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'Archimedes of Syracuse' (Greek: ''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 well ahead of their time. 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 today.^[1]
The 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.^[2]
The death of Archimedes occurred during the Siege of Syracuse, when he was killed by a Roman soldier despite orders that he should not be harmed. Carl Friedrich Gauss is said to have remarked that Archimedes was one of the three epoch-making mathematicians, with the others being Sir Isaac Newton and Ferdinand Eisenstein.^[3]

Contents

Biography

Discoveries and inventions

Mathematics

Writings by Archimedes

Apocryphal works

The Archimedes Palimpsest

Legacy

See also

Notes and references

Notes

References

Further reading

External links

Biography

Archimedes was born ''c''. 287 BC in the seaport city of Syracuse, Sicily, which was then a colony of Magna Graecia. The date of his birth is based on an assertion by the Byzantine Greek historian John Tzetzes that he lived for 75 years.^[4] 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 Hieron II, the ruler of Syracuse.^[5] A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure.^[6] 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'') contain introductions addressed to Eratosthenes. Archimedes used his correspondence with the scholars in Alexandria as a way of making his results known.
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.^[7]
The last words attributed to Archimedes are "Do not disturb my circles" (Greek: 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 Latin as "''Noli turbare circulos meos''", but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch.

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.^[8]
The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege of Syracuse given by Polybius in his ''Universal History'' was written around seventy years after his death, and was used subsequently as a source by Plutarch and Livy. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city.^[9]

Discoveries and inventions

The most commonly related anecdote about Archimedes tells how he discovered the principle of buoyancy. According to Vitruvius, a new crown in the shape of a laurel wreath had been made for King Hieron, and Archimedes was asked to determine whether it was of solid gold, or whether silver had been added by a dishonest goldsmith. ^[10] Archimedes had to solve the problem without damaging the crown, so he could not melt it down in order to measure its density 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 volume 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 "Eureka!" "I have found it!" (Greek: "εύρηκα!")^[11]
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 hydrostatics as Archimedes' Principle. This states that a body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid.^[12]

Another invention bearing his name is the Archimedes' screw, a machine with a revolving screw shaped blade inside a cylinder. It was turned by hand, and used to drain ships or 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.^[13][14][15]
While Archimedes did not invent the lever, he wrote the earliest known rigorous explanation of the principle involved. According to Pappus of Alexandria, his work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." (Greek: "δος μοι πα στω και ταν γαν κινάσω")^[16] Plutarch describes how Archimedes designed block and tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move.^[17]
A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of Naucratis described how King Hieron II commissioned Archimedes to design a huge ship, the ''Syracusia'', which could be used for luxury travel, carrying supplies, and as a naval warship. The ''Syracusia'' is said to have been the largest ship built in classical antiquity. According to Athenaeus, it was capable of carrying 600 people and contained garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite. 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.^[18]

Lucian wrote that during the Siege of Syracuse (''c.'' 214-212 BC), Archimedes repelled an attack by Roman forces with a burning glass.^[19] 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.^[20] René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes.
It has been suggested that a large array of highly polished bronze or copper shields acting as mirrors could have been employed to focus sunlight on to a ship. This would have used the principle of the parabolic reflector in a manner similar to a solar furnace. In October 2005 a group of students from the Massachusetts Institute of Technology carried out an experiment with 127 one foot (30 cm) square mirror tiles, focused on a 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 ''MythBusters'', using a wooden fishing boat in San Francisco 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 "busted" due to the length of time and ideal weather conditions required for combustion to occur. Critics of the MIT experiments have argued that the moisture content of the wood needs to be taken into consideration. However, the flash point of wood is around 300 degrees Celsius (572 degrees Fahrenheit), and this is hotter than the maximum temperature produced by domestic ovens.^[21][22][23]
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 Skaramagas naval base outside Athens. On this occasion 70 mirrors were used, each with a copper coating and a size of around five by three feet (1.5 by 1 m). The mirrors were pointed at a plywood 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 tar paint, which is flammable and may have aided combustion.^[24]
The Claw of Archimedes 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. As with the "Archimedes death ray" 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.^[25][26]
Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.^[27]
Cicero wrote that after the capture of Syracuse, General Marcellus took two mechanical devices back to Rome that were used as aids in astronomy. He credits Thales and Eudoxus of Cnidus with constructing these devices. The motions of the Sun, Moon and five planets were shown by one device, and it was demonstrated to Cicero some 150 years later by a man named Gallus. Cicero described the event as follows:
The device described by Cicero is a planetarium or orrery. Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the construction of these devices entitled ''On Sphere-Making''. Modern research in this area has been focused on the Antikythera mechanism, another device from classical antiquity that was probably designed for the same purpose. Constructing devices of this kind would have required a sophisticated knowledge of differential gearing. 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.^[28][29]

Mathematics

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.”^[30]

Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. By assuming a proposition to be true and showing that this would lead to a contradiction, he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π (Pi). He did this by drawing a larger polygon outside a circle, 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 3 + 1/7 (approximately 3.1429) and 3 + 10/71 (approximately 3.1408). This was a remarkable achievement, since the ancient Greek numerals did not use the positional notation system of today. He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle.
In ''The Measurement of a Circle'', Archimedes gives the value of the square root of 3 as being more than 265/153 (approximately 1.732) and less than 1351/780 (approximately 1.7320512). The modern 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 John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results."^[31]


In ''The Quadrature of the Parabola'', Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He expressed the solution to the problem as a geometric progression that summed to infinity with the ratio 1/4:
:$sum\_\{n=0\}^infty\; 4^\{-n\}\; =\; 1\; +\; 4^\{-1\}\; +\; 4^\{-2\}\; +\; 4^\{-3\}\; +\; cdots\; =\; \{4over\; 3\}\; ;\; .$
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 secant lines, and so on. This proof is a variation of the infinite series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.
In ''The Sand Reckoner'', Archimedes set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote: "There are some, King Gelon (Gelon II, son of Hieron II), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited." To solve the problem, Archimedes devised a system of counting based around the myriad. This was a word used to mean infinity, based on the Greek word for uncountable, ''murious''. The word myriad 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 in modern notation.^[32]

Writings by Archimedes

★ ''On the Equilibrium of Planes'' (Two volumes)
: The first book is in fifteen propositions with seven postulates, while the second book is in ten propositions. In this work Archimedes explains the ''Law of the Lever'', stating:
:Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, paraboloids, and hemispheres.

★ ''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 Conon of Samos. In Proposition II, Archimedes shows that the value of π (Pi) is greater than 223/71 and less than 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''
Main articles: Archimedean spiral

:This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in polar coordinates (''r'', θ) it can be described by the equation
:$,\; r=a+b\; heta$
:with real numbers ''a'' and ''b''. This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.

★ ''On the Sphere and the Cylinder'' (Two volumes)
:In this treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter. The volume is '4/3πr³' for the sphere, and '2πr³' for the cylinder; the surface area is '4πr²' for the sphere, and '6πr²' for the cylinder. Archimedes therefore shows that the sphere will have two thirds of the volume and surface area of the cylinder. A carving of this proof was used on the tomb of Archimedes at his request.

★ ''On Conoids and Spheroids''
:A work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.

★ ''On Floating Bodies'' (Two volumes)
:In the first part of this treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not self-gravitating, since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.

:In the second part, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, similar to the way that icebergs float. Archimedes' principle of buoyancy is given in the work, stated as follows:

★ ''The Quadrature of the Parabola''
Main articles: The Quadrature of the Parabola

:A work of 24 propositions addressed to Dositheus. In this treatise Archimedes proves by two methods that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He achieves this by calculating the value of a geometric progression that sums to infinity with the ratio 1/4.

★ ''Stomachion''
:This is a dissection puzzle similar to a Tangram, and the treatise describing it was found in more complete form in the Archimedes Palimpsest. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Research published by Dr. Reviel Netz of Stanford University in 2003 argued that Archimedes was attempting to determine how many ways the pieces of paper could be assembled into the shape of a square. The figure given by Dr. Netz is that the pieces can be made into a square in 17,152 ways. The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded. The ''Stomachion'' represents an example of an early problem in combinatorics. For reasons that are unclear, ''Stomachion'' is derived from the Greek word for stomach, στομάχ.^[33][34]

★ ''Archimedes' Cattle Problem''
Main articles: Archimedes' Cattle Problem

:This work was discovered by Gotthold Ephraim Lessing in a Greek manuscript containing 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.^[35]

★ ''The Sand Reckoner''
Main articles: The Sand Reckoner

: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 in modern notation. The introductory letter contains the information 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.^[36]

★ ''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 containing 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. Archimedes is quoted in the work, suggesting modification by another author. The ''Lemmas'' may be based on an earlier work by Archimedes that is now lost.^[37]
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.^[38]

The Archimedes Palimpsest

Main articles: Archimedes Palimpsest

The written work of Archimedes has not survived as well as that of Euclid, and seven of his treatises are known to exist only through references made to them by other authors. Pappus of Alexandria mentions ''On Sphere-Making'' and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the now-lost ''Catoptrica''. The writings of Archimedes were collected by the Byzantine architect Isidore of Miletus (''c''. 530 AD), while translations into Arabic and Latin made during the Middle Ages helped to keep his work alive.^[39] Archimedes' work was translated into Arabic by Thābit ibn Qurra (836-901 AD), and Latin by Gerard of Cremona (''c.'' 1114-1187 AD). During the Renaissance, the ''Editio Princeps'' (First Edition) was published in Basel in 1544 by Johann Herwagen and contained the works of Archimedes in Greek and Latin.^[40] Around the year 1586 Galileo Galilei invented a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes.^[41]
The foremost document containing Archimedes' work is the Archimedes Palimpsest. A palimpsest is a document written on vellum that has been 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 Johan Ludvig Heiberg realized that a goatskin parchment containing 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 Constantinople before being sold to a private collector in the 1920s. On October 29, 1998 it was sold at auction to an anonymous buyer for $2 million at Christie's in London. The palimpsest contains seven treatises, including the only surviving copy of ''On Floating Bodies'' in the original Greek. It contains 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, containing a more complete analysis of the puzzle than had been found in previous texts. The palimpsest is now stored at the Walters Art Museum in Baltimore, Maryland, where it has been subjected to a range of modern tests including the use of ultraviolet and x-ray light to read the overwritten text.^[42]
The treatises contained 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 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).^[43][44]
Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982) and Spain (1963). ^[45]
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.

See also

★ Archimedes' Axiom

★ Archimedes number

★ Archimedes Paradox

★ Archimedean property

★ Archimedean solid

★ Archimedes' use of infinitesimals

★ Zhang Heng

Notes and references

Notes

'a.' In the preface to ''On Spirals'' addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived ''c.'' 280-220 BC, suggesting that Archimedes may have been an older man when writing some of his works.
'b.' The treatises by Archimedes known to exist only through references in the works of other authors are: ''On Sphere-Making'' and a work on polyhedra mentioned by Pappus of Alexandria; ''Catoptrica'', a work on optics mentioned by Theon of Alexandria; ''Principles'', addressed to Zeuxippus and explaining the number system used in ''The Sand Reckoner''; ''On Balances and Levers''; ''On Centers of Gravity''; ''On the Calendar''. Of the surviving works by Archimedes, T. L. Heath offers the following suggestion as to the order in which they were written: ''On the Equilibrium of Planes I'', ''The Quadrature of the Parabola'', ''On the Equilibrium of Planes II'', ''On the Sphere and the Cylinder I, II'', ''On Spirals'', ''On Conoids and Spheroids'', ''On Floating Bodies I, II'', ''On the Measurement of a Circle'', ''The Sand Reckoner''.
'c.'Boyer, Carl Benjamin ''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 chord' [...] Archimedes is reported by the Arabs to have given several proofs of the theorem."

References

1. A history of calculus O'Connor, J.J. and Robertson, E.F.
2. Galileo, Archimedes, and Renaissance engineers Bursill-Hall, Piers
3. Review of ''Archimedes: What Did He Do Besides Cry Eureka?'' Sandifer, Ed
4. John Tzetzes. Chiliades, II.35,105.
5. ''Parallel Lives'' Complete e-text from Gutenberg.org Plutarch
6. Archimedes of Syracuse O'Connor, J.J. and Robertson, E.F.
7. Death of Archimedes: Sources
8. Tomb of Archimedes: Sources
9. Siege of Syracuse
10. ''De Architectura'', Book IX, paragraphs 9-12, text in English and Latin Vitruvius
11. Buoyancy
12. ''Archimedes' Principle''
13. ''Sennacherib, Archimedes, and the Water Screw: The Context of Invention in the Ancient World'' Dalley, Stephanie. Oleson, John Peter
14. Archimedes Screw - Optimal Design Rorres, Chris
15. Watch an animation of an Archimedes' Screw
16. Quoted by Pappus of Alexandria in ''Synagoge'', Book VIII
17. Pulleys Dougherty, F. C.; Macari, J.; Okamoto, C.
18. The Syracusia Ship Lahanas, Michael
19. ''Hippias'', C.2.
20. Archimedes and his Burning Mirrors Lahanas, Michael
21. How Wildfires Work Bonsor, Kevin
22. Archimedes Death Ray: Idea Feasibility Testing
23. Archimedes Death Ray: Testing with MythBusters
24. Archimedes' Weapon
25. Archimedes' Claw - Illustrations and Animations - a range of possible designs for the claw
26. Archimedes' Claw - watch an animation
27. Gears by Archimedes Lahanas, Michael
28. Spheres and Planetaria
29. Ancient Moon 'computer' revisited
30. Extract from ''Parallel Lives'' Plutarch
31. Quoted in T L Heath, ''Works of Archimedes'', Dover Publications, ISBN 0-486-42084-1.
32. The Sand Reckoner
33. In Archimedes' Puzzle, a New Eureka Moment Kolata, Gina
34. Stomachion
35. Archimedes' Cattle Problem
36. English translation of ''The Sand Reckoner''
37. Archimedes' Book of Lemmas
38. Heron's Formula
39. Archimedes Sources Lahanas, Michael
40. Editions of Archimedes' Work
41. Hydrostatic balance Van Helden, Albert
42. X-rays reveal Archimedes' secrets
43. Fields Medal
44. Fields Medal
45. Stamps of Archimedes

Further reading

★ Dijksterhuis, E.J., ''Archimedes'', 1987, Princeton University Press, Princeton, ISBN 0-691-08421-1. Republished translation of the 1938 study of Archimedes and his works by an historian of science.

★ Gow, Mary, ''Archimedes: Mathematical Genius of the Ancient World'', 2005, Enslow Publishers, Inc, ISBN 0-7660-2502-0

★ Hasan, Heather, ''Archimedes: The Father of Mathematics'', 2005, Rosen Central, ISBN 978-1404207745

★ Heath, T.L., ''Works of Archimedes'', 1897, Dover Publications, ISBN 0-486-42084-1. Complete works of Archimedes in English.

★ Netz, Reviel and Noel, William ''The Archimedes Codex'', 2007, Orion Publishing Group, ISBN 0-297-64547-1

★ Simms, Dennis L., ''Archimedes the Engineer'', 1995, Continuum International Publishing Group Ltd , ISBN 0-720-12284-8

★ Stein, Sherman, ''Archimedes: What Did He Do Besides Cry Eureka?'', 1999, Mathematical Association of America, ISBN 0-88385-718-9

External links

★ Archimedes—The Greek mathematician and his Eureka moments—BBC Radio 4 discussion from ''In Our Time'', broadcast January 25, 2007 (requires RealPlayer)

★ The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland

★ New techniques used to examine the Archimedes Palimpsest

★ NOVA program on the Archimedes Palimpsest

★ New commentary on Aristotle found by researchers in Archimedes Palimpsest BBC News April 27, 2007

★ Archimedes Home Page - Site maintained by Chris Rorres at the Courant Institute of Mathematical Sciences

★ Archimedes—The Golden Crown argues that Archimedes may have used a more subtle method than the one in the classic version of the story

★ ''Archimedes and the Crown'' Illustrated version of the story by Doug Craigen

★ The Mathematical Achievements and Methodologies of Archimedes

★ Article examining how Archimedes may have calculated the square root of 3

★ Archimedes' ''Quadrature Of The Parabola'' Translated by T. L. Heath

★ Archimedes' ''On The Measurement Of The Circle'' Translated by T. L. Heath

★ ''The Works Of Archimedes'' (1897) by T. L. Heath Download the unabridged work from Archive.org in PDF form (19 MB)

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★ Photograph of the Sakkas experiment in 1973

★ Diagram of the Claw of Archimedes

★ Archimedes and the Rhombicuboctahedron by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas

★ The Archimedes Water Clock and other examples of ancient technology

★ Archimedes—The Greatest Scientist Ever Article by Jürgen Schmidhuber