1.
Turin
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Turin is a city and an important business and cultural centre in northern Italy, capital of the Piedmont region and was the first capital city of Italy. The city is located mainly on the bank of the Po River, in front of Susa Valley and surrounded by the western Alpine arch. The population of the city proper is 892,649 while the population of the area is estimated by Eurostat to be 1.7 million inhabitants. The Turin metropolitan area is estimated by the OECD to have a population of 2.2 million, in 1997 a part of the historical center of Torino was inscribed in the World Heritage List under the name Residences of the Royal House of Savoy. Turin is well known for its Renaissance, Baroque, Rococo, Neo-classical, many of Turins public squares, castles, gardens and elegant palazzi such as Palazzo Madama, were built between the 16th and 18th centuries. This was after the capital of the Duchy of Savoy was moved to Turin from Chambery as part of the urban expansion, the city used to be a major European political center. Turin was Italys first capital city in 1861 and home to the House of Savoy, from 1563, it was the capital of the Duchy of Savoy, then of the Kingdom of Sardinia ruled by the Royal House of Savoy and finally the first capital of the unified Italy. Turin is sometimes called the cradle of Italian liberty for having been the birthplace and home of notable politicians and people who contributed to the Risorgimento, such as Cavour. The city currently hosts some of Italys best universities, colleges, academies, lycea and gymnasia, such as the University of Turin, founded in the 15th century, in addition, the city is home to museums such as the Museo Egizio and the Mole Antonelliana. Turins attractions make it one of the worlds top 250 tourist destinations, Turin is ranked third in Italy, after Milan and Rome, for economic strength. With a GDP of $58 billion, Turin is the worlds 78th richest city by purchasing power, as of 2010, the city has been ranked by GaWC as a Gamma World city. Turin is also home to much of the Italian automotive industry, the Taurini were an ancient Celto-Ligurian Alpine people, who occupied the upper valley of the Po River, in the center of modern Piedmont. In 218 BC, they were attacked by Hannibal as he was allied with their long-standing enemies, the Taurini chief town was captured by Hannibals forces after a three-day siege. As a people they are mentioned in history. It is believed that a Roman colony was established in 27 BC under the name of Castra Taurinorum, both Livy and Strabo mention the Taurinis country as including one of the passes of the Alps, which points to a wider use of the name in earlier times. In the 1st century BC, the Romans created a military camp, the typical Roman street grid can still be seen in the modern city, especially in the neighborhood known as the Quadrilatero Romano. Via Garibaldi traces the path of the Roman citys decumanus which began at the Porta Decumani. The Porta Palatina, on the side of the current city centre, is still preserved in a park near the Cathedral
Turin
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From top to bottom, left to right: panorama of the Mole Antonelliana, Valentino Park with the medieval village, Piazza Castello with Palazzo Reale and Palazzo Madama, San Carlo Plaza with the Caval ëd Bronz, the Arco Olimpico and the Lingotto, the sarcophagus of Oki at the Egyptian Museum, a view of the hills, the Po, the Gran Madre, the Monte of Cappuccini and Palatine Towers.
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The Roman Palatine Towers.
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Siege of Turin
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Turin in the 17th century.
2.
Paris
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Paris is the capital and most populous city of France. It has an area of 105 square kilometres and a population of 2,229,621 in 2013 within its administrative limits, the agglomeration has grown well beyond the citys administrative limits. By the 17th century, Paris was one of Europes major centres of finance, commerce, fashion, science, and the arts, and it retains that position still today. The aire urbaine de Paris, a measure of area, spans most of the Île-de-France region and has a population of 12,405,426. It is therefore the second largest metropolitan area in the European Union after London, the Metropole of Grand Paris was created in 2016, combining the commune and its nearest suburbs into a single area for economic and environmental co-operation. Grand Paris covers 814 square kilometres and has a population of 7 million persons, the Paris Region had a GDP of €624 billion in 2012, accounting for 30.0 percent of the GDP of France and ranking it as one of the wealthiest regions in Europe. The city is also a rail, highway, and air-transport hub served by two international airports, Paris-Charles de Gaulle and Paris-Orly. Opened in 1900, the subway system, the Paris Métro. It is the second busiest metro system in Europe after Moscow Metro, notably, Paris Gare du Nord is the busiest railway station in the world outside of Japan, with 262 millions passengers in 2015. In 2015, Paris received 22.2 million visitors, making it one of the top tourist destinations. The association football club Paris Saint-Germain and the rugby union club Stade Français are based in Paris, the 80, 000-seat Stade de France, built for the 1998 FIFA World Cup, is located just north of Paris in the neighbouring commune of Saint-Denis. Paris hosts the annual French Open Grand Slam tennis tournament on the red clay of Roland Garros, Paris hosted the 1900 and 1924 Summer Olympics and is bidding to host the 2024 Summer Olympics. The name Paris is derived from its inhabitants, the Celtic Parisii tribe. Thus, though written the same, the name is not related to the Paris of Greek mythology. In the 1860s, the boulevards and streets of Paris were illuminated by 56,000 gas lamps, since the late 19th century, Paris has also been known as Panam in French slang. Inhabitants are known in English as Parisians and in French as Parisiens and they are also pejoratively called Parigots. The Parisii, a sub-tribe of the Celtic Senones, inhabited the Paris area from around the middle of the 3rd century BC. One of the areas major north-south trade routes crossed the Seine on the île de la Cité, this place of land and water trade routes gradually became a town
Paris
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In the 1860s Paris streets and monuments were illuminated by 56,000 gas lamps, making it literally "The City of Light."
Paris
Paris
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Gold coins minted by the Parisii (1st century BC)
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The Palais de la Cité and Sainte-Chapelle, viewed from the Left Bank, from the Très Riches Heures du duc de Berry (month of June) (1410)
3.
Greater French Empire
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The First French Empire, Note 1 was the empire of Napoleon Bonaparte of France and the dominant power in much of continental Europe at the beginning of the 19th century. Its name was a misnomer, as France already had colonies overseas and was short lived compared to the Colonial Empire, a series of wars, known collectively as the Napoleonic Wars, extended French influence over much of Western Europe and into Poland. The plot included Bonapartes brother Lucien, then serving as speaker of the Council of Five Hundred, Roger Ducos, another Director, on 9 November 1799 and the following day, troops led by Bonaparte seized control. They dispersed the legislative councils, leaving a rump legislature to name Bonaparte, Sieyès, although Sieyès expected to dominate the new regime, the Consulate, he was outmaneuvered by Bonaparte, who drafted the Constitution of the Year VIII and secured his own election as First Consul. He thus became the most powerful person in France, a power that was increased by the Constitution of the Year X, the Battle of Marengo inaugurated the political idea that was to continue its development until Napoleons Moscow campaign. Napoleon planned only to keep the Duchy of Milan for France, setting aside Austria, the Peace of Amiens, which cost him control of Egypt, was a temporary truce. He gradually extended his authority in Italy by annexing the Piedmont and by acquiring Genoa, Parma, Tuscany and Naples, then he laid siege to the Roman state and initiated the Concordat of 1801 to control the material claims of the pope. Napoleon would have ruling elites from a fusion of the new bourgeoisie, on 12 May 1802, the French Tribunat voted unanimously, with exception of Carnot, in favour of the Life Consulship for the leader of France. This action was confirmed by the Corps Législatif, a general plebiscite followed thereafter resulting in 3,653,600 votes aye and 8,272 votes nay. On 2 August 1802, Napoleon Bonaparte was proclaimed Consul for life, pro-revolutionary sentiment swept through Germany aided by the Recess of 1803, which brought Bavaria, Württemberg and Baden to Frances side. The memories of imperial Rome were for a time, after Julius Caesar and Charlemagne. The Treaty of Pressburg, signed on 26 December 1805, did little other than create a more unified Germany to threaten France. On the other hand, Napoleons creation of the Kingdom of Italy, the occupation of Ancona, to create satellite states, Napoleon installed his relatives as rulers of many European states. The Bonapartes began to marry into old European monarchies, gaining sovereignty over many nations, in addition to the vassal titles, Napoleons closest relatives were also granted the title of French Prince and formed the Imperial House of France. Met with opposition, Napoleon would not tolerate any neutral power, Prussia had been offered the territory of Hanover to stay out of the Third Coalition. With the diplomatic situation changing, Napoleon offered Great Britain the province as part of a peace proposal and this, combined with growing tensions in Germany over French hegemony, Prussia responded by forming an alliance with Russia and sending troops into Bavaria on 1 October 1806. In this War of the Fourth Coalition, Napoleon destroyed the armies of Frederick William at Jena-Auerstedt, the Eylau and the Friedland against the Russians finally ruined Frederick the Greats formerly mighty kingdom, obliging Russia and Prussia to make peace with France at Tilsit. The Treaties of Tilsit ended the war between Russia and the French Empire and began an alliance between the two empires that held power of much of the rest of Europe, the two empires secretly agreed to aid each other in disputes
Greater French Empire
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The Battle of Austerlitz
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Flag
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The Arc de Triomphe, ordered by Napoleon in honour of his Grande Armée, is one of the several landmarks whose construction was started in Paris during the First French Empire.
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Napoleon reviews the Imperial Guard before the Battle of Jena, 1806
4.
Prussia
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Prussia was a historic state originating out of the Duchy of Prussia and the Margraviate of Brandenburg, and centred on the region of Prussia. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually well-organised, Prussia, with its capital in Königsberg and from 1701 in Berlin, shaped the history of Germany. In 1871, German states united to create the German Empire under Prussian leadership, in November 1918, the monarchies were abolished and the nobility lost its political power during the German Revolution of 1918–19. The Kingdom of Prussia was thus abolished in favour of a republic—the Free State of Prussia, from 1933, Prussia lost its independence as a result of the Prussian coup, when the Nazi regime was successfully establishing its Gleichschaltung laws in pursuit of a unitary state. Prussia existed de jure until its liquidation by the Allied Control Council Enactment No.46 of 25 February 1947. The name Prussia derives from the Old Prussians, in the 13th century, the Teutonic Knights—an organized Catholic medieval military order of German crusaders—conquered the lands inhabited by them. In 1308, the Teutonic Knights conquered the region of Pomerelia with Gdańsk and their monastic state was mostly Germanised through immigration from central and western Germany and in the south, it was Polonised by settlers from Masovia. The Second Peace of Thorn split Prussia into the western Royal Prussia, a province of Poland, and the part, from 1525 called the Duchy of Prussia. The union of Brandenburg and the Duchy of Prussia in 1618 led to the proclamation of the Kingdom of Prussia in 1701, Prussia entered the ranks of the great powers shortly after becoming a kingdom, and exercised most influence in the 18th and 19th centuries. During the 18th century it had a say in many international affairs under the reign of Frederick the Great. During the 19th century, Chancellor Otto von Bismarck united the German principalities into a Lesser Germany which excluded the Austrian Empire. At the Congress of Vienna, which redrew the map of Europe following Napoleons defeat, Prussia acquired a section of north western Germany. The country then grew rapidly in influence economically and politically, and became the core of the North German Confederation in 1867, and then of the German Empire in 1871. The Kingdom of Prussia was now so large and so dominant in the new Germany that Junkers and other Prussian élites identified more and more as Germans and less as Prussians. In the Weimar Republic, the state of Prussia lost nearly all of its legal and political importance following the 1932 coup led by Franz von Papen. East Prussia lost all of its German population after 1945, as Poland, the main coat of arms of Prussia, as well as the flag of Prussia, depicted a black eagle on a white background. The black and white colours were already used by the Teutonic Knights. The Teutonic Order wore a white coat embroidered with a cross with gold insert
Prussia
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... during the Renaissance period
Prussia
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Flag (1892–1918)
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... according to the design of 1702
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Prussian King's Crown (Hohenzollern Castle Collection)
5.
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
Mathematics
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Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.
Mathematics
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Greek mathematician Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the Pythagorean theorem
Mathematics
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Leonardo Fibonacci, the Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
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Carl Friedrich Gauss, known as the prince of mathematicians
6.
Alma mater
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Alma mater is an allegorical Latin phrase for a university or college. In modern usage, it is a school or university which an individual has attended, the phrase is variously translated as nourishing mother, nursing mother, or fostering mother, suggesting that a school provides intellectual nourishment to its students. Before its modern usage, Alma mater was a title in Latin for various mother goddesses, especially Ceres or Cybele. The source of its current use is the motto, Alma Mater Studiorum, of the oldest university in continuous operation in the Western world and it is related to the term alumnus, denoting a university graduate, which literally means a nursling or one who is nourished. The phrase can also denote a song or hymn associated with a school, although alma was a common epithet for Ceres, Cybele, Venus, and other mother goddesses, it was not frequently used in conjunction with mater in classical Latin. Alma Redemptoris Mater is a well-known 11th century antiphon devoted to Mary, the earliest documented English use of the term to refer to a university is in 1600, when University of Cambridge printer John Legate began using an emblem for the universitys press. In English etymological reference works, the first university-related usage is often cited in 1710, many historic European universities have adopted Alma Mater as part of the Latin translation of their official name. The University of Bologna Latin name, Alma Mater Studiorum, refers to its status as the oldest continuously operating university in the world. At least one, the Alma Mater Europaea in Salzburg, Austria, the College of William & Mary in Williamsburg, Virginia, has been called the Alma Mater of the Nation because of its ties to the founding of the United States. At Queens University in Kingston, Ontario, and the University of British Columbia in Vancouver, British Columbia, the ancient Roman world had many statues of the Alma Mater, some still extant. Modern sculptures are found in prominent locations on several American university campuses, outside the United States, there is an Alma Mater sculpture on the steps of the monumental entrance to the Universidad de La Habana, in Havana, Cuba. Media related to Alma mater at Wikimedia Commons The dictionary definition of alma mater at Wiktionary Alma Mater Europaea website
Alma mater
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The Alma Mater statue by Mario Korbel, at the entrance of the University of Havana in Cuba.
Alma mater
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John Legate's Alma Mater for Cambridge in 1600
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Alma Mater (1929, Lorado Taft), University of Illinois at Urbana–Champaign
7.
Joseph Fourier
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The Fourier transform and Fouriers law are also named in his honour. Fourier is also credited with the discovery of the greenhouse effect. Fourier was born at Auxerre, the son of a tailor and he was orphaned at age nine. Fourier was recommended to the Bishop of Auxerre, and through this introduction, the commissions in the scientific corps of the army were reserved for those of good birth, and being thus ineligible, he accepted a military lectureship on mathematics. He took a prominent part in his own district in promoting the French Revolution and he was imprisoned briefly during the Terror but in 1795 was appointed to the École Normale, and subsequently succeeded Joseph-Louis Lagrange at the École Polytechnique. Fourier accompanied Napoleon Bonaparte on his Egyptian expedition in 1798, as scientific adviser, cut off from France by the English fleet, he organized the workshops on which the French army had to rely for their munitions of war. He also contributed several papers to the Egyptian Institute which Napoleon founded at Cairo. After the British victories and the capitulation of the French under General Menou in 1801, in 1801, Napoleon appointed Fourier Prefect of the Department of Isère in Grenoble, where he oversaw road construction and other projects. However, Fourier had previously returned home from the Napoleon expedition to Egypt to resume his academic post as professor at École Polytechnique when Napoleon decided otherwise in his remark. The Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place, hence being faithful to Napoleon, he took the office of Prefect. It was while at Grenoble that he began to experiment on the propagation of heat and he presented his paper On the Propagation of Heat in Solid Bodies to the Paris Institute on December 21,1807. He also contributed to the monumental Description de lÉgypte, Fourier moved to England in 1816. Later, he returned to France, and in 1822 succeeded Jean Baptiste Joseph Delambre as Permanent Secretary of the French Academy of Sciences, in 1830, he was elected a foreign member of the Royal Swedish Academy of Sciences. In 1830, his health began to take its toll, Fourier had already experienced, in Egypt and Grenoble. At Paris, it was impossible to be mistaken with respect to the cause of the frequent suffocations which he experienced. A fall, however, which he sustained on the 4th of May 1830, while descending a flight of stairs, shortly after this event, he died in his bed on 16 May 1830. His name is one of the 72 names inscribed on the Eiffel Tower, a bronze statue was erected in Auxerre in 1849, but it was melted down for armaments during World War II. Joseph Fourier University in Grenoble is named after him and this book was translated, with editorial corrections, into English 56 years later by Freeman
Joseph Fourier
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Jean-Baptiste Joseph Fourier
Joseph Fourier
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1820 watercolor caricatures of French mathematicians Adrien-Marie Legendre (left) and Joseph Fourier (right) by French artist Julien-Leopold Boilly, watercolor portrait numbers 29 and 30 of Album de 73 Portraits-Charge Aquarelle’s des Membres de I’Institute.
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Sketch of Fourier, circa 1820.
Joseph Fourier
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Bust of Fourier in Grenoble
8.
Giovanni Antonio Amedeo Plana
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Giovanni Antonio Amedeo Plana was an Italian astronomer and mathematician. Plana was born in Voghera, Italy to Antonio Maria Plana and Giovanna Giacoboni, at the age of 15 he was sent to live with his uncles in Grenoble to complete his education. In 1800 he entered the École Polytechnique, and was one of the students of Joseph-Louis Lagrange, in 1811 he was appointed to the chair of astronomy at the University of Turin thanks to the influence of Lagrange. He spent the remainder of his teaching at that institution. Planas contributions included work on the motions of the Moon, as well as integrals, elliptic functions, heat, electrostatics, and geodesy. In 1820 he was one of the winners of an awarded by the Académie des Sciences in Paris based on the construction of lunar tables using the law of gravity. In 1832 he published the Théorie du mouvement de la lune, in 1834 he was awarded with the Copley Medal by the Royal Society for his studies on lunar motion. He became astronomer royal, and then in 1844 a Baron, at the age of 80 he was granted membership in the prestigious Académie des Sciences. He is considered one of the premiere Italian scientists of his age, the crater Plana on the Moon is named in his honor. Biography and a source for this page, oConnor, John J. Robertson, Edmund F. Giovanni Antonio Amedeo Plana, MacTutor History of Mathematics archive, University of St Andrews
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Giovanni Antonio Amedeo Plana.
9.
Enlightenment Era
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The Enlightenment was an intellectual movement which dominated the world of ideas in Europe during the 18th century, The Century of Philosophy. In France, the doctrines of les Lumières were individual liberty and religious tolerance in opposition to an absolute monarchy. French historians traditionally place the Enlightenment between 1715, the year that Louis XIV died, and 1789, the beginning of the French Revolution, some recent historians begin the period in the 1620s, with the start of the scientific revolution. Les philosophes of the widely circulated their ideas through meetings at scientific academies, Masonic lodges, literary salons, coffee houses. The ideas of the Enlightenment undermined the authority of the monarchy and the Church, a variety of 19th-century movements, including liberalism and neo-classicism, trace their intellectual heritage back to the Enlightenment. The Age of Enlightenment was preceded by and closely associated with the scientific revolution, earlier philosophers whose work influenced the Enlightenment included Francis Bacon, René Descartes, John Locke, and Baruch Spinoza. The major figures of the Enlightenment included Cesare Beccaria, Voltaire, Denis Diderot, Jean-Jacques Rousseau, David Hume, Adam Smith, Benjamin Franklin visited Europe repeatedly and contributed actively to the scientific and political debates there and brought the newest ideas back to Philadelphia. Thomas Jefferson closely followed European ideas and later incorporated some of the ideals of the Enlightenment into the Declaration of Independence, others like James Madison incorporated them into the Constitution in 1787. The most influential publication of the Enlightenment was the Encyclopédie, the ideas of the Enlightenment played a major role in inspiring the French Revolution, which began in 1789. After the Revolution, the Enlightenment was followed by an intellectual movement known as Romanticism. René Descartes rationalist philosophy laid the foundation for enlightenment thinking and his attempt to construct the sciences on a secure metaphysical foundation was not as successful as his method of doubt applied in philosophic areas leading to a dualistic doctrine of mind and matter. His skepticism was refined by John Lockes 1690 Essay Concerning Human Understanding and his dualism was challenged by Spinozas uncompromising assertion of the unity of matter in his Tractatus and Ethics. Both lines of thought were opposed by a conservative Counter-Enlightenment. In the mid-18th century, Paris became the center of an explosion of philosophic and scientific activity challenging traditional doctrines, the political philosopher Montesquieu introduced the idea of a separation of powers in a government, a concept which was enthusiastically adopted by the authors of the United States Constitution. Francis Hutcheson, a philosopher, described the utilitarian and consequentialist principle that virtue is that which provides, in his words. Much of what is incorporated in the method and some modern attitudes towards the relationship between science and religion were developed by his protégés David Hume and Adam Smith. Hume became a figure in the skeptical philosophical and empiricist traditions of philosophy. Immanuel Kant tried to reconcile rationalism and religious belief, individual freedom and political authority, as well as map out a view of the sphere through private
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German philosopher Immanuel Kant
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History of Western philosophy
Enlightenment Era
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Cesare Beccaria, father of classical criminal theory (1738–1794)
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Like other Enlightenment philosophers, Rousseau was critical of the Atlantic slave trade.
10.
Astronomer
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An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside of the scope of Earth. They look at stars, planets, moons, comets and galaxies, as well as other celestial objects — either in observational astronomy. Examples of topics or fields astronomers work on include, planetary science, solar astronomy, there are also related but distinct subjects like physical cosmology which studies the Universe as a whole. Astronomers usually fit into two types, Observational astronomers make direct observations of planets, stars and galaxies, and analyze the data, theoretical astronomers create and investigate models of things that cannot be observed. They use this data to create models or simulations to theorize how different celestial bodies work, there are further subcategories inside these two main branches of astronomy such as planetary astronomy, galactic astronomy or physical cosmology. Today, that distinction has disappeared and the terms astronomer. Professional astronomers are highly educated individuals who typically have a Ph. D. in physics or astronomy and are employed by research institutions or universities. They spend the majority of their time working on research, although quite often have other duties such as teaching, building instruments. The number of astronomers in the United States is actually quite small. The American Astronomical Society, which is the organization of professional astronomers in North America, has approximately 7,000 members. This number includes scientists from other such as physics, geology. The International Astronomical Union comprises almost 10,145 members from 70 different countries who are involved in research at the Ph. D. level. Before CCDs, photographic plates were a method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year, analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers time. Astronomers who serve as faculty spend much of their time teaching undergraduate and graduate classes, most universities also have outreach programs including public telescope time and sometimes planetariums as a public service to encourage interest in the field. Those who become astronomers usually have a background in maths, sciences. Taking courses that teach how to research, write and present papers are also invaluable, in college/university most astronomers get a Ph. D. in astronomy or physics. Keeping in mind how few there are it is understood that graduate schools in this field are very competitive
Astronomer
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The Astronomer by Johannes Vermeer
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Galileo is often referred to as the Father of modern astronomy
Astronomer
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Guy Consolmagno (Vatikan observatory), analyzing a meteorite, 2014
Astronomer
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Emily Lakdawalla at the Planetary Conference 2013
11.
Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. Newtons Principia formulated the laws of motion and universal gravitation that dominated scientists view of the universe for the next three centuries. Newtons work on light was collected in his influential book Opticks. He also formulated a law of cooling, made the first theoretical calculation of the speed of sound. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He was knighted by Queen Anne in 1705 and he spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint and his father, also named Isaac Newton, had died three months before. Born prematurely, he was a child, his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Newtons mother had three children from her second marriage. From the age of twelve until he was seventeen, Newton was educated at The Kings School, Grantham which taught Latin and Greek. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, Henry Stokes, master at the Kings School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a bully, he became the top-ranked student. In June 1661, he was admitted to Trinity College, Cambridge and he started as a subsizar—paying his way by performing valets duties—until he was awarded a scholarship in 1664, which guaranteed him four more years until he would get his M. A. He set down in his notebook a series of Quaestiones about mechanical philosophy as he found it, in 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his B. A. degree in August 1665, in April 1667, he returned to Cambridge and in October was elected as a fellow of Trinity. Fellows were required to become ordained priests, although this was not enforced in the restoration years, however, by 1675 the issue could not be avoided and by then his unconventional views stood in the way. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II. A and he was elected a Fellow of the Royal Society in 1672. Newtons work has been said to distinctly advance every branch of mathematics then studied and his work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers
Isaac Newton
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Portrait of Isaac Newton in 1689 (age 46) by Godfrey Kneller
Isaac Newton
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Newton in a 1702 portrait by Godfrey Kneller
Isaac Newton
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Isaac Newton (Bolton, Sarah K. Famous Men of Science. NY: Thomas Y. Crowell & Co., 1889)
Isaac Newton
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Replica of Newton's second Reflecting telescope that he presented to the Royal Society in 1672
12.
French First Republic
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In the history of France, the First Republic, officially the French Republic, was founded on 21 September 1792 during the French Revolution. The First Republic lasted until the declaration of the First Empire in 1804 under Napoleon, under the Legislative Assembly, which was in power before the proclamation of the First Republic, France was engaged in war with Prussia and Austria. The foreign threat exacerbated Frances political turmoil amid the French Revolution and deepened the passion, in the violence of 10 August 1792, citizens stormed the Tuileries Palace, killing six hundred of the Kings Swiss guards and insisting on the removal of the king. A renewed fear of action prompted further violence, and in the first week of September 1792, mobs of Parisians broke into the citys prisons. This included nobles, clergymen, and political prisoners, but also numerous common criminals, such as prostitutes and petty thieves, many murdered in their cells—raped, stabbed and this became known as the September Massacres. The resulting Convention was founded with the purpose of abolishing the monarchy. The Conventions first act, on 10 August 1792, was to establish the French First Republic, the King, by then a private citizen bearing his family name of Capet, was subsequently put on trial for crimes of high treason starting in December 1792. On 16 January 1793 he was convicted, and on 21 January, throughout the winter of 1792 and spring of 1793, Paris was plagued by food riots and mass hunger. The new Convention did little to remedy the problem until late spring of 1793, despite growing discontent with the National Convention as a ruling body, in June the Convention drafted the Constitution of 1793, which was ratified by popular vote in early August. The Committees laws and policies took the revolution to unprecedented heights, after the arrest and execution of Robespierre in July 1794, the Jacobin club was closed, and the surviving Girondins were reinstated. A year later, the National Convention adopted the Constitution of the Year III and they reestablished freedom of worship, began releasing large numbers of prisoners, and most importantly, initiated elections for a new legislative body. On 3 November 1795, the Directory was established, the period known as the French Consulate began with the coup of 18 Brumaire in 1799. Members of the Directory itself planned the coup, indicating clearly the failing power of the Directory, Napoleon Bonaparte was a co-conspirator in the coup, and became head of the government as the First Consul. He would later proclaim himself Emperor of the French, ending the First French Republic and ushering in the French First Empire
French First Republic
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Napoleon Bonaparte seizes power during the Coup of 18 Brumaire
French First Republic
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13.
Differential calculus
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In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two divisions of calculus, the other being integral calculus. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, the derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation, geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus and integral calculus are connected by the theorem of calculus. Differentiation has applications to nearly all quantitative disciplines, for example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. The derivative of the momentum of a body equals the applied to the body. The reaction rate of a reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials, derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena, derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra. Suppose that x and y are real numbers and that y is a function of x and this relationship can be written as y = f. If f is the equation for a line, then there are two real numbers m and b such that y = mx + b. In this slope-intercept form, the m is called the slope and can be determined from the formula, m = change in y change in x = Δ y Δ x. It follows that Δy = m Δx, a general function is not a line, so it does not have a slope. Geometrically, the derivative of f at the point x = a is the slope of the tangent line to the function f at the point a and this is often denoted f ′ in Lagranges notation or dy/dx|x = a in Leibnizs notation. Since the derivative is the slope of the approximation to f at the point a. If every point a in the domain of f has a derivative, for example, if f = x2, then the derivative function f ′ = dy/dx = 2x
Differential calculus
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The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.
14.
Probability theory
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Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. It is not possible to predict precisely results of random events, two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, a great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory and this culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of space, introduced by Richard von Mises. This became the mostly undisputed axiomatic basis for modern probability theory, most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment. The power set of the space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results, one collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the set of the sample space of die rolls. In this case, is the event that the die falls on some odd number, If the results that actually occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one, the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6 and this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, and the event has a probability of 1, discrete probability theory deals with events that occur in countable sample spaces. Modern definition, The modern definition starts with a finite or countable set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by Ω
Probability theory
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The normal distribution, a continuous probability distribution.
Probability theory
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The Poisson distribution, a discrete probability distribution.
15.
Three-body problem
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The three-body problem is a special case of the n-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, in an extended modern sense, a three-body problem is a class of problems in classical or quantum mechanics that model the motion of three particles. The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia. The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei, however the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth. They submitted their competing first analyses to the Académie Royale des Sciences in 1747 and it was in connection with these researches, in Paris, in the 1740s, that the name three-body problem began to be commonly used. An account published in 1761 by Jean le Rond dAlembert indicates that the name was first used in 1747, in 1887, mathematicians Heinrich Bruns and Henri Poincaré showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases, a prominent example of the classical three-body problem is the movement of a planet with a satellite around a star. In this case, the problem is simplified to two instances of the two-body problem, however, the effect of the star on the movement of the satellite around the planet can be considered as a perturbation. While a spaceflight involving a gravity assist tends to be at least a problem, once far away from the Earth when Earths gravity becomes negligible. The general statement for the three body problem is as follows, in the circular restricted three-body problem, two massive bodies move in circular orbits around their common center of mass, and the third mass is negligible with respect to the other two. It can be useful to consider the effective potential, in 1767 Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. In 1772 Lagrange found a family of solutions in which the three form an equilateral triangle at each instant. Together, these form the central configurations for the three-body problem. These solutions are valid for any mass ratios, and the move on Keplerian ellipses. These five families are the only solutions for which there are explicit analytic formulae. In 1893 Meissel stated what is called the Pythagorean three-body problem. Burrau further investigated this problem in 1913, in 1967 Victor Szebehely and coworkers established eventual escape for this problem using numerical integration, while at the same time finding a nearby periodic solution. In 1911, United States scientist William Duncan MacMillan found one special solution, in 1961, Russian mathematician Sitnikov improved this solution
Three-body problem
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Figure 1: Configuration of the Sitnikov Problem
Three-body problem
16.
Lagrangian mechanics
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Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. No new physics is introduced in Lagrangian mechanics compared to Newtonian mechanics, Newtons laws can include non-conservative forces like friction, however, they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system, dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler-Lagrange equations. Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, as a special case of Noethers theorem. Lagrangian mechanics is important not just for its applications. It can also be applied to systems by analogy, for instance to coupled electric circuits with inductances and capacitances. Lagrangian mechanics is used to solve mechanical problems in physics. Lagrangian mechanics applies to the dynamics of particles, fields are described using a Lagrangian density, Lagranges equations are also used in optimisation problems of dynamic systems. In mechanics, Lagranges equations of the second kind are used more than those of the first kind. Suppose we have a bead sliding around on a wire, or a simple pendulum. This choice eliminates the need for the constraint force to enter into the resultant system of equations, there are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment. For a wide variety of systems, if the size and shape of a massive object are negligible. For a system of N point particles with masses m1, m2, MN, each particle has a position vector, denoted r1, r2. Cartesian coordinates are often sufficient, so r1 =, r2 =, in three dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system. These are all points in space to locate the particles. The velocity of particle is how fast the particle moves along its path of motion. In Newtonian mechanics, the equations of motion are given by Newtons laws, the second law net force equals mass times acceleration, Σ F = m d2r/dt2, applies to each particle. For an N particle system in 3d, there are 3N second order differential equations in the positions of the particles to solve for
Lagrangian mechanics
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Joseph-Louis Lagrange (1736—1813)
Lagrangian mechanics
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Isaac Newton (1642—1726)
Lagrangian mechanics
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Jean d'Alembert (1717—1783)
17.
Kingdom of France
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The Kingdom of France was a medieval and early modern monarchy in Western Europe. It was one of the most powerful states in Europe and a great power since the Late Middle Ages and it was also an early colonial power, with possessions around the world. France originated as West Francia, the half of the Carolingian Empire. A branch of the Carolingian dynasty continued to rule until 987, the territory remained known as Francia and its ruler as rex Francorum well into the High Middle Ages. The first king calling himself Roi de France was Philip II, France continued to be ruled by the Capetians and their cadet lines—the Valois and Bourbon—until the monarchy was overthrown in 1792 during the French Revolution. France in the Middle Ages was a de-centralised, feudal monarchy, in Brittany and Catalonia the authority of the French king was barely felt. Lorraine and Provence were states of the Holy Roman Empire and not yet a part of France, during the Late Middle Ages, the Kings of England laid claim to the French throne, resulting in a series of conflicts known as the Hundred Years War. Subsequently, France sought to extend its influence into Italy, but was defeated by Spain in the ensuing Italian Wars, religiously France became divided between the Catholic majority and a Protestant minority, the Huguenots, which led to a series of civil wars, the Wars of Religion. France laid claim to large stretches of North America, known collectively as New France, Wars with Great Britain led to the loss of much of this territory by 1763. French intervention in the American Revolutionary War helped secure the independence of the new United States of America, the Kingdom of France adopted a written constitution in 1791, but the Kingdom was abolished a year later and replaced with the First French Republic. The monarchy was restored by the great powers in 1814. During the later years of the elderly Charlemagnes rule, the Vikings made advances along the northern and western perimeters of the Kingdom of the Franks, after Charlemagnes death in 814 his heirs were incapable of maintaining political unity and the empire began to crumble. The Treaty of Verdun of 843 divided the Carolingian Empire into three parts, with Charles the Bald ruling over West Francia, the nucleus of what would develop into the kingdom of France. Viking advances were allowed to increase, and their dreaded longboats were sailing up the Loire and Seine rivers and other waterways, wreaking havoc. During the reign of Charles the Simple, Normans under Rollo from Norway, were settled in an area on either side of the River Seine, downstream from Paris, that was to become Normandy. With its offshoots, the houses of Valois and Bourbon, it was to rule France for more than 800 years. Henry II inherited the Duchy of Normandy and the County of Anjou, and married Frances newly divorced ex-queen, Eleanor of Aquitaine, after the French victory at the Battle of Bouvines in 1214, the English monarchs maintained power only in southwestern Duchy of Guyenne. The death of Charles IV of France in 1328 without male heirs ended the main Capetian line, under Salic law the crown could not pass through a woman, so the throne passed to Philip VI, son of Charles of Valois
Kingdom of France
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The Kingdom of France in 1789. Ancien Régime provinces in 1789.
Kingdom of France
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Royal Standarda
Kingdom of France
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Henry IV, by Frans Pourbus the younger, 1610.
Kingdom of France
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Louis XIII, by Philippe de Champaigne, 1647.
18.
Roman Catholic
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The Catholic Church, also known as the Roman Catholic Church or the Universal Church, is the largest Christian church, with more than 1.28 billion members worldwide. As one of the oldest religious institutions in the world, it has played a prominent role in the history, headed by the Bishop of Rome, known as the Pope, the churchs doctrines are summarised in the Nicene Creed and the Apostles Creed. Its central administration is located in Vatican City, enclaved within Rome, the Catholic Church is notable within Western Christianity for its sacred tradition and seven sacraments. It teaches that it is the one church founded by Jesus Christ, that its bishops are the successors of Christs apostles. The Catholic Church maintains that the doctrine on faith and morals that it declares as definitive is infallible. The Latin Church, the Eastern Catholic Churches, as well as such as mendicant orders and enclosed monastic orders. Among the sacraments, the one is the Eucharist, celebrated liturgically in the Mass. The church teaches that through consecration by a priest the sacrificial bread and wine become the body, the Catholic Church practises closed communion, with only baptised members in a state of grace ordinarily permitted to receive the Eucharist. The Virgin Mary is venerated in the Catholic Church as Queen of Heaven and is honoured in numerous Marian devotions. The Catholic Church has influenced Western philosophy, science, art and culture, Catholic spiritual teaching includes spreading the Gospel while Catholic social teaching emphasises support for the sick, the poor and the afflicted through the corporal and spiritual works of mercy. The Catholic Church is the largest non-government provider of education and medical services in the world, from the late 20th century, the Catholic Church has been criticised for its doctrines on sexuality, its refusal to ordain women and its handling of sexual abuse cases. Catholic was first used to describe the church in the early 2nd century, the first known use of the phrase the catholic church occurred in the letter from Saint Ignatius of Antioch to the Smyrnaeans, written about 110 AD. In the Catechetical Discourses of Saint Cyril of Jerusalem, the name Catholic Church was used to distinguish it from other groups that call themselves the church. The use of the adjective Roman to describe the Church as governed especially by the Bishop of Rome became more widespread after the Fall of the Western Roman Empire and into the Early Middle Ages. Catholic Church is the name used in the Catechism of the Catholic Church. The Catholic Church follows an episcopal polity, led by bishops who have received the sacrament of Holy Orders who are given formal jurisdictions of governance within the church. Ultimately leading the entire Catholic Church is the Bishop of Rome, commonly called the pope, in parallel to the diocesan structure are a variety of religious institutes that function autonomously, often subject only to the authority of the pope, though sometimes subject to the local bishop. Most religious institutes only have male or female members but some have both, additionally, lay members aid many liturgical functions during worship services
Roman Catholic
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Saint Peter's Basilica, Vatican City
Roman Catholic
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St. Peter's Basilica, Vatican City
Roman Catholic
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Pope Francis, elected in the papal conclave, 2013
Roman Catholic
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Traditional graphic representation of the Trinity: The earliest attested version of the diagram, from a manuscript of Peter of Poitiers ' writings, c. 1210
19.
Charles Emmanuel III of Sardinia
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Charles Emmanuel III was the Duke of Savoy and King of Sardinia from 1730 until his death. He was born a Prince of Savoy in Turin to Victor Amadeus II of Savoy and his maternal grandparents were Prince Philippe of France and his first wife Princess Henrietta Anne, the youngest daughter of King Charles I of England and Henrietta Maria of France. From his birth he was styled as the Duke of Aosta, at the time of his birth, Charles Emmanuel was not the heir to the Duchy of Savoy, his older brother Prince Victor Amadeus John Philip, Prince of Piedmont, was the heir apparent. Charles Emmanuel was the second of three males that would be born to his parents and his older brother died in 1715 and Charles Emmanuel then became heir apparent. As a result of his aid in the War of the Spanish Succession, Victor Amadeus was forced to exchange Sicily for the less important kingdom of Sardinia in 1720 after objections from an alliance of four nations, including several of his former allies. Yet he retained his new title of King, however, Victor Amadeus in his late years was dominated by shyness and sadness, probably under the effect of some mental illness. In the end, on 3 September 1730, he abdicated and he was not loved by Victor Amadeus, and consequently received an incomplete education. He however acquired noteworthy knowledge in the field along his father. In summer,1731, after having recovered from a fatal illness. The old king was confined to the Castle of Rivoli, where he died without any further harm to Charles. In the War of the Polish Succession Charles Emmanuel sided with the French- backed king Stanislaw I, after the treaty of alliance signed in Turin, on 28 October 1733 he marched on Milan and occupied Lombardy without significant losses. However, when France tried to convince Philip V of Spain to join the coalition, he asked to receive Milan and this was not acceptable for Charles Emmanuel, as it would recreate a Spanish domination in Italy as it had been in the previous centuries. While negotiations continued about the matter, the Savoy-French-Spanish troops attacked Mantua under the command of Charles Emmanuel himself. Sure that in the end Mantua would be assigned to Spain, the Franco-Piedmontese army was victorious in two battles at Crocetta and Guastalla. In the end, when Austria and France signed a peace, in exchange, he was given some territories, including Langhe, Tortona and Novara. Charles Emmanuel was involved in the War of the Austrian Succession, in which he sided with Maria Theresa of Austria, with financial and naval support from England. After noteworthy but inconclusive initial successes, he had to face the French-Spanish invasion of Savoy and, after a failed allied attempt to conquer the Kingdom of Naples, when the enemy army invaded Piedmont, in 1744 he personally defended Cuneo against the Spanish-French besiegers. The following year, with some 20,000 men, he was facing an invasion of two armies with a total of some 60,000 troops, the important strongholds of Alessandria, Asti and Casale fell
Charles Emmanuel III of Sardinia
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Charles Emmanuel III
Charles Emmanuel III of Sardinia
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The young Charles Emmanuel as duke of Aosta, by an unknown artist.
Charles Emmanuel III of Sardinia
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A portrait of a young Charles Emmanuel
Charles Emmanuel III of Sardinia
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The children of Charles and his second wife; (L-R) Eleonora; Victor Amadeus; Maria Felicita and Maria Luisa Gabriella.
20.
Edmond Halley
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Edmond Halley, FRS was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of Halleys Comet. He was the second Astronomer Royal in Britain, succeeding John Flamsteed, Halley was born in Haggerston, in east London. His father, Edmond Halley Sr. came from a Derbyshire family and was a wealthy soap-maker in London, as a child, Halley was very interested in mathematics. He studied at St Pauls School, and from 1673 at The Queens College, while still an undergraduate, Halley published papers on the Solar System and sunspots. While at Oxford University, Halley was introduced to John Flamsteed, influenced by Flamsteeds project to compile a catalog of northern stars, Halley proposed to do the same for the Southern Hemisphere. In 1676, Halley visited the south Atlantic island of Saint Helena, while there he observed a transit of Mercury, and realised that a similar transit of Venus could be used to determine the absolute size of the Solar System. He returned to England in May 1678, in the following year he went to Danzig on behalf of the Royal Society to help resolve a dispute. Because astronomer Johannes Hevelius did not use a telescope, his observations had been questioned by Robert Hooke, Halley stayed with Hevelius and he observed and verified the quality of Hevelius observations. In 1679, Halley published the results from his observations on St. Helena as Catalogus Stellarum Australium which included details of 341 southern stars and these additions to contemporary star maps earned him comparison with Tycho Brahe, e. g. the southern Tycho as described by Flamsteed. Halley was awarded his M. A. degree at Oxford, in 1686, Halley published the second part of the results from his Helenian expedition, being a paper and chart on trade winds and monsoons. The symbols he used to represent trailing winds still exist in most modern day weather chart representations, in this article he identified solar heating as the cause of atmospheric motions. He also established the relationship between pressure and height above sea level. His charts were an important contribution to the field of information visualisation. Halley spent most of his time on lunar observations, but was interested in the problems of gravity. One problem that attracted his attention was the proof of Keplers laws of planetary motion, Halleys first calculations with comets were thereby for the orbit of comet Kirch, based on Flamsteeds observations in 1680-1. Although he was to calculate the orbit of the comet of 1682. They indicated a periodicity of 575 years, thus appearing in the years 531 and 1106 and it is now known to have an orbital period of circa 10,000 years. In 1691, Halley built a bell, a device in which the atmosphere was replenished by way of weighted barrels of air sent down from the surface
Edmond Halley
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Bust of Halley (Royal Observatory, Greenwich)
Edmond Halley
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Portrait by Thomas Murray, c. 1687
Edmond Halley
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Halley's grave
Edmond Halley
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Plaque in South Cloister of Westminster Abbey
21.
Tautochrone
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A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. The curve is a cycloid, and the time is equal to π times the root of the radius over the acceleration of gravity. The tautochrone curve is the same as the curve for any given starting point. The tautochrone problem, the attempt to identify this curve, was solved by Christiaan Huygens in 1659 and he proved geometrically in his Horologium Oscillatorium, originally published in 1673, that the curve was a cycloid. Huygens also proved that the time of descent is equal to the time a body takes to fall vertically the same distance as the diameter of the circle that generates the cycloid, multiplied by π/2. In modern terms, this means that the time of descent is π r / g, where r is the radius of the circle which generates the cycloid and this solution was later used to attack the problem of the brachistochrone curve. Jakob Bernoulli solved the problem using calculus in a paper that saw the first published use of the term integral and these attempts proved to not be useful for a number of reasons. First, the bending of the string causes friction, changing the timing, second, there were much more significant sources of timing errors that overwhelmed any theoretical improvements that traveling on the tautochrone curve helps. Finally, the error of a pendulum decreases as length of the swing decreases. Later, the mathematicians Joseph Louis Lagrange and Leonhard Euler provided a solution to the problem. If the particles position is parametrized by the s from the lowest point. The potential energy is proportional to the height y, one way the curve can be an isochrone is if the Lagrangian is that of a simple harmonic oscillator, the height of the curve must be proportional to the arclength squared. Y = s 2, where the constant of proportionality has been set to 1 by changing units of length. The differential form of relation is d y =2 s d s, d y 2 =4 s 2 d s 2 =4 y, which eliminates s. To find the solution, integrate for x in terms of y, d x d y =1 −4 y 2 y, x = ∫1 −4 u 2 d u, where u = y. This integral is the area under a circle, which can be cut into a triangle. To see that this is a strangely parametrized cycloid, change variables to disentangle the transcendental, the simplest solution to the tautochrone problem is to note a direct relation between the angle of an incline and the gravity felt by a particle on the incline. A particle on a 90° vertical incline feels the effect of gravity
Tautochrone
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Schematic of a cycloidal pendulum.
Tautochrone
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Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points' acceleration along the curve. On the top is the time-position diagram.
22.
Pierre Louis Maupertuis
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Pierre Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, Maupertuis made an expedition to Lapland to determine the shape of the Earth. He is often credited with having invented the principle of least action and his work in natural history is interesting in relation to modern science, since he touched on aspects of heredity and the struggle for life. Maupertuis was born at Saint-Malo, France, to a wealthy family of merchant-corsairs. His father, Renė, had involved in a number of enterprises that were central to the monarchy so that he thrived socially and politically. The son was educated in mathematics by a tutor, Nicolas Guisnée. In 1723 he was admitted to the Académie des Sciences and his early mathematical work revolved around the vis viva controversy, for which Maupertuis developed and extended the work of Isaac Newton and argued against the waning Cartesian mechanics. In the 1730s, the shape of the Earth became a flashpoint in the battle among rival systems of mechanics, Maupertuis, based on his exposition of Newton predicted that the Earth should be oblate, while his rival Jacques Cassini measured it astronomically to be prolate. In 1736 Maupertuis acted as chief of the French Geodesic Mission sent by King Louis XV to Lapland to measure the length of a degree of arc of the meridian and his results, which he published in a book detailing his procedures, essentially settled the controversy in his favor. The book included a narrative of the expedition, and an account of the Käymäjärvi Inscriptions. On his return home he became a member of almost all the societies of Europe. He also expanded into the realm, anonymously publishing a book that was part popular science, part philosophy. In 1740 Maupertuis went to Berlin at the invitation of Frederick II of Prussia, and took part in the Battle of Mollwitz, where he was taken prisoner by the Austrians. On his release he returned to Berlin, and thence to Paris, where he was elected director of the Academy of Sciences in 1742, and in the following year was admitted into the Académie française. His position became extremely awkward with the outbreak of the Seven Years War between his country and his patrons, and his reputation suffered in both Paris and Berlin. Finding his health declining, he retired in 1757 to the south of France, but went in 1758 to Basel, Maupertuis difficult disposition involved him in constant quarrels, of which his controversies with Samuel König and Voltaire during the latter part of his life are examples. The brilliance of much of what he did was undermined by his tendency to leave work unfinished and it was the insight of genius that led him to least-action principle, but a lack of intellectual energy or rigour that prevented his giving it the mathematical foundation that Lagrange would provide. He reveals remarkable powers of perception in heredity, in understanding the mechanism by which developed, even in immunology
Pierre Louis Maupertuis
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Maupertuis, wearing " lapmudes " from his Lapland expedition.
Pierre Louis Maupertuis
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Lettres
23.
Vibrating string
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A vibration in a string is a wave. Resonance causes a string to produce a sound with constant frequency. If the length or tension of the string is correctly adjusted, vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. Let Δ x be the length of a piece of string, m its mass, and μ its linear density. If the horizontal component of tension in the string is a constant, T, if both angles are small, then the tensions on either side are equal and the net horizontal force is zero. This is the equation for y, and the coefficient of the second time derivative term is equal to v −2, thus v = T μ. Once the speed of propagation is known, the frequency of the produced by the string can be calculated. The speed of propagation of a wave is equal to the wavelength λ divided by the period τ, or multiplied by the frequency f, v = λ τ = λ f. If the length of the string is L, the harmonic is the one produced by the vibration whose nodes are the two ends of the string, so L is half of the wavelength of the fundamental harmonic. Hence one obtains Mersennes laws, f = v 2 L =12 L T μ where T is the tension, μ is the linear density, and L is the length of the vibrating part of the string. This effect is called the effect, and the rate at which the string seems to vibrate is the difference between the frequency of the string and the refresh rate of the screen. The same can happen with a fluorescent lamp, at a rate that is the difference between the frequency of the string and the frequency of the alternating current. In daylight and other non-oscillating light sources, this effect does not occur and the string appears still but thicker, a similar but more controllable effect can be obtained using a stroboscope. This device allows matching the frequency of the flash lamp to the frequency of vibration of the string. In a dark room, this shows the waveform. Otherwise, one can use bending or, perhaps more easily, by adjusting the machine heads, to obtain the same, or a multiple, of the AC frequency to achieve the same effect. For example, in the case of a guitar, the 6th string pressed to the third gives a G at 97.999 Hz. A slight adjustment can alter it to 100 Hz, exactly one octave above the current frequency in Europe and most countries in Africa
Vibrating string
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Vibrating string
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Vibration, standing waves in a string. The fundamental and the first 5 overtones in the harmonic series.
24.
Beat (acoustics)
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In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies. When tuning instruments that can produce sustained tones, beats can readily be recognized, tuning two tones to a unison will present a peculiar effect, when the two tones are close in pitch but not identical, the difference in frequency generates the beating. The volume varies like in a tremolo as the sounds alternately interfere constructively and destructively, as the two tones gradually approach unison, the beating slows down and may become so slow as to be imperceptible. It can be proven that the envelope of the maxima and minima form a wave frequency is half the difference between the frequencies of the two original waves. Instead, it is perceived as a variation in the amplitude of the first term in the expression above. It can be said that the lower frequency cosine term is an envelope for the higher frequency one, the frequency of the modulation is f1 + f2/2, that is, the average of the two frequencies. It can be noted that every second burst in the pattern is inverted. Each peak is replaced by a trough and vice versa, however, because the human ear is not sensitive to the phase of a sound, only its amplitude or intensity, only the magnitude of the envelope is heard. A physical interpretation is that when cos =1 the two waves are in phase and they interfere constructively, when it is zero, they are out of phase and interfere destructively. Beats occur also in more complex sounds, or in sounds of different volumes, beating can also be heard between notes that are near to, but not exactly, a harmonic interval, due to some harmonic of the first note beating with a harmonic of the second note. For example, in the case of perfect fifth, the harmonic of the bass note beats with the second harmonic of the other note. Musicians commonly use interference beats to objectively check tuning at the unison, perfect fifth, piano and organ tuners even use a method involving counting beats, aiming at a particular number for a specific interval. The composer Alvin Lucier has written many pieces that feature interference beats as their main focus, composer Phill Niblocks music is entirely based on beating caused by microtonal differences. Binaural beats are heard when the right ear listens to a different tone than the left ear. Here, the tones do not interfere physically, but are summed by the brain in the olivary nucleus and this effect is related to the brains ability to locate sounds in three dimensions. Combination tone Gamelan tuning Heterodyne Consonance and dissonance Moiré pattern, a form of interference that generates new frequencies
Beat (acoustics)
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Diagram of beat frequency
25.
Probability
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Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, the higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair coin, since the coin is unbiased, the two outcomes are both equally probable, the probability of head equals the probability of tail. Since no other outcomes are possible, the probability is 1/2 and this type of probability is also called a priori probability. Probability theory is used to describe the underlying mechanics and regularities of complex systems. For example, tossing a coin twice will yield head-head, head-tail, tail-head. The probability of getting an outcome of head-head is 1 out of 4 outcomes or 1/4 or 0.25 and this interpretation considers probability to be the relative frequency in the long run of outcomes. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, subjectivists assign numbers per subjective probability, i. e. as a degree of belief. The degree of belief has been interpreted as, the price at which you would buy or sell a bet that pays 1 unit of utility if E,0 if not E. The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as data to produce probabilities. The expert knowledge is represented by some prior probability distribution and these data are incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a probability distribution that incorporates all the information known to date. The scientific study of probability is a development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, there are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the study of probability. According to Richard Jeffrey, Before the middle of the century, the term probable meant approvable. A probable action or opinion was one such as people would undertake or hold. However, in legal contexts especially, probable could also apply to propositions for which there was good evidence, the sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes
Probability
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Christiaan Huygens probably published the first book on probability
Probability
–
Gerolamo Cardano
Probability
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Carl Friedrich Gauss
26.
Integral calculus
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
Integral calculus
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A definite integral of a function can be represented as the signed area of the region bounded by its graph.
27.
N-body problem
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In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, in the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is more difficult to solve. Having done so, he and others soon discovered over the course of a few years, Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits. Thus came the awareness and rise of the problem in the early 17th century. Ironically, this conformity led to the wrong approach, after Newtons time the n-body problem historically was not stated correctly because it did not include a reference to those gravitational interactive forces. Newton does not say it directly but implies in his Principia the n-body problem is unsolvable because of gravitational interactive forces. Newton said in his Principia, paragraph 21, And hence it is that the force is found in both bodies. The Sun attracts Jupiter and the planets, Jupiter attracts its satellites. Two bodies can be drawn to other by the contraction of rope between them. Newton concluded via his third law of motion according to this Law all bodies must attract each other. This last statement, which implies the existence of gravitational forces, is key. The problem of finding the solution of the n-body problem was considered very important. Indeed, in the late 19th century King Oscar II of Sweden, advised by Gösta Mittag-Leffler, in case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was awarded to Poincaré, even though he did not solve the original problem, the version finally printed contained many important ideas which led to the development of chaos theory. The problem as stated originally was finally solved by Karl Fritiof Sundman for n =3. The n-body problem considers n point masses mi, i =1,2, …, n in a reference frame in three dimensional space ℝ3 moving under the influence of mutual gravitational attraction. Each mass mi has a position vector qi, Newtons second law says that mass times acceleration mi d2qi/dt2 is equal to the sum of the forces on the mass
N-body problem
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The Real Motion v.s. Kepler's Apparent Motion
N-body problem
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Restricted 3-Body Problem
28.
Libration
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Lunar libration is distinct from the slight changes in the Moons visual size as seen from Earth. Although this appearance can also be described as a motion, libration is caused by actual changes in the physical distance of the Moon. The Moon generally has one hemisphere facing the Earth, due to tidal locking, therefore, humans first view of the far side of the Moon resulted from lunar exploration on October 7,1959. However, this picture is only approximately true, over time. Libration is manifested as a slow rocking back and forth of the Moon as viewed from Earth, libration in latitude results from a slight inclination between the Moons axis of rotation and the normal to the plane of its orbit around Earth. Its origin is analogous to how the seasons arise from Earths revolution about the Sun. In 1772 Lagranges analyses determined that small bodies can stably share the orbit as a planet if they remain near Lagrange points. Such ‘trojan asteroids’ have been found co-orbiting with Earth, Jupiter, Mars, trojan asteroids associated with Earth are difficult to observe in the visible spectrum, as their libration paths are such that they would be visible primarily in the daylight sky. In 2010, however, using infrared observation techniques, the asteroid 2010 TK7 was found to be a companion of the Earth, it librates around the leading Lagrange point, L4
Libration
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Theoretical extent of visible lunar surface due to libration in Winkel tripel projection
Libration
29.
Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
Euler
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Portrait by Jakob Emanuel Handmann (1756)
Euler
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1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Euler
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Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as " v − e + f = 2".
Euler
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Euler's grave at the Alexander Nevsky Monastery
30.
Frederick the Great
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Frederick II was King of Prussia from 1740 until 1786, the longest reign of any Hohenzollern king. Frederick was the last titled King in Prussia and declared himself King of Prussia after achieving full sovereignty for all historical Prussian lands, Prussia had greatly increased its territories and became a leading military power in Europe under his rule. He became known as Frederick the Great and was affectionately nicknamed Der Alte Fritz by the Prussian, in his youth, Frederick was more interested in music and philosophy than the art of war. Upon ascending to the Prussian throne, he attacked Austria and claimed Silesia during the Silesian Wars, winning acclaim for himself. Near the end of his life, Frederick physically connected most of his realm by conquering Polish territories in the First Partition of Poland and he was an influential military theorist whose analysis emerged from his extensive personal battlefield experience and covered issues of strategy, tactics, mobility and logistics. Considering himself the first servant of the state, Frederick was a proponent of enlightened absolutism and he modernized the Prussian bureaucracy and civil service and pursued religious policies throughout his realm that ranged from tolerance to segregation. He reformed the system and made it possible for men not of noble stock to become judges. Frederick also encouraged immigrants of various nationalities and faiths to come to Prussia, some critics, however, point out his oppressive measures against conquered Polish subjects during the First Partition. Frederick supported arts and philosophers he favored, as well as allowing complete freedom of the press, Frederick is buried at his favorite residence, Sanssouci in Potsdam. Because he died childless, Frederick was succeeded by his nephew, Frederick William II, son of his brother, historian Leopold von Ranke was unstinting in his praise of Fredericks Heroic life, inspired by great ideas, filled with feats of arms. Immortalized by the raising of the Prussian state to the rank of a power, Johann Gustav Droysen was even more extolling. However, by the 21st century, a re-evaluation of his legacy as a great warrior, Frederick, the son of Frederick William I and his wife, Sophia Dorothea of Hanover, was born in Berlin on 24 January 1712. The birth of Frederick was welcomed by his grandfather, Frederick I, with more than usual pleasure, with the death of his father in 1713, Frederick William became King of Prussia, thus making young Frederick the crown prince. The new king wished for his sons and daughters to be educated not as royalty and he had been educated by a Frenchwoman, Madame de Montbail, who later became Madame de Rocoulle, and he wished that she educate his children. However, he possessed a violent temper and ruled Brandenburg-Prussia with absolute authority. As Frederick grew, his preference for music, literature and French culture clashed with his fathers militarism, in contrast, Fredericks mother Sophia was polite, charismatic and learned. Her father, George Louis of Brunswick-Lüneburg, succeeded to the British throne as King George I in 1714, Frederick was brought up by Huguenot governesses and tutors and learned French and German simultaneously. Although Frederick William I was raised a Calvinist, he feared he was not of the elect, to avoid the possibility of Frederick being motivated by the same concerns, the king ordered that his heir not be taught about predestination
Frederick the Great
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Portrait of Frederick the Great; By Anton Graff, 1781
Frederick the Great
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Baptism of Frederick, 1712 (Harper's Magazine, 1870)
Frederick the Great
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Frederick as Crown Prince (1739)
Frederick the Great
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Rheinsberg Palace, Frederick's residence 1736-1740
31.
Spain
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By population, Spain is the sixth largest in Europe and the fifth in the European Union. Spains capital and largest city is Madrid, other urban areas include Barcelona, Valencia, Seville, Bilbao. Modern humans first arrived in the Iberian Peninsula around 35,000 years ago, in the Middle Ages, the area was conquered by Germanic tribes and later by the Moors. Spain is a democracy organised in the form of a government under a constitutional monarchy. It is a power and a major developed country with the worlds fourteenth largest economy by nominal GDP. Jesús Luis Cunchillos argues that the root of the span is the Phoenician word spy. Therefore, i-spn-ya would mean the land where metals are forged, two 15th-century Spanish Jewish scholars, Don Isaac Abravanel and Solomon ibn Verga, gave an explanation now considered folkloric. Both men wrote in two different published works that the first Jews to reach Spain were brought by ship by Phiros who was confederate with the king of Babylon when he laid siege to Jerusalem. This man was a Grecian by birth, but who had given a kingdom in Spain. He became related by marriage to Espan, the nephew of king Heracles, Heracles later renounced his throne in preference for his native Greece, leaving his kingdom to his nephew, Espan, from whom the country of España took its name. Based upon their testimonies, this eponym would have already been in use in Spain by c.350 BCE, Iberia enters written records as a land populated largely by the Iberians, Basques and Celts. Early on its coastal areas were settled by Phoenicians who founded Western Europe´s most ancient cities Cadiz, Phoenician influence expanded as much of the Peninsula was eventually incorporated into the Carthaginian Empire, becoming a major theater of the Punic Wars against the expanding Roman Empire. After an arduous conquest, the peninsula came fully under Roman Rule, during the early Middle Ages it came under Germanic rule but later, much of it was conquered by Moorish invaders from North Africa. In a process took centuries, the small Christian kingdoms in the north gradually regained control of the peninsula. The last Moorish kingdom fell in the same year Columbus reached the Americas, a global empire began which saw Spain become the strongest kingdom in Europe, the leading world power for a century and a half, and the largest overseas empire for three centuries. Continued wars and other problems led to a diminished status. The Napoleonic invasions of Spain led to chaos, triggering independence movements that tore apart most of the empire, eventually democracy was peacefully restored in the form of a parliamentary constitutional monarchy. Spain joined the European Union, experiencing a renaissance and steady economic growth
Spain
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Lady of Elche
Spain
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Flag
Spain
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Altamira Cave paintings, in Cantabria.
Spain
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Celtic castro in A Guarda, Galicia.
32.
Louis XVI of France
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Louis XVI, born Louis-Auguste, was the last King of France and Navarre before the French Revolution, during which he was also known as Louis Capet. In 1765, at the death of his father, Louis, Dauphin of France, son and heir apparent of Louis XV of France, Louis XVI was guillotined on 21 January 1793. The first part of his reign was marked by attempts to reform France in accordance with Enlightenment ideas and these included efforts to abolish serfdom, remove the taille, and increase tolerance toward non-Catholics. The French nobility reacted to the reforms with hostility. Louis implemented deregulation of the market, advocated by his liberal minister Turgot. In periods of bad harvests, it would lead to food scarcity which would prompt the masses to revolt, from 1776, Louis XVI actively supported the North American colonists, who were seeking their independence from Great Britain, which was realized in the 1783 Treaty of Paris. The ensuing debt and financial crisis contributed to the unpopularity of the Ancien Régime and this led to the convening of the Estates-General of 1789. In 1789, the storming of the Bastille during riots in Paris marked the beginning of the French Revolution. Louiss indecisiveness and conservatism led some elements of the people of France to view him as a symbol of the tyranny of the Ancien Régime. The credibility of the king was deeply undermined, and the abolition of the monarchy, Louis XVI was the only King of France ever to be executed, and his death brought an end to more than a thousand years of continuous French monarchy. Louis-Auguste de France, who was given the title Duc de Berry at birth, was born in the Palace of Versailles. Out of seven children, he was the son of Louis, the Dauphin of France. His mother was Marie-Josèphe of Saxony, the daughter of Frederick Augustus II of Saxony, Prince-Elector of Saxony and King of Poland. A strong and healthy boy, but very shy, Louis-Auguste excelled in his studies and had a taste for Latin, history, geography, and astronomy. He enjoyed physical activities such as hunting with his grandfather, and rough-playing with his brothers, Louis-Stanislas, comte de Provence. From an early age, Louis-Auguste had been encouraged in another of his hobbies, locksmithing, upon the death of his father, who died of tuberculosis on 20 December 1765, the eleven-year-old Louis-Auguste became the new Dauphin. His mother never recovered from the loss of her husband, and died on 13 March 1767, throughout his education, Louis-Auguste received a mixture of studies particular to religion, morality, and humanities. His instructors may have also had a hand in shaping Louis-Auguste into the indecisive king that he became
Louis XVI of France
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King Louis XVI by Antoine-François Callet
Louis XVI of France
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Marie Antoinette Queen of France with her three eldest children, Marie-Thérèse, Louis-Charles and Louis-Joseph. By Marie Louise Élisabeth Vigée-Lebrun
Louis XVI of France
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Louis XVI at the age of 20
Louis XVI of France
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Louis-Charles, the dauphin of France and future Louis XVII. By Marie Louise Élisabeth Vigée-Lebrun.
33.
Pierre Charles Le Monnier
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Pierre Charles Le Monnier was a French astronomer. His name is given as Lemonnier. Le Monnier was born in Paris, where his father Pierre, in 1738, shortly after his return, he explained, in a memoir read before the Academy, the advantages of John Flamsteeds mode of determining right ascensions. His persistent recommendation of British methods and instruments contributed effectively to the reform of French practical astronomy, and constituted the most eminent of his services to science. He corresponded with James Bradley, was the first to represent the effects of nutation in the tables, and introduced, in 1741. He visited England in 1748, and, in company with the Earl of Morton and James Shore the optician, continued his journey to Scotland, where he observed the annular eclipse of 25 July. The liberality of King Louis XV of France, in whose favour Le Monnier stood high, furnished him with the means of procuring the best instruments, many made in Britain. In his lectures at the Collège de France he first publicly expounded the theory of gravitation. Le Monniers temper and hasty speech resulted in arguments and grudges. He fell out with Lalande during a revolution of the moons nodes. His career was ended by late in 1791, and a repetition of the stroke terminated his life. He died at Héril near Bayeux, by his marriage with Mademoiselle de Cussy he left three daughters, one of whom became the wife of J. L. Lagrange. Le Monnier was admitted on 5 April 1739 to the Royal Society, on 29 January 1745 he also became a member of the Prussian Academy of Sciences. The crater Le Monnier on the Moon is named after him
Pierre Charles Le Monnier
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Latin and French inscriptions at the base of the obelisk of the Gnomon of Saint-Sulpice, mentioning Pierre Charles Claude Le Monnier.
34.
Reign of Terror
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The Reign of Terror or The Terror, is the label given by some historians to a period of violence during the French Revolution. Different historians place the date at either 5 September 1793 or June 1793 or March 1793 or September 1792 or July 1789. Between June 1793 and the end of July 1794, there were 16,594 official death sentences in France, but the total number of deaths in France in 1793–96 in only the civil war in the Vendée is estimated at 250,000 counter-revolutionaries and 200,000 republicans. During 1794, revolutionary France was beset with conspiracies by internal, within France, the revolution was opposed by the French nobility, which had lost its inherited privileges. The Catholic Church opposed the revolution, which had turned the clergy into employees of the state, in addition, the French First Republic was engaged in a series of wars with neighboring powers, and parts of France were engaging in civil war against the loyalist regime. The latter were grouped in the parliamentary faction called the Mountain. Through the Revolutionary Tribunal, the Terrors leaders exercised broad powers, the Reign was a manifestation of the strong strain on centralized power. Many historians have debated the reasons the French Revolution took such a turn during the Reign of Terror of 1793–94. The public was frustrated that the equality and anti-poverty measures that the revolution originally promised were not materializing. Jacques Rouxs Manifesto of the Enraged on 25 June 1793, describes the extent to which, four years into the revolution, the foundation of the Terror is centered on the April 1793 creation of the Committee of Public Safety and its militant Jacobin delegates. Those in power believed the Committee of Public Safety was an unfortunate, according to Mathiez, they touched only with trepidation and reluctance the regime established by the Constituent Assembly so as not to interfere with the early accomplishments of the revolution. Similar to Mathiez, Richard Cobb introduced competing circumstances of revolt, counter-revolutionary rebellions taking place in Lyon, Brittany, Vendée, Nantes, and Marseille were threatening the revolution with royalist ideas. Cobb writes, the revolutionaries themselves, living as if in combat… were easily persuaded that only terror, Terror was used in these rebellions both to execute inciters and to provide a very visible example to those who might be considering rebellion. Cobb agrees with Mathiez that the Terror was simply a response to circumstances, at the same time, Cobb rejects Mathiezs Marxist interpretation that elites controlled the Reign of Terror to the significant benefit of the bourgeoisie. Instead, Cobb argues that social struggles between the classes were seldom the reason for actions and sentiments. Widespread terror and a consequent rise in executions came after external and internal threats were vastly reduced, with the backing of the national guard, they persuaded the convention to arrest 29 Girondist leaders, including Jacques Pierre Brissot. On 13 July the assassination of Jean-Paul Marat – a Jacobin leader, georges Danton, the leader of the August 1792 uprising against the king, was removed from the committee. The Jacobins identified themselves with the movement and the sans-culottes
Reign of Terror
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Nine emigrants are executed by guillotine, 1793
Reign of Terror
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Heads of aristocrats, on spikes (pikes)
Reign of Terror
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Maximilien Robespierre had others executed via his role on the Revolutionary Tribunal and the Committee of Public Safety
Reign of Terror
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A satirical engraving of Robespierre guillotining the executioner after having guillotined everyone else in France
35.
Lavoisier
–
Antoine-Laurent de Lavoisier was a French nobleman and chemist central to the 18th-century chemical revolution and had a large influence on both the history of chemistry and the history of biology. He is widely considered in popular literature as the father of modern chemistry and it is generally accepted that Lavoisiers great accomplishments in chemistry largely stem from his changing the science from a qualitative to a quantitative one. Lavoisier is most noted for his discovery of the role oxygen plays in combustion and he recognized and named oxygen and hydrogen and opposed the phlogiston theory. Lavoisier helped construct the system, wrote the first extensive list of elements. He predicted the existence of silicon and was also the first to establish that sulfur was an element rather than a compound and he discovered that, although matter may change its form or shape, its mass always remains the same. Lavoisier was a member of a number of aristocratic councils. All of these political and economic activities enabled him to fund his scientific research, at the height of the French Revolution, he was accused by Jean-Paul Marat of selling adulterated tobaccoand of other crimes, and was eventually guillotined a year after Marats death. Antoine-Laurent Lavoisier was born to a family of the nobility in Paris on 26 August 1743. The son of an attorney at the Parliament of Paris, he inherited a fortune at the age of five with the passing of his mother. Lavoisier began his schooling at the Collège des Quatre-Nations, University of Paris in Paris in 1754 at the age of 11, in his last two years at the school, his scientific interests were aroused, and he studied chemistry, botany, astronomy, and mathematics. Lavoisier entered the school of law, where he received a degree in 1763. Lavoisier received a law degree and was admitted to the bar, however, he continued his scientific education in his spare time. Lavoisiers education was filled with the ideals of the French Enlightenment of the time and he attended lectures in the natural sciences. Lavoisiers devotion and passion for chemistry were largely influenced by Étienne Condillac and his first chemical publication appeared in 1764. From 1763 to 1767, he studied geology under Jean-Étienne Guettard, in collaboration with Guettard, Lavoisier worked on a geological survey of Alsace-Lorraine in June 1767. In 1768 Lavoisier received an appointment to the Academy of Sciences. In 1769, he worked on the first geological map of France, on behalf of the Ferme générale Lavoisier commissioned the building of a wall around Paris so that customs duties could be collected from those transporting goods into and out of the city. Lavoisier attempted to introduce reforms in the French monetary and taxation system to help the peasants, Lavoisier consolidated his social and economic position when, in 1771 at age 28, he married Marie-Anne Pierrette Paulze, the 13-year-old daughter of a senior member of the Ferme générale
Lavoisier
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Line engraving by Louis Jean Desire Delaistre, after a design by Julien Leopold Boilly
Lavoisier
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Antoine-Laurent Lavoisier by Jules Dalou 1866
Lavoisier
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Portrait of Antoine-Laurent Lavoisier and his wife by Jacques-Louis David, ca. 1788
Lavoisier
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Joseph Priestley, an English chemist known for isolating oxygen, which he termed "dephlogisticated air."
36.
Order of the Reunion
–
It was established in 1811 and abolished in 1815. It was set up on 11 or 18 October 1811 by Napoleon I and it was set up as an order of merit to replace Louis Bonapartes Order of the Union. It had three ranks and Napoleon himself was its Grand Master, the knights of the order were authorised to bear their old decorations until 1 April or exchange them for ones of the new order. Within the First French Empires hierarchy of orders it was only to the Légion dhonneur. Napoleon disliked the idea of a nobility and so assigned 500,000 francs annually to provide pensions to the orders members. This great event that truly characterises the Empire, could be called the Order of the Union, Napoleon reserved himself the exclusive right to exclude someone from the order or nominate them to it - Napoleon felt his brother Louis had been too generous in giving out medals. Charles-François Lebrun, duc de Plaisance and Napoleons representative in Amsterdam as Prins-stadhouder, oversaw the order, Louis continued to wear ‘his’ Order of the Union throughout his life and old-established nobles did not receive the Order of the Reunion. The Dutch statesmen Godert van der Capellen, Anton Reinhard Falck and Vischer did not accept the Order of the Reunion, Van Capellen noted that “the oath was of such a nature to me that I forever refused it, with better opportunities to cooperate in restoring our independence. All the other Grand Crosses, Commanders and Knights of the Dutch Order of the Union thought the new Order was just under a different name, Knights of the new Order were appointed right up to the end of the First Empire in 1814. On their initial restoration in 1814 the Bourbons neither abolished nor awarded the Order of the Reunion and Napoleon awarded it during the Hundred Days. On 28 July 1815 Louis XVIII of France abolished it, asking its knights to return their gold and silver badges to the chancellory of the Legion d’Honneur. The target number of members for the order was at least 10,000 knights,2,000 commanders and 500 grand cross members, though in the end it only reached 527,90 and 64 respectively. According to a statement by Van der Goes Dirxland,11 great crosses,36 commanders’ crosses and 59 knights’ crosses were handed in, the French state replaced them, though it was usually paid for by the recipient himself, honouring the awards of the Order of the Reunion. An official statement said that by its end the order had been awarded 1,622 times,614 of these cases involved a foreigner, that is those who were not subjects of Napoleon. Since the order began as a replacement for the Order of the Union,681 recipients had previously borne the Order of the Union, the medal of the Order of the Reunion was a gold enamelled twelve-pointed star with a ball on each point. Between each point was a bundle of golden spears, at its centre was a circle surrounded in gold and blue, encircled by a gold laurel wreath and bearing a gold ‘N’ on a gold ground. On the blue circle was written ‘A JAMAIS’, the reverse is similar to the obverse but bears an empty throne instead of the imperial monogram. In front of the throne is the Capitoline Wolf suckly Romulus and Remus, on the surrounding circlet is ‘TOUT POUR L’EMPIRE’
Order of the Reunion
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Insignia of the Order
37.
Astronomy
–
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
Astronomy
–
A star -forming region in the Large Magellanic Cloud, an irregular galaxy.
Astronomy
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A giant Hubble mosaic of the Crab Nebula, a supernova remnant
Astronomy
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19th century Sydney Observatory, Australia (1873)
Astronomy
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19th century Quito Astronomical Observatory is located 12 minutes south of the Equator in Quito, Ecuador.
38.
Jupiter
–
Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a giant planet with a mass one-thousandth that of the Sun, Jupiter and Saturn are gas giants, the other two giant planets, Uranus and Neptune are ice giants. Jupiter has been known to astronomers since antiquity, the Romans named it after their god Jupiter. Jupiter is primarily composed of hydrogen with a quarter of its mass being helium and it may also have a rocky core of heavier elements, but like the other giant planets, Jupiter lacks a well-defined solid surface. Because of its rotation, the planets shape is that of an oblate spheroid. The outer atmosphere is visibly segregated into several bands at different latitudes, resulting in turbulence, a prominent result is the Great Red Spot, a giant storm that is known to have existed since at least the 17th century when it was first seen by telescope. Surrounding Jupiter is a faint planetary ring system and a powerful magnetosphere, Jupiter has at least 67 moons, including the four large Galilean moons discovered by Galileo Galilei in 1610. Ganymede, the largest of these, has a greater than that of the planet Mercury. Jupiter has been explored on several occasions by robotic spacecraft, most notably during the early Pioneer and Voyager flyby missions and later by the Galileo orbiter. In late February 2007, Jupiter was visited by the New Horizons probe, the latest probe to visit the planet is Juno, which entered into orbit around Jupiter on July 4,2016. Future targets for exploration in the Jupiter system include the probable ice-covered liquid ocean of its moon Europa, Earth and its neighbor planets may have formed from fragments of planets after collisions with Jupiter destroyed those super-Earths near the Sun. Astronomers have discovered nearly 500 planetary systems with multiple planets, Jupiter moving out of the inner Solar System would have allowed the formation of inner planets, including Earth. Jupiter is composed primarily of gaseous and liquid matter and it is the largest of the four giant planets in the Solar System and hence its largest planet. It has a diameter of 142,984 km at its equator, the average density of Jupiter,1.326 g/cm3, is the second highest of the giant planets, but lower than those of the four terrestrial planets. Jupiters upper atmosphere is about 88–92% hydrogen and 8–12% helium by percent volume of gas molecules, a helium atom has about four times as much mass as a hydrogen atom, so the composition changes when described as the proportion of mass contributed by different atoms. Thus, Jupiters atmosphere is approximately 75% hydrogen and 24% helium by mass, the atmosphere contains trace amounts of methane, water vapor, ammonia, and silicon-based compounds. There are also traces of carbon, ethane, hydrogen sulfide, neon, oxygen, phosphine, the outermost layer of the atmosphere contains crystals of frozen ammonia. The interior contains denser materials - by mass it is roughly 71% hydrogen, 24% helium, through infrared and ultraviolet measurements, trace amounts of benzene and other hydrocarbons have also been found
Jupiter
–
Jupiter in natural color, photographed by the Cassini spacecraft in 2001
Jupiter
Jupiter
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Jupiter's diameter is one order of magnitude smaller (×0.10045) than the Sun, and one order of magnitude larger (×10.9733) than the Earth. The Great Red Spot is roughly the same size as the Earth.
Jupiter
–
This view of Jupiter's Great Red Spot and its surroundings was obtained by Voyager 1 on February 25, 1979, when the spacecraft was 9.2 million km (5.7 million mi) from Jupiter. The white oval storm directly below the Great Red Spot is approximately the same diameter as Earth.
39.
Acceleration
–
Acceleration, in physics, is the rate of change of velocity of an object with respect to time. An objects acceleration is the net result of any and all forces acting on the object, the SI unit for acceleration is metre per second squared. Accelerations are vector quantities and add according to the parallelogram law, as a vector, the calculated net force is equal to the product of the objects mass and its acceleration. For example, when a car starts from a standstill and travels in a line at increasing speeds. If the car turns, there is an acceleration toward the new direction, in this example, we can call the forward acceleration of the car a linear acceleration, which passengers in the car might experience as a force pushing them back into their seats. When changing direction, we call this non-linear acceleration, which passengers might experience as a sideways force. If the speed of the car decreases, this is an acceleration in the direction from the direction of the vehicle. Passengers may experience deceleration as a force lifting them forwards, mathematically, there is no separate formula for deceleration, both are changes in velocity. Each of these accelerations might be felt by passengers until their velocity matches that of the car, an objects average acceleration over a period of time is its change in velocity divided by the duration of the period. Mathematically, a ¯ = Δ v Δ t, instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimal interval of time. The SI unit of acceleration is the metre per second squared, or metre per second per second, as the velocity in metres per second changes by the acceleration value, every second. An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, in this case it is said to be undergoing centripetal acceleration. Proper acceleration, the acceleration of a relative to a free-fall condition, is measured by an instrument called an accelerometer. As speeds approach the speed of light, relativistic effects become increasingly large and these components are called the tangential acceleration and the normal or radial acceleration. Geometrical analysis of space curves, which explains tangent, normal and binormal, is described by the Frenet–Serret formulas. Uniform or constant acceleration is a type of motion in which the velocity of an object changes by an amount in every equal time period. A frequently cited example of uniform acceleration is that of an object in free fall in a gravitational field. The acceleration of a body in the absence of resistances to motion is dependent only on the gravitational field strength g
Acceleration
–
Components of acceleration for a curved motion. The tangential component a t is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector (or in the opposite direction). The normal component (also called centripetal component for circular motion) a c is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.
Acceleration
–
Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as time interval Δt → 0 of Δ v / Δt
40.
Angular momentum
–
In physics, angular momentum is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque. The definition of momentum for a point particle is a pseudovector r×p. This definition can be applied to each point in continua like solids or fluids, unlike momentum, angular momentum does depend on where the origin is chosen, since the particles position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object via the moment of inertia I. However, while ω always points in the direction of the rotation axis, Angular momentum is additive, the total angular momentum of a system is the vector sum of the angular momenta. For continua or fields one uses integration, torque can be defined as the rate of change of angular momentum, analogous to force. Applications include the gyrocompass, control moment gyroscope, inertial systems, reaction wheels, flying discs or Frisbees. In general, conservation does limit the motion of a system. In quantum mechanics, angular momentum is an operator with quantized eigenvalues, Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the spin of elementary particles does not correspond to literal spinning motion, Angular momentum is a vector quantity that represents the product of a bodys rotational inertia and rotational velocity about a particular axis. Angular momentum can be considered an analog of linear momentum. Thus, where momentum is proportional to mass m and linear speed v, p = m v, angular momentum is proportional to moment of inertia I. Unlike mass, which only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation. Unlike linear speed, which occurs in a line, angular speed occurs about a center of rotation. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center and this simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case, L = r m v ⊥, where v ⊥ = v sin θ is the component of the motion. It is this definition, × to which the moment of momentum refers
Angular momentum
–
This gyroscope remains upright while spinning due to the conservation of its angular momentum.
Angular momentum
–
An ice skater conserves angular momentum – her rotational speed increases as her moment of inertia decreases by drawing in her arms and legs.
41.
D'Alembert's principle
–
DAlemberts principle, also known as the Lagrange–dAlembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond dAlembert and it is the dynamic analogue to the principle of virtual work for applied forces in a static system and in fact is more general than Hamiltons principle, avoiding restriction to holonomic systems. A holonomic constraint depends only on the coordinates and time and it does not depend on the velocities. The principle does not apply for irreversible displacements, such as sliding friction, DAlemberts contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish. That is to say that the generalized forces Q j need not include constraint forces and it is equivalent to the somewhat more cumbersome Gausss principle of least constraint. The general statement of dAlemberts principle mentions the time derivatives of the momenta of the system. The momentum of the mass is the product of its mass and velocity, p i = m i v i. In many applications, the masses are constant and this reduces to p i ˙ = m i v ˙ i = m i a i. However, some applications involve changing masses and in those cases both terms m ˙ i v i and m i v ˙ i have to remain present, to date, nobody has shown that DAlemberts principle is equivalent to Newtons Second Law. This is true only for very special cases e. g. rigid body constraints. However, a solution to this problem does exist. Consider Newtons law for a system of particles, i, if arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces, the constraint forces do no work. Such displacements are said to be consistent with the constraints and this leads to the formulation of dAlemberts principle, which states that the difference of applied forces and inertial forces for a dynamic system does no virtual work. There is also a principle for static systems called the principle of virtual work for applied forces. DAlembert showed that one can transform an accelerating rigid body into an equivalent static system by adding the so-called inertial force, the inertial force must act through the center of mass and the inertial torque can act anywhere. The system can then be analyzed exactly as a static system subjected to this force and moment. The advantage is that, in the equivalent static system one can take moments about any point and this often leads to simpler calculations because any force can be eliminated from the moment equations by choosing the appropriate point about which to apply the moment equation. Even in the course of Fundamentals of Dynamics and Kinematics of machines, in textbooks of engineering dynamics this is sometimes referred to as dAlemberts principle
D'Alembert's principle
–
Jean d'Alembert (1717—1783)
D'Alembert's principle
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Free body diagram of a wire pulling on a mass with weight W, showing the d’Alembert inertia “force” ma.
D'Alembert's principle
–
Free body diagram depicting an inertia moment and an inertia force on a rigid body in free fall with an angular velocity.
42.
Mass
–
In physics, mass is a property of a physical body. It is the measure of a resistance to acceleration when a net force is applied. It also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram, Mass is not the same as weight, even though mass is often determined by measuring the objects weight using a spring scale, rather than comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity and this is because weight is a force, while mass is the property that determines the strength of this force. In Newtonian physics, mass can be generalized as the amount of matter in an object, however, at very high speeds, special relativity postulates that energy is an additional source of mass. Thus, any body having mass has an equivalent amount of energy. In addition, matter is a defined term in science. There are several distinct phenomena which can be used to measure mass, active gravitational mass measures the gravitational force exerted by an object. Passive gravitational mass measures the force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force, according to Newtons second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A bodys mass also determines the degree to which it generates or is affected by a gravitational field and this is sometimes referred to as gravitational mass. The standard International System of Units unit of mass is the kilogram, the kilogram is 1000 grams, first defined in 1795 as one cubic decimeter of water at the melting point of ice. Then in 1889, the kilogram was redefined as the mass of the prototype kilogram. As of January 2013, there are proposals for redefining the kilogram yet again. In this context, the mass has units of eV/c2, the electronvolt and its multiples, such as the MeV, are commonly used in particle physics. The atomic mass unit is 1/12 of the mass of a carbon-12 atom, the atomic mass unit is convenient for expressing the masses of atoms and molecules. Outside the SI system, other units of mass include, the slug is an Imperial unit of mass, the pound is a unit of both mass and force, used mainly in the United States
Mass
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Depiction of early balance scales in the Papyrus of Hunefer (dated to the 19th dynasty, ca. 1285 BC). The scene shows Anubis weighing the heart of Hunefer.
Mass
–
The kilogram is one of the seven SI base units and one of three which is defined ad hoc (i.e. without reference to another base unit).
Mass
–
Galileo Galilei (1636)
Mass
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Distance traveled by a freely falling ball is proportional to the square of the elapsed time
43.
Power (physics)
–
In physics, power is the rate of doing work. It is the amount of energy consumed per unit time, having no direction, it is a scalar quantity. In the SI system, the unit of power is the joule per second, known as the watt in honour of James Watt, another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written, because this integral depends on the trajectory of the point of application of the force and torque, this calculation of work is said to be path dependent. As a physical concept, power requires both a change in the universe and a specified time in which the change occurs. This is distinct from the concept of work, which is measured in terms of a net change in the state of the physical universe. The output power of a motor is the product of the torque that the motor generates. The power involved in moving a vehicle is the product of the force of the wheels. The dimension of power is divided by time. The SI unit of power is the watt, which is equal to one joule per second, other units of power include ergs per second, horsepower, metric horsepower, and foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the required to lift 550 pounds by one foot in one second. Other units include dBm, a logarithmic measure with 1 milliwatt as reference, food calories per hour, Btu per hour. This shows how power is an amount of energy consumed per unit time. If ΔW is the amount of work performed during a period of time of duration Δt and it is the average amount of work done or energy converted per unit of time. The average power is simply called power when the context makes it clear. The instantaneous power is then the value of the average power as the time interval Δt approaches zero. P = lim Δ t →0 P a v g = lim Δ t →0 Δ W Δ t = d W d t. In the case of constant power P, the amount of work performed during a period of duration T is given by, W = P t
Power (physics)
–
Ansel Adams photograph of electrical wires of the Boulder Dam Power Units, 1941–1942
44.
Speed
–
In everyday use and in kinematics, the speed of an object is the magnitude of its velocity, it is thus a scalar quantity. Speed has the dimensions of distance divided by time, the SI unit of speed is the metre per second, but the most common unit of speed in everyday usage is the kilometre per hour or, in the US and the UK, miles per hour. For air and marine travel the knot is commonly used, the fastest possible speed at which energy or information can travel, according to special relativity, is the speed of light in a vacuum c =299792458 metres per second. Matter cannot quite reach the speed of light, as this would require an amount of energy. In relativity physics, the concept of rapidity replaces the classical idea of speed, italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time, in equation form, this is v = d t, where v is speed, d is distance, and t is time. A cyclist who covers 30 metres in a time of 2 seconds, objects in motion often have variations in speed. If s is the length of the path travelled until time t, in the special case where the velocity is constant, this can be simplified to v = s / t. The average speed over a time interval is the total distance travelled divided by the time duration. Speed at some instant, or assumed constant during a short period of time, is called instantaneous speed. By looking at a speedometer, one can read the speed of a car at any instant. A car travelling at 50 km/h generally goes for less than one hour at a constant speed, if the vehicle continued at that speed for half an hour, it would cover half that distance. If it continued for one minute, it would cover about 833 m. Different from instantaneous speed, average speed is defined as the distance covered divided by the time interval. For example, if a distance of 80 kilometres is driven in 1 hour, likewise, if 320 kilometres are travelled in 4 hours, the average speed is also 80 kilometres per hour. When a distance in kilometres is divided by a time in hours, average speed does not describe the speed variations that may have taken place during shorter time intervals, and so average speed is often quite different from a value of instantaneous speed. If the average speed and the time of travel are known, using this equation for an average speed of 80 kilometres per hour on a 4-hour trip, the distance covered is found to be 320 kilometres. Linear speed is the distance travelled per unit of time, while speed is the linear speed of something moving along a circular path
Speed
–
Speed can be thought of as the rate at which an object covers distance. A fast-moving object has a high speed and covers a relatively large distance in a given amount of time, while a slow-moving object covers a relatively small amount of distance in the same amount of time.
45.
Time
–
Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future. Time is often referred to as the dimension, along with the three spatial dimensions. Time has long been an important subject of study in religion, philosophy, and science, nevertheless, diverse fields such as business, industry, sports, the sciences, and the performing arts all incorporate some notion of time into their respective measuring systems. Two contrasting viewpoints on time divide prominent philosophers, one view is that time is part of the fundamental structure of the universe—a dimension independent of events, in which events occur in sequence. Isaac Newton subscribed to this realist view, and hence it is referred to as Newtonian time. This second view, in the tradition of Gottfried Leibniz and Immanuel Kant, holds that time is neither an event nor a thing, Time in physics is unambiguously operationally defined as what a clock reads. Time is one of the seven fundamental physical quantities in both the International System of Units and International System of Quantities, Time is used to define other quantities—such as velocity—so defining time in terms of such quantities would result in circularity of definition. The operational definition leaves aside the question there is something called time, apart from the counting activity just mentioned, that flows. Investigations of a single continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy. Furthermore, it may be there is a subjective component to time. Temporal measurement has occupied scientists and technologists, and was a motivation in navigation. Periodic events and periodic motion have long served as standards for units of time, examples include the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum, and the beat of a heart. Currently, the unit of time, the second, is defined by measuring the electronic transition frequency of caesium atoms. Time is also of significant social importance, having economic value as well as value, due to an awareness of the limited time in each day. In day-to-day life, the clock is consulted for periods less than a day whereas the calendar is consulted for periods longer than a day, increasingly, personal electronic devices display both calendars and clocks simultaneously. The number that marks the occurrence of an event as to hour or date is obtained by counting from a fiducial epoch—a central reference point. Artifacts from the Paleolithic suggest that the moon was used to time as early as 6,000 years ago. Lunar calendars were among the first to appear, either 12 or 13 lunar months, without intercalation to add days or months to some years, seasons quickly drift in a calendar based solely on twelve lunar months
Time
–
The flow of sand in an hourglass can be used to keep track of elapsed time. It also concretely represents the present as being between the past and the future.
Time
Time
–
Horizontal sundial in Taganrog
Time
–
A contemporary quartz watch
46.
Newton's laws of motion
–
Newtons laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a measure of the force. These three laws have been expressed in different ways, over nearly three centuries, and can be summarised as follows. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation. Newtons laws are applied to objects which are idealised as single point masses, in the sense that the size and this can be done when the object is small compared to the distances involved in its analysis, or the deformation and rotation of the body are of no importance. In this way, even a planet can be idealised as a particle for analysis of its orbital motion around a star, in their original form, Newtons laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newtons laws of motion for rigid bodies called Eulers laws of motion, if a body is represented as an assemblage of discrete particles, each governed by Newtons laws of motion, then Eulers laws can be derived from Newtons laws. Eulers laws can, however, be taken as axioms describing the laws of motion for extended bodies, Newtons laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Other authors do treat the first law as a corollary of the second, the explicit concept of an inertial frame of reference was not developed until long after Newtons death. In the given mass, acceleration, momentum, and force are assumed to be externally defined quantities. This is the most common, but not the interpretation of the way one can consider the laws to be a definition of these quantities. Newtonian mechanics has been superseded by special relativity, but it is useful as an approximation when the speeds involved are much slower than the speed of light. The first law states that if the net force is zero, the first law can be stated mathematically when the mass is a non-zero constant, as, ∑ F =0 ⇔ d v d t =0. Consequently, An object that is at rest will stay at rest unless a force acts upon it, an object that is in motion will not change its velocity unless a force acts upon it. This is known as uniform motion, an object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest, if an object is moving, it continues to move without turning or changing its speed
Newton's laws of motion
–
Newton's First and Second laws, in Latin, from the original 1687 Principia Mathematica.
Newton's laws of motion
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Isaac Newton (1643–1727), the physicist who formulated the laws
47.
Damping ratio
–
In engineering, the damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium, a mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system is trying to return to its equilibrium position, sometimes losses damp the system and can cause the oscillations to gradually decay in amplitude towards zero or attenuate. The damping ratio is a measure of describing how rapidly the oscillations decay from one bounce to the next, where the spring–mass system is completely lossless, the mass would oscillate indefinitely, with each bounce of equal height to the last. This hypothetical case is called undamped, If the system contained high losses, for example if the spring–mass experiment were conducted in a viscous fluid, the mass could slowly return to its rest position without ever overshooting. Commonly, the mass tends to overshoot its starting position, and then return, with each overshoot, some energy in the system is dissipated, and the oscillations die towards zero. Between the overdamped and underdamped cases, there exists a level of damping at which the system will just fail to overshoot. This case is called critical damping, the key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time. The damping ratio is a parameter, usually denoted by ζ and it is particularly important in the study of control theory. It is also important in the harmonic oscillator, the damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. This equation can be solved with the approach, X = C e s t, where C and s are both complex constants. That approach assumes a solution that is oscillatory and/or decaying exponentially, using it in the ODE gives a condition on the frequency of the damped oscillations, s = − ω n. Undamped, Is the case where ζ →0 corresponds to the simple harmonic oscillator. Underdamped, If s is a number, then the solution is a decaying exponential combined with an oscillatory portion that looks like exp . This case occurs for ζ <1, and is referred to as underdamped, overdamped, If s is a real number, then the solution is simply a decaying exponential with no oscillation. This case occurs for ζ >1, and is referred to as overdamped, critically damped, The case where ζ =1 is the border between the overdamped and underdamped cases, and is referred to as critically damped. This turns out to be an outcome in many cases where engineering design of a damped oscillator is required. The factors Q, damping ratio ζ, and exponential decay rate α are related such that ζ =12 Q = α ω0, a lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times
Damping ratio
–
The effect of varying damping ratio on a second-order system.
48.
Displacement (vector)
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A displacement is a vector that is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a line from the initial position to the final position of the point. The velocity then is distinct from the speed which is the time rate of change of the distance traveled along a specific path. The velocity may be defined as the time rate of change of the position vector. For motion over an interval of time, the displacement divided by the length of the time interval defines the average velocity. In dealing with the motion of a body, the term displacement may also include the rotations of the body. In this case, the displacement of a particle of the body is called linear displacement, for a position vector s that is a function of time t, the derivatives can be computed with respect to t. These derivatives have common utility in the study of kinematics, control theory, vibration sensing and other sciences, by extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the displacement function. Such higher-order terms are required in order to represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering. The fourth order derivative is called jounce
Displacement (vector)
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Displacement versus distance traveled along a path
49.
Equations of motion
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In mathematical physics, equations of motion are equations that describe the behaviour of a physical system in terms of its motion as a function of time. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system, the functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions to the equations describing the motion of the dynamics. There are two descriptions of motion, dynamics and kinematics. Dynamics is general, since momenta, forces and energy of the particles are taken into account, in this instance, sometimes the term refers to the differential equations that the system satisfies, and sometimes to the solutions to those equations. However, kinematics is simpler as it concerns only variables derived from the positions of objects, equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the types of motion are translations, rotations, oscillations. A differential equation of motion, usually identified as some physical law, solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the initial values, to state this formally, in general an equation of motion M is a function of the position r of the object, its velocity, and its acceleration, and time t. Euclidean vectors in 3D are denoted throughout in bold and this is equivalent to saying an equation of motion in r is a second order ordinary differential equation in r, M =0, where t is time, and each overdot denotes one time derivative. The initial conditions are given by the constant values at t =0, r, r ˙, the solution r to the equation of motion, with specified initial values, describes the system for all times t after t =0. Sometimes, the equation will be linear and is likely to be exactly solvable. In general, the equation will be non-linear, and cannot be solved exactly so a variety of approximations must be used, the solutions to nonlinear equations may show chaotic behavior depending on how sensitive the system is to the initial conditions. Despite the great strides made in the development of geometry made by Ancient Greeks and surveys in Rome, the exposure of Europe to Arabic numerals and their ease in computations encouraged first the scholars to learn them and then the merchants and invigorated the spread of knowledge throughout Europe. These studies led to a new body of knowledge that is now known as physics, thomas Bradwardine, one of those scholars, extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested a law involving force, resistance, distance, velocity. Nicholas Oresme further extended Bradwardines arguments, for writers on kinematics before Galileo, since small time intervals could not be measured, the affinity between time and motion was obscure. They used time as a function of distance, and in free fall, de Sotos comments are shockingly correct regarding the definitions of acceleration and the observation that during the violent motion of ascent acceleration would be negative
Equations of motion
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Kinematic quantities of a classical particle of mass m: position r, velocity v, acceleration a.
50.
Friction
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Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction, Dry friction resists relative lateral motion of two surfaces in contact. Dry friction is subdivided into static friction between non-moving surfaces, and kinetic friction between moving surfaces, fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other. Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces, skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body. Internal friction is the force resisting motion between the making up a solid material while it undergoes deformation. When surfaces in contact move relative to other, the friction between the two surfaces converts kinetic energy into thermal energy. This property can have consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to thermal energy whenever motion with friction occurs, another important consequence of many types of friction can be wear, which may lead to performance degradation and/or damage to components. Friction is a component of the science of tribology, Friction is not itself a fundamental force. Dry friction arises from a combination of adhesion, surface roughness, surface deformation. The complexity of interactions makes the calculation of friction from first principles impractical and necessitates the use of empirical methods for analysis. Friction is a non-conservative force - work done against friction is path dependent, in the presence of friction, some energy is always lost in the form of heat. Thus mechanical energy is not conserved, the Greeks, including Aristotle, Vitruvius, and Pliny the Elder, were interested in the cause and mitigation of friction. They were aware of differences between static and kinetic friction with Themistius stating in 350 A. D. that it is easier to further the motion of a moving body than to move a body at rest. The classic laws of sliding friction were discovered by Leonardo da Vinci in 1493, a pioneer in tribology and these laws were rediscovered by Guillaume Amontons in 1699. Amontons presented the nature of friction in terms of surface irregularities, the understanding of friction was further developed by Charles-Augustin de Coulomb. Coulomb further considered the influence of sliding velocity, temperature and humidity, the distinction between static and dynamic friction is made in Coulombs friction law, although this distinction was already drawn by Johann Andreas von Segner in 1758. Leslie was equally skeptical about the role of adhesion proposed by Desaguliers, in Leslies view, friction should be seen as a time-dependent process of flattening, pressing down asperities, which creates new obstacles in what were cavities before
Friction
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When the mass is not moving, the object experiences static friction. The friction increases as the applied force increases until the block moves. After the block moves, it experiences kinetic friction, which is less than the maximum static friction.
51.
Harmonic oscillator
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If a frictional force proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the coefficient, the system can, Oscillate with a frequency lower than in the non-damped case. Decay to the position, without oscillations. The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a value of the friction coefficient and is called critically damped. If an external time dependent force is present, the oscillator is described as a driven oscillator. Mechanical examples include pendulums, masses connected to springs, and acoustical systems, other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many devices, such as clocks. They are the source of virtually all sinusoidal vibrations and waves, a simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a m, which experiences a single force, F, which pulls the mass in the direction of the point x=0 and depends only on the masss position x. Balance of forces for the system is F = m a = m d 2 x d t 2 = m x ¨ = − k x. Solving this differential equation, we find that the motion is described by the function x = A cos , the motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude, A. The position at a time t also depends on the phase, φ. The period and frequency are determined by the size of the mass m, the velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the direction as the displacement. The potential energy stored in a harmonic oscillator at position x is U =12 k x 2. In real oscillators, friction, or damping, slows the motion of the system, due to frictional force, the velocity decreases in proportion to the acting frictional force. While simple harmonic motion oscillates with only the force acting on the system
Harmonic oscillator
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Another damped harmonic oscillator
Harmonic oscillator
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Dependence of the system behavior on the value of the damping ratio ζ
52.
Mechanics of planar particle motion
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This article describes a particle in planar motion when observed from non-inertial reference frames. The most famous examples of motion are related to the motion of two spheres that are gravitationally attracted to one another, and the generalization of this problem to planetary motion. See centrifugal force, two-body problem, orbit and Keplers laws of planetary motion and those problems fall in the general field of analytical dynamics, the determination of orbits from given laws of force. This article is focused more on the issues surrounding planar motion. The Lagrangian approach to fictitious forces is introduced, unlike real forces such as electromagnetic forces, fictitious forces do not originate from physical interactions between objects. The appearance of fictitious forces normally is associated with use of a frame of reference. For solving problems of mechanics in non-inertial reference frames, the advice given in textbooks is to treat the fictitious forces like real forces, elaboration of this point and some citations on the subject follow. Examples are Cartesian coordinates, polar coordinates and curvilinear coordinates, or as seen from a rotating frame. A time-dependent description of observations does not change the frame of reference in which the observations are made, in discussion of a particle moving in a circular orbit, in an inertial frame of reference one can identify the centripetal and tangential forces. It then seems to be no problem to switch hats, change perspective and that switch is unconscious, but real. Suppose we sit on a particle in planar motion. What analysis underlies a switch of hats to introduce fictitious centrifugal, to explore that question, begin in an inertial frame of reference. In Figure 1, the arc length s is the distance the particle has traveled along its path in time t, the path r with components x, y in Cartesian coordinates is described using arc length s as, r =. One way to look at the use of s is to think of the path of the particle as sitting in space, like the left by a skywriter. Any position on this path is described by stating its distance s from some starting point on the path, then an incremental displacement along the path ds is described by, d r = = d s, where primes are introduced to denote derivatives with respect to s. The magnitude of this displacement is ds, showing that, =1, the unit magnitude of these vectors is a consequence of Eq.1. As an aside, notice that the use of vectors that are not aligned along the Cartesian xy-axes does not mean we are no longer in an inertial frame. All it means is that we are using unit vectors that vary with s to describe the path, the radius of curvature is introduced completely formally as,1 ρ = d θ d s
Mechanics of planar particle motion
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The arc length s(t) measures distance along the skywriter's trail. Image from NASA ASRS
Mechanics of planar particle motion
Mechanics of planar particle motion
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Figure 2: Two coordinate systems differing by a displacement of origin. Radial motion with constant velocity v in one frame is not radial in the other frame. Angular rate, but
53.
Motion (physics)
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In physics, motion is a change in position of an object over time. Motion is described in terms of displacement, distance, velocity, acceleration, time, motion of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame. If the position of a body is not changing with respect to a frame of reference. An objects motion cannot change unless it is acted upon by a force, momentum is a quantity which is used for measuring motion of an object. As there is no frame of reference, absolute motion cannot be determined. Thus, everything in the universe can be considered to be moving, more generally, motion is a concept that applies to objects, bodies, and matter particles, to radiation, radiation fields and radiation particles, and to space, its curvature and space-time. One can also speak of motion of shapes and boundaries, so, the term motion in general signifies a continuous change in the configuration of a physical system. For example, one can talk about motion of a wave or about motion of a quantum particle, in physics, motion is described through two sets of apparently contradictory laws of mechanics. Motions of all large scale and familiar objects in the universe are described by classical mechanics, whereas the motion of very small atomic and sub-atomic objects is described by quantum mechanics. It produces very accurate results within these domains, and is one of the oldest and largest in science, engineering, classical mechanics is fundamentally based on Newtons laws of motion. These laws describe the relationship between the acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica and his three laws are, A body either is at rest or moves with constant velocity, until and unless an outer force is applied to it. An object will travel in one direction only until an outer force changes its direction, whenever one body exerts a force F onto a second body, the second body exerts the force −F on the first body. F and −F are equal in magnitude and opposite in sense, so, the body which exerts F will go backwards. Newtons three laws of motion, along with his Newtons law of motion, which were the first to provide a mathematical model for understanding orbiting bodies in outer space. This explanation unified the motion of bodies and motion of objects on earth. Classical mechanics was later enhanced by Albert Einsteins special relativity. Motion of objects with a velocity, approaching the speed of light
Motion (physics)
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Motion involves a change in position, such as in this perspective of rapidly leaving Yongsan Station.
54.
Newton's law of universal gravitation
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This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning. It is a part of classical mechanics and was formulated in Newtons work Philosophiæ Naturalis Principia Mathematica, in modern language, the law states, Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them, the first test of Newtons theory of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798. It took place 111 years after the publication of Newtons Principia, Newtons law of gravitation resembles Coulombs law of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Both are inverse-square laws, where force is proportional to the square of the distance between the bodies. Coulombs law has the product of two charges in place of the product of the masses, and the constant in place of the gravitational constant. Newtons law has since been superseded by Albert Einsteins theory of general relativity, at the same time Hooke agreed that the Demonstration of the Curves generated thereby was wholly Newtons. In this way, the question arose as to what, if anything and this is a subject extensively discussed since that time and on which some points, outlined below, continue to excite controversy. And that these powers are so much the more powerful in operating. Thus Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, Hookes statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hookes gravitation was also not yet universal, though it approached universality more closely than previous hypotheses and he also did not provide accompanying evidence or mathematical demonstration. It was later on, in writing on 6 January 1679|80 to Newton, Newton, faced in May 1686 with Hookes claim on the inverse square law, denied that Hooke was to be credited as author of the idea. Among the reasons, Newton recalled that the idea had been discussed with Sir Christopher Wren previous to Hookes 1679 letter, Newton also pointed out and acknowledged prior work of others, including Bullialdus, and Borelli. D T Whiteside has described the contribution to Newtons thinking that came from Borellis book, a copy of which was in Newtons library at his death. Newton further defended his work by saying that had he first heard of the inverse square proportion from Hooke, Hooke, without evidence in favor of the supposition, could only guess that the inverse square law was approximately valid at great distances from the center. Thus Newton gave a justification, otherwise lacking, for applying the inverse square law to large spherical planetary masses as if they were tiny particles, after his 1679-1680 correspondence with Hooke, Newton adopted the language of inward or centripetal force. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s, the lesson offered by Hooke to Newton here, although significant, was one of perspective and did not change the analysis. This background shows there was basis for Newton to deny deriving the inverse square law from Hooke, on the other hand, Newton did accept and acknowledge, in all editions of the Principia, that Hooke had separately appreciated the inverse square law in the solar system
Newton's law of universal gravitation
55.
Vibration
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Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem, the oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road. Vibration can be desirable, for example, the motion of a fork, the reed in a woodwind instrument or harmonica. In many cases, however, vibration is undesirable, wasting energy, for example, the vibrational motions of engines, electric motors, or any mechanical device in operation are typically unwanted. Such vibrations could be caused by imbalances in the parts, uneven friction. Careful designs usually minimize unwanted vibrations, the studies of sound and vibration are closely related. Sound, or pressure waves, are generated by vibrating structures, hence, attempts to reduce noise are often related to issues of vibration. Free vibration occurs when a system is set in motion with an initial input. Examples of this type of vibration are pulling a child back on a swing and letting go, or hitting a tuning fork, the mechanical system vibrates at one or more of its natural frequencies and damps down to motionlessness. Forced vibration is when a disturbance is applied to a mechanical system. The disturbance can be a periodic and steady-state input, a transient input, the periodic input can be a harmonic or a non-harmonic disturbance. Damped vibration, When the energy of a system is gradually dissipated by friction and other resistances. The vibrations gradually reduce or change in frequency or intensity or cease, Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. Alternately, a DUT is attached to the table of a shaker, Vibration testing is performed to examine the response of a device under test to a defined vibration environment. The measured response may be life, resonant frequencies or squeak. Squeak and rattle testing is performed with a type of quiet shaker that produces very low sound levels while under operation. For relatively low frequency forcing, servohydraulic shakers are used, for higher frequencies, electrodynamic shakers are used. Generally, one or more input or control points located on the DUT-side of a fixture is kept at a specified acceleration, other response points experience maximum vibration level or minimum vibration level
Vibration
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Car Suspension: designing vibration control is undertaken as part of acoustic, automotive or mechanical engineering.
Vibration
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One of the possible modes of vibration of a circular drum (see other modes).
56.
Circular motion
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In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, the rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body, since the objects velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, in physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times, though the bodys speed is constant, its velocity is not constant, velocity, a vector quantity, depends on both the bodys speed and its direction of travel. This changing velocity indicates the presence of an acceleration, this acceleration is of constant magnitude. This acceleration is, in turn, produced by a force which is also constant in magnitude. For motion in a circle of radius r, the circumference of the circle is C = 2π r, the axis of rotation is shown as a vector ω perpendicular to the plane of the orbit and with a magnitude ω = dθ / dt. The direction of ω is chosen using the right-hand rule, likewise, the acceleration is given by a = ω × v = ω ×, which is a vector perpendicular to both ω and v of magnitude ω |v| = ω2 r and directed exactly opposite to r. In the simplest case the speed, mass and radius are constant, consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second. The speed is one metre per second, the inward acceleration is one metre per square second, v2/r. It is subject to a force of one kilogram metre per square second. The momentum of the body is one kg·m·s−1, the moment of inertia is one kg·m2. The angular momentum is one kg·m2·s−1, the kinetic energy is 1/2 joule. The circumference of the orbit is 2π metres, the period of the motion is 2π seconds per turn. It is convenient to introduce the unit vector orthogonal to u ^ R as well and it is customary to orient u ^ θ to point in the direction of travel along the orbit. The velocity is the derivative of the displacement, v → = d d t r → = d R d t u ^ R + R d u ^ R d t. Because the radius of the circle is constant, the component of the velocity is zero
Circular motion
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Figure 1: Velocity v and acceleration a in uniform circular motion at angular rate ω; the speed is constant, but the velocity is always tangent to the orbit; the acceleration has constant magnitude, but always points toward the center of rotation
57.
Centripetal force
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A centripetal force is a force that makes a body follow a curved path. Its direction is orthogonal to the motion of the body. Isaac Newton described it as a force by which bodies are drawn or impelled, or in any way tend, in Newtonian mechanics, gravity provides the centripetal force responsible for astronomical orbits. One common example involving centripetal force is the case in which a body moves with uniform speed along a circular path, the centripetal force is directed at right angles to the motion and also along the radius towards the centre of the circular path. The mathematical description was derived in 1659 by the Dutch physicist Christiaan Huygens, the direction of the force is toward the center of the circle in which the object is moving, or the osculating circle. The speed in the formula is squared, so twice the speed needs four times the force, the inverse relationship with the radius of curvature shows that half the radial distance requires twice the force. Expressed using the orbital period T for one revolution of the circle, the rope example is an example involving a pull force. The centripetal force can also be supplied as a push force, newtons idea of a centripetal force corresponds to what is nowadays referred to as a central force. Another example of centripetal force arises in the helix that is traced out when a particle moves in a uniform magnetic field in the absence of other external forces. In this case, the force is the centripetal force that acts towards the helix axis. Below are three examples of increasing complexity, with derivations of the formulas governing velocity and acceleration, uniform circular motion refers to the case of constant rate of rotation. Here are two approaches to describing this case, assume uniform circular motion, which requires three things. The object moves only on a circle, the radius of the circle r does not change in time. The object moves with constant angular velocity ω around the circle, therefore, θ = ω t where t is time. Now find the velocity v and acceleration a of the motion by taking derivatives of position with respect to time, consequently, a = − ω2 r. negative shows that the acceleration is pointed towards the center of the circle, hence it is called centripetal. While objects naturally follow a path, this centripetal acceleration describes the circular motion path caused by a centripetal force. The image at right shows the relationships for uniform circular motion. In this subsection, dθ/dt is assumed constant, independent of time, consequently, d r d t = lim Δ t →0 r − r Δ t = d ℓ d t
Centripetal force
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A body experiencing uniform circular motion requires a centripetal force, towards the axis as shown, to maintain its circular path.
58.
Centrifugal force
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In Newtonian mechanics, the centrifugal force is an inertial force directed away from the axis of rotation that appears to act on all objects when viewed in a rotating reference frame. When they are analyzed in a coordinate system. The term has also been used for the force that is a reaction to a centripetal force. The centrifugal force is an outward force apparent in a reference frame. All measurements of position and velocity must be relative to some frame of reference. An inertial frame of reference is one that is not accelerating, the use of an inertial frame of reference, which will be the case for all elementary calculations, is often not explicitly stated but may generally be assumed unless stated otherwise. In terms of a frame of reference, the centrifugal force does not exist. All calculations can be performed using only Newtons laws of motion, in its current usage the term centrifugal force has no meaning in an inertial frame. In an inertial frame, an object that has no acting on it travels in a straight line. When measurements are made with respect to a reference frame, however. If it is desired to apply Newtons laws in the frame, it is necessary to introduce new, fictitious. Consider a stone being whirled round on a string, in a horizontal plane, the only real force acting on the stone in the horizontal plane is the tension in the string. There are no forces acting on the stone so there is a net force on the stone in the horizontal plane. In an inertial frame of reference, were it not for this net force acting on the stone, in order to keep the stone moving in a circular path, this force, known as the centripetal force, must be continuously applied to the stone. As soon as it is removed the stone moves in a straight line, in this inertial frame, the concept of centrifugal force is not required as all motion can be properly described using only real forces and Newtons laws of motion. In a frame of reference rotating with the stone around the axis as the stone. However, the tension in the string is still acting on the stone, if Newtons laws were applied in their usual form, the stone would accelerate in the direction of the net applied force, towards the axis of rotation, which it does not do. With this new the net force on the stone is zero, with the addition of this extra inertial or fictitious force Newtons laws can be applied in the rotating frame as if it were an inertial frame
Centrifugal force
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The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.
59.
Reactive centrifugal force
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In classical mechanics, a reactive centrifugal force forms part of an action–reaction pair with a centripetal force. In accordance with Newtons first law of motion, an object moves in a line in the absence of any external forces acting on the object. A curved path may however ensue when a physical acts on it, the two forces will only have the same magnitude in the special cases where circular motion arises and where the axis of rotation is the origin of the rotating frame of reference. It is the force that is the subject of this article. Any force directed away from a center can be called centrifugal, centrifugal simply means directed outward from the center. Similarly, centripetal means directed toward the center, the reactive centrifugal force discussed in this article is not the same thing as the centrifugal pseudoforce, which is usually whats meant by the term centrifugal force. The figure at right shows a ball in circular motion held to its path by a massless string tied to an immovable post. The figure is an example of a real force. In this system a centripetal force upon the ball provided by the string maintains the motion. In this model, the string is assumed massless and the rotational motion frictionless, the string transmits the reactive centrifugal force from the ball to the fixed post, pulling upon the post. Again according to Newtons third law, the post exerts a reaction upon the string, labeled the post reaction, the two forces upon the string are equal and opposite, exerting no net force upon the string, but placing the string under tension. It should be noted, however, that the reason the post appears to be immovable is because it is fixed to the earth. If the rotating ball was tethered to the mast of a boat, for example, even though the reactive centrifugal is rarely used in analyses in the physics literature, the concept is applied within some mechanical engineering concepts. An example of this kind of engineering concept is an analysis of the stresses within a rapidly rotating turbine blade, the blade can be treated as a stack of layers going from the axis out to the edge of the blade. Each layer exerts a force on the immediately adjacent, radially inward layer. At the same time the inner layer exerts a centripetal force on the middle layer, while and the outer layer exerts an elastic centrifugal force. It is the stresses in the blade and their causes that mainly interest mechanical engineers in this situation, another example of a rotating device in which a reactive centrifugal force can be identified used to describe the system behavior is the centrifugal clutch. A centrifugal clutch is used in small engine-powered devices such as saws, go-karts
Reactive centrifugal force
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A two-shoe centrifugal clutch. The motor spins the input shaft that makes the shoes go around, and the outer drum (removed) turns the output power shaft.
60.
Coriolis force
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In physics, the Coriolis force is an inertial force that acts on objects that are in motion relative to a rotating reference frame. In a reference frame with clockwise rotation, the acts to the left of the motion of the object. In one with anticlockwise rotation, the acts to the right. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology, deflection of an object due to the Coriolis force is called the Coriolis effect. Newtons laws of motion describe the motion of an object in a frame of reference. When Newtons laws are transformed to a frame of reference. Both forces are proportional to the mass of the object, the Coriolis force is proportional to the rotation rate and the centrifugal force is proportional to its square. The Coriolis force acts in a perpendicular to the rotation axis. The centrifugal force acts outwards in the direction and is proportional to the distance of the body from the axis of the rotating frame. These additional forces are termed inertial forces, fictitious forces or pseudo forces and they allow the application of Newtons laws to a rotating system. They are correction factors that do not exist in a non-accelerating or inertial reference frame, a commonly encountered rotating reference frame is the Earth. The Coriolis effect is caused by the rotation of the Earth, such motions are constrained by the surface of the Earth, so only the horizontal component of the Coriolis force is generally important. This force causes moving objects on the surface of the Earth to be deflected to the right in the Northern Hemisphere, the horizontal deflection effect is greater near the poles, since the effective rotation rate about a local vertical axis is largest there, and smallest at the equator. This effect is responsible for the rotation of large cyclones, riccioli, Grimaldi, and Dechales all described the effect as part of an argument against the heliocentric system of Copernicus. In other words, they argued that the Earths rotation should create the effect, the effect was described in the tidal equations of Pierre-Simon Laplace in 1778. Gaspard-Gustave Coriolis published a paper in 1835 on the yield of machines with rotating parts. That paper considered the forces that are detected in a rotating frame of reference. Coriolis divided these forces into two categories
Coriolis force
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This low-pressure system over Iceland spins counter-clockwise due to balance between the Coriolis force and the pressure gradient force.
Coriolis force
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Coordinate system at latitude φ with x -axis east, y -axis north and z -axis upward (that is, radially outward from center of sphere).
Coriolis force
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Cloud formations in a famous image of Earth from Apollo 17, makes similar circulation directly visible
Coriolis force
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A carousel is rotating counter-clockwise. Left panel: a ball is tossed by a thrower at 12:00 o'clock and travels in a straight line to the center of the carousel. While it travels, the thrower circles in a counter-clockwise direction. Right panel: The ball's motion as seen by the thrower, who now remains at 12:00 o'clock, because there is no rotation from their viewpoint.
61.
Galileo Galilei
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Galileo Galilei was an Italian polymath, astronomer, physicist, engineer, philosopher, and mathematician. He played a role in the scientific revolution of the seventeenth century. Galileo also worked in applied science and technology, inventing an improved military compass, Galileos championing of heliocentrism and Copernicanism was controversial during his lifetime, when most subscribed to either geocentrism or the Tychonic system. He met with opposition from astronomers, who doubted heliocentrism because of the absence of a stellar parallax. He was tried by the Inquisition, found vehemently suspect of heresy and he spent the rest of his life under house arrest. He has been called the father of observational astronomy, the father of modern physics, the father of scientific method, and the father of science. Galileo was born in Pisa, Italy, on 15 February 1564, the first of six children of Vincenzo Galilei, a famous lutenist, composer, and music theorist, and Giulia, three of Galileos five siblings survived infancy. The youngest, Michelangelo, also became a noted lutenist and composer although he contributed to financial burdens during Galileos young adulthood, Michelangelo was unable to contribute his fair share of their fathers promised dowries to their brothers-in-law, who would later attempt to seek legal remedies for payments due. Michelangelo would also occasionally have to borrow funds from Galileo to support his musical endeavours and these financial burdens may have contributed to Galileos early fire to develop inventions that would bring him additional income. When Galileo Galilei was eight, his family moved to Florence and he then was educated in the Vallombrosa Abbey, about 30 km southeast of Florence. Galileo Bonaiuti was buried in the church, the Basilica of Santa Croce in Florence. It was common for mid-sixteenth century Tuscan families to name the eldest son after the parents surname, hence, Galileo Galilei was not necessarily named after his ancestor Galileo Bonaiuti. The Italian male given name Galileo derives from the Latin Galilaeus, meaning of Galilee, the biblical roots of Galileos name and surname were to become the subject of a famous pun. In 1614, during the Galileo affair, one of Galileos opponents, in it he made a point of quoting Acts 1,11, Ye men of Galilee, why stand ye gazing up into heaven. Despite being a genuinely pious Roman Catholic, Galileo fathered three children out of wedlock with Marina Gamba and they had two daughters, Virginia and Livia, and a son, Vincenzo. Their only worthy alternative was the religious life, both girls were accepted by the convent of San Matteo in Arcetri and remained there for the rest of their lives. Virginia took the name Maria Celeste upon entering the convent and she died on 2 April 1634, and is buried with Galileo at the Basilica of Santa Croce, Florence. Livia took the name Sister Arcangela and was ill for most of her life, Vincenzo was later legitimised as the legal heir of Galileo and married Sestilia Bocchineri
Galileo Galilei
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Portrait of Galileo Galilei by Giusto Sustermans
Galileo Galilei
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Galileo's beloved elder daughter, Virginia (Sister Maria Celeste), was particularly devoted to her father. She is buried with him in his tomb in the Basilica of Santa Croce, Florence.
Galileo Galilei
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Galileo Galilei. Portrait by Leoni
Galileo Galilei
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Cristiano Banti 's 1857 painting Galileo facing the Roman Inquisition
62.
Johannes Kepler
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Johannes Kepler was a German mathematician, astronomer, and astrologer. A key figure in the 17th-century scientific revolution, he is best known for his laws of motion, based on his works Astronomia nova, Harmonices Mundi. These works also provided one of the foundations for Isaac Newtons theory of universal gravitation, Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague and he was also a mathematics teacher in Linz, and an adviser to General Wallenstein. Kepler lived in an era when there was no distinction between astronomy and astrology, but there was a strong division between astronomy and physics. Kepler was born on December 27, the feast day of St John the Evangelist,1571 and his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline and his father, Heinrich Kepler, earned a precarious living as a mercenary, and he left the family when Johannes was five years old. He was believed to have died in the Eighty Years War in the Netherlands and his mother Katharina Guldenmann, an innkeepers daughter, was a healer and herbalist. Born prematurely, Johannes claimed to have weak and sickly as a child. Nevertheless, he often impressed travelers at his grandfathers inn with his phenomenal mathematical faculty and he was introduced to astronomy at an early age, and developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, in 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being called outdoors to see it and that the moon appeared quite red. However, childhood smallpox left him with vision and crippled hands. In 1589, after moving through grammar school, Latin school, there, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who also taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590. He proved himself to be a mathematician and earned a reputation as a skilful astrologer. Under the instruction of Michael Maestlin, Tübingens professor of mathematics from 1583 to 1631 and he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and he accepted the position in April 1594, at the age of 23. Keplers first major work, Mysterium Cosmographicum, was the first published defense of the Copernican system
Johannes Kepler
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A 1610 portrait of Johannes Kepler by an unknown artist
Johannes Kepler
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Birthplace of Johannes Kepler in Weil der Stadt
Johannes Kepler
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Portraits of Kepler and his wife in oval medallions
Johannes Kepler
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House of Johannes Kepler and Barbara Müller in Gössendorf near Graz (1597–1599)
63.
Alexis Clairaut
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Alexis Claude Clairaut was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles, Clairaut was one of the key figures in the expedition to Lapland that helped to confirm Newtons theory for the figure of the Earth. In that context, Clairaut worked out a mathematical result now known as Clairauts theorem and he also tackled the gravitational three-body problem, being the first to obtain a satisfactory result for the apsidal precession of the Moons orbit. In mathematics he is credited with Clairauts equation and Clairauts relation. Clairaut was born in Paris, France, to Jean-Babtiste and Catherine Petit Clairaut, the couple had 20 children, however only a few of them survived childbirth. Alexis was a prodigy — at the age of ten he began studying calculus, Clairaut was unmarried, and known for leading an active social life. Though he led a social life, he was very prominent in the advancement of learning in young mathematicians. He was elected a Fellow of the Royal Society of London in November,1737, Clairaut died in Paris in 1765. In 1736, together with Pierre Louis Maupertuis, he took part in the expedition to Lapland, the goal of the excursion was to geometrically calculate the shape of the Earth, which Sir Issac Newton theorized in his book Principia was an ellipsoid shape. They sought to prove if Newtons theory and calculations were correct or not, before the expedition team returned to Paris, Clairaut sent his calculations to the Royal Society of London. The writing was published by the society in the 1736-37 volume of Philosophical Transactions. Initially, Clairaut disagrees with Newtons theory on the shape of the Earth, in the article, he outlines several key problems that effectively disprove Newtons calculations, and provides some solutions to the complications. The issues addressed include calculating gravitational attraction, the rotation of an ellipsoid on its axis, and this conclusion suggests not only that the Earth is of an oblate ellipsoid shape, but it is flattened more at the poles and is wider at the center. His article in Philosophical Transactions created much controversy, as he addressed the problems of Newtons theory, after his return, he published his treatise Théorie de la figure de la terre. This proved Sir Issac Newtons theory that the shape of the Earth was an oblate ellipsoid, in 1849 Stokes showed that Clairauts result was true whatever the interior constitution or density of the Earth, provided the surface was a spheroid of equilibrium of small ellipticity. In 1741, Alexis Clairaut wrote a book called Èléments de Géométrie, the book outlines the basic concepts of geometry. Geometry in the 1700s was complex to the average learner and it was considered to be a dry subject. Clairaut saw this trend, and wrote the book in an attempt to make the more interesting for the average learner
Alexis Clairaut
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Alexis Claude Clairaut
64.
Pierre-Simon Laplace
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Pierre-Simon, marquis de Laplace was an influential French scholar whose work was important to the development of mathematics, statistics, physics and astronomy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste and this work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace, Laplace formulated Laplaces equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is named after him. Laplace is remembered as one of the greatest scientists of all time, sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries. Laplace became a count of the Empire in 1806 and was named a marquis in 1817, Laplace was born in Beaumont-en-Auge, Normandy on 23 March 1749, a village four miles west of Pont lEveque in Normandy. According to W. W. Rouse Ball, His father, Pierre de Laplace and his great-uncle, Maitre Oliver de Laplace, had held the title of Chirurgien Royal. It would seem that from a pupil he became an usher in the school at Beaumont, however, Karl Pearson is scathing about the inaccuracies in Rouse Balls account and states, Indeed Caen was probably in Laplaces day the most intellectually active of all the towns of Normandy. It was here that Laplace was educated and was provisionally a professor and it was here he wrote his first paper published in the Mélanges of the Royal Society of Turin, Tome iv. 1766–1769, at least two years before he went at 22 or 23 to Paris in 1771, thus before he was 20 he was in touch with Lagrange in Turin. He did not go to Paris a raw self-taught country lad with only a peasant background, the École Militaire of Beaumont did not replace the old school until 1776. His parents were from comfortable families and his father was Pierre Laplace, and his mother was Marie-Anne Sochon. The Laplace family was involved in agriculture until at least 1750, Pierre Simon Laplace attended a school in the village run at a Benedictine priory, his father intending that he be ordained in the Roman Catholic Church. At sixteen, to further his fathers intention, he was sent to the University of Caen to read theology, at the university, he was mentored by two enthusiastic teachers of mathematics, Christophe Gadbled and Pierre Le Canu, who awoke his zeal for the subject. Here Laplaces brilliance as a mathematician was recognised and while still at Caen he wrote a memoir Sur le Calcul integral aux differences infiniment petites et aux differences finies. About this time, recognizing that he had no vocation for the priesthood, in this connection reference may perhaps be made to the statement, which has appeared in some notices of him, that he broke altogether with the church and became an atheist. Laplace did not graduate in theology but left for Paris with a letter of introduction from Le Canu to Jean le Rond dAlembert who at time was supreme in scientific circles. According to his great-great-grandson, dAlembert received him rather poorly, and to get rid of him gave him a mathematics book
Pierre-Simon Laplace
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Pierre-Simon Laplace (1749–1827). Posthumous portrait by Jean-Baptiste Paulin Guérin, 1838.
Pierre-Simon Laplace
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Laplace's house at Arcueil.
Pierre-Simon Laplace
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Laplace.
Pierre-Simon Laplace
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Tomb of Pierre-Simon Laplace
65.
Daniel Bernoulli
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Daniel Bernoulli FRS was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, Daniel Bernoulli was born in Groningen, in the Netherlands, into a family of distinguished mathematicians. The Bernoulli family came originally from Antwerp, at time in the Spanish Netherlands. After a brief period in Frankfurt the family moved to Basel, Daniel was the son of Johann Bernoulli, nephew of Jacob Bernoulli. He had two brothers, Niklaus and Johann II, Daniel Bernoulli was described by W. W. Rouse Ball as by far the ablest of the younger Bernoullis. He is said to have had a bad relationship with his father, Johann Bernoulli also plagiarized some key ideas from Daniels book Hydrodynamica in his own book Hydraulica which he backdated to before Hydrodynamica. Despite Daniels attempts at reconciliation, his father carried the grudge until his death, around schooling age, his father, Johann, encouraged him to study business, there being poor rewards awaiting a mathematician. However, Daniel refused, because he wanted to study mathematics and he later gave in to his fathers wish and studied business. His father then asked him to study in medicine, and Daniel agreed under the condition that his father would teach him mathematics privately, Daniel studied medicine at Basel, Heidelberg, and Strasbourg, and earned a PhD in anatomy and botany in 1721. He was a contemporary and close friend of Leonhard Euler and he went to St. Petersburg in 1724 as professor of mathematics, but was very unhappy there, and a temporary illness in 1733 gave him an excuse for leaving St. Petersberg. He returned to the University of Basel, where he held the chairs of medicine, metaphysics. In May,1750 he was elected a Fellow of the Royal Society and his earliest mathematical work was the Exercitationes, published in 1724 with the help of Goldbach. Two years later he pointed out for the first time the frequent desirability of resolving a compound motion into motions of translation and motion of rotation, together Bernoulli and Euler tried to discover more about the flow of fluids. In particular, they wanted to know about the relationship between the speed at which blood flows and its pressure, soon physicians all over Europe were measuring patients blood pressure by sticking point-ended glass tubes directly into their arteries. It was not until about 170 years later, in 1896 that an Italian doctor discovered a less painful method which is still in use today. However, Bernoullis method of measuring pressure is used today in modern aircraft to measure the speed of the air passing the plane. Taking his discoveries further, Daniel Bernoulli now returned to his work on Conservation of Energy. It was known that a moving body exchanges its kinetic energy for energy when it gains height
Daniel Bernoulli
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Daniel Bernoulli
66.
Johann Bernoulli
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Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to calculus and educating Leonhard Euler in the pupils youth. Johann was born in Basel, the son of Nicolaus Bernoulli, an apothecary, however, Johann Bernoulli did not enjoy medicine either and began studying mathematics on the side with his older brother Jacob. Throughout Johann Bernoulli’s education at Basel University the Bernoulli brothers worked together spending much of their time studying the newly discovered infinitesimal calculus and they were among the first mathematicians to not only study and understand calculus but to apply it to various problems. After graduating from Basel University Johann Bernoulli moved to teach differential equations, later, in 1694, he married Dorothea Falkner and soon after accepted a position as the professor of mathematics at the University of Groningen. At the request of Johann Bernoulli’s father-in-law, Johann Bernoulli began the voyage back to his town of Basel in 1705. Just after setting out on the journey he learned of his brother’s death to tuberculosis, Johann Bernoulli had planned on becoming the professor of Greek at Basel University upon returning but instead was able to take over as professor of mathematics, his older brother’s former position. As a student of Leibniz’s calculus, Johann Bernoulli sided with him in 1713 in the Newton–Leibniz debate over who deserved credit for the discovery of calculus, Johann Bernoulli defended Leibniz by showing that he had solved certain problems with his methods that Newton had failed to solve. Johann Bernoulli also promoted Descartes’ vortex theory over Newton’s theory of gravitation and this ultimately delayed acceptance of Newton’s theory in continental Europe. In consequence he was disqualified for the prize, which was won by Maclaurin, however, Bernoullis paper was subsequently accepted in 1726 when the Académie considered papers regarding elastic bodies, for which the prize was awarded to Pierre Mazière. Bernoulli received a mention in both competitions. Although Jacob and Johann worked together before Johann graduated from Basel University, shortly after this, Johann was jealous of Jacobs position and the two often attempted to outdo each other. After Jacobs death Johanns jealousy shifted toward his own talented son, in 1738 the father–son duo nearly simultaneously published separate works on hydrodynamics. Johann Bernoulli attempted to take precedence over his son by purposely predating his work two prior to his son’s. Johann married Dorothea Falkner, daughter of an Alderman of Basel and he was the father of Nicolaus II Bernoulli, Daniel Bernoulli and Johann II Bernoulli and uncle of Nicolaus I Bernoulli. The Bernoulli brothers often worked on the problems, but not without friction. In 1697 Jacob offered a reward for its solution, a protracted, bitter dispute then arose when Jacob challenged the solution and proposed his own. The dispute marked the origin of a new discipline, the calculus of variations, Bernoulli was hired by Guillaume de lHôpital for tutoring in mathematics
Johann Bernoulli
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Johann Bernoulli (portrait by Johann Rudolf Huber, circa 1740)
67.
Augustin-Louis Cauchy
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Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician who made pioneering contributions to analysis. He was one of the first to state and prove theorems of calculus rigorously and he almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had an influence over his contemporaries. His writings range widely in mathematics and mathematical physics, more concepts and theorems have been named for Cauchy than for any other mathematician. Cauchy was a writer, he wrote approximately eight hundred research articles. Cauchy was the son of Louis François Cauchy and Marie-Madeleine Desestre, Cauchy married Aloise de Bure in 1818. She was a relative of the publisher who published most of Cauchys works. By her he had two daughters, Marie Françoise Alicia and Marie Mathilde, Cauchys father was a high official in the Parisian Police of the New Régime. He lost his position because of the French Revolution that broke out one month before Augustin-Louis was born, the Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris, there Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, the famous mathematician Lagrange was also a friend of the Cauchy family. On Lagranges advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, most of the curriculum consisted of classical languages, the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of successes, Augustin-Louis chose an engineering career. In 1805 he placed second out of 293 applicants on this exam, one of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young, nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées. He graduated in engineering, with the highest honors. After finishing school in 1810, Cauchy accepted a job as an engineer in Cherbourg. Cauchys first two manuscripts were accepted, the one was rejected
Augustin-Louis Cauchy
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Cauchy around 1840. Lithography by Zéphirin Belliard after a painting by Jean Roller.
Augustin-Louis Cauchy
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The title page of a textbook by Cauchy.
Augustin-Louis Cauchy
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Leçons sur le calcul différentiel, 1829
68.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
Algebra
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A page from Al-Khwārizmī 's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Algebra
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Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
69.
Lagrange's theorem (group theory)
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Lagranges theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange and this can be shown using the concept of left cosets of H in G. If we can show that all cosets of H have the number of elements. This map is bijective because its inverse is given by f −1 = a b −1 y and this proof also shows that the quotient of the orders |G| / |H| is equal to the index. If we allow G and H to be infinite, and write this statement as | G | = ⋅ | H |, then, seen as a statement about cardinal numbers, it is equivalent to the axiom of choice. A consequence of the theorem is that the order of any element a of a group divides the order of that group. If the group has n elements, it follows a n = e and this can be used to prove Fermats little theorem and its generalization, Eulers theorem. These special cases were known long before the general theorem was proved, the theorem also shows that any group of prime order is cyclic and simple. This in turn can be used to prove Wilsons theorem, that if p is prime p is a factor of. Hence p < q, contradicting the assumption that p is the largest prime, Lagranges theorem raises the converse question as to whether every divisor of the order of a group is the order of some subgroup. This does not hold in general, given a finite group G, the smallest example is the alternating group G = A4, which has 12 elements but no subgroup of order 6. A CLT group is a group with the property that for every divisor of the order of the group. It is known that a CLT group must be solvable and that every group is a CLT group. There are partial converses to Lagranges theorem, for solvable groups, Halls theorems assert the existence of a subgroup of order equal to any unitary divisor of the group order. Lagrange did not prove Lagranges theorem in its general form, the number of such polynomials is the index in the symmetric group Sn of the subgroup H of permutations that preserve the polynomial. So the size of H divides n, with the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name. In his Disquisitiones Arithmeticae in 1801, Carl Friedrich Gauss proved Lagranges theorem for the case of Z*, the multiplicative group of nonzero integers modulo p. In 1844, Augustin-Louis Cauchy proved Lagranges theorem for the symmetric group Sn, camille Jordan finally proved Lagranges theorem for the case of any permutation group in 1861
Lagrange's theorem (group theory)
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G is the group, the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to. There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is an additive group). Together they partition the entire group G into equal-size, non-overlapping sets. Thus the index [G: H] is 4.
70.
Galois theory
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In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, originally, Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, further abstraction of Galois theory is achieved by the theory of Galois connections. Further, it gives a clear, and often practical. Galois theory also gives an insight into questions concerning problems in compass. It gives an elegant characterisation of the ratios of lengths that can be constructed with this method, for instance, = x2 – x + ab, where 1, a + b and ab are the elementary polynomials of degree 0,1 and 2 in two variables. This was first formalized by the 16th-century French mathematician François Viète, in Viètes formulas, the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation, see Discriminant, Nature of the roots for details. This solution was then rediscovered independently in 1535 by Niccolò Fontana Tartaglia, Cardano then extended this to numerous other cases, using similar arguments, see more details at Cardanos method. After the discovery of Ferros work, he felt that Tartaglias method was no longer secret and his student Lodovico Ferrari solved the quartic polynomial, his solution was also included in Ars Magna. With the benefit of modern notation and complex numbers, the formulae in this book do work in the general case and it was Rafael Bombelli who managed to understand how to work with complex numbers in order to solve all forms of cubic equation. Crucially, however, he did not consider composition of permutations, lagranges method did not extend to quintic equations or higher, because the resolvent had higher degree. The quintic was almost proven to have no general solutions by radicals by Paolo Ruffini in 1799, whose key insight was to use permutation groups, not just a single permutation. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, prior to this publication, Liouville announced Galois result to the Academy in a speech he gave on 4 July 1843. According to Allan Clark, Galoiss characterization dramatically supersedes the work of Abel, Galois theory was notoriously difficult for his contemporaries to understand, especially to the level where they could expand on it. For example, in his 1846 commentary, Liouville completely missed the core of Galois method. Joseph Alfred Serret who attended some of Liouvilles talks, included Galois theory in his 1866 of his textbook Cours dalgèbre supérieure, serrets pupil, Camille Jordan had an even better understanding reflected in his 1870 book Traité des substitutions et des équations algébriques. Outside France Galois theory remained more obscure for a longer period, in Britain, Cayley failed to grasp its depth and popular British algebra textbooks didnt even mention Galois theory until well after the turn of the century
Galois theory
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Évariste Galois (1811–1832)
71.
Jacobian matrix and determinant
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In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a matrix, both the matrix and its determinant are referred to as the Jacobian in literature. Suppose f, ℝn → ℝm is a function takes as input the vector x ∈ ℝn. Then the Jacobian matrix J of f is an m×n matrix, usually defined and arranged as follows, J = = or, component-wise and this matrix, whose entries are functions of x, is also denoted by Df, Jf, and ∂/∂. This linear map is thus the generalization of the notion of derivative. If m = n, the Jacobian matrix is a matrix, and its determinant. It carries important information about the behavior of f. In particular, the f has locally in the neighborhood of a point x an inverse function that is differentiable if. The Jacobian determinant also appears when changing the variables in multiple integrals, if m =1, f is a scalar field and the Jacobian matrix is reduced to a row vector of partial derivatives of f—i. e. the gradient of f. These concepts are named after the mathematician Carl Gustav Jacob Jacobi, the Jacobian generalizes the gradient of a scalar-valued function of multiple variables, which itself generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian for a multivariate function is the gradient. The Jacobian can also be thought of as describing the amount of stretching, rotating or transforming that a transformation imposes locally, for example, if = f is used to transform an image, the Jacobian Jf, describes how the image in the neighborhood of is transformed. If p is a point in ℝn and f is differentiable at p, compare this to a Taylor series for a scalar function of a scalar argument, truncated to first order, f = f + f ′ + o. The Jacobian of the gradient of a function of several variables has a special name, the Hessian matrix. If m=n, then f is a function from ℝn to itself and we can then form its determinant, known as the Jacobian determinant. The Jacobian determinant is occasionally referred to as the Jacobian, the Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the differentiable function f is invertible near a point p ∈ ℝn if the Jacobian determinant at p is non-zero. This is the inverse function theorem, furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p, if it is negative, f reverses orientation
Jacobian matrix and determinant
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A nonlinear map f: R 2 → R 2 sends a small square to a distorted parallelepiped close to the image of the square under the best linear approximation of f near the point.
72.
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
Volume
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A measuring cup can be used to measure volumes of liquids. This cup measures volume in units of cups, fluid ounces, and millilitres.
73.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
Tetrahedron
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(Click here for rotating model)
Tetrahedron
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4-sided die
74.
Determinant
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In linear algebra, the determinant is a useful value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det, detA and it can be viewed as the scaling factor of the transformation described by the matrix. In the case of a 2 ×2 matrix, the formula for the determinant. Each determinant of a 2 ×2 matrix in this equation is called a minor of the matrix A, the same sort of procedure can be used to find the determinant of a 4 ×4 matrix, the determinant of a 5 ×5 matrix, and so forth. The use of determinants in calculus includes the Jacobian determinant in the change of rule for integrals of functions of several variables. Determinants are also used to define the characteristic polynomial of a matrix, in analytical geometry, determinants express the signed n-dimensional volumes of n-dimensional parallelepipeds. Sometimes, determinants are used merely as a notation for expressions that would otherwise be unwieldy to write down. When the entries of the matrix are taken from a field, it can be proven that any matrix has an inverse if. There are various equivalent ways to define the determinant of a square matrix A, i. e. one with the number of rows. Another way to define the determinant is expressed in terms of the columns of the matrix and these properties mean that the determinant is an alternating multilinear function of the columns that maps the identity matrix to the underlying unit scalar. These suffice to uniquely calculate the determinant of any square matrix, provided the underlying scalars form a field, the definition below shows that such a function exists, and it can be shown to be unique. Assume A is a matrix with n rows and n columns. The entries can be numbers or expressions, the definition of the determinant depends only on the fact that they can be added and multiplied together in a commutative manner. The determinant of a 2 ×2 matrix is defined by | a b c d | = a d − b c. If the matrix entries are numbers, the matrix A can be used to represent two linear maps, one that maps the standard basis vectors to the rows of A. In either case, the images of the vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the rows of the matrix is the one with vertices at. The absolute value of ad − bc is the area of the parallelogram, the absolute value of the determinant together with the sign becomes the oriented area of the parallelogram
Determinant
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The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.
75.
Pell's equation
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These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y. His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was translated into Latin in 1126. Bhaskara II in the 12th century and Narayana Pandit in the 14th century both found general solutions to Pells equation and other quadratic indeterminate equations, the name of Pells equation arose from Leonhard Eulers mistakenly attributing Lord Brounckers solution of the equation to John Pell. Indeed, if x and y are positive integers satisfying this equation, then x/y is an approximation of √2. Similarly, Baudhayana discovered that x =17, y =12 and x =577, later, Archimedes approximated the square root of 3 by the rational number 1351/780. Although he did not explain his methods, this approximation may be obtained in the same way, Archimedes cattle problem involves solving a Pellian equation. It is now accepted that this problem is due to Archimides. Around AD250, Diophantus considered the equation a 2 x 2 + c = y 2 and this equation is different in form from Pells equation but equivalent to it. Diophantus solved the equation for equal to, and, al-Karaji, a 10th-century Persian mathematician, worked on similar problems to Diophantus. In Indian mathematics, Brahmagupta discovered that =2 − N2 =2 − N2, using this, he was able to compose triples and that were solutions of x 2 − N y 2 = k, to generate the new triples and. For instance, for N =92, Brahmagupta composed the triple with itself to get the new triple, dividing throughout by 64 gave the triple, which when composed with itself gave the desired integer solution. Brahmagupta solved many Pell equations with this method, in particular he showed how to obtain solutions starting from a solution of x 2 − N y 2 = k for k = ±1, ±2. The first general method for solving the Pell equation was given by Bhaskara II in 1150, called the chakravala method, it starts by composing any triple with the trivial triple to get the triple, which can be scaled down to. When m is chosen so that / k is an integer, among such m, the method chooses one that minimizes / k, and repeats the process. This method always terminates with a solution, Bhaskara used it to give the solution x =1766319049, y =226153980 to the notorious N =61 case. Several European mathematicians rediscovered how to solve Pells equation in the 17th century, Fermat found how to solve the equation and in a 1657 letter issued it as a challenge to English mathematicians. In a letter to Digby, Bernard Frénicle de Bessy said that Fermat found the smallest solution for N up to 150, both Wallis and Lord Brouncker gave solutions to these problems, though Wallis suggests in a letter that the solution was due to Brouncker. Pells connection with the equation is that he revised Thomas Brankers translation of Johann Rahns 1659 book Teutsche Algebra into English, euler mistakenly thought that this solution was due to Pell, as a result of which he named the equation after Pell
Pell's equation
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Pell's equation for n = 2 and six of its integer solutions
76.
Quadric
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In mathematics, a quadric or quadric surface, is a generalization of conic sections. It is an hypersurface in a space, and is defined as the zero set of an irreducible polynomial of degree two in D +1 variables. When the defining polynomial is not absolutely irreducible, the set is generally not considered as a quadric. The values Q, P and R are often taken to be real numbers or complex numbers. A quadric is an algebraic variety, or, if it is reducible. Quadrics may also be defined in spaces, see Quadric. Quadrics in the Euclidean plane are those of dimension D =1, in this case, one talks of conic sections, or conics. In three-dimensional Euclidean space, quadrics have dimension D =2 and they are classified and named by their orbits under affine transformations. More precisely, if an affine transformation maps a quadric onto another one, they belong to the same class, each of these 17 normal forms correspond to a single orbit under affine transformations. In three cases there are no points, ε1 = ε2 =1, ε1 =0, ε2 =1. In one case, the cone, there is a single point. If ε4 =1, one has a line, for ε4 =0, one has a double plane. For ε4 =1, one has two intersecting planes and it remains nine true quadrics, a cone and three cylinders and five non-degenerated quadrics, which are detailed in the following table. In a three-dimensional Euclidean space there are 17 such normal forms, of these 16 forms, five are nondegenerate, and the remaining are degenerate forms. Degenerate forms include planes, lines, points or even no points at all, the quadrics can be treated in a uniform manner by introducing homogeneous coordinates on a Euclidean space, thus effectively regarding it as a projective space. Thus if the coordinates on RD+1 are one introduces new coordinates on RD+2 related to the original coordinates by x i = X i / X0. In the new variables, every quadric is defined by an equation of the form Q = ∑ i j a i j X i X j =0 where the coefficients aij are symmetric in i and j. Regarding Q =0 as an equation in projective space exhibits the quadric as an algebraic variety
Quadric
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Ellipse (e = 1/2), parabola (e =1) and hyperbola (e = 2) with fixed focus F and directrix.
77.
Orbit
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In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet about a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating path around a body, to a close approximation, planets and satellites follow elliptical orbits, with the central mass being orbited at a focal point of the ellipse, as described by Keplers laws of planetary motion. For ease of calculation, in most situations orbital motion is adequately approximated by Newtonian Mechanics, historically, the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres. This model posited the existence of perfect moving spheres or rings to which the stars and it assumed the heavens were fixed apart from the motion of the spheres, and was developed without any understanding of gravity. After the planets motions were accurately measured, theoretical mechanisms such as deferent. Originally geocentric it was modified by Copernicus to place the sun at the centre to help simplify the model, the model was further challenged during the 16th century, as comets were observed traversing the spheres. The basis for the understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. Second, he found that the speed of each planet is not constant, as had previously been thought. Third, Kepler found a relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 AU distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter,5. 23/11.862, is equal to that for Venus,0. 7233/0.6152. Idealised orbits meeting these rules are known as Kepler orbits, isaac Newton demonstrated that Keplers laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections. Newton showed that, for a pair of bodies, the sizes are in inverse proportion to their masses. Where one body is more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, in a dramatic vindication of classical mechanics, in 1846 le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits, in relativity theory, orbits follow geodesic trajectories which are usually approximated very well by the Newtonian predictions but the differences are measurable. Essentially all the evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy
Orbit
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The International Space Station orbits above Earth.
Orbit
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Planetary orbits
Orbit
Orbit
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Conic sections describe the possible orbits (yellow) of small objects around the earth. A projection of these orbits onto the gravitational potential (blue) of the earth makes it possible to determine the orbital energy at each point in space.
78.
Comet
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Community of Metros is a system of international railway benchmarking. CoMET consists of metro systems from around the world. Each metro has a volume of at least 500 million passengers annually, the four main objectives of CoMET are, To build measures to establish metro best practice. To provide comparative information both for the board and the government. To introduce a system of measures for management and these objectives were discussed in detail at the CoMET Annual Meeting 2016, hosted by SMRT Trains of SMRT Corporation. The meeting was held at Singapore in November 2016, in the UITP conference of 1982, London Underground and Hamburger Hochbahn decided to create a benchmarking exercise to compare their two railways with additional data for other 24 metro systems. The project was successful despite the fact that metros were very different in sizes, structures, however, CoMET used the Key Performance Indicator innovatively to solve the problem. In 1994, the Mass Transit Railway of Hong Kong proposed to London Underground, Berlin U-Bahn, New York City Subway, thus, the metros can exchange performance data and investigate best practice amongst similar heavy metros. These five metros are later known as the Group of Five, over time, other large transit systems joined the group. For example, Mexico City Metro, São Paulo Metro and Tokyo Metro joined in 1996, with eight members in total, the group became known as the Community of Metros. Following the success of the CoMET, the Nova group was created in 1998 as another benchmarking association, the Nova is currently consisted of 14 metro systems from around the world. Later, Moscow Metro joined the CoMET in 1999, madrid Metro transferred from Nova to CoMET in 2004. Santiago Metro and Beijing Subway joined in 2008, taipei Metro was the last member to join the CoMET which also joined in 2010
Comet
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"CoMET" redirects here. For the geoprofession, see Geoprofessions § Construction-materials engineering and testing (CoMET).
Comet
79.
Orbital elements
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Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in classical two-body systems. There are many different ways to describe the same orbit. A real orbit changes over time due to perturbations by other objects. A Keplerian orbit is merely an idealized, mathematical approximation at a particular time, the traditional orbital elements are the six Keplerian elements, after Johannes Kepler and his laws of planetary motion. When viewed from a frame, two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the center of mass. When viewed from a non-inertial frame centred on one of the bodies, only the trajectory of the body is apparent. An orbit has two sets of Keplerian elements depending on which body is used as the point of reference, the reference body is called the primary, the other body is called the secondary. The primary does not necessarily possess more mass than the secondary, and even when the bodies are of equal mass, the orbital elements depend on the choice of the primary. The main two elements that define the shape and size of the ellipse, Eccentricity —shape of the ellipse, semimajor axis —the sum of the periapsis and apoapsis distances divided by two. For circular orbits, the axis is the distance between the centers of the bodies, not the distance of the bodies from the center of mass. For paraboles or hyperboles, this is infinite, tilt angle is measured perpendicular to line of intersection between orbital plane and reference plane. Any three points on an ellipse will define the ellipse orbital plane, the plane and the ellipse are both two-dimensional objects defined in three-dimensional space. Longitude of the ascending node —horizontally orients the ascending node of the ellipse with respect to the reference frames vernal point, and finally, Argument of periapsis defines the orientation of the ellipse in the orbital plane, as an angle measured from the ascending node to the periapsis. True anomaly at epoch defines the position of the body along the ellipse at a specific time. The mean anomaly is a mathematically convenient angle which varies linearly with time and it can be converted into the true anomaly ν, which does represent the real geometric angle in the plane of the ellipse, between periapsis and the position of the orbiting object at any given time. Thus, the anomaly is shown as the red angle ν in the diagram
Orbital elements
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In this diagram, the orbital plane (yellow) intersects a reference plane (gray). For earth-orbiting satellites, the reference plane is usually the Earth's equatorial plane, and for satellites in solar orbits it is the ecliptic plane. The intersection is called the line of nodes, as it connects the center of mass with the ascending and descending nodes. This plane, together with the Vernal Point (♈), establishes a reference frame.
80.
Urbain Le Verrier
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Urbain Jean Joseph Le Verrier was a French mathematician who specialized in celestial mechanics and is best known for predicting the existence and position of Neptune using only mathematics. The calculations were made to explain discrepancies with Uranuss orbit and the laws of Kepler, Le Verrier sent the coordinates to Johann Gottfried Galle in Berlin, asking him to verify. Galle found Neptune in the night he received Le Verriers letter. The discovery of Neptune is widely regarded as a validation of celestial mechanics. Le Verrier was born at Saint-Lô, Manche, France, and he briefly studied chemistry under Gay-Lussac, writing papers on the combinations of phosphorus and hydrogen, and phosphorus and oxygen. He then switched to astronomy, particularly celestial mechanics, and accepted a job at the Paris Observatory and he spent most of his professional life there, and eventually became that institutions Director, from 1854 to 1870 and again from 1873 to 1877. In 1846, Le Verrier became a member of the French Academy of Sciences, Le Verriers name is one of the 72 names inscribed on the Eiffel Tower. Le Verriers first work in astronomy was presented to the Académie des Sciences in September 1839 and this work addressed the then most-important question in astronomy, the stability of the Solar System, first investigated by Laplace. He was able to some important limits on the motions of the system. From 1844 to 1847, Le Verrier published a series of works on periodic comets, in particular those of Lexell, Faye and DeVico. He was able to some interesting interactions with the planet Jupiter. Le Verriers most famous achievement is his prediction of the existence of the unknown planet Neptune, using only mathematics. At the same time, but unknown to Le Verrier, similar calculations were made by John Couch Adams in England, Le Verrier transmitted his own prediction by 18 September in a letter to Johann Galle of the Berlin Observatory. There was, and to an extent still is, controversy over the apportionment of credit for the discovery, there is no ambiguity to the discovery claims of Le Verrier, Galle, and dArrest. Adamss work was earlier than Le Verriers but was finished later and was unrelated to the actual discovery. Not even the briefest account of Adamss predicted orbital elements was published more than a month after Berlins visual confirmation. Galle, so that the facts stated above cannot detract, in the slightest degree, early in the 19th century, the methods of predicting the motions of the planets were somewhat scattered, having been developed over decades by many different researchers. In 1847, Le Verrier took on the task to, embrace in a single work the entire planetary system, put everything in harmony if possible, otherwise, declare with certainty that there are as yet unknown causes of perturbations
Urbain Le Verrier
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Urbain Le Verrier
Urbain Le Verrier
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Signature of M. LeVerrier
Urbain Le Verrier
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The grave of Urbain Le Verrier.
81.
Mechanics
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Mechanics is an area of science concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. The scientific discipline has its origins in Ancient Greece with the writings of Aristotle, during the early modern period, scientists such as Khayaam, Galileo, Kepler, and Newton, laid the foundation for what is now known as classical mechanics. It is a branch of physics that deals with particles that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as a branch of science which deals with the motion of, historically, classical mechanics came first, while quantum mechanics is a comparatively recent invention. Classical mechanics originated with Isaac Newtons laws of motion in Philosophiæ Naturalis Principia Mathematica, both are commonly held to constitute the most certain knowledge that exists about physical nature. Classical mechanics has especially often been viewed as a model for other so-called exact sciences, essential in this respect is the relentless use of mathematics in theories, as well as the decisive role played by experiment in generating and testing them. Quantum mechanics is of a scope, as it encompasses classical mechanics as a sub-discipline which applies under certain restricted circumstances. According to the principle, there is no contradiction or conflict between the two subjects, each simply pertains to specific situations. The correspondence principle states that the behavior of systems described by quantum theories reproduces classical physics in the limit of quantum numbers. Quantum mechanics has superseded classical mechanics at the level and is indispensable for the explanation and prediction of processes at the molecular, atomic. However, for macroscopic processes classical mechanics is able to solve problems which are difficult in quantum mechanics and hence remains useful. Modern descriptions of such behavior begin with a definition of such quantities as displacement, time, velocity, acceleration, mass. Until about 400 years ago, however, motion was explained from a different point of view. He showed that the speed of falling objects increases steadily during the time of their fall and this acceleration is the same for heavy objects as for light ones, provided air friction is discounted. The English mathematician and physicist Isaac Newton improved this analysis by defining force and mass, for objects traveling at speeds close to the speed of light, Newton’s laws were superseded by Albert Einstein’s theory of relativity. For atomic and subatomic particles, Newton’s laws were superseded by quantum theory, for everyday phenomena, however, Newton’s three laws of motion remain the cornerstone of dynamics, which is the study of what causes motion. In analogy to the distinction between quantum and classical mechanics, Einsteins general and special theories of relativity have expanded the scope of Newton, the differences between relativistic and Newtonian mechanics become significant and even dominant as the velocity of a massive body approaches the speed of light. Relativistic corrections are also needed for quantum mechanics, although general relativity has not been integrated, the two theories remain incompatible, a hurdle which must be overcome in developing a theory of everything
Mechanics
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Arabic Machine Manuscript. Unknown date (at a guess: 16th to 19th centuries).
82.
Adrien-Marie Legendre
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Adrien-Marie Legendre was a French mathematician. Legendre made numerous contributions to mathematics, well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family and he received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780, at the same time, he was associated with the Bureau des Longitudes. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media and this treatise also brought him to the attention of Lagrange. The Académie des Sciences made Legendre an adjoint member in 1783, in 1789 he was elected a Fellow of the Royal Society. He assisted with the Anglo-French Survey to calculate the distance between the Paris Observatory and the Royal Greenwich Observatory by means of trigonometry. To this end in 1787 he visited Dover and London together with Dominique, comte de Cassini, the three also visited William Herschel, the discoverer of the planet Uranus. Legendre lost his fortune in 1793 during the French Revolution. That year, he also married Marguerite-Claudine Couhin, who helped him put his affairs in order, in 1795 Legendre became one of six members of the mathematics section of the reconstituted Académie des Sciences, renamed the Institut National des Sciences et des Arts. Later, in 1803, Napoleon reorganized the Institut National, and his pension was partially reinstated with the change in government in 1828. In 1831 he was made an officer of the Légion dHonneur, Legendre died in Paris on 10 January 1833, after a long and painful illness, and Legendres widow carefully preserved his belongings to memorialize him. Upon her death in 1856, she was buried next to her husband in the village of Auteuil, where the couple had lived, Legendres name is one of the 72 names inscribed on the Eiffel Tower. Today, the term least squares method is used as a translation from the French méthode des moindres carrés. Around 1811 he named the gamma function and introduced the symbol Γ normalizing it to Γ = n, in 1830 he gave a proof of Fermats last theorem for exponent n =5, which was also proven by Lejeune Dirichlet in 1828. In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss, in connection to this and he also did pioneering work on the distribution of primes, and on the application of analysis to number theory. His 1798 conjecture of the prime number theorem was proved by Hadamard. He is known for the Legendre transformation, which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics, in thermodynamics it is also used to obtain the enthalpy and the Helmholtz and Gibbs energies from the internal energy
Adrien-Marie Legendre
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1820 watercolor caricature of Adrien-Marie Legendre by French artist Julien-Leopold Boilly (see portrait debacle), the only existing portrait known
Adrien-Marie Legendre
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1820 watercolor caricatures of the French mathematicians Adrien-Marie Legendre (left) and Joseph Fourier (right) by French artist Julien-Leopold Boilly, watercolor portrait numbers 29 and 30 of Album de 73 portraits-charge aquarellés des membres de I’Institut.
Adrien-Marie Legendre
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Side view sketching of French politician Louis Legendre (1752–1797), whose portrait has been mistakenly used, for nearly 200 years, to represent French mathematician Adrien-Marie Legendre, i.e. up until 2005 when the mistake was discovered.
83.
Augustin Louis Cauchy
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Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician who made pioneering contributions to analysis. He was one of the first to state and prove theorems of calculus rigorously and he almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had an influence over his contemporaries. His writings range widely in mathematics and mathematical physics, more concepts and theorems have been named for Cauchy than for any other mathematician. Cauchy was a writer, he wrote approximately eight hundred research articles. Cauchy was the son of Louis François Cauchy and Marie-Madeleine Desestre, Cauchy married Aloise de Bure in 1818. She was a relative of the publisher who published most of Cauchys works. By her he had two daughters, Marie Françoise Alicia and Marie Mathilde, Cauchys father was a high official in the Parisian Police of the New Régime. He lost his position because of the French Revolution that broke out one month before Augustin-Louis was born, the Cauchy family survived the revolution and the following Reign of Terror by escaping to Arcueil, where Cauchy received his first education, from his father. After the execution of Robespierre, it was safe for the family to return to Paris, there Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power, Louis-François Cauchy was further promoted, the famous mathematician Lagrange was also a friend of the Cauchy family. On Lagranges advice, Augustin-Louis was enrolled in the École Centrale du Panthéon, most of the curriculum consisted of classical languages, the young and ambitious Cauchy, being a brilliant student, won many prizes in Latin and Humanities. In spite of successes, Augustin-Louis chose an engineering career. In 1805 he placed second out of 293 applicants on this exam, one of the main purposes of this school was to give future civil and military engineers a high-level scientific and mathematical education. The school functioned under military discipline, which caused the young, nevertheless, he finished the Polytechnique in 1807, at the age of 18, and went on to the École des Ponts et Chaussées. He graduated in engineering, with the highest honors. After finishing school in 1810, Cauchy accepted a job as an engineer in Cherbourg. Cauchys first two manuscripts were accepted, the one was rejected
Augustin Louis Cauchy
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Cauchy around 1840. Lithography by Zéphirin Belliard after a painting by Jean Roller.
Augustin Louis Cauchy
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The title page of a textbook by Cauchy.
Augustin Louis Cauchy
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Leçons sur le calcul différentiel, 1829
84.
Karl Weierstrass
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Karl Theodor Wilhelm Weierstrass was a German mathematician often cited as the father of modern analysis. Despite leaving university without a degree, he studied mathematics and trained as a teacher, eventually teaching mathematics, physics, botany, Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, a government official and his interest in mathematics began while he was a gymnasium student at the Theodorianum in Paderborn. He was sent to the University of Bonn upon graduation to prepare for a government position, because his studies were to be in the fields of law, economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his course of study. The outcome was to leave the university without a degree, after that he studied mathematics at the Münster Academy and his father was able to obtain a place for him in a teacher training school in Münster. Later he was certified as a teacher in that city, during this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions. In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg, besides mathematics he also taught physics, botanics and gymnastics. Weierstrass may have had a child named Franz with the widow of his friend Carl Wilhelm Borchardt. After 1850 Weierstrass suffered from a period of illness, but was able to publish papers that brought him fame. The University of Königsberg conferred an honorary degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut, which became the Technical University of Berlin. In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which became the Humboldt Universität zu Berlin. He was immobile for the last three years of his life, and died in Berlin from pneumonia, delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval, notably, in his 1821 Cours danalyse, Cauchy argued that the limit of continuous functions was itself continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the limit of continuous functions is continuous. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus, using this definition, he proved the Intermediate Value Theorem. He also proved the Bolzano–Weierstrass theorem and used it to study the properties of functions on closed and bounded intervals
Karl Weierstrass
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Karl Theodor Wilhelm Weierstrass (Weierstraß)
85.
Fermat's little theorem
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Fermats little theorem states that if p is a prime number, then for any integer a, the number a p − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as a p ≡ a, for example, if a =2 and p =7,27 =128, and 128 −2 =7 ×18 is an integer multiple of 7. If a is not divisible by p, Fermats little theorem is equivalent to the statement that a p −1 −1 is a multiple of p. For example, if a =2 and p =7 then 26 =64 and 64 −1 =63 is thus a multiple of 7, Fermats little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640 and it is called the little theorem to distinguish it from Fermats last theorem. Pierre de Fermat first stated the theorem in a letter dated October 18,1640, to his friend, an early use in English occurs in A. A. Albert, Modern Higher Algebra, which refers to the so-called little Fermat theorem on page 206, some mathematicians independently made the related hypothesis that 2p ≡2 if and only if p is a prime. Indeed, the if part is true, and is a case of Fermats little theorem. However, the if part of this hypothesis is false, for example,2341 ≡2. Several proofs of Fermats little theorem are known and it is frequently proved as a corollary of Eulers theorem. Fermats little theorem is a case of Eulers theorem, for any modulus n and any integer a coprime to n, we have a φ ≡1. Eulers theorem is indeed a generalization, because if n = p is a prime number, then φ = p −1. A slight generalization of Eulers theorem, which follows from it, is, if a, n, x, y are integers with n positive. This follows as x is of the form y + φk, in this form, the theorem finds many uses in cryptography and, in particular, underlies the computations used in the RSA public key encryption method. The special case with n a prime may be considered a consequence of Fermats little theorem, Fermats little theorem is also related to the Carmichael function and Carmichaels theorem, as well as to Lagranges theorem in group theory. The algebraic setting of Fermats little theorem can be generalized to finite fields, the converse of Fermats little theorem is not generally true, as it fails for Carmichael numbers. However, a stronger form of the theorem is true. The theorem is as follows, If there exists an a such that a p −1 ≡1 and this theorem forms the basis for the Lucas–Lehmer test, an important primality test
Fermat's little theorem
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Pierre de Fermat
86.
Royal Society
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Founded in November 1660, it was granted a royal charter by King Charles II as The Royal Society. The society is governed by its Council, which is chaired by the Societys President, according to a set of statutes and standing orders. The members of Council and the President are elected from and by its Fellows, the members of the society. As of 2016, there are about 1,600 fellows, allowed to use the postnominal title FRS, there are also royal fellows, honorary fellows and foreign members, the last of which are allowed to use the postnominal title ForMemRS. The Royal Society President is Venkatraman Ramakrishnan, who took up the post on 30 November 2015, since 1967, the society has been based at 6–9 Carlton House Terrace, a Grade I listed building in central London which was previously used by the Embassy of Germany, London. The Royal Society started from groups of physicians and natural philosophers, meeting at variety of locations and they were influenced by the new science, as promoted by Francis Bacon in his New Atlantis, from approximately 1645 onwards. A group known as The Philosophical Society of Oxford was run under a set of rules still retained by the Bodleian Library, after the English Restoration, there were regular meetings at Gresham College. It is widely held that these groups were the inspiration for the foundation of the Royal Society, I will not say, that Mr Oldenburg did rather inspire the French to follow the English, or, at least, did help them, and hinder us. But tis well known who were the men that began and promoted that design. This initial royal favour has continued and, since then, every monarch has been the patron of the society, the societys early meetings included experiments performed first by Hooke and then by Denis Papin, who was appointed in 1684. These experiments varied in their area, and were both important in some cases and trivial in others. The Society returned to Gresham in 1673, there had been an attempt in 1667 to establish a permanent college for the society. Michael Hunter argues that this was influenced by Solomons House in Bacons New Atlantis and, to a lesser extent, by J. V. The first proposal was given by John Evelyn to Robert Boyle in a letter dated 3 September 1659, he suggested a scheme, with apartments for members. The societys ideas were simpler and only included residences for a handful of staff and these plans were progressing by November 1667, but never came to anything, given the lack of contributions from members and the unrealised—perhaps unrealistic—aspirations of the society. During the 18th century, the gusto that had characterised the early years of the society faded, with a number of scientific greats compared to other periods. The pointed lightning conductor had been invented by Benjamin Franklin in 1749, during the same time period, it became customary to appoint society fellows to serve on government committees where science was concerned, something that still continues. The 18th century featured remedies to many of the early problems
Royal Society
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The entrance to the Royal Society in Carlton House Terrace, London
Royal Society
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The President, Council, and Fellows of the Royal Society of London for Improving Natural Knowledge
Royal Society
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John Evelyn, who helped to found the Royal Society
Royal Society
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Mace granted by Charles II
87.
Royal Swedish Academy of Sciences
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The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. Every year the Academy awards the Nobel Prizes in Physics and Chemistry, the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, the Crafoord Prize, the Academy has elected about 1.700 Swedish and 1.200 foreign members since it was founded in 1739. Hansson, appointed from 1 July 2015 The transactions of the Academy were published as its main series between 1739 and 1974, in parallel, other major series have appeared and gone, Öfversigt af Kungl. These lasted into the 1860s, when they were replaced by the single Bihang series, further restructuring of their topics occurred in 1949 and 1974. The purpose of the academy was to focus on practically useful knowledge, the academy was intended to be different from the Royal Society of Sciences in Uppsala, which had been founded in 1719 and published in Latin. The location close to the activities in Swedens capital was also intentional. The academy was modeled after the Royal Society of London and Academie Royale des Sciences in Paris, France, members of the Royal Swedish Academy of Sciences Official website Royal Swedish Academy of Sciences video site
Royal Swedish Academy of Sciences
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Main building of the Royal Swedish Academy of Sciences in Stockholm.
Royal Swedish Academy of Sciences
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Kongl. Svenska Vetenskaps-Academiens handlingar, volume XI (1750).
Royal Swedish Academy of Sciences
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The Royal Swedish Academy of Sciences
88.
Count of the Empire
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Napoleon I created titles of nobility to institute a stable elite in the First French Empire, after the instability resulting from the French Revolution. Like many others, both before and since, Napoleon found that the ability to confer titles was also a tool of patronage which cost the state little treasure. The Grand Dignitaries of the Empire ranked, regardless of noble title, enoblement started in 1804 with the creation of the princely title for members of Napoleons imperial family. In 1806 ducal titles were created and in 1808 those of count, baron, Napoleon founded the concept of nobility of Empire by an imperial decree on 1 March 1808. The purpose of creation was to amalgamate the old nobility. This step, which aimed at the introduction of a elite, is fully in line with the creation of the Legion of Honour. A council of the seals and the titles was created and charged with establishing armorial bearings. These creations are to be distinguished from an order such as the Order of the Bath. These titles of nobility did not have any true privileges, with two exceptions, right of bearing, the lands granted with the title were held in a majorat. This nobility is essentially a nobility of service, to a large extent made up of soldiers, some civil servants, there were 239 remaining families belonging to the First Empire nobility in 1975. Of those, perhaps about 135 were titled, only one princely title and seven ducal titles remain today. Along with a new system of titles of nobility, the First French Empire also introduced a new system of heraldry, Napoleonic heraldry was based on traditional heraldry but was characterised by a stronger sense of hierarchy. It employed a system of additional marks in the shield to indicate official functions and positions. Another notable difference from traditional heraldry was the toques, which replaced coronets, the toques were surmounted by ostrich feathers, dukes had 7, counts had 5, barons 3, knights 1. The number of lambrequins was also regulated,3,2,1, as many grantees were new men, and the arms often alluded to their life or specific actions, many new or unusual charges were also introduced. The most characteristic mark of Napoleonic heraldry was the additional marks in the shield to indicate official functions and positions and these came in the form of quarters in various colours, and would be differenced further by marks of the specific rank or function. The said marks of the rank or function as used by Barons. A decree of 3 March 1810 states, The name, arms and this provision applied only to the bearers of Napoleonic titles
Count of the Empire
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Imperial coat of arms
89.
Satellites of Jupiter
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There are 67 known moons of Jupiter. This gives Jupiter the largest number of moons with reasonably stable orbits of any planet in the Solar System. The Galilean moons are by far the largest and most massive objects to orbit Jupiter, with the remaining 63 moons, of Jupiters moons, eight are regular satellites with prograde and nearly circular orbits that are not greatly inclined with respect to Jupiters equatorial plane. The Galilean satellites are nearly spherical in shape due to their planetary mass, the other four regular satellites are much smaller and closer to Jupiter, these serve as sources of the dust that makes up Jupiters rings. The remainder of Jupiters moons are irregular satellites whose prograde and retrograde orbits are farther from Jupiter and have high inclinations. These moons were probably captured by Jupiter from solar orbits, sixteen irregular satellites have been discovered since 2003 and have not yet been named. The physical and orbital characteristics of the moons vary widely, all other Jovian moons are less than 250 kilometres in diameter, with most barely exceeding 5 kilometres. Their orbital shapes range from perfectly circular to highly eccentric and inclined. Orbital periods range from seven hours, to three thousand times more. Jupiters regular satellites are believed to have formed from a circumplanetary disk and they may be the remnants of a score of Galilean-mass satellites that formed early in Jupiters history. Simulations suggest that, while the disk had a high mass at any given moment. However, only 2% the proto-disk mass of Jupiter is required to explain the existing satellites, thus there may have been several generations of Galilean-mass satellites in Jupiters early history. Each generation of moons might have spiraled into Jupiter, due to drag from the disk, by the time the present generation formed, the disk had thinned to the point that it no longer greatly interfered with the moons orbits. The current Galilean moons were still affected, falling into and being protected by an orbital resonance with each other, which still exists for Io, Europa. Ganymedes larger mass means that it would have migrated inward at a faster rate than Europa or Io, many broke up due to the mechanical stresses of capture, or afterward by collisions with other small bodies, producing the moons we see today. The first claimed observation of one of Jupiters moons is that of Chinese astronomer Gan De around 364 BC, however, the first certain observations of Jupiters satellites were those of Galileo Galilei in 1609. By January 1610, he had sighted the four massive Galilean moons with his 30× magnification telescope, no additional satellites were discovered until E. E. Barnard observed Amalthea in 1892. With the aid of photography, further discoveries followed quickly over the course of the twentieth century
Satellites of Jupiter
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A montage of Jupiter and its four largest moons (distance and sizes not to scale)
Satellites of Jupiter
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Jupiter and the Galilean moons through a 10" (25 cm) Meade LX200 telescope
Satellites of Jupiter
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The Galilean moons. From left to right, in order of increasing distance from Jupiter: Io, Europa, Ganymede, Callisto
Satellites of Jupiter
90.
Eiffel Tower
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The Eiffel Tower is a wrought iron lattice tower on the Champ de Mars in Paris, France. It is named after the engineer Gustave Eiffel, whose company designed, the Eiffel Tower is the most-visited paid monument in the world,6.91 million people ascended it in 2015. The tower is 324 metres tall, about the height as an 81-storey building. Its base is square, measuring 125 metres on each side, due to the addition of a broadcasting aerial at the top of the tower in 1957, it is now taller than the Chrysler Building by 5.2 metres. Excluding transmitters, the Eiffel Tower is the second-tallest structure in France after the Millau Viaduct, the tower has three levels for visitors, with restaurants on the first and second levels. The top levels upper platform is 276 m above the ground – the highest observation deck accessible to the public in the European Union, tickets can be purchased to ascend by stairs or lift to the first and second levels. The climb from ground level to the first level is over 300 steps, although there is a staircase to the top level, it is usually only accessible by lift. Eiffel openly acknowledged that inspiration for a tower came from the Latting Observatory built in New York City in 1853, sauvestre added decorative arches to the base of the tower, a glass pavilion to the first level, and other embellishments. Little progress was made until 1886, when Jules Grévy was re-elected as president of France and Édouard Lockroy was appointed as minister for trade. On 12 May, a commission was set up to examine Eiffels scheme and its rivals, which, after some debate about the exact location of the tower, a contract was signed on 8 January 1887. Eiffel was to all income from the commercial exploitation of the tower during the exhibition. He later established a company to manage the tower, putting up half the necessary capital himself. The proposed tower had been a subject of controversy, drawing criticism from those who did not believe it was feasible and these objections were an expression of a long-standing debate in France about the relationship between architecture and engineering. And for twenty years … we shall see stretching like a blot of ink the hateful shadow of the column of bolted sheet metal. Gustave Eiffel responded to criticisms by comparing his tower to the Egyptian pyramids. Will it not also be grandiose in its way, and why would something admirable in Egypt become hideous and ridiculous in Paris. Indeed, Garnier was a member of the Tower Commission that had examined the various proposals, some of the protesters changed their minds when the tower was built, others remained unconvinced. Guy de Maupassant supposedly ate lunch in the restaurant every day because it was the one place in Paris where the tower was not visible
Eiffel Tower
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The Eiffel Tower as seen from the Champ de Mars
Eiffel Tower
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First drawing of the Eiffel Tower by Maurice Koechlin including size comparison with other Parisian landmarks such as Notre Dame de Paris, the Statue of Liberty and the Vendôme Column
Eiffel Tower
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A calligram by Guillaume Apollinaire
91.
Lunar crater
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Lunar craters are impact craters on Earths Moon. The Moons surface has many craters, almost all of which were formed by impacts, the word crater was adopted by Galileo from the Greek word for vessel -. Galileo built his first telescope in late 1609, and turned it to the Moon for the first time on November 30,1609. He discovered that, contrary to general opinion at that time, the Moon was not a perfect sphere, scientific opinion as to the origin of craters swung back and forth over the ensuing centuries. The formation of new craters is studied in the lunar impact monitoring program at NASA, the biggest recorded creation was caused by an impact recorded on March 17,2013. Visible to the eye, the impact is believed to be from an approximately 40 kg meteoroid striking the surface at a speed of 90,000 km/h. Because of the Moons lack of water, and atmosphere, or tectonic plates, there is little erosion, the age of large craters is determined by the number of smaller craters contained within it, older craters generally accumulating more small, contained craters. The smallest craters found have been microscopic in size, found in rocks returned to Earth from the Moon, the largest crater called such is about 290 kilometres across in diameter, located near the lunar South Pole. However, it is believed many of the lunar maria were formed by giant impacts. In 1978, Chuck Wood and Leif Andersson of the Lunar & Planetary Lab devised a system of categorization of lunar impact craters and they used a sampling of craters that were relatively unmodified by subsequent impacts, then grouped the results into five broad categories. These successfully accounted for about 99% of all lunar impact craters, the LPC Crater Types were as follows, ALC — small, cup-shaped craters with a diameter of about 10 km or less, and no central floor. The archetype for this category is Albategnius C, BIO — similar to an ALC, but with small, flat floors. Typical diameter is about 15 km, the lunar crater archetype is Biot. SOS — the interior floor is wide and flat, with no central peak, the inner walls are not terraced. The diameter is normally in the range of 15–25 km, TRI — these complex craters are large enough so that their inner walls have slumped to the floor. They can range in size from 15–50 km in diameter, TYC — these are larger than 50 km, with terraced inner walls and relatively flat floors. They frequently have large central peak formations, tycho is the archetype for this class. Beyond a couple of hundred kilometers diameter, the peak of the TYC class disappear
Lunar crater
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Side view of the Moltke crater taken from Apollo 11.
Lunar crater
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Webb crater, as seen from Lunar Orbiter 1. Several smaller craters can be seen in and around Webb crater.
Lunar crater
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Lunar craters as captured through the backyard telescope of an amateur astronomer, partially illuminated by the sun on a waning crescent moon.
Lunar crater
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Albategnius
92.
Public domain
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The term public domain has two senses of meaning. Anything published is out in the domain in the sense that it is available to the public. Once published, news and information in books is in the public domain, in the sense of intellectual property, works in the public domain are those whose exclusive intellectual property rights have expired, have been forfeited, or are inapplicable. Examples for works not covered by copyright which are therefore in the domain, are the formulae of Newtonian physics, cooking recipes. Examples for works actively dedicated into public domain by their authors are reference implementations of algorithms, NIHs ImageJ. The term is not normally applied to situations where the creator of a work retains residual rights, as rights are country-based and vary, a work may be subject to rights in one country and be in the public domain in another. Some rights depend on registrations on a basis, and the absence of registration in a particular country, if required. Although the term public domain did not come into use until the mid-18th century, the Romans had a large proprietary rights system where they defined many things that cannot be privately owned as res nullius, res communes, res publicae and res universitatis. The term res nullius was defined as not yet appropriated. The term res communes was defined as things that could be enjoyed by mankind, such as air, sunlight. The term res publicae referred to things that were shared by all citizens, when the first early copyright law was first established in Britain with the Statute of Anne in 1710, public domain did not appear. However, similar concepts were developed by British and French jurists in the eighteenth century, instead of public domain they used terms such as publici juris or propriété publique to describe works that were not covered by copyright law. The phrase fall in the domain can be traced to mid-nineteenth century France to describe the end of copyright term. In this historical context Paul Torremans describes copyright as a coral reef of private right jutting up from the ocean of the public domain. Because copyright law is different from country to country, Pamela Samuelson has described the public domain as being different sizes at different times in different countries. According to James Boyle this definition underlines common usage of the public domain and equates the public domain to public property. However, the usage of the public domain can be more granular. Such a definition regards work in copyright as private property subject to fair use rights, the materials that compose our cultural heritage must be free for all living to use no less than matter necessary for biological survival
Public domain
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Newton's own copy of his Principia, with hand-written corrections for the second edition
Public domain
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L.H.O.O.Q. (1919). Derivative work by the Dadaist Marcel Duchamp based on the Mona Lisa.
93.
Rouse History of Mathematics
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Walter William Rouse Ball, known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge from 1878 to 1905. He was also an amateur magician, and the founding president of the Cambridge Pentacle Club in 1919. Ball was the son and heir of Walter Frederick Ball, of 3, St Johns Park Villas, South Hampstead, London. Educated at University College School, he entered Trinity College, Cambridge in 1870, became a scholar and first Smiths Prizeman and he became a Fellow of Trinity in 1875, and remained one for the rest of his life. He is buried at the Parish of the Ascension Burial Ground in Cambridge and he is commemorated in the naming of the small pavilion, now used as changing rooms and toilets, on Jesus Green in Cambridge. A History of the Study of Mathematics at Cambridge, Cambridge University Press,1889 A Short Account of the History of Mathematics at Project Gutenberg, dover 1960 republication of fourth edition. Mathematical Recreations and Essays at Project Gutenberg A History of the First Trinity Boat Club Cambridge Papers at Project Gutenberg, string Figures, Cambridge, W. Heffer & Sons Rouse Ball Professor of Mathematics Rouse Ball Professor of English Law Martin Gardner, another author of recreational mathematics. Singmaster, David,1892 Walter William Rouse Ball, Mathematical recreations and problems of past and present times, in Grattan-Guinness, W. W. Rouse Ball at the Mathematics Genealogy Project W. W. Rouse Ball at Find a Grave
Rouse History of Mathematics
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W.W. Rouse Ball
94.
Merriam-Webster Dictionary
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Merriam-Webster, Incorporated, is an American company that publishes reference books, especially known for its dictionaries. In 1831, George and Charles Merriam founded the company as G & C Merriam Co. in Springfield, in 1843, after Noah Webster died, the company bought the rights to An American Dictionary of the English Language from Websters estate. All Merriam-Webster dictionaries trace their lineage to this source, in 1964, Encyclopædia Britannica, Inc. acquired Merriam-Webster, Inc. as a subsidiary. The company adopted its current name in 1982, in 1806, Webster published his first dictionary, A Compendious Dictionary of the English Language. In 1807 Webster started two decades of work to expand his publication into a fully comprehensive dictionary, An American Dictionary of the English Language. To help him trace the etymology of words, Webster learned 26 languages, Webster hoped to standardize American speech, since Americans in different parts of the country used somewhat different vocabularies and spelled, pronounced, and used words differently. Webster completed his dictionary during his year abroad in 1825 in Paris and his 1820s book contained 70,000 words, of which about 12,000 had never appeared in a dictionary before. He also added American words, including skunk and squash, that did not appear in British dictionaries, at the age of 70 in 1828, Webster published his dictionary, it sold poorly, with only 2,500 copies putting him in debt. However, in 1840, he published the edition in two volumes with much greater success. He shows ways that American poetry inherited Websters ideas and draws on his lexicography to develop the language, in 1843, after Websters death, George Merriam and Charles Merriam secured publishing and revision rights to the 1840 edition of the dictionary. They published a revision in 1847, which did not change any of the text but merely added new sections. This began a series of revisions that were described as being unabridged in content, in 1884 it contained 118,000 words,3000 more than any other English dictionary. With the edition of 1890, the dictionary was retitled Websters International, the Collegiate Dictionary was introduced in 1898 and the series is now in its eleventh edition. Following the publication of Websters International in 1890, two Collegiate editions were issued as abridgments of each of their Unabridged editions, with the ninth edition, the Collegiate adopted changes which distinguish it as a separate entity rather than merely an abridgment of the Third New International. Some proper names were returned to the word list, including names of Knights of the Round Table, the most notable change was the inclusion of the date of the first known citation of each word, to document its entry into the English language. The eleventh edition includes more than 225,000 definitions, a CD-ROM of the text is sometimes included. This dictionary is preferred as a source for general matters of spelling by the influential The Chicago Manual of Style, the Chicago Manual states that it normally opts for the first spelling listed. Merriam overhauled the dictionary again with the 1961 Websters Third New International under the direction of Philip B, gove, making changes that sparked public controversy
Merriam-Webster Dictionary
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Merriam-Webster's eleventh edition of the Collegiate Dictionary
Merriam-Webster Dictionary
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Merriam-Webster
95.
Isoperimetric
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In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume. The equality holds when S is a ball in R n, on a plane, i. e. when n =2, the isoperimetric inequality relates square of the circumference of a closed curve and the area of a plane region it encloses. Isoperimetric literally means having the same perimeter, the isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length. The closely related Didos problem asks for a region of the area bounded by a straight line. It is named after Dido, the founder and first queen of Carthage. The solution to the problem is given by a circle and was known already in Ancient Greece. However, the first mathematically rigorous proof of this fact was obtained only in the 19th century, since then, many other proofs have been found. The isoperimetric problem has been extended in multiple ways, for example, to curves on surfaces, perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will assume a symmetric round shape. Since the amount of water in a drop is fixed, surface forces the drop into a shape which minimizes the surface area of the drop. The classical isoperimetric problem dates back to antiquity, the problem can be stated as follows, Among all closed curves in the plane of fixed perimeter, which curve maximizes the area of its enclosed region. This question can be shown to be equivalent to the problem, Among all closed curves in the plane enclosing a fixed area. German astronomer and astrologer Johannes Kepler invoked the principle in discussing the morphology of the solar system. Although the circle appears to be a solution to the problem. The first progress toward the solution was made by Swiss geometer Jakob Steiner in 1838, Steiner showed that if a solution existed, then it must be the circle. Steiners proof was completed later by other mathematicians. It can further be shown that any closed curve which is not fully symmetrical can be tilted so that it encloses more area. The one shape that is convex and symmetrical is the circle, although this, in itself
Isoperimetric
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If a region is not convex, a "dent" in its boundary can be "flipped" to increase the area of the region while keeping the perimeter unchanged.
96.
Ivor Grattan-Guinness
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Ivor Owen Grattan-Guinness was a historian of mathematics and logic. Grattan-Guinness was born in Bakewell, England, his father was a mathematics teacher and he gained his bachelor degree as a Mathematics Scholar at Wadham College, Oxford, and an MSc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966. He gained both the doctorate in 1969, and higher doctorate in 1978, in the History of Science at the University of London and he was Emeritus Professor of the History of Mathematics and Logic at Middlesex University, and a Visiting Research Associate at the London School of Economics. He was awarded the Kenneth O, in 2010, he was elected an Honorary Member of the Bertrand Russell Society. Grattan-Guinness spent much of his career at Middlesex University and he was a fellow at the Institute for Advanced Study in Princeton, New Jersey, United States, and a member of the International Academy of the History of Science. From 1974 to 1981, Grattan-Guinness was editor of the history of science journal Annals of Science, in 1979 he founded the journal History and Philosophy of Logic, and edited it until 1992. He was an editor of Historia Mathematica for twenty years from its inception in 1974. He also acted as editor to the editions of the writings of C. S. Peirce and Bertrand Russell. He was a member of the Executive Committee of the International Commission on the History of Mathematics from 1977 to 1993, Grattan-Guinness gave over 570 invited lectures to organisations and societies, or to conferences and congresses, in over 20 countries around the world. These lectures include tours undertaken in Australia, New Zealand, Italy, South Africa, from 1986 to 1988, Grattan-Guinness was the President of the British Society for the History of Mathematics, and for 1992 the Vice-President. In 1991, he was elected a member of the Académie Internationale dHistoire des Sciences. He was the Associate Editor for mathematicians and statisticians for the Oxford Dictionary of National Biography, Grattan-Guinness took an interest in the phenomenon of coincidence and has written on it for the Society for Psychical Research. He died of heart failure on 12 December 2014, aged 73 and he was especially interested in characterising how past thinkers, far removed from us in time, view their findings differently from the way we see them now. He has emphasised the importance of ignorance as a notion in this task. He did extensive research with original sources both published and unpublished, thanks to his reading and spoken knowledge of the main European languages, the Development of the Foundations of Mathematical Analysis from Euler to Riemann. Dear Russell—Dear Jourdain, a Commentary on Russells Logic, Based on His Correspondence with Philip Jourdain, from the Calculus to Set Theory, 1630–1910, An Introductory History. Psychical Research, A Guide to Its History, Principles & Practices - in celebration of 100 years of the Society for Psychical Research, Aquarian Press, convolutions in French Mathematics, 1800–1840 in 3 Vols. The Rainbow of Mathematics, A History of the Mathematical Sciences, from the Calculus to Set Theory 1630–1910, An Introductory History
Ivor Grattan-Guinness
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Ivor Grattan-Guinness in 2003.
97.
Cambridge University Press
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Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the worlds oldest publishing house and it also holds letters patent as the Queens Printer. The Presss mission is To further the Universitys mission by disseminating knowledge in the pursuit of education, learning, Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global presence, publishing hubs, and offices in more than 40 countries. Its publishing includes journals, monographs, reference works, textbooks. Cambridge University Press is an enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press and it originated from Letters Patent granted to the University of Cambridge by Henry VIII in 1534, and has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses, authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, and Stephen Hawking. In 1591, Thomass successor, John Legate, printed the first Cambridge Bible, the London Stationers objected strenuously, claiming that they had the monopoly on Bible printing. The universitys response was to point out the provision in its charter to print all manner of books. In July 1697 the Duke of Somerset made a loan of £200 to the university towards the house and presse and James Halman, Registrary of the University. It was in Bentleys time, in 1698, that a body of scholars was appointed to be responsible to the university for the Presss affairs. The Press Syndicates publishing committee still meets regularly, and its role still includes the review, John Baskerville became University Printer in the mid-eighteenth century. Baskervilles concern was the production of the finest possible books using his own type-design, a technological breakthrough was badly needed, and it came when Lord Stanhope perfected the making of stereotype plates. This involved making a mould of the surface of a page of type. The Press was the first to use this technique, and in 1805 produced the technically successful, under the stewardship of C. J. Clay, who was University Printer from 1854 to 1882, the Press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks, during Clays administration, the Press also undertook a sizable co-publishing venture with Oxford, the Revised Version of the Bible, which was begun in 1870 and completed in 1885. It was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories, the Cambridge Modern History was published between 1902 and 1912
Cambridge University Press
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The University Printing House, on the main site of the Press
Cambridge University Press
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The letters patent of Cambridge University Press by Henry VIII allow the Press to print "all manner of books". The fine initial with the king's portrait inside it and the large first line of script are still discernible.
Cambridge University Press
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The Pitt Building in Cambridge, which used to be the headquarters of Cambridge University Press, and now serves as a conference centre for the Press.
Cambridge University Press
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On the main site of the Press
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Project Gutenberg
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Project Gutenberg is a volunteer effort to digitize and archive cultural works, to encourage the creation and distribution of eBooks. It was founded in 1971 by Michael S. Hart and is the oldest digital library, most of the items in its collection are the full texts of public domain books. The project tries to make these as free as possible, in long-lasting, as of 3 October 2015, Project Gutenberg reached 50,000 items in its collection. The releases are available in plain text but, wherever possible, other formats are included, such as HTML, PDF, EPUB, MOBI, most releases are in the English language, but many non-English works are also available. There are multiple affiliated projects that are providing additional content, including regional, Project Gutenberg is also closely affiliated with Distributed Proofreaders, an Internet-based community for proofreading scanned texts. Project Gutenberg was started by Michael Hart in 1971 with the digitization of the United States Declaration of Independence, Hart, a student at the University of Illinois, obtained access to a Xerox Sigma V mainframe computer in the universitys Materials Research Lab. Through friendly operators, he received an account with an unlimited amount of computer time. Hart has said he wanted to back this gift by doing something that could be considered to be of great value. His initial goal was to make the 10,000 most consulted books available to the public at little or no charge and this particular computer was one of the 15 nodes on ARPANET, the computer network that would become the Internet. Hart believed that computers would one day be accessible to the general public and he used a copy of the United States Declaration of Independence in his backpack, and this became the first Project Gutenberg e-text. He named the project after Johannes Gutenberg, the fifteenth century German printer who propelled the movable type printing press revolution, by the mid-1990s, Hart was running Project Gutenberg from Illinois Benedictine College. More volunteers had joined the effort, all of the text was entered manually until 1989 when image scanners and optical character recognition software improved and became more widely available, which made book scanning more feasible. Hart later came to an arrangement with Carnegie Mellon University, which agreed to administer Project Gutenbergs finances, as the volume of e-texts increased, volunteers began to take over the projects day-to-day operations that Hart had run. Starting in 2004, an online catalog made Project Gutenberg content easier to browse, access. Project Gutenberg is now hosted by ibiblio at the University of North Carolina at Chapel Hill, Italian volunteer Pietro Di Miceli developed and administered the first Project Gutenberg website and started the development of the Project online Catalog. In his ten years in this role, the Project web pages won a number of awards, often being featured in best of the Web listings, Hart died on 6 September 2011 at his home in Urbana, Illinois at the age of 64. In 2000, a corporation, the Project Gutenberg Literary Archive Foundation. Long-time Project Gutenberg volunteer Gregory Newby became the foundations first CEO, also in 2000, Charles Franks founded Distributed Proofreaders, which allowed the proofreading of scanned texts to be distributed among many volunteers over the Internet
Project Gutenberg
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Michael Hart (left) and Gregory Newby (right) of Project Gutenberg, 2006
Project Gutenberg
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Project Gutenberg
Project Gutenberg
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Formats
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Internet Archive
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The Internet Archive is a San Francisco–based nonprofit digital library with the stated mission of universal access to all knowledge. As of October 2016, its collection topped 15 petabytes, in addition to its archiving function, the Archive is an activist organization, advocating for a free and open Internet. Its web archive, the Wayback Machine, contains over 150 billion web captures, the Archive also oversees one of the worlds largest book digitization projects. Founded by Brewster Kahle in May 1996, the Archive is a 501 nonprofit operating in the United States. It has a budget of $10 million, derived from a variety of sources, revenue from its Web crawling services, various partnerships, grants, donations. Its headquarters are in San Francisco, California, where about 30 of its 200 employees work, Most of its staff work in its book-scanning centers. The Archive has data centers in three Californian cities, San Francisco, Redwood City, and Richmond, the Archive is a member of the International Internet Preservation Consortium and was officially designated as a library by the State of California in 2007. Brewster Kahle founded the Archive in 1996 at around the time that he began the for-profit web crawling company Alexa Internet. In October 1996, the Internet Archive had begun to archive and preserve the World Wide Web in large quantities, the archived content wasnt available to the general public until 2001, when it developed the Wayback Machine. In late 1999, the Archive expanded its collections beyond the Web archive, Now the Internet Archive includes texts, audio, moving images, and software. It hosts a number of projects, the NASA Images Archive, the contract crawling service Archive-It. According to its web site, Most societies place importance on preserving artifacts of their culture, without such artifacts, civilization has no memory and no mechanism to learn from its successes and failures. Our culture now produces more and more artifacts in digital form, the Archives mission is to help preserve those artifacts and create an Internet library for researchers, historians, and scholars. In August 2012, the Archive announced that it has added BitTorrent to its file download options for over 1.3 million existing files, on November 6,2013, the Internet Archives headquarters in San Franciscos Richmond District caught fire, destroying equipment and damaging some nearby apartments. The nonprofit Archive sought donations to cover the estimated $600,000 in damage, in November 2016, Kahle announced that the Internet Archive was building the Internet Archive of Canada, a copy of the archive to be based somewhere in the country of Canada. The announcement received widespread coverage due to the implication that the decision to build an archive in a foreign country was because of the upcoming presidency of Donald Trump. Kahle was quoted as saying that on November 9th in America and it was a firm reminder that institutions like ours, built for the long-term, need to design for change. For us, it means keeping our cultural materials safe, private and it means preparing for a Web that may face greater restrictions
Internet Archive
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Since 2009, headquarters have been at 300 Funston Avenue in San Francisco, a former Christian Science Church
Internet Archive
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Internet Archive
Internet Archive
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Mirror of the Internet Archive in the Bibliotheca Alexandrina
Internet Archive
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From 1996 to 2009, headquarters were in the Presidio of San Francisco, a former U.S. military base
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Virtual International Authority File
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The Virtual International Authority File is an international authority file. It is a joint project of national libraries and operated by the Online Computer Library Center. The project was initiated by the US Library of Congress, the German National Library, the National Library of France joined the project on October 5,2007. The project transitions to a service of the OCLC on April 4,2012, the aim is to link the national authority files to a single virtual authority file. In this file, identical records from the different data sets are linked together, a VIAF record receives a standard data number, contains the primary see and see also records from the original records, and refers to the original authority records. The data are available online and are available for research and data exchange. Reciprocal updating uses the Open Archives Initiative Protocol for Metadata Harvesting protocol, the file numbers are also being added to Wikipedia biographical articles and are incorporated into Wikidata. VIAFs clustering algorithm is run every month, as more data are added from participating libraries, clusters of authority records may coalesce or split, leading to some fluctuation in the VIAF identifier of certain authority records
Virtual International Authority File
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Screenshot 2012
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National Library of Australia
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In 2012–2013, the National Library collection comprised 6,496,772 items, and an additional 15,506 metres of manuscript material. In 1901, a Commonwealth Parliamentary Library was established to serve the newly formed Federal Parliament of Australia, from its inception the Commonwealth Parliamentary Library was driven to development of a truly national collection. The present library building was opened in 1968, the building was designed by the architectural firm of Bunning and Madden. The foyer is decorated in marble, with windows by Leonard French. In 2012–2013 the Library collection comprised 6,496,772 items, the Librarys collections of Australiana have developed into the nations single most important resource of materials recording the Australian cultural heritage. Australian writers, editors and illustrators are actively sought and well represented—whether published in Australia or overseas, approximately 92. 1% of the Librarys collection has been catalogued and is discoverable through the online catalogue. The Library has digitized over 174,000 items from its collection and, the Library is a world leader in digital preservation techniques, and maintains an Internet-accessible archive of selected Australian websites called the Pandora Archive. A core Australiana collection is that of John A. Ferguson, the Library has particular collection strengths in the performing arts, including dance. The Librarys considerable collections of general overseas and rare materials, as well as world-class Asian. The print collections are further supported by extensive microform holdings, the Library also maintains the National Reserve Braille Collection. The Library has acquired a number of important Western and Asian language scholarly collections from researchers, williams Collection The Asian Collections are searchable via the National Librarys catalogue. The National Library holds a collection of pictures and manuscripts. The manuscript collection contains about 26 million separate items, covering in excess of 10,492 meters of shelf space, the collection relates predominantly to Australia, but there are also important holdings relating to Papua New Guinea, New Zealand and the Pacific. The collection also holds a number of European and Asian manuscript collections or single items have received as part of formed book collections. Examples are the papers of Alfred Deakin, Sir John Latham, Sir Keith Murdoch, Sir Hans Heysen, Sir John Monash, Vance Palmer and Nettie Palmer, A. D. Hope, Manning Clark, David Williamson, W. M. The Library has also acquired the records of many national non-governmental organisations and they include the records of the Federal Secretariats of the Liberal party, the A. L. P, the Democrats, the R. S. L. Finally, the Library holds about 37,000 reels of microfilm of manuscripts and archival records, mostly acquired overseas and predominantly of Australian, the National Librarys Pictures collection focuses on Australian people, places and events, from European exploration of the South Pacific to contemporary events. Art works and photographs are acquired primarily for their informational value, media represented in the collection include photographs, drawings, watercolours, oils, lithographs, engravings, etchings and sculpture/busts
National Library of Australia
National Library of Australia
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National Library of Australia as viewed from Lake Burley Griffin, Canberra
National Library of Australia
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The original National Library building on Kings Avenue, Canberra, was designed by Edward Henderson. Originally intended to be several wings, only one wing was completed and was demolished in 1968. Now the site of the Edmund Barton Building.
National Library of Australia
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The library seen from Lake Burley Griffin in autumn.
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National Diet Library
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The National Diet Library is the only national library in Japan. It was established in 1948 for the purpose of assisting members of the National Diet of Japan in researching matters of public policy, the library is similar in purpose and scope to the United States Library of Congress. The National Diet Library consists of two facilities in Tokyo and Kyoto, and several other branch libraries throughout Japan. The Diets power in prewar Japan was limited, and its need for information was correspondingly small, the original Diet libraries never developed either the collections or the services which might have made them vital adjuncts of genuinely responsible legislative activity. Until Japans defeat, moreover, the executive had controlled all political documents, depriving the people and the Diet of access to vital information. The U. S. occupation forces under General Douglas MacArthur deemed reform of the Diet library system to be an important part of the democratization of Japan after its defeat in World War II. In 1946, each house of the Diet formed its own National Diet Library Standing Committee, hani Gorō, a Marxist historian who had been imprisoned during the war for thought crimes and had been elected to the House of Councillors after the war, spearheaded the reform efforts. Hani envisioned the new body as both a citadel of popular sovereignty, and the means of realizing a peaceful revolution, the National Diet Library opened in June 1948 in the present-day State Guest-House with an initial collection of 100,000 volumes. The first Librarian of the Diet Library was the politician Tokujirō Kanamori, the philosopher Masakazu Nakai served as the first Vice Librarian. In 1949, the NDL merged with the National Library and became the national library in Japan. At this time the collection gained a million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, in 1986, the NDLs Annex was completed to accommodate a combined total of 12 million books and periodicals. The Kansai-kan, which opened in October 2002 in the Kansai Science City, has a collection of 6 million items, in May 2002, the NDL opened a new branch, the International Library of Childrens Literature, in the former building of the Imperial Library in Ueno. This branch contains some 400,000 items of literature from around the world. Though the NDLs original mandate was to be a library for the National Diet. In the fiscal year ending March 2004, for example, the library reported more than 250,000 reference inquiries, in contrast, as Japans national library, the NDL collects copies of all publications published in Japan. The NDL has an extensive collection of some 30 million pages of documents relating to the Occupation of Japan after World War II. This collection include the documents prepared by General Headquarters and the Supreme Commander of the Allied Powers, the Far Eastern Commission, the NDL maintains a collection of some 530,000 books and booklets and 2 million microform titles relating to the sciences
National Diet Library
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Tokyo Main Library of the National Diet Library
National Diet Library
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Kansai-kan of the National Diet Library
National Diet Library
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The National Diet Library
National Diet Library
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Main building in Tokyo
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National Library of the Czech Republic
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The National Library of the Czech Republic is the central library of the Czech Republic. It is directed by the Ministry of Culture, the librarys main building is located in the historical Clementinum building in Prague, where approximately half of its books are kept. The other half of the collection is stored in the district of Hostivař, the National Library is the biggest library in the Czech Republic, in its funds there are around 6 million documents. The library has around 60,000 registered readers, as well as Czech texts, the library also stores older material from Turkey, Iran and India. The library also houses books for Charles University in Prague, the library won international recognition in 2005 as it received the inaugural Jikji Prize from UNESCO via the Memory of the World Programme for its efforts in digitising old texts. The project, which commenced in 1992, involved the digitisation of 1,700 documents in its first 13 years, the most precious medieval manuscripts preserved in the National Library are the Codex Vyssegradensis and the Passional of Abbes Kunigunde. In 2006 the Czech parliament approved funding for the construction of a new building on Letna plain. In March 2007, following a request for tender, Czech architect Jan Kaplický was selected by a jury to undertake the project, later in 2007 the project was delayed following objections regarding its proposed location from government officials including Prague Mayor Pavel Bém and President Václav Klaus. Later in 2008, Minister of Culture Václav Jehlička announced the end of the project, the library was affected by the 2002 European floods, with some documents moved to upper levels to avoid the excess water. Over 4,000 books were removed from the library in July 2011 following flooding in parts of the main building, there was a fire at the library in December 2012, but nobody was injured in the event. List of national and state libraries Official website
National Library of the Czech Republic
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Baroque library hall in the National Library of the Czech Republic
National Library of the Czech Republic
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General reading room (former refectory of the Jesuit residence in Clementinum)