Editing Archimedes (section)

==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 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 the 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 |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,<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]],"{{efn|{{langx|la|Noli turbare circulos meos}}; {{langx|el|μὴ μου τοὺς κύκλους τάραττε}})}} which otherwise 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 |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><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>