Classical Greek philosophy
Ancient philosophy
Archimedes Thoughtful by Fetti (1620)
Name
Archimedes of Syracuse (Greek: Άρχιμήδης)
Birthc. Domenico Fetti (also spelled Feti c 1589 – 1623 was an Italian Baroque painter active mainly in Rome, Mantua and Venice. 287 BC (Syracuse, Sicily, Magna Graecia)
Deathc. Syracuse (Siracusa Sicilian: Sarausa, Classical Greek: / transliterated Syrakousai) is a historic City in 212 BC (Syracuse)
Natural philosophy
Main interestsmathematics, physics, engineering, astronomy, invention
Notable ideashydrostatics, levers,
infinitesimals

Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time. A mathematician is a person whose primary area of study and research is the field of Mathematics. [2][3] He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of Pi. 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 is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems [4] He also defined the spiral bearing his name, formulas for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers. The Archimedean spiral (also known as the arithmetic spiral) is a Spiral named after the 3rd century BC Greek Mathematician The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically A surface of revolution is a Surface created by rotating a Curve lying on some plane (the Generatrix) around a Straight line (the Axis

Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. The Roman Republic was the phase of the ancient Roman civilization characterized by a Republican form of government a period which began with the overthrow of the Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere inscribed within a cylinder. Marcus Tullius Cicero ( Classical Latin ˈkikeroː usually ˈsɪsərəʊ in English January 3, 106 BC &ndash December 7, 43 BC was a Roman "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Geometry, an inscribed Planar Shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis Archimedes had proved that the sphere has two thirds of the volume and surface area of the cylinder (including the bases of the latter), and regarded this as the greatest of his mathematical achievements.

Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was made only by Isidore of Miletus (c. Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια Isidore of Miletus (Ισίδωρος ο Μιλήσιοςin Greek) was one of the two Greek Architects (the other being Anthemius 530 AD), while commentaries on the works of Archimedes written by Eutocius in the sixth century AD opened them to wider readership for the first time. Eutocius of Ascalon (ca 480 &ndash ca 540 was a Greek Mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian Conics. The relatively few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance,[5] while the discovery in 1906 of previously unknown works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere The Archimedes Palimpsest is a Palimpsest on Parchment in the form of a Codex which originally was a copy of an otherwise unknown work of the ancient [6]

## Biography

This bronze statue of Archimedes is at the Archenhold Observatory in Berlin. The Archenhold Observatory, named in honor of Friedrich Simon Archenhold, is an Observatory in Berlin-Treptow. Berlin is the capital city and one of sixteen states of Germany. It was sculpted by Gerhard Thieme and unveiled in 1972.

Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a colony of Magna Graecia. Syracuse (Siracusa Sicilian: Sarausa, Classical Greek: / transliterated Syrakousai) is a historic City in The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years. Byzantine Greeks or Byzantines or Romaioi, is a conventional term used by modern historians to refer to the medieval Greek or Hellenized citizens John (Johannes Tzetzes (Ιωάννης Τζέτζης (c 1110 &ndash 1180 was a Byzantine Poet and Grammarian known to have lived at Constantinople [7] In The Sand Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing is known. The Sand Reckoner ( Greek: Ψαμμίτης Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number Historically Astronomy was more concerned with the classification and description of phenomena in the sky while Astrophysics attempted to explain these phenomena Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse. Lucius Mestrius Plutarchus ( Greek: Μέστριος Πλούταρχος c Plutarch 's Lives of the Noble Greeks and Romans, commonly called Parallel Lives or Plutarch's Lives, is a series of Hieron II, king of Syracuse from 270 to 215 BC was the illegitimate son of a Syracusan noble Hierocles, who claimed descent from Gelon He was a former [8] A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. [9] It is unknown, for instance, whether he ever married or had children. During his youth Archimedes may have studied in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene were contemporaries. Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια Ancient Egypt was an Ancient Civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now Conon of Samos (ca 280 BC - ca 220 BC was a Greek astronomer and Mathematician. Eratosthenes of Cyrene ( Greek; 276 BC - 194 BC was a Greek Mathematician, Poet, athlete, Geographer and He referred to Conon of Samos as his friend, while two of his works (The Sand Reckoner and the Cattle Problem) have introductions addressed to Eratosthenes. The Sand Reckoner ( Greek: Ψαμμίτης Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number Archimedes' cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of Polynomial equations [a]

Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. The Second Punic War (referred to as "The War Against Hannibal" by the Romans lasted from 218 to 201 BC and involved combatants in the western Marcus Claudius Marcellus (ca 268 BC-208 BC was a Roman general one of the commanders of the Roman Army during the Second Punic War and the conqueror of Syracuse According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. Lucius Mestrius Plutarchus ( Greek: Μέστριος Πλούταρχος c 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 lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he not be harmed. [10]

The last words attributed to Archimedes are "Do not disturb my circles" (Greek: μή μου τούς κύκλους τάραττε), a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly This quote is often given in Latin as "Noli turbare circulos meos", but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. [10]

The sphere has 2/3 the volume and surface area of the circumscribing cylinder. A sphere and cylinder were placed on the tomb of Archimedes at his request.

The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same height and diameter. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis Archimedes had proved that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. In 75 BC, 137 years after his death, the Roman orator Cicero was serving as quaestor in Sicily. Marcus Tullius Cicero ( Classical Latin ˈkikeroː usually ˈsɪsərəʊ in English January 3, 106 BC &ndash December 7, 43 BC was a Roman Quaestors were originally appointed by the Consuls to investigate criminal acts and determine if the consul needed to take public action Sicily ( Italian and Sicilian: Sicilia) is an autonomous region of Italy. He had heard stories about the tomb of Archimedes, but none of the locals was able to give him the location. Eventually he found the tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription. [11]

The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege of Syracuse given by Polybius in his Universal History was written around seventy years after Archimedes' death, and was used subsequently as a source by Plutarch and Livy. Polybius (ca 203 &ndash 120 BC, Greek) was a Greek historian of the Hellenistic Period noted for his book called The Histories Titus Livius (traditionally 59 BC &ndash AD 17 known as Livy in English, was a Roman historian who wrote a monumental history of Rome 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. [12]

## Discoveries and inventions

The story about the golden crown does not appear in the known works of Archimedes, but in his treatise On Floating Bodies he gives the principle known in hydrostatics as Archimedes' Principle. Eureka ( Greek "I have found it" is an exclamation used as an Interjection to celebrate a discovery Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Fluid statics (also called hydrostatics) is the Science of Fluids at rest and is a sub-field within Fluid mechanics. In Physics, buoyancy ( BrE IPA: /ˈbɔɪənsi/ is the upward Force on an object produced by the surrounding liquid or gas in which it is This states that a body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid. [16]

While Archimedes did not invent the lever, he wrote the earliest known rigorous explanation of the principle involved. According to Pappus of Alexandria, his work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth. Pappus of Alexandria ( Greek) (c 290 &ndash c 350 was one of the last great Greek mathematicians of antiquity known for his Synagoge or Collection " (Greek: "δος μοι πα στω και ταν γαν κινάσω")[17] Plutarch describes how Archimedes designed block and tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly A block and tackle is a system of two or more Pulleys with a Rope or Cable threaded between them usually used to lift or pull heavy loads A pulley (also called a sheave or block) is a Wheel with a groove between two Flanges around its Circumference [18]

The Archimedes' screw can raise water efficiently. The Archimedes' screw, Archimedean screw, or screwpump is a Machine historically used for transferring water from a low-lying body of water into Irrigation

A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of Naucratis described how King Hieron II commissioned Archimedes to design a huge ship, the Syracusia, which could be used for luxury travel, carrying supplies, and as a naval warship. Athenaeus ( Ancient Greek - Athếnaios Naukratios Latin Athenaeus Naucratita of Naucratis in Egypt Greek rhetorician and grammarian flourished The Syracusia was an ancient Greek ship With a length of 55 meters (180 The Syracusia is said to have been the largest ship built in classical antiquity. [19] According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite among its facilities. The gymnasium in Ancient Greece functioned as a training facility for competitors in public Games It was also a place for socializing and engaging in intellectual Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes screw was purportedly developed in order to remove the bilge water. The Archimedes' screw, Archimedean screw, or screwpump is a Machine historically used for transferring water from a low-lying body of water into Irrigation 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 low-lying body of water into irrigation canals. The Archimedes screw is still in use today for pumping liquids and semifluid 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. Marcus Vitruvius Pollio (born c 80–70 BC died after c 15 BC was a Roman Writer, Architect and Engineer (possibly praefectus fabrum [20][21][22]

The Claw of Archimedes is another weapon that he is said to have designed in order to defend the city of Syracuse. The Claw ( harpágē, "snatcher" of Archimedes was an ancient weapon devised by Archimedes to defend the seaward portion 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 on to 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. [23][24]

Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War. A catapult is any one of a number of non-handheld mechanical devices used to throw a Projectile a great distance without the aid of an explosive substance—particularly various An odometer (often known colloquially as a mileometer or milometer) is a device used for indicating Distance traveled by an Automobile or other The First Punic War ( 264 to 241 BC) was the first of three major wars fought between Carthage and the Roman Republic. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled. [25]

Cicero (106 BC–43 BC) mentions Archimedes briefly in his dialogue De re publica, which portrays a fictional conversation taking place in 129 BC. Marcus Tullius Cicero ( Classical Latin ˈkikeroː usually ˈsɪsərəʊ in English January 3, 106 BC &ndash December 7, 43 BC was a Roman A dialogue (sometimes spelled dialog) is a reciprocal Conversation between two or more entities. De re publica ( On the commonwealth, see below) is a dialogue by Cicero, written in six Books between 54 and After the capture of Syracuse c. 212 BC, General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Marcus Claudius Marcellus (ca 268 BC-208 BC was a Roman general one of the commanders of the Roman Army during the Second Punic War and the conqueror of Syracuse Cicero mentions similar mechanisms designed by Thales of Miletus and Eudoxus of Cnidus. Thales of Miletus According to Bertrand Russell, "Philosophy begins with Thales Eudoxus of Cnidus ( Greek Εὔδοξος ὁ Κνίδιος (410 or 408 BC &ndash 355 or 347 BC was a Greek Astronomer, Mathematician 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:

 “ Hanc sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in aere illo quot diebus in ipso caelo succederet, ex quo et in caelo sphaera solis fieret eadem illa defectio, et incideret luna tum in eam metam quae esset umbra terrae, cum sol e regione. Gaius Sulpicius Gallus, Roman general statesman and orator Under Lucius Aemilius Paulus, his intimate friend he commanded the 2nd legion in the campaign against Lucius Furius Philus was a Consul of Ancient Rome in 136 BC. He was a member of the Scipionic circle and particularly close to Scipio Aemilianus — When Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that position which was its shadow on the Earth, when the Sun was in line. [26][27] ”

This is a description of a planetarium or orrery. A planetarium is a Theatre built primarily for presenting educational and entertaining shows about Astronomy and the night sky or for training in Celestial navigation This article is on the mechanical device For the British peerage see Earl of Orrery. Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the construction of these mechanisms entitled On Sphere-Making. Pappus of Alexandria ( Greek) (c 290 &ndash c 350 was one of the last great Greek mathematicians of antiquity known for his Synagoge or Collection On Sphere-Making is the title of a Lost work by Archimedes, mentioned by Pappus of Alexandria. Modern research in this area has been focused on the Antikythera mechanism, another device from classical antiquity that was probably designed for the same purpose. The Antikythera mechanism (ˌæntɪkɪˈθɪərə an-ti-ki- theer -uh is an ancient mechanical Calculator (also described as the first known " mechanical Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing. This article deals with the concept of a differential in mechanical engineering. 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. [28][29]

### The Archimedes Heat Ray - myth or reality?

There is debate over whether or not Archimedes may have used mirrors acting as a parabolic reflector to burn ships attacking Syracuse

The 2nd century AD historian Lucian wrote that during the Siege of Syracuse (c. A parabolic reflector (or dish or mirror) is a Parabola -shaped reflective device used to collect or distribute Energy such as Syracuse (Siracusa Sicilian: Sarausa, Classical Greek: / transliterated Syrakousai) is a historic City in The 2nd century is the period from 101 to 200 in accordance with the Julian calendar in the Christian / Common Era. Lucian of Samosata (Λουκιανός ὁ Σαμοσατεύς Lucianus c 214–212 BC), Archimedes repelled an attack by Roman soldiers with a burning-glass. A burning-glass 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 [30] The device was used to focus sunlight on to the approaching ships, causing them to catch fire. This claim, sometimes called the "Archimedes heat ray", has been the subject of ongoing debate about its credibility since the Renaissance. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes. [31] 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 on to a ship. Bronze is any of a broad range of Copper alloys, usually with Tin as the main additive but sometimes with other elements such as Phosphorus Copper (ˈkɒpɚ is a Chemical element with the symbol Cu (cuprum and Atomic number 29 This would have used the principle of the parabolic reflector in a manner similar to a solar furnace. A parabolic reflector (or dish or mirror) is a Parabola -shaped reflective device used to collect or distribute Energy such as For the cooking apparatus see Solar cooker. For electricity generation see Solar updraft tower.

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. Nearest places Aspropyrgos, north Chaidari, east Nikaia, southeast Keratsini, Athens (ˈæθənz Αθήνα Athina,) the Capital and largest city of Greece, dominates the Attica periphery as one of the world's On this occasion 70 mirrors were used, each with a copper coating and a size of around five by three feet (1. 5 by 1 m). The mirrors were pointed at a plywood mock-up of a Roman warship at a distance of around 160 feet (50 m). When the mirrors were focused accurately, the ship burst into flames within a few seconds. The plywood ship had a coating of tar paint, which may have aided combustion. Bitumen is a mixture of organic Liquids that are highly Viscous, black sticky entirely soluble in Carbon disulfide, and composed primarily [32]

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 mocked-up wooden ship at a range of around 100 feet (30 m). Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. It was concluded that the weapon was a feasible device 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. MythBusters is a Popular science Television program produced by Australian firm Beyond Television Productions originally for the The City and County of San Francisco is the fourth most populous city Again some charring occurred, along with a small amount of flame. In order to catch fire, wood needs to reach its flash point, which is around 300 degrees Celsius (570 °F). The flash point of a flammable liquid is the lowest Temperature at which it can form an ignitable mixture in air [33]

When MythBusters broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "busted" (or failed) because of the length of time and the ideal weather conditions required for combustion to occur. It was also pointed out that since Syracuse faces the sea towards the east, the Roman fleet would have had to attack during the morning for optimal gathering of light by the mirrors. MythBusters also pointed out that conventional weaponry, such as flaming arrows or bolts from a catapult, would have been a far easier way of setting a ship on fire at short distances. [1]

## Mathematics

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life. Lucius Mestrius Plutarchus ( Greek: Μέστριος Πλούταρχος c[34]

Archimedes used the method of exhaustion to approximate the value of π. 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 IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems

In The Measurement of a Circle, Archimedes gives the value of the square root of 3 as being more than 265/153 (approximately 1. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose 7320261) and less than 1351/780 (approximately 1. 7320512). The actual value is approximately 1. 7320508, making this a very accurate estimate. He introduced this result without offering any explanation of the method used to obtain it. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results. John Wallis ( November 23, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the "[35]

In The Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. The Quadrature of the Parabola is a treatise on Geometry, written by Archimedes in the 3rd century B In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line He expressed the solution to the problem as a geometric series that summed to infinity with the ratio 1/4:

$\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. \;$

If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and so on. In Mathematics, a geometric series is a series with a constant ratio between successive terms. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with A ratio is an expression which compares quantities relative to each other A secant line of a Curve is a line that (locally intersects two points on the curve This proof is a variation of the infinite series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with

In The Sand Reckoner, Archimedes set out to calculate the number of grains of sand that the universe could contain. The Sand Reckoner ( Greek: Ψαμμίτης Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number 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. Hieron II, king of Syracuse from 270 to 215 BC was the illegitimate son of a Syracusan noble Hierocles, who claimed descent from Gelon He was a former " To solve the problem, Archimedes devised a system of counting based on the myriad. Myriad is a classical Greek name for the Number 104 = 10000. In modern English the word refers to an unspecified large quantity The word is from the Greek μυριάς murias, for the number 10,000. He proposed a number system using powers of a myriad of myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8×1063, which can also be expressed as eight vigintillion. Names of numbers larger than a quadrillion are almost never used for reasons discussed further below [36]

## Writings

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. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Pappus of Alexandria mentions On Sphere-Making and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the now-lost Catoptrica. Pappus of Alexandria ( Greek) (c 290 &ndash c 350 was one of the last great Greek mathematicians of antiquity known for his Synagoge or Collection On Sphere-Making is the title of a Lost work by Archimedes, mentioned by Pappus of Alexandria. What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions Theon ( Greek: Θέων ca 335 - ca 405 AD was a Greek (or as some scholars contend an Egyptian) Scholar and Mathematician who lived Refraction is the change in direction of a Wave due to a change in its Speed. [b] During his lifetime, Archimedes made his work known through correspondence with the mathematicians in Alexandria. Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια The writings of Archimedes were collected by the Byzantine architect Isidore of Miletus (c. Isidore of Miletus (Ισίδωρος ο Μιλήσιοςin Greek) was one of the two Greek Architects (the other being Anthemius 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. Eutocius of Ascalon (ca 480 &ndash ca 540 was a Greek Mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian Conics. Archimedes' work was translated into Arabic by Thābit ibn Qurra (836–901 AD), and Latin by Gerard of Cremona (c. (836 in Harran, Mesopotamia &ndash February 18, 901 in Baghdad) was an Arab astronomer, mathematician Gerard of Cremona ( Italian: Gerardo da Cremona; Latin: Gerardus Cremonensis; c 1114–1187 AD). During the Renaissance, the Editio Princeps (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere "Basilia" redirects here For the Fly Genus, see Basilia (fly. [37] Around the year 1586 Galileo Galilei invented a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher [38]

### Surviving works

Archimedes is said to have remarked about the lever: "Give me a place to stand on, and I will move the Earth. "
• On the Equilibrium of Planes (two volumes)
The first book is in fifteen propositions with seven postulates, while the second book is in ten propositions. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In this work Archimedes explains the Law of the Lever, stating:
 “ Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance. ”
Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, paraboloids, and hemispheres. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Mathematics, a paraboloid is a Quadric surface of special kind "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe [39]
• On the Measurement of the 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. Conon of Samos (ca 280 BC - ca 220 BC was a Greek astronomer and Mathematician. In Proposition II, Archimedes shows that the value of π (Pi) is greater than 223/71 and less than 22/7. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The latter figure was used as an approximation of π throughout the Middle Ages and is still used today when a rough figure is required.
• On Spirals
This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. The Archimedean spiral (also known as the arithmetic spiral) is a Spiral named after the 3rd century BC Greek Mathematician It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property Do not confuse with Angular frequency The unit for angular velocity is rad/s Equivalently, in polar coordinates (r, θ) it can be described by the equation
$\, r=a+b\theta$
with real numbers a and b. Coordinates are numbers which describe the location of points in a plane or in space In Mathematics, the real numbers may be described informally in several different ways This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume
• On the Sphere and the Cylinder (two volumes)
In this treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the The volume is $\tfrac{4}{3}\pi r^3$ for the sphere, and r3 for the cylinder. The surface area is r2 for the sphere, and r2 for the cylinder (including its two bases), where r is the radius of the sphere and cylinder. The sphere has a volume and surface area two-thirds that of the cylinder. A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.
• On Conoids and Spheroids
This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids. In Geometry, a cross section is the intersection of a body in 2-dimensional space with a line or of a body in 3-dimensional space with a plane etc A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex
• On Floating Bodies (two volumes)
In the first part of this treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. Eratosthenes of Cyrene ( Greek; 276 BC - 194 BC was a Greek Mathematician, Poet, athlete, Geographer and The fluids described by Archimedes are not self-gravitating, since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.
Archimedes is commemorated on a Greek postage stamp from 1983.
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. ”
In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. The Quadrature of the Parabola is a treatise on Geometry, written by Archimedes in the 3rd century B In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line He achieves this by calculating the value of a geometric series that sums to infinity with the ratio 1/4. In Mathematics, a geometric series is a series with a constant ratio between successive terms. A ratio is an expression which compares quantities relative to each other
This is a dissection puzzle similar to a Tangram, and the treatise describing it was found in more complete form in the Archimedes Palimpsest. Ostomachion is a mathematical treatise attributed to Archimedes. A dissection puzzle, also called a transformation puzzle is a Tiling puzzle where a solver is given a set of pieces that can be assembled in different ways to produce Tangram ( is a Dissection puzzle. It consists of seven pieces called tans, which fit together to form a shape of some sort The Archimedes Palimpsest is a Palimpsest on Parchment in the form of a Codex which originally was a copy of an otherwise unknown work of the ancient Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides 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. Leland Stanford Junior University, commonly known as Stanford University or simply Stanford, is a private Research university located in The figure given by Dr. Netz is that the pieces can be made into a square in 17,152 ways. [40] The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded. [41] The puzzle represents an example of an early problem in combinatorics. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects
The origin of the puzzle's name is unclear, and it has been suggested it is taken from the Ancient Greek word for throat or gullet, stomachos (στόμαχος). The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c [42] Ausonius refers to the puzzle as Ostomachion, a Greek compound word formed from the roots of ὀστέον (osteon - bone) and μάχη (machē - fight). This article is about the Roman poet Ausonius For John Ausonius the Swedish murderer see John Ausonius. The puzzle is also known as the Loculus of Archimedes or Archimedes' Box. [43]
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. Archimedes' cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of Polynomial equations Gotthold Ephraim Lessing ( 22 January, 1729 15 February, 1781) was a German Writer, Philosopher, Dramatist Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. It is addressed to Eratosthenes and the mathematicians in Alexandria. Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only There is a more difficult version of the problem in which some of the answers are required to be square numbers. In Mathematics, a square number, sometimes also called a Perfect square, is an Integer that can be written as the square of some other This version of the problem was first solved by A. Amthor[44] in 1880, and the answer is a very large number, approximately 7. 760271×10206544. [45]
In this treatise, Archimedes counts the number of grains of sand that will fit inside the universe. The Sand Reckoner ( Greek: Ψαμμίτης Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos, contemporary ideas about the size of the Earth and the distance between various celestial bodies. In Astronomy, heliocentrism is the theory that the Sun is at the center of the Solar System. The Solar System consists of the Sun and those celestial objects bound to it by Gravity. Aristarchus (Ἀρίσταρχος 310 BC - ca 230 BC) was a Greek Astronomer and Mathematician, born on the island of 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 8×1063 in modern notation. Myriad is a classical Greek name for the Number 104 = 10000. In modern English the word refers to an unspecified large quantity The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner or Psammites is the only surviving work in which Archimedes discusses his views on astronomy. [46]
• The Method of Mechanical Theorems
This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. The Archimedes Palimpsest is a Palimpsest on Parchment in the form of a Codex which originally was a copy of an otherwise unknown work of the ancient In this work Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes' use of infinitesimals is the first attested explicit use Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. 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 As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria. Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια

### 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. Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language 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. Sir Thomas Little Heath ( October 5, 1861 &ndash March 16, 1940) was a British civil servant Mathematician, classical Marshall Clagett ( January 23, 1916 - October 21, 2005) was an American scholar who specialized in the history of Science The Lemmas may be based on an earlier work by Archimedes that is now lost. [47]

It has also been claimed by the Arab scholar Abu'l Raihan Muhammed al-Biruni that Heron's formula for calculating the area of a triangle from the length of its sides was known to Archimedes. In Geometry, Heron's (or Hero's formula states that the Area (A of a Triangle whose sides have lengths a, b, and [c] However, the first reliable reference to the formula is given by Heron of Alexandria in the 1st century  AD. Hero (or Heron) of Alexandria ( Ήρων ο Αλεξανδρεύς) (c [48]

## Archimedes Palimpsest

Main article: Archimedes Palimpsest

The foremost document containing the work of Archimedes is the Archimedes Palimpsest. The Archimedes Palimpsest is a Palimpsest on Parchment in the form of a Codex which originally was a copy of an otherwise unknown work of the ancient A dissection puzzle, also called a transformation puzzle is a Tiling puzzle where a solver is given a set of pieces that can be assembled in different ways to produce The Archimedes Palimpsest is a Palimpsest on Parchment in the form of a Codex which originally was a copy of an otherwise unknown work of the ancient The Archimedes Palimpsest is a Palimpsest on Parchment in the form of a Codex which originally was a copy of an otherwise unknown work of the ancient In 1906, the Danish professor Johan Ludvig Heiberg realized that a 174-page goatskin parchment of prayers written in the 13th century AD was in fact a palimpsest: the text was written over erased older work, which he identified as copies, written in the 10th century AD, of previously unknown treatises by Archimedes. Johan Ludvig Heiberg (1854&ndash1928 was a Danish Philologist and Historian. A palimpsest is a Manuscript page whether from scroll or Book that has been written on scraped off and used again [49] The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. Constantinople (Κωνσταντινούπολις Konstantinoúpolis, or gr ἡ Πόλις hē Polis, Latin: la CONSTANTINOPOLIS On October 29, 1998 it was sold at auction to an anonymous buyer for $2 million at Christie's in New York. Events 437 - Valentinian III, Western Roman Emperor, marries Licinia Eudoxia, daughter of his cousin Theodosius II Year 1998 ( MCMXCVIII) was a Common year starting on Thursday (link will display full 1998 Gregorian calendar) Christie's is a leading art business and a fine arts Auction house The City of New York [50] The palimpsest holds seven treatises, including the only surviving copy of On Floating Bodies in the original Greek. It is the only known source of the Method of Mechanical Theorems, referred to by Suidas and thought to have been lost forever. Stomachion was also discovered in the palimpsest, with a more complete analysis of the puzzle than had been found in previous texts. The palimpsest is now stored at the Walters Art Museum in Baltimore, Maryland, where it has been subjected to a range of modern tests including the use of ultraviolet and x-ray light to read the overwritten text. The Walters Art Museum located in Baltimore Maryland 's Mount Vernon neighborhood is a public art museum founded in 1934 Ultraviolet ( UV) light is Electromagnetic radiation with a Wavelength shorter than that of Visible light, but longer than X-rays X-radiation (composed of X-rays) is a form of Electromagnetic radiation. Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 [51] The treatises in the Archimedes Palimpsest are: On the Equilibrium of Planes, On Spirals, The Measurement of the Circle, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems and Stomachion. ## Legacy The Fields Medal carries a portrait of Archimedes. The Fields Medal is a prize awarded to two three or four Mathematicians not over 40 years of age at each International Congress of the International Mathematical There is a crater on the Moon named Archimedes (29. In the broadest sense the term impact crater can be applied to any depression natural or manmade resulting from the high velocity impact of a projectile with larger body Archimedes is a large lunar Impact crater on the eastern edges of the Mare Imbrium. 7° N, 4. 0° W) in his honor, and a lunar mountain range, the Montes Archimedes (25. Montes Archimedes is a Mountain range on the Moon. They were named for the Archimedes crater that lies to the north which in turn has an Eponym 3° N, 4. 6° W). [52] The asteroid 3600 Archimedes is named after him. Asteroids, sometimes called Minor planets or planetoids', are bodies—primarily of the inner Solar System —that are smaller than planets but 3600 Archimedes is a small main belt Asteroid, belonging to the Rafita family. [53] The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with his proof concerning the sphere and the cylinder. The Fields Medal is a prize awarded to two three or four Mathematicians not over 40 years of age at each International Congress of the International Mathematical 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). [54] Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963). The German Democratic Republic ( GDR; Deutsche Demokratische Republik DDR; commonly known in English as East Germany) was a Socialist state Greece (Ελλάδα transliterated: Elláda, historically, Ellás,) officially the Hellenic Republic (Ελληνική Δημοκρατία Italy (Italia officially the Italian Republic, (Repubblica Italiana is located on the Italian Peninsula in Southern Europe, and on the two largest Nicaragua (ˌnɪkəˈrɑgwə officially the Republic of Nicaragua () is a representative democratic republic and the largest nation in Central America The Most Serene Republic of San Marino (Serenissima Repubblica di San Marino is a country in the Apennine Mountains. Spain () or the Kingdom of Spain (Reino de España is a country located mostly in southwestern Europe on the Iberian Peninsula. [55] The exclamation of Eureka! attributed to Archimedes is the state motto of California. Eureka ( Greek "I have found it" is an exclamation used as an Interjection to celebrate a discovery California ( is a US state on the West Coast of the United States, along the Pacific Ocean. In this instance the word refers to the discovery of gold near Sutter's Mill in 1848 which sparked the California Gold Rush. Sutter's Mill was a Sawmill owned by 19th century pioneer John Sutter. The California Gold Rush (1848&ndash1855 began on January 24 1848 when Gold was discovered by James Marshall at Sutter's Mill in Coloma, California [56] A movement for civic engagement targeting universal access to health care in the US state of Oregon has been named the "Archimedes Movement", headed by former Oregon Governor John Kitzhaber. Oregon ( is a state in the Pacific Northwest region of the United States. John Albert Kitzhaber (born March 5 1947 in Colfax, Washington) is a Physician, member of the Democratic Party and [57] ## See also ## Notes and references ### Notes a. The axiom of Archimedes can be stated in modern notation as follows Let x be any real number An Archimedes number (not to be confused with Archimedes' constant, π) named after the ancient Greek scientist Archimedes —used to determine the motion The Archimedes paradox, named after Archimedes of Syracuse states that an object can float in a quantity of water that has less volume than the object itself if its average In Abstract algebra, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some groups The Archimedes' screw, Archimedean screw, or screwpump is a Machine historically used for transferring water from a low-lying body of water into Irrigation In Geometry an Archimedean solid is a highly symmetric semi-regular convex Polyhedron composed of two or more types of Regular polygons meeting Archimedes' use of infinitesimals is the first attested explicit use Diocles (ca 240 BCE - ca 180 BCE was a Greek Mathematician and Geometer. This article presents and explains several methods which can be used to calculate Square roots Exponential identity Pocket calculators typically implement good Pseudo-Archimedes is a name given to an unknown source quoted by various sources of the Islamic Golden Age such as Al-Jazari as a reference for the construction of The salinon (meaning "salt-cellar" in Greek is a Geometrical figure that consists of four Semicircles It was first introduced by Archimedes Marcus Vitruvius Pollio (born c 80–70 BC died after c 15 BC was a Roman Writer, Architect and Engineer (possibly praefectus fabrum Zhang Heng ( (CE 78–139 was an astronomer, mathematician, inventor, geographer, cartographer, artist, poet ^ 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. Conon of Samos (ca 280 BC - ca 220 BC was a Greek astronomer and Mathematician. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works. b. ^ The treatises by Archimedes known to exist only through references in the works of other authors are: On Sphere-Making and a work on polyhedra mentioned by Pappus of Alexandria; Catoptrica, a work on optics mentioned by Theon of Alexandria; Principles, addressed to Zeuxippus and explaining the number system used in The Sand Reckoner; On Balances and Levers; On Centers of Gravity; On the Calendar. On Sphere-Making is the title of a Lost work by Archimedes, mentioned by Pappus of Alexandria. Theon ( Greek: Θέων ca 335 - ca 405 AD was a Greek (or as some scholars contend an Egyptian) Scholar and Mathematician who lived The Sand Reckoner ( Greek: Ψαμμίτης Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number 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. Sir Thomas Little Heath ( October 5, 1861 &ndash March 16, 1940) was a British civil servant Mathematician, classical c. ^ Boyer, Carl Benjamin A History of Mathematics (1991) ISBN 0471543977 "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula — k = √(s(s − a)(s − b)(s − c)), where s is the semiperimeter — was known to Archimedes several centuries before Heron lived. Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he Arabic scholars also attribute to Archimedes the 'theorem on the broken chord' … Archimedes is reported by the Arabs to have given several proofs of the theorem. A chord of a Curve is a geometric Line segment whose endpoints both lie on the curve " ### References 1. ^ a b Archimedes Death Ray: Testing with MythBusters. MIT. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 2. ^ Calinger, Ronald (1999). A Contextual History of Mathematics. Prentice-Hall, 150. ISBN 0-02-318285-7. “Shortly after Euclid, compiler of the definitive textbook, came Archimedes of Syracuse (ca. 287–212 B. C. ), the most original and profound mathematician of antiquity. ” 3. ^ Archimedes of Syracuse. The MacTutor History of Mathematics archive (January 1999). Retrieved on 2008-06-09. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 53 - Roman Emperor Nero marries Claudia Octavia 62 - Claudia Octavia commits 4. ^ O'Connor, J. J. and Robertson, E. F. (February 1996). A history of calculus. University of St Andrews. The University of St Andrews is the oldest University in Scotland and third oldest in the English-speaking world, having been founded between Retrieved on 2007-08-07. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 322 BC - Battle of Crannon between Athens and Macedon following the death of Alexander the Great. 5. ^ Bursill-Hall, Piers. Galileo, Archimedes, and Renaissance engineers. sciencelive with the University of Cambridge. Retrieved on 2007-08-07. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 322 BC - Battle of Crannon between Athens and Macedon following the death of Alexander the Great. 6. ^ Archimedes - The Palimpsest. Walters Art Museum. The Walters Art Museum located in Baltimore Maryland 's Mount Vernon neighborhood is a public art museum founded in 1934 Retrieved on 2007-10-14. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1066 - Norman Conquest: Battle of Hastings - In England on Senlac Hill seven miles from Hastings, the forces 7. ^ T. L. Heath, Works of Archimedes, 1897 8. ^ Plutarch. Sir Thomas Little Heath ( October 5, 1861 &ndash March 16, 1940) was a British civil servant Mathematician, classical Lucius Mestrius Plutarchus ( Greek: Μέστριος Πλούταρχος c Parallel Lives Complete e-text from Gutenberg.org. Project Gutenberg. Project Gutenberg, abbreviated as PG, is a volunteer effort to Digitize, archive and distribute Cultural works Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 9. ^ O'Connor, J. J. and Robertson, E. F. . Archimedes of Syracuse. University of St Andrews. Retrieved on 2007-01-02. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 366 - The Alamanni cross the frozen Rhine River in large numbers invading the Roman Empire. 10. ^ a b Rorres, Chris. Death of Archimedes: Sources. Courant Institute of Mathematical Sciences. The Courant Institute of Mathematical Sciences (CIMS is a division of New York University (NYU and serves as a center for research and advanced training in computer science Retrieved on 2007-01-02. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 366 - The Alamanni cross the frozen Rhine River in large numbers invading the Roman Empire. 11. ^ Rorres, Chris. Tomb of Archimedes: Sources. Courant Institute of Mathematical Sciences. Retrieved on 2007-01-02. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 366 - The Alamanni cross the frozen Rhine River in large numbers invading the Roman Empire. 12. ^ Rorres, Chris. Siege of Syracuse. Courant Institute of Mathematical Sciences. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 13. ^ Vitruvius. Marcus Vitruvius Pollio (born c 80–70 BC died after c 15 BC was a Roman Writer, Architect and Engineer (possibly praefectus fabrum De Architectura, Book IX, paragraphs 9–12, text in English and Latin. University of Chicago. The University of Chicago is a Private university located principally in the Hyde Park neighborhood of Chicago. Retrieved on 2007-08-30. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1363 - Beginning date of the Battle of Lake Poyang; the forces of two Chinese rebel leaders— Chen Youliang and 14. ^ Incompressibility of Water. Harvard University. Retrieved on 2008-02-27. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 1560 - The Treaty of Berwick, which would expel the French from Scotland, is signed by England and the Congregation 15. ^ HyperPhysics. HyperPhysics is an educational resource about Physics topics The Information architecture of the website is based on trees that organize topics from general Buoyancy. Georgia State University. Georgia State University ( GSU) is an urban Research University in Downtown Atlanta, Georgia, USA. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 16. ^ Carroll, Bradley W. Archimedes' Principle. Weber State University. Weber State University (ˈwiːbər is a public university located in the city of Ogden in Weber County Utah USA Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 17. ^ Quoted by Pappus of Alexandria in Synagoge, Book VIII 18. ^ Pulleys. Pappus of Alexandria ( Greek) (c 290 &ndash c 350 was one of the last great Greek mathematicians of antiquity known for his Synagoge or Collection Society of Women Engineers. The Society of Women Engineers (SWE founded in 1950 is a Not-for-profit educational and service organization Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 19. ^ Casson, Lionel (1971). Ships and Seamanship in the Ancient World. Princeton University Press. ISBN 0691035369. 20. ^ Dalley, Stephanie. Oleson, John Peter. Sennacherib, Archimedes, and the Water Screw: The Context of Invention in the Ancient World. Technology and Culture Volume 44, Number 1, January 2003 (PDF). Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 21. ^ Rorres, Chris. Archimedes Screw - Optimal Design. Courant Institute of Mathematical Sciences. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 22. ^ Watch an animation of an Archimedes screw. Wikimedia Commons. Wikimedia Commons (or simply Commons) is a repository of Free content images sound and other multimedia files Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 23. ^ Rorres, Chris. Archimedes' Claw - Illustrations and Animations - a range of possible designs for the claw. Courant Institute of Mathematical Sciences. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 24. ^ Carroll, Bradley W. Archimedes' Claw - watch an animation. Weber State University. Retrieved on 2007-08-12. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1099 - First Crusade: Battle of Ascalon - Crusaders under the command of Godfrey of Bouillon defeat Fatimid 25. ^ Ancient Greek Scientists: Hero of Alexandria. Technology Museum of Thessaloniki. Retrieved on 2007-09-14. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 81 - Domitian becomes Emperor of the Roman Empire upon the death of his brother Titus. 26. ^ Cicero. Marcus Tullius Cicero ( Classical Latin ˈkikeroː usually ˈsɪsərəʊ in English January 3, 106 BC &ndash December 7, 43 BC was a Roman De re publica 1.xiv §21. thelatinlibrary. com. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 27. ^ Cicero. Marcus Tullius Cicero ( Classical Latin ˈkikeroː usually ˈsɪsərəʊ in English January 3, 106 BC &ndash December 7, 43 BC was a Roman De re publica Complete e-text in English from Gutenberg.org. Project Gutenberg. Project Gutenberg, abbreviated as PG, is a volunteer effort to Digitize, archive and distribute Cultural works Retrieved on 2007-09-18. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 96 - Nerva is proclaimed Roman Emperor after Domitian is assassinated 28. ^ Rorres, Chris. Spheres and Planetaria. Courant Institute of Mathematical Sciences. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 29. ^ Ancient Moon 'computer' revisited. BBC News (November 29, 2006). Events 1777 - San Jose California, is founded as el Pueblo de San José de Guadalupe Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 30. ^ Hippias, C. 2. 31. ^ John Wesley. John Wesley (ˈwɛslɪ ( – March 2, 1791) was an Anglican cleric and Christian theologian who was the founder of the (Evangelical A Compendium of Natural Philosophy (1810) Chapter XII, Burning Glasses. Online text at Wesley Center for Applied Theology. Retrieved on 2007-09-14. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 81 - Domitian becomes Emperor of the Roman Empire upon the death of his brother Titus. 32. ^ Archimedes' Weapon. Time Magazine (November 26, 1973). Time (trademarked in capitals as TIME) is a weekly American Newsmagazine, similar to Newsweek and Events 43 BC - The Second Triumvirate alliance of Gaius Julius Caesar Octavianus ("Octavian" later "Caesar Augustus" Year 1973 ( MCMLXXIII) was a Common year starting on Monday (link will display full calendar of the 1973 Gregorian calendar. Retrieved on 2007-08-12. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1099 - First Crusade: Battle of Ascalon - Crusaders under the command of Godfrey of Bouillon defeat Fatimid 33. ^ Bonsor, Kevin. How Wildfires Work. HowStuffWorks. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 34. ^ Plutarch. Lucius Mestrius Plutarchus ( Greek: Μέστριος Πλούταρχος c Extract from Parallel Lives. fullbooks. com. Retrieved on 2007-08-07. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 322 BC - Battle of Crannon between Athens and Macedon following the death of Alexander the Great. 35. ^ Quoted in T. L. Heath, Works of Archimedes, Dover Publications, ISBN 0-486-42084-1. 36. ^ Carroll, Bradley W. The Sand Reckoner. Weber State University. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 37. ^ Editions of Archimedes' Work. Brown University Library. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 38. ^ Van Helden, Al. The Galileo Project: Hydrostatic Balance. Rice University. William Marsh Rice University (commonly called Rice University and opened in 1912 as The William Marsh Rice Institute for the Advancement of Letters Science and Art Retrieved on 2007-09-14. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 81 - Domitian becomes Emperor of the Roman Empire upon the death of his brother Titus. 39. ^ Heath,T. L. . The Works of Archimedes (1897). The unabridged work in PDF form (19 MB). Archive.org. The Internet Archive ( IA) is a Nonprofit organization dedicated to maintaining an on-line Library and archive of Web and Retrieved on 2007-10-14. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1066 - Norman Conquest: Battle of Hastings - In England on Senlac Hill seven miles from Hastings, the forces 40. ^ Kolata, Gina (December 14, 2003). Events 1287 - St Lucia's flood: The Zuider Zee sea wall in the Netherlands collapses killing over 50000 people Year 2003 ( MMIII) was a Common year starting on Wednesday of the Gregorian calendar. In Archimedes' Puzzle, a New Eureka Moment. The New York Times. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 41. ^ Ed Pegg Jr. (November 17, 2003). Events 284 - Diocletian is proclaimed emperor by his soldiers Year 2003 ( MMIII) was a Common year starting on Wednesday of the Gregorian calendar. The Loculus of Archimedes, Solved. Mathematical Association of America. The Mathematical Association of America ( MAA) is a professional society that focuses on Mathematics accessible at the undergraduate level Retrieved on 2008-05-18. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 1152 - Henry II of England marries Eleanor of Aquitaine. 42. ^ Rorres, Chris. Archimedes' Stomachion. Courant Institute of Mathematical Sciences. Retrieved on 2007-09-14. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 81 - Domitian becomes Emperor of the Roman Empire upon the death of his brother Titus. 43. ^ Graeco Roman Puzzles. Gianni A. Sarcone and Marie J. Waeber. Retrieved on 2008-05-09. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 1457 BC - Battle of Megiddo (15th century BC between Thutmose III and a large Canaanite coalition under the King of 44. ^ B. Krumbiegel, A. Amthor, Das Problema Bovinum des Archimedes, Historisch-literarische Abteilung der Zeitschrift Für Mathematik und Physik 25 (1880) 121-136, 153-171. 45. ^ Calkins, Keith G. Archimedes' Problema Bovinum. Andrews University. Andrews University is a Seventh-day Adventist university in Berrien Springs Michigan. Retrieved on 2007-09-14. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 81 - Domitian becomes Emperor of the Roman Empire upon the death of his brother Titus. 46. ^ English translation of The Sand Reckoner. University of Waterloo. The University of Waterloo (also referred to as UW and Waterloo) is a comprehensive Public university in the city of Waterloo, Ontario Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France. 47. ^ Archimedes' Book of Lemmas. cut-the-knot. Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics. Retrieved on 2007-08-07. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 322 BC - Battle of Crannon between Athens and Macedon following the death of Alexander the Great. 48. ^ Wilson, James W. Problem Solving with Heron's Formula. University of Georgia. The University of Georgia ( UGA) is a public research University located in Athens, Georgia, the oldest and largest of the Retrieved on 2007-09-14. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 81 - Domitian becomes Emperor of the Roman Empire upon the death of his brother Titus. 49. ^ Miller, Mary K. (March, 2007). Reading Between the Lines. Smithsonian Magazine. Smithsonian is a monthly Magazine published by the Smithsonian Institution in Washington D Retrieved on 2008-01-24. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 41 - Gaius Caesar (Caligula, known for his eccentricity and cruel Despotism, is Assassinated by his disgruntled 50. ^ Rare work by Archimedes sells for$2 million. CNN (October 29, 1998). Cable News Network, usually referred to by its Initialism CNN, is a major English language Television network founded in 1980 by Ted Turner Events 437 - Valentinian III, Western Roman Emperor, marries Licinia Eudoxia, daughter of his cousin Theodosius II Year 1998 ( MCMXCVIII) was a Common year starting on Thursday (link will display full 1998 Gregorian calendar) Retrieved on 2008-01-15. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 588 BC - Nebuchadrezzar II of Babylon lays siege to Jerusalem under Zedekiah 's reign
51. ^ X-rays reveal Archimedes' secrets. BBC News (August 2, 2006). Events 338 BC - A Macedonian army led by Philip II defeated the combined forces of Athens and Thebes in the Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France.
52. ^ Friedlander, Jay and Williams, Dave. Oblique view of Archimedes crater on the Moon. NASA. The National Aeronautics and Space Administration ( NASA, ˈnæsə is an agency of the United States government, responsible for the nation's public space program Retrieved on 2007-09-13. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 509 BC - The Temple of Jupiter on Rome 's Capitoline Hill is dedicated on the ides of September
53. ^ Planetary Data System. NASA. Retrieved on 2007-09-13. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 509 BC - The Temple of Jupiter on Rome 's Capitoline Hill is dedicated on the ides of September
54. ^ Fields Medal. International Mathematical Union. The International Mathematical Union is an international Non-governmental organization devoted to international cooperation in the field of Mathematics. Retrieved on 2007-07-23. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1632 - Three hundred colonists bound for New France depart from Dieppe France.
55. ^ Rorres, Chris. Stamps of Archimedes. Courant Institute of Mathematical Sciences. Retrieved on 2007-08-25. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1248 - The Dutch city of Ommen receives city rights and fortification rights from Otto III the
56. ^ California Symbols. California State Capitol Museum. Retrieved on 2007-09-14. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 81 - Domitian becomes Emperor of the Roman Empire upon the death of his brother Titus.
57. ^ The Archimedes Movement