Everything about Archimedes totally explained
Archimedes of Syracuse (
Greek:
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 a statement 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. During his youth Archimedes may have studied 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'd to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a 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'd ordered him not to be harmed.
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's no reliable evidence that Archimedes uttered these words and they don't appear in the account given by Plutarch.
The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a
sphere and a
cylinder of the same height and diameter. Archimedes had proved that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. 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.
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 Archimedes' 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's said to have built in order to defend the city.
Discoveries and inventions
The most commonly related
anecdote about Archimedes tells how he invented a method for measuring the volume of an object with an irregular shape. According to
Vitruvius, a new crown in the shape of a
laurel wreath had been made for
King Hiero II, and Archimedes was asked to determine whether it was of solid
gold, or whether
silver had been added by a dishonest goldsmith. Archimedes had to solve the problem without damaging the crown, so he couldn't 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. For practical purposes water is incompressible, so the crown would displace an amount of water equal to its own volume. By dividing the weight of the crown by the volume of water displaced, its density could be obtained. The density of the crown would be lower if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he'd forgotten to dress, crying "
Eureka!" "I have found it!" (
Greek: "εύρηκα!")
The story about the golden crown doesn't 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.
While Archimedes didn't 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'll move the Earth." (
Greek: "δος μοι πα στω και ταν γαν κινάσω") 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.
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 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. Archimedes' machine was a device with a revolving screw shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a body of water into irrigation canals. The Archimedes screw is still in use today for pumping liquids and semifluid solids such as coal and grain.
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.
The
Claw of Archimedes is another weapon that he's 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.
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.
Cicero (106 BC–43 BC) mentions Archimedes briefly in his
dialogue De re publica, which portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse
c. 212 BC, General
Marcus Claudius Marcellus 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
Thales of Miletus and
Eudoxus of Cnidus. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by
Gaius Sulpicius Gallus to
Lucius Furius Philus, who described it thus:
This is a description of a
planetarium or
orrery.
Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled . 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 mechanisms 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.
The Archimedes Heat Ray - myth or reality?
The
2nd century AD historian
Lucian wrote that during the
Siege of Syracuse (
c. 214–212 BC), Archimedes repelled an attack by Roman soldiers with a
burning-glass. The device was used to focus sunlight on to the approaching ships, causing them to catch fire. This claim, sometimes called the "Archimedes heat ray", has been the subject of ongoing debate about its credibility since the Renaissance.
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.
A test of the Archimedes heat 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 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 may have aided combustion.
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 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. In order to catch fire, wood needs to reach its
flash point, which is around 300 degrees Celsius (570 °F).
When
MythBusters broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (or failed) because of the length of time and the ideal weather conditions required for combustion to occur. It was also pointed out that since Syracuse faces the sea towards the east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors.
MythBusters also pointed out that conventional weaponry, such as flaming arrows or bolts from a catapult, would have been a far easier way of setting a ship on fire at short distances.
Mathematics
While he's 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.”
Archimedes was able to use
infinitesimals in a way that's 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 (approximately 3.1429) and (approximately 3.1408). 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 (approximately 1.7320261) and less than (approximately 1.7320512). The actual value is approximately 1.7320508, 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'd grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results."
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 series that
summed to infinity with the
ratio :
» for the sphere, and
for the cylinder. The surface area is
for the sphere, and
for the cylinder (including its two bases), where
is the radius of the sphere and cylinder. The sphere has a volume and surface area that of the cylinder. A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.
On Conoids and Spheroids » This is 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 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, 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 » In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He achieves this by calculating the value of a geometric series that sums to infinity with the ratio .
(O)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 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 puzzle represents an example of an early problem in combinatorics.
» The origin of the puzzle's name is unclear, and it has been suggested it's taken from the Ancient Greek word for throat or gullet, stomachos (στόμαχος). Ausonius refers to the puzzle as Ostomachion, a Greek compound word formed from the roots of ὀστέον (osteon - bone) and μάχη (machē - fight). The puzzle is also known as the Loculus of Archimedes or Archimedes' Box.
Archimedes' cattle problem » 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 in 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. Amthor in 1880, and the answer is a very large number, approximately 7.760271.
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, 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 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 can't 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's 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.
Archimedes Palimpsest
The foremost document containing the work of Archimedes is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg realized that a 174-page goatskin parchment of prayers written in the 13th century AD was in fact a palimpsest: the text was written over erased older work, which he identified as copies, written in the 10th century AD, of previously unknown treatises 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 New York. 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 Walters Art Museum in Baltimore, Maryland, where it has been subjected to a range of modern tests including the use of ultraviolet and light to read the overwritten text.
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.
A movement for civic engagement targeting universal access to health care in the US state of Oregon has been named the "Archimedes Movement", headed by former Oregon Governor John Kitzhaber.
Further Information
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