Archimedes of Syracuse (;; ) was a

Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
* Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancesto ...

mathematician, physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate caus ...

, engineer
Engineers, as practitioners of engineering, are professionals who invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the limit ...

, astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...

, and inventor from the ancient city of Syracuse in Sicily
(man) it, Siciliana (woman)
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, demographics_type1 = Ethnicity
, demographics1_footnotes =
, demographi ...

. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history
Ancient history is a time period from the beginning of writing and recorded human history to as far as late antiquity. The span of recorded history is roughly 5,000 years, beginning with the Sumerian cuneiform script. Ancient history c ...

, and one of the greatest of all time,*
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* Archimedes anticipated modern calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arith ...

and analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

s. These include the area of a circle, the surface area and volume of a sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

, the area of an ellipse, the area under a parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...

, the volume of a segment of a paraboloid of revolution
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
Every plan ...

, the volume of a segment of a hyperboloid of revolution
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...

, and the area of a spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are: Heath, Thomas L. 1897. ''Works of Archimedes''.
Archimedes' other mathematical achievements include deriving an approximation of pi, defining and investigating the

Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...

; ''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''.
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

parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...

and a straight line is 4/3 times the area of a

^{3} for the sphere, and 2^{3} for the cylinder. The surface area is 4^{2} for the sphere, and 6^{2} 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).

Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a const ...

, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena
Physical may refer to:
*Physical examination
In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally con ...

, founding hydrostatics
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...

and statics
Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with ...

. Archimedes' achievements in this area include a proof of the principle of the lever
A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or ''fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is divi ...

, the widespread use of the concept of center of gravity
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force ma ...

, and the enunciation of the law of buoyancy. He is also credited with designing innovative machine
A machine is a physical system using power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolecul ...

s, such as his screw pump
A screw pump is a positive-displacement pump that use one or several screws to move fluid solids or liquids along the screw(s) axis.
Three principal forms exist; In its simplest form (the Archimedes' screw pump or 'water screw'), a single scr ...

, 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
Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the esta ...

describes visiting Archimedes' tomb, which was surmounted by a sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

and a cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...

that Archimedes requested be placed there to represent his mathematical discoveries.
Unlike his inventions, Archimedes' mathematical writings 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
la, Constantinopolis ota, قسطنطينيه
, alternate_name = Byzantion (earlier Greek name), Nova Roma ("New Rome"), Miklagard/Miklagarth (Old Norse), Tsargrad ( Slavic), Qustantiniya ( Arabic), Basileuousa ("Queen of Cities"), Megalopolis ( ...

, while commentaries on the works of Archimedes by Eutocius
Eutocius of Ascalon (; el, Εὐτόκιος ὁ Ἀσκαλωνίτης; 480s – 520s) was a Palestinian-Greek mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian ''Conics''.
Life and work
Little is ...

in the 6th century opened them to wider readership for the first time. The relatively few copies of Archimedes' written work that survived through the Middle Ages
In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire ...

were an influential source of ideas for scientists during the Renaissance and again in the 17th century, while the discovery in 1906 of previously lost works by Archimedes in the Archimedes Palimpsest
The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. It contains two works of Archimedes that were thought to have been lost (the '' Ostomachion'' and the ...

has provided new insights into how he obtained mathematical results.
Biography

Archimedes was born c. 287 BC in the seaport city of Syracuse,Sicily
(man) it, Siciliana (woman)
, population_note =
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, at that time a self-governing colony in Magna Graecia
Magna Graecia (, ; , , grc, Μεγάλη Ἑλλάς, ', it, Magna Grecia) was the name given by the Romans to the coastal areas of Southern Italy in the present-day Italian regions of Calabria, Apulia, Basilicata, Campania and Sicily; the ...

. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes
John Tzetzes ( grc-gre, Ἰωάννης Τζέτζης, Iōánnēs Tzétzēs; c. 1110, Constantinople – 1180, Constantinople) was a Byzantine poet and grammarian who is known to have lived at Constantinople in the 12th century.
He was able to pr ...

that Archimedes lived for 75 years before his death in 212 BC. In the '' Sand-Reckoner'', Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known. 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, or if he ever visited Alexandria, Egypt, during his youth. From his surviving written works, it is clear that he maintained collegiate relations with scholars based there, including his friend Conon of Samos
Conon of Samos ( el, Κόνων ὁ Σάμιος, ''Konōn ho Samios''; c. 280 – c. 220 BC) was a Greek astronomer and mathematician. He is primarily remembered for naming the constellation Coma Berenices.
Life and work
Conon was born on Samos ...

and the head librarian Eratosthenes of Cyrene.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
Conon of Samos ( el, Κόνων ὁ Σάμιος, ''Konōn ho Samios''; c. 280 – c. 220 BC) was a Greek astronomer and mathematician. He is primarily remembered for naming the constellation Coma Berenices.
Life and work
Conon was born on Samos ...

lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works.
The standard versions of Archimedes' life were written long after his death by Greek and Roman historians. The earliest reference to Archimedes occurs in '' The Histories'' by Polybius ( 200–118 BC), written about 70 years after his death. 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 from the Romans. Polybius remarks how, during the Second Punic War, Syracuse switched allegiances from Rome
, established_title = Founded
, established_date = 753 BC
, founder = King Romulus (Romulus and Remus, legendary)
, image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg
...

to Carthage, resulting in a military campaign to take the city under the command of Marcus Claudius Marcellus and Appius Claudius Pulcher, which lasted from 213 to 212 BC. He notes that the Romans underestimated Syracuse's defenses, and mentions several machines Archimedes designed, including improved catapults, cranelike machines that could be swung around in an arc, and stone-throwers. Although the Romans ultimately captured the city, they suffered considerable losses due to Archimedes' inventiveness.
Cicero
Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the esta ...

(106–43 BC) mentions Archimedes in some of his works. While serving as a quaestor
A ( , , ; "investigator") was a public official in Ancient Rome. There were various types of quaestors, with the title used to describe greatly different offices at different times.
In the Roman Republic, quaestors were elected officials wh ...

in Sicily, Cicero found what was presumed to be Archimedes' 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 tomb carried a sculpture illustrating Archimedes' favorite mathematical proof, that the volume and surface area of the sphere are two-thirds that of the cylinder including its bases. He also mentions that Marcellus brought to Rome two planetariums Archimedes built. The Roman historian Livy (59 BC–17 AD) retells Polybius' story of the capture of Syracuse and Archimedes' role in it.
Plutarch
Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for h ...

(45–119 AD) wrote in his ''Parallel Lives
Plutarch's ''Lives of the Noble Greeks and Romans'', commonly called ''Parallel Lives'' or ''Plutarch's Lives'', is a series of 48 biographies of famous men, arranged in pairs to illuminate their common moral virtues or failings, probably writ ...

'' that Archimedes was related to King Hiero II, the ruler of Syracuse. He also provides at least two accounts on how Archimedes died after the city was taken. According to the most popular account, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet Marcellus, but he declined, saying that he had to finish working on the problem. This enraged the soldier, who killed Archimedes with his sword. Another story has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items. Marcellus was reportedly angered by Archimedes' death, as he considered him a valuable scientific asset (he called Archimedes "a geometrical Briareus") and had ordered that he should not be harmed.
The last words attributed to Archimedes are " Do not disturb my circles" (Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of t ...

, "''Noli turbare circulos meos''"; Katharevousa Greek, "μὴ μου τοὺς κύκλους τάραττε"), a reference to the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. There is no reliable evidence that Archimedes uttered these words and they do not appear in Plutarch's account. A similar quotation is found in the work of Valerius Maximus (fl. 30 AD), who wrote in ''Memorable Doings and Sayings'', "" ("... but protecting the dust with his hands, said 'I beg of you, do not disturb this).
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 toVitruvius
Vitruvius (; c. 80–70 BC – after c. 15 BC) was a Roman architect and engineer during the 1st century BC, known for his multi-volume work entitled ''De architectura''. He originated the idea that all buildings should have three attribute ...

, 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
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematica ...

.
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 crown's volume. 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 on the crown was conducted successfully, proving that silver had indeed been mixed in.
The story of the golden crown does not appear anywhere in Archimedes' known works. The practicality of the method it describes has been called into question due to the extreme accuracy that would be required while measuring the water displacement. Archimedes may have instead sought a solution that applied the principle known in hydrostatics
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...

as Archimedes' principle
Archimedes' principle (also spelled Archimedes's principle) states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces. Archimedes' ...

, which he describes in his treatise ''On Floating Bodies
''On Floating Bodies'' ( el, Περὶ τῶν ἐπιπλεόντων σωμάτων) is a Greek-language work consisting of two books written by Archimedes of Syracuse (287 – c. 212 BC), one of the most important mathematicians, physicis ...

''. This principle states that a body immersed in a fluid experiences a buoyant force
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the p ...

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
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...

, who in 1586 invented a hydrostatic balance for weighing metals in air and water inspired by the work of Archimedes, 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."
Archimedes' screw

A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of Syracuse. The Greek writerAthenaeus of Naucratis
Athenaeus of Naucratis (; grc, Ἀθήναιος ὁ Nαυκρατίτης or Nαυκράτιος, ''Athēnaios Naukratitēs'' or ''Naukratios''; la, Athenaeus Naucratita) was a Greek rhetorician and grammarian, flourishing about the end of th ...

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
Aphrodite ( ; grc-gre, Ἀφροδίτη, Aphrodítē; , , ) is an ancient Greek religion, ancient Greek goddess associated with love, lust, beauty, pleasure, passion (emotion), passion, and procreation. She was syncretized with the Roman god ...

among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, 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. Archimedes' screw is still in use today for pumping liquids and granulated solids such as coal and grain. Described in Roman times by Vitruvius, Archimedes' screw may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon
The Hanging Gardens of Babylon were one of the Seven Wonders of the Ancient World listed by Hellenic culture. They were described as a remarkable feat of engineering with an ascending series of tiered gardens containing a wide variety of tr ...

. The world's first seagoing steamship
A steamship, often referred to as a steamer, is a type of steam-powered vessel, typically ocean-faring and seaworthy, that is propelled by one or more steam engines that typically move (turn) propellers or paddlewheels. The first steamships c ...

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.
Archimedes' claw

Archimedes is said to have designed aclaw
A claw is a curved, pointed appendage found at the end of a toe or finger in most amniotes (mammals, reptiles, birds). Some invertebrates such as beetles and spiders have somewhat similar fine, hooked structures at the end of the leg or tarsus ...

as a weapon 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
A grappling hook or grapnel is a device that typically has multiple hooks (known as ''claws'' or ''flukes'') attached to a rope; it is thrown, dropped, sunk, projected, or fastened directly by hand to where at least one hook may catch and hol ...

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 aparabolic reflector
A parabolic (or paraboloid or paraboloidal) reflector (or dish or mirror) is a reflective surface used to collect or project energy such as light, sound, or radio waves. Its shape is part of a circular paraboloid, that is, the surface generated ...

to burn ships attacking Syracuse. The 2nd-century author Lucian
Lucian of Samosata, '; la, Lucianus Samosatensis ( 125 – after 180) was a Hellenized Syrian satirist, rhetorician and pamphleteer who is best known for his characteristic tongue-in-cheek style, with which he frequently ridiculed superstiti ...

wrote that during the siege of Syracuse (c. 214–212 BC), Archimedes destroyed enemy ships with fire. Centuries later, Anthemius of Tralles mentions burning-glass
A burning glass or burning lens is a large convex lens that can concentrate the sun's rays onto a small area, heating up the area and thus resulting in ignition of the exposed surface. Burning mirrors achieve a similar effect by using reflecting s ...

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
A heliostat (from '' helios'', the Greek word for ''sun'', and ''stat'', as in stationary) is a device that includes a mirror, usually a plane mirror, which turns so as to keep reflecting sunlight toward a predetermined target, compensating ...

or solar furnace.
This purported weapon has been the subject of an 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, mostly with negative results. 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, but the overall effect would have been blinding, dazzling, or distracting the crew of the ship rather than fire.
Lever

While Archimedes did not invent thelever
A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or ''fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is divi ...

, he gave a mathematical proof of the principle involved in his work '' On the Equilibrium of Planes''. Earlier descriptions of the lever are found in the Peripatetic school
The Peripatetic school was a school of philosophy in Ancient Greece. Its teachings derived from its founder, Aristotle (384–322 BC), and ''peripatetic'' is an adjective ascribed to his followers.
The school dates from around 335 BC when Aristo ...

of the followers of Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phi ...

, and are sometimes attributed to Archytas
Archytas (; el, Ἀρχύτας; 435/410–360/350 BC) was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder ...

. There are several, often conflicting, reports regarding Archimedes' feats using the lever to lift very heavy objects. Plutarch describes how Archimedes designed block-and-tackle pulley
A pulley is a wheel on an axle or shaft that is designed to support movement and change of direction of a taut cable or belt, or transfer of power between the shaft and cable or belt. In the case of a pulley supported by a frame or shell that d ...

systems, allowing sailors to use the principle of lever
A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or ''fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is divi ...

age to lift objects that would otherwise have been too heavy to move. According to Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...

, 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). Olympiodorus later attributed the same boast to Archimedes' invention of the ''baroulkos'', a kind of windlass, rather than the lever.
Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer
An odometer or odograph is an instrument used for measuring the distance traveled by a vehicle, such as a bicycle or car. The device may be electronic, mechanical, or a combination of the two (electromechanical). The noun derives from ancient ...

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.
Astronomical instruments

Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus' heliocentric model of the universe, in the ''Sand-Reckoner''. Without the use of either trigonometry or 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' solstice observations in the ''Almagest''. This would make Archimedes the first known Greek to have recorded multiple solstice dates and times in successive years. Cicero's '' De re publica'' portrays a fictional conversation taking place in 129 BC, after the Second Punic War. General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms after capturing Syracuse in 212 BC, which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets. Cicero also mentions similar mechanisms designed by Thales of Miletus andEudoxus of Cnidus
Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...

. 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 small planetarium
A planetarium ( planetariums or ''planetaria'') is a theatre built primarily for presenting educational and entertaining shows about astronomy and the night sky, or for training in celestial navigation.
A dominant feature of most planetariu ...

. Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...

reports on a treatise by Archimedes (now lost) dealing with the construction of these mechanisms entitled ''On Sphere-Making''. Modern research in this area has been focused on the Antikythera mechanism
The Antikythera mechanism ( ) is an Ancient Greek hand-powered orrery, described as the oldest example of an analogue computer used to predict astronomical positions and eclipses decades in advance. It could also be used to track the four-ye ...

, 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.
Mathematics

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field ofmathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

. Plutarch
Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for h ...

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 (a precursor toinfinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refer ...

s) in a way that is similar to modern integral calculus
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with dif ...

. Through proof by contradiction (''reductio ad absurdum
In logic, ( Latin for "reduction to absurdity"), also known as ( Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to ab ...

''), 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 ($\backslash pi\; r^2$).
Archimedean property

In ''On the Sphere and Cylinder
''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of th ...

'', 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.
Archimedes gives the value of the square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...

of 3 as lying between (approximately 1.7320261) and (approximately 1.7320512) in ''Measurement of a Circle''. 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
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...

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
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

procedure to calculate these values.
The infinite series

In '' Quadrature of the Parabola'', Archimedes proved that the area enclosed by aparabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...

and a straight line is times the area of a corresponding inscribed triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collin ...

as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio :
:$\backslash sum\_^\backslash infty\; 4^\; =\; 1\; +\; 4^\; +\; 4^\; +\; 4^\; +\; \backslash cdots\; =\; .\; \backslash ;$
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
Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to:
* a secant line, in geometry
* the secant variety, in algebraic geometry
* secant (trigonometry) (Latin: secans), the multiplicative inverse (or recipr ...

s, and whose third vertex is where the line that is parallel to the parabola's axis and that passes through the midpoint of the base intersects the parabola, 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
A myriad (from Ancient Greek grc, μυριάς, translit=myrias, label=none) is technically the number 10,000 (ten thousand); in that sense, the term is used in English almost exclusively for literal translations from Greek, Latin or Sinosphe ...

. 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.
Writings

The works of Archimedes were written inDoric Greek
Doric or Dorian ( grc, Δωρισμός, Dōrismós), also known as West Greek, was a group of Ancient Greek dialects; its varieties are divided into the Doric proper and Northwest Doric subgroups. Doric was spoken in a vast area, that include ...

, the dialect of ancient Syracuse. Many written works by Archimedes have not survived or are only extant in heavily edited fragments; at least seven of his treatises are known to have existed due to references made by other authors. Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...

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 Eutocius
Eutocius of Ascalon (; el, Εὐτόκιος ὁ Ἀσκαλωνίτης; 480s – 520s) was a Palestinian-Greek mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian ''Conics''.
Life and work
Little is ...

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 into Latin via Arabic by Gerard of Cremona (c. 1114–1187). Direct Greek to Latin translations were later done by William of Moerbeke
William of Moerbeke, O.P. ( nl, Willem van Moerbeke; la, Guillelmus de Morbeka; 1215–35 – 1286), was a prolific medieval translator of philosophical, medical, and scientific texts from Greek language into Latin, enabled by the period ...

(c. 1215–1286) and Iacobus Cremonensis (c. 1400–1453).
During the Renaissance, the ''Editio princeps In classical scholarship, the ''editio princeps'' (plural: ''editiones principes'') of a work is the first printed edition of the work, that previously had existed only in manuscripts, which could be circulated only after being copied by hand.
Fo ...

'' (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin.
Surviving works

The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).''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 ofConon of Samos
Conon of Samos ( el, Κόνων ὁ Σάμιος, ''Konōn ho Samios''; c. 280 – c. 220 BC) was a Greek astronomer and mathematician. He is primarily remembered for naming the constellation Coma Berenices.
Life and work
Conon was born on Samos ...

. In Proposition II, Archimedes gives an approximation
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...

of the value of pi (), showing that it is greater than and less than .
''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 theheliocentric
Heliocentrism (also known as the Heliocentric model) is the astronomical model in which the Earth and planets revolve around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed the Ear ...

theory of the solar system
The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...

proposed by Aristarchus of Samos, as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies
An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often us ...

. By using a system of numbers based on powers of the myriad
A myriad (from Ancient Greek grc, μυριάς, translit=myrias, label=none) is technically the number 10,000 (ten thousand); in that sense, the term is used in English almost exclusively for literal translations from Greek, Latin or Sinosphe ...

, 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.
''On the Equilibrium of Planes''

There are two books to ''On the Equilibrium of Planes'': the first contains seven postulates and fifteenproposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

s, while the second book contains ten propositions. In the first work, Archimedes proves the ''Law of the lever
A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '' fulcrum''. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is d ...

'', which states that:
Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collin ...

s, parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

s and parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...

s.
''Quadrature of the Parabola''

In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by atriangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collin ...

with equal base and height. He achieves this by calculating the value of a geometric series that sums to infinity with the ratio .
''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 asphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

and a circumscribe
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...

d cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...

of the same height and diameter. The volume is ''On Spirals''

This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called theArchimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a const ...

. It is the locus
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* ''Locus'' (magazine), science fiction and fantasy magazine
** '' Locus Awar ...

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
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an object ...

. Equivalently, in polar coordinates (, ) it can be described by the equation $\backslash ,\; r=a+b\backslash theta$ with real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...

s and .
This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.
''On Conoids and Spheroids''

This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes ofsections
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...

of cones, 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 asEratosthenes
Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandr ...

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. Archimedes' principle of buoyancy is given in this work, stated as follows:Any body wholly or partially immersed in fluid experiences an upthrust equal to, but opposite in sense to, the weight of the fluid displaced.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.

''Ostomachion''

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
The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. It contains two works of Archimedes that were thought to have been lost (the '' Ostomachion'' and the ...

''. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Reviel Netz of Stanford University
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is conside ...

argued in 2003 that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. 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 are 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 calls the puzzle , a Greek compound word formed from the roots of () and ().
The cattle problem

Gotthold Ephraim Lessing
Gotthold Ephraim Lessing (, ; 22 January 1729 – 15 February 1781) was a philosopher, dramatist, publicist and art critic, and a representative of the Enlightenment era. His plays and theoretical writings substantially influenced the developme ...

discovered this work in a Greek manuscript consisting of a 44-line poem in the Herzog August Library
The Herzog August Library (german: link=no, Herzog August Bibliothek — "HAB"), in Wolfenbüttel, Lower Saxony, known also as ''Bibliotheca Augusta'', is a library of international importance for its collection from the Middle Ages and ea ...

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 number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The u ...

s. A. Amthor first solved this version of the problem in 1880, and the answer is a very large number, approximately 7.760271.
''The Method of Mechanical Theorems''

This treatise was thought lost until the discovery of theArchimedes Palimpsest
The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. It contains two works of Archimedes that were thought to have been lost (the '' Ostomachion'' and the ...

in 1906. In this work Archimedes uses indivisibles, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. He 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
Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandr ...

in Alexandria.
Apocryphal works

Archimedes' ''Book of Lemmas
The ''Book of Lemmas'' or ''Book of Assumptions'' (Arabic ''Maʾkhūdhāt Mansūba ilā Arshimīdis'') is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositi ...

'' or ''Liber Assumptorum'' is a treatise with 15 propositions on the nature of circles. The earliest known copy of the text is in Arabic. 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 the formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

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 — $k\; =\; \backslash sqrt$, 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." though its first appearance is in the work of Heron of Alexandria in the 1st century AD. Other questionable attributions to Archimedes' work include the Latin poem '' Carmen de ponderibus et mensuris'' (4th or 5th century), which describes the use of a hydrostatic balance to solve the problem of the crown, and the 12th-century text '' Mappae clavicula'', which contains instructions on how to perform assaying of metals by calculating their specific gravities. Dilke, Oswald A. W. 1990. ntitled ''Gnomon
A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields.
History
A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the ol ...

'' 62(8):697–99. .
Archimedes Palimpsest

The foremost document containing Archimedes' work is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg visitedConstantinople
la, Constantinopolis ota, قسطنطينيه
, alternate_name = Byzantion (earlier Greek name), Nova Roma ("New Rome"), Miklagard/Miklagarth (Old Norse), Tsargrad ( Slavic), Qustantiniya ( Arabic), Basileuousa ("Queen of Cities"), Megalopolis ( ...

to examined a 174-page goatskin parchment
Parchment is a writing material
Writing material refers to the materials that provide the surfaces on which humans use writing instruments to inscribe writings. The same materials can also be used for symbolic or representational drawings. Bui ...

of prayers, written in the 13th century, after reading a short transcription published seven years earlier by Papadopoulos-Kerameus. He confirmed that it was indeed a palimpsest
In textual studies, a palimpsest () is a manuscript page, either from a scroll or a book, from which the text has been scraped or washed off so that the page can be reused for another document. Parchment was made of lamb, calf, or kid skin an ...

, 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, a common practice in the Middle Ages, as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th-century copies of previously lost 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.
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 was stored at the Walters Art Museum
The Walters Art Museum, located in Mount Vernon-Belvedere, Baltimore, Maryland, United States, is a public art museum founded and opened in 1934. It holds collections established during the mid-19th century. The museum's collection was amassed ...

in Baltimore
Baltimore ( , locally: or ) is the most populous city in the U.S. state of Maryland, fourth most populous city in the Mid-Atlantic, and the 30th most populous city in the United States with a population of 585,708 in 2020. Baltimore wa ...

, Maryland
Maryland ( ) is a state in the Mid-Atlantic region of the United States. It shares borders with Virginia, West Virginia, and the District of Columbia to its south and west; Pennsylvania to its north; and Delaware and the Atlantic Ocean to ...

, where it was subjected to a range of modern tests including the use of ultraviolet and light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 ter ...

to read the overwritten text. It has since returned to its anonymous owner.
The treatises in the Archimedes Palimpsest include:
* '' On the Equilibrium of Planes''
* ''On Spirals
''On Spirals'' ( el, Περὶ ἑλίκων) is a treatise by Archimedes, written around 225 BC. Notably, Archimedes employed the Archimedean spiral in this book to square the circle and trisect an angle.
Contents
Preface
Archimedes begins '' ...

''
* '' Measurement of a Circle''
* ''On the Sphere and Cylinder
''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of th ...

''
* ''On Floating Bodies
''On Floating Bodies'' ( el, Περὶ τῶν ἐπιπλεόντων σωμάτων) is a Greek-language work consisting of two books written by Archimedes of Syracuse (287 – c. 212 BC), one of the most important mathematicians, physicis ...

''
* '' The Method of Mechanical Theorems''
* ''Stomachion
''Ostomachion'', also known as ''loculus Archimedius'' (Archimedes' box in Latin) and also as ''syntomachion'', is a mathematical treatise attributed to Archimedes. This work has survived fragmentarily in an Arabic version and a copy, the '' ...

''
* Speeches by the 4th century BC politician Hypereides
Hypereides or Hyperides ( grc-gre, Ὑπερείδης, ''Hypereidēs''; c. 390 – 322 BC; English pronunciation with the stress variably on the penultimate or antepenultimate syllable) was an Athenian logographer (speech writer). He was one ...

* A commentary on Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phi ...

's ''Categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
* Categories (Peirce)
...

''
* Other works
Legacy

Sometimes called the father of mathematics andmathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...

, Archimedes had a wide influence on mathematics and science.
* father of mathematics: Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, p 19.
* father of mathematical physics: James H. Williams Jr., Fundamentals of Applied Dynamics, p 30., Carl B. Boyer, Uta C. Merzbach, A History of Mathematics, p 111., Stuart Hollingdale, Makers of Mathematics, p 67., Igor Ushakov, In the Beginning, Was the Number (2), p 114.
Mathematics and physics

Historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity. Eric Temple Bell, for instance, wrote: Likewise,Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applica ...

and George F. Simmons said of Archimedes:
Reviel Netz, Suppes Professor in Greek Mathematics and Astronomy at Stanford University
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is conside ...

and an expert in Archimedes notes:
Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested on ...

repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes. Galileo called him "superhuman" and "my master", while Huygens
Huygens (also Huijgens, Huigens, Huijgen/Huygen, or Huigen) is a Dutch patronymic surname, meaning "son of Hugo". Most references to "Huygens" are to the polymath Christiaan Huygens. Notable people with the surname include:
* Jan Huygen (1563–1 ...

said, "I think Archimedes is comparable to no one" and modeled his work after him. Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...

said, "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times." Gauss's heroes were Archimedes and Newton, and Moritz Cantor, who studied under Gauss in the University of Göttingen, reported that he once remarked in conversation that “there had been only three epoch-making mathematicians: Archimedes, Newton, and Eisenstein."
The inventor Nikola Tesla
Nikola Tesla ( ; ,"Tesla"

''

''

There is a

Heiberg's Edition of Archimedes

'' Texts in Classical Greek, with some in English. * * * * *

The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland

* *

{{DEFAULTSORT:Archimedes 3rd-century BC Greek people 3rd-century BC writers People from Syracuse, Sicily Ancient Greek engineers Ancient Greek inventors Ancient Greek geometers Ancient Greek physicists Hellenistic-era philosophers Doric Greek writers Sicilian Greeks Mathematicians from Sicily Scientists from Sicily Ancient Greeks who were murdered Ancient Syracusans Fluid dynamicists Buoyancy 280s BC births 210s BC deaths Year of birth uncertain Year of death uncertain 3rd-century BC mathematicians 3rd-century BC Syracusans

crater
Crater may refer to:
Landforms
*Impact crater, a depression caused by two celestial bodies impacting each other, such as a meteorite hitting a planet
* Explosion crater, a hole formed in the ground produced by an explosion near or below the surfa ...

on the Moon named Archimedes () in his honor, as well as a lunar mountain range
A mountain range or hill range is a series of mountains or hills arranged in a line and connected by high ground. A mountain system or mountain belt is a group of mountain ranges with similarity in form, structure, and alignment that have ari ...

, the Montes Archimedes ().
The Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...

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 1st century AD poet Manilius, 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
Greece,, or , romanized: ', officially the Hellenic Republic, is a country in Southeast Europe. It is situated on the southern tip of the Balkans, and is located at the crossroads of Europe, Asia, and Africa. Greece shares land borders with ...

(1983), Italy
Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the Italy (geographical region) ...

(1983), Nicaragua
Nicaragua (; ), officially the Republic of Nicaragua (), is the largest country in Central America, bordered by Honduras to the north, the Caribbean to the east, Costa Rica to the south, and the Pacific Ocean to the west. Managua is the cou ...

(1971), San Marino (1982), and Spain
, image_flag = Bandera de España.svg
, image_coat = Escudo de España (mazonado).svg
, national_motto = '' Plus ultra'' ( Latin)(English: "Further Beyond")
, national_anthem = (English: "Royal March")
, ...

(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
Sutter's Mill was a water-powered sawmill on the bank of the South Fork American River in the foothills of the Sierra Nevada in California. It was named after its owner John Sutter. A worker constructing the mill, James W. Marshall, found go ...

in 1848 which sparked the California Gold Rush.See also

Concepts

*Arbelos
In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that con ...

* Archimedean point
* Archimedes' axiom
* Archimedes number
In viscous fluid dynamics, the Archimedes number (Ar), is a dimensionless number used to determine the motion of fluids due to density differences, named after the ancient Greek scientist and mathematician Archimedes.
It is the ratio of gravitat ...

* Archimedes paradox
* Archimedean solid
* Archimedes' twin circles
* Methods of computing square roots
Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted \sqrt, \sqrt /math>, or S^) of a real number. Arithmetically, it means given S, a procedure for fi ...

* Salinon
* Steam cannon
* Trammel of ArchimedesPeople

* Diocles *Pseudo-Archimedes Pseudo-Archimedes is a name given to pseudo-anonymous authors writing under the name of 'Archimedes' as quoted by various sources of the Islamic Golden Age such as Al-Jazari for the construction of water clocks. Archimedes himself is not known to ...

* Zhang Heng
References

Notes

Citations

Further reading

* Boyer, Carl Benjamin. 1991. '' A History of Mathematics''. New York: Wiley. . * Clagett, Marshall. 1964–1984. ''Archimedes in the Middle Ages'' 1–5. Madison, WI:University of Wisconsin Press
The University of Wisconsin Press (sometimes abbreviated as UW Press) is a non-profit university press publishing peer-reviewed books and journals. It publishes work by scholars from the global academic community; works of fiction, memoir and po ...

.
* Dijksterhuis, Eduard J. 9381987. ''Archimedes'', translated. Princeton: Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financia ...

. .
* Gow, Mary. 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.
* Netz, Reviel, and William Noel. 2007. ''The Archimedes Codex''. Orion Publishing Group
Orion Publishing Group Ltd. is a UK-based book publisher. It was founded in 1991 and acquired Weidenfeld & Nicolson the following year. The group has published numerous bestselling books by notable authors including Ian Rankin, Michael Connel ...

. .
* Pickover, Clifford A. 2008. ''Archimedes to Hawking: Laws of Science and the Great Minds Behind Them''. Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...

. .
*Simms, Dennis L. 1995. ''Archimedes the Engineer''. Continuum International Publishing Group
Continuum International Publishing Group was an academic publisher of books with editorial offices in London and New York City. It was purchased by Nova Capital Management in 2005. In July 2011, it was taken over by Bloomsbury Publishing. , all ...

. .
* Stein, Sherman. 1999. '' Archimedes: What Did He Do Besides Cry Eureka?''. Mathematical Association of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

. .
External links

*Heiberg's Edition of Archimedes

'' Texts in Classical Greek, with some in English. * * * * *

The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland

* *

{{DEFAULTSORT:Archimedes 3rd-century BC Greek people 3rd-century BC writers People from Syracuse, Sicily Ancient Greek engineers Ancient Greek inventors Ancient Greek geometers Ancient Greek physicists Hellenistic-era philosophers Doric Greek writers Sicilian Greeks Mathematicians from Sicily Scientists from Sicily Ancient Greeks who were murdered Ancient Syracusans Fluid dynamicists Buoyancy 280s BC births 210s BC deaths Year of birth uncertain Year of death uncertain 3rd-century BC mathematicians 3rd-century BC Syracusans