Archimedes of Syracuse (
Greek: ;
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 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.
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.
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. 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, 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 colony of Magna Graecia. The date of birth is
based on a statement by the
Byzantine
Greek historian
John Tzetzes that
Archimedes lived for 75 years. In
The Sand Reckoner, Archimedes gives
his father's name as Phidias, an
astronomer about whom nothing is known.
Plutarch wrote in his
Parallel Lives that Archimedes was
related to King
Hiero II, the
ruler of Syracuse. A biography of Archimedes was written by his
friend Heracleides but this work has been lost, leaving the details
of his life obscure. It is unknown, for instance, whether he ever
married or had children.
During his youth Archimedes may have studied
in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene were
contemporaries. He referred to Conon of Samos as his friend,
while two of his works (
The 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 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.
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.
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.
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
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 new crown in the shape of a
laurel wreath had been made for
King Hiero II, and Archimedes was asked
to determine whether it was of solid
gold, or
whether
silver had been added by a dishonest
goldsmith. Archimedes had to solve the problem without damaging the
crown, so he 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 weight 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 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 prohibitive amount of
accuracy required in measuring 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 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
A large part of Archimedes' work in engineering arose from
fulfilling the needs of his home city of Syracuse. The Greek writer
Athenaeus of Naucratis described how King
Hieron II commissioned Archimedes to design a huge ship, the
Syracusia, which could be used
for luxury travel, carrying supplies, and as a naval warship. The
Syracusia is said to have been the largest ship built in
classical antiquity. According to Athenaeus, it was capable of
carrying 600 people and included garden decorations, a
gymnasium and a temple dedicated
to the goddess
Aphrodite among its
facilities. Since a ship of this size would leak a considerable
amount of water through the hull, the
Archimedes screw was purportedly developed
in order to remove the bilge water. Archimedes' machine was a
device with a revolving screw-shaped blade inside a cylinder. It
was turned by hand, and could also be used to transfer water from a
body of water into irrigation canals. The Archimedes screw is still
in use today for pumping liquids and 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?
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.
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 of a Roman warship at a distance of
around 160 feet (50 m). When the mirrors were focused
accurately, the ship burst into flames within a few seconds. The
plywood ship had a coating of
tar paint,
which may have aided combustion.
In October
2005 a group of students from the Massachusetts
Institute of Technology carried out an experiment with 127 one-foot
(30 cm) square mirror tiles, focused on a wooden ship at a
range of around 100 feet (30 m). Flames broke out
on a patch of the ship, but only after the sky had been cloudless
and the ship had remained stationary for around ten minutes. It was
concluded that the 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
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." ( ) 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
Lucius Furius Philus, who
described it thus:
This is a description of a
planetarium
or
orrery.
Pappus of Alexandria stated that
Archimedes had written a manuscript (now lost) on the construction
of these mechanisms entitled . Modern research in this area has
been focused on the
Antikythera
mechanism, another device from classical antiquity that was
probably designed for the same purpose. Constructing mechanisms of
this kind would have required a sophisticated knowledge of
differential gearing. This
was once thought to have been beyond the range of the technology
available in ancient times, but the discovery of the Antikythera
mechanism in 1902 has confirmed that devices of this kind were
known to the ancient Greeks.
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 (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 (approximately 3.1429) and 3
(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.
In
Measurement of a
Circle, Archimedes gives the value of the
square root of 3 as lying between (approximately
1.7320261) and (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."
In
The Quadrature of
the Parabola, Archimedes proved that the area enclosed by
a
parabola and a straight line is 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 :
- \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 which sums to .
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
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 .
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
Catoptrica.
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. 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.
- 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 and less than . 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
- :\, 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 πr^{3} for
the sphere, and 2πr^{3} for the cylinder. The
surface area is 4πr^{2} for the sphere, and
6πr^{2} 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 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 , 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:
- 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 .
- 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. 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 ( ). Ausonius refers to the
puzzle as Ostomachion, a Greek compound word formed from
the roots of (osteon, bone) and (machē - fight). The
puzzle is also known as the Loculus of Archimedes or Archimedes'
Box.
- This
work was discovered by Gotthold
Ephraim Lessing in a Greek manuscript consisting of a poem of
44 lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773. It is addressed to Eratosthenes and
the mathematicians in Alexandria. Archimedes challenges them to
count the numbers of cattle in the Herd of the Sun by solving a
number of simultaneous Diophantine
equations. There is a more difficult version of the problem in
which some of the answers are required to be square numbers. This version of the problem
was first solved by A. Amthor in 1880, and the answer is a very
large number, approximately 7.760271 .
- 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 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.
- 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.
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 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
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.
See also
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 , 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."
References
- T. L. Heath,
Works of Archimedes, 1897
- Hippias, 2 (cf. Galen, On temperaments 3.2, who mentions
pyreia, "torches"); Anthemius of Tralles, On
miraculous engines 153 [Westerman].
- Quoted by Pappus of Alexandria in
Synagoge, Book VIII
- Quoted in T. L. Heath, Works of Archimedes, Dover
Publications, ISBN 0-486-42084-1.
- Encyclopedia of ancient Greece By Nigel Guy Wilson Page 77 ISBN 0794502253 (2006)
- B. Krumbiegel, A. Amthor, Das Problema Bovinum des
Archimedes, Historisch-literarische Abteilung der Zeitschrift
Für Mathematik und Physik 25 (1880) 121-136, 153-171.
Further reading
- Republished translation of the 1938 study of Archimedes and his
works by an historian of science.
- Complete works of Archimedes in English.
The Works of Archimedes online
External links