Archimedes of Syracuse (; Greek: Ἀρχιμήδης;
c. 287 – 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. Generally considered
the greatest mathematician of antiquity and
one of the greatest of all time, Archimedes
anticipated modern calculus and analysis by
applying concepts of infinitesimals and the
method of exhaustion to derive and rigorously
prove a range of geometrical theorems, including
the area of a circle, the surface area and
volume of a sphere, and the area under a parabola.Other
mathematical achievements include deriving
an accurate approximation of pi, defining
and investigating the spiral bearing his name,
and creating a system using exponentiation
for expressing very large numbers. He was
also one of the first to apply mathematics
to physical phenomena, founding hydrostatics
and statics, including an explanation of the
principle of the lever. He is credited with
designing innovative machines, such as his
screw pump, compound pulleys, and defensive
war machines to protect his native Syracuse
from invasion.
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 and a cylinder,
which Archimedes had requested be placed on
his tomb to represent his mathematical discoveries.
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 in Byzantine Constantinople, 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, 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 ==
Archimedes was born c. 287 BC in the seaport
city of Syracuse, Sicily, at that time a self-governing
colony in Magna Graecia, located along the
coast of Southern Italy. 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 else 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.
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 must
not be harmed. Marcellus called Archimedes
"a geometrical Briareus".The last words attributed
to Archimedes are "Do not disturb my circles",
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.
Valerius Maximus, writing in Memorable Doings
and Sayings in the 1st century AD, gives the
phrase as "...sed protecto manibus puluere
'noli' inquit, 'obsecro, istum disturbare'"
– "... but protecting the dust with his
hands, said 'I beg of you, do not disturb
this.'" The phrase is also given in Katharevousa
Greek as "μὴ μου τοὺς κύκλους
τάραττε!" (Mē mou tous kuklous taratte!).
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
were 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. A tomb discovered
in the courtyard of the Hotel Panorama in
Syracuse in the early 1960s was claimed to
be that of Archimedes, but there was no compelling
evidence for this and the location of his
tomb today is unknown.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 The Histories 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 ==
=== 
Archimedes' principle ===
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 of Syracuse, 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 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,
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, heúrēka!", 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 this principle,
it would have been possible to compare the
density of the crown to that of pure gold
by balancing the crown on a scale with a pure
gold reference sample of the same weight,
then immersing the apparatus in water. The
difference in density between the two samples
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." In a 12th-century text titled Mappae
clavicula there are instructions on how to
perform the weighings in the water in order
to calculate the percentage of silver used,
and thus solve the problem. The Latin poem
Carmen de ponderibus et mensuris of the 4th
or 5th century describes the use of a hydrostatic
balance to solve the problem of the crown,
and attributes the method to Archimedes.
=== 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 Hiero 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 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 world's first seagoing steamship with
a screw propeller was the SS Archimedes, which
was launched in 1839 and named in honor of
Archimedes and his work on the screw.
=== 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.
=== Heat ray ===
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.
The device, sometimes called the "Archimedes
heat ray", was used to focus sunlight onto
approaching ships, causing them to catch fire.
In the modern era, similar devices have been
constructed and may be referred to as a heliostat
or solar furnace.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.
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. A coating of tar would have been
commonplace on ships in the classical era.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).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.In December 2010,
MythBusters again looked at the heat ray story
in a special edition entitled "President's
Challenge". Several experiments were carried
out, including a large scale test with 500
schoolchildren aiming mirrors at a mock-up
of a Roman sailing ship 400 feet (120 m) away.
In all of the experiments, the sail failed
to reach the 210 °C (410 °F) required to
catch fire, and the verdict was again "busted".
The show concluded that a more likely effect
of the mirrors would have been blinding, dazzling,
or distracting the crew of the ship.
=== 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. 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) 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,
constructed by Archimedes and 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
On Sphere-Making. Modern research in this
area has been focused on the Antikythera mechanism,
another device built c. 100 BC 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.
== 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. 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. In Measurement of a Circle
he did this by drawing a larger regular hexagon
outside a circle and a smaller regular hexagon
inside the circle, and progressively doubling
the number of sides of each regular polygon,
calculating the length of a side of each polygon
at each step. As the number of sides increases,
it becomes a more accurate approximation of
a circle. After four such steps, when the
polygons had 96 sides each, he was able to
determine 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
(πr2). 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.
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 how he had obtained 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." It is possible that he used an iterative
procedure to calculate these values.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:
∑
n
=
0
∞
4
−
n
=
1
+
4
−
1
+
4
−
2
+
4
−
3
+
⋯
=
4
3
.
{\displaystyle \sum _{n=0}^{\infty }4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots
={4 \over 3}.\;}
If the 
first term in this series is the area of the
triangle, then the second is the sum of the
areas of two triangles whose bases are the
two smaller secant lines, and so on. This
proof 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.
== Writings ==
The works of Archimedes were written in Doric
Greek, the dialect of ancient Syracuse. 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. During his lifetime,
Archimedes made his work known through correspondence
with the mathematicians in Alexandria. The
writings of Archimedes were first collected
by the Byzantine Greek 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. 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.
=== Surviving works ===
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.On the Measurement
of a CircleThis 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 gives an approximation
of the value of pi (π), showing that it is
greater than 223/71 and less than 22/7.On
SpiralsThis work of 28 propositions is also
addressed to Dositheus. The treatise defines
what is now called the Archimedean spiral.
It is the locus of points corresponding to
the locations over time of a point moving
away from a fixed point with a constant speed
along a line which rotates with constant angular
velocity. Equivalently, in polar coordinates
(r, θ) it can be described by the equation
r
=
a
+
b
θ
{\displaystyle \,r=a+b\theta }
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 two-thirds that of
the circumscribed cylinder. Similarly, the
sphere has an area two-thirds that of the
cylinder (including the bases). A sculpted
sphere and cylinder were placed on the tomb
of Archimedes at his request.On Conoids and
SpheroidsThis 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 ParabolaIn 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.
(O)stomachionThis 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. 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 that it is 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 problemThis 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
7.760271×10206544.The Sand ReckonerIn 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.The Method
of Mechanical TheoremsThis 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.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. 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 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 of ultraviolet
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 Cylinder"
"On Floating Bodies"
"The Method of Mechanical Theorems"
"Stomachion"
Speeches by the 4th century BC politician
Hypereides
A commentary on Aristotle's Categories by
Alexander of Aphrodisias
Other works
== 
Legacy ==
Galileo praised Archimedes many times, and
referred to him as a "superhuman". Leibniz
said "He who understands Archimedes and Apollonius
will admire less the achievements of the foremost
men of later times."
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 Fields Medal for outstanding achievement
in mathematics carries a portrait of Archimedes,
along with a carving illustrating his proof
on 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.
== See also ==
Archimedes portal
Arbelos
Archimedes' axiom
Archimedes number
Archimedes paradox
Archimedean solid
Archimedes' twin circles
Diocles
List of things named after Archimedes
Methods of computing square roots
Pseudo-Archimedes
Salinon
Steam cannon
Zhang Heng
== 
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 0-471-54397-7 "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."
d. ^ "It was usual to smear the seams or even
the whole hull with pitch or with pitch and
wax". In Νεκρικοὶ Διάλογοι
(Dialogues of the Dead), Lucian refers to
coating the seams of a skiff with wax, a reference
to pitch (tar) or wax
