TheInfoList

Archimedes of Syracuse (; grc, ; ; ) was a
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ...
,
physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In classical antiquity, there was no real ancient analog of a modern sci ...

,
engineer Engineers, as practitioners of engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The dis ...

,
inventor An invention is a unique or novel A novel is a relatively long work of narrative fiction, typically written in prose and published as a book. The present English word for a long work of prose fiction derives from the for "new", "news", or ...

, and
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, ga ...
. Although few details of his life are known, he is regarded as one of the leading
scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the development of science since at least the 17th century. It involves ...

s in
classical antiquity Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history History (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ...
. Considered to be the greatest mathematician of
antiquity Antiquity or Antiquities may refer to Historical objects or periods Artifacts * Antiquities, objects or artifacts surviving from ancient cultures Eras Any period before the European Middle Ages In the history of Europe, the Middle Ages ...

, and one of the greatest scientists of all time, Archimedes anticipated modern
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

and
analysis Analysis is the process of breaking a complexity, complex topic or Substance theory, 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 Ari ...
by applying the concept of the
infinitely small In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
and the
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface o ...

to derive and rigorously prove a range of
geometrical Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ...

theorem In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, 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 the ...
s, including: the
area of a circle In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

; the
surface area of radius has surface area . The surface area of a Solid geometry, solid object is a measure of the total area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface ...

and
volume Volume is a expressing the of enclosed by a . For example, the space that a substance (, , , or ) or occupies or contains. Volume is often quantified numerically using the , the . The volume of a container is generally understood to be the ...

of a
sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

; area of an
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

; the area under a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

; the volume of a segment of a
paraboloid of revolution In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...
; 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 the ...

; and the area of a
spiral In , a spiral is a which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the are:
. His other mathematical achievements include deriving an accurate approximation of pi; defining and investigating the ; and creating 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 An integer (from the Latin wikt: ...
for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, founding
hydrostatics Fluid statics or hydrostatics is the branch of fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical object ...
and
statics Statics is the branch of mechanics that is concerned with the analysis of (force and torque, torque, or "moment") acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with their enviro ...

, including an explanation of the principle of the
lever A lever ( or ) is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '':wikt:fulcrum, 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 effo ...

. He is credited with designing innovative
machine A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine A molecular machine, nanite, or nanomachine is a molecular component that produce ...

s, such as his
screw pump upPrinciple of screw pump (Saugseite = intake, Druckseite = outflow) A screw pump, also known as a water screw, is a positive-displacement (PD) pump that use one or several screws to move fluid solids or liquids along the screw(s) axis. In its si ...
, compound pulleys, and defensive war machines to protect his native
Syracuse Syracuse may refer to: Places Italy *Syracuse, Sicily Syracuse ( ; it, Siracusa , or scn, Seragusa, label=none ; lat, Syrācūsae ; grc-att, wikt:Συράκουσαι, Συράκουσαι, Syrákousai ; grc-dor, wikt:Συράκοσ ...

from invasion. Archimedes died during the siege of Syracuse, where 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 Roman or Romans usually refers to: *Rome, the capital city of Italy *Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people ...

describes visiting the tomb of Archimedes, which was surmounted by a
sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

and a
cylinder A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditio ...

, which Archimedes had requested be placed on his tomb to represent his mathematical discoveries. Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from
Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; : Rakodī; el, Αλεξάνδρεια ''Alexandria'') is the in after and , in , and a major economic centre. With a total population of 5,200,000, Alexandria is the ...

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 Medieval Greek (also known as Middle Greek or Byzantine Greek) is the stage of the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, anc ...
in
Byzantine The Byzantine Empire, also referred to as the Eastern Roman Empire, or Byzantium, was the continuation of the Roman Empire in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinople. It surviv ...
Constantinople la, Constantinopolis ota, قسطنطينيه , alternate_name = Byzantion (earlier Greek name), Nova Roma ("New Rome"), Miklagard/Miklagarth (), Tsargrad (), Qustantiniya (), Basileuousa ("Queen of Cities"), Megalopolis ("the Great City"), Πό ...

, while commentaries on the works of Archimedes written by Eutocius in the 6th century AD opened them to wider readership for the first time. The relatively few copies of Archimedes' written work that survived through the
Middle Ages In the history of Europe The history of Europe concerns itself with the discovery and collection, the study, organization and presentation and the interpretation of past events and affairs of the people of Europe since the beginning of w ...
were an influential source of ideas for scientists during the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in m ...

and again , while the discovery in 1906 of previously unknown works by Archimedes in the
Archimedes Palimpsest Discovery reported in the New York Times on July 16, 1907 The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. All images and transcriptions are now free ...

has provided new insights into how he obtained mathematical results.

# Biography

Archimedes was born in the seaport city of
Syracuse, Sicily Syracuse ( ; it, Siracusa , or scn, Seragusa, label=none ; lat, Syrācūsae ; grc-att, wikt:Συράκουσαι, Συράκουσαι, Syrákousai ; grc-dor, wikt:Συράκοσαι, Συράκοσαι, Syrā́kosai ; grc-x-medieval, ...

, at that time a self-governing
colony In , a colony is a subject to a form of foreign rule. Though dominated by the foreign colonizers, colonies remain separate from the administration of the original country of the colonizers, the ' (or "mother country"). This administrative co ...
in
Magna Graecia Magna Graecia (, ; Latin meaning "Greater Greece", grc, Μεγάλη Ἑλλάς, ', it, Magna Grecia) was the name given by the Roman people, Romans to the coastal areas of Southern Italy in the present-day regions of Campania, Apulia, Basilicat ...

. The date of birth is based on a statement by the
Byzantine Greek Medieval Greek (also known as Middle Greek or Byzantine Greek) is the stage of the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is an independent branch of the Indo-European languages, ...
historian
John Tzetzes John Tzetzes ( gr, Ἰωάννης Τζέτζης, Iōánnēs Tzétzēs; c. 1110, Constantinople la, Constantinopolis ota, قسطنطينيه , alternate_name = Byzantion (earlier Greek name), Nova Roma ("New Rome"), Miklagard/Miklagarth (Old No ...
that Archimedes lived for 75 years. Heath, Thomas L. 1897. ''Works of Archimedes''. In ''
The Sand Reckoner ''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathemat ...
'', Archimedes gives his father's name as Phidias, an
astronomer An astronomer is a in the field of who focuses their studies on a specific question or field outside the scope of . They observe s such as s, s, , s and – in either (by analyzing the data) or . Examples of topics or fields astronomers stud ...

about whom nothing else is known.
Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; AD 46 – after AD 119) was a Greek Middle Platonist Middle Platonism is the modern name given to a stage in the development of Platonic philosophy, lasting from about 90 BC&nbs ...

wrote in his ''
Parallel Lives Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; AD 46–after AD 119) was a Greek Middle Platonism, Middle Platonist philosopher, historian, Biography, biographer, essayist, and priest at the Temple of Apollo (Delphi ...
'' that Archimedes was related to King
Hiero II Image:HieroII syracusa.jpg, Zeus' sacrificial altar built by Hiëro II in Syracuse Hiero II ( el, Ἱέρων Β΄; c. 308 BC – 215 BC) was the Greek tyrant of Syracuse, Sicily, Syracuse from 270 to 215 BC, and the illegitimate son of a Syracusan ...
, the ruler of Syracuse. A biography of Archimedes was written by his friend Heracleides, but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children. During his youth, Archimedes may have studied in
Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; : Rakodī; el, Αλεξάνδρεια ''Alexandria'') is the in after and , in , and a major economic centre. With a total population of 5,200,000, Alexandria is the ...

,
Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a spanning the and the of . It is bordered by the to , the () and to , the to the east, to , and to . In the northeast, the , which is the northern arm of the R ...

, where Conon of Samos and
Eratosthenes of Cyrene Eratosthenes of Cyrene (; el, Ἐρατοσθένης ὁ Κυρηναῖος, translit=Eratosthénēs ho Kurēnaĩos, ;  – ) was a Greek polymath A polymath ( el, πολυμαθής, ', "having learned much"; Latin Latin (, or , ...

were contemporaries. He referred to Conon of Samos as his friend, while two of his works (''
The Method of Mechanical Theorems ''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is considered one of the major surviving works of the ancient Greek ...
'' and the '' Cattle Problem'') have introductions addressed to Eratosthenes.In the preface to ''On Spirals'' addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works. Archimedes died during the
Second Punic War The Second Punic War, which lasted from 218 to 201BC, was the second of three wars fought between Carthage Carthage was the capital city of the ancient Ancient Carthage, Carthaginian civilization, on the eastern side of the Lake of Tunis in ...
, when Roman forces under General
Marcus Claudius Marcellus Marcus Claudius Marcellus (; 270 – 208 BC), five times elected as Roman consul, 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 ...

captured the city of Syracuse after a two-year-long
siege A siege is a military blockade of a city, or fortress, with the intent of conquering by attrition, or a well-prepared assault. This derives from la, sedere, lit=to sit. Siege warfare is a form of constant, low-intensity conflict characteri ...

. According to the popular account given by
Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; AD 46 – after AD 119) was a Greek Middle Platonist Middle Platonism is the modern name given to a stage in the development of Platonic philosophy, lasting from about 90 BC&nbs ...

, Archimedes was contemplating a
mathematical diagram, ms. from Lüneburg, A.D. 1200 Mathematical diagrams, such as chart A chart is a graphical representation for data visualization, in which "the data Data are units of information Information can be thought of as the resolution of u ...
when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he must not be harmed. Marcellus called Archimedes "a geometrical
Briareus In Greek mythology Greek mythology is the body of myths originally told by the Ancient Greece, ancient Greeks, and a genre of Ancient Greek folklore. These stories concern the Cosmogony, origin and Cosmology#Metaphysical cosmology, nature of th ...
." The last words attributed to Archimedes are "Do not disturb my circles", a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is often given in
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became the dominant la ...

as "'' Noli turbare circulos meos''," but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch.
Valerius Maximus Valerius Maximus () was a 1st-century Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through t ...

, writing in ''Memorable Doings and Sayings'' in the 1st century AD, gives the phrase as "" ("…but protecting the dust with his hands, said 'I beg of you, do not disturb this). The phrase is also given in
Katharevousa Greek Katharevousa ( el, Καθαρεύουσα, , literally "purifying anguage) is a conservative Conservatism is a Political philosophy, political and social philosophy promoting traditional social institutions. The central tenets of conservati ...
as "" (). The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a
sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

and a
cylinder A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditio ...

of the same height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. In 75 BC, 137 years after his death, the Roman
orator An orator, or oratist, is a public speaker, especially one who is eloquent or skilled. Etymology Recorded in English c. 1374, with a meaning of "one who pleads or argues for a cause", from Anglo-French ''oratour'', Old French ''orateur'' (14th c ...
Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman Roman or Romans usually refers to: *Rome, the capital city of Italy *Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people ...

was serving as
quaestor A ( , ; "investigator") was a public official in Ancient Rome. The position served different functions depending on the period. In the Roman Kingdom, ' (quaestors with judicial powers) were appointed by the king to investigate and handle murders. ...
in
Sicily Sicily ( it, Sicilia ; scn, Sicilia ) is the in the and one of the 20 of . It is one of the five and is officially referred to as ''Regione Siciliana''. The region has 5 million inhabitants. Its is . Sicily is in the central Mediterranean ...

. He had heard stories about the tomb of Archimedes, but none of the locals were able to give him the location. Eventually he found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription. A tomb discovered in the courtyard of the Hotel Panorama in Syracuse in the early 1960s was claimed to be that of Archimedes, but there was no compelling evidence for this and the location of his tomb today is unknown. The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege of Syracuse given by
Polybius Polybius (; grc-gre, Πολύβιος, ; ) was a Greek historian of the Hellenistic period The Hellenistic period covers the period of Mediterranean history between the death of Alexander the Great in 323 BC and the emergence of the ...

in his '' The Histories'' was written around seventy years after Archimedes' death, and was used subsequently as a source by Plutarch and
Livy Titus Livius (; 59 BC – AD 17), known in English as Livy ( ), was a historian. He wrote a monumental history of and the Roman people, titled , covering the period from the earliest legends of Rome before the traditional founding in 753 BC th ...
. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city.

# Discoveries and inventions

## Archimedes' principle

The most widely known
anecdote An ''anecdote'' is a brief, revealing account of an individual person or an incident: "a story with a point," such as to communicate an abstract idea about a person, place, or thing through the concrete details of a short narrative or to character ...

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 (''On architecture'', published as ''Ten Books on Architecture'') i ...

, a
votive crown A votive crown is a votive offering Bronze animal statuettes from Olympia, votive offerings, 8th–7th century BC. A votive offering or votive deposit is one or more objects displayed or deposited, without the intention of recovery or use, ...

for a temple had been made for King Hiero II of Syracuse, who had supplied the pure
gold Gold is a chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the same numb ...

to be used; Archimedes was asked to determine whether some
silver Silver is a chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the same n ...

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 The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

. In Vitruvius' account, Archimedes noticed, while taking a bath, that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the
volume Volume is a expressing the of enclosed by a . For example, the space that a substance (, , , or ) or occupies or contains. Volume is often quantified numerically using the , the . The volume of a container is generally understood to be the ...

of the crown. For practical purposes water is incompressible, so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "
Eureka Eureka often refers to: * Eureka (word) file:Eureka! Archimede.jpg, Archimedes exclaiming ''Eureka''. In his excitement, he forgets to dress and runs nude in the streets straight out of his bath ''Eureka'' ( grc, εὕρηκα) is an interjection ...
!" ( el, "εὕρηκα, ''heúrēka''!, ). The test was conducted successfully, proving that silver had indeed been mixed in. The story of the golden crown does not appear anywhere in the known works of Archimedes. Moreover, the practicality of the method it describes has been called into question, due to the extreme accuracy with which one would have to measure the water displacement. Archimedes may have instead sought a solution that applied the principle known in
hydrostatics Fluid statics or hydrostatics is the branch of fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical object ...
as
Archimedes' principle Archimedes' principle states that the upward buoyant force that is exerted on a body immersed in a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external f ...
, which he describes in his treatise ''
On Floating Bodies ''On Floating Bodies'' ( el, Περὶ τῶν ἐπιπλεόντων σωμάτων) is a Greek-language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is an independent branch of the Indo-European langua ...
''. This principle states that a body immersed in a fluid experiences a
buoyant force . Buoyancy (), or upthrust, is an upward force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies mat ...

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 An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the ...

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."

### Influence

In a 12th-century text titled ''
Mappae clavicula The Mappae clavicula is a medieval Latin text containing manufacturing recipes for crafts materials, including for metals, glass, mosaics, and dyes and tints for materials. The information and style in the recipes is very terse. Each recipe consists ...
'' there are instructions on how to perform the weighings in the water in order to calculate the percentage of silver used, and thus solve the problem. Dilke, Oswald A. W. 1990. ntitled ''
Gnomon A gnomon (, from Greek , ''gnōmōn'', literally: "one that knows or examines") 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 2 ...
'' 62(8):697–99. .
The Latin poem '' Carmen de ponderibus et mensuris'' of the 4th or 5th century describes the use of a hydrostatic balance to solve the problem of the crown, and attributes the method to Archimedes.

## Archimedes' screw

A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of
Syracuse Syracuse may refer to: Places Italy *Syracuse, Sicily Syracuse ( ; it, Siracusa , or scn, Seragusa, label=none ; lat, Syrācūsae ; grc-att, wikt:Συράκουσαι, Συράκουσαι, Syrákousai ; grc-dor, wikt:Συράκοσ ...

. The Greek writer
Athenaeus of Naucratis Athenaeus of Naucratis (; grc, Ἀθήναιος ὁ Nαυκρατίτης or Nαυκράτιος, ''Athēnaios Naukratitēs'' or ''Naukratios''; la, Athenaeus Naucratita) was a Greeks, Greek rhetorician and grammarian, flourishing about the end ...
described how King Hiero II commissioned Archimedes to design a huge ship, the ''
Syracusia ''Syracusia'' ( el, Συρακουσία, ''syrakousía'', literally "of Syracuse, Sicily, Syracuse") was an ancient Greeks, ancient Greek ship sometimes claimed to be the List of world's largest wooden ships, largest transport ship of antiquity. S ...
'', 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 Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history History (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ...
. According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a
gymnasium Gymnasium may refer to: *Gymnasium (ancient Greece), educational and sporting institution *Gymnasium (school), type of secondary school that prepares students for higher education **Gymnasium (Denmark) **Gymnasium (Germany) **Gymnasium UNT, high ...
and a temple dedicated to the goddess
Aphrodite Aphrodite; , , ) is an ancient Greek goddess associated with love Love encompasses a range of strong and positive emotional and mental states, from the most sublime virtue or good habit, the deepest Interpersonal relationship, interpe ...

among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the
Archimedes' screw Archimedes' screw, also known as the water screw, screw pump upPrinciple of screw pump (Saugseite = intake, Druckseite = outflow) A screw pump, also known as a water screw, is a positive-displacement (PD) pump that use one or several screws to mo ...
was purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a body of water into irrigation canals. The Archimedes' screw is still in use today for pumping liquids and granulated solids such as coal and grain. The Archimedes' screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the
Hanging Gardens of Babylon The Hanging Gardens of Babylon were one of the Seven Wonders of the Ancient World 324px, Timeline and map of the Seven Wonders. Dates in bold green and dark red are of their construction and destruction, respectively. The Seven Wonders of ...

. The world's first seagoing
steamship A steamship, often referred to as a steamer, is a type of steam-powered vessel Steam-powered vessels include steamboats and steamships. Smaller steamboats were developed first. They were replaced by larger steamships which were often ocean-going. ...

with a
screw propeller . A propeller is a device with a rotating hub and radiating blades that are set at a pitch to form a helical spiral, that, when rotated, exerts linear thrust Thrust is a reaction (physics), reaction force (physics), force described quantitat ...

was the SS ''Archimedes'', which was launched in 1839 and named in honor of Archimedes and his work on the screw.

## Claw of Archimedes

The Claw of Archimedes is a weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker," the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped onto an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it. There have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitled ''Superweapons of the Ancient World'' built a version of the claw and concluded that it was a workable device.

## Heat ray

Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse. The 2nd century AD author Lucian wrote that during the siege of Syracuse (c. 214–212 BC), Archimedes destroyed enemy ships with fire. Centuries later, Anthemius of Tralles mentions burning-glasses as Archimedes' weapon. The device, sometimes called the "Archimedes heat ray," was used to focus sunlight onto approaching ships, causing them to catch fire. In the modern era, similar devices have been constructed and may be referred to as a heliostat or solar furnace. This purported weapon has been the subject of ongoing debate about its credibility since the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in m ...

. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes. It has been suggested that a large array of highly polished bronze or copper shields acting as mirrors could have been employed to focus sunlight onto a ship.

### Modern tests

A test of the Archimedes heat ray was carried out in 1973 by the Greek scientist Ioannis Sakkas. The experiment took place at the Skaramagas naval base outside Athens. On this occasion 70 mirrors were used, each with a copper coating and a size of around . The mirrors were pointed at a plywood of a Roman warship at a distance of around . When the mirrors were focused accurately, the ship burst into flames within a few seconds. The plywood ship had a coating of bitumen, tar paint, which may have aided combustion. A coating of tar would have been commonplace on ships in the classical era.Lionel Casson, Casson, Lionel. 1995
''Ships and seamanship in the ancient world''
Baltimore: Johns Hopkins University Press. pp. 211–12. : "It was usual to smear the seams or even the whole hull with pitch or with pitch and wax". In Νεκρικοὶ Διάλογοι (''Dialogues of the Dead''), Lucian refers to coating the seams of a skiff with wax, a reference to pitch (tar) or wax.
In October 2005 a group of students from the Massachusetts Institute of Technology carried out an experiment with 127 one-foot (30 cm) square mirror tiles, focused on a wooden ship at a range of around . Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. It was concluded that the device was a feasible weapon under these conditions. The MIT group repeated the experiment for the television show ''MythBusters'', using a wooden fishing boat in San Francisco as the target. Again some charring occurred, along with a small amount of flame. In order to catch fire, wood needs to reach its autoignition temperature, which is around . When ''MythBusters'' broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (i.e. failed) because of the length of time and the ideal weather conditions required for combustion to occur. It was also pointed out that since Syracuse faces the sea towards the east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors. ''MythBusters'' also pointed out that conventional weaponry, such as flaming arrows or bolts from a catapult, would have been a far easier way of setting a ship on fire at short distances. In December 2010, ''MythBusters'' again looked at the heat ray story in a special edition entitled "MythBusters (2010 season)#Episode 157 – "President's Challenge", President's Challenge". Several experiments were carried out, including a large scale test with 500 schoolchildren aiming mirrors at a of a Roman sailing ship away. In all of the experiments, the sail failed to reach the required to catch fire, and the verdict was again "busted". The show concluded that a more likely effect of the mirrors would have been blinding, Glare (vision), dazzling, or distracting the crew of the ship.

## Lever

While Archimedes did not invent the
lever A lever ( or ) is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '':wikt:fulcrum, 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 effo ...

, he gave an explanation of the principle involved in his work ''On the Equilibrium of Planes''. Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas. According to Pappus of Alexandria, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth" ( el, δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω). Plutarch describes how Archimedes designed block and tackle, block-and-tackle pulley systems, allowing sailors to use the principle of
lever A lever ( or ) is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '':wikt:fulcrum, 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 effo ...

age to lift objects that would otherwise have been too heavy to move. Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.

## Astronomical instruments

Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus of Samos, Aristarchus' heliocentric model of the universe, in the ''Sand-reckoner''. Despite a lack of trigonometry and a table of chords, Archimedes describes the procedure and instrument used to make observations (a straight rod with pegs or grooves), applies correction factors to these measurements, and finally gives the result in the form of upper and lower bounds to account for observational error. Ptolemy, quoting Hipparchus, also references Archimedes's solstice observations in the ''Almagest''. This would make Archimedes the first known Greek to have recorded multiple solstice dates and times in successive years.
Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman Roman or Romans usually refers to: *Rome, the capital city of Italy *Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people ...

(106–43 BC) mentions Archimedes briefly in his dialogue, ''De re publica'', which portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse c. 212 BC, General
Marcus Claudius Marcellus Marcus Claudius Marcellus (; 270 – 208 BC), five times elected as Roman consul, 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 ...

is said to have taken back to Rome two mechanisms, constructed by Archimedes and used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Cicero mentions similar mechanisms designed by Thales, Thales of Miletus and Eudoxus of Cnidus. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius Philus, who described it thus: This is a description of a planetarium or orrery. Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled ''On Sphere-Making''. Modern research in this area has been focused on the Antikythera mechanism, another device built  BC that was probably designed for the same purpose. Constructing mechanisms of this kind would have required a sophisticated knowledge of Differential (mechanical device), differential gearing. This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.

# Mathematics

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics.
Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; AD 46 – after AD 119) was a Greek Middle Platonist Middle Platonism is the modern name given to a stage in the development of Platonic philosophy, lasting from about 90 BC&nbs ...

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 Cavalieri's principle, indivisibles (an early form of infinitesimals) in a way that is similar to modern Integral, integral calculus. Through proof by contradiction (''reductio ad absurdum''), he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface o ...

, and he employed it to approximate the areas of figures and the value of Pi, 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.

### Archimedean property

He also proved that the
area of a circle In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

was equal to π multiplied by the square of the radius of the circle ($\pi r^2$). In ''On the Sphere and Cylinder'', Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. Today this is known as the Archimedean property of real numbers. In ''Measurement of a Circle'', Archimedes gives the value of the square root of 3 as lying between (approximately 1.7320261) and (approximately 1.7320512). The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this result without offering any explanation of how he had obtained it. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results." It is possible that he used an iteration, iterative procedure to calculate these values.

## The infinite series

In ''The Quadrature of the Parabola'', Archimedes proved that the area enclosed by a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

and a straight line is times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an Series (mathematics)#History of the theory of infinite series, infinite geometric series with the Geometric series#Common ratio, 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 lines, and so on. This proof uses a variation of the series which sums to .

In ''
The Sand Reckoner ''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathemat ...
'', Archimedes set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote:
There are some, King Gelo (Gelo II, son of Hiero II), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited.
To solve the problem, Archimedes devised a system of counting based on the myriad. The word itself derives from the Greek , for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million, i.e., 10,000 x 10,000) and concluded that the number of grains of sand required to fill the universe would be 8 Names of large numbers, vigintillion, or 8.

# Writings

The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse. The written work of Archimedes has not survived as well as that of Euclid, and seven of his treatises are known to have existed only through references made to them by other authors. Pappus of Alexandria mentions ''On Sphere-Making'' and another work on polyhedron, 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 Polyhedron, polyhedra mentioned by Pappus of Alexandria; ''Catoptrica'', a work on optics mentioned by Theon of Alexandria; ''Principles'', addressed to Zeuxippus and explaining the number system used in ''
The Sand Reckoner ''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathemat ...
''; ''On Balances and Levers''; ''On Centers of Gravity''; ''On the Calendar''. Of the surviving works by Archimedes, T.L. Heath offers the following suggestion as to the order in which they were written: ''On the Equilibrium of Planes I'', ''The Quadrature of the Parabola'', ''On the Equilibrium of Planes II'', ''On the Sphere and the Cylinder I, II'', ''On Spirals'', ''On Conoids and Spheroids'', ''On Floating Bodies I, II'', ''On the Measurement of a Circle'', ''The Sand Reckoner''.
During his lifetime, Archimedes made his work known through correspondence with the mathematicians in
Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; : Rakodī; el, Αλεξάνδρεια ''Alexandria'') is the in after and , in , and a major economic centre. With a total population of 5,200,000, Alexandria is the ...

. The writings of Archimedes were first collected by the Byzantine Empire, Byzantine Greek architect
Isidore of Miletus Isidore of Miletus ( el, Ἰσίδωρος ὁ Μιλήσιος; Medieval Greek Medieval Greek (also known as Middle Greek or Byzantine Greek) is the stage of the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, anc ...
(c. 530 AD), while commentaries on the works of Archimedes written by Eutocius in the sixth century AD helped to bring his work a wider audience. Archimedes' work was translated into Arabic by Thābit ibn Qurra (836–901 AD), and Latin by Gerard of Cremona (c. 1114–1187 AD). During the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in m ...

, the ''Editio Princeps'' (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin. Around the year 1586, Galileo invented a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes.

## Surviving works

### ''On the Equilibrium of Planes''

There are two volumes to ''On the Equilibrium of Planes'': the being is in fifteen propositions with seven Axiom, postulates, while the second book is in ten propositions. In this work Archimedes explains the ''Torque, Law of the Lever'', stating, "Magnitude (mathematics), Magnitudes are in equilibrium at distances reciprocally proportional to their weights." Archimedes uses the principles derived to calculate the areas and center of mass, centers of gravity of various geometric figures including triangles, parallelograms and
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

s.

### ''Measurement of a Circle''

This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of Conon of Samos. In Proposition II, Archimedes gives an Approximations of π, approximation of the value of pi (), showing that it is greater than and less than .

### ''On Spirals''

This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. 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 the Sphere and Cylinder''

In this two-volume treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a
sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

and a circumscribed
cylinder A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditio ...

of the same height and diameter. The volume is 3 for the sphere, and 23 for the cylinder. The surface area is 42 for the sphere, and 62 for the cylinder (including its two bases), where is the radius of the sphere and cylinder. The sphere has a volume that of the circumscribed cylinder. Similarly, the sphere has an area that of the cylinder (including the bases). A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.

### ''On Conoids and Spheroids''

This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of 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. In the second part, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, similar to the way that icebergs float. Archimedes' principle of buoyancy is given in the work, stated as follows:
Any body wholly or partially immersed in a fluid experiences an upthrust equal to, but opposite in sense to, the weight of the fluid displaced.

### ''The Quadrature of the Parabola''

In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He achieves this by calculating the value of a geometric series that sums to infinity with the ratio .

### ''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 Discovery reported in the New York Times on July 16, 1907 The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. All images and transcriptions are now free ...

''. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Research published by Dr. Reviel Netz of Stanford University in 2003 argued that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. Dr. Netz calculates that the pieces can be made into a square 17,152 ways. The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded. The puzzle represents an example of an early problem in combinatorics. The origin of the puzzle's name is unclear, and it has been suggested that it is taken from the Ancient Greek word for 'throat' or 'gullet', ''stomachos'' (). Ausonius refers to the puzzle as , a Greek compound word formed from the roots of () and ().

### The cattle problem

This work was discovered by Gotthold Ephraim Lessing in a Greek manuscript consisting of a poem of 44 lines, in the Herzog August Library 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. This version of the problem was first solved by A. Amthor in 1880, and the answer is a very large number, approximately 7.760271.

### ''The Sand Reckoner''

In this treatise, also known as ''Psammites'', Archimedes counts the number of 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 myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8 in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. ''The Sand Reckoner'' is the only surviving work in which Archimedes discusses his views on astronomy.

### ''The Method of Mechanical Theorems''

This treatise was thought lost until the discovery of the
Archimedes Palimpsest Discovery reported in the New York Times on July 16, 1907 The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. All images and transcriptions are now free ...

in 1906. In this work Archimedes uses Archimedes' use of infinitesimals, infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface o ...

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
Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; : Rakodī; el, Αλεξάνδρεια ''Alexandria'') is the in after and , in , and a major economic centre. With a total population of 5,200,000, Alexandria is the ...

.

## Apocryphal works

Archimedes' ''Book of Lemmas'' or ''Liber Assumptorum'' is a treatise with fifteen propositions on the nature of circles. The earliest known copy of the text is in Arabic language, Arabic. The scholars T. L. Heath and Marshall Clagett argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. The ''Lemmas'' may be based on an earlier work by Archimedes that is now lost. It has also been claimed that Heron's formula for calculating the area of a triangle from the length of its sides was known to Archimedes.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." However, the first reliable reference to the formula is given by Hero of Alexandria, Heron of Alexandria in the 1st century AD.

## Archimedes Palimpsest

The foremost document containing the work of Archimedes is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg (historian), Johan Ludvig Heiberg visited
Constantinople la, Constantinopolis ota, قسطنطينيه , alternate_name = Byzantion (earlier Greek name), Nova Roma ("New Rome"), Miklagard/Miklagarth (), Tsargrad (), Qustantiniya (), Basileuousa ("Queen of Cities"), Megalopolis ("the Great City"), Πό ...

and examined a 174-page Goatskin (material), goatskin parchment of prayers written in the 13th century AD. He discovered that it was a palimpsest, a document with text that had been written over an erased older work. Palimpsests were created by scraping the ink from existing works and reusing them, which was a common practice in the Middle Ages as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th century AD copies of previously unknown treatises by Archimedes. The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. On 29 October 1998, it was sold at auction to an anonymous buyer for \$2 million at Christie's in New York City, New York. The palimpsest holds seven treatises, including the only surviving copy of ''On Floating Bodies'' in the original Greek. It is the only known source of ''The Method of Mechanical Theorems'', referred to by 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 is now stored at the Walters Art Museum in Baltimore, Maryland, where it has been subjected to a range of modern tests including the use of ultraviolet and light to read the overwritten text. The treatises in the Archimedes Palimpsest include: * ''On the Equilibrium of Planes'' * ''On Spirals'' * ''Measurement of a Circle'' * ''On the Sphere and Cylinder'' * ''
On Floating Bodies ''On Floating Bodies'' ( el, Περὶ τῶν ἐπιπλεόντων σωμάτων) is a Greek-language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is an independent branch of the Indo-European langua ...
'' * ''The Method of Mechanical Theorems'' * ''Stomachion'' * Speeches by the 4th century BC politician Hypereides * A commentary on Aristotle's ''Categories (Aristotle), Categories'' * Other works

# Legacy

* Galileo Galilei, Galileo praised Archimedes many times, and referred to him as a "superhuman" and as "his master". 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." * 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 him which reads in Latin: ''Transire suum pectus mundoque potiri'' ('Rise above oneself and grasp the world)'. * Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963). * The exclamation of Eureka (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.

* Arbelos * Archimedean point * Axiom of Archimedes, Archimedes' axiom * Archimedes number * Archimedes paradox * Archimedean solid * Archimedes' circles, Archimedes' twin circles * Diocles (mathematician), Diocles * List of things named after Archimedes * Methods of computing square roots * Pseudo-Archimedes * Salinon * Steam cannon * Zhang Heng

# References

## Citations

*Carl Benjamin Boyer, Boyer, Carl Benjamin. 1991. ''iarchive:historyofmathema00boye, A History of Mathematics''. New York: Wiley. . *Marshall Clagett, Clagett, Marshall. 1964–1984. ''Archimedes in the Middle Ages'' 1–5. Madison, WI: University of Wisconsin Press. *Eduard Jan Dijksterhuis, Dijksterhuis, Eduard J. [1938] 1987. ''Archimedes'', translated. Princeton: Princeton University Press. . *Mary Gow, Gow, Mary. 2005. ''iarchive:archimedesmathem0000gowm, Archimedes: Mathematical Genius of the Ancient World''. Enslow Publishing. . *Hasan, Heather. 2005. ''iarchive:archimedesfather00hasa, Archimedes: The Father of Mathematics''. Rosen Central. . * Heath, Thomas L. 1897. iarchive:worksofarchimede029517mbp, ''Works of Archimedes''. Dover Publications. . Complete works of Archimedes in English. *Reviel Netz, Netz, Reviel, and William Noel. 2007. ''The Archimedes Codex''. Orion Publishing Group. . *Clifford A. Pickover, Pickover, Clifford A. 2008. ''Archimedes to Hawking: Laws of Science and the Great Minds Behind Them''. Oxford University Press. . *Simms, Dennis L. 1995. ''Archimedes the Engineer''. Continuum International Publishing Group. . *Sherman K. Stein, Stein, Sherman. 1999. ''iarchive:archimedeswhatdi00stei, Archimedes: What Did He Do Besides Cry Eureka?''. Mathematical Association of America. .

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Heiberg's Edition of Archimedes
'' Texts in Classical Greek, with some in English. *iarchive:geometricalsolu00smitgoog, ''The Method of Mechanical Theorems'', translated by L.G. Robinson * * * * * *
The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
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Photograph of the Sakkas experiment in 1973

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