1.
Domenico Fetti
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Domenico Fetti was an Italian Baroque painter who had been active mainly in Rome, Mantua and Venice. Born in Rome to a painter, Pietro Fetti, Domenico is said to have apprenticed initially under Ludovico Cigoli. He then worked in Mantua from 1613 to 1622, patronized by the Cardinal, in the Ducal Palace, he painted the Miracle of the Loaves and Fishes. The series of representations of New Testament parables he carried out for his patrons studiolo gave rise to a popular specialty, and he and his studio often repeated his compositions. In August or September 1622, his feuds with some prominent Mantuans led him to move to Venice, into this mix, in the 1620s–30s, three foreigners—Fetti and his younger contemporaries Bernardo Strozzi and Jan Lys—breathed the first influences of Roman Baroque style. They adapted some of the coloration of Venice but adapted it to Caravaggio-influenced realism. In Venice, where he remained despite pleas from the Duke to return to Mantua, Fetti changed his style, in addition, he devoted attention to smaller cabinet pieces that adapt genre imaging to religious stories. His group of paintings entitled Parables, which represent New Testament scenes, are at the Dresden Gemäldegalerie and his painting style appears to have been influenced by Rubens. He would likely have continued to find excellent patronage in Venice had he not died there in 1623 or 1624, Jan Lys, eight years younger, but who had arrived in Venice nearly contemporaneously, died during the plague of 1629–30. Subsequently, Fettis style would influence the Venetians Pietro della Vecchia and his pupils in Mantua were Francesco Bernardi and Dionisio Guerri. Fettis works include, The Good Samaritan Melancholy Emperor Domitian Eve and Laboring Adam Angel in the Garden Jacobs Dream Portrait of an Actor Wittkower, pelican History of Art, Art and Architecture Italy, 1600–1750. Web Gallery of Art, paintings by Domenico Feti
2.
Syracuse, Sicily
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Syracuse is a historic city in Sicily, the capital of the province of Syracuse. The city is notable for its rich Greek history, culture, amphitheatres, architecture and 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, the city was founded by Ancient Greek Corinthians and Teneans and became a very powerful city-state. Syracuse was allied with Sparta and Corinth and exerted influence over the entirety of Magna Graecia, described by Cicero as the greatest Greek city and the most beautiful of them all, it equaled Athens in size during the fifth century BC. It later became part of the Roman Republic and Byzantine Empire, after this Palermo overtook it in importance, as the capital of the Kingdom of Sicily. Eventually the kingdom would be united with the Kingdom of Naples to form the Two Sicilies until the Italian unification of 1860, in the modern day, the city is listed by UNESCO as a World Heritage Site along with the Necropolis of Pantalica. In the central area, the city itself has a population of around 125,000 people, the inhabitants are known as Siracusans. Syracuse is mentioned in the Bible in the Acts of the Apostles book at 28,12 as Paul stayed there, the patron saint of the city is Saint Lucy, she was born in Syracuse and her feast day, Saint Lucys Day, is celebrated on 13 December. Syracuse was founded in 734 or 733 BC by Greek settlers from Corinth and Tenea, there are many attested variants of the name of the city including Συράκουσαι Syrakousai, Συράκοσαι Syrakosai and Συρακώ Syrako. The nucleus of the ancient city was the island of Ortygia. The settlers found the fertile and the native tribes to be reasonably well-disposed to their presence. The city grew and prospered, and for some time stood as the most powerful Greek city anywhere in the Mediterranean, colonies were founded at Akrai, Kasmenai, Akrillai, Helorus and Kamarina. The descendants of the first colonists, called Gamoroi, held power until they were expelled by the Killichiroi, the former, however, returned to power in 485 BC, thanks to the help of Gelo, ruler of Gela. Gelo himself became the despot of the city, and moved many inhabitants of Gela, Kamarina and Megera to Syracuse, building the new quarters of Tyche, the enlarged power of Syracuse made unavoidable the clash against the Carthaginians, who ruled western Sicily. In the Battle of Himera, Gelo, who had allied with Theron of Agrigento, a temple dedicated to Athena, was erected in the city to commemorate the event. Syracuse grew considerably during this time and its walls encircled 120 hectares in the fifth century, but as early as the 470s BC the inhabitants started building outside the walls. The complete population of its territory approximately numbered 250,000 in 415 BC, Gelo was succeeded by his brother Hiero, who fought against the Etruscans at Cumae in 474 BC. His rule was eulogized by poets like Simonides of Ceos, Bacchylides and Pindar, a democratic regime was introduced by Thrasybulos
3.
Magna Graecia
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The settlers who began arriving in the 8th century BC brought with them their Hellenic civilization, which was to leave a lasting imprint in 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, according to Strabo, Magna Graecias colonization started already at the time of the Trojan War and lasted for several centuries. Also during that period, Greek colonies were established in places as widely separated as the eastern coast of the Black Sea, Eastern Libya and they included settlements in Sicily and the southern part of the Italian Peninsula. The Romans called the area of Sicily and the foot of Italy Magna Graecia since it was so densely inhabited by the Greeks, the ancient geographers differed on whether the term included Sicily or merely Apulia and Calabria, Strabo being the most prominent advocate of the wider definitions. With colonization, Greek culture was exported to Italy, in its dialects of the Ancient Greek language, its religious rites, an original Hellenic civilization soon developed, later interacting with the native Italic civilisations. Many of the new Hellenic cities became very rich and powerful, like Neapolis, Syracuse, Acragas Paestum, other cities in Magna Graecia included Tarentum, Epizephyrian Locri, Rhegium, Croton, Thurii, Elea, Nola, Ancona, Syessa, Bari and others. Following the Pyrrhic War in the 3rd century BC, Magna Graecia was absorbed into the Roman Republic, a remarkable example of the influence is the Griko-speaking minority that still exists today in the Italian regions of Calabria and Apulia. Griko is the name of a language combining ancient Doric, Byzantine Greek, there is a rich oral tradition and Griko folklore, limited now but once numerous, to around 30,000 people, most of them having become absorbed into the surrounding Italian element. Some scholars, such as Gerhard Rohlfs, argue that the origins of Griko may ultimately be traced to the colonies of Magna Graecia, one example is the Griko people, some of whom still maintain their Greek language and customs. For example, Greeks re-entered the region in the 16th and 17th century in reaction to the conquest of the Peloponnese by the Ottoman Empire, especially after the end of the Siege of Coron, large numbers of Greeks took refuge in the areas of Calabria, Salento and Sicily. Greeks from Coroni, the so-called Coronians, were nobles, who brought with them substantial movable property and they were granted special privileges and tax exemptions. Other Greeks who moved to Italy came from the Mani Peninsula of the Peloponnese, the Maniots were known for their proud military traditions and for their bloody vendettas, many of which still continue today. Another group of Maniot Greeks moved to Corsica, Ancient Greek dialects Greeks in Italy Italiotes Graia Graïke Graecus Griko people Griko language Hellenic civilization Names of the Greeks Cerchiai L. Jannelli L. Longo F. The Greek Cities of Magna Graecia and Sicily, in Dictionary of Greek and Roman Geography. 21 June,2005,17,19 GMT18,19 UK, salentinian Peninsula, Greece and Greater Greece. Traditional Griko song performed by Ghetonia, traditional Griko song performed by amateur local group. Second Interdisciplinary Symposium on the Hellenic Heritage of Southern Italy, the Greeks in the West, genetic signatures of the Hellenic colonisation in southern Italy and Sicily
4.
Archimedes' principle
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Archimedes principle is a law of physics fundamental to fluid mechanics. It was formulated by Archimedes of Syracuse, in On Floating Bodies, Archimedes suggested that, Practically, Archimedes principle allows the buoyancy of an object partially or fully immersed in a liquid to be calculated. The downward force on the object is simply its weight, the upward, or buoyant, force on the object is that stated by Archimedes principle, above. Thus, the net force on the object is the difference between the buoyant force and its weight. If this net force is positive, the object rises, if negative, the sinks, and if zero. Consider a cube immersed in a fluid, with its sides parallel to the direction of gravity, the fluid will exert a normal force on each face, and therefore only the forces on the top and bottom faces will contribute to buoyancy. The pressure difference between the bottom and the top face is directly proportional to the height, multiplying the pressure difference by the area of a face gives the net force on the cube – the buoyancy, or the weight of the fluid displaced. By extending this reasoning to irregular shapes, we can see that, whatever the shape of the submerged body, the buoyant force is equal to the weight of the fluid displaced. Apparent loss in weight of water = weight of object in air − weight of object in water The weight of the fluid is directly proportional to the volume of the displaced fluid. The weight of the object in the fluid is reduced, because of the acting on it. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy, suppose a rocks weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting on it. Suppose that, when the rock is lowered into water, it displaces water of weight 3 newtons, the force it then exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyant force,10 −3 =7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor and it is generally easier to lift an object up through the water than it is to pull it out of the water. Example, If you drop wood into water, buoyancy will keep it afloat, example, A helium balloon in a moving car. When increasing speed or driving in a curve, the air moves in the direction to the cars acceleration. However, due to buoyancy, the balloon is pushed out of the way by the air, and will actually drift in the same direction as the cars acceleration. When an object is immersed in a liquid, the liquid exerts a force, which is known as the buoyant force. The sum force acting on the object, then, is equal to the difference between the weight of the object and the weight of displaced liquid, equilibrium, or neutral buoyancy, is achieved when these two weights are equal
5.
Archimedes' screw
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The Archimedes screw, also called the Archimedean screw or screwpump, is a machine historically used for transferring water from a low-lying body of water into irrigation ditches. Water is pumped by turning a screw-shaped surface inside a pipe, the screw pump is commonly attributed to Archimedes on the occasion of his visit to Egypt. This tradition may reflect only that the apparatus was unknown to the Greeks before Hellenistic times and was introduced in Archimedess lifetime by unknown Greek engineers, some writers have suggested the device may have been in use in Assyria some 350 years earlier. The Archimedes screw consists of a screw inside a hollow pipe, the screw is turned usually by a windmill or by manual labour or by cattle. As the shaft turns, the bottom end scoops up a volume of water and this water is then pushed up the tube by the rotating helicoid until finally it pours out from the top of the tube. If water from one section leaks into the lower one. In some designs, the screw is fused to the casing, a screw could be sealed with pitch resin or other adhesive to its casing, or cast as a single piece in bronze. Some researchers have postulated this as being the device used to irrigate the Hanging Gardens of Babylon, the design of the everyday Greek and Roman water screw, in contrast to the heavy bronze device of Sennacherib, with its problematic drive chains, has a powerful simplicity. A double or triple helix was built of wood strips around a wooden pole. It was used for draining land that was underneath the sea in the Netherlands, Archimedes screws are used in sewage treatment plants because they cope well with varying rates of flow and with suspended solids. An auger in a snow blower or grain elevator is essentially an Archimedes screw, many forms of axial flow pump basically contain an Archimedes screw. The principle is found in pescalators, which are Archimedes screws designed to lift fish safely from ponds. This technology is used primarily at fish hatcheries, where it is desirable to minimize the handling of fish. An Archimedes screw was used in the successful 2001 stabilization of the Leaning Tower of Pisa, small amounts of subsoil saturated by groundwater were removed from far below the north side of the Tower, and the weight of the tower itself corrected the lean. Archimedes screws are used in chocolate fountains. The invention of the screw is credited to Greek polymath Archimedes of Syracuse in the 3rd century BC. A cuneiform inscription of Assyrian king Sennacherib has been interpreted by Stephanie Dalley to describe casting water screws in bronze some 350 years earlier and this is consistent with classical author Strabo, who describes the Hanging Gardens as watered by screws. A contrary view is expressed by Dalley and Oleson, german engineer Konrad Kyeser equips the Archimedes screw with a crank mechanism in his Bellifortis
6.
Fluid statics
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Fluid statics or hydrostatics is the branch of fluid mechanics that studies incompressible fluids at rest. It encompasses the study of the conditions under which fluids are at rest in stable equilibrium as opposed to fluid dynamics, hydrostatics are categorized as a part of the fluid statics, which is the study of all fluids, incompressible or not, at rest. Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids and it is also relevant to geophysics and astrophysics, to meteorology, to medicine, and many other fields. Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats, cisterns, aqueducts and fountains. Archimedes is credited with the discovery of Archimedes Principle, which relates the force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The fair cup or Pythagorean cup, which dates from about the 6th century BC, is a technology whose invention is credited to the Greek mathematician. It was used as a learning tool, the cup consists of a line carved into the interior of the cup, and a small vertical pipe in the center of the cup that leads to the bottom. The height of this pipe is the same as the line carved into the interior of the cup, the cup may be filled to the line without any fluid passing into the pipe in the center of the cup. However, when the amount of fluid exceeds this fill line, due to the drag that molecules exert on one another, the cup will be emptied. Herons fountain is a device invented by Heron of Alexandria that consists of a jet of fluid being fed by a reservoir of fluid. The fountain is constructed in such a way that the height of the jet exceeds the height of the fluid in the reservoir, the device consisted of an opening and two containers arranged one above the other. The intermediate pot, which was sealed, was filled with fluid, trapped air inside the vessels induces a jet of water out of a nozzle, emptying all water from the intermediate reservoir. Pascal made contributions to developments in both hydrostatics and hydrodynamics, due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface, if a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force, thus, the pressure on a fluid at rest is isotropic, i. e. it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes, i. e. a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe. This principle was first formulated, in an extended form, by Blaise Pascal. In a fluid at rest, all frictional and inertial stresses vanish, when this condition of V =0 is applied to the Navier-Stokes equation, the gradient of pressure becomes a function of body forces only
7.
Lever
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A lever is a machine consisting of a beam or rigid rod pivoted at a fixed hinge, or fulcrum. A lever is a body capable of rotating on a point on itself. On the basis of the location of fulcrum, load and effort and it is one of the six simple machines identified by Renaissance scientists. A lever amplifies an input force to provide an output force. The ratio of the force to the input force is the mechanical advantage of the lever. The word lever entered English about 1300 from Old French, in which the word was levier and this sprang from the stem of the verb lever, meaning to raise. The verb, in turn, goes back to the Latin levare, itself from the adjective levis, the words primary origin is the Proto-Indo-European stem legwh-, meaning light, easy or nimble, among other things. The PIE stem also gave rise to the English word light, the earliest remaining writings regarding levers date from the 3rd century BC and were provided by Archimedes. Give me a place to stand, and I shall move the Earth with it is a remark of Archimedes who formally stated the correct mathematical principle of levers. The distance required to do this might be exemplified in astronomical terms as the distance to the Circinus galaxy - about 9 million light years. It is assumed that in ancient Egypt, constructors used the lever to move, a lever is a beam connected to ground by a hinge, or pivot, called a fulcrum. The ideal lever does not dissipate or store energy, which there is no friction in the hinge or bending in the beam. This is known as the law of the lever, the mechanical advantage of a lever can be determined by considering the balance of moments or torque, T, about the fulcrum. T1 = F1 a, T2 = F2 b where F1 is the force to the lever. The distances a and b are the distances between the forces and the fulcrum. Since the moments of torque must be balanced, T1 = T2, the mechanical advantage of the lever is the ratio of output force to input force, M A = F2 F1 = a b. Levers are classified by the positions of the fulcrum, effort. It is common to call the force the effort and the output force the load or the resistance
8.
Archimedes' use of infinitesimals
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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 librarian at the Library of Alexandria. The work was thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. Archimedes did not admit the method of indivisibles as part of rigorous mathematics, in these treatises, he proves the same theorems by exhaustion, finding rigorous upper and lower bounds which both converge to the answer required. Nevertheless, the method was what he used to discover the relations for which he later gave rigorous proofs. To explain Archimedes method today, it is convenient to use of a little bit of Cartesian geometry. His idea is to use the law of the lever to determine the areas of figures from the center of mass of other figures. The simplest example in modern language is the area of the parabola. Archimedes uses an elegant method, but in Cartesian language, his method is calculating the integral ∫01 x 2 d x =13. The idea is to balance the parabola with a certain triangle that is made of the same material. The parabola is the region in the x-y plane between the x-axis and y = x2 as x varies from 0 to 1, the triangle is the region in the x-y plane between the x-axis and the line y = x, also as x varies from 0 to 1. Slice the parabola and triangle into vertical slices, one for each value of x, imagine that the x-axis is a lever, with a fulcrum at x =0. The law of the states that two objects on opposite sides of the fulcrum will balance if each has the same torque. Since each pair of balances, moving the whole parabola to x = −1 would balance the whole triangle. This means that if the original uncut parabola is hung by a hook from the point x = −1, the center of mass of a triangle can be easily found by the following method, also due to Archimedes. If a median line is drawn from any one of the vertices of a triangle to the opposite edge E, the triangle will balance on the median, considered as a fulcrum. The reason is if the triangle is divided into infinitesimal line segments parallel to E, each segment has equal length on opposite sides of the median. This argument can be made rigorous by exhaustion by using little rectangles instead of infinitesimal lines
9.
Neusis construction
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The neusis is a geometric construction method that was used in antiquity by Greek mathematicians. The neusis construction consists of fitting a line element of length in between two given lines, in such a way that the line element, or its extension. That is, one end of the element has to lie on l. A neusis construction might be performed by means of a neusis ruler, in the figure one end of the ruler is marked with a yellow eye with crosshairs, this is the origin of the scale division on the ruler. A second marking on the ruler indicates the distance a from the origin, the yellow eye is moved along line l, until the blue eye coincides with line m. The position of the element thus found is shown in the figure as a dark blue bar. Point P is called the pole of the neusis, line l the directrix, or guiding line, length a is called the diastema. Neuseis have been important because they provide a means to solve geometric problems that are not solvable by means of compass. Examples are the trisection of any angle in three parts, the doubling of the cube, and the construction of a regular heptagon, nonagon. Mathematicians such as Archimedes of Syracuse and Pappus of Alexandria freely used neuseis, Sir Isaac Newton followed their line of thought, nevertheless, gradually the technique dropped out of use. Modified by the recent finding by Benjamin and Snyder that the regular hendecagon is neusis-constructible, T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides was the first to put compass-and-straightedge constructions above neuseis. One hundred years after him Euclid too shunned neuseis in his influential textbook. The next attack on the neusis came when, from the fourth century BC, under its influence a hierarchy of three classes of geometrical constructions was developed. In the end the use of neusis was deemed acceptable only when the two other, higher categories of constructions did not offer a solution, Neusis became a kind of last resort that was invoked only when all other, more respectable, methods had failed. Using neusis where other methods might have been used was branded by the late Greek mathematician Pappus of Alexandria as a not inconsiderable error. R. Boeker, Neusis, in, Paulys Realencyclopädie der Classischen Altertumswissenschaft, the most comprehensive survey, however, the author sometimes has rather curious opinions. T. L. Heath, A history of Greek Mathematics, H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum. MathWorld page Angle Trisection by Paper Folding
10.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
11.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
12.
Engineering
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The term Engineering is derived from the Latin ingenium, meaning cleverness and ingeniare, meaning to contrive, devise. Engineering has existed since ancient times as humans devised fundamental inventions such as the wedge, lever, wheel, each of these inventions is essentially consistent with the modern definition of engineering. The term engineering is derived from the engineer, which itself dates back to 1390 when an engineer originally referred to a constructor of military engines. In this context, now obsolete, a referred to a military machine. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, the word engine itself is of even older origin, ultimately deriving from the Latin ingenium, meaning innate quality, especially mental power, hence a clever invention. The earliest civil engineer known by name is Imhotep, as one of the officials of the Pharaoh, Djosèr, he probably designed and supervised the construction of the Pyramid of Djoser at Saqqara in Egypt around 2630–2611 BC. Ancient Greece developed machines in both civilian and military domains, the Antikythera mechanism, the first known mechanical computer, and the mechanical inventions of Archimedes are examples of early mechanical engineering. In the Middle Ages, the trebuchet was developed, the first steam engine was built in 1698 by Thomas Savery. The development of this gave rise to the Industrial Revolution in the coming decades. With the rise of engineering as a profession in the 18th century, similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. The inventions of Thomas Newcomen and the Scottish engineer James Watt gave rise to mechanical engineering. The development of specialized machines and machine tools during the revolution led to the rapid growth of mechanical engineering both in its birthplace Britain and abroad. John Smeaton was the first self-proclaimed civil engineer and is regarded as the father of civil engineering. He was an English civil engineer responsible for the design of bridges, canals, harbours and he was also a capable mechanical engineer and an eminent physicist. Smeaton designed the third Eddystone Lighthouse where he pioneered the use of hydraulic lime and his lighthouse remained in use until 1877 and was dismantled and partially rebuilt at Plymouth Hoe where it is known as Smeatons Tower. The United States census of 1850 listed the occupation of engineer for the first time with a count of 2,000, there were fewer than 50 engineering graduates in the U. S. before 1865. In 1870 there were a dozen U. S. mechanical engineering graduates, in 1890 there were 6,000 engineers in civil, mining, mechanical and electrical. There was no chair of applied mechanism and applied mechanics established at Cambridge until 1875, the theoretical work of James Maxwell and Heinrich Hertz in the late 19th century gave rise to the field of electronics
13.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
14.
Invention
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An invention is a unique or novel device, method, composition or process. The invention process is a process within an overall engineering and product development process and it may be an improvement upon a machine or product or a new process for creating an object or a result. An invention that achieves a unique function or result may be a radical breakthrough. Such works are novel and not obvious to others skilled in the same field, an inventor may be taking a big step in success or failure. A patent legally protects the property rights of the inventor. The rules and requirements for patenting an invention vary from country to country, another meaning of invention is cultural invention, which is an innovative set of useful social behaviours adopted by people and passed on to others. The Institute for Social Inventions collected many such ideas in magazines, Invention is also an important component of artistic and design creativity. Inventions often extend the boundaries of knowledge, experience or capability. Brainstorming also can spark new ideas for an invention, collaborative creative processes are frequently used by engineers, designers, architects and scientists. Co-inventors are frequently named on patents, in addition, many inventors keep records of their working process - notebooks, photos, etc. including Leonardo da Vinci, Galileo Galilei, Evangelista Torricelli, Thomas Jefferson and Albert Einstein. In the process of developing an invention, the idea may change. The invention may become simpler, more practical, it may expand, working on one invention can lead to others too. History shows that turning the concept of an invention into a device is not always swift or direct. Inventions may also more useful after time passes and other changes occur. For example, the became more useful once powered flight was a reality. Invention is often a creative process, an open and curious mind allows an inventor to see beyond what is known. Seeing a new possibility, connection, or relationship can spark an invention, inventive thinking frequently involves combining concepts or elements from different realms that would not normally be put together. Sometimes inventors disregard the boundaries between distinctly separate territories or fields, several concepts may be considered when thinking about invention
15.
Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and 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. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
16.
Greeks
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The Greeks or Hellenes are an ethnic group native to Greece, Cyprus, southern Albania, Turkey, Sicily, 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, many of these regions coincided to a large extent with the borders of the Byzantine Empire of the late 11th century and the Eastern Mediterranean areas of ancient Greek colonization. The cultural centers of the Greeks have included Athens, Thessalonica, Alexandria, Smyrna, most ethnic Greeks live nowadays within the borders of the modern Greek state and Cyprus. The Greek genocide and population exchange between Greece and Turkey nearly ended the three millennia-old Greek presence in Asia Minor, other longstanding Greek populations can be found from southern Italy to the Caucasus and southern Russia and Ukraine and in the Greek diaspora communities in a number of other countries. Today, most Greeks are officially registered as members of the Greek Orthodox Church, the Greeks speak the Greek language, which forms its own unique branch within the Indo-European family of languages, the Hellenic. They are part of a group of ethnicities, described by Anthony D. Smith as an archetypal diaspora people. Both migrations occur at incisive periods, the Mycenaean at the transition to the Late Bronze Age, the Mycenaeans quickly penetrated the Aegean Sea and, by the 15th century BC, had reached Rhodes, Crete, Cyprus and the shores of Asia Minor. Around 1200 BC, the Dorians, another Greek-speaking people, followed from Epirus, the Dorian invasion was followed by a poorly attested period of migrations, appropriately called the Greek Dark Ages, but by 800 BC the landscape of Archaic and Classical Greece was discernible. The Greeks of classical antiquity idealized their Mycenaean ancestors and the Mycenaean period as an era of heroes, closeness of the gods. The Homeric Epics were especially and generally accepted as part of the Greek past, as part of the Mycenaean heritage that survived, the names of the gods and goddesses of Mycenaean Greece became major figures of the Olympian Pantheon of later antiquity. The ethnogenesis of the Greek nation is linked to the development of Pan-Hellenism in the 8th century BC, the works of Homer and Hesiod were written in the 8th century BC, becoming the basis of the national religion, ethos, history and mythology. The Oracle of Apollo at Delphi was established in this period, the classical period of Greek civilization covers a time spanning from the early 5th century BC to the death of Alexander the Great, in 323 BC. It is so named because it set the standards by which Greek civilization would be judged in later eras, the Peloponnesian War, the large scale civil war between the two most powerful Greek city-states Athens and Sparta and their allies, left both greatly weakened. Many Greeks settled in Hellenistic cities like Alexandria, Antioch and Seleucia, two thousand years later, there are still communities in Pakistan and Afghanistan, like the Kalash, who claim to be descended from Greek settlers. The Hellenistic civilization was the period of Greek civilization, the beginnings of which are usually placed at Alexanders death. This Hellenistic age, so called because it saw the partial Hellenization of many non-Greek cultures and this age saw the Greeks move towards larger cities and a reduction in the importance of the city-state. These larger cities were parts of the still larger Kingdoms of the Diadochi, Greeks, however, remained aware of their past, chiefly through the study of the works of Homer and the classical authors. An important factor in maintaining Greek identity was contact with barbarian peoples and this led to a strong desire among Greeks to organize the transmission of the Hellenic paideia to the next generation
17.
Greek mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed 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, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, 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. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
18.
Inventor
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An invention is a unique or novel device, method, composition or process. The invention process is a process within an overall engineering and product development process and it may be an improvement upon a machine or product or a new process for creating an object or a result. An invention that achieves a unique function or result may be a radical breakthrough. Such works are novel and not obvious to others skilled in the same field, an inventor may be taking a big step in success or failure. A patent legally protects the property rights of the inventor. The rules and requirements for patenting an invention vary from country to country, another meaning of invention is cultural invention, which is an innovative set of useful social behaviours adopted by people and passed on to others. The Institute for Social Inventions collected many such ideas in magazines, Invention is also an important component of artistic and design creativity. Inventions often extend the boundaries of knowledge, experience or capability. Brainstorming also can spark new ideas for an invention, collaborative creative processes are frequently used by engineers, designers, architects and scientists. Co-inventors are frequently named on patents, in addition, many inventors keep records of their working process - notebooks, photos, etc. including Leonardo da Vinci, Galileo Galilei, Evangelista Torricelli, Thomas Jefferson and Albert Einstein. In the process of developing an invention, the idea may change. The invention may become simpler, more practical, it may expand, working on one invention can lead to others too. History shows that turning the concept of an invention into a device is not always swift or direct. Inventions may also more useful after time passes and other changes occur. For example, the became more useful once powered flight was a reality. Invention is often a creative process, an open and curious mind allows an inventor to see beyond what is known. Seeing a new possibility, connection, or relationship can spark an invention, inventive thinking frequently involves combining concepts or elements from different realms that would not normally be put together. Sometimes inventors disregard the boundaries between distinctly separate territories or fields, several concepts may be considered when thinking about invention
19.
Scientist
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A scientist is a person engaging in a systematic activity to acquire knowledge that describes and predicts the natural world. In a more restricted sense, a scientist may refer to an individual who uses the scientific method, the person may be an expert in one or more areas of science. The term scientist was coined by the theologian, philosopher and historian of science William Whewell and this article focuses on the more restricted use of the word. Scientists perform research toward a comprehensive understanding of nature, including physical, mathematical and social realms. Philosophers aim to provide an understanding of fundamental aspects of reality and experience, often pursuing inquiries with conceptual, rather than empirical. When science is done with a goal toward practical utility, it is called applied science, an applied scientist may not be designing something in particular, but rather is conducting research with the aim of developing new technologies and practical methods. When science seeks to answer questions about aspects of reality it is sometimes called natural philosophy. Science and technology have continually modified human existence through the engineering process, as a profession the scientist of today is widely recognized. Jurisprudence and mathematics are often grouped with the sciences, some of the greatest physicists have also been creative mathematicians and lawyers. There is a continuum from the most theoretical to the most empirical scientists with no distinct boundaries, in terms of personality, interests, training and professional activity, there is little difference between applied mathematicians and theoretical physicists. Scientists can be motivated in several ways, many have a desire to understand why the world is as we see it and how it came to be. They exhibit a strong curiosity about reality, other motivations are recognition by their peers and prestige, or the desire to apply scientific knowledge for the benefit of peoples health, the nations, the world, nature or industries. Scientists tend to be motivated by direct financial reward for their work than other careers. As a result, scientific researchers often accept lower average salaries when compared with other professions which require a similar amount of training. The number of scientists is vastly different from country to country, for instance, there are only 4 full-time scientists per 10,000 workers in India while this number is 79 for the United Kingdom and the United States. According to the US National Science Foundation 4.7 million people with science degrees worked in the United States in 2015, across all disciplines, the figure included twice as many men as women. Of that total, 17% worked in academia, that is, at universities and undergraduate institutions, 5% of scientists worked for the federal government and about 3. 5% were self-employed. Of the latter two groups, two-thirds were men, 59% of US scientists were employed in industry or business, and another 6% worked in non-profit positions
20.
Classical antiquity
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It is the period in which Greek and Roman society flourished and wielded great influence throughout Europe, North Africa and Southwestern Asia. Conventionally, it is taken to begin with the earliest-recorded Epic Greek poetry of Homer, and continues through the emergence of Christianity and it ends with the dissolution of classical culture at the close of Late Antiquity, blending into the Early Middle Ages. Such a wide sampling of history and territory covers many disparate cultures, Classical antiquity may refer also to an idealised vision among later people of what was, in Edgar Allan Poes words, the glory that was Greece, and the grandeur that was Rome. The culture of the ancient Greeks, together with influences from the ancient Near East, was the basis of art, philosophy, society. The earliest period of classical antiquity takes place before the background of gradual re-appearance of historical sources following the Bronze Age collapse, the 8th and 7th centuries BC are still largely proto-historical, with the earliest Greek alphabetic inscriptions appearing in the first half of the 8th century. Homer is usually assumed to have lived in the 8th or 7th century BC, in the same period falls the traditional date for the establishment of the Ancient Olympic Games, in 776 BC. The Phoenicians originally expanded from Canaan ports, by the 8th century dominating trade in the Mediterranean, carthage was founded in 814 BC, and the Carthaginians by 700 BC had firmly established strongholds in Sicily, Italy and Sardinia, which created conflicts of interest with Etruria. The Etruscans had established control in the region by the late 7th century BC, forming the aristocratic. According to legend, Rome was founded on April 21,753 BC by twin descendants of the Trojan prince Aeneas, Romulus and Remus. As the city was bereft of women, legend says that the Latins invited the Sabines to a festival and stole their unmarried maidens, leading to the integration of the Latins and the Sabines. Archaeological evidence indeed shows first traces of settlement at the Roman Forum in the mid-8th century BC, the seventh and final king of Rome was Tarquinius Superbus. As the son of Tarquinius Priscus and the son-in-law of Servius Tullius, Superbus was of Etruscan birth and it was during his reign that the Etruscans reached their apex of power. Superbus removed and destroyed all the Sabine shrines and altars from the Tarpeian Rock, the people came to object to his rule when he failed to recognize the rape of Lucretia, a patrician Roman, at the hands of his own son. Lucretias kinsman, Lucius Junius Brutus, summoned the Senate and had Superbus, after Superbus expulsion, the Senate voted to never again allow the rule of a king and reformed Rome into a republican government in 509 BC. In fact the Latin word Rex meaning King became a dirty and hated throughout the Republic. In 510, Spartan troops helped the Athenians overthrow the tyrant Hippias, cleomenes I, king of Sparta, put in place a pro-Spartan oligarchy conducted by Isagoras. Greece entered the 4th century under Spartan hegemony, but by 395 BC the Spartan rulers removed Lysander from office, and Sparta lost her naval supremacy. Athens, Argos, Thebes and Corinth, the two of which were formerly Spartan allies, challenged Spartan dominance in the Corinthian War, which ended inconclusively in 387 BC
21.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
22.
Mathematical analysis
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Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are studied in the context of real and complex numbers. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis, analysis may be distinguished from geometry, however, it can be applied to any space of mathematical objects that has a definition of nearness or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, in Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieris principle to find the volume of a sphere in the 5th century, the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of analysis were established in 17th century Europe. During this period, calculus techniques were applied to approximate discrete problems by continuous ones, in the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the definition of continuity in 1816. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations, the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Riemann introduced his theory of integration, the last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the size of the set of discontinuities of real functions, also, monsters began to be investigated
23.
Method of exhaustion
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The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes small, the possible values for the area of the shape are systematically exhausted by the lower bound areas successively established by the sequence members. The method of exhaustion typically required a form of proof by contradiction and this amounts to finding an area of a region by first comparing it to the area of a second region. The idea originated in the late 5th century BC with Antiphon, the theory was made rigorous a few decades later by Eudoxus of Cnidus, who used it to calculate areas and volumes. It was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle, the first use of the term was in 1647 by Grégoire de Saint-Vincent in Opus geometricum quadraturae circuli et sectionum. The method of exhaustion is seen as a precursor to the methods of calculus, the development of analytical geometry and rigorous integral calculus in the 17th-19th centuries subsumed the method of exhaustion so that it is no longer explicitly used to solve problems. Euclid used the method of exhaustion to prove the following six propositions in the book 12 of his Elements, proposition 2 The area of a circle is proportional to the square of its radius. Proposition 5 The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases, proposition 10 The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. Proposition 11 The volume of a cone of the height is proportional to the area of the base. Proposition 12 The volume of a cone that is the similar to another is proportional to the cube of the ratio of the diameters of the bases, proposition 18 The volume of a sphere is proportional to the cube of its diameter. Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater area and greater number of sides. He also provided the bounds 3 + 10/71 < π <3 + 10/70, the Method of Mechanical Theorems The Quadrature of the Parabola Trapezoidal rule
24.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
25.
Theorem
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In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a consequence of the axioms. The proof of a theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises, however, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a symbolic form, for example, within the propositional calculus. In some cases, a picture alone may be sufficient to prove a theorem, because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being trivial, or difficult, or deep and these subjective judgments vary not only from person to person, but also with time, for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a theorem may be simply stated. Fermats Last Theorem is a particularly well-known example of such a theorem, logically, many theorems are of the form of an indicative conditional, if A, then B. Such a theorem does not assert B, only that B is a consequence of A. In this case A is called the hypothesis of the theorem and B the conclusion. The theorem If n is an natural number then n/2 is a natural number is a typical example in which the hypothesis is n is an even natural number. To be proved, a theorem must be expressible as a precise, nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader can produce a formal statement from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and these hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of known as proof theory studies formal languages, axioms. Some theorems are trivial, in the sense that they follow from definitions, axioms, a theorem might be simple to state and yet be deep
26.
Area of a circle
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In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents a constant, approximately equal to 3.14159, one method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons. The area of a polygon is half its perimeter multiplied by the distance from its center to its sides. Therefore, the area of a disk is the precise phrase for the area enclosed by a circle. Modern mathematics can obtain the area using the methods of calculus or its more sophisticated offspring. However the area of a disk was studied by the Ancient Greeks, eudoxus of Cnidus in the fifth century B. C. had found that the area of a disk is proportional to its radius squared. The circumference is 2πr, and the area of a triangle is half the times the height. A variety of arguments have been advanced historically to establish the equation A = π r 2 of varying degrees of mathematical rigor, the area of a regular polygon is half its perimeter times the apothem. As the number of sides of the regular polygon increases, the polygon tends to a circle, and this suggests that the area of a disk is half the circumference of its bounding circle times the radius. Following Archimedes, compare the area enclosed by a circle to a triangle whose base has the length of the circles circumference. If the area of the circle is not equal to that of the triangle and we eliminate each of these by contradiction, leaving equality as the only possibility. We use regular polygons in the same way, suppose that the area C enclosed by the circle is greater than the area T = 1⁄2cr of the triangle. Let E denote the excess amount, inscribe a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments, if the total area of those gaps, G4, is greater than E, split each arc in half. This makes the square into an inscribed octagon, and produces eight segments with a smaller total gap. Continue splitting until the total gap area, Gn, is less than E, now the area of the inscribed polygon, Pn = C − Gn, must be greater than that of the triangle. E = C − T > G n P n = C − G n > C − E P n > T But this forces a contradiction, as follows. Draw a perpendicular from the center to the midpoint of a side of the polygon, its length, also, let each side of the polygon have length s, then the sum of the sides, ns, is less than the circle circumference
27.
Surface area
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The surface area of a solid object is a measure of the total area that the surface of the object occupies. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces and this definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of area was sought by Henri Lebesgue. Their work led to the development of measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface, while the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a great deal of care. This should provide a function S ↦ A which assigns a real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the area is its additivity. More rigorously, if a surface S is a union of many pieces S1, …, Sr which do not overlap except at their boundaries. Surface areas of polygonal shapes must agree with their geometrically defined area. Since surface area is a notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface. This means that surface area is invariant under the group of Euclidean motions and these properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of many pieces that can be represented in the parametric form S D, r → = r →, ∈ D with a continuously differentiable function r →. The area of a piece is defined by the formula A = ∬ D | r → u × r → v | d u d v. Thus the area of SD is obtained by integrating the length of the vector r → u × r → v to the surface over the appropriate region D in the parametric uv plane. The area of the surface is then obtained by adding together the areas of the pieces. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f and surfaces of revolution. It was demonstrated by Hermann Schwarz that already for the cylinder, various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a natural notion of surface area, if a surface is very irregular, or rough
28.
Volume
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Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
29.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3
30.
Parabola
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A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point and a line, the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus, a parabola is a graph of a quadratic function, y = x2, for example. The line perpendicular to the directrix and passing through the focus is called the axis of symmetry, the point on the parabola that intersects the axis of symmetry is called the vertex, and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the focal length, the latus rectum is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy and this reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from an antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas, the earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas, the name parabola is due to Apollonius who discovered many properties of conic sections. It means application, referring to application of concept, that has a connection with this curve. The focus–directrix property of the parabola and other conics is due to Pappus, Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a reflector could produce an image was already well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes, solving for y yields y =14 f x 2. The length of the chord through the focus is called latus rectum, one half of it semi latus rectum
31.
Pi
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The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
32.
Archimedes spiral
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The Archimedean spiral is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in polar coordinates it can be described by the equation r = a + b θ with real numbers a and b, changing the parameter a will turn the spiral, while b controls the distance between successive turnings. Archimedes described such a spiral in his book On Spirals, the Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance, hence the name arithmetic spiral. In contrast to this, in a logarithmic spiral these distances, as well as the distances of the intersection points measured from the origin, the Archimedean spiral has two arms, one for θ >0 and one for θ <0. The two arms are connected at the origin. Only one arm is shown on the accompanying graph, taking the mirror image of this arm across the y-axis will yield the other arm. Some sources describe the Archimedean spiral as a spiral with a constant separation distance between successive turns, there is a curve slightly different from the Archimedean spiral, the involute of a circle, whose turns have constant separation distance in the latter sense of parallel curves. Sometimes the term Archimedean spiral is used for the general group of spirals r = a + b θ1 / c. The normal Archimedean spiral occurs when c =1, other spirals falling into this group include the hyperbolic spiral, Fermats spiral, and the lituus. Virtually all static spirals appearing in nature are logarithmic spirals, not Archimedean ones, one method of squaring the circle, due to Archimedes, makes use of an Archimedean spiral. Archimedes also showed how the spiral can be used to trisect an angle, both approaches relax the traditional limitation on the use of straightedge and compass. The Archimedean spiral has a variety of real-world applications, scroll compressors, made from two interleaved involutes of a circle of the same size that almost resemble Archimedean spirals, are used for compressing liquids and gases. Asking for a patient to draw an Archimedean spiral is a way of quantifying human tremor, additionally, Archimedean spirals are used in food microbiology to quantify bacterial concentration through a spiral platter. They are also used to model the pattern occurs in a roll of paper or tape of constant thickness wrapped around a cylinder. Page with Java application to interactively explore the Archimedean spiral and its related curves Online exploration using JSXGraph Archimedean spiral at mathcurve
33.
Exponentiation
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Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5
34.
Statics
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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 Newtons second law to a system gives, F = m a, where bold font indicates a vector that has magnitude and direction. F is the total of the acting on the system, m is the mass of the system. The summation of forces will give the direction and the magnitude of the acceleration will be proportional to the mass. The assumption of static equilibrium of a =0 leads to, the summation of forces, one of which might be unknown, allows that unknown to be found. Likewise the application of the assumption of zero acceleration to the summation of moments acting on the leads to. The summation of moments, one of which might be unknown and these two equations together, can be applied to solve for as many as two loads acting on the system. From Newtons first law, this implies that the net force, the net forces equaling zero is known as the first condition for equilibrium, and the net torque equaling zero is known as the second condition for equilibrium. A scalar is a quantity which only has a magnitude, such as mass or temperature, a vector has a magnitude and a direction. There are several notations to identify a vector, including, A bold faced character V An underlined character V A character with an arrow over it V →, vectors are added using the parallelogram law or the triangle law. Vectors contain components in orthogonal bases, unit vectors i, j, and k are, by convention, along the x, y, and z axes, respectively. Force is the action of one body on another, a force is either a push or a pull. A force tends to move a body in the direction of its action, the action of a force is characterized by its magnitude, by the direction of its action, and by its point of application. Thus, force is a quantity, because its effect depends on the direction as well as on the magnitude of the action. Forces are classified as either contact or body forces, a contact force is produced by direct physical contact, an example is the force exerted on a body by a supporting surface. A body force is generated by virtue of the position of a body within a field such as a gravitational, electric. An example of a force is the weight of a body in the Earths gravitational field. In addition to the tendency to move a body in the direction of its application, the axis may be any line which neither intersects nor is parallel to the line of action of the force
35.
Machine
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Gary DellAbate, also known as Baba Booey, is an American radio producer, and has been the executive producer of The Howard Stern Show since 1984. His autobiography, They Call Me Baba Booey, was released on November 2,2010, DellAbate was born and raised in Uniondale, New York, on Long Island. He comes from a large Italian-American family and his father, Salvatore DellAbate, was a salesman for Hagen Daz ice cream, while his mother was a food demonstartor at Macys in New York City and Fortunoffs on Long Island. DellAbate attended Adelphi University, receiving the Richard F. Clemo Award his senior year, while interning with Roz Frank, a traffic reporter on WNBC, he came into contact with Howard Stern. DellAbate has worked on The Howard Stern Show since September 4,1984, originally on 66 WNBC, then syndicated through K-Rock in New York City, DellAbate was originally hired for $150 a week, with duties including getting Sterns lunch and scheduling guests for the show. Previously nicknamed Boy Gary, DellAbates Baba Booey moniker originated on The Howard Stern Show on July 26,1990, in the course of discussing a Quick Draw McGraw cel he might purchase, he misstated the name of McGraws sidekick Baba Looey as Baba Booey. Speaking to Howard at the end of the show, DellAbate said, Howard Stern replied, Gary, weve only scratched the surface of this. DellAbate remains Baba Booey to this day, eventually he titled his autobiography They Call Me Baba Booey. DellAbate later recalled that when he watched the cartoon as a child, Quick Draw would often call Baba Looey Baba Boy, Quick Draws drawn-out pronunciation of boy often sounded like booey, which led DellAbate to think that the characters name was actually Baba Booey. Baba Booey as a term has become a mantra for fans of The Howard Stern Show and is used during crank calls to live network television or radio broadcasts. For example, a made to ABC News during the low-speed police chase of O. J. Simpsons Ford Bronco through the streets of Brentwood, Los Angeles, California. Sportscaster Al Michaels explained to anchor Peter Jennings the significance of the phrase, more recently, Freestyle wrestler Jake Herbert yelled Baba Booey as the camera panned over Team USA during the opening ceremony of the 2012 Olympic Games. Beginning in 2013, there have been hundreds of instances on the PGA Tour in which spectators have shouted Baba Booey. as soon as a golfer has completed his swing, DellAbate has been married since 1992 to Mary DellAbate. They have two sons, Lucas and Jackson, after DellAbates brother Steven died of AIDS in January 1991, DellAbate became a supporter of LIFEbeat The Music Industry Fights AIDS and serves as the charitys president. DellAbate wrote a biography titled They Call Me Baba Booey which was a New York Times best seller and it debuted at #6 on the NY Times Best Seller list in November of 2010. In the book DellAbate tells the story of his brothers death from AIDS and he was featured in the May 2007 issue of Sound and Vision magazine where his home theater was profiled. DellAbate is an avid New York Mets fan, and, on May 9,2009 he threw out the ceremonial first pitch at a Mets game, the pitch did not go well, as it landed down the third-base line and hit an umpire. DellAbate is also a fan of the New York Islanders hockey team, DellAbates thoughts and memories of following the Islanders are featured throughout the book, as well as several photos of DellAbate with Islanders players at a Stanley Cup victory parade in 1980
36.
Block and tackle
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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. The pulleys are assembled together to form blocks and then blocks are paired so that one is fixed, the rope is threaded, or rove, through the pulleys to provide mechanical advantage that amplifies that force applied to the rope. Hero of Alexandria described cranes formed from assemblies of pulleys in the first century, illustrated versions of Heros book on raising heavy weights show early block and tackle systems. A block is a set of pulleys or sheaves mounted on a single axle, an assembly of blocks with a rope threaded through the pulleys is called tackle. A block and tackle system amplifies the force in the rope to lift heavy loads. They are common on boats and sailing ships, where tasks are performed manually. A block and tackle is characterized by the use of a continuous rope to transmit a tension force around one or more pulleys to lift or move a load. Its mechanical advantage is the number of parts of the rope that act on the load, the mechanical advantage of a tackle dictates how much easier it is to haul or lift the load. An ideal block and tackle with a moving block supported by n rope sections has the advantage, M A = F B F A = n. Consider the set of pulleys that form the block and the parts of the rope that support this block. If there are n of these parts of the supporting the load FB. This means the force on the rope is FA=FB/n. Thus, the block and tackle reduces the force by the factor n. Ideal mechanical advantage correlates directly with velocity ratio, the velocity ratio of a tackle is the ratio between the velocity of the hauling line to that of the hauled load. A line with an advantage of 4 has a velocity ratio of 4,1. In other words, to raise a load at 1 metre per second, therefore, the mechanical advantage of a double tackle is 4. In this case the block and tackle is said to be rove to advantage, rove to advantage – where the pull on the rope is in the same direction as that in which the load is to be moved. The hauling part is pulled from the moving block, rove to disadvantage – where the pull on the rope is in the opposite direction to that in which the load is to be moved
37.
Roman Republic
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It was during this period that Romes control expanded from the citys immediate surroundings to hegemony over the entire Mediterranean world. During the first two centuries of its existence, the Roman Republic expanded through a combination of conquest and alliance, by the following century, it included North Africa, most of the Iberian Peninsula, and what is now southern France. Two centuries after that, towards the end of the 1st century BC, it included the rest of modern France, Greece, and much of the eastern Mediterranean. By this time, internal tensions led to a series of wars, culminating with the assassination of Julius Caesar. The exact date of transition can be a matter of interpretation, Roman government was headed by two consuls, elected annually by the citizens and advised by a senate composed of appointed magistrates. Over time, the laws that gave exclusive rights to Romes highest offices were repealed or weakened. The leaders of the Republic developed a tradition and morality requiring public service and patronage in peace and war, making military. Many of Romes legal and legislative structures can still be observed throughout Europe and much of the world in modern nation states, the exact causes and motivations for Romes military conflicts and expansions during the republic are subject to wide debate. While they can be seen as motivated by outright aggression and imperialism and they argue that Romes expansion was driven by short-term defensive and inter-state factors, and the new contingencies that these decisions created. In its early history, as Rome successfully defended itself against foreign threats in central and then northern Italy, with some important exceptions, successful wars in early republican Rome generally led not to annexation or military occupation, but to the restoration of the way things were. But the defeated city would be weakened and thus able to resist Romanizing influences. It was also able to defend itself against its non-Roman enemies. It was, therefore, more likely to seek an alliance of protection with Rome and this growing coalition expanded the potential enemies that Rome might face, and moved Rome closer to confrontation with major powers. The result was more alliance-seeking, on the part of both the Roman confederacy and city-states seeking membership within that confederacy. While there were exceptions to this, it was not until after the Second Punic War that these alliances started to harden into something more like an empire and this shift mainly took place in parts of the west, such as the southern Italian towns that sided with Hannibal. In contrast, Roman expansion into Spain and Gaul occurred as a mix of alliance-seeking, in the 2nd century BC, Roman involvement in the Greek east remained a matter of alliance-seeking, but this time in the face of major powers that could rival Rome. This had some important similarities to the events in Italy centuries earlier, with some major exceptions of outright military rule, the Roman Republic remained an alliance of independent city-states and kingdoms until it transitioned into the Roman Empire. It was not until the time of the Roman Empire that the entire Roman world was organized into provinces under explicit Roman control
38.
Cicero
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Marcus Tullius Cicero was a Roman philosopher, politician, lawyer, orator, political theorist, consul, and constitutionalist. He came from a wealthy family of the Roman equestrian order. According to Michael Grant, the influence of Cicero upon the history of European literature, Cicero introduced the Romans to the chief schools of Greek philosophy and created a Latin philosophical vocabulary distinguishing himself as a translator and philosopher. Though he was an orator and successful lawyer, Cicero believed his political career was his most important achievement. During the chaotic latter half of the 1st century BC marked by civil wars, following Julius Caesars death, Cicero became an enemy of Mark Antony in the ensuing power struggle, attacking him in a series of speeches. His severed hands and head were then, as a revenge of Mark Antony. Petrarchs rediscovery of Ciceros letters is often credited for initiating the 14th-century Renaissance in public affairs, humanism, according to Polish historian Tadeusz Zieliński, the Renaissance was above all things a revival of Cicero, and only after him and through him of the rest of Classical antiquity. Cicero was born in 106 BC in Arpinum, a hill town 100 kilometers southeast of Rome and his father was a well-to-do member of the equestrian order and possessed good connections in Rome. However, being a semi-invalid, he could not enter public life, although little is known about Ciceros mother, Helvia, it was common for the wives of important Roman citizens to be responsible for the management of the household. Ciceros brother Quintus wrote in a letter that she was a thrifty housewife, Ciceros cognomen, or personal surname, comes from the Latin for chickpea, cicer. Plutarch explains that the name was given to one of Ciceros ancestors who had a cleft in the tip of his nose resembling a chickpea. However, it is likely that Ciceros ancestors prospered through the cultivation. Romans often chose down-to-earth personal surnames, the family names of Fabius, Lentulus, and Piso come from the Latin names of beans, lentils. Plutarch writes that Cicero was urged to change this name when he entered politics. During this period in Roman history, cultured meant being able to speak both Latin and Greek, Cicero used his knowledge of Greek to translate many of the theoretical concepts of Greek philosophy into Latin, thus translating Greek philosophical works for a larger audience. It was precisely his broad education that tied him to the traditional Roman elite, according to Plutarch, Cicero was an extremely talented student, whose learning attracted attention from all over Rome, affording him the opportunity to study Roman law under Quintus Mucius Scaevola. Ciceros fellow students were Gaius Marius Minor, Servius Sulpicius Rufus, the latter two became Ciceros friends for life, and Pomponius would become, in Ciceros own words, as a second brother, with both maintaining a lifelong correspondence. Cicero wanted to pursue a career in politics along the steps of the Cursus honorum
39.
Cylinder (geometry)
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In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. If the ends are open, it is called an open cylinder, if the ends are closed by flat surfaces it is called a solid cylinder. The formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h
40.
Alexandria
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Alexandria is the second largest city and a major economic centre in Egypt, extending about 32 km along the coast of the Mediterranean Sea in the north central part of the country. Its low elevation on the Nile delta makes it vulnerable to rising sea levels. Alexandria is Egypts largest seaport, serving approximately 80% of Egypts imports and exports and it is an important industrial center because of its natural gas and oil pipelines from Suez. Alexandria is also an important tourist destination, Alexandria was founded around a small Ancient Egyptian town c.331 BC by Alexander the Great. Alexandria was the second most powerful city of the ancient world after Rome, Alexandria is believed to have been founded by Alexander the Great in April 331 BC as Ἀλεξάνδρεια. Alexanders chief architect for the project was Dinocrates, Alexandria was intended to supersede Naucratis as a Hellenistic center in Egypt, and to be the link between Greece and the rich Nile valley. The city and its museum attracted many of the greatest scholars, including Greeks, Jews, the city was later plundered and lost its significance. Just east of Alexandria, there was in ancient times marshland, as early as the 7th century BC, there existed important port cities of Canopus and Heracleion. The latter was rediscovered under water. An Egyptian city, Rhakotis, already existed on the shore also and it continued to exist as the Egyptian quarter of the city. A few months after the foundation, Alexander left Egypt and never returned to his city, after Alexanders departure, his viceroy, Cleomenes, continued the expansion. Although Cleomenes was mainly in charge of overseeing Alexandrias continuous development, the Heptastadion, inheriting the trade of ruined Tyre and becoming the center of the new commerce between Europe and the Arabian and Indian East, the city grew in less than a generation to be larger than Carthage. In a century, Alexandria had become the largest city in the world and and it became Egypts main Greek city, with Greek people from diverse backgrounds. Alexandria was not only a center of Hellenism, but was home to the largest urban Jewish community in the world. The Septuagint, a Greek version of the Tanakh, was produced there, in AD115, large parts of Alexandria were destroyed during the Kitos War, which gave Hadrian and his architect, Decriannus, an opportunity to rebuild it. On 21 July 365, Alexandria was devastated by a tsunami, the Islamic prophet, Muhammads first interaction with the people of Egypt occurred in 628, during the Expedition of Zaid ibn Haritha. He sent Hatib bin Abi Baltaeh with a letter to the king of Egypt and Alexandria called Muqawqis In the letter Muhammad said, I invite you to accept Islam, Allah the sublime, shall reward you doubly. But if you refuse to do so, you bear the burden of the transgression of all the Copts
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Isidore of Miletus
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Isidore of Miletus was one of the two main Byzantine Greek architects that Emperor Justinian I commissioned to design the church of Hagia Sophia in Constantinople from 532-537. He also created the first comprehensive compilation of Archimedes works. ”Isidore is also renowned for producing the first comprehensive compilation of Archimedes work, one copy of which survived to the present. Emperor Justinian I appointed his architects to rebuild the Hagia Sophia following his victory over protesters within the city of his Roman Empire, Constantinople. ”The warring factions of Byzantine society, the Blues. During the Nika Riot, more than thirty people died. ”The Hagia Sophia was repeatedly cracked by earthquakes and was quickly repaired. Isidore of Miletus’ nephew, Isidore the Younger, introduced the new design that can be viewed in the Hagia Sophia in present-day Istanbul. After a great earthquake in 989 ruined the dome of Hagia Sophia, the restored dome was completed by 994. Cakmak, AS, Taylor, RM, Durukal, E, the Structural Configuration of the First Dome of Justinians Hagia Sophia, An Investigation Based on Structural and Literary Analysis. The Art of the Byzantine Empire, 312-1453, Sources and Documents, the Architect Trdat, Building Practices and Cross-Cultural Exchange in Byzantium and Armenia. The Journal of the Society of Architectural Historians, the Secret History, With Related Texts
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Byzantine
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It survived the fragmentation and fall of the Western Roman Empire in the 5th century AD and continued to exist for an additional thousand years until it fell to the Ottoman Turks in 1453. During most of its existence, the empire was the most powerful economic, cultural, several signal events from the 4th to 6th centuries mark the period of transition during which the Roman Empires Greek East and Latin West divided. Constantine I reorganised the empire, made Constantinople the new capital, under Theodosius I, Christianity became the Empires official state religion and other religious practices were proscribed. Finally, under the reign of Heraclius, the Empires military, the borders of the Empire evolved significantly over its existence, as it went through several cycles of decline and recovery. During the reign of Maurice, the Empires eastern frontier was expanded, in a matter of years the Empire lost its richest provinces, Egypt and Syria, to the Arabs. This battle opened the way for the Turks to settle in Anatolia, the Empire recovered again during the Komnenian restoration, such that by the 12th century Constantinople was the largest and wealthiest European city. Despite the eventual recovery of Constantinople in 1261, the Byzantine Empire remained only one of several small states in the area for the final two centuries of its existence. Its remaining territories were annexed by the Ottomans over the 15th century. The Fall of Constantinople to the Ottoman Empire in 1453 finally ended the Byzantine Empire, the term comes from Byzantium, the name of the city of Constantinople before it became Constantines capital. This older name of the city would rarely be used from this point onward except in historical or poetic contexts. The publication in 1648 of the Byzantine du Louvre, and in 1680 of Du Canges Historia Byzantina further popularised the use of Byzantine among French authors, however, it was not until the mid-19th century that the term came into general use in the Western world. The Byzantine Empire was known to its inhabitants as the Roman Empire, the Empire of the Romans, Romania, the Roman Republic, Graikia, and also as Rhōmais. The inhabitants called themselves Romaioi and Graikoi, and even as late as the 19th century Greeks typically referred to modern Greek as Romaika and Graikika. The authority of the Byzantine emperor as the legitimate Roman emperor was challenged by the coronation of Charlemagne as Imperator Augustus by Pope Leo III in the year 800. No such distinction existed in the Islamic and Slavic worlds, where the Empire was more seen as the continuation of the Roman Empire. In the Islamic world, the Roman Empire was known primarily as Rûm, the Roman army succeeded in conquering many territories covering the entire Mediterranean region and coastal regions in southwestern Europe and north Africa. These territories were home to different cultural groups, both urban populations and rural populations. The West also suffered heavily from the instability of the 3rd century AD
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Middle Ages
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In the history of Europe, the Middle Ages or Medieval Period lasted from the 5th to the 15th century. It began with the fall of the Western Roman Empire and merged into the Renaissance, the Middle Ages is the middle period of the three traditional divisions of Western history, classical antiquity, the medieval period, and the modern period. The medieval period is subdivided into the Early, High. Population decline, counterurbanisation, invasion, and movement of peoples, the large-scale movements of the Migration Period, including various Germanic peoples, formed new kingdoms in what remained of the Western Roman Empire. In the seventh century, North Africa and the Middle East—once part of the Byzantine Empire—came under the rule of the Umayyad Caliphate, although there were substantial changes in society and political structures, the break with classical antiquity was not complete. The still-sizeable Byzantine Empire survived in the east and remained a major power, the empires law code, the Corpus Juris Civilis or Code of Justinian, was rediscovered in Northern Italy in 1070 and became widely admired later in the Middle Ages. In the West, most kingdoms incorporated the few extant Roman institutions, monasteries were founded as campaigns to Christianise pagan Europe continued. The Franks, under the Carolingian dynasty, briefly established the Carolingian Empire during the later 8th, the Crusades, first preached in 1095, were military attempts by Western European Christians to regain control of the Holy Land from Muslims. Kings became the heads of centralised nation states, reducing crime and violence, intellectual life was marked by scholasticism, a philosophy that emphasised joining faith to reason, and by the founding of universities. Controversy, heresy, and the Western Schism within the Catholic Church paralleled the conflict, civil strife. Cultural and technological developments transformed European society, concluding the Late Middle Ages, the Middle Ages is one of the three major periods in the most enduring scheme for analysing European history, classical civilisation, or Antiquity, the Middle Ages, and the Modern Period. Medieval writers divided history into periods such as the Six Ages or the Four Empires, when referring to their own times, they spoke of them as being modern. In the 1330s, the humanist and poet Petrarch referred to pre-Christian times as antiqua, leonardo Bruni was the first historian to use tripartite periodisation in his History of the Florentine People. Bruni and later argued that Italy had recovered since Petrarchs time. The Middle Ages first appears in Latin in 1469 as media tempestas or middle season, in early usage, there were many variants, including medium aevum, or middle age, first recorded in 1604, and media saecula, or middle ages, first recorded in 1625. The alternative term medieval derives from medium aevum, tripartite periodisation became standard after the German 17th-century historian Christoph Cellarius divided history into three periods, Ancient, Medieval, and Modern. The most commonly given starting point for the Middle Ages is 476, for Europe as a whole,1500 is often considered to be the end of the Middle Ages, but there is no universally agreed upon end date. English historians often use the Battle of Bosworth Field in 1485 to mark the end of the period
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Renaissance
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The Renaissance was a period in European history, from the 14th to the 17th century, regarded as the cultural bridge between the Middle Ages and modern history. It started as a movement in Italy in the Late Medieval period and later spread to the rest of Europe. This new thinking became manifest in art, architecture, politics, science, Early examples were the development of perspective in oil painting and the recycled knowledge of how to make concrete. Although the invention of movable type sped the dissemination of ideas from the later 15th century. In politics, the Renaissance contributed to the development of the customs and conventions of diplomacy, the Renaissance began in Florence, in the 14th century. Other major centres were northern Italian city-states such as Venice, Genoa, Milan, Bologna, the word Renaissance, literally meaning Rebirth in French, first appeared in English in the 1830s. The word also occurs in Jules Michelets 1855 work, Histoire de France, the word Renaissance has also been extended to other historical and cultural movements, such as the Carolingian Renaissance and the Renaissance of the 12th century. The Renaissance was a movement that profoundly affected European intellectual life in the early modern period. Renaissance scholars employed the humanist method in study, and searched for realism, however, a subtle shift took place in the way that intellectuals approached religion that was reflected in many other areas of cultural life. In addition, many Greek Christian works, including the Greek New Testament, were back from Byzantium to Western Europe. Political philosophers, most famously Niccolò Machiavelli, sought to describe life as it really was. Others see more competition between artists and polymaths such as Brunelleschi, Ghiberti, Donatello, and Masaccio for artistic commissions as sparking the creativity of the Renaissance. Yet it remains much debated why the Renaissance began in Italy, accordingly, several theories have been put forward to explain its origins. During the Renaissance, money and art went hand in hand, Artists depended entirely on patrons while the patrons needed money to foster artistic talent. Wealth was brought to Italy in the 14th, 15th, and 16th centuries by expanding trade into Asia, silver mining in Tyrol increased the flow of money. Luxuries from the Eastern world, brought home during the Crusades, increased the prosperity of Genoa, unlike with Latin texts, which had been preserved and studied in Western Europe since late antiquity, the study of ancient Greek texts was very limited in medieval Western Europe. One of the greatest achievements of Renaissance scholars was to bring this entire class of Greek cultural works back into Western Europe for the first time since late antiquity, Arab logicians had inherited Greek ideas after they had invaded and conquered Egypt and the Levant. Their translations and commentaries on these ideas worked their way through the Arab West into Spain and Sicily and this work of translation from Islamic culture, though largely unplanned and disorganized, constituted one of the greatest transmissions of ideas in history
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Archimedes Palimpsest
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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 Archimedes work is now readable after scientific and scholarly work from 1998 to 2008 on images produced by ultraviolet, infrared, visible and raking light, and X-ray. The Palimpsest is the only copy of Stomachion and The Method of Mechanical Theorems. Archimedes lived in the 3rd century BC and wrote his proofs as letters in Doric Greek addressed to contemporaries, a copy of this text was made around 950 AD, again in the Byzantine Empire, by an anonymous scribe. This medieval Byzantine manuscript then traveled to Jerusalem, likely sometime after the Crusader sack of Constantinople in 1204, there, in 1229, the original Archimedes codex was unbound, scraped and washed, along with at least six other parchment manuscripts, including one with works of Hypereides. The parchment leaves were folded in half and reused for a Christian liturgical text of 177 pages, the palimpsest remained near Jerusalem through at least the 16th century at the isolated Greek Orthodox monastery of Mar Saba. At some point before 1840 the palimpsest was brought back by the Greek Orthodox Patriarchate of Jerusalem to their library in Constantinople, in 1899 the Greek scholar Papadopoulos-Kerameus produced a catalog of the librarys manuscripts and included a transcription of several lines of the partially visible underlying text. Upon seeing these lines Johan Heiberg, the authority on Archimedes. When Heiberg studied the palimpsest in Constantinople in 1906, he confirmed that the palimpsest included works by Archimedes thought to have been lost. Heiberg was permitted to take photographs of the palimpsests pages. Shortly thereafter Archimedes Greek text was translated into English by T. L. Heath, before that it was not widely known among mathematicians, physicists or historians. The manuscript was still in the Greek Orthodox Patriarchate of Jerusalems library in Constantinople in 1920, sometime between 1923 and 1930 the palimpsest was acquired by Marie Louis Sirieix, a businessman and traveler to the Orient who lived in Paris. Though Sirieix claimed to have bought it from a monk, who would not in any case have had the authority to sell it, stored secretly for years in Sirieixs cellar, the palimpsest suffered damage from water and mold. These gold leaf portraits nearly obliterated the text underneath them, Sirieix died in 1956, and in 1970 his daughter began attempting quietly to sell the manuscript. Unable to sell it privately in 1998 she finally turned to Christies to sell it in a public auction, indeed, the ownership of the palimpsest was immediately contested in federal court in New York in the case of the Greek Orthodox Patriarchate of Jerusalem v. Christies, Inc. The plaintiff contended that the palimpsest had been stolen from its library in Constantinople in the 1920s, judge Kimba Wood decided in favor of Christies Auction House on laches grounds, and the palimpsest was bought for $2 million by an anonymous buyer. Simon Finch, who represented the anonymous buyer, stated that the buyer was a private American who worked in the high-tech industry, at the Walters Art Museum in Baltimore, the palimpsest was the subject of an extensive imaging study from 1999 to 2008, and conservation. This was directed by Dr. Will Noel, curator of manuscripts at the Walters Art Museum, Toth of R. B. Toth Associates, with Dr. Abigail Quandt performing the conservation of the manuscript
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Colonies in antiquity
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Colonies in antiquity were 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, however, unlike in the period of European colonialism during the early and late modern era, ancient colonies were usually sovereign and self-governing from their inception. An Egyptian colony that was stationed in southern Canaan dates to slightly before the First Dynasty, narmer had Egyptian pottery produced in Canaan and exported back to Egypt, from regions such as Arad, En Besor, Rafiah, and Tel ʿErani. Shipbuilding was known to the ancient Egyptians as early as 3000 BC, the Archaeological Institute of America reports that the earliest dated ship—75 feet long, dating to 3000 BC – may have possibly belonged to Pharaoh Aha. Egypt at its height controlled Crete across the Mediterranean Sea, the Phoenicians were the major trading power in the Mediterranean in the early part of the first millennium BC. They had trading contacts in Egypt and Greece, and established colonies as far west as modern Spain, from Gadir the Phoenicians controlled access to the Atlantic Ocean and the trade routes to Britain. The most famous and successful of Phoenician colonies was founded by settlers from Tyre in 814–813 BC and called Kart-Hadasht (Qart-ḥadašt, the Carthaginians later founded their own colony in the southeast of Spain, Carthago Nova, which was eventually conquered by their enemy, Rome. But in most cases the motivation was to establish and facilitate relations of trade with foreign countries, colonies were established in Ionia and Thrace as early as the 8th century BC. There were two types of colony, one known as an ἀποικία - apoikia and the other as an ἐμπορίov - emporion. The first type of colony was a city-state on its own, through this Greek expansion the use of coins flourished throughout the Mediterranean Basin. The Greeks also colonised modern-day Crimea on the Black Sea, among the settlements they established there was the city of Chersonesos, at the site of modern-day Sevastopol. Another area with significant Greek colonies was the coast of ancient Illyria on the Adriatic Sea, the extensive Greek colonization is remarked upon by Cicero when noting that It were as though a Greek fringe has been woven about the shores of the barbarians. Several formulae were generally adhered to on the solemn and sacred occasions when a new colony set forth, if a Greek city was sending out a colony, an oracle, especially one such as the Oracle of Delphi, was almost invariably consulted beforehand. A person of distinction was selected to guide the emigrants and make the necessary arrangements and it was usual to honor these founders of colonies, after their death, as heroes. Some of the fire was taken from the public hearth in the Prytaneum. After the conquests of the Macedonian Kingdom and Alexander the Great, the relation between colony and mother-city, known literally as the metropolis, was viewed as one of mutual affection. Any differences that arose were resolved by peaceful means whenever possible and it is worth noting that the Peloponnesian War was in part a result of a dispute between Corinth and her colony of Corcyra. The charter of foundation contained general provisions for the arrangement of the affairs of the colony, the constitution of the mother-city was usually adopted by the colony, but the new city remained politically independent
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Southern Italy
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It generally coincides with the administrative regions of Abruzzo, Apulia, Basilicata, Campania, Calabria, Molise, Sicily, and Sardinia. Some also include the most southern and eastern parts of Lazio within the Mezzogiorno, Southern Italy carries a unique legacy of culture. It features many major tourist attractions, such as the Palace of Caserta, there are also many ancient Greek cities in Southern Italy, such as Sybaris, which were founded several centuries before the start of the Roman Republic. These same subdivisions are at the bottom of the Italian First level NUTS of the European Union, the term Mezzogiorno first came into use in the 18th century and is an Italian rendition of meridies. The term was popularised by Giuseppe Garibaldi and it eventually came into vogue after the Italian unification. In a similar manner, Southern France is colloquially known as le Midi, Southern Italy forms the lower part of the Italian boot, containing the ankle, the toe, the arch, and the heel, along with the island of Sicily. Separating the heel and the boot is the Gulf of Taranto, named after the city of Taranto and it is an arm of the Ionian Sea. The island of Sardinia, right below the French island of Corsica, on the eastern coast is the Adriatic Sea, leading into the rest of the Mediterranean through the Strait of Otranto. Along the northern coast of the Salernitan Gulf and on the south of the Sorrentine Peninsula runs the Amalfi Coast, off the tip of the peninsula is the isle of Capri. The climate is mainly Mediterranean, except at the highest elevations and the eastern stretches in Apulia, along the Ionian Sea in Calabria. The largest city of Southern Italy is Naples, a name from the Greek that it has maintained for millennia. Bari, Taranto, Reggio Calabria, Foggia, and Salerno are the next largest cities in the area. The region is very active and highly seismic, the 1980 Irpinia earthquake killed 2,914 people, injured more than 10,000. Also during this period, Greek colonies were established in places as widely separated as the eastern coast of the Black Sea, Eastern Libya and they included settlements in Sicily and the southern part of the Italian Peninsula. The Romans called the area of Sicily and the foot of Italy, Magna Graecia, the ancient geographers differed on whether the term included Sicily or merely Apulia and Calabria—Strabo being the most prominent advocate of the wider definitions. With this colonisation, Greek culture was exported to Italy, in its dialects of the Ancient Greek language, its religious rites, an original Hellenic civilization soon developed, later interacting with the native Italic and Latin civilisations. Many of the new Hellenic cities became very rich and powerful, like Neapolis, Syracuse, Acragas, other cities in Magna Graecia included Tarentum, Epizephyrian Locri, Rhegium, Croton, Thurii, Elea, Nola, Syessa, Bari, and others. After Pyrrhus of Epirus failed in his attempt to stop the spread of Roman hegemony in 282 BCE, from then to the Norman conquest of the 11th century, the south of the peninsula was constantly plunged into wars between Greece, Lombardy, and the Islamic Caliphate