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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]] (1620)
| birth_date        = {{circa|287{{nbsp}}BC}}
| birth_place       = [[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 />[[List of things named after Archimedes|List of other namesakes]]|}}
}}

'''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|kih|MEE|deez}}; {{circa|287|212 BC}}) was an [[Ancient Greek mathematics|Ancient Greek mathematician]], [[History of physics#Greek concept|physicist]], [[Ancient Greek technology|engineer]], [[Ancient Greek astronomy|astronomer]], and inventor from the city of [[Syracuse, Sicily|Syracuse]] in [[History of Greek and Hellenistic Sicily|Sicily]]. Although few details of his life are known, based on his surviving work, he is considered one of the leading scientists in [[classical antiquity]], and one of the greatest mathematicians of all time. Archimedes anticipated modern [[calculus]] and [[mathematical analysis|analysis]] by applying the concept of the [[Cavalieri's principle|infinitesimals]] and the [[method of exhaustion]] to derive and rigorously prove many [[geometry|geometrical]] [[theorem|theorems]], including 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]].

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]], the widespread use of the concept of [[Center of mass#History|center of gravity]], and the enunciation of the law of [[buoyancy]] known as [[Archimedes' principle]]. In astronomy, he made measurements of the apparent diameter of the [[Sun]] and the size of the [[universe]]. 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]]. 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. In the [[Middle Ages]], Archimedes' work was translated into Arabic in the 9th century and then into Latin in the 12th century, and were an influential source of ideas for scientists during the [[History of science in the Renaissance|Renaissance]] and in the [[Scientific Revolution]]. The discovery in 1906 of works by Archimedes in the [[Archimedes Palimpsest]] has provided new insights into how he obtained mathematical results.

==Biography==
[[File:Cicero Discovering the Tomb of Archimedes by Benjamin West.jpeg|thumb|right|''[[Cicero]] Discovering the Tomb of Archimedes'' (1805) by [[Benjamin West]]]]
The details of Archimedes's life are obscure; a biography of Archimedes mentioned by [[Eutocius]] was allegedly written by his friend [[Heraclides Lembus]], but this work has been lost, and modern scholarship is doubtful that it was written by Heraclides to begin with.<ref>Commentarius in dimensionem circuli (Archimedis opera omnia ed. Heiberg-Stamatis (1915), vol. 3, p. 228); Commentaria in conica (Apollonii Pergaei quae Graece exstant, ed. Heiberg (1893) vol. 2, p. 168: "Hērakleios"</ref> 

Based on a statement by the Byzantine Greek scholar [[John Tzetzes]] that Archimedes lived for 75 years before his death in 212 BC, Archimedes is estimated to have been born c. 287 BC in the seaport city of [[Syracuse, Sicily|Syracuse]], [[Sicily]], at that time a self-governing colony in [[Magna Graecia]]. In the ''[[The Sand Reckoner|Sand-Reckoner]]'', Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known; [[Plutarch]] wrote in his ''[[Parallel Lives]]''<ref>[[Plutarch]], ''Life of Marcellus''</ref> that Archimedes was related to King [[Hiero II of Syracuse|Hiero II]], the ruler of Syracuse, although [[Cicero]] and [[Silius Italicus]] suggest he was of humble origin.{{sfn|Dijksterhuis|1987|p=10}} It is also unknown whether he ever married or had children, or if he ever visited [[Alexandria]], Egypt, during his youth;<ref name="Acerbi2008">{{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> though his surviving written works, addressed to Dositheus of Pelusium, a student of the Alexandrian astronomer [[Conon of Samos]], and to the head librarian [[Eratosthenes|Eratosthenes of Cyrene]], suggested that he maintained collegial relations with scholars based there.{{sfn|Dijksterhuis|1987|pp=11-12}} In the preface to ''On Spirals'' addressed to Dositheus, 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.{{cn|date=April 2025}}

===Golden wreath===
[[File:Displacement-measurement.svg|thumb|Measurement of volume (a) before and (b) after an object has been submerged, with (∆V) indicating the rising amount of liquid is equal to the volume of the object]]
Another story of a problem that Archimedes is credited solving with in service of Hiero II is the "wreath problem."{{sfn|Dijksterhuis|1987|p=18}} According to [[Vitruvius]], writing about two centuries after Archimedes' death, [[Hiero II of Syracuse|King Hiero II of Syracuse]] had commissioned a golden wreath for a temple to the immortal gods, and had supplied pure gold to be used by the goldsmith.<ref>[[Vitruvius]], ''De Architectura'', Book IX, 3</ref> However, the king had begun to suspect that the goldsmith had substituted some cheaper silver and kept some of the pure gold for himself, and, unable to make the smith confess, asked Archimedes to investigate.{{sfn|Dijksterhuis|1987|p=19}} Later, while stepping into a bath, Archimedes allegedly noticed that the level of the water in the tub rose more the lower he sank in the tub and, realizing that this effect could be used to determine the golden crown's [[volume]], was so excited that he took to the streets naked, having forgotten to dress, crying "[[Eureka (word)|Eureka]]!"{{efn|{{langx|el|"εὕρηκα|heúrēka}}}}, meaning "I have found [it]!"{{sfn|Dijksterhuis|1987|p=19}} According to Vitruvius, Archimedes then took a lump of gold and a lump of silver that were each equal in weight to the wreath, and, placing each in the bathtub, showed that the wreath displaced more water than the gold and less than the silver, demonstrating that the wreath was gold mixed with silver.{{sfn|Dijksterhuis|1987|p=19}}

A different account is given in the ''Carmen de Ponderibus'',<ref>''Metrologicorum Scriptorum reliquiae'', ed. F. Hultsch (Leipzig 1864), II, 88</ref> an anonymous 5th century Latin didactic poem on weights and measures once attributed to the grammarian [[Priscian]].{{sfn|Dijksterhuis|1987|p=19}} In this poem, the lumps of gold and silver were placed on the scales of a balance, and then the entire apparatus was immersed in water; the difference in density between the gold and the silver, or between the gold and the crown, causes the scale to tip accordingly.<ref>''Carmen de Ponderibus'', lines 124-162</ref> Unlike the more famous bathtub account given by Vitruvius, this poetic account uses the [[hydrostatics]] principle now known as [[Archimedes's principle|Archimedes' principle]] that is found in his treatise ''[[On Floating Bodies]]'', where a body immersed in a fluid experiences a [[buoyancy|buoyant force]] equal to the weight of the fluid it displaces.{{sfn|Dijksterhuis|1987|pp=20-21}} [[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>

===Launching the ''Syracusia''===
A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of [[Syracuse, Sicily|Syracuse]].{{sfn|Dijksterhuis|1987|p=14}} [[Athenaeus|Athenaeus of Naucratis]] in his ''[[Deipnosophistae]]'' quotes a certain Moschion for a description on how King Hiero II commissioned the design of a huge ship, the ''[[Syracusia]]'', which is said to have been the largest ship built in [[classical antiquity]] and, according to Moschion's account, it was launched by Archimedes.<ref>Athenaeus, Deipnosophistae, V.40-45</ref> Plutarch tells a slightly different account,<ref>Plutarch, Life of Marcellus 7-8</ref> relating that Archimedes boasted to Hiero that he was able to move any large weight, at which point Hiero challenged him to move a ship.{{sfn|Dijksterhuis|1987|p=15}} These accounts contain many fantastic details that are historically implausible, and the authors of these stories provide conflicting about how this task was accomplished:{{sfn|Dijksterhuis|1987|p=15}} Plutarch states that Archimedes constructed a [[block and tackle|block-and-tackle]] [[pulley]] system, while [[Hero of Alexandria]] attributed the same boast to Archimedes' invention of the ''baroulkos'', a kind of [[windlass]].<ref> Heronis Opera Vol II, 1, 256, III 306</ref> [[Pappus of Alexandria]] attributed this feat, instead, to Archimedes' use of [[mechanical advantage]],{{sfn|Dijksterhuis|1987|p=15}} the principle of [[lever]]age to lift objects that would otherwise have been too heavy to move, attributing to him the oft-quoted remark: "Give me a place to stand on, and I will move the Earth."{{efn|{{langx|el|δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω}}}}<ref>[[Pappus of Alexandria]], ''Synagoge'' Book VIII</ref> 

Athenaeus, likely garbling the details of Hero's account of the baroulkos,{{sfn|Dijksterhuis|1987|p=16}} also mentions that Archimedes used a "screw" in order to remove any potential water leaking through the hull of the ''Syracusia''. Although this device is sometimes referred to as [[Archimedes' screw]], it likely predates him by a significant amount, and none of his closest contemporaries who describe its use ([[Philo of Byzantium]], [[Strabo]], and [[Vitruvius]]) credit him with its use.{{sfn|Dijksterhuis|1987|p=15}}

===War machines===
[[File:Archimedes Heat Ray conceptual diagram.svg|thumb|Mirrors placed as a [[parabolic reflector]] to attack upcoming ships]]
The greatest reputation Archimedes earned during antiquity was for the defense of his city from the Romans during the [[Siege of Syracuse (213–212 BC)|Siege of Syracuse]].{{sfn|Dijksterhuis|1987|pp=28-29}} According to Plutarch,<ref>Life of Marcellus, 25-27</ref> Archimedes had constructed war machines for Hiero II, but had never been given an opportunity to use them during Hiero's lifetime. In 214 BC, however, during the [[Second Punic War]], when Syracuse switched allegiances from [[Roman Republic|Rome]] to [[Carthage]], the Roman army under [[Marcus Claudius Marcellus]] attempted to take the city, Archimedes allegedly personally oversaw the use of these war machines in the defense of the city, greatly delaying the Romans, who were only able to capture the city after a long siege.{{sfn|Dijksterhuis|1987|pp=26,28}} Three different historians, [[Plutarch]], [[Livy]], and [[Polybius]] provide testimony about these war machines, describing improved [[catapults]], cranes that dropped heavy pieces of lead on the Roman ships or which used an iron [[Claw of Archimedes|claw]] to lift them out of the water, dropping them back in so that they sank.{{efn|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>}}{{sfn|Dijksterhuis|1987|p=27}} 

A much more improbable account, not found in any of the three earliest accounts (Plutarch, Polybius, or Livy) describes how Archimedes used "burning mirrors" to focus the sun's rays onto the attacking Roman ships, setting them on fire.{{sfn|Dijksterhuis|1987|pp=28-29}} The earliest account to mention ships being set on fire, by the 2nd century CE satirist [[Lucian of Samosata]],<ref>Lucian, ''Hippias'', [https://archive.org/details/lucianha01luciuoft/lucianha01luciuoft/page/36/ ¶ 2], in ''Lucian'', vol. 1, ed. A. M. Harmon, Harvard, 1913, {{pgs|36–37}}</ref> does not mention mirrors, and only says the ships were set on fire by artificial means, which may imply that burning projectiles were used.{{sfn|Dijksterhuis|1987|pp=28-29}} The first author to mention mirrors is [[Galen]], writing later in the same century.<ref>[[Galen]], ''On temperaments'' 3.2</ref> Nearly four hundred years after Lucian and Galen, [[Anthemius of Tralles|Anthemius]], despite skepticism, tried to reconstruct Archimedes' hypothetical reflector geometry.<ref>[[Anthemius of Tralles]], ''On miraculous engines'' 153.</ref><ref>{{Cite journal |last=Knorr |first=Wilbur |date=1983 |title=The Geometry of Burning-Mirrors in Antiquity |journal=Isis |volume=74 |issue=1 |pages=53–73 |doi=10.1086/353176 }}</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,<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> while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes, with mixed results.<ref name="death ray">{{cite book
 | last = Jaeger | first = Mary
 | editor-last = Rorres | editor-first = Chris
 | contribution = Archimedes in the 21st century imagination
 | doi = 10.1007/978-3-319-58059-3_8
 | isbn = 9783319580593
 | pages = 143–152
 | publisher = Birkhäuser
 | series = Trends in the History of Science
 | title = Archimedes in the 21st Century: Proceedings of a World Conference at the Courant Institute of Mathematical Sciences
 | year = 2017}} See p. 144.</ref>

===Death===
[[File:Death of Archimedes (1815) by Thomas Degeorge.png|thumb|''The Death of Archimedes'' (1815) by [[Thomas Degeorge]]]]
There are several divergent accounts of Archimedes' death during the sack of Syracuse after it fell to the Romans:{{sfn|Dijksterhuis|1987|pp=30-31}} The oldest account, from [[Livy]],<ref>Livy, ''Ab Urbe Condita'' Book XXV, 31</ref> says that, while drawing figures in the dust, Archimedes was killed by a Roman soldier who did not know he was Archimedes. According to Plutarch,<ref>Life of Marcellus, XIX, 1</ref> the soldier demanded that Archimedes come with him, but Archimedes declined, saying that he had to finish working on the problem, and the soldier killed Archimedes with his sword. Another story from Plutarch has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items.{{sfn|Dijksterhuis|1987|pp=30-31}} Another Roman writer, [[Valerius Maximus]] (fl. 30 AD), wrote in ''Memorable Doings and Sayings'' that Archimedes' last words as the soldier killed him were "... but protecting the dust with his hands, said 'I beg of you, do not disturb this." which is similar to the last words now commonly attributed to him, "[[Do not disturb my circles]]," which do not appear in any ancient sources.{{sfn|Dijksterhuis|1987|pp=30-31}}

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>Plutarch, Parallel Lives</ref><ref>Jaeger, Mary. ''Archimedes and the Roman Imagination''. p. 113.</ref> [[Cicero]] (106–43 BC) mentions that Marcellus brought to Rome two planetariums Archimedes built,{{sfn|Dijksterhuis|1987|pp=23-25}} which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets, one of which he donated to the [[Temple of Honor and Virtue|Temple of Virtue]] in Rome, and the other he allegedly kept as his only personal loot from Syracuse."<ref>Cicero, ''[https://www.gutenberg.org/ebooks/14988 De republica]''</ref> [[Pappus of Alexandria]] reports on a now lost treatise by Archimedes ''On Sphere-Making'', which may have dealt with the construction of these mechanisms.{{sfn|Dijksterhuis|1987|p=27}} Constructing mechanisms of this kind would have required a sophisticated knowledge of [[Differential (mechanical device)|differential gearing]], which was once thought to have been beyond the range of the technology available in ancient times, but the discovery in 1902 of the Antikythera mechanism, another device built {{circa|100}} BC designed with a similar purpose, has confirmed that devices of this kind were known to the ancient Greeks,<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> with some scholars regarding Archimedes' device as a precursor.<ref>{{Cite journal |last=Freeth |first=Tony |date=2022 |title=Wonder of the Ancient World |journal=Scientific American |volume=32 |issue=1 |pages=24 |doi=10.1038/scientificamerican0122-24 |doi-access=free |pmid=39016582 }}</ref><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>

While serving as a [[quaestor]] in Sicily, Cicero himself found what was presumed to be Archimedes' 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. 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>

==Mathematics==
While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of [[Greek mathematics|mathematics]], both in applying the techniques of his predecessors to obtain new results, and developing new methods of his own.

=== Method of exhaustion ===
{{main|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]]
In ''[[Quadrature of the Parabola]]'', Archimedes states that a certain proposition in [[Euclid's Elements|Euclid's ''Elements'']] demonstrating that the area of a [[circle]] is proportional to its diameter was proven using a lemma now known as the [[Archimedean property]], that “the excess by which the greater of two unequal regions exceed the lesser, if added to itself, can exceed any given bounded region.” Prior to Archimedes, [[Eudoxus of Cnidus]] and other earlier mathematicians{{efn|While Eudoxus is often credited for the whole of Euclid's Book XII, Archimedes explicitly credits him only with the proofs of XII.7 and XII.10, that the volume of a pyramid and a cone are, respectively, one-third of the volume of the rectangular prism and cone with the same base and height.{{harvnb|Acerbi|2018|p=279}}}} applied this lemma, a technique now referred to as the "method of exhaustion," to find the volume of a [[tetrahedron]], [[cylinder]], [[cone]], and [[sphere]], for which proofs are given in book XII of [[Euclid's Elements]].{{sfn|Acerbi|2018|p=279}}

In ''[[Measurement of a Circle]]'', Archimedes employed this method to show that the area of a circle is the same as a right triangle whose base and height are equal to its radius and circumference.{{sfn|Acerbi|2018|p=280}} He then approximated the ratio between the radius and the circumference, the value of [[Pi|{{pi}}]], 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=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> In the same treatise, he also asserts that the value of the [[square root]] of 3 as lying between {{sfrac|265|153}} (approximately 1.7320261) and {{sfrac|1351|780}} (approximately 1.7320512), which he may have derived from a similar method.<ref>{{cite journal |last=Miel |first=G |year=1983 |title=Of Calculations Past and Present: The Archimedean Algorithm |journal=The American Mathematical Monthly |volume=90 |number=1 |pages=17–35 |doi=10.1080/00029890.1983.11971147 |jstor=2975687 |url=https://www.maa.org/programs/maa-awards/writing-awards/of-calculations-past-and-present-the-archimedean-algorithm |archive-url=https://web.archive.org/web/20150905063500/http://www.maa.org/programs/maa-awards/writing-awards/of-calculations-past-and-present-the-archimedean-algorithm |url-status=dead |archive-date=2015-09-05 }}</ref>

[[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 used this technique to prove 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, expressing 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}}:{{sfn|Netz|2022|p=139}}

:<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}}.

He also used this technique in order to measure the surface areas of a sphere and cone,<ref>''[[On the Sphere and Cylinder]]'' 13-14, 33-34, 42, 44</ref> to calculate the area of an ellipse,<ref>''[[On Conoids and Spheroids]]'' 4</ref> and to find the area contained within an [[Archimedean spiral]].<ref>''[[On Spirals]]'', 24-25</ref>{{sfn|Acerbi|2018|p=280}}

=== Mechanical method === 
{{blockquote
|text=For it is more feasible, having already in one’s possession, through the method, of a knowledge of some sort of the matters under investigation, to provide the proof – rather than investigating it, knowing nothing.
|author=Archimedes
|title=''[[The Method of Mechanical Theorems]]''<ref>Netz, Noel, Tchernetska, Wilson. "Archimedes Palimpsest" p. 71</ref>
}}
In addition to developing on the works of earlier mathematicians with the method of exhaustion, Archimedes also pioneered a novel technique using the [[law of the lever]] in order to measure the area and volume of shapes using physical means. He first gives an outline of this proof in ''Quadrature of the Parabola'' alongside the geometric proof, but he gives a fuller explanation in ''[[The Method of Mechanical Theorems]]''.{{sfn|Netz|2022|p=139}} According to Archimedes, he proved the results in his mathematical treatises first using this method, and then worked backwards, applying the method of exhaustion only after he had already calculated an approximate value for the answer.{{sfn|Netz|2022|p=187}}

=== Large numbers ===
Archimedes also developed methods for representing large numbers.

In ''[[The Sand Reckoner]]'', Archimedes devised a system of counting based on the [[myriad]],{{efn|{{Langx|grc|μυριάς|translit=murias|label=none}}}} the Greek term for the number 10,000, in order to calculate a number that was greater than the grains of sand needed to fill the universe. 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> In doing so, he demonstrated that mathematics could represent arbitrarily large numbers.

In the ''[[Archimedes' cattle problem|Cattle Problem]]'', Archimedes challenges the mathematicians at the [[Library of Alexandria]] to count the numbers of cattle in the [[The Cattle of Helios|Herd of the Sun]], which involves solving a number of simultaneous [[Diophantine equation]]s.  A more difficult version of the problem in which some of the answers are required to be [[square number]]s, 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>

=== Archimedean solid ===
{{main|Archimedean solid}}
In a lost work described by [[Pappus of Alexandria]], Archimedes proved that there are exactly thirteen semiregular polyhedra.{{sfn|Netz|2022|p=133}}

==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)]]
Archimedes made his work known through correspondence with mathematicians in [[Alexandria]],{{sfn|Acerbi|2018}} which were originally written in [[Doric Greek]], the dialect of ancient Syracuse.<ref>{{cite book |last1=Wilson |first1=Nigel Guy |title=Encyclopedia of Ancient Greece |date=2006 |publisher=Psychology Press |page=77 |url=https://books.google.com/books?id=-aFtPdh6-2QC&pg=PA77 |access-date=29 April 2025 |language=en |isbn=978-0-7945-0225-6}}</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]] 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>{{Cite journal |last=Osborne |first=Catherine |date=1983 |title=Archimedes on the Dimensions of the Cosmos |journal=Isis |volume=74 |issue=2 |pages=234–242 |doi=10.1086/353246 |jstor=233105 }}</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 |arxiv=1609.02710 |last2=Kosovichev |first2=Alexander G. |last3=Kilcik |first3=Ali}}</ref> 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>

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''.<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 for measuring the sun's apparent diameter; this was a rod along which different coloured pegs could be moved."}}</ref> 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}}</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>{{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> 

[[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="Acerbi2008" />

==== ''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]],<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> which states that:

{{Blockquote|text=[[Magnitude (mathematics)|Magnitudes]] are in equilibrium at distances reciprocally proportional to their weights.}}

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 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>

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 by two different methods: first by applying the [[law of the lever]], and by calculating the value of a [[geometric series]] that sums to infinity with the [[ratio]] 1/4.<ref>{{cite book|title=The works of Archimedes|editor=Heath|year=1897|publisher=Dover|author=Archimedes}}, page 233</ref>

==== ''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.<ref>{{cite book|title=The works of Archimedes|editor=Heath|year=1897|publisher=Dover|author=Archimedes}}, page 1</ref><ref>{{cite book |editor-last=Netz |editor-first=Reviel |editor-link=Reviel Netz |title=The Works of Archimedes |volume=I: The Two Books On the Sphere and the Cylinder |year=2004 |publisher=Cambridge University Press |doi=10.1017/CBO9780511482557 |isbn=0-521-66160-9 }}</ref>

==== ''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]].<ref>{{cite book |editor-last=Netz |editor-first=Reviel |editor-link=Reviel Netz |title=The Works of Archimedes: Translation and Commentary |volume=II: On Spirals |year=2017 |publisher=Cambridge University Press |doi=10.1017/9781139019279 |isbn=978-0-521-66145-4 }}</ref> 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}}.<ref>{{cite book|title=The works of Archimedes|editor=Heath|year=1897|publisher=Dover|author=Archimedes}}, page 151</ref>

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.<ref>{{cite book|title=The works of Archimedes|editor=Heath|year=1897|publisher=Dover|author=Archimedes}}, page 99</ref><ref>{{cite journal |last=Saito |first=Ken |year=2013 |title=Archimedes and double contradiction proof |journal=Lettera Matematica International Edition |volume=1 |pages=97–104 |doi= 10.1007/s40329-013-0017-x }}</ref>

==== ''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.<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>

[[Archimedes' principle]] of buoyancy is given in this work, stated as follows:<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}}
In this work, 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]], which involves solving a number of simultaneous [[Diophantine equation]]s. [[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. 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}}
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]].

In this work Archimedes uses a novel method, an early form of [[Cavalieri's principle]],<ref name="ArchimedesCalc">{{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}} {{pb}} {{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}} {{pb}} {{MacTutor |class=HistTopics |id=The_rise_of_calculus |title=A history of the calculus |date=February 1996 }} {{pb}} {{Cite journal |last=Kirfel |first=Christoph |date=2013 |title=A generalisation of Archimedes' method |journal=The Mathematical Gazette |volume=97 |issue=538 |pages=43–52 |doi=10.1017/S0025557200005416 |jstor=24496758}}</ref> to rederive the results from the treatises sent to Dositheus (''Quadrature of the Parabola'', ''On the Sphere and Cylinder'', ''On Spirals'', ''On Conoids and Spheroids'') that he had previously used the [[method of exhaustion]] to prove,{{sfn|Netz|2022|p=131}} using the [[law of the lever]] he applied in ''On the Equilbrium of Planes'' in order to find the [[center of gravity]] of an object first, and reasoning geometrically from there in order to more easily derive the volume of an object.{{sfn|Netz|2022|pp=187-193}} Archimedes states that he used this method to derive the results in the treatises sent to Dositheus before he proved them more rigorously with the method of exhaustion, stating that it is useful to know that a result is true before proving it rigorously, much as [[Eudoxus of Cnidus]] was aided in proving that the volume of a cone is one-third the volume of cylinder by knowing that [[Democritus]] had already asserted it to be true on the argument that this is true by the fact that the pyramid has one-third the rectangular prism of the same base.{{sfn|Netz|2022|p=150-151}}

This treatise was thought lost until the discovery of the [[Archimedes Palimpsest]] in 1906.<ref>{{cite book |last1=Smith |first1=David Eugene |title=Geometrical Solutions Derived from Mechanics: A Treatise of Archimedes |date=1909 |publisher=Open Court Publishing Company |url=https://archive.org/details/geometricalsolu00smitgoog/mode/2up |access-date=4 May 2025 |language=English}}</ref>

===Apocryphal works===
{{main|Pseudo-Archimedes}}
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,{{cn|date=April 2025}} since it quotes Archimedes, suggesting modification by another author. The ''Lemmas'' may be based on an earlier work by Archimedes that is now lost.{{cn|date=April 2025}}

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>

===Lost works===
Many written works by Archimedes have not survived or are only extant in heavily edited fragments:{{sfn|Netz|2022|p=133-135}} [[Pappus of Alexandria]] mentions ''On Sphere-Making'', as well as a work on [[semiregular polyhedra]], and another work on spirals, while [[Theon of Alexandria]] quotes a remark about [[refraction]] from the {{nowrap|now-lost}} ''Catoptrica''. ''Principles'', addressed to Zeuxippus, explained the number system used in ''[[The Sand Reckoner]]''; there are also ''On Balances''; ''On Centers of Gravity.''{{sfn|Netz|2022|p=133-135}}

Scholars in the medieval Islamic world also attribute to Archimedes a formula for calculating the area of a triangle from the length of its sides, which today is known as [[Heron's formula]] due to its first known appearance in the work of [[Hero of Alexandria|Heron of Alexandria]] in the 1st century AD, and may have been proven in a lost work of Archimedes that is no longer extant.{{sfn|Netz|2022|p=135-136}}

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

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 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 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

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 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=https://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>

==Legacy==
[[File:Gerhard Thieme Archimedes.jpg|thumb| Bronze statue of Archimedes in Berlin]]
{{further|List of things named after Archimedes}}
Sometimes called the father of mathematics<ref>Father of mathematics: Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, p 19.</ref> and [[mathematical physics]],<ref>[[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> historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity.<ref> {{MacTutor |id=Archimedes |title= Archimedes of Syracuse |date=January 1999 }} {{pb}} {{cite book |last=Hirshfeld |first=Alan |url=https://books.google.com/books?id=zbcfLoZKDl8C&pg=PA206 |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"}}</ref> 

===Classical antiquity===
The reputation that Archimedes had for mechanical inventions in classical antiquity is well-documented;<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> [[Athenaeus]]<ref>''[[Deipnosophistae]]'', V, 206d)</ref> recounts in his ''[[Deipnosophistae]]'' how Archimedes supervised the construction of the largest known ship in antiquity, the [[Syracusia]], while [[Apuleius]]<ref>''Apologia'', 16</ref> talks about his work in [[catoptrics]].<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}}</ref> [[Plutarch]]<ref>Plutarch, Parallel lives</ref> had claimed that Archimedes disdained mechanics and focused primarily on pure [[geometry]], but this is generally considered to be a mischaracterization by modern scholarship, fabricated to bolster Plutarch's own [[Platonism|Platonist]] values rather than to an accurate presentation of Archimedes,<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}}</ref> and, unlike his inventions, Archimedes' mathematical writings were little known in antiquity outside of the works of [[Alexandria|Alexandrian]] mathematicians.{{cn|date=April 2025}} The first comprehensive compilation was not made until {{circa|530{{nbsp}}AD}} by [[Isidore of Miletus]] in [[Byzantine]] [[Constantinople]],{{cn|date=April 2025}} while [[Eutocius of Ascalon|Eutocius]]' commentaries on Archimedes' works earlier in the same century opened them to wider readership for the first time.{{cn|date=April 2025}}

===Middle ages===
{{Expand section|date=April 2025}}<!--- This can be expanded from the ten volume work by Marshall Clagett on Archimedes in the middle ages --->
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> 

===Renaissance and early modern Europe===
[[File:Archimedes bronze coin.jpg|thumb|1612 drawing of a now-lost bronze coin depicting Archimedes]]
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> which 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> 

[[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> 

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> 

=== In modern mathematics ===
[[File:FieldsMedalFront.jpg|thumb|The [[Fields Medal]] carries a portrait of Archimedes.|190x190px]]
[[Gauss]]'s heroes were Archimedes and [[Isaac Newton|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> Likewise, [[Alfred North Whitehead]] said that "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> The historian of mathematics [[Reviel Netz]],<ref>Reviel Netz, William Noel, The Archimedes Codex: Revealing The Secrets of the World's Greatest Palimpsest</ref> echoing Whitehead's proclamation on [[Plato]] and [[philosophy]], said that "Western science is but a series of footnotes to Archimedes," calling him "the most important scientist who ever lived." and [[Eric Temple Bell]],<ref>E.T. Bell, Men of Mathematics, p 20.</ref> wrote that "Any list of the three 'greatest' mathematicians of all history would include the name of Archimedes. The other two usually associated with him are Newton and 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." 

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>

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>

===Cultural influence===
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 has also 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>

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>

==See also==
{{Portal|Biography|Mathematics|Physics}}
* [[Arbelos]]
* [[Archimedean point]]
* [[Archimedes number]]
* [[Archimedes paradox]]
* [[Methods of computing square roots]]
* [[Salinon]]
* [[Steam cannon]]
* [[Twin circles]]
* [[Zhang Heng]]

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

=== Citations ===
{{Reflist|25em}}

== References ==
=== Ancient testimony ===
* Plutarch, Life of Marcellus
* {{Cite web |title=Athenaeus, Deipnosophistae |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}}

=== Modern sources ===
* {{Cite encyclopedia|last=Acerbi|first=Fabio |editor1-first=Paul T|editor1-last=Keyser|editor2-first=John| editor2-last=Scarborough| date=2018| title=Hellenistic Mathematics | encyclopedia = Oxford Handbook of Science and Medicine in the Classical World | url=https://www.academia.edu/36286615 |access-date=2021-05-26| pages=268–292| doi=10.1093/oxfordhb/9780199734146.013.69| isbn=978-0-19-973414-6}}
* {{cite book |last1=Dijksterhuis |first1=E. J. (Eduard Jan) |title=Archimedes |date=1987 |publisher=Princeton, N.J. : Princeton University Press |isbn=978-0-691-08421-3 |url=https://archive.org/details/archimedes0000dijk_m6s3 |access-date=30 April 2025}}
* {{cite book |last1=Netz |first1=Reviel |title=A New History of Greek Mathematics |date=2022 |publisher=Cambridge University Press |isbn=978-1-108-83384-4}}

==Further reading==
{{EB1911 poster|Archimedes}}
*[[Marshall Clagett|Clagett, Marshall]]. 1964–1984. ''Archimedes in the Middle Ages'' 5 volumes. 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}}.
*[[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}}.
*[[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 }}

{{Archimedes}}
{{Ancient Greek mathematics}}
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[[Category:3rd-century BC Greek writers]]
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[[Category:Doric Greek writers]]
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[[Category:Fluid dynamicists]]
[[Category:Buoyancy]]
[[Category:280s BC births]]
[[Category:210s BC deaths]]
[[Category:Year of birth uncertain]]
[[Category:Year of death uncertain]]
[[Category:3rd-century BC Greek mathematicians]]
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