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 Archimedes

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كاتب الموضوعرسالة
ام الياس
عضو محترف
عضو محترف
ام الياس


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مُساهمةموضوع: Archimedes   Archimedes Emptyالجمعة فبراير 06, 2009 11:50 pm

Archimedes
of Syracuse
(Greek: ρχιμήδης
) (c.
287 BC – c. 212 BC)
Archimedes Archimedes-OHe was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are
known, he is regarded as one of the leading scientists in classical antiquity.
Among his advances in physics are the foundations of
hydrostatics, statics and the explanation of the principle of
the lever. He is credited with designing innovative machines, including siege engines and the screw pump that
bears his name. Modern experiments have tested claims that Archimedes designed
machines capable of lifting attacking ships out of the water and setting ships
on fire using an array of mirrors.[1]


Archimedes
is generally considered to be the greatest mathematician of antiquity and one of the greatest
of all time.[2][3] He used the method of exhaustion
to calculate the area under the arc of a parabola with the summation of an infinite
series
, and gave a remarkably accurate approximation of pi.[4] He also defined the spiral bearing his name, formulas for the volumes of surfaces of revolution
and an ingenious system for expressing very large numbers.


Archimedes
died during the Siege of
Syracuse
when he was killed by a Roman soldier despite orders that he should not be
harmed. Cicero describes visiting the tomb of Archimedes,
which was surmounted by a sphere inscribed within a cylinder.
Archimedes had proven that the sphere has two thirds of the volume and surface
area of the cylinder (including the bases of the latter), and regarded this as
the greatest of his mathematical achievements.


Unlike
his inventions, the mathematical writings of Archimedes were little known in
antiquity. Mathematicians from Alexandria read and quoted
him, but the first comprehensive compilation was not made until c.
AD 530 by Isidore of Miletus,
while commentaries on the works of Archimedes written by Eutocius in the
sixth century AD opened them to wider readership for the first time. The
relatively few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas
for scientists during the Renaissance,[5] while the discovery in
1906 of previously unknown works by Archimedes in the Archimedes Palimpsest
has provided new insights into how he obtained mathematical results.[6]



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ام الياس
عضو محترف
عضو محترف
ام الياس


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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.[7] 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.[8] A biography of Archimedes was written by
his friend Heracleides but this work has been lost, leaving the details of his
life obscure.[9] 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 Method of
Mechanical Theorems
and the Cattle Problem)
have introductions addressed to Eratosthenes.[a]


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 considered him a valuable scientific asset and had ordered that he not be
harmed.[10]


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.[10]


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.[11]


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 is said to have
built in order to defend the city.[12]
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ام الياس
عضو محترف
عضو محترف
ام الياس


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مُساهمةموضوع: رد: Archimedes   Archimedes Emptyالجمعة فبراير 06, 2009 11:55 pm

Discoveries and inventions


The Golden Crown


The most
widely known anecdote about Archimedes tells how he invented a
method for determining 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.[13] Archimedes had to solve the problem
without damaging the crown, so he could not melt it down into a regularly
shaped body in order to calculate its density. While taking a bath, he noticed that the
level of the water in the tub rose as he got in, and realized that this effect
could be used to determine the volume of the crown. For
practical purposes water is incompressible,[14] so the submerged 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, the density of the crown could be obtained.
The density of the crown would be lower than that of gold if cheaper and less
dense metals had been added. Archimedes then took to the streets naked, so
excited by his discovery that he had forgotten to dress, crying "Eureka!" (Greek: "ε
ρηκα!," meaning "I
have found it!")[15]



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.[16]



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:
δς μοι π στ κα τν γν κινάσω)[17] 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.[18]



The
Archimedes' Screw


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.[19] 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 low-lying
body of water into irrigation canals. The Archimedes' screw is still in use
today for pumping liquids and granulated 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
.[20][21][22]


The Claw of Archimedes



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. 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.[23][24]


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.[25] 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.[26] 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 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 may have aided combustion.[27]



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. In order to catch fire, wood needs to reach its flash point, which is around 300 degrees Celsius
(570 °F).[28]



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.[1]
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ام الياس
عضو محترف
عضو محترف
ام الياس


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مُساهمةموضوع: رد: Archimedes   Archimedes Emptyالجمعة فبراير 06, 2009 11:57 pm






Other inventions


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.[29]



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:





Hanc
sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in
aere illo quot diebus in ipso caelo succederet, ex quo et in caelo sphaera
solis fieret eadem illa defectio, et incideret luna tum in eam metam quae
esset umbra terrae, cum sol e regione. — 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.[30][31]





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 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 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.[32][33]


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


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). He also proved that the area
of a circle was equal to π multiplied by the square of the radius of the circle.


In Measurement of a Circle,
Archimedes gives the value of the square root of 3 as being more than 265/153
(approximately 1.7320261) and less than 1351/780 (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
had grudged posterity the secret of his method of inquiry while he wished to
extort from them assent to his results."[35]



In The Quadrature
of the Parabola
, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 times the area
of a corresponding inscribed triangle as shown in the
figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio 1/4:



Archimedes Clip_image001


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 uses a
variation of the 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 Gelo (Gelo II, son of
Hiero 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 on the myriad. The word is from the Greek μυριάς murias, for
the number 10,000. He proposed a number system using powers of a myriad of
myriads (100 million) and concluded that the number of grains of sand required
to fill the universe would be 8 vigintillion,
or 8×1063.[36]





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