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 ancestor ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

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 l ...

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, natural satellite, moons, comets and galaxy, g ...

, and

inventor An invention is a unique or novel device, method, composition, idea or process. An invention may be an improvement upon a machine, product, or process for increasing efficiency or lowering cost. It may also be an entirely new concept. If an ...

from the ancient city of

Syracuse Syracuse may refer to: Places Italy *Syracuse, Sicily, or spelled as ''Siracusa'' *Province of Syracuse United States *Syracuse, New York **East Syracuse, New York **North Syracuse, New York *Syracuse, Indiana * Syracuse, Kansas *Syracuse, Miss ...

Sicily (man) it, Siciliana (woman) , population_note = , population_blank1_title = , population_blank1 = , 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 Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th century AD centred on the Mediterranean Sea, comprising the interlocking civilizations of ...

. 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 cove ...

, and one of the greatest of all time,* * * * * * * * * * 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 arithm ...

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 The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area bet ...

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 ca ...

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 In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159. One method of deriving this formula, which origi ...

, the

surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...

and

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

of a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, the area of an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

, 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 plane ...

, the volume of a segment of a

hyperboloid of revolution In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface (mathematics), surface generated by rotating a hyperbola around one of its Hyperbola#Nomenclature and features, principal axes. A hyperboloid is th ...

, 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

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 con ...

, and devising a system using

exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...

for expressing very large numbers. He was also one of the first to apply mathematics to

physical phenomena Physical may refer to: *Physical examination, a regular overall check-up with a doctor *Physical (Olivia Newton-John album), ''Physical'' (Olivia Newton-John album), 1981 **Physical (Olivia Newton-John song), "Physical" (Olivia Newton-John song) *P ...

, 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 imm ...

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 div ...

, 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 weight function, weighted relative position (vector), position of the distributed mass sums to zero. Thi ...

, and the enunciation of the law of buoyancy. He is also credited with designing innovative

machine A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to na ...

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 sc ...

, compound pulleys, and defensive war machines to protect his native

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 estab ...

describes visiting Archimedes' tomb, which was surmounted by a

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 Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria ...

read and quoted him, but the first comprehensive compilation was not made until by

Isidore of Miletus Isidore of Miletus ( el, Ἰσίδωρος ὁ Μιλήσιος; Medieval Greek pronunciation: ; la, Isidorus Miletus) was one of the two main Byzantine Greek architects (Anthemius of Tralles was the other) that Emperor Justinian I commissioned ...

Byzantine The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinopl ...

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 a ...

were an influential source of ideas for scientists during the

Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...

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

Death of Archimedes (1815) by Thomas Degeorge

Archimedes was born c. 287 BC in the seaport city of

Sicily (man) it, Siciliana (woman) , population_note = , population_blank1_title = , population_blank1 = , demographics_type1 = Ethnicity , demographics1_footnotes = , demographi ...

, 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; these re ...

. 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 p ...

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

, 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 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 Alexandria ...

.In the preface to ''On Spirals'' addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death."

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 Polybius (; grc-gre, Πολύβιος, ; ) was a Greek historian of the Hellenistic period. He is noted for his work , which covered the period of 264–146 BC and the Punic Wars in detail. Polybius is important for his analysis of the mixed ...

( 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 The Second Punic War (218 to 201 BC) was the second of three wars fought between Carthage and Rome, the two main powers of the western Mediterranean in the 3rd century BC. For 17 years the two states struggled for supremacy, primarily in Ital ...

, Syracuse switched allegiances from

Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus (legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption ...

Carthage Carthage was the capital city of Ancient Carthage, on the eastern side of the Lake of Tunis in what is now Tunisia. Carthage was one of the most important trading hubs of the Ancient Mediterranean and one of the most affluent cities of the classi ...

, resulting in a military campaign to take the city under the command of

Marcus Claudius Marcellus Marcus Claudius Marcellus (; 270 – 208 BC), five times elected as consul of the Roman Republic, was an important Roman military leader during the Gallic War of 225 BC and the Second Punic War. Marcellus gained the most prestigious award a Roma ...

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 Discovering the Tomb of Archimedes by Benjamin West

Cicero Discovering the Tomb of Archimedes by Benjamin West

(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 who ...

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 Titus Livius (; 59 BC – AD 17), known in English as Livy ( ), was a Ancient Rome, Roman historian. He wrote a monumental history of Rome and the Roman people, titled , covering the period from the earliest legends of Rome before the traditiona ...

(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 his ''P ...

(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 writt ...

'' that Archimedes was related to King

Hiero II Hiero II ( el, Ἱέρων Β΄; c. 308 BC – 215 BC) was the Greek tyrant of Syracuse from 275 to 215 BC, and the illegitimate son of a Syracusan noble, Hierocles, who claimed descent from Gelon. He was a former general of Pyrrhus of Epirus a ...

, 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 In Greek mythology, the Hecatoncheires ( grc-gre, Ἑκατόγχειρες, , Hundred-Handed Ones), or Hundred-Handers, also called the Centimanes, (; la, Centimani), named Cottus, Briareus (or Aegaeon) and Gyges (or Gyes), were three monstrous ...

") and had ordered that he should not be harmed. The last words attributed to Archimedes are "

Do not disturb my circles "''Nōlī turbāre circulōs meōs!''" is a Latin phrase, meaning "Do not disturb my circles!". It is said to have been uttered by Archimedes—in reference to a geometric figure he had outlined on the sand—when he was confronted by a Roman sold ...

" (

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 the ...

, "''Noli turbare circulos meos''";

Katharevousa Greek Katharevousa ( el, Καθαρεύουσα, , literally "purifying anguage) is a conservative form of the Modern Greek language conceived in the late 18th century as both a literary language and a compromise between Ancient Greek and the contempor ...

, "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 Valerius Maximus () was a 1st-century Latin writer and author of a collection of historical anecdotes: ''Factorum ac dictorum memorabilium libri IX'' ("Nine books of memorable deeds and sayings", also known as ''De factis dictisque memorabilibus'' ...

(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 to

Vitruvius 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 A votive crown is a votive offering in the form of a crown, normally in precious metals and often adorned with jewels. Especially in the Early Middle Ages, they are of a special form, designed to be suspended by chains at an altar, shrine or imag ...

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. Mathematical ...

. 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

. 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 Eureka (often abbreviated as E!, or Σ!) is an intergovernmental organisation for research and development funding and coordination. Eureka is an open platform for international cooperation in innovation. Organisations and companies applying th ...

!" ( 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

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. Archimede ...

, 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 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

. The Greek writer

Athenaeus 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 ''Syracusia'' ( el, Συρακουσία, ''syrakousía'', literally "of Syracuse") was an ancient Greek ship sometimes claimed to be the largest transport ship of antiquity. She was reportedly too big for any port in Sicily, and thus only sailed ...

'', 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

. 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 goddess associated with love, lust, beauty, pleasure, passion, and procreation. She was syncretized with the Roman goddess . Aphrodite's major symbols include ...

among its facilities. Since a ship of this size would leak a considerable amount of water through the hull,

Archimedes' screw The Archimedes screw, also known as the Archimedean screw, hydrodynamic screw, water screw or Egyptian screw, is one of the earliest hydraulic machines. Using Archimedes screws as water pumps (Archimedes screw pump (ASP) or screw pump) dates back ...

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 tre ...

. 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 ...

with a

screw propeller A propeller (colloquially often called a screw if on a ship or an airscrew if on an aircraft) is a device with a rotating hub and radiating blades that are set at a pitch to form a helical spiral which, when rotated, exerts linear thrust upon ...

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 a

claw 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 a

parabolic 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 Pamphleteer is a historical term for someone who creates or distributes pamphlets, unbound (and therefore ...

wrote that during the siege of Syracuse (c. 214–212 BC), Archimedes destroyed enemy ships with fire. Centuries later,

Anthemius of Tralles Anthemius of Tralles ( grc-gre, Ἀνθέμιος ὁ Τραλλιανός, Medieval Greek: , ''Anthémios o Trallianós''; – 533 558) was a Greek from Tralles who worked as a geometer and architect in Constantinople, the capit ...

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 ...

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 ...

solar furnace A solar furnace is a structure that uses concentrated solar power to produce high temperatures, usually for industry. Parabolic mirrors or heliostats concentrate light (Insolation) onto a focal point. The temperature at the focal point may rea ...

. This purported weapon has been the subject of an ongoing debate about its credibility since the

René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...

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 Bronze is an alloy consisting primarily of copper, commonly with about 12–12.5% tin and often with the addition of other metals (including aluminium, manganese, nickel, or zinc) and sometimes non-metals, such as phosphorus, or metalloids such ...

copper Copper is a chemical element with the symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pinkis ...

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 the

, he gave a mathematical proof of the principle involved in his work ''

On the Equilibrium of Planes ''On the Equilibrium of Planes'' ( grc, Περὶ ἐπιπέδων ἱσορροπιῶν, translit=perí epipédōn isorropiôn) is a treatise by Archimedes in two volumes. The first book contains a proof of the law of the lever and culminate ...

''. 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 phil ...

, 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 ...

systems, allowing sailors to use the principle of

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 The windlass is an apparatus for moving heavy weights. Typically, a windlass consists of a horizontal cylinder (barrel), which is rotated by the turn of a crank or belt. A winch is affixed to one or both ends, and a cable or rope is wound arou ...

, rather than the lever. Archimedes has also been credited with improving the power and accuracy of the

catapult A catapult is a ballistic device used to launch a projectile a great distance without the aid of gunpowder or other propellants – particularly various types of ancient and medieval siege engines. A catapult uses the sudden release of stored p ...

, 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 Gr ...

during the

First Punic War The First Punic War (264–241 BC) was the first of three wars fought between Rome and Carthage, the two main powers of the western Mediterranean in the early 3rd century BC. For 23 years, in the longest continuous conflict and grea ...

. 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 Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...

, 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 ''De re publica'' (''On the Commonwealth''; see below) is a dialogue on Roman politics by Cicero, written in six books between 54 and 51 BC. The work does not survive in a complete state, and large parts are missing. The surviving sections derive ...

'' portrays a fictional conversation taking place in 129 BC, after the

. General

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 Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him ...

and

Eudoxus 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 Gaius Sulpicius Gallus or Galus () was a general, statesman and orator of the Roman Republic. In 169 BC, he served as ''praetor urbanus''.Livy xliii.14 Under Lucius Aemilius Paulus, his intimate friend, he commanded the 2nd legion in the campaign ...

Lucius Furius Philus Lucius Furius Philus was a Roman statesman who became consul of ancient Rome in 136 BC. He was a member of the Scipionic Circle, and particularly close to Scipio Aemilianus. As proconsul, his allotted province was Spain. The consul of the previous ...

, 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 planetarium ...

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 Greece, Ancient Greek hand-powered orrery, described as the oldest example of an analogue computer used to predict astronomy, astronomical positions and eclipses decades in advance. It could also be ...

, 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 of

mathematics 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 ...

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 In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...

(a precursor to

infinitesimal 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 referr ...

s) in a way that is similar to modern

integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...

. 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 absu ...

''), 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

, and he employed it to approximate the areas of figures and the value of π. In ''

Measurement of a Circle ''Measurement of a Circle'' or ''Dimension of the Circle'' (Greek: , ''Kuklou metrēsis'') is a treatise that consists of three propositions by Archimedes, ca. 250 BCE. The treatise is only a fraction of what was a longer work. Propositions Prop ...

'', he did this by drawing a larger

regular hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

outside a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

, 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

was equal to π multiplied by the

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

of the

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

of the circle (

\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 t ...

'', Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. Today this is known as the

Archimedean property In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typical ...

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 . E ...

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

Parabolic segment and inscribed triangle

In ''

Quadrature of the Parabola ''Quadrature of the Parabola'' ( el, Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions rega ...

'', Archimedes proved that the area enclosed by a

and a straight line is times the area of a corresponding inscribed

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...

as shown in the figure at right. He expressed the solution to the problem as an

infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music * Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...

geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each succ ...

with the common ratio : :

\sum_^\infty 4^ = 1 + 4^ + 4^ + 4^ + \cdots = . \;

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 reciproc ...

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 ''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes, an Ancient Greek mathematician of the 3rd century BC, in which he set out to determine an upper bound for the number of grains of sand that fit into the unive ...

'', 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 Sinospher ...

. 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 in

Doric 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 included ...

, 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.

mentions ''On Sphere-Making'' and another work on

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...

, 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 Polyhedron, polyhedra mentioned by

; ''Catoptrica'', a work on optics mentioned by Theon of Alexandria; ''Principles'', addressed to Zeuxippus and explaining the number system used in ''

''; ''On Balances and Levers''; ''On Centers of Gravity''; ''On the Calendar''. Archimedes made his work known through correspondence with the mathematicians in

. The writings of Archimedes were first collected by the Byzantine Empire, Byzantine Greek architect

(c. 530 AD), while commentaries on the works of Archimedes written by

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 (c. 1215–1286) and Iacopo da San Cassiano, Iacobus Cremonensis (c. 1400–1453). During the

, the ''Editio princeps'' (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 of

. In Proposition II, Archimedes gives an Approximations of π, approximation 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 Sand, grains of sand that will fit inside the universe. This book mentions the Heliocentrism, heliocentric theory of the Solar System, 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

, 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 Axiom, postulates and fifteen propositions, while the second book contains ten propositions. In the first work, Archimedes proves the ''Torque, Law of the lever'', which states that: Archimedes uses the principles derived to calculate the areas and center of mass, centers of gravity of various geometric figures including

s, parallelograms and

''Quadrature of the Parabola''

In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by a

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

with equal base and height. He achieves this by calculating the value of a

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 a

and a circumscribed

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).

''On Spirals''

This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the

. It is the locus (mathematics), 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 coordinate system, polar coordinates (, ) it can be described by the equation

\, r=a+b\theta

with real numbers and . This is an early example of a Curve, mechanical curve (a curve traced by a moving point (geometry), 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 of cross section (geometry), sections of Cone (geometry), cones, spheres, and paraboloids.

''On Floating Bodies''

In the first part of this two-volume treatise, Archimedes spells out the law of wikt:equilibrium, 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. Archimedes' Buoyancy, 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 calculates the areas of the 14 pieces which can be assembled to form a

. Reviel Netz of Stanford University 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 discovered this work in a Greek manuscript consisting of a 44-line poem in the Herzog August Library in Wolfenbüttel, Germany in 1773. It is addressed to Eratosthenes and the mathematicians in Alexandria. Archimedes challenges them to count the numbers of cattle in the The Cattle of Helios, Herd of the Sun by solving a number of simultaneous Diophantine equations. There is a more difficult version of the problem in which some of the answers are required to be square numbers. 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 the

in 1906. In this work Archimedes uses

, 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

to derive the results. As with ''Archimedes's cattle problem, The Cattle Problem'', ''The Method of Mechanical Theorems'' was written in the form of a letter to Eratosthenes in

Apocryphal works

Archimedes' ''Book of Lemmas'' or ''Liber Assumptorum'' is a treatise with 15 propositions on the nature of circles. The earliest known copy of the text is in Arabic language, 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 Heron's formula, formula for calculating the area of a triangle from the length of its sides was known to Archimedes,Carl Benjamin Boyer, 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 = \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 (geometry), chord' ... Archimedes is reported by the Arabs to have given several proofs of the theorem." though its first appearance is in the work of Hero of Alexandria, 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 Assay, assaying of metals by calculating their specific gravities.Oswald A. W. Dilke, Dilke, Oswald A. W. 1990. [Untitled]. ''Gnomon (journal), Gnomon'' 62(8):697–99. .

Archimedes Palimpsest

The foremost document containing Archimedes' work is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg (historian), Johan Ludvig Heiberg visited

to examined a 174-page Goatskin (material), goatskin parchment of prayers, written in the 13th century, after reading a short transcription published seven years earlier by Athanasios Papadopoulos-Kerameus, Papadopoulos-Kerameus. He confirmed that it was indeed 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, 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 Suda, 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 in Baltimore, Maryland, where it was subjected to a range of modern tests including the use of ultraviolet and light to read the overwritten text. It has since returned to its anonymous owner. The treatises in the Archimedes Palimpsest include: * ''

'' * ''On Spirals'' * ''

'' * ''

'' * ''The Method of Mechanical Theorems'' * ''Stomachion'' * Speeches by the 4th century BC politician Hypereides * A commentary on

's ''Categories (Aristotle), Categories'' * Other works

Legacy

Sometimes called the father of mathematics and mathematical physics, 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 and George F. Simmons said of Archimedes: Reviel Netz, Suppes Professor in Greek Mathematics and Astronomy at Stanford University and an expert in Archimedes notes: Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes. Galileo Galilei, Galileo called him "superhuman" and "my master", while Christiaan Huygens, Huygens said, "I think Archimedes is comparable to no one" and modeled his work after him. Gottfried Wilhelm Leibniz, Leibniz said, "He who understands Archimedes and Apollonius of Perga, Apollonius will admire less the achievements of the foremost men of later times." Carl Friedrich Gauss, 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, Isaac Newton, Newton, and Gotthold Eisenstein, Eisenstein." The inventor Nikola Tesla praised him, saying:

Honors and commemorations

There is a impact crater, crater on the Moon named Archimedes (crater), Archimedes () 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 1st century AD poet Marcus Manilius, 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 (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963). The exclamation of Eureka (word), 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.

References

Notes

Citations

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 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

Biography

Discoveries and inventions

Archimedes' principle

Archimedes' screw

Archimedes' claw

Heat ray

Lever

Astronomical instruments

Mathematics

Method of exhaustion

Archimedean property

The infinite series

Myriad of myriads

Writings

Surviving works

''Measurement of a Circle''

''The Sand Reckoner''

''On the Equilibrium of Planes''

''Quadrature of the Parabola''

''On the Sphere and Cylinder''

''On Spirals''

''On Conoids and Spheroids''

''On Floating Bodies''

''Ostomachion''

The cattle problem

''The Method of Mechanical Theorems''

Apocryphal works

Archimedes Palimpsest

Legacy

Mathematics and physics

Honors and commemorations

See also

Concepts

People

References

Notes

Citations

Further reading

External links