Archimedes of Syracuse (;
grc| ; ; ) was a Greek mathematician
, and astronomer
. Although few details of his life are known, he is regarded as one of the leading scientist
s in classical antiquity
. Considered to be the greatest mathematician of antiquity
, and one of the greatest scientists of all time,
Archimedes anticipated modern calculus
by applying the concept of the infinitely small
and the method of exhaustion
to derive and rigorously prove a range of geometrical theorem
s, including: the area of a circle
; the surface area
of a sphere
; area of an ellipse
; the area under a parabola
; the volume of a segment of a paraboloid of revolution
; the volume of a segment of a hyperboloid of revolution
; and the area of a spiral
His other mathematical achievements include deriving an accurate approximation of pi
; defining and investigating the spiral that now bears 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
, including an explanation of the principle of the lever
. He is credited with designing innovative machine
s, such as his screw pump
, compound pulleys
, and defensive war machines to protect his native Syracuse
Archimedes died during the siege of Syracuse
, where 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 by Isidore of Miletus
in Byzantine Constantinople
, while commentaries on the works of Archimedes written by Eutocius
in the 6th 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
and again in the 17th century
, 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.
Archimedes was born 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.
[Heath, Thomas L. 1897. ''Works of Archimedes''.]
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
, 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.
[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.]
Archimedes died 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 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 "" ("…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 "" ().
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
. 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
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; 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
In Vitruvius' account, Archimedes noticed, while taking a bath, 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
!" ( el|"εὕρηκα, ''heúrēka''!, ).
The test was conducted successfully, proving that silver had indeed been mixed in.
The story of the golden crown does not appear anywhere 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 Galilei
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.
[Dilke, Oswald A. W. 1990. ntitled ''Gnomon'' 62(8):697–99. .]
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.
A large part of Archimedes' work in engineering probably 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 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.
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
es 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
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 . The mirrors were pointed at a plywood of a Roman warship at a distance of around . 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.
[Casson, Lionel. 1995]
''Ships and seamanship in the ancient world''
Baltimore: Johns Hopkins University Press. pp. 211–12. : "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.
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 . 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 .
When ''MythBusters'' broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (i.e. 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 of a Roman sailing ship away. In all of the experiments, the sail failed to reach the 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.
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" ( el|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 lever
age 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.
Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus
' heliocentric model of the universe, in the ''Sand-reckoner''. Despite a lack of trigonometry and a table of chords, Archimedes describes the procedure and instrument used to make observations (a straight rod with pegs or grooves), applies correction factors to these measurements, and finally gives the result in the form of upper and lower bounds to account for observational error. Ptolemy
, quoting Hipparchus, also references Archimedes's solstice observations in the ''Almagest''. This would make Archimedes the first known Greek to have recorded multiple solstice dates and times in successive years.
(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
. 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 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.
While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch
wrote that Archimedes "placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life," though some scholars believe this may be a mischaracterization.
Method of exhaustion
Archimedes was able to use indivisibles
(an early form of infinitesimal
s) 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 areas of figures and the value of π
In ''Measurement of a Circle
'', he did this by drawing a larger regular hexagon
outside a circle
then 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 3 (approx. 3.1429) and 3 (approx. 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. Today this is known as 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 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.
The infinite series
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
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 line
s, and so on. This proof uses a variation of the series which sums to .
Myriad of myriads
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 itself derives from the Greek , for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million, i.e., 10,000 x 10,000) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion
, or 8.
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''.
[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''.]
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 invented a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes.
''On the Equilibrium of Planes''
There are two volumes to ''On the Equilibrium of Planes'': the being is in fifteen proposition
s 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 triangle
s and parabola
''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 gives an approximation
of the value of pi (), showing that it is greater than and less than .
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
(, ) it can be described by the equation
with real number
s and .
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 Cylinder''
In this two-volume treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere
and a circumscribe
of the same height and diameter
. The volume is 3
for the sphere, and 23
for the cylinder. The surface area is 42
for the sphere, and 62
for the cylinder (including its two bases), where is the radius of the sphere and cylinder. The sphere has a volume that of the circumscribed cylinder. Similarly, the sphere has an area 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 Spheroids''
This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of sections
, spheres, and paraboloids.
''On Floating Bodies''
In the first part of this two-volume 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:
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
Also known as Loculus of Archimedes or Archimedes' Box,
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 , a Greek compound word formed from the roots of () and ().
The 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
, 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 equation
s. There is a more difficult version of the problem in which some of the answers are required to be square number
s. This version of the problem was first solved by A. Amthor in 1880, and the answer is a very large number
, approximately 7.760271.
''The Sand Reckoner''
In this treatise, also known as ''Psammites'', 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'' is the only surviving work in which Archimedes discusses his views on astronomy.
''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
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.
[Boyer, Carl Benjamin. 1991. ''A History of Mathematics''. : "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula — , where 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."]
However, the first reliable reference to the formula is given by Heron of Alexandria
in the 1st century AD.
The foremost document containing the work of Archimedes is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg
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 29 October 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
, where it has been subjected to a range of modern tests including the use of ultraviolet
to read the overwritten text.
The treatises in the Archimedes Palimpsest include:
* ''On the Equilibrium of Planes
* ''On Spirals
* ''Measurement of a Circle
* ''On the Sphere and Cylinder
* ''On Floating Bodies
* ''The Method of Mechanical Theorems
* Speeches by the 4th century BC politician Hypereides
* A commentary on Aristotle
* Other works
praised Archimedes many times, and referred to him as a "superhuman" and as "his master". 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
() in his honor, as well as a lunar mountain range
, the Montes Archimedes
* 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
(1971), San Marino
(1982), and Spain
* 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
* Archimedean point
* Archimedes' axiom
* Archimedes number
* Archimedes paradox
* Archimedean solid
* Archimedes' twin circles
* List of things named after Archimedes
* Methods of computing square roots
* Steam cannon
* Zhang Heng
*Boyer, Carl Benjamin
. 1991. ''A History of Mathematics
''. New York: Wiley. .
. 1964–1984. ''Archimedes in the Middle Ages'' 1–5. Madison, WI: University of Wisconsin Press
*Dijksterhuis, Eduard J. 938
1987. ''Archimedes'', translated. Princeton: Princeton University Press
. 2005. ''Archimedes: Mathematical Genius of the Ancient World
''. Enslow Publishing
*Hasan, Heather. 2005. ''Archimedes: The Father of Mathematics
''. Rosen Central. .
*Heath, Thomas L.
1897. ''Works of Archimedes''
. Dover Publications
. . Complete works of Archimedes in English.
, and William Noel. 2007. ''The Archimedes Codex''. Orion Publishing Group
*Pickover, Clifford A.
2008. ''Archimedes to Hawking: Laws of Science and the Great Minds Behind Them''. Oxford University Press
*Simms, Dennis L. 1995. ''Archimedes the Engineer''. Continuum International Publishing Group
. 1999. ''Archimedes: What Did He Do Besides Cry Eureka?
''. Mathematical Association of America
*Heiberg's Edition of Archimedes
'' Texts in Classical Greek, with some in English.
*''The Method of Mechanical Theorems''
, translated by L.G. Robinson
The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
Photograph of the Sakkas experiment in 1973Archimedes Palimpsest reveals insights centuries ahead of its time
Category:3rd-century BC Greek people
Category:3rd-century BC writers
Category:People from Syracuse, Sicily
Category:Ancient Greek engineers
Category:Ancient Greek inventors
Category:Ancient Greek mathematicians
Category:Ancient Greek physicists
Category:Doric Greek writers
Category:Mathematicians from Sicily
Category:Scientists from Sicily
Category:Ancient Greeks who were murdered
Category:280s BC births
Category:210s BC deaths
Category:Year of birth uncertain
Category:Year of death uncertain