]]<br />[[Archimedes' screw]]<br />[[Center of mass#History|Center of gravity]]<br />[[Statics]]<br />[[Fluid statics|Hydrostatics]]<br />[[Lever|Law of the lever]]<br />[[Archimedes' use of infinitesimals|Indivisibles]]<br />[[Neusis construction|Neuseis constructions]]<ref>{{cite journal |last=Knorr |first=Wilbur R. |title=Archimedes and the spirals: The heuristic background |journal=[[Historia Mathematica]] |year=1978 |volume=5 |issue=1 |pages=43–75 |quote="To be sure, Pappus does twice mention the theorem on the tangent to the spiral [IV, 36, 54]. But in both instances the issue is Archimedes' inappropriate use of a 'solid neusis,' that is, of a construction involving the sections of solids, in the solution of a plane problem. Yet Pappus' own resolution of the difficulty [IV, 54] is by his own classification a 'solid' method, as it makes use of conic sections." (p. 48) |doi=10.1016/0315-0860(78)90134-9 |doi-access=free}}</ref><br>[[List of things named after Archimedes|List of other things named after him]]|}}
}}
'''Archimedes of Syracuse'''{{Efn|{{langx|grc-x-doric|{{wikt-lang|grc|Ἀρχιμήδης}}}}, {{IPA|grc-x-doric|arkʰimɛːdɛ̂ːs|pron}}.}} ({{IPAc-en|ˌ|ɑːr|k|ᵻ|ˈ|m|iː|d|iː|z}} {{respell|AR|kim|EE|deez}};<ref name="Collins">{{cite web |url=http://www.collinsdictionary.com/dictionary/english/archimedes?showCookiePolicy=true |title=Archimedes |access-date=25 September 2014 |publisher=Collins Dictionary |date=n.d.}}</ref> {{circa|287|212 BC}}) was an [[Ancient Greece|Ancient Greek]] [[Greek mathematics|mathematician]], [[physicist]], [[engineer]], [[astronomer]], and [[Invention|inventor]] from the ancient city of [[Syracuse, Sicily|Syracuse]] in [[History of Greek and Hellenistic Sicily|Sicily]].<ref>{{cite web |title=Archimedes (c. 287 – c. 212 BC) |url=https://www.bbc.co.uk/history/historic_figures/archimedes.shtml |publisher=BBC History |access-date=7 June 2012}}</ref> Although few details of his life are known, he is considered one of the leading scientists in [[classical antiquity]]. Regarded as the greatest mathematician of [[ancient history]], and one of the greatest of all time,<ref name="LitList">
{{cite book |author=John M. Henshaw |url=https://books.google.com/books?id=-0ljBAAAQBAJ&pg=PA68 |title=An Equation for Every Occasion: Fifty-Two Formulas and Why They Matter |date=2014 |publisher=JHU Press |isbn=978-1-4214-1492-8 |page=68 |quote="Archimedes is on most lists of the greatest mathematicians of all time and is considered the greatest mathematician of antiquity."}}
{{pb}} {{cite book |last=Calinger |first=Ronald |title=A Contextual History of Mathematics |publisher=Prentice-Hall |year=1999 |isbn=978-0-02-318285-3 |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."}}
{{pb}} {{cite web |date=January 1999 |title=Archimedes of Syracuse |url=http://www-history.mcs.st-and.ac.uk/Biographies/Archimedes.html |access-date=9 June 2008 |publisher=The MacTutor History of Mathematics archive}}
{{pb}} {{cite book |author=Sadri Hassani |url=https://books.google.com/books?id=GWPgBwAAQBAJ&pg=PA81 |title=Mathematical Methods: For Students of Physics and Related Fields |date=2013 |publisher=Springer Science & Business Media |isbn=978-0-387-21562-4 |page=81 |quote="Archimedes is arguably believed to be the greatest mathematician of antiquity."}}
{{pb}} {{cite book |author=Hans Niels Jahnke |url=https://books.google.com/books?id=CVRZEXFVsZkC&pg=PA21 |title=A History of Analysis |publisher=American Mathematical Soc. |isbn=978-0-8218-9050-9 |page=21 |quote="Archimedes was the greatest mathematician of antiquity and one of the greatest of all times"}}
{{pb}} {{cite book |author=Stephen Hawking |url=https://books.google.com/books?id=eU_RzM7OoI4C&pg=PT12 |title=God Created The Integers: The Mathematical Breakthroughs that Changed History |date=2007 |publisher=Running Press |isbn=978-0-7624-3272-1 |page=12 |quote="Archimedes, the greatest mathematician of antiquity"}}
{{pb}} {{cite book |author=Hirshfeld |first= Alan |url=https://books.google.com/books?id=zbcfLoZKDl8C |title=Eureka Man: The Life and Legacy of Archimedes |date=2009 |publisher=Bloomsbury Publishing |isbn= 978-0-8027-1979-9|pages=206 |language=en |quote="the Archimedes Palimpsest has ridden the roiling waves of circumstance to become the most celebrated link to antiquity's greatest mathematician-inventor"}}{{pb}}

{{cite news |last1=Vallianatos |first1=Evaggelos |date=27 July 2014 |title=Archimedes: The Greatest Scientist Who Ever Lived |url=https://www.huffpost.com/entry/archimedes-the-greatest-scientist-who-ever-lived_b_5390263 |access-date=17 April 2021 |newspaper=HuffPost}}
{{pb}} {{cite news |last1=Kiersz. |first1=Andy |date=2 July 2014 |title=The 12 mathematicians who unlocked the modern world |url=https://www.businessinsider.com/12-classic-mathematicians-2014-7#archimedes-c-287-212-bc-3 |access-date=3 May 2021 |newspaper=Business Insider}}
{{pb}} {{Cite web |title=Archimedes |url=https://www.math.wichita.edu/history/Men/archimedes.html |access-date=3 May 2021}}
{{pb}} {{cite news |last1=Livio |first1=Mario |date=6 December 2017 |title=Who's the Greatest Mathematician of Them All? |url=https://www.huffpost.com/entry/whos-the-greatest-mathematician-of-them-all_b_5526648 |access-date=7 May 2021 |newspaper=HuffPost}}
</ref> Archimedes anticipated modern [[calculus]] and [[mathematical analysis|analysis]] by applying the concept of the [[Cavalieri's principle|infinitely small]] and the [[method of exhaustion]] to derive and rigorously prove many [[geometry|geometrical]] [[theorem]]s.<ref>{{Cite journal |last=Kirfel |first=Christoph |date=2013 |title=A generalisation of Archimedes' method |url=https://www.jstor.org/stable/24496758 |journal=The Mathematical Gazette |volume=97 |issue=538 |pages=43–52 |doi=10.1017/S0025557200005416 |jstor=24496758 |issn=0025-5572}}</ref><ref name=":2" /><ref name=":9">{{Citation |last=Jullien |first=V. |title=Archimedes and Indivisibles |date=2015 |work=Seventeenth-Century Indivisibles Revisited |volume=49 |pages=451–457 |editor-last=J. |editor-first=Vincent |series=Science Networks. Historical Studies |place=Cham |publisher=Springer International Publishing |doi=10.1007/978-3-319-00131-9_18 |isbn=978-3-319-00131-9}}</ref> These include the [[area of a circle]], the [[surface area]] and [[volume]] of a [[sphere]], the area of an [[ellipse]], the area under a [[parabola]], the volume of a segment of a [[paraboloid of revolution]], the volume of a segment of a [[hyperboloid of revolution]], and the area of a [[spiral]].<ref>{{cite web |title=A history of calculus |author1=O'Connor, J.J. |author2=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 |access-date=7 August 2007}}</ref><ref name=":0">[[Thomas Heath (classicist)|Heath, Thomas L.]] 1897. ''Works of Archimedes''.</ref>

Archimedes' other mathematical achievements include deriving an [[Approximations of π|approximation of pi ({{pi}})]], defining and investigating the [[Archimedean spiral]], and devising a system using [[exponentiation]] for expressing [[large numbers|very large numbers]]. He was also one of the first to [[Applied mathematics|apply mathematics]] to [[Physics|physical phenomena]], working on [[statics]] and [[hydrostatics]]. Archimedes' achievements in this area include a proof of the law of the [[lever]],<ref>{{Cite journal |last=Goe |first=G. |date=1972 |title=Archimedes' theory of the lever and Mach's critique |journal=Studies in History and Philosophy of Science Part A |volume=2 |issue=4 |pages=329–345 |doi=10.1016/0039-3681(72)90002-7 |bibcode=1972SHPSA...2..329G}}</ref> the widespread use of the concept of [[Center of mass#History|center of gravity]],<ref>{{Cite journal |last=Berggren |first=J. L. |date=1976 |title=Spurious Theorems in Archimedes' Equilibrium of Planes: Book I |journal=Archive for History of Exact Sciences |volume=16 |issue=2 |pages=87–103 |doi=10.1007/BF00349632 |jstor=41133463}}</ref> and the enunciation of the law of [[buoyancy]] known as [[Archimedes' principle]].{{r|graf}} In astronomy, he made measurements of the apparent diameter of the [[Sun]] and the size of the [[universe]].<ref>{{Cite journal |last=Osborne |first=Catherine |date=1983 |title=Archimedes on the Dimensions of the Cosmos |url=https://www.jstor.org/stable/233105 |journal=Isis |volume=74 |issue=2 |pages=234–242 |doi=10.1086/353246 |jstor=233105 |issn=0021-1753}}</ref><ref name=":7">{{Citation |last1=Rozelot |first1=Jean Pierre |title=A brief history of the solar diameter measurements: a critical quality assessment of the existing data |date=2016 |url=https://arxiv.org/abs/1609.02710 |access-date=2025-01-28 |arxiv=1609.02710 |last2=Kosovichev |first2=Alexander G. |last3=Kilcik |first3=Ali}}</ref> He is also said to have built a [[planetarium]] device that demonstrated the movements of the known celestial bodies, and may have been a precursor to the [[Antikythera mechanism]].<ref name=":11">{{Cite journal |last=Freeth |first=Tony |date=2022 |title=Wonder of the Ancient World |url=https://www.scientificamerican.com/article/an-ancient-greek-astronomical-calculation-machine-reveals-new-secrets/ |journal=Scientific American |volume=32 |issue=1 |pages=24 |doi=10.1038/scientificamerican0122-24 |pmid=39016582 |issn=0036-8733}}</ref> He is also credited with designing innovative [[machine]]s, such as his [[Archimedes' screw|screw pump]], [[block and tackle|compound pulleys]], and defensive war machines to protect his native [[Syracuse, Sicily|Syracuse]] from invasion.

Archimedes died during the [[Siege of Syracuse (213–212 BC)|siege of Syracuse]], when he was killed by a Roman soldier despite orders that he should not be harmed. [[Cicero]] describes visiting Archimedes' tomb, which was surmounted by a [[sphere]] and a [[cylinder (geometry)|cylinder]] that Archimedes requested be placed there to represent his most valued mathematical discovery.

Unlike his inventions, Archimedes' mathematical writings were little known in antiquity. [[Alexandria|Alexandrian]] mathematicians read and quoted him, but the first comprehensive compilation was not made until {{circa|530{{nbsp}}AD}} by [[Isidore of Miletus]] in [[Byzantine]] [[Constantinople]], while [[Eutocius of Ascalon|Eutocius]]' commentaries on Archimedes' works in the same century 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 [[History of science in the Renaissance|Renaissance]] and again [[Scientific Revolution|in the 17th century]],<ref>
{{cite journal |last=Høyrup |first=Jens |year=2017 |title=Archimedes: Knowledge and Lore from Latin Antiquity to the Outgoing European Renaissance |journal=Gaņita Bhāratī |volume=39 |number=1 |pages=1–22 |url=http://webhotel4.ruc.dk/~jensh/Publications/2017%7Bg%7D_Archimedes%20--%20%20Knowledge%20and%20Lore_S.pdf}} Reprinted in {{Cite book |last=Hoyrup |first=J. |year=2019 |title=Selected Essays on Pre- and Early Modern Mathematical Practice |pages=459–477 |doi=10.1007/978-3-030-19258-7_17}}</ref><ref>{{Cite journal |last=Leahy |first=A. |date=2018 |title=The method of Archimedes in the seventeenth century. |journal=The American Monthly |volume=125 |issue=3 |pages=267–272 |doi=10.1080/00029890.2018.1413857}}</ref> while the discovery in 1906 of previously lost works by Archimedes in the [[Archimedes Palimpsest]] has provided new insights into how he obtained mathematical results.<ref>{{cite web |title=Works, Archimedes |date=23 June 2015 |publisher=University of Oklahoma |url=https://galileo.ou.edu/exhibits/works-archimedes |access-date=18 June 2019}}</ref><ref>{{cite book |title=The Genius of Archimedes – 23 Centuries of Influence on Mathematics, Science and Engineering: Proceedings of an International Conference held at Syracuse, Italy |date=8–10 June 2010 |series=History of Mechanism and Machine Science |volume=11 |publisher=Springer |editor1=Paipetis, Stephanos A. |editor2=Ceccarelli, Marco |isbn=978-90-481-9091-1 |doi=10.1007/978-90-481-9091-1}}</ref><ref>{{cite web |title=Archimedes&nbsp;– The Palimpsest |publisher=[[Walters Art Museum]] |url=http://www.archimedespalimpsest.org/palimpsest_making1.html |access-date=14 October 2007 |archive-url=https://web.archive.org/web/20070928102802/http://www.archimedespalimpsest.org/palimpsest_making1.html |archive-date=28 September 2007 |url-status=dead}}</ref><ref>{{cite news |url=https://www.theguardian.com/books/2011/oct/26/archimedes-palimpsest-ahead-of-time |title=Archimedes Palimpsest reveals insights centuries ahead of its time |newspaper=The Guardian |first1=Alison |last1=Flood |access-date=10 February 2017}}</ref>

==Biography==
=== Early life ===
[[File:Cicero Discovering the Tomb of Archimedes by Benjamin West.jpeg|thumb|right|''[[Cicero]] Discovering the Tomb of Archimedes'' (1805) by [[Benjamin West]]]]
Archimedes was born c. 287 BC in the seaport city of [[Syracuse, Sicily|Syracuse]], [[Sicily]], at that time a self-governing colony in [[Magna Graecia]]. The date of birth is based on a statement by the Byzantine Greek scholar [[John Tzetzes]] that Archimedes lived for 75 years before his death in 212 BC.<ref name=":0" /> [[Plutarch]] wrote in his ''[[Parallel Lives]]'' that Archimedes was related to King [[Hiero II of Syracuse|Hiero II]], the ruler of Syracuse, although [[Cicero]] suggests he was of humble origin.<ref>{{cite book |author=[[Plutarch]] |url=https://www.gutenberg.org/ebooks/674 |title=''Parallel Lives'' Complete e-text from Gutenberg.org |date=October 1996 |via=[[Project Gutenberg]]}}</ref><ref name="dijksterhuis" /> In the ''[[The Sand Reckoner|Sand-Reckoner]]'', Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known.<ref name="dijksterhuis">[[Eduard Jan Dijksterhuis|Dijksterhuis, Eduard J.]] [1938] 1987. ''Archimedes'', translated. Princeton: [[Princeton University Press]]. {{ISBN|978-0-691-08421-3}}.</ref><ref name=":3">{{Cite journal |last=Shapiro |first=A. E. |date=1975 |title=Archimedes's measurement of the Sun's apparent diameter. |journal=Journal for the History of Astronomy |volume=6 |issue=2 |pages=75–83 |bibcode=1975JHA.....6...75S |doi=10.1177/002182867500600201}}</ref> A biography of Archimedes was written by his friend Heracleides, but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children, or if he ever visited [[Alexandria]], Egypt, during his youth.<ref name=":4">{{Cite book |last=Acerbi |first=Fabio |chapter=Archimedes |title=New Dictionary of Scientific Biography |volume=I |pages=85–91 |publisher=Scribner |location=Detroit |year=2008}}</ref> From his surviving written works, it is clear that he maintained collegial relations with scholars based there, including his friend [[Conon of Samos]] and the head librarian [[Eratosthenes|Eratosthenes of Cyrene]].<ref group="lower-alpha">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.</ref>

=== Career ===
The standard versions of Archimedes' life were written long after his death by Greek and Roman historians. The earliest reference to Archimedes occurs in the ''[[Histories (Polybius)|Histories]]'' by [[Polybius]] ({{circa}} 200–118 BC), written about 70 years after his death.<ref name="dijksterhuis"/> 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 from the Romans.<ref name="death">{{cite web |last=Rorres |first=Chris |title=Death of Archimedes: Sources |url=http://www.math.nyu.edu/~crorres/Archimedes/Death/Histories.html |access-date=2 January 2007 |publisher=[[Courant Institute of Mathematical Sciences]]}}</ref> Polybius remarks how, during the [[Second Punic War]], Syracuse switched allegiances from [[Roman Republic|Rome]] to [[Carthage]], resulting in a military campaign under the command of [[Marcus Claudius Marcellus]] and [[Appius Claudius Pulcher (consul 212 BC)|Appius Claudius Pulcher]], who besieged the city from 213 to 212 BC. He notes that the Romans underestimated Syracuse's defenses, and mentions several machines Archimedes designed, including improved [[catapult]]s, crane-like machines that could be swung around in an arc, and other [[Lithobolos|stone-throwers]]. Although the Romans ultimately captured the city, they suffered considerable losses due to Archimedes' inventiveness.<ref>{{cite web |last=Rorres |first=Chris |title=Siege of Syracuse |url=http://www.math.nyu.edu/~crorres/Archimedes/Siege/Polybius.html |access-date=23 July 2007 |publisher=Courant Institute of Mathematical Sciences}}</ref>

Cicero (106–43 BC) mentions Archimedes in some of his works.<ref name="dijksterhuis"/> While serving as a [[quaestor]] in Sicily, Cicero found what was presumed to be Archimedes' tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes.<ref name=":0"/><ref>{{cite web |url=https://math.nyu.edu/Archimedes/Tomb/Cicero.html |title=Tomb of Archimedes |publisher=Courant Institute of Mathematical Sciences |access-date=2024-06-29}}</ref> 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. The tomb carried a sculpture illustrating Archimedes' [[On the Sphere and Cylinder|favorite mathematical proof]], that the volume and surface area of the sphere are two-thirds that of an enclosing cylinder including its bases.<ref>{{cite web |last=Rorres |first=Chris |title=Tomb of Archimedes: Sources |url=http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html |access-date=2 January 2007 |publisher=Courant Institute of Mathematical Sciences}}</ref><ref>{{cite web |last=Rorres |first=Chris |title=Tomb of Archimedes&nbsp;– Illustrations |url=http://www.math.nyu.edu/~crorres/Archimedes/Tomb/TombIllus.html |access-date=15 March 2011 |publisher=Courant Institute of Mathematical Sciences}}</ref> He also mentions that Marcellus brought to Rome two planetariums Archimedes built.<ref name=":5">{{Cite web |title=The Planetarium of Archimedes |url=https://studylib.net/doc/8971077/the-planetarium-of-archimedes |access-date=14 April 2021 |website=studylib.net}}</ref> The Roman historian [[Livy]] (59 BC–17 AD) retells Polybius' story of the capture of Syracuse and Archimedes' role in it.<ref name="death" />

=== Death ===
[[File:Death of Archimedes (1815) by Thomas Degeorge.png|thumb|''The Death of Archimedes'' (1815) by [[Thomas Degeorge]]<ref>{{cite web |title=The Death of Archimedes: Illustrations |url=https://www.math.nyu.edu/~crorres/Archimedes/Death/DeathIllus.html |website=math.nyu.edu |publisher=[[New York University]] |access-date=13 December 2017}}</ref>]]
Plutarch (45–119 AD) provides at least two accounts on how Archimedes died after Syracuse was taken.<ref name="dijksterhuis" /> According to the most popular account, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet Marcellus, but he declined, saying that he had to finish working on the problem. This enraged the soldier, who killed Archimedes with his sword. Another story has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items. Marcellus was reportedly angered by Archimedes' death, as he considered him a valuable scientific asset (he called Archimedes "a geometrical [[Hecatoncheires|Briareus]]") and had ordered that he should not be harmed.<ref name=":6" /><ref>Jaeger, Mary. ''Archimedes and the Roman Imagination''. p. 113.</ref>

The last words attributed to Archimedes are "[[Do not disturb my circles]]" ({{langx|la|Noli turbare circulos meos}}; {{langx|el|μὴ μου τοὺς κύκλους τάραττε}}), a reference to the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier.<ref name="dijksterhuis"/> There is no reliable evidence that Archimedes uttered these words and they do not appear in Plutarch's account. A similar quotation is found in the work of [[Valerius Maximus]] (fl. 30 AD), who wrote in ''Memorable Doings and Sayings'', "{{Langx|la|... sed protecto manibus puluere 'noli' inquit, 'obsecro, istum disturbare'|label=none}}" ("... but protecting the dust with his hands, said 'I beg of you, do not disturb this{{'"}}).<ref name="death" />

==Discoveries and inventions==

===Archimedes' principle===
{{main|Archimedes' principle}}
[[File:Displacement-measurement.svg|thumb|Measurement of volume by displacement, (a) before and (b) after an object has been submerged; the amount by which the liquid rises in the cylinder (∆V) is equal to the volume of the object]]
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 crown for a temple had been made for [[Hiero II of Syracuse|King Hiero II of Syracuse]], who supplied the pure gold to be used. The crown was likely made in the shape of a [[Wreaths and crowns in antiquity|votive wreath]].<ref name = "crown"/> Archimedes was asked to determine whether some silver had been substituted by the goldsmith without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its [[density]].<ref name="vitruvius">{{cite book |author=[[Vitruvius]] |url=http://www.gutenberg.org/files/20239/20239-h/20239-h.htm |title=''De Architectura'', Book IX, Introduction, paragraphs 9–12 |date=31 December 2006 |via=[[Project Gutenberg]]}}</ref>

In this account, Archimedes noticed while taking a bath 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 golden crown's [[volume]]. Archimedes was so excited by this discovery that he took to the streets naked, having forgotten to dress, crying "[[Eureka (word)|Eureka]]!" ({{langx|el|"εὕρηκα}}, ''heúrēka''!, {{Literal translation|I have found [it]!}}). 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 |access-date=27 February 2008}}</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, its density could be obtained; if cheaper and less dense metals had been added, the density would be lower than that of gold. Archimedes found that this is what had happened, proving that silver had been mixed in.<ref name = "crown">{{cite web |editor-last=Rorres |editor-first=Chris |title=The Golden Crown: Sources |publisher=[[New York University]] |url=https://www.math.nyu.edu/~crorres/Archimedes/Crown/Vitruvius.html |access-date=6 April 2021}}
{{bulleted list
|{{cite book |last=Morgan |first=Morris Hicky |author-link=Morris H. Morgan |year=1914 |title=Vitruvius: The Ten Books on Architecture |publisher=Harvard University Press |location=Cambridge |pages=253–254 |quote="Finally, filling the vessel again and dropping the crown itself into the same quantity of water, he found that more water ran over the crown than for the mass of gold of the same weight. Hence, reasoning from the fact that more water was lost in the case of the crown than in that of the mass, he detected the mixing of silver with the gold, and made the theft of the contractor perfectly clear."}}
|{{cite book |author=Vitruvius |author-link=Vitruvius |date=1567 |title=[[De architectura|De Architetura libri decem]] |publisher=Daniele Barbaro |location=Venice |pages=270–271 |quote="''Postea vero repleto vase in eadem aqua ipsa corona demissa, invenit plus aquae defluxisse in coronam, quàm in auream eodem pondere massam, et ita ex eo, quod plus defluxerat aquae in corona, quàm in massa, ratiocinatus, deprehendit argenti in auro mixtionem, et manifestum furtum redemptoris.''"}}
}}</ref><ref name="vitruvius" />

The story of the golden crown does not appear anywhere in Archimedes' known works. The practicality of the method described has been called into question due to the extreme accuracy that would be required to measure [[Displacement (fluid)|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]] |access-date=24 March 2009}}</ref> Archimedes may have instead sought a solution that applied the [[hydrostatics]] principle known as [[Archimedes's principle|Archimedes' principle]], found in his treatise ''[[On Floating Bodies]]'': 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 |access-date=23 July 2007}}</ref> Using this principle, it would have been possible to compare the density of the crown to that of pure gold by balancing it on a scale with a pure gold reference sample of the same weight, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly.<ref name=graf>{{cite journal |last1=Graf |first1=Erlend H. |title=Just What Did Archimedes Say About Buoyancy? |journal=The Physics Teacher |year=2004 |volume=42 |number=5 |pages=296–299 |doi=10.1119/1.1737965 |bibcode=2004PhTea..42..296G}}</ref> [[Galileo Galilei]], who invented a [[hydrostatic equilibrium|hydrostatic balance]] in 1586 inspired by Archimedes' work, 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>{{cite web |author=Van Helden, Al |title=The Galileo Project: Hydrostatic Balance |url=http://galileo.rice.edu/sci/instruments/balance.html |access-date=14 September 2007 |publisher=[[Rice University]]}}</ref><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]] |access-date=24 March 2009}}</ref>

===Law of the lever===
While Archimedes did not invent the [[lever]], he gave a mathematical proof of the principle involved in his work ''[[On the Equilibrium of Planes]]''.<ref>Finlay, M. (2013). ''[https://theses.gla.ac.uk/5129/ Constructing ancient mechanics] {{Webarchive|url=https://web.archive.org/web/20210414075253/https://theses.gla.ac.uk/5129/ |date=14 April 2021 }}'' [Master's thesis]. University of Glassgow.</ref> Earlier descriptions of the principle of the lever are found in a work by [[Euclid]] and in the ''[[Mechanics (Aristotle)|Mechanical Problems]],'' belonging to the [[Peripatetic school]] of the followers of [[Aristotle]], the authorship of which has been attributed by some 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]] |access-date=20 March 2010 |archive-url=https://web.archive.org/web/20130927050651/http://www.math.nyu.edu/~crorres/Archimedes/Lever/LeverLaw.html |archive-date=27 September 2013 |url-status=dead}}</ref><ref name="lever clagett">{{cite book |first=Marshall |last=Clagett |url=https://books.google.com/books?id=mweWMAlf-tEC&q=archytas%20lever&pg=PA72 |title=Greek Science in Antiquity |publisher=Dover Publications |isbn=978-0-486-41973-2 |year=2001}}</ref>

There are several, often conflicting, reports regarding Archimedes' feats using the lever to lift very heavy objects. 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 |author1=Dougherty, F.C. |author2=Macari, J. |author3=Okamoto, C. |title=Pulleys |url=http://www.swe.org/iac/lp/pulley_03.html |url-status=dead |archive-url=https://web.archive.org/web/20070718031943/http://www.swe.org/iac/LP/pulley_03.html |archive-date=18 July 2007 |access-date=23 July 2007 |publisher=[[Society of Women Engineers]]}}</ref> According to [[Pappus of Alexandria]], Archimedes' work on levers and his understanding of [[mechanical advantage]] caused him to remark: "Give me a place to stand on, and I will move the Earth" ({{langx|el|δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω}}).<ref>Quoted by [[Pappus of Alexandria]] in ''Synagoge'', Book VIII</ref> [[Olympiodorus the Younger|Olympiodorus]] later attributed the same boast to Archimedes' invention of the ''baroulkos'', a kind of [[windlass]], rather than the lever.<ref name=":10">{{Cite journal |last=Berryman |first=S. |date=2020 |title=How Archimedes Proposed to Move the Earth |journal=Isis |volume=111 |issue=3 |pages=562–567 |doi=10.1086/710317}}</ref>

===Archimedes' screw===
{{main|Archimedes' screw}}
[[File:Archimedes-screw one-screw-threads with-ball 3D-view animated small.gif|thumb|The [[Archimedes' screw]] can raise water efficiently]]

A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of [[Syracuse, Sicily|Syracuse]]. [[Athenaeus|Athenaeus of Naucratis]] quotes a certain Moschion in a description on how King Hiero II commissioned the design of a huge ship, the ''[[Syracusia]]'', which could be used for luxury travel, carrying supplies, and as a display of [[Navy|naval power]].<ref>{{cite book |last=Casson |first=Lionel |author-link=Lionel Casson |title=Ships and Seamanship in the Ancient World |year=1971 |publisher=Princeton University Press |isbn=978-0-691-03536-9 |url-access=registration |url=https://archive.org/details/shipsseamanshipi0000cass}}</ref> The ''Syracusia'' is said to have been the largest ship built in [[classical antiquity]] and, according to Moschion's account, it was launched by Archimedes.<ref name=":10" /> The ship presumably 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.<ref>{{Cite web |title=Athenaeus, The Deipnosophists, BOOK V., chapter 40 |url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:2013.01.0003:book=5:chapter=pos=377 |access-date=7 March 2023 |website=perseus.tufts.edu}}</ref> The account also mentions that, in order to remove any potential water leaking through the hull, a device with a revolving screw-shaped blade inside a cylinder was designed by Archimedes.

Archimedes' screw was turned by hand, and could also be used to transfer water from a {{nowrap|low-lying}} body of water into irrigation canals. The screw is still in use today for pumping liquids and granulated solids such as coal and grain. Described by [[Vitruvius]], Archimedes' device may have been an improvement on a screw pump that was used to irrigate the [[Hanging Gardens of Babylon]].<ref>{{cite journal |title=Sennacherib, Archimedes, and the Water Screw: The Context of Invention in the Ancient World |author=[[Stephanie Dalley|Dalley, Stephanie]] |author2=[[John Peter Oleson|Oleson, John Peter]] |journal=Technology and Culture |volume=44 |number=1 |year=2003 |url=http://muse.jhu.edu/journals/technology_and_culture/toc/tech44.1.html}}</ref><ref>{{cite web |title=Archimedes's 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 |access-date=23 July 2007}}</ref> The world's first seagoing [[steamboat|steamship]] with a [[propeller|screw propeller]] was the [[SS Archimedes|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 |access-date=22 January 2011}}</ref>

===Archimedes' claw===
Archimedes is said to have designed a [[Claw of Archimedes|claw]] as a weapon to defend the city of Syracuse. Also known as "{{visible anchor|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.<ref>{{cite web |first=Chris |last=Rorres |title=Archimedes's 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 |access-date=23 July 2007}}</ref> 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 |title=Archimedes' Claw: watch an animation |first=Bradley W |last=Carroll |publisher=Weber State University |url=http://physics.weber.edu/carroll/Archimedes/claw.htm |access-date=12 August 2007}}</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 journal |last=Sleeswyk |first=André Wegener |date=1981 |title=Vitruvius' Odometer |url=https://www.jstor.org/stable/24964584 |journal=Scientific American |volume=245 |issue=4 |pages=188–201 |doi=10.1038/scientificamerican1081-188 |jstor=24964584 |bibcode=1981SciAm.245d.188S |issn=0036-8733}}</ref><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 |access-date=14 September 2007 |archive-url=https://archive.today/20070905125400/http://www.tmth.edu.gr/en/aet/5/55.html |archive-date=5 September 2007 |url-status=dead}}</ref>

===Heat ray===
{{main|Archimedes' heat ray}}
[[File:Archimedes Heat Ray conceptual diagram.svg|thumb|Mirrors placed as a [[parabolic reflector]] to attack upcoming ships]]

As legend has it, Archimedes arranged mirrors as a [[parabolic reflector]] to burn ships attacking Syracuse using focused sunlight. While there is no extant contemporary evidence of this feat and modern scholars believe it did not happen, Archimedes may have written a work on mirrors entitled ''Catoptrica'',<ref name=":0" group="lower-alpha" /> and [[Lucian]] and [[Galen]], writing in the second century AD, mentioned that during the [[Siege of Syracuse (213–212 BC)|siege of Syracuse]] Archimedes had burned enemy ships. Nearly four hundred years later, [[Anthemius of Tralles|Anthemius]], despite skepticism, tried to reconstruct Archimedes' hypothetical reflector geometry.<ref>Archimedes's contemporary [[Diocles (mathematician)|Diocles]] made no mention of Archimedes or burning ships in his treatise about focusing reflectors. Diocles, ''On Burning Mirors'', ed. G. J. Toomer, Berlin: Springer, 1976. {{pb}} Lucian, ''Hippias'', [https://archive.org/details/lucianha01luciuoft/lucianha01luciuoft/page/36/ ¶ 2], in ''Lucian'', vol. 1, ed. A. M. Harmon, Harvard, 1913, {{pgs|36–37}}, says Archimedes burned ships with his ''[[techne]]'', "skill". {{pb}} [[Galen]], ''On temperaments'' 3.2, mentions ''pyreia'', "torches". {{pb}} [[Anthemius of Tralles]], ''On miraculous engines'' 153 [Westerman]. {{pb}} {{Cite journal |last=Knorr |first=Wilbur |date=1983 |title=The Geometry of Burning-Mirrors in Antiquity |url=http://dx.doi.org/10.1086/353176 |journal=Isis |volume=74 |issue=1 |pages=53–73 |doi=10.1086/353176 |issn=0021-1753}}</ref>

The purported device, sometimes called "[[Archimedes' heat ray]]", has been the subject of an ongoing debate about its credibility since the [[Renaissance]].<ref>{{cite journal |last=Simms |first=D. L. |title=Archimedes and the Burning Mirrors of Syracuse |journal=Technology and Culture |year=1977 |volume=18 |number=1 |pages=1–24 |doi=10.2307/3103202 |jstor=3103202}}</ref> [[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, mostly with negative results.<ref name="death ray">{{cite web |title=Archimedes Death Ray: Testing with MythBusters |url=http://web.mit.edu/2.009/www//experiments/deathray/10_Mythbusters.html |access-date=23 July 2007 |publisher=MIT |archive-url=https://web.archive.org/web/20061120030352/http://web.mit.edu/2.009/www//experiments/deathray/10_Mythbusters.html |archive-date=20 November 2006 |url-status=dead}}</ref><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 |access-date=14 September 2007 |archive-url=https://web.archive.org/web/20071012154432/http://wesley.nnu.edu/john_wesley/wesley_natural_philosophy/duten12.htm |archive-date=12 October 2007}}</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, but the overall effect would have been blinding, [[Glare (vision)|dazzling]], or distracting the crew of the ship rather than fire.<ref name="death ray2">{{cite web |date=13 December 2010 |title=TV Review: MythBusters 8.27&nbsp;– President's Challenge |url=http://fandomania.com/tv-review-mythbusters-8-27-presidents-challenge/ |access-date=18 December 2010}}</ref> Using modern materials and larger scale, sunlight-concentrating [[solar furnace]]s can reach very high temperatures, and are sometimes [[Concentrated solar power|used for generating electricity]].<ref>{{cite web |title=World's Largest Solar Furnace |website=Atlas Obscura |url=http://www.atlasobscura.com/places/worlds-largest-solar-furnace |access-date=6 November 2016}}</ref>

===Astronomical instruments===
Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as [[Aristarchus of Samos|Aristarchus]]' heliocentric model of the universe, in the ''Sand-Reckoner''. Without the use of either trigonometry or a table of chords, Archimedes determines the Sun's apparent diameter by first describing the procedure and instrument used to make observations (a straight rod with pegs or grooves),<ref>{{Cite journal |last=Evans |first=James |date=1 August 1999 |title=The Material Culture of Greek Astronomy |journal=Journal for the History of Astronomy |volume=30 |issue=3 |pages=238–307 |bibcode=1999JHA....30..237E |doi=10.1177/002182869903000305 |quote="But even before Hipparchus, Archimedes had described a similar instrument in his Sand-Reckoner. A fuller description of the same sort of instrument is given by Pappus of Alexandria&nbsp;... Figure 30 is based on Archimedes and Pappus. Rod R has a groove that runs its whole length&nbsp;... A cylinder or prism C is fixed to a small block that slides freely in the groove (p. 281)."}}</ref><ref>{{Cite encyclopedia |last1=Toomer |first1=G. J. |last2=Jones |first2=Alexander |date=7 March 2016 |title=Astronomical Instruments |encyclopedia=Oxford Research Encyclopedia of Classics |doi=10.1093/acrefore/9780199381135.013.886 |isbn=9780199381135 |quote="Perhaps the earliest instrument, apart from sundials, of which we have a detailed description is the device constructed by Archimedes (Sand-Reckoner 11-15) for measuring the sun's apparent diameter; this was a rod along which different coloured pegs could be moved."}}</ref> applying correction factors to these measurements, and finally giving the result in the form of upper and lower bounds to account for observational error.<ref name=":3" /> [[Ptolemy]], quoting Hipparchus, also references Archimedes' [[solstice]] observations in the ''Almagest''. This would make Archimedes the first known Greek to have recorded multiple solstice dates and times in successive years.<ref name=":4" />

Cicero's ''[[De re publica]]'' portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse in the [[Second Punic War]], [[Marcus Claudius Marcellus|Marcellus]] is said to have taken back to Rome two mechanisms which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets. Cicero also 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 Honor and Virtue|Temple of Virtue]] in Rome. Marcellus's mechanism was demonstrated, according to Cicero, by [[Gaius Sulpicius Gallus]] to [[Lucius Furius Philus]], who described it thus:<ref>{{cite web |author=[[Cicero]] |title=''De re publica'' 1.xiv §21 |url=http://www.thelatinlibrary.com/cicero/repub1.shtml#21 |access-date=23 July 2007 |publisher=thelatinlibrary.com}}</ref><ref>{{cite book |author=[[Cicero]] |url=https://www.gutenberg.org/ebooks/14988 |title=''De re publica'' Complete e-text in English from Gutenberg.org |date=9 February 2005 |via=[[Project Gutenberg]] |access-date=18 September 2007}}</ref>

{{Blockquote|text={{lang|la|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.}} {{pb}}
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.}}

This is a description of a small [[planetarium]]. [[Pappus of Alexandria]] reports on a now lost treatise by Archimedes dealing with the construction of these mechanisms entitled ''On Sphere-Making''.<ref name=":5" /><ref>{{cite book |last=Wright |first=Michael T. |chapter=Archimedes, Astronomy, and the Planetarium |date=2017 |title=Archimedes in the 21st Century: Proceedings of a World Conference at the Courant Institute of Mathematical Sciences |series=Trends in the History of Science |pages=125–141 |editor-last=Rorres |editor-first=Chris |place=Cham |publisher=Springer |doi=10.1007/978-3-319-58059-3_7 |isbn=978-3-319-58059-3}}</ref> Modern research in this area has been focused on the [[Antikythera mechanism]], another device built {{circa|100}} BC designed with a similar purpose, with some scholars regarding Archimedes' device as a precursor.<ref name=":11" /><ref>{{cite web |title=The Antikythera Mechanism II |url=http://www.math.sunysb.edu/~tony/whatsnew/column/antikytheraII-0500/diff4.html |access-date=25 December 2013 |publisher=[[Stony Brook University]] |archive-date=12 December 2013 |archive-url=https://web.archive.org/web/20131212212956/http://www.math.sunysb.edu/~tony/whatsnew/column/antikytheraII-0500/diff4.html |url-status=dead}}</ref> Constructing mechanisms of this kind would have required a sophisticated knowledge of [[Differential (mechanical device)|differential gearing]].<ref>{{cite news |last=Noble Wilford |first=John |date=31 July 2008 |title=Discovering How Greeks Computed in 100 B.C. |newspaper=[[The New York Times]] |url=https://www.nytimes.com/2008/07/31/science/31computer.html?_r=0 |access-date=25 December 2013}}</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 news |title=Ancient Moon 'computer' revisited |publisher=BBC News |date=29 November 2006 |url=http://news.bbc.co.uk/1/hi/sci/tech/6191462.stm |access-date=23 July 2007}}</ref><ref>{{cite web |last=Rorres |first=Chris |title=Spheres and Planetaria |url=http://www.math.nyu.edu/~crorres/Archimedes/Sphere/SphereIntro.html |access-date=23 July 2007 |publisher=Courant Institute of Mathematical Sciences}}</ref>

==Mathematics==
While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of [[Greek mathematics|mathematics]]. [[Plutarch]] wrote that Archimedes "placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life",<ref name=":6">{{cite book |title=Extract from ''Parallel Lives'' |author=[[Plutarch]] |publisher=fulltextarchive.com |url=https://www.fulltextarchive.com/page/Plutarch-s-Lives10/#p35 |access-date=10 August 2009}}</ref> though some scholars believe this may be a mischaracterization.<ref>{{Cite journal |last=Russo |first=L. |date=2013 |title=Archimedes between legend and fact. |journal=Lettera Matematica |volume=1 |issue=3 |pages=91–95 |doi=10.1007/s40329-013-0016-y |doi-access=free |quote="It is amazing that for a long time Archimedes's attitude towards the applications of science was deduced from the acritical acceptance of the opinion of Plutarch: a polygraph who lived centuries later, in a cultural climate that was completely different, certainly could not have known the intimate thoughts of the scientist. On the other hand, the dedication with which Archimedes developed applications of all kinds is well documented: of catoptrica, as Apuleius tells in the passage already cited (Apologia, 16), of hydrostatics (from the design of clocks to naval engineering: we know from Athenaeus (Deipnosophistae, V, 206d) that the largest ship in Antiquity, the Syracusia, was constructed under his supervision), and of mechanics (from machines to hoist weights to those for raising water and devices of war)."}}</ref><ref>{{Cite journal |last=Drachmann |first=A. G. |author-link=A. G. Drachmann |date=1968 |title=Archimedes and the Science of Physics |journal=Centaurus |volume=12 |issue=1 |pages=1–11 |doi=10.1111/j.1600-0498.1968.tb00074.x |bibcode=1968Cent...12....1D}}</ref><ref>{{Cite thesis |title=Attitudes toward the natural philosopher in the early Roman empire (100 B.C. to 313 A.D.) |url=https://clio.columbia.edu/catalog/8602980 |date=2008 |first=Richard |last=Carrier |access-date=6 April 2021}} "Hence Plutarch's conclusion that Archimedes disdained all mechanics, shop work, or anything useful as low and vulgar, and only directed himself to geometric theory, is obviously untrue. Thus, as several scholars have now concluded, his account of Archimedes appears to be a complete fabrication, invented to promote the Platonic values it glorifies by attaching them to a much-revered hero." (p.444)</ref>

=== Method of exhaustion ===
[[File:PiArchimede4.svg|thumb|Archimedes calculates 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 [[Cavalieri's principle|indivisibles]] (a precursor to [[infinitesimal]]s) in a way that is similar to modern [[Integral|integral calculus]].<ref name=":2">{{Cite web |last=Powers |first=J. |date=2020 |title=Did Archimedes do calculus? |url=https://old.maa.org/sites/default/files/images/upload_library/46/HOMSIGMAA/2020-Jeffery%20Powers.pdf |access-date=14 April 2021 |website=maa.org}}</ref> 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 areas of figures and the value of [[Pi|{{pi}}]].

In ''[[Measurement of a Circle]]'', he did this by drawing a larger [[regular hexagon]] outside a [[circle]] then 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 {{pi}} lay between 3{{sfrac|1|7}} (approx. 3.1429) and 3{{sfrac|10|71}} (approx. 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 |access-date=30 October 2012}}</ref> He also proved that the [[area of a circle]] was equal to {{pi}} multiplied by the [[square]] of the [[radius]] of the circle (<math>\pi r^2</math>).

=== Archimedean property ===
In ''[[On the Sphere and Cylinder]]'', Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. Today this is known as 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 |access-date=7 November 2009 |archive-url=https://web.archive.org/web/20090316065753/http://web.mat.bham.ac.uk/R.W.Kaye/seqser/archfields |archive-date=16 March 2009 |url-status=dead}}</ref>

Archimedes gives the value of the [[square root]] of 3 as lying between {{sfrac|265|153}} (approximately 1.7320261) and {{sfrac|1351|780}} (approximately 1.7320512) in ''Measurement of a Circle''. 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|978-0-486-42084-4}}.</ref> It is possible that he used an [[iteration|iterative]] procedure to calculate these values.<ref>{{Cite web |title=Of Calculations Past and Present: The Archimedean Algorithm |url=https://www.maa.org/programs/maa-awards/writing-awards/of-calculations-past-and-present-the-archimedean-algorithm |access-date=14 April 2021 |website=maa.org |publisher=Mathematical Association of America}}</ref><ref>{{cite web |title=The Computation of Pi by Archimedes |author=McKeeman, Bill |author-link=William M. McKeeman |website=Matlab Central |url=http://www.mathworks.com/matlabcentral/fileexchange/29504-the-computation-of-pi-by-archimedes/content/html/ComputationOfPiByArchimedes.html#37 |access-date=30 October 2012}}</ref>

=== The infinite series ===
[[File:Parabolic segment and inscribed triangle.svg|thumb|upright=.8|A proof that the area of the [[parabola|parabolic]] segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure from ''[[Quadrature of the Parabola]]'']]
In ''[[Quadrature of the Parabola]]'', Archimedes proved that the area enclosed by a [[parabola]] and a straight line is {{sfrac|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 r|common ratio]] {{sfrac|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 whose third vertex is where the line that is parallel to the parabola's axis and that passes through the midpoint of the base intersects the parabola, and so on. This proof uses a variation of the series {{nowrap|[[1/4 + 1/16 + 1/64 + 1/256 + · · ·]]}} which sums to&nbsp;{{sfrac|1|3}}.

=== Myriad of myriads ===
In ''[[The Sand Reckoner]]'', Archimedes set out to calculate a number that was greater than the grains of sand needed to fill the universe. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote:<blockquote>There are some, King [[Gelo, son of Hiero II|Gelo]], 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.</blockquote>To solve the problem, Archimedes devised a system of counting based on the [[myriad]]. The word itself derives from the Greek {{Langx|grc|μυριάς|translit=murias|label=none}}, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million, i.e., 10,000 x 10,000) 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 |access-date=23 July 2007}}</ref>

==Writings==
[[File:Archimedes – Opere, 1615 – BEIC 9741168.jpg|thumb|upright=.8|Front page of Archimedes' ''Opera'', in Greek and Latin, edited by [[David Rivault de Flurence|David Rivault]] (1615)]]
The works of Archimedes were written in [[Doric Greek]], the dialect of ancient Syracuse.<ref>Encyclopedia of ancient Greece By Wilson, Nigel Guy [https://books.google.com/books?id=-aFtPdh6-2QC&pg=PA77 p. 77] {{Webarchive|url=https://web.archive.org/web/20160508081544/https://books.google.com/books?id=-aFtPdh6-2QC&pg=PA77 |date=8 May 2016 }} {{ISBN|978-0-7945-0225-6}} (2006)</ref> Many written works by Archimedes have not survived or are only extant in heavily edited fragments; at least seven of his treatises are known to have existed due to references made by other authors.<ref name=":0" /> [[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 group="lower-alpha" name=":0">The treatises by Archimedes known to exist only through references in the works of other authors are: ''On Sphere-Making'' and a work on [[Polyhedron|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'' or ''On Levers''; ''On Centers of Gravity''; ''On the Calendar''.</ref>

Archimedes made his work known through correspondence with mathematicians in [[Alexandria]]. The writings of Archimedes were first collected by the [[Byzantine Empire|Byzantine]] Greek architect [[Isidore of Miletus]] ({{Circa|530&nbsp;AD}}), while commentaries on the works of Archimedes written by [[Eutocius of Ascalon|Eutocius]] in the same century helped bring his work to a wider audience. Archimedes' work was translated into Arabic by [[Thābit ibn Qurra]] (836–901 AD), and into Latin via Arabic by [[Gerard of Cremona]] (c. 1114–1187). Direct Greek to Latin translations were later done by [[William of Moerbeke]] (c. 1215–1286) and [[Iacopo da San Cassiano|Iacobus Cremonensis]] (c. 1400–1453).<ref>{{Cite journal |last=Clagett |first=Marshall |date=1982 |title=William of Moerbeke: Translator of Archimedes |journal=Proceedings of the American Philosophical Society |volume=126 |issue=5 |pages=356–36 6 |jstor=986212}}</ref><ref>{{Cite journal |last=Clagett |first=Marshall |date=1959 |title=The Impact of Archimedes on Medieval Science |journal=Isis |volume=50 |issue=4 |pages=419–429 |doi=10.1086/348797}}</ref>

During the [[History of science in the Renaissance|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's Work |publisher=Brown University Library |year=1999 |url=https://library.brown.edu/exhibits/archive/math/wholefr.html}}</ref>

===Surviving works===
The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).<ref>{{Cite journal |last=Knorr |first=W. R. |date=1978 |title=Archimedes and the Elements: Proposal for a Revised Chronological Ordering of the Archimedean Corpus |journal=Archive for History of Exact Sciences |volume=19 |issue=3 |pages=211–290 |doi=10.1007/BF00357582 |jstor=41133526}}</ref><ref>{{Cite journal |last=Sato |first=T. |date=1986 |title=A Reconstruction of The Method Proposition 17, and the Development of Archimedes' Thought on Quadrature...Part One |journal=Historia scientiarum: International journal of the History of Science Society of Japan}}</ref>

==== ''Measurement of a Circle'' ====
{{Main|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 approximation of the value of pi ({{pi}}), showing that it is greater than {{sfrac|223|71}} (3.1408...) and less than {{sfrac|22|7}} (3.1428...).

==== ''The Sand Reckoner'' ====
{{Main|The Sand Reckoner}}
In this treatise, also known as '''''Psammites''''', Archimedes finds a number that is greater than the [[Sand|grains of sand]] needed to fill 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]], and attempts to measure the apparent diameter of the [[Sun]].<ref name=":7" /> 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'' is the only surviving work in which Archimedes discusses his views on astronomy.<ref>{{cite web |year=2002 |title=English translation of ''The Sand Reckoner'' |publisher=[[University of Waterloo]] |url=http://www.math.uwaterloo.ca/navigation/ideas/reckoner.shtml |archive-date=2002-06-01 |archive-url=https://web.archive.org/web/20020601231141/https://www.math.uwaterloo.ca/navigation/ideas/reckoner.shtml |url-status=dead}} Adapted from {{cite book |last=Newman |first=James R. |title=The World of Mathematics |volume=1 |publisher=Simon & Schuster |location=New York |year=1956}}</ref>

==== ''On the Equilibrium of Planes'' ====
{{Main|On the Equilibrium of Planes}}
There are two books to ''On the Equilibrium of Planes'': the first contains seven [[Axiom|postulates]] and fifteen [[proposition]]s, while the second book contains ten propositions. In the first book, Archimedes proves the law of the [[lever]], which states that:

{{Blockquote|text=[[Magnitude (mathematics)|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 book |author=Heath, T.L. |url=https://archive.org/details/worksofarchimede029517mbp |title=The Works of Archimedes |year=1897 |publisher=Cambridge University Press}}</ref>

==== ''Quadrature of the Parabola'' ====
{{Main|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 the area of a [[triangle]] with equal base and height. He achieves this in one of his proofs by calculating the value of a [[geometric series]] that sums to infinity with the [[ratio]] 1/4.

==== ''On the Sphere and Cylinder'' ====
{{Main|On the Sphere and Cylinder}}
[[File:Esfera Arquímedes.svg|thumb|A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases|209x209px]]
In this two-volume 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 {{sfrac|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.

==== ''On Spirals'' ====
{{Main|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 modern [[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 Conoids and Spheroids'' ====
{{Main|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'' ====
{{Main|On Floating Bodies}}
There are two books of ''On Floating Bodies''. In the first book, 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. [[Archimedes' principle]] of buoyancy is given in this work, stated as follows:{{r|graf}}<ref>{{cite book |last=Netz |first=Reviel |chapter=Archimedes' Liquid Bodies |title=ΣΩΜΑ: Körperkonzepte und körperliche Existenz in der antiken Philosophie und Literatur |year=2017 |pages=287–322 |editor1-first=Thomas |editor1-last=Buchheim |editor2-first=David |editor2-last=Meißner |editor3-first=Nora |editor3-last=Wachsmann |isbn=978-3-7873-2928-1 |place=Hamburg |publisher=Felix Meiner |chapter-url=https://books.google.com/books?id=rQ2KDwAAQBAJ&pg=PA287 |chapter-url-access=limited}}</ref>

<blockquote>Any body wholly or partially immersed in fluid experiences an upthrust equal to, but opposite in direction to, the weight of the fluid displaced.</blockquote>

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.<ref>{{cite book |last=Stein |first=Sherman |chapter=Archimedes and his floating paraboloids |editor1-first=David F. |editor1-last=Hayes |editor2-first=Tatiana |editor2-last=Shubin |title=Mathematical Adventures for Students and Amateurs |publisher=Mathematical Association of America |place=Washington |year=2004 |pages=219–231 |isbn=0-88385-548-8 |chapter-url=https://archive.org/details/mathematicaladve0000unse/page/219 |chapter-url-access=limited}} {{pb}} {{cite journal |last=Rorres |first=Chris |year=2004 |title=Completing Book II of Archimedes's on Floating Bodies |journal=The Mathematical Intelligencer |volume=26 |number=3 |pages=32–42 |doi=10.1007/bf02986750}} {{pb}} {{cite journal |last1=Girstmair |first1=Kurt |last2=Kirchner |first2=Gerhard |title=Towards a completion of Archimedes' treatise on floating bodies |journal=Expositiones Mathematicae |volume=26 |number=3 |year=2008 |pages=219–236 |doi=10.1016/j.exmath.2007.11.002 |doi-access=free}}</ref>

==== ''Ostomachion'' ====
{{Main|Ostomachion}}
[[File:Stomachion.JPG|thumb|''[[Ostomachion]]'' is a [[dissection puzzle]] found in the [[Archimedes Palimpsest]]|200x200px]]
Also known as '''Loculus of Archimedes''' or '''Archimedes' Box''',<ref name=":1" /> 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]]. [[Reviel Netz]] of [[Stanford University]] argued in 2003 that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. 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 |newspaper=[[The New York Times]] |date=14 December 2003 |url=https://query.nytimes.com/gst/fullpage.html?res=9D00E6DD133CF937A25751C1A9659C8B63&sec=&spon=&pagewanted=all |access-date=23 July 2007}}</ref> The number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded.<ref>{{cite web |title=The Loculus of Archimedes, Solved |author=Ed Pegg Jr. |publisher=[[Mathematical Association of America]] |date=17 November 2003 |url=http://www.maa.org/editorial/mathgames/mathgames_11_17_03.html |access-date=18 May 2008}}</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 |access-date=14 September 2007}}</ref> [[Ausonius]] calls the puzzle {{Langx|grc|Ostomachion|label=none|italic=yes}}, a Greek compound word formed from the roots of {{Langx|grc|osteon|label=none|italic=yes}} ({{Langx|grc|ὀστέον|label=none|lit=bone}}) and {{Langx|grc|machē|label=none|italic=yes}} ({{Langx|grc|μάχη|label=none|lit=fight}}).<ref name=":1">{{cite web |url=http://www.archimedes-lab.org/latin.html#archimede |title=Graeco Roman Puzzles |publisher=Gianni A. Sarcone and Marie J. Waeber |access-date=9 May 2008}}</ref>

==== The cattle problem ====
{{Main|Archimedes's cattle problem|l1 = Archimedes' cattle problem}}
[[Gotthold Ephraim Lessing]] discovered this work in a Greek manuscript consisting of a 44-line poem 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 [[The Cattle of Helios|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. A. Amthor first solved this version of the problem<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]] |access-date=14 September 2007 |archive-url=https://web.archive.org/web/20071012171254/http://andrews.edu/~calkins/profess/cattle.htm |archive-date=12 October 2007}}</ref>

==== ''The Method of Mechanical Theorems'' ====
{{Main|The Method of Mechanical Theorems}}
This treatise was thought lost until the discovery of the [[Archimedes Palimpsest]] in 1906. In this work Archimedes uses [[Cavalieri's principle|indivisibles]],<ref name=":2" /><ref name=":9" /> and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. He may have considered this method lacking in formal rigor, so he also used the [[method of exhaustion]] to derive the results. As with ''[[Archimedes's cattle problem|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 15 propositions on the nature of circles. The earliest known copy of the text is in [[Arabic language|Arabic]]. [[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 |website=[[cut-the-knot]] |url=http://www.cut-the-knot.org/Curriculum/Geometry/BookOfLemmas/index.shtml |access-date=7 August 2007}}</ref>

It has also been claimed that the [[Heron's formula|formula]] for calculating the area of a triangle from the length of its sides was known to Archimedes,<ref group="lower-alpha">[[Carl Benjamin Boyer|Boyer, Carl Benjamin]]. 1991. ''A History of Mathematics''. {{ISBN|978-0-471-54397-8}}: "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula – <math>\textstyle k = \sqrt{s(s-a)(s-b)(s-c)}</math>, where <math>s</math> 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."</ref> though its first appearance is in the work of [[Hero of Alexandria|Heron of Alexandria]] in the 1st century AD.<ref>{{cite web |title=Heron of Alexandria |author1=O'Connor, J.J. |author2=Robertson, E.F. |publisher=[[University of St Andrews]] |url=http://www-history.mcs.st-and.ac.uk/Biographies/Heron.html |date=April 1999 |access-date=17 February 2010}}</ref> Other questionable attributions to Archimedes' work include the Latin poem ''[[Carmen de ponderibus et mensuris]]'' (4th or 5th century), which describes the use of a [[hydrostatic equilibrium|hydrostatic balance]], to solve the problem of the crown, and the 12th-century text ''[[Mappae clavicula]]'', which contains instructions on how to perform [[assay]]ing of metals by calculating their specific gravities.<ref name="kingcrown">[[Oswald A. W. Dilke|Dilke, Oswald A. W.]] 1990. [Untitled]. ''[[Gnomon (journal)|Gnomon]]'' 62(8):697–99. {{JSTOR|27690606}}.</ref><ref>Berthelot, Marcel. 1891. "Sur l histoire de la balance hydrostatique et de quelques autres appareils et procédés scientifiques." ''[[Annales de chimie et de physique|Annales de Chimie et de Physique]]'' 6(23):475–85.</ref>

===Archimedes Palimpsest===
{{main|Archimedes Palimpsest}}
[[File:Archimedes Palimpsest.jpg|thumb|In 1906, the Archimedes Palimpsest revealed works by Archimedes thought to have been lost]]

The foremost document containing Archimedes' work is the Archimedes Palimpsest. In 1906, the Danish professor [[Johan Ludvig Heiberg (historian)|Johan Ludvig Heiberg]] visited [[Constantinople]] to examine a 174-page [[Goatskin (material)|goatskin]] [[parchment]] of prayers, written in the 13th century, after reading a short transcription published seven years earlier by [[Athanasios Papadopoulos-Kerameus|Papadopoulos-Kerameus]].<ref name=":8">{{Cite journal |last=Wilson |first=Nigel |date=2004 |title=The Archimedes Palimpsest: A Progress Report |journal=The Journal of the Walters Art Museum |volume=62 |pages=61–68 |jstor=20168629}}</ref><ref>{{Cite journal |last1=Easton |first1=R. L. |last2=Noel |first2=W. |date=2010 |title=Infinite Possibilities: Ten Years of Study of the Archimedes Palimpsest |journal=Proceedings of the American Philosophical Society |volume=154 |issue=1 |pages=50–76 |jstor=20721527}}</ref> He confirmed that it was indeed 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, a common practice in the Middle Ages, as [[vellum]] was expensive. The older works in the palimpsest were identified by scholars as 10th-century copies of previously lost treatises by Archimedes.<ref name=":8" /><ref>{{cite magazine |title=Reading Between the Lines |author=Miller, Mary K. |magazine=[[Smithsonian (magazine)|Smithsonian]] |date=March 2007 |url=https://www.smithsonianmag.com/science-nature/reading-between-the-lines-148131057/}}</ref> The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. On 29 October 1998, it was sold at auction to an anonymous buyer for a total of $2.2 million.<ref>{{cite news |title=Rare work by Archimedes sells for $2 million |publisher=[[CNN]] |date=29 October 1998 |url=http://edition.cnn.com/books/news/9810/29/archimedes/ |access-date=15 January 2008 |archive-url=https://web.archive.org/web/20080516000109/http://edition.cnn.com/books/news/9810/29/archimedes/ |archive-date=16 May 2008}}</ref><ref>[https://www.christies.com/results/printauctionresults.aspx?saleid=8685&lid=1 Christie's (n.d). ''Auction results'']</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 was stored at the [[Walters Art Museum]] in [[Baltimore]], [[Maryland]], where it was 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=2 August 2006 |url=http://news.bbc.co.uk/1/hi/sci/tech/5235894.stm |access-date=23 July 2007}}</ref> It has since returned to its anonymous owner.<ref>{{Cite journal |last1=Piñar |first1=G. |last2=Sterflinger |first2=K. |last3=Ettenauer |first3=J. |last4=Quandt |first4=A. |last5=Pinzari |first5=F. |date=2015 |title=A Combined Approach to Assess the Microbial Contamination of the Archimedes Palimpsest |journal=Microbial Ecology |volume=69 |issue=1 |pages=118–134 |pmid=25135817 |doi=10.1007/s00248-014-0481-7 |pmc=4287661 |bibcode=2015MicEc..69..118P}}</ref><ref>{{Cite journal |last=Acerbi |first=F. |date=2013 |title=Review: R. Netz, W. Noel, N. Tchernetska, N. Wilson (eds.), ''The Archimedes Palimpsest'', 2001 |url=https://www.academia.edu/8016340 |journal=Aestimatio |volume=10 |pages=34–46}}</ref>

The treatises in the Archimedes Palimpsest include:
* ''[[On the Equilibrium of Planes]]''
* ''[[On Spirals]]''
* ''[[Measurement of a Circle]]''
* ''[[On the Sphere and Cylinder]]''
* ''[[On Floating Bodies]]''
* ''[[The Method of Mechanical Theorems]]''
* ''[[Stomachion]]''
* Speeches by the 4th century BC politician [[Hypereides]]
* A commentary on [[Aristotle]]'s ''[[Categories (Aristotle)|Categories]]''
* Other works

==Legacy==
{{further|List of things named after Archimedes|Eureka (disambiguation){{!}}Eureka}}
Sometimes called the father of mathematics and [[mathematical physics]], Archimedes had a wide influence on mathematics and science.<ref name="LitList2">
Father of mathematics: Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, p 19. {{pb}}
Father of mathematical physics: [[James H. Williams Jr.]], Fundamentals of Applied Dynamics, p 30., Carl B. Boyer, Uta C. Merzbach, A History of Mathematics, p 111., Stuart Hollingdale, Makers of Mathematics, p 67., Igor Ushakov, In the Beginning, Was the Number (2), p 114.</ref>

=== Mathematics and physics ===
[[File:Gerhard Thieme Archimedes.jpg|thumb| Bronze statue of Archimedes in [[Berlin]]]]
Historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity. [[Eric Temple Bell]], for instance, wrote: 
{{blockquote|Any list of the three "greatest" mathematicians of all history would include the name of Archimedes. The other two usually associated with him are [[Isaac Newton|Newton]] and [[Carl Friedrich Gauss|Gauss]]. Some, considering the relative wealth—or poverty—of mathematics and physical science in the respective ages in which these giants lived, and estimating their achievements against the background of their times, would put Archimedes first.<ref>E.T. Bell, Men of Mathematics, p 20.</ref>}}

Likewise, [[Alfred North Whitehead]] and [[George F. Simmons]] said of Archimedes:
{{blockquote|... in the year 1500 Europe knew less than Archimedes who died in the year 212 BC ...<ref>{{cite web |author=Alfred North Whitehead |title=The Influence of Western Medieval Culture Upon the Development of Modern Science |url=https://inters.org/Whitehead-Western-Development-Science |access-date=4 April 2022}}</ref>}}{{blockquote|If we consider what all other men accomplished in mathematics and physics, on every continent and in every civilization, from the beginning of time down to the seventeenth century in Western Europe, the achievements of Archimedes outweighs it all. He was a great civilization all by himself.<ref>George F. Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, p 43.</ref>}}
[[Reviel Netz]], Suppes Professor in Greek Mathematics and Astronomy at [[Stanford University]] and an expert in Archimedes notes:
{{blockquote|And so, since Archimedes led more than anyone else to the formation of the calculus and since he was the pioneer of the application of mathematics to the physical world, it turns out that Western science is but a series of footnotes to Archimedes. Thus, it turns out that Archimedes is the most important scientist who ever lived.<ref>Reviel Netz, William Noel, The Archimedes Codex: Revealing The Secrets of the World's Greatest Palimpsest</ref>}}

[[Leonardo da Vinci]] repeatedly expressed admiration for Archimedes, and attributed his invention [[Architonnerre]] to Archimedes.<ref name="Nelson Examiner">{{cite news |url=http://paperspast.natlib.govt.nz/cgi-bin/paperspast?a=d&d=NENZC18420521.2.11 |title=The Steam-Engine |volume=I |issue=11 |date=21 May 1842 |work=Nelson Examiner and New Zealand Chronicle |publisher=National Library of New Zealand |page=43 |access-date=14 February 2011 |location=Nelson}}</ref><ref name="PennyM">{{cite book |title=The Steam Engine |url=https://books.google.com/books?id=E1oFAAAAQAAJ&pg=RA1-PA104 |year=1838 |publisher=The Penny Magazine |page=104}}</ref><ref name="Thurston1996">{{cite book |author=Robert Henry Thurston |title=A History of the Growth of the Steam-Engine |year=1996 |url=https://books.google.com/books?id=KCMUmXV1C1gC |publisher=Elibron |isbn=1-4021-6205-7 |page=12}}</ref> [[Galileo Galilei]] called him "superhuman" and "my master",<ref>Matthews, Michael. ''Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion Can Contribute to Science Literacy''. p. 96.</ref><ref>{{Cite web |title=Archimedes – Galileo Galilei and Archimedes |url=https://exhibits.museogalileo.it/archimedes/section/GalileoGalileiArchimedes.html |access-date=16 June 2021 |website=exhibits.museogalileo.it}}</ref> while [[Christiaan Huygens]] said, "I think Archimedes is comparable to no one", consciously emulating him in his early work.<ref>{{Cite web |last=Yoder |first=J. |date=1996 |title=Following in the footsteps of geometry: the mathematical world of Christiaan Huygens |url=https://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0009.php |website=De Zeventiende Eeuw. Jaargang 12}}</ref> [[Gottfried Wilhelm Leibniz]] said, "He who understands Archimedes and [[Apollonius of Perga|Apollonius]] will admire less the achievements of the foremost men of later times".<ref>[[Carl Benjamin Boyer|Boyer, Carl B.]], and [[Uta Merzbach|Uta C. Merzbach]]. 1968. ''A History of Mathematics''. ch. 7.</ref> Gauss's heroes were Archimedes and Newton,<ref>Jay Goldman, The Queen of Mathematics: A Historically Motivated Guide to Number Theory, p 88.</ref> and [[Moritz Cantor]], who studied under Gauss in the [[University of Göttingen]], reported that he once remarked in conversation that "there had been only three epoch-making mathematicians: Archimedes, Newton, and [[Gotthold Eisenstein|Eisenstein]]".<ref>E.T. Bell, Men of Mathematics, p 237</ref>

The inventor [[Nikola Tesla]] praised him, saying:

{{blockquote|Archimedes was my ideal. I admired the works of artists, but to my mind, they were only shadows and semblances. The inventor, I thought, gives to the world creations which are palpable, which live and work.<ref>W. Bernard Carlson, Tesla: Inventor of the Electrical Age, p 57</ref>}}

=== Honors and commemorations ===
[[File:Archimedes bronze coin.jpg|thumb|1612 drawing of a now-lost bronze coin depicting Archimedes]]
Italian numismatist and archaeologist Filippo Paruta (1552–1629) and [[Leonardo Agostini]] (1593–1676) reported on a bronze coin in Sicily with the portrait of Archimedes on the obverse and a cylinder and sphere with the monogram ARMD in Latin on the reverse.<ref>{{Cite book |last1=Paruta |first1=Filippo |url=https://dn720400.ca.archive.org/0/items/lasiciliadifilip00paru/lasiciliadifilip00paru.pdf |title=La Sicilia descritta con medaglie |last2=Agostini |first2=Leonardo |publisher=Marco Maier |year=1697 |publication-date=1697 |pages=73, 326 |language=it |trans-title=Sicily described by medals |access-date=2025-01-20}}</ref> Although the coin is now lost and its date is not precisely known, [[Ivo Schneider]] described the reverse as "a sphere resting on a base – probably a rough image of one of the planetaria created by Archimedes," and suggested it might have been minted in Rome for Marcellus who "according to ancient reports, brought two spheres of Archimedes with him to Rome".<ref>{{Cite book |last=Schneider |first=Ivo |title=Archimedes. Ingenieur, Naturwissenschaftler und Mathematiker |publisher=Wissenschaftliche Buchgesellschaft |year=1979 |isbn=3-534-06844-0 |location=Darmstadt |pages=19, 23 |language=de |trans-title=Archimedes. Engineer, natural scientist and mathematician}}</ref>

There is a [[impact crater|crater]] on the [[Moon]] named [[Archimedes (crater)|Archimedes]] ({{Coord|29.7|-4.0|display=}}) in his honor, as well as a lunar [[mountain range]], the [[Montes Archimedes]] ({{Coord|25.3|-4.6|display=}}).<ref>{{cite web |title=Oblique view of Archimedes crater on the Moon |author1=Friedlander, Jay |author2=Williams, Dave |publisher=[[NASA]] |url=http://nssdc.gsfc.nasa.gov/imgcat/html/object_page/a15_m_1541.html |access-date=13 September 2007}}</ref>

[[File:FieldsMedalFront.jpg|thumb|The [[Fields Medal]] carries a portrait of Archimedes|190x190px]]
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 1st century AD poet [[Marcus Manilius|Manilius]], which reads in Latin: ''Transire suum pectus mundoque potiri'' ("Rise above oneself and grasp the world").<ref>{{Cite journal |last=Riehm |first=C. |date=2002 |title=The early history of the Fields Medal |url=https://www.ams.org/notices/200207/comm-riehm.pdf |journal=Notices of the AMS |volume=49 |issue=7 |pages=778–782 |quote="The Latin inscription from the Roman poet Manilius surrounding the image may be translated 'To pass beyond your understanding and make yourself master of the universe.' The phrase comes from Manilius's Astronomica 4.392 from the first century A.D. (p. 782)."}}</ref><ref>{{Cite web |date=5 February 2015 |title=The Fields Medal |url=http://www.fields.utoronto.ca/about/fields-medal |access-date=23 April 2021 |website=Fields Institute for Research in Mathematical Sciences}}</ref><ref>{{cite web |title=Fields Medal |url=https://www.mathunion.org/imu-awards/fields-medal |access-date=23 April 2021 |publisher=[[International Mathematical Union]]}}</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 |access-date=25 August 2007}}</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 |access-date=14 September 2007 |archive-url=https://web.archive.org/web/20071012123245/http://capitolmuseum.ca.gov/VirtualTour.aspx?content1=1278&Content2=1374&Content3=1294 |archive-date=12 October 2007 |url-status=dead}}</ref>

==See also==
{{Portal|Biography|Mathematics|Physics}}

===Concepts===
* [[Arbelos]]
* [[Archimedean point]]
* [[Axiom of Archimedes|Archimedes' axiom]]
* [[Archimedes number]]
* [[Archimedes paradox]]
* [[Archimedean solid]]
* [[Archimedes' circles|Archimedes' twin circles]]
* [[Methods of computing square roots]]
* [[Salinon]]
* [[Steam cannon]]

===People===
* [[Diocles (mathematician)|Diocles]]
* [[Pseudo-Archimedes]]
* [[Zhang Heng]]

== References ==

=== Notes ===
{{Reflist|35em|group=lower-alpha}}

=== Citations ===
{{Reflist}}

==Further reading==
{{EB1911 poster|Archimedes}}
*[[Carl Benjamin Boyer|Boyer, Carl Benjamin]]. 1991. ''[[iarchive:historyofmathema00boye|A History of Mathematics]]''. New York: Wiley. {{ISBN|978-0-471-54397-8}}.
*[[Marshall Clagett|Clagett, Marshall]]. 1964–1984. ''Archimedes in the Middle Ages'' 1–5. Madison, WI: [[University of Wisconsin Press]].
*Clagett, Marshall. 1970. [[iarchive:dictionaryofscie0001unse/page/213/mode/1up|"Archimedes"]]. In [[Charles Coulston Gillispie]], ed. ''[[Dictionary of Scientific Biography]]''. Vol. 1 (Abailard–Berg). New York: Charles Scribner's Sons. {{pgs|213–231}}.
*[[Eduard Jan Dijksterhuis|Dijksterhuis, Eduard J.]] 1956. ''Archimedes''. Translated by C. Dikshoorn. Copenhagen: Ejnar Munksgaard. Chapters 1–5 were translated from ''Archimedes'' (in Dutch). Groningen: Noordhoff. 1938. Later chapters appeared in ''Euclides'' Vols. 15–17, 20. 1938–1944. Reprinted 1987 by Princeton University Press. {{isbn|0-691-08421-1}}
*[[Mary Gow|Gow, Mary]]. 2005. ''[[iarchive:archimedesmathem0000gowm|Archimedes: Mathematical Genius of the Ancient World]]''. [[Enslow Publishing]]. {{ISBN|978-0-7660-2502-8}}.
*Hasan, Heather. 2005. ''[[iarchive:archimedesfather00hasa|Archimedes: The Father of Mathematics]]''. Rosen Central. {{ISBN|978-1-4042-0774-5}}.
*[[Thomas Heath (classicist)|Heath, Thomas L.]] 1897. [[iarchive:worksofarchimede029517mbp|''Works of Archimedes'']]. [[Dover Publications]]. {{ISBN|978-0-486-42084-4}}. Complete works of Archimedes in English.
*[[Reviel Netz|Netz, Reviel]]. 2004–2017. ''The Works of Archimedes: Translation and Commentary''. 1–2. Cambridge University Press. Vol. 1: "The Two Books on the Sphere and the Cylinder". {{isbn|978-0-521-66160-7}}. Vol. 2: "On Spirals". {{isbn|978-0-521-66145-4}}.
* Netz, Reviel, and William Noel. 2007. ''The Archimedes Codex''. [[Orion Publishing Group]]. {{ISBN|978-0-297-64547-4}}.
*[[Clifford A. Pickover|Pickover, Clifford A.]] 2008. ''Archimedes to Hawking: Laws of Science and the Great Minds Behind Them''. [[Oxford University Press]]. {{ISBN|978-0-19-533611-5}}.
*Simms, Dennis L. 1995. ''Archimedes the Engineer''. [[Continuum International Publishing Group]]. {{ISBN|978-0-7201-2284-8}}.
*[[Sherman K. Stein|Stein, Sherman]]. 1999. ''[[iarchive:archimedeswhatdi00stei|Archimedes: What Did He Do Besides Cry Eureka?]]''. [[Mathematical Association of America]]. {{ISBN|978-0-88385-718-2}}.

== 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}}
* ''[https://www.wilbourhall.org/index.html#archimedes Heiberg's Edition of Archimedes].'' Texts in Classical Greek, with some in English.
* {{In Our Time|Archimedes|b00773bv|Archimedes}}
* {{Gutenberg author | id=2545| name=Archimedes}}
* {{Internet Archive author}}
* {{InPho|thinker|2546}}
* {{PhilPapers|search|archimedes}}
* [http://www.archimedespalimpsest.org/ The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland]
* {{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://web.mit.edu/2.009/www/experiments/steamCannon/ArchimedesSteamCannon.html Testing the Archimedes steam cannon] {{Webarchive|url=https://web.archive.org/web/20100329220142/http://web.mit.edu/2.009/www/experiments/steamCannon/ArchimedesSteamCannon.html |date=29 March 2010 }}

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