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| region = Classical Greek philosophy | | |||
| era = Ancient philosophy | | |||
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| '''Archimedes of Syracuse''' (]: {{polytonic|Άρχιμήδης}} ''c''. 287 BC – ''c.'' 212 BC) was an ] ], ] and ]. Although little is known of his life, he is regarded as one of the leading ]s in ]. In addition to making discoveries in the fields of ] and ], he is credited with designing ]s that were innovative. He laid the foundations of ], and explained the principle of the ], the device on which ] is based. His early advances in ] included the first known ] of an ] with a method that is still used todayThe historians of ] showed a strong interest in Archimedes and wrote accounts of his life and works, while the relatively few copies of his treatises that survived through the ] were an influential source of ideas for scientists during the ].<ref>{{cite web | title = Galileo, Archimedes, and Renaissance engineers | |||
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| Archimedes died during the ], when he was killed by a Roman soldier despite orders that he should not be harmed. At his request, his tomb carried a carving of his favorite mathematical proof. Modern experiments have tested claims that he built a "death ray" capable of setting ships on fire at a distance, and that he constructed a device that could sink ships by lifting them out of the water. <ref>{{cite web | title = Archimedes Death Ray: Idea Feasibility Testing|author= | publisher = ]| url = ] is said to have remarked that Archimedes was one of the three {{nowrap|epoch-making}} mathematicians, with the others being ] and ].<ref>{{cite web | title = Review of ''Archimedes: What Did He Do Besides Cry Eureka?'' |author=Sandifer, Ed| publisher =] | url = http://www.maa.org/reviews/archim.html|accessdate=2007-07-23}}</ref> | |||
| image_name = Domenico-Fetti Archimedes 1620.jpg | | |||
| image_caption = ''Archimedes Thoughtful'' by ] (1620) | | |||
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| name = Archimedes of Syracuse (Greek: Άρχιμήδης) | | |||
| birth = ''c''. 287 BC (], ]) | | |||
| death = ''c''. 212 BC (Syracuse) | | |||
| school_tradition = ]<br /> ] | | |||
| main_interests = ], ], ], ]| | |||
| influences = | | |||
| influenced = | | |||
| notable_ideas = ], ]s, <br />] | | |||
| }} | |||
| '''Archimedes of Syracuse''' (]: {{polytonic|Άρχιμήδης}} ''c''. 287 BC – ''c.'' 212 BC) was an ] ], ] and ]. Although little is known of his life, he is regarded as one of the leading ]s in ]. In addition to making discoveries in the fields of ] and ], he is credited with designing ]s that were innovative. He laid the foundations of ], and explained the principle of the ], the device on which ] is based. His early advances in ] included the first known ] of an ] with a method that is still used today.<ref>{{cite web | title = A history of calculus |author=O'Connor, J.J. and Robertson, E.F. | publisher = ]| url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html |date= February 1996|accessdate= 2007-08-07}}</ref> | |||
| The historians of ] showed a strong interest in Archimedes and wrote accounts of his life and works, while the relatively few copies of his treatises that survived through the ] were an influential source of ideas for scientists during the ].<ref>{{cite web | title = Galileo, Archimedes, and Renaissance engineers |author=Bursill-Hall, Piers | publisher = sciencelive with the University of Cambridge| url = http://www.sciencelive.org/component/option,com_mediadb/task,view/idstr,CU-MMP-PiersBursillHall/Itemid,30|accessdate= 2007-08-07 }}</ref> | |||
| Archimedes died during the ], when he was killed by a Roman soldier despite orders that he should not be harmed. At his request, his tomb carried a carving of his favorite mathematical proof. Modern experiments have tested claims that he built a "death ray" capable of setting ships on fire at a distance, and that he constructed a device that could sink ships by lifting them out of the water. <ref>{{cite web | title = Archimedes Death Ray: Idea Feasibility Testing|author= | publisher = ]| url = http://web.mit.edu/2.009/www//experiments/deathray/10_ArchimedesResult.html|accessdate=2007-07-23}}</ref> The discovery of previously unknown works by Archimedes in the ] has provided new insights into how he obtained mathematical results. <ref>{{cite web | title = Archimedes - The Palimpsest|author=| publisher =] | url = http://www.archimedespalimpsest.org/palimpsest_making1.html|accessdate=2007-10-14}}</ref> | |||
| ] is said to have remarked that Archimedes was one of the three {{nowrap|epoch-making}} mathematicians, with the others being ] and ].<ref>{{cite web | title = Review of ''Archimedes: What Did He Do Besides Cry Eureka?'' |author=Sandifer, Ed| publisher =] | url = http://www.maa.org/reviews/archim.html|accessdate=2007-07-23}}</ref> | |||
| ==Biography== | ==Biography== | ||
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| Archimedes died ''c''. 212 BC during the ], when Roman forces under General ] captured the city of Syracuse after a two year long ]. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a {{nowrap|lesser-known}} account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he had ordered him not to be harmed.<ref name="death">{{cite web |first=Chris |last=Rorres | url = http://www.math.nyu.edu/~crorres/Archimedes/Death/Histories.html | title = Death of Archimedes: Sources | publisher = ] | accessdate = 2007-01-02 }}</ref> | Archimedes died ''c''. 212 BC during the ], when Roman forces under General ] captured the city of Syracuse after a two year long ]. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a {{nowrap|lesser-known}} account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he had ordered him not to be harmed.<ref name="death">{{cite web |first=Chris |last=Rorres | url = http://www.math.nyu.edu/~crorres/Archimedes/Death/Histories.html | title = Death of Archimedes: Sources | publisher = ] | accessdate = 2007-01-02 }}</ref> | ||
| The last words attributed to Archimedes are "Do not disturb my circles" (]: μή μου τούς κύκλους τάραττε), a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is often given in ] as "Noli turbare circulos meos", but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch.<ref name="death">{{cite web |first=Chris |last=Rorres | url = http://www.math.nyu.edu/~crorres/Archimedes/Death/Histories.html | title = Death of Archimedes: Sources | publisher = Courant Institute of Mathematical Sciences | accessdate = 2007-01-02 }}</ref> | |||
| {{listen | |||
| |filename=Archimedes circles.ogg | |||
| |title=Μη μου τους κύκλους τάραττε – ''Do not disturb my circles'' | |||
| |description=Listen to the last words attributed to Archimedes. | |||
| }} | |||
| ] | ] | ||
| The tomb of Archimedes had a carving of his favorite mathematical diagram, which was a ] inside a ] of the same height and diameter. Archimedes had proved that the volume and surface area of the sphere would be two thirds that of the cylinder. In 75 BC, 137 years after his death, the Roman ] ] was serving as ] in ]. He had heard stories about the tomb of Archimedes, but none of the locals was able to give him the location. Eventually he found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription.<ref>{{cite web |
The tomb of Archimedes had a carving of his favorite mathematical diagram, which was a ] inside a ] of the same height and diameter. Archimedes had proved that the volume and surface area of the sphere would be two thirds that of the cylinder. In 75 BC, 137 years after his death, the Roman ] ] was serving as ] in ]. He had heard stories about the tomb of Archimedes, but none of the locals was able to give him the location. Eventually he found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription.<ref>{{cite web | ||
| The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege of Syracuse given by ] in his ''Universal History'' was written around seventy years after Archimedes' death, and was used subsequently as a source by Plutarch and ]. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city.<ref>{{cite web |name | siege| first=Chris |last=Rorres | url = https://www.math.nyu.edu/~crorres/Archimedes/Siege/Polybius.html | title = Siege of Syracuse| publisher = Courant Institute of Mathematical Sciences | accessdate = 2007-07-23 }}</ref> | |||
| ==Discoveries and inventions== | ==Discoveries and inventions== | ||
| The most commonly related ] about Archimedes tells how he invented a method for measuring the volume of an object with an irregular shape. According to ], a new crown in the shape of a ] had been made for ], and Archimedes was asked to determine whether it was of solid ], or whether ] had been added by a dishonest goldsmith |
The most commonly related ] about Archimedes tells how he invented a method for measuring the volume of an object with an irregular shape. According to ], a new crown in the shape of a ] had been made for ], and Archimedes was asked to determine whether it was of solid ], or whether ] had been added by a dishonest goldsmith> Archimedes had to solve the problem without damaging the crown, so he could not melt it down in order to measure its ] as a cube, which would have been the simplest solution. While taking a bath, he noticed that the level of the water rose as he got in. He realized that this effect could be used to determine the ] of the crown, and therefore its density after weighing it. The density of the crown would be lower if cheaper and less dense metals had been added. He then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "]!" "I have found it!" | ||
| The story about the golden crown does not appear in the known works of Archimedes, but in his treatise ''On Floating Bodies'' he gives the principle known in ] as ]. This states that a body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid | |||
| While Archimedes did not invent the ], he wrote the earliest known rigorous explanation of the principle involved. According to ], his work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." (] to lift objects that would otherwise have been too heavy to move.<ref>{{cite web | author=Dougherty, F. C.; Macari, J.; ] was operated by hand and could raise water efficiently.]] | |||
| The story about the golden crown does not appear in the known works of Archimedes, but in his treatise ''On Floating Bodies'' he gives the principle known in ] as ]. This states that a body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid.<ref>{{cite web | title = ''Archimedes' Principle''|first=Bradley W |last=Carroll |publisher=] | url =http://www.physics.weber.edu/carroll/Archimedes/principle.htm|accessdate=2007-07-23}}</ref> | |||
| A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer ] described how King Hieron II commissioned Archimedes to design a huge ship, the '']'', which could be used for luxury travel, carrying supplies, and as a naval warship. The ''Syracusia'' is said to have been the largest ship built in classical antiquity.<ref>{{cite book |last=Casson|first= Lionel|authorlink= |coauthors= |title=''Ships and Seamanship in the Ancient World'' |year=1971 |publisher= Princeton University Press |location= |isbn=0691035369 }}</ref> According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a ] and a temple dedicated to the goddess ] among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the ] was purportedly developed in order to remove the bilge water. The screw was a machine with a revolving screw shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a {{nowrap|low-lying}} body of water into irrigation canals. Versions of the Archimedes' screw are still in use today in developing countries. The Archimedes' screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the ].<ref>{{cite web | title = ''Sennacherib, Archimedes, and the Water Screw: The Context of Invention in the Ancient | |||
| The ] is another weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker", the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped on to an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it. There have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitled ''Superweapons of the Ancient World'' built a version of the claw and concluded that it was a workable device.<ref>{{cite web |first=Chris |last=Rorres | title = Archimedes' Claw - Illustrations and Animations - a range of possible designs for the claw| | |||
| While Archimedes did not invent the ], he wrote the earliest known rigorous explanation of the principle involved. According to ], his work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." (]: "δος μοι πα στω και ταν γαν κινάσω")<ref>Quoted by ] in ''Synagoge'', Book VIII</ref> Plutarch describes how Archimedes designed ] ] systems, allowing sailors to use the principle of ] to lift objects that would otherwise have been too heavy to move.<ref>{{cite web | author=Dougherty, F. C.; Macari, J.; Okamoto, C.|title = Pulleys |author= | publisher=] | url = http://www.swe.org/iac/lp/pulley_03.html|accessdate=2007-07-23}}</ref> | |||
| Archimedes has also been credited with improving the power and accuracy of the ], and with inventing the ] during the ]. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.<ref | |||
| ] was operated by hand and could raise water efficiently.]] | |||
| A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer ] described how King Hieron II commissioned Archimedes to design a huge ship, the '']'', which could be used for luxury travel, carrying supplies, and as a naval warship. The ''Syracusia'' is said to have been the largest ship built in classical antiquity.<ref>{{cite book |last=Casson|first= Lionel|authorlink= |coauthors= |title=''Ships and Seamanship in the Ancient World'' |year=1971 |publisher= Princeton University Press |location= |isbn=0691035369 }}</ref> According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a ] and a temple dedicated to the goddess ] among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the ] was purportedly developed in order to remove the bilge water. The screw was a machine with a revolving screw shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a {{nowrap|low-lying}} body of water into irrigation canals. Versions of the Archimedes' screw are still in use today in developing countries. The Archimedes' screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the ].<ref>{{cite web | title = ''Sennacherib, Archimedes, and the Water Screw: The Context of Invention in the Ancient World''|author=Dalley, Stephanie. Oleson, John Peter| publisher = ''Technology and Culture'' Volume 44, Number 1, January 2003 (PDF)| url =http://muse.jhu.edu/journals/technology_and_culture/toc/tech44.1.html|accessdate=2007-07-23}}</ref><ref>{{cite web | title = Archimedes Screw - Optimal Design|author=Rorres, Chris| publisher =Courant Institute of Mathematical Sciences | url =http://www.mcs.drexel.edu/~crorres/Archimedes/Screw/optimal/optimal.html |accessdate=2007-07-23}}</ref><ref>{{cite web | title = Watch an animation of an Archimedes' Screw|author=| publisher =] | url =http://commons.wikimedia.org/Image:Archimedes-screw_one-screw-threads_with-ball_3D-view_animated.gif|accessdate=2007-07-23}}</ref> | |||
| The ] is another weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker", the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped on to an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it. There have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitled ''Superweapons of the Ancient World'' built a version of the claw and concluded that it was a workable device.<ref>{{cite web |first=Chris |last=Rorres | title = Archimedes' Claw - Illustrations and Animations - a range of possible designs for the claw| publisher = Courant Institute of Mathematical Sciences | url = https://www.math.nyu.edu/~crorres/Archimedes/Claw/illustrations.html|accessdate=2007-07-23}}</ref><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|accessdate=2007-08-12}}</ref> | |||
| Archimedes has also been credited with improving the power and accuracy of the ], and with inventing the ] during the ]. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.<ref>{{cite web |first= |last= | url = http://www.tmth.edu.gr/en/aet/5/55.html| title = Ancient Greek Scientists: Hero of Alexandria | publisher = Technology Museum of Thessaloniki | accessdate = 2007-09-14 }}</ref> | |||
| ] (106 BC–43 BC) mentions Archimedes briefly in his ] '']'', which portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse ''c.'' 212 BC, General ] is said to have taken back to Rome two mechanisms used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Cicero mentions similar mechanisms designed by ] and ]. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by ] to ], who described it thus: | ] (106 BC–43 BC) mentions Archimedes briefly in his ] '']'', which portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse ''c.'' 212 BC, General ] is said to have taken back to Rome two mechanisms used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Cicero mentions similar mechanisms designed by ] and ]. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by ] to ], who described it thus: | ||
| ; When Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that position which was its shadow on the Earth, when the Sun was in line.<ref>{{cite web | title = ''De re publica'' 1.xiv §21|author= | |||
| This is a description of a ] or ]. ] stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled {{nowrap|'']''}}. Modern research in this area has been focused on the ], another device from classical antiquity that was probably designed for the same purpose. Constructing mechanisms of this kind would have required a sophisticated knowledge of ]. This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.<ref>{{cite web | title = Spheres and Planetaria |first=Chris |last=Rorres | publisher = Courant Institute of Mathematical Sciences | url = | |||
| This is a description of a ] or ]. ] stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled {{nowrap|'']''}}. Modern research in this area has been focused on the ], another device from classical antiquity that was probably designed for the same purpose. Constructing mechanisms of this kind would have required a sophisticated knowledge of ]. This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.<ref>{{cite web | title = Spheres and Planetaria |first=Chris |last=Rorres | publisher = Courant Institute of Mathematical Sciences | url = https://www.math.nyu.edu/~crorres/Archimedes/Sphere/SphereIntro.html|accessdate=2007-07-23}}</ref><ref>{{cite web | title = Ancient Moon 'computer' revisited|author= | publisher = BBC News|date = ], ]| url = http://news.bbc.co.uk/1/hi/sci/tech/6191462.stm|accessdate=2007-07-23}}</ref> | |||
| ==="Death ray"=== | ==="Death ray"=== | ||
| Archimedes may have used mirrors acting as a ] to burn ships attacking , Archimedes repelled an attack by Roman forces with a ].<ref>''Hippias'', The device was used to focus sunlight on to the approaching ships, causing them to catch fire. This claim, sometimes called the "Archimedes death ray", has been the subject of ongoing debate about its credibility since the Renaissance. ] rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes | |||
| ] to burn ships attacking ]]] | |||
| ] wrote that during the ] (''c.'' 214–212 BC), Archimedes repelled an attack by Roman forces with a ].<ref>''Hippias'', C.2.</ref> The device was used to focus sunlight on to the approaching ships, causing them to catch fire. This claim, sometimes called the "Archimedes death ray", has been the subject of ongoing debate about its credibility since the Renaissance. ] rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes.<ref>{{cite web |author=] |last= | url = http://wesley.nnu.edu/john_wesley/wesley_natural_philosophy/duten12.htm| title = ''A Compendium of Natural Philosophy'' (1810) Chapter XII, ''Burning Glasses'' | publisher = Online text at Wesley Center for Applied Theology | accessdate = 2007-09-14 }}</ref> | |||
| It has been suggested that a large array of highly polished ] or ] shields acting as mirrors could have been employed to focus sunlight on to a ship. This would have used the principle of the ] in a manner similar to a ]. | |||
| In October 2005 a group of students from the ] carried out an experiment with 127 one-foot (30 cm) square mirror tiles, focused on a {{nowrap|mocked-up}} wooden ship at a range of around 100 feet (30 m). Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. It was concluded that the weapon was a feasible device under these conditions. The MIT group repeated the experiment for the television show '']'', using a wooden fishing boat in ] as the target. Again some charring occurred, along with a small amount of flame. When ''Mythbusters'' broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of ']' (a slang expression used here in the sense of 'invalidated') because of the length of time and ideal weather conditions required for combustion to occur. <ref>{{cite web | title = Archimedes Death Ray: Testing with MythBusters|author= | publisher = MIT| url = http://web.mit.edu/2.009/www//experiments/deathray/10_Mythbusters.html|accessdate=2007-07-23}}</ref> Critics of the MIT experiments have argued that the moisture content of the wood needs to be taken into consideration. However, the ] of wood is around 300 degrees Celsius (572 degrees F), and this is hotter than the maximum temperature produced by domestic ovens.<ref>{{cite web | title = How Wildfires Work|author= Bonsor, Kevin| publisher = ]| url = http://science.howstuffworks.com/wildfire.htm|accessdate=2007-07-23}}</ref> | |||
| A similar test of the "Archimedes death ray" was carried out in 1973 by the Greek scientist Ioannis Sakkas. The experiment took place at the ] naval base outside ]. On this occasion 70 mirrors were used, each with a copper coating and a size of around five by three feet (1.5 by 1 m). The mirrors were pointed at a plywood {{nowrap|mock-up}} of a Roman warship at a distance of around 160 feet (50 m). When the mirrors were focused accurately, the ship burst into flames within a few seconds. The plywood ship had a coating of ] paint, which is flammable and may have aided combustion.<ref>{{cite web | title = Archimedes' Weapon| publisher = ]|date = ], ]| url = http://www.time.com/time/magazine/article/0,9171,908175,00.html?promoid=googlep|accessdate=2007-08-12}}</ref> | |||
| ==Mathematics== | ==Mathematics== | ||
| While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. ] wrote: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of |
While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. ] wrote: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of lifeArchimedes was able to use ] in a way that is similar to modern ]. By assuming a proposition to be true and showing that this would lead to a ], he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the ], and he employed it to approximate the value of ] (Pi). He did this by drawing a larger ] outside a ] and a smaller polygon inside the circle. As the number of sides of the polygon increases, it becomes a more accurate approximation of a circle. When the polygons had 96 sides each, he calculated the lengths of their sides and showed that the value of π lay between {{nowrap|3 + 1/7}} (approximately 3.1429) and {{nowrap|3 + 10/71}} (approximately 3.1408). He also proved that the ] of a circle was equal to π multiplied by the ] of the ] of the circle. | ||
| ] to approximate the value of ].]] | |||
| Archimedes was able to use ] in a way that is similar to modern ]. By assuming a proposition to be true and showing that this would lead to a ], he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the ], and he employed it to approximate the value of ] (Pi). He did this by drawing a larger ] outside a ] and a smaller polygon inside the circle. As the number of sides of the polygon increases, it becomes a more accurate approximation of a circle. When the polygons had 96 sides each, he calculated the lengths of their sides and showed that the value of π lay between {{nowrap|3 + 1/7}} (approximately 3.1429) and {{nowrap|3 + 10/71}} (approximately 3.1408). He also proved that the ] of a circle was equal to π multiplied by the ] of the ] of the circle. | |||
| In ''The Measurement of a Circle'', Archimedes gives the value of the ] of 3 as being more than {{nowrap|265/153}} (approximately 1.732) and less than {{nowrap|1351/780}} (approximately 1.7320512). The actual value is around 1.7320508076, making this a very accurate estimate. He introduced this result without offering any explanation of the method used to obtain it. This aspect of the work of Archimedes caused ] to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his |
In ''The Measurement of a Circle'', Archimedes gives the value of the ] of 3 as being more than {{nowrap|265/153}} (approximately 1.732) and less than {{nowrap|1351/780}} (approximately 1.7320512). The actual value is around 1.7320508076, making this a very accurate estimate. He introduced this result without offering any explanation of the method used to obtain it. This aspect of the work of Archimedes caused ] to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his resultsIn '']'', Archimedes proved that the area enclosed by a ] and a straight line is {{nowrap|4/3}} multiplied by the area of a ] with equal base and height. He expressed the solution to the problem as a ] that summed to ] with the ] {{nowrap|1/4}}: | ||
| <div style="float:right;padding:5px;text-align:center">]<br /></div> | |||
| In '']'', Archimedes proved that the area enclosed by a ] and a straight line is {{nowrap|4/3}} multiplied by the area of a ] with equal base and height. He expressed the solution to the problem as a ] that summed to ] with the ] {{nowrap|1/4}}: | |||
| :<math>\begin{smallmatrix} \sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. \; \end{smallmatrix}</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 ]s, and so on. This proof is a variation of the ] {{nowrap|]}} which sums to {{nowrap|1/3}}. | If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller ]s, and so on. This proof is a variation of the ] {{nowrap|]}} which sums to {{nowrap|1/3}}. | ||
| In '']'', Archimedes set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote: "There are some, King Gelo (Gelo II, son of ]), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited." To solve the problem, Archimedes devised a system of counting based around the ]. The word is based on the Greek for uncountable, ''murious'', and was also used to denote the number 10,000. He proposed a number system using powers of myriad myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8{{e|63}} in modern notation.<ref>{{cite web | title = |
In '']'', Archimedes set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote: "There are some, King Gelo (Gelo II, son of ]), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited." To solve the problem, Archimedes devised a system of counting based around the ]. The word is based on the Greek for uncountable, ''murious'', and was also used to denote the number 10,000. He proposed a number system using powers of myriad myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8{{e|63}} in modern notation.<ref>{{cite web | title ==Writings == | ||
| ==Writings == | |||
| ] | ] | ||
| * ''On the Equilibrium of Planes'' (two volumes) | * ''On the Equilibrium of Planes'' (two volumes) | ||
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| {{cquote|Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.}} | {{cquote|Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.}} | ||
| :Archimedes uses the principles derived to calculate the areas and ] of various geometric figures including ], ]s, and ]. <ref name="works">{{cite web |first= |last=Heath,T.L. | url = |
:Archimedes uses the principles derived to calculate the areas and ] of various geometric figures including ], ]s, and ]. <ref name="works">{{cite web |first= |last=Heath,T.L. | url = * ''On the Measurement of the Circle'' | ||
| * ''On the Measurement of the Circle'' | |||
| :This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of ]. In Proposition II, Archimedes shows that the value of ] (Pi) is greater than {{nowrap|223/71}} and less than {{nowrap|22/7}}. The latter figure was used as an approximation of π throughout the Middle Ages and is still used today when a rough figure is required. | :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 ]. In Proposition II, Archimedes shows that the value of ] (Pi) is greater than {{nowrap|223/71}} and less than {{nowrap|22/7}}. The latter figure was used as an approximation of π throughout the Middle Ages and is still used today when a rough figure is required. | ||
| * ''On Spirals'' | * ''On Spirals'' | ||
| :This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the ]. It is the ] 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 ]. Equivalently, in ] (''r'', θ) it can be described by the equation | :This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the ]. It is the ] 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 ]. Equivalently, in ] (''r'', θ) it can be described by the equation | ||
| ::<math>\begin{smallmatrix}r\ =\ a\ +\ b\theta\end{smallmatrix}</math> | |||
| :with ] ''a'' and ''b''. This is an early example of a ] (a curve traced by a moving ]) considered by a Greek mathematician. | :with ] ''a'' and ''b''. This is an early example of a ] (a curve traced by a moving ]) considered by a Greek mathematician. | ||
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| It has also been claimed by the Arab scholar Abu'l Raihan Muhammed al-Biruni that ] for calculating the area of a triangle from the length of its sides was known to Archimedes.{{Ref_label|C|c|none}} However, the first reliable reference to the formula is given by ] in the 1st century AD.<ref>{{cite web |first=James W |last=Wilson | url = http://jwilson.coe.uga.edu/emt725/Heron/Heron.html | title = Problem Solving with Heron's Formula| publisher = ] | accessdate = 2007-09-14 }}</ref> | It has also been claimed by the Arab scholar Abu'l Raihan Muhammed al-Biruni that ] for calculating the area of a triangle from the length of its sides was known to Archimedes.{{Ref_label|C|c|none}} However, the first reliable reference to the formula is given by ] in the 1st century AD.<ref>{{cite web |first=James W |last=Wilson | url = http://jwilson.coe.uga.edu/emt725/Heron/Heron.html | title = Problem Solving with Heron's Formula| publisher = ] | accessdate = 2007-09-14 }}</ref> | ||
| ==Archimedes Palimpsest== | |||
| ] in the ]]] | |||
| {{main|Archimedes Palimpsest}} | |||
| The written work of Archimedes has not survived as well as that of ], and seven of his treatises are known to exist only through references made to them by other authors. ] mentions '']'' and another work on ], while ] quotes a remark about ] from the {{nowrap|now-lost}} ''Catoptrica''.{{Ref_label|B|b|none}} The writings of Archimedes were collected by the ] architect ] (''c''. 530 AD), while translations into Arabic and Latin made during the Middle Ages helped to keep his work alive. Archimedes' work was translated into Arabic by ] (836–901 AD), and Latin by ] (''c.'' 1114–1187 AD). During the ], the ''Editio Princeps'' (First Edition) was published in ] in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin.<ref>{{cite web | title = Editions of Archimedes' Work|author= | publisher = Brown University Library| url = http://www.brown.edu/Facilities/University_Library/exhibits/math/wholefr.html|accessdate=2007-07-23}}</ref> Around the year 1586 ] invented a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes.<ref>{{cite web | title = The Galileo Project: Hydrostatic Balance|author=Van Helden, Al | publisher = ]| url = http://galileo.rice.edu/sci/instruments/balance.html|accessdate=2007-09-14}}</ref> | |||
| The foremost document containing the work of Archimedes is the ]. A ] is a document written on ] that has been {{nowrap|re-used}} by scraping off the ink of an older text and writing new text in its place. This was often done during the Middle Ages since animal skin parchments were expensive. In 1906, the Danish professor ] realized that a goatskin parchment of prayers written in the 13th century AD also carried an older work written in the 10th century AD, which he identified as previously unknown copies of works by Archimedes. The parchment spent hundreds of years in a monastery library in ] before being sold to a private collector in the 1920s. On ], ] it was sold at auction to an anonymous buyer for $2 million at ] in ]. 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 Suidas and thought to have been lost forever. ''Stomachion'' was also discovered in the palimpsest, with a more complete analysis of the puzzle than had been found in previous texts. The palimpsest is now stored at the ] in ], ], where it has been subjected to a range of modern tests including the use of ] and {{nowrap|]}} ] to read the overwritten text.<ref>{{cite web | title = X-rays reveal Archimedes' secrets|author= | publisher = BBC News|date = ], ]| url = http://news.bbc.co.uk/1/hi/sci/tech/5235894.stm|accessdate=2007-07-23}}</ref> | |||
| The treatises in the Archimedes Palimpsest are: ''On the Equilibrium of Planes, On Spirals, The Measurement of the Circle, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems'' and ''Stomachion''. | The treatises in the Archimedes Palimpsest are: ''On the Equilibrium of Planes, On Spirals, The Measurement of the Circle, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems'' and ''Stomachion''. | ||
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| The exclamation of ] attributed to Archimedes is the state motto of ]. In this instance the word refers to the discovery of gold near ] in 1848 which sparked the ].<ref>{{cite web | title = California Symbols|author= | publisher =California State Capitol Museum| url =http://www.capitolmuseum.ca.gov/VirtualTour.aspx?content1=1278&Content2=1374&Content3=1294|accessdate=2007-09-14}}</ref> | The exclamation of ] attributed to Archimedes is the state motto of ]. In this instance the word refers to the discovery of gold near ] in 1848 which sparked the ].<ref>{{cite web | title = California Symbols|author= | publisher =California State Capitol Museum| url =http://www.capitolmuseum.ca.gov/VirtualTour.aspx?content1=1278&Content2=1374&Content3=1294|accessdate=2007-09-14}}</ref> | ||
| ==See also== | |||
| <div style="-moz-column-count:2; column-count:2;"> | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| * ] | |||
| </div> | |||
| ==Notes and references== | |||
| ===Notes=== | |||
| '''a.''' {{Note_label|A|a|none}}In the preface to ''On Spirals'' addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death." ] lived {{nowrap|''c.'' 280–220 BC}}, suggesting that Archimedes may have been an older man when writing some of his works. | |||
| '''b.''' {{Note_label|B|b|none}}The treatises by Archimedes known to exist only through references in the works of other authors are: '']'' and a work on polyhedra mentioned by Pappus of Alexandria; ''Catoptrica'', a work on optics mentioned by ]; ''Principles'', addressed to Zeuxippus and explaining the number system used in '']''; ''On Balances and Levers''; ''On Centers of Gravity''; ''On the Calendar''. Of the surviving works by Archimedes, ] offers the following suggestion as to the order in which they were written: ''On the Equilibrium of Planes I'', ''The Quadrature of the Parabola'', ''On the Equilibrium of Planes II'', ''On the Sphere and the Cylinder I, II'', ''On Spirals'', ''On Conoids and Spheroids'', ''On Floating Bodies I, II'', ''On the Measurement of a Circle'', ''The Sand Reckoner''. | |||
| '''c.''' {{Note_label|C|c|none}}] ''A History of Mathematics'' (1991) ISBN 0471543977 "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula—k=sqrt(s(s-a)(s-b)(s-c)), where s is the semiperimeter—was known to Archimedes several centuries before Heron lived. Arabic scholars also attribute to Archimedes the 'theorem on the broken ]' Archimedes is reported by the Arabs to have given several proofs of the theorem." | |||
| ===References=== | |||
| <div class="references-small" style="-moz-column-count:2; column-count:2;"> | |||
| <references/> | |||
| </div> | |||
| ==Further reading== | |||
| *{{cite book |last=] |first= |authorlink= |coauthors= |title=''A History of Mathematics'' |year=1991|publisher= Wiley|location= New York|isbn=0-471-54397-7 }} | |||
| *{{cite book |last=] |first= |authorlink= |coauthors= |title=''Archimedes'' |year=1987 |publisher= Princeton University Press, Princeton|location= |isbn=0-691-08421-1 }} Republished translation of the 1938 study of Archimedes and his works by an historian of science. | |||
| *{{cite book |last=Gow |first=Mary |authorlink= |coauthors= |title=''Archimedes: Mathematical Genius of the Ancient World'' |year=2005|publisher=Enslow Publishers, Inc |location= |isbn=0-7660-2502-0 }} | |||
| *{{cite book |last=Hasan |first=Heather |authorlink= |coauthors= |title=''Archimedes: The Father of Mathematics'' |year= 2005|publisher=Rosen Central |location= |isbn=978-1404207745 }} | |||
| *{{cite book |last=] |first= |authorlink= |coauthors= |title=''Works of Archimedes'' |year=1897 |publisher=Dover Publications |location= |isbn=0-486-42084-1 }} Complete works of Archimedes in English. | |||
| *{{cite book |last=Netz, Reviel and Noel, William |first= |authorlink= |coauthors= |title=''The Archimedes Codex'' |year=2007|publisher=Orion Publishing Group|location= |isbn= 0-297-64547-1 }} | |||
| *{{cite book |last=Simms, Dennis L. |first= |authorlink= |coauthors= |title=''Archimedes the Engineer'' |year=1995 |publisher= Continuum International Publishing Group Ltd |location= |isbn=0-720-12284-8 }} | |||
| *{{cite book |last=Stein, Sherman |first= |authorlink= |coauthors= |title=''Archimedes: What Did He Do Besides Cry Eureka?'' |year=1999 |publisher= Mathematical Association of America|location= |isbn=0-88385-718-9 }} | |||
| == External links == | |||
| {{Commonscat|Archimedes}} | |||
| {{wikiquote}} | |||
| *—BBC Radio 4 discussion from '']'', broadcast ], ] (requires ]) | |||
| * | |||
| * | |||
| * | |||
| * (Text in Classical Greek) | |||
| * | |||
| {{Greek mathematics}} | |||
| {{Ancient_Greece}} | |||
| <!-- Metadata: see ] --> | |||
| {{Persondata | |||
| |NAME=Archimedes | |||
| |ALTERNATIVE NAMES= | |||
| |SHORT DESCRIPTION=ancient Greek mathematician, physicist and engineer | |||
| |DATE OF BIRTH=circa 287 BC | |||
| |PLACE OF BIRTH=], ] | |||
| |DATE OF DEATH=circa 212 BC | |||
| |PLACE OF DEATH=], ] | |||
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Revision as of 01:07, 23 October 2007
Archimedes of Syracuse (Greek: Template:Polytonic c. 287 BC – c. 212 BC) was an ancient Greek mathematician, physicist and engineer. Although little is known of his life, he is regarded as one of the leading scientists in classical antiquity. In addition to making discoveries in the fields of mathematics and geometry, he is credited with designing machines that were innovative. He laid the foundations of hydrostatics, and explained the principle of the lever, the device on which mechanics is based. His early advances in calculus included the first known summation of an infinite series with a method that is still used todayThe historians of Ancient Rome showed a strong interest in Archimedes and wrote accounts of his life and works, while the relatively few copies of his treatises that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance.Cite error: A <ref> tag is missing the closing </ref> (see the help page).
Biography
Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a colony of Magna Graecia. The date of birth is based on an assertion by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years. In The Sand Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse. A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children. Archimedes probably spent part of his youth in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene were contemporaries. He referred to Conon of Samos as his friend, while two of his works (The Sand Reckoner and the Cattle Problem) have introductions addressed to Eratosthenes.
Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two year long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he had ordered him not to be harmed.
The tomb of Archimedes had a carving of his favorite mathematical diagram, which was a sphere inside a cylinder of the same height and diameter. Archimedes had proved that the volume and surface area of the sphere would be two thirds that of the cylinder. In 75 BC, 137 years after his death, the Roman orator Cicero was serving as quaestor in Sicily. He had heard stories about the tomb of Archimedes, but none of the locals was able to give him the location. Eventually he found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription.Cite error: A <ref> tag is missing the closing </ref> (see the help page). According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes' Screw was purportedly developed in order to remove the bilge water. The screw was a machine with a revolving screw shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a low-lying body of water into irrigation canals. Versions of the Archimedes' screw are still in use today in developing countries. The Archimedes' screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon.Cite error: A <ref> tag is missing the closing </ref> (see the help page).
- This work was discovered by Gotthold Ephraim Lessing in a Greek manuscript consisting of a poem of 44 lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773. It is addressed to Eratosthenes and the mathematicians at the University of Alexandria. Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations. There is a more difficult version of the problem in which some of the answers are required to be square numbers. This version of the problem was first solved by a computer in 1965, and the answer is a very large number, approximately 7.760271×10.
- In this treatise, Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos (concluding that "this is impossible"), contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8×10 in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner or Psammites is the only surviving work in which Archimedes discusses his views on astronomy.
- The Method of Mechanical Theorems
- This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.
Apocryphal works
Archimedes' Book of Lemmas or Liber Assumptorum is a treatise with fifteen propositions on the nature of circles. The earliest known copy of the text is in Arabic. The scholars T. L. Heath and Marshall Clagett argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. The Lemmas may be based on an earlier work by Archimedes that is now lost.
It has also been claimed by the Arab scholar Abu'l Raihan Muhammed al-Biruni that Heron's formula for calculating the area of a triangle from the length of its sides was known to Archimedes. However, the first reliable reference to the formula is given by Heron of Alexandria in the 1st century AD.
The treatises in the Archimedes Palimpsest are: On the Equilibrium of Planes, On Spirals, The Measurement of the Circle, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems and Stomachion.
Legacy
There is a crater on the Moon named Archimedes (29.7° N, 4.0° W) in his honor, and a lunar mountain range, the Montes Archimedes (25.3° N, 4.6° W). The asteroid 3600 Archimedes is named after him.
The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with his proof concerning the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to him which reads in Latin: "Transire suum pectus mundoque potiri" (Rise above oneself and grasp the world).
Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982) and Spain (1963).
The exclamation of Eureka! attributed to Archimedes is the state motto of California. In this instance the word refers to the discovery of gold near Sutter's Mill in 1848 which sparked the California gold rush.
- T. L. Heath, Works of Archimedes, 1897
- Plutarch. "Parallel Lives Complete e-text from Gutenberg.org". Project Gutenberg. Retrieved 2007-07-23.
{{cite web}}: Text "lives" ignored (help); Text "name" ignored (help) - O'Connor, J.J. and Robertson, E.F. "Archimedes of Syracuse". University of St Andrews. Retrieved 2007-01-02.
{{cite web}}: Text "andrews" ignored (help); Text "name" ignored (help)CS1 maint: multiple names: authors list (link) - Rorres, Chris. "Death of Archimedes: Sources". Courant Institute of Mathematical Sciences. Retrieved 2007-01-02.
- Rorres, Chris. "Archimedes' Stomachion". Courant Institute of Mathematical Sciences. Retrieved 2007-09-14.
- Calkins, Keith G. "Archimedes' Problema Bovinum". Andrews University. Retrieved 2007-09-14.
- "English translation of The Sand Reckoner". University of Waterloo. Retrieved 2007-07-23.
- "Archimedes' Book of Lemmas". cut-the-knot. Retrieved 2007-08-07.
- Wilson, James W. "Problem Solving with Heron's Formula". University of Georgia. Retrieved 2007-09-14.
- Friedlander, Jay and Williams, Dave. "Oblique view of Archimedes crater on the Moon". NASA. Retrieved 2007-09-13.
{{cite web}}: CS1 maint: multiple names: authors list (link) - "Planetary Data System". NASA. Retrieved 2007-09-13.
- "Fields Medal". International Mathematical Union. Retrieved 2007-07-23.
- Rorres, Chris. "Stamps of Archimedes". Courant Institute of Mathematical Sciences. Retrieved 2007-08-25.
- "California Symbols". California State Capitol Museum. Retrieved 2007-09-14.