Archimedes of Syracuse (Greek: Ἀρχιμήδης; 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.[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,
formulae 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. 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.[6]
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 in the 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.[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]
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: 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 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
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]
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.[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 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.[15]
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.[16]
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.[17] Using this principle, 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."[18]
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
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 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.[23][24]
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 of Tralles mentions burning-glasses as Archimedes'
weapon.[25] 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.[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 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.[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 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 autoignition temperature, which is
around 300 °C (570 °F).[28][29]
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]
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 in the Peripatetic school
of the followers of Aristotle, and are sometimes attributed to
Archytas.[30][31] 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: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω)[32]
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.[33]
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.[34]
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 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.[35][36]
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.[37][38]
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."[39]
Archimedes used the method of exhaustion to approximate the value of π.
Archimedes
was able to use infinitesimals in a way that is similar to modern
integral calculus. Through proof by contradiction (reductio ad
absurdum), 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 31⁄7 (approximately 3.1429) and 310⁄71 (approximately
3.1408), consistent with its actual value of approximately 3.1416. He
also proved that the area of a circle was equal to π multiplied by the
square of the radius of the circle. In On the Sphere and Cylinder,
Archimedes postulates that any magnitude when added to itself enough
times will exceed any given magnitude. This is the Archimedean property
of real numbers.[40]
In Measurement of a Circle,
Archimedes gives the value of the square root of 3 as lying between
265⁄153 (approximately 1.7320261) and 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."[41]
As
proven by Archimedes, the area of the parabolic segment in the upper
figure is equal to 4/3 that of the inscribed triangle in the lower
figure.
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:
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.[42]
Writings
The
works of Archimedes were written in Doric Greek, the dialect of ancient
Syracuse.[43] The written work of Archimedes has not survived as well
as that of Euclid, and seven of his treatises are known to have existed
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.[b]
During his lifetime, Archimedes made his work known through
correspondence with the mathematicians in Alexandria. The writings of
Archimedes were collected by the Byzantine architect Isidore of Miletus
(c. 530 AD), while commentaries on the works of Archimedes
written by Eutocius in the sixth century AD helped to bring his work a
wider audience. 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 with
the works of Archimedes in Greek and Latin.[44] 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.[45]
Surviving works
Archimedes is said to have remarked of the lever: Give me a place to stand on, and I will move the Earth.
- 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, "Magnitudes are in equilibrium
at distances reciprocally proportional to their weights."Archimedes
uses the principles derived to calculate the areas and centers of
gravity of various geometric figures including triangles,
parallelograms and parabolas.[46]
- On the Measurement of a 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 only a rough figure is required.
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
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 (including its two bases), where r
is the radius of the sphere and cylinder. The sphere has a volume and
surface area two-thirds that of the cylinder. A sculpted sphere and
cylinder were placed on the tomb of Archimedes at his request.
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
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:
Any body wholly or partially
immersed in a fluid experiences an upthrust equal to, but opposite in
sense to, the weight of the fluid displaced.
- 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 1⁄4.
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. Dr. Netz calculates that the pieces can be made into a
square 17,152 ways.[47] The number of arrangements is 536 when
solutions that are equivalent by rotation and reflection have been
excluded.[48] 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 that it is taken from the Ancient Greek word for throat
or gullet, stomachos (στόμαχος).[49] 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.[50]
- 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[51]
in 1880, and the answer is a very large number, approximately
7.760271 × 10206,544.[52]
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, as well as
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 × 1063 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.[53]
- The Method of Mechanical Theorems
This
treatise was thought lost until the discovery of the Archimedes
Palimpsest in 1906. In this work Archimedes uses infinitesimals, and
shows how breaking up a figure into an infinite number of infinitely
small parts can be used to determine its area or volume. Archimedes may
have considered this method lacking in formal rigor, so he also used
the method of exhaustion to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.
Apocryphal works
Archimedes' Book of Lemmas or Liber Assumptorum
is a treatise with fifteen propositions on the nature of circles. The
earliest known copy of the text is in Arabic. The scholars T. L. Heath
and Marshall Clagett argued that it cannot have been written by
Archimedes in its current form, since it quotes Archimedes, suggesting
modification by another author. The Lemmas may be based on an earlier work by Archimedes that is now lost.[54]
It
has also been claimed that Heron's formula for calculating the area of a
triangle from the length of its sides was known to Archimedes.[c]
However, the first reliable reference to the formula is given by Heron
of Alexandria in the 1st century AD.[55]
Archimedes Palimpsest
Main article: Archimedes Palimpsest
Stomachion is a dissection puzzle in the Archimedes 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.[56] 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.[57] 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 x-ray light to read the overwritten text.[58]
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).[59]
The asteroid 3600 Archimedes is named after him.[60]
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).[61]
Archimedes
has appeared on postage stamps issued by East Germany (1973), Greece
(1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain
(1963).[62]
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.[63]
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.[64]
See also
- Arbelos
- Archimedes' axiom
- Archimedes number
- Archimedes paradox
- Archimedes' screw
- Archimedean solid
- Archimedes' twin circles
- Archimedes' use of infinitesimals
- Archytas
- Diocles
- Methods of computing square roots
- Pseudo-Archimedes
- Salinon
- Steam cannon
- Syracusia
- Vitruvius
- 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 = √(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."
引用出處：
http://en.wikipedia.org/wiki/Archimedes
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