Archimedes
Archimedes of Syracuse
(Greek: ἈρχιμI®δης) |
Archimedes Thoughtful by Fetti (1620) |
Born | c. 287 BC
Syracuse, Sicily
Magna Graecia |
Died | c. 212 BC (aged around 75)
Syracuse |
Residence | Syracuse, Sicily |
Fields | Mathematics, Physics, Engineering, Astronomy, Invention |
Known for | Archimedes' Principle, Archimedes' screw, Hydrostatics,
Levers, Infinitesimals |
Archimedes of Syracuse (Greek: ἈρχιμI®δης;
c. 287 BC – c. 212 BC) 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 an 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.
Archimedes is generally considered to be the greatest mathematician of
antiquity and one of the greatest of all time. 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, formulae
for thevolumes 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. 530 AD 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.
Biography
This bronze statue of Archimedes is at the Archenhold Observatory in Berlin. It was sculpted by Gerhard Thieme and unveiled in 1972.
Archimedes was born c. 287 BC inthe seaport city
of Syracuse, Sicily,
at that time a self-governing colony in 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 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 killedArchimedes 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.
A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A
sphere and cylinder were placed on the tomb of Archimedes at his request.
The last words attributed to Archimedes are 'Do not disturb my
circles' (Greek: μI®
τI¬ραττε), 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 carried a sculpture illustrating his favorite
mathematical proof, consisting of a sphere and a cylinder of the same height
and diameter. Archimedes had proven 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 asquaestor 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 is said to have
built in order to defend the city.
Discoveries and inventions
The Golden Crown
Archimedes may have used his principle of buoyancy to determine whether the
golden crown was less dense than solid gold.
The most widely known anecdote about Archimedes tells of how he invented a
method for determining the volume of an object with an irregular shape.
According to Vitruvius, a votive crown for a temple
had been made for King Hiero II, who had supplied the
pure gold to be used, and Archimedes was asked to determine whether some silver
had been substituted by the dishonest goldsmith. Archimedes had to solve the
problem withoutdamaging 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, so the submerged
crown would displace an amount of water equal to its own volume. By dividing
the mass of the crown by the volume of water displaced, the density of the
crown could be obtained. This density 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: 'εὕρηκα '
meaning 'I have found it!'). The test was conducted successfully,
proving that silver had indeed been mixed in.
The story of the golden crown does not appear in the
known works of Archimedes. Moreover, the practicality of the method it
describes has been called into question, due to the extreme accuracy with which
one would have to measure the water displacement. Archimedes may have instead
sought a solution that applied the principle known in hydrostatics as Archimedes'
Principle, which he describes in his treatise On Floating Bodies. This
principle states that a body immersed in a fluid experiences a buoyant force
equal to the weight of the fluid it displaces. Using thisprinciple,
it would have been possible to compare the density of the golden crown to that
of solid gold by balancing the crown on a scale with a gold reference sample,
then immersing the apparatus in water. If the crown was less dense than gold,
it would displace more water due to its larger volume, and thus experience a
greater buoyant force than the reference sample. This difference in buoyancy
would cause the scale to tip accordingly. Galileo considered it 'probable
that this method is the same that Archimedes followed, since, besides being
very accurate, it is based on demonstrations found by Archimedes himself.'
The Archimedes Screw
The Archimedes screw can 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 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 thebilge 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.
The Claw of Archimedes
The Claw of Archimedes is a 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 onto 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.
The Archimedes Heat Ray – myth or reality?
Archimedes may have used mirrors acting collectively as a parabolic reflector
to burn ships attacking Syracuse.
The 2nd century AD author Lucian wrote that during the Siege of Syracuse (c.
214–212 BC), Archimedes destroyed enemy ships with fire. Centuries later, Anthemius ofTralles mentions
burning-glasses as Archimedes' weapon. The device, sometimes called the
'Archimedes heat ray', was used to focus sunlight onto approaching
ships, causing them to catch fire.
This purported weapon 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 onto 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.
In October 2005 a group of students from the Massachusetts Institute of
Technology carried out an experiment with 127 one-foot (30 cm) square mirrortiles, focused on a mock-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 device was a feasible weapon
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.
Other discoveries and inventions
While Archimedes did not invent the lever, he gave an
explanation of the principle involved in his work On the Equilibrium of Planes.
Earlier descriptions of the lever are found inthe
Peripatetic school of the followers of Aristotle, and are sometimes attributed
to Archytas.[29][30] According to Pappus
of Alexandria, Archimedes' work on levers caused him to remark: 'Give me a
place to stand on, and I will move the Earth.' (Greek: s¶ς
πᾶ s¶
καὶ τá½°ν γᾶν κινI¬σω)
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. 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 (106–43 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 LuciusFurius 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.
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.
Archimedes Palimpsest
Main article: Archimedes Palimpsest
Stomachion is a dissection puzzle in theArchimedes Palimpsest.
The foremost document containing the work of Archimedes is the Archimedes
Palimpsest. In 1906, the Danish professor Johan Ludvig
Heiberg visited Constantinople
and examined a 174-page goatskin parchment of prayers written in the 13th
century AD. He discovered that it was a palimpsest, a document with text that
had been written over an erased older work. Palimpsests were created by
scraping the ink from existing works and reusing them, which was a common
practice in the Middle Ages as vellum was expensive. The older works in the
palimpsest were identified by scholars as 10th century AD copies 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 ofultraviolet and x-ray light to
read the overwritten text.
The treatises in the Archimedes Palimpsest are: On the Equilibrium of Planes,
On Spirals, Measurement of a Circle, On the Sphere and the Cylinder, On
Floating Bodies, The Method of Mechanical Theorems and Stomachion.
Legacy
The Fields Medal carries a portrait of Archimedes.
There is a crater on the Moon named Archimedes (29.7° N, 4.0° W) in his honor,
as well as 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.
