Archimedes

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Archimedes of Syracuse
Domenico-Fetti Archimedes 1620.jpg
Archimedes Thoughtful
by Domenico Fetti (1620)
Native name
Ἀρχιμήδης
Bornc.287 BC
Diedc.212 BC (aged around 75)
Known for
Scientific career
Fields

Archimedes of Syracuse ( /ˌɑːrkɪˈmdz/ ; [2] Greek : Ἀρχιμήδης ; c.287 – c.212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer. [3] Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Generally considered the greatest mathematician of antiquity and one of the greatest of all time, [4] [5] [6] [7] [8] [9] Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola. [10]

Greek language Language spoken in Greece, Cyprus and Southern Albania

Greek is an independent branch of the Indo-European family of languages, native to Greece, Cyprus and other parts of the Eastern Mediterranean and the Black Sea. It has the longest documented history of any living Indo-European language, spanning more than 3000 years of written records. Its writing system has been the Greek alphabet for the major part of its history; other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic, and many other writing systems.

The Greeks or Hellenes are an ethnic group native to Greece, Cyprus, southern Albania, Italy, Turkey, Egypt and, to a lesser extent, other countries surrounding the Mediterranean Sea. They also form a significant diaspora, with Greek communities established around the world.

Greek mathematics Mathematics texts written by and ideas stemming from Ancient Greek mathematicians

Greek mathematics refers to mathematics texts written by and ideas stemming from the Classical and Hellinistic (i.e. post-Alexander the Great Greek culture, extant from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture and language. The word "mathematics" itself derives from the Ancient Greek: μάθημα, romanized: máthēmaAttic Greek: [má.tʰɛː.ma]Koine Greek: [ˈma.θi.ma], meaning "subject of instruction". The study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations.

Contents

Other mathematical achievements include deriving an accurate approximation of pi, defining and investigating the spiral bearing his name, and creating a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics, including an explanation of the principle of the lever. He is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.

The number π is a mathematical constant. Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes spelled out as "pi". It is also called Archimedes' constant.

Exponentiation Mathematical operation

Exponentiation is a mathematical operation, written as bn, involving two numbers, the baseb and the exponent or powern. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:

Statics is the branch of mechanics that is concerned with the analysis of loads acting on physical systems that do not experience an acceleration (a=0), but rather, are in static equilibrium with their environment. When in static equilibrium, the acceleration of the system is zero and the system is either at rest, or its center of mass moves at constant velocity. The application of Newton's second law to a system gives:

Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, which Archimedes had requested be placed on his tomb to represent his mathematical discoveries.

Siege of Syracuse (213–212 BC) siege of the Second Punic War ending in the fall of the Hellenistic city of Syracuse to the Roman Republic

The Siege of Syracuse by the Roman Republic took place in 213–212 BC, at the end of which the Hellenistic city of Syracuse, located on the east coast of Sicily, fell. The Romans stormed the city after a protracted siege giving them control of the entire island of Sicily. During the siege, the city was protected by weapons developed by Archimedes. Archimedes, the great inventor and polymath, was slain at the conclusion of the siege by a Roman soldier, in contravention of the Roman proconsul Marcellus' instructions to spare his life.

Roman Republic Period of ancient Roman civilization (509–27 BC)

The Roman Republic was the era of classical Roman civilization beginning with the overthrow of the Roman Kingdom, traditionally dated to 509 BC, and ending in 27 BC with the establishment of the Roman Empire. It was during this period that Rome's control expanded from the city's immediate surroundings to hegemony over the entire Mediterranean world.

Cicero 1st-century BC Roman lawyer, orator, philosopher and statesman

Marcus Tullius Cicero was a Roman statesman, orator, lawyer and philosopher, who served as consul in the year 63 BC. He came from a wealthy municipal family of the Roman equestrian order, and is considered one of Rome's greatest orators and prose stylists.

Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c.530 AD by Isidore of Miletus in Byzantine Constantinople, while commentaries on the works of Archimedes written by Eutocius in the sixth century AD opened them to wider readership for the first time. The relatively few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance, [11] [12] 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. [13]

Alexandria Metropolis in Egypt

Alexandria is the second-largest city in Egypt and a major economic centre, extending about 32 km (20 mi) along the coast of the Mediterranean Sea in the north central part of the country. Its low elevation on the Nile delta makes it highly vulnerable to rising sea levels. Alexandria is an important industrial center because of its natural gas and oil pipelines from Suez. Alexandria is also a popular tourist destination.

Isidore of Miletus Byzantine Greek architect

Isidore of Miletus was one of the two main Byzantine Greek architects that Emperor Justinian I commissioned to design the cathedral Hagia Sophia in Constantinople from 532 to 537. The creation of an important compilation of Archimedes' works has been attributed to him. The spurious Book XV from Euclid's Elements has been partly attributed to Isidore of Miletus.

Eutocius of Ascalon was a Greek mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian Conics.

Biography

Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years. [14] In The Sand Reckoner , Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse. [15] A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. [16] 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. He referred to Conon of Samos as his friend, while two of his works ( The Method of Mechanical Theorems and the Cattle Problem ) have introductions addressed to Eratosthenes. [a]

Syracuse, Sicily Comune in Sicily, Italy

Syracuse is a historic city on the island of Sicily, the capital of the Italian province of Syracuse. The city is notable for its rich Greek history, culture, amphitheatres, architecture, and as the birthplace of the preeminent mathematician and engineer Archimedes. This 2,700-year-old city played a key role in ancient times, when it was one of the major powers of the Mediterranean world. Syracuse is located in the southeast corner of the island of Sicily, next to the Gulf of Syracuse beside the Ionian Sea.

Colonies in antiquity Aspect of history

Colonies in antiquity were post-Iron Age city-states founded from a mother-city, not from a territory-at-large. Bonds between a colony and its metropolis remained often close, and took specific forms during the period of classical antiquity. Generally, colonies founded by the ancient Phoenicians, Carthage, Rome, Alexander the Great and his successors remained tied to their metropolis, but Greek colonies of the Archaic and Classical eras were sovereign and self-governing from their inception. While Greek colonies were often founded to solve social unrest in the mother-city, by expelling a part of the population, Hellenistic, Roman, Carthaginian, and Han Chinese colonies were used for expansion and empire-building.

Magna Graecia Ancient Greek colonies in southern Italy

Magna Graecia was the name given by the Romans to the coastal areas of Southern Italy in the present-day regions of Campania, Apulia, Basilicata, Calabria and Sicily; these regions were extensively populated by Greek settlers, particularly the Achaean settlements of Croton, and Sybaris, and to the north, the settlements of Cumae and Neapolis. The settlers who began arriving in the 8th century BC brought with them their Hellenic civilization, which was to leave a lasting imprint on Italy, such as in the culture of ancient Rome. Most notably the Roman poet Ovid referred to the south of Italy as Magna Graecia in his poem Fasti.

The Death of Archimedes (1815) by Thomas Degeorge Death of Archimedes.png
The Death of Archimedes (1815) by Thomas Degeorge

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. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a 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 must not be harmed. [18] Marcellus called Archimedes "a geometrical Briareus". [19]

Second Punic War second war between the Roman Republic and Carthage, fought between 218 and 201 BCE

The Second Punic War, also referred to as The Hannibalic War and by the Romans the War Against Hannibal, was the second of three Punic Wars between Ancient Carthage and the Roman Republic, with the participation of Greek polities and Numidian and Iberian forces on both sides. It was one of the deadliest human conflicts of ancient times. Fought across the entire Western Mediterranean region for 17 years and regarded by ancient historians as the greatest war in history, it was waged with unparalleled resources, skill, and hatred. It saw hundreds of thousands killed, some of the most lethal battles in military history, the destruction of cities, and massacres and enslavements of civilian populations and prisoners of war by both sides.

Marcus Claudius Marcellus Roman military leader and Second Punic War general

Marcus Claudius Marcellus, five times elected as consul of the Roman Republic, was an important Roman military leader during the Gallic War of 225 BC and the Second Punic War. Marcellus gained the most prestigious award a Roman general could earn, the spolia opima, for killing the Gallic military leader and king Viridomarus in hand-to-hand combat in 222 BC at the Battle of Clastidium. Furthermore, he is noted for having conquered the fortified city of Syracuse in a protracted siege during which Archimedes, the famous mathematician, scientist and inventor, was killed. Marcus Claudius Marcellus died in battle in 208 BC, leaving behind a legacy of military conquests and a reinvigorated Roman legend of the spolia opima.

Plutarch Ancient Greek historian and philosopher

Plutarch, later named, upon becoming a Roman citizen, Lucius Mestrius Plutarchus, was a Greek biographer and essayist, known primarily for his Parallel Lives and Moralia. He is classified as a Middle Platonist. Plutarch's surviving works were written in Greek, but intended for both Greek and Roman readers.

The last words attributed to Archimedes are "Do not disturb my circles", a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is often given in Latin as " Noli turbare circulos meos ," but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch. Valerius Maximus, writing in Memorable Doings and Sayings in the 1st century AD, gives the phrase as "...sed protecto manibus puluere 'noli' inquit, 'obsecro, istum disturbare'" – "... but protecting the dust with his hands, said 'I beg of you, do not disturb this.'" The phrase is also given in Katharevousa Greek as "μὴ μου τοὺς κύκλους τάραττε!" (Mē mou tous kuklous taratte!). [18]

Cicero Discovering the Tomb of Archimedes (1805) by Benjamin West Cicero Discovering the Tomb of Archimedes by Benjamin West.jpeg
Cicero Discovering the Tomb of Archimedes (1805) by Benjamin West

The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. In 75 BC, 137 years after his death, the Roman orator Cicero was serving as quaestor in Sicily. 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. [20] 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. [21]

The standard versions of the life of Archimedes were written long after his death by the historians of Ancient Rome. The account of the siege of Syracuse given by Polybius in his The Histories was written around seventy years after Archimedes' death, and was used subsequently as a source by Plutarch and Livy. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city. [22]

Discoveries and inventions

Archimedes' principle

By placing a metal bar in a container with water on a scale, the bar displaces as much water as its own volume, increasing its mass and weighing down the scale.

The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. According to Vitruvius, a votive crown for a temple had been made for King Hiero II of Syracuse, who had supplied the pure gold to be used, and Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith. [23] 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. While taking a bath, he noticed that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the volume of the crown. For practical purposes water is incompressible, [24] 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!" (Greek : "εὕρηκα, heúrēka!", meaning "I have found [it]!"). [23] The test was conducted successfully, proving that silver had indeed been mixed in. [25]

The story of the golden crown does not appear 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. [26] Archimedes may have instead sought a solution that applied the principle known in hydrostatics as Archimedes' principle, which he describes in his treatise On Floating Bodies. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. [27] 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 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." [28] In a 12th-century text titled Mappae clavicula there are instructions on how to perform the weighings in the water in order to calculate the percentage of silver used, and thus solve the problem. [29] [30] 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. [29]

Archimedes' screw

The Archimedes' screw can raise water efficiently. Archimedes-screw one-screw-threads with-ball 3D-view animated small.gif
The Archimedes' screw can raise water efficiently.

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 Hiero II commissioned Archimedes to design a huge ship, the Syracusia , which could be used for luxury travel, carrying supplies, and as a naval warship. The Syracusia is said to have been the largest ship built in classical antiquity. [31] According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes' screw was purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a low-lying 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. [32] [33] [34] The world's first seagoing steamship with a screw propeller was the SS Archimedes, which was launched in 1839 and named in honor of Archimedes and his work on the screw. [35]

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. [36] [37]

Heat ray

Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse. Archimedes Heat Ray conceptual diagram.svg
Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse.
Artistic interpretation of Archimedes' mirror used to burn Roman ships. Painting by Giulio Parigi, c. 1599 Archimedes-Mirror by Giulio Parigi.jpg
Artistic interpretation of Archimedes' mirror used to burn Roman ships. Painting by Giulio Parigi, c. 1599

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. [38] 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. [39]

This purported weapon has been the subject of ongoing debate about its credibility since the Renaissance. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes. [40] 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.

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 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. [41] A coating of tar would have been commonplace on ships in the classical era. [d]

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 mock-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 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 300 °C (570 °F). [42] [43]

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. [44]

In December 2010, MythBusters again looked at the heat ray story in a special edition entitled "President's Challenge". Several experiments were carried out, including a large scale test with 500 schoolchildren aiming mirrors at a mock-up of a Roman sailing ship 400 feet (120 m) away. In all of the experiments, the sail failed to reach the 210 °C (410 °F) required to catch fire, and the verdict was again "busted". The show concluded that a more likely effect of the mirrors would have been blinding, dazzling, or distracting the crew of the ship. [45]

Other discoveries and inventions

While Archimedes did not invent the lever, he gave an explanation of the principle involved in his work On the Equilibrium of Planes. Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas. [46] [47] 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." (Greek : δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω) [48] 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. [49] 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. [50]

Cicero (106–43 BC) mentions Archimedes briefly in his dialogue De re publica , which portrays a fictional conversation taking place in 129 BC. After the capture of Syracuse c. 212 BC, General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms, constructed by Archimedes and used as aids in astronomy, which showed the motion of the Sun, Moon and five planets. Cicero mentions similar mechanisms designed by Thales of Miletus and Eudoxus of Cnidus. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to Lucius Furius Philus, who described it thus:

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.
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. [51] [52]

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 c.100 BC that was probably designed for the same purpose. [53] Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing. [54] 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. [55] [56]

Mathematics

Archimedes used Pythagoras' Theorem to calculate the side of the 12-gon from that of the hexagon and for each subsequent doubling of the sides of the regular polygon. PiArchimede4.svg
Archimedes used Pythagoras' Theorem to calculate the side of the 12-gon from that of the hexagon and for each subsequent doubling of the sides of the regular polygon.

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." [57] Archimedes was able to use infinitesimals in a way that is similar to modern integral calculus. Through proof by contradiction (reductio ad absurdum), he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π. In Measurement of a Circle he did this by drawing a larger regular hexagon outside a circle and 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 31/7 (approximately 3.1429) and 310/71 (approximately 3.1408), consistent with its actual value of approximately 3.1416. [58] He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle (πr2). In On the Sphere and Cylinder , Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. This is the Archimedean property of real numbers. [59]

As proven by Archimedes, the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure. Parabolic segment and inscribed triangle.svg
As proven by Archimedes, the area of the parabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure.

In Measurement of a Circle, Archimedes gives the value of the square root of 3 as lying between 265/153 (approximately 1.7320261) and 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 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." [60] It is possible that he used an iterative procedure to calculate these values. [61]

In The Quadrature of the Parabola , Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio 1/4:

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 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.

In The Sand Reckoner , Archimedes set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote: "There are some, King Gelo (Gelo II, son of Hiero II), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited." To solve the problem, Archimedes devised a system of counting based on the myriad. The word is from the Greek μυριάςmurias, 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 vigintillion, or 8×1063. [62]

Writings

The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse. [63] The written work of Archimedes has not survived as well as that of Euclid, and seven of his treatises are known to have existed only through references made to them by other authors. Pappus of Alexandria mentions On Sphere-Making and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the now-lostCatoptrica. [b] During his lifetime, Archimedes made his work known through correspondence with the mathematicians in Alexandria. The writings of Archimedes were first collected by the Byzantine Greek architect Isidore of Miletus (c. 530 AD), while commentaries on the works of Archimedes written by Eutocius in the sixth century AD helped to bring his work a wider audience. Archimedes' work was translated into Arabic by Thābit ibn Qurra (836–901 AD), and Latin by Gerard of Cremona (c. 1114–1187 AD). During the Renaissance, the Editio Princeps (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin. [64] 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. [65]

Surviving works

The first book is in fifteen propositions with seven postulates, while the second book is in ten propositions. In this work Archimedes explains the Law of the Lever , stating, "Magnitudes are in equilibrium at distances reciprocally proportional to their weights."
Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, parallelograms and parabolas. [66]
This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of Conon of Samos. In Proposition II, Archimedes gives an approximation of the value of pi (π), showing that it is greater than 223/71 and less than 22/7.
This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation
with real numbers a and b. This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases. A sphere and cylinder were placed on the tomb of Archimedes at his request. (see also: Equiareal map) Esfera Arquimedes.svg
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases. A sphere and cylinder were placed on the tomb of Archimedes at his request. (see also: Equiareal map)
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. The volume is 4/3πr3 for the sphere, and 2πr3 for the cylinder. The surface area is 4πr2 for the sphere, and 6πr2 for the cylinder (including its two bases), where r is the radius of the sphere and cylinder. The sphere has a volume two-thirds that of the circumscribed cylinder. Similarly, the sphere has an area two-thirds that of the cylinder (including the bases). A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.
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 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. 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.
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. He achieves this by calculating the value of a geometric series that sums to infinity with the ratio 1/4.
Stomachion is a dissection puzzle in the Archimedes Palimpsest. Stomachion.JPG
Stomachion is a dissection puzzle in the Archimedes Palimpsest.
This is a dissection puzzle similar to a Tangram, and the treatise describing it was found in more complete form in the Archimedes Palimpsest. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Research published by Dr. Reviel Netz of Stanford University in 2003 argued that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. Dr. Netz calculates that the pieces can be made into a square 17,152 ways. [67] The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded. [68] 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 (στόμαχος). [69] Ausonius refers to the puzzle as Ostomachion, a Greek compound word formed from the roots of ὀστέον (osteon, bone) and μάχη (machē, fight). The puzzle is also known as the Loculus of Archimedes or Archimedes' Box. [70]
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 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 [71] in 1880, and the answer is a very large number, approximately 7.760271×10206544. [72]
In this treatise, Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos, as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8×1063 in modern notation. 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. [73]
This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.

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. 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. [74]

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. [c] However, the first reliable reference to the formula is given by Heron of Alexandria in the 1st century AD. [75]

Archimedes Palimpsest

In 1906, The Archimedes Palimpsest revealed works by Archimedes thought to have been lost. Archimedes Palimpsest.jpg
In 1906, The Archimedes Palimpsest revealed works by Archimedes thought to have been lost.

The foremost document containing the work of Archimedes is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople and examined a 174-page goatskin parchment of prayers written in the 13th century AD. He discovered that it was a palimpsest, a document with text that had been written over an erased older work. Palimpsests were created by scraping the ink from existing works and reusing them, which was a common practice in the Middle Ages as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th century AD copies of previously unknown treatises by Archimedes. [76] The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. On October 29, 1998 it was sold at auction to an anonymous buyer for $2 million at Christie's in New York. [77] 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. [78]

The treatises in the Archimedes Palimpsest are:

Legacy

The Fields Medal carries a portrait of Archimedes. FieldsMedalFront.jpg
The Fields Medal carries a portrait of Archimedes.

See also

Notes

a. ^ 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.

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. 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.

c. ^ Boyer, Carl Benjamin A History of Mathematics (1991) ISBN   0-471-54397-7 "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. 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."

d. ^ "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. [85]

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<i>Archimedes Palimpsest</i> manuscript prayer book written over a work by Archimedes

The Archimedes Palimpsest is a parchment codex palimpsest, which originally was a 10th-century Byzantine Greek copy of an otherwise unknown work of Archimedes of Syracuse and other authors. It was overwritten with a Christian religious text by 13th-century monks. The erasure was incomplete, and most of the text, still visible, was published by Johan Heiberg in 1915. The manuscript went missing in the early 20th century, and a forger added pictures to some of its pages to increase its value. The text under these forged pictures, as well as additional text previously unreadable, has been revealed by scientific and scholarly work from 1998 to 2008 on images produced by ultraviolet, infrared, visible and raking light, and X-ray. All images and scholarly transcriptions with metadata are now freely available on the web at the Digital Palimpsest, now hosted on OPenn and other web sites for free use under the Creative Commons License CC BY.

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The Method of Mechanical Theorems, also referred to as The Method, is considered one of the major surviving works of the ancient Greek polymath Archimedes. The Method takes the form of a letter from Archimedes to Eratosthenes, the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles. The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. The palimpsest includes Archimedes' account of the "mechanical method", so-called because it relies on the law of the lever, which was first demonstrated by Archimedes, and of the center of mass, which he had found for many special shapes.

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Abu-Abdullah Muhammad ibn Īsa Māhānī was a Persian Muslim mathematician and astronomer born in Mahan, and active in Baghdad, Abbasid Caliphate. His known mathematical works included his commentaries on Euclid's Elements, Archimedes' On the Sphere and Cylinder and Menelaus' Sphaerica, as well as two independent treatises. He unsuccessfully tried to solve a problem posed by Archimedes of cutting a sphere into two volumes of a given ratio, which was later solved by 10th century mathematician Abū Ja'far al-Khāzin. His only known surviving work on astronomy was on the calculation of azimuths. He was also known to make astronomical observations, and claimed his estimates of the start times of three consecutive lunar eclipses were accurate to within half an hour.

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Johan Ludvig Heiberg (historian) Danish philologist and historian (1854–1928)

Johan Ludvig Heiberg was a Danish philologist and historian. He is best known for his discovery of previously unknown texts in the Archimedes Palimpsest, and for his edition of Euclid's Elements that T. L. Heath translated into English. He also published an edition of Ptolemy's Almagest.

<i>Ostomachion</i>

Ostomachion, also known as loculus Archimedius and also as syntomachion, is a mathematical treatise attributed to Archimedes. This work has survived fragmentarily in an Arabic version and a copy, the Archimedes Palimpsest, of the original ancient Greek text made in Byzantine times. The word Ostomachion has as its roots in the Greek Ὀστομάχιον, which means "bone-fight", from ὀστέον (osteon), "bone" and μάχη (mache), "fight, battle, combat". Note that the manuscripts refer to the word as "Stomachion", an apparent corruption of the original Greek. Ausonius gives us the correct name "Ostomachion". The Ostomachion which he describes was a puzzle similar to tangrams and was played perhaps by several persons with pieces made of bone. It is not known which is older, Archimedes' geometrical investigation of the figure, or the game. Victorinus, Bassus Ennodius and Lucretius have talked about the game too.

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

<i>On the Sphere and Cylinder</i>

On the Sphere and Cylinder is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.

Measurement of a Circle or Dimension of the Circle is a treatise that consists of three propositions by Archimedes, ca. 250 BCE. The treatise is only a fraction of what was a longer work.

On Spirals is a treatise by Archimedes, written around 225 BC. Although Archimedes did not discover the Archimedean spiral, he employed it in this book to square the circle and trisect an angle.

<i>On Floating Bodies</i>

On Floating Bodies is a Greek-language work consisting of two books written by Archimedes of Syracuse, one of the most important mathematicians, physicists, and engineers of antiquity. On Floating Bodies, which is thought to have been written around 250 BC, survives only partly in Greek, the rest in medieval Latin translation from the Greek. It is the first known work on hydrostatics, of which Archimedes is recognized as the founder.

Reviel Netz is an Israeli scholar of the history of pre-modern mathematics, who is currently a professor of classics and of philosophy at Stanford University.

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Further reading

The Works of Archimedes online