1.
Kingdom of Sardinia
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The Kingdom of Sardinia was a state in Southern Europe which existed from the early 14th until the mid-19th century. It was the state of todays Italy. When it was acquired by the Duke of Savoy in 1720, however, the Savoyards united it with their possessions on the Italian mainland and, by the time of the Crimean War in 1853, had built the resulting kingdom into a strong power. The formal name of the entire Savoyard state was the States of His Majesty the King of Sardinia and its final capital was Turin, the capital of Savoy since the Middle Ages. Beginning in 1324, James and his successors conquered the island of Sardinia, in 1420 the last competing claim to the island was bought out. After the union of the crowns of Aragon and Castile, Sardinia became a part of the burgeoning Spanish Empire, in 1720 it was ceded by the Habsburg and Bourbon claimants to the Spanish throne to Duke Victor Amadeus II of Savoy. While in theory the traditional capital of the island of Sardinia and seat of its viceroys was Cagliari, the Congress of Vienna, which restructured Europe after Napoleons defeat, returned to Savoy its mainland possessions and augmented them with Liguria, taken from the Republic of Genoa. In 1847–48, in a fusion, the various Savoyard states were unified under one legal system, with the capital in Turin, and granted a constitution. There followed the annexation of Lombardy, the central Italian states and the Two Sicilies, Venetia, in 238 BC Sardinia became, along with Corsica, a province of the Roman Empire. The Romans ruled the island until the middle of the 5th century, when it was occupied by the Vandals, in 534 AD it was reconquered by the Romans, but now from the Eastern Roman Empire, Byzantium. It remained a Byzantine province until the Arab conquest of Sicily in the 9th century, after that, communications with Constantinople became very difficult, and powerful families of the island assumed control of the land. Starting from 705–706, Saracens from north Africa harassed the population of the coastal cities, information about the Sardinian political situation in the following centuries is scarce. There is a record of another massive Saracen sea attack in 1015–16 from the Balearics, the Saracen attempt to invade the island was stopped by the Judicatus with the support of the fleets of the maritime republics of Pisa and Genoa, free cities of the Holy Roman Empire. Pope Benedict VIII also requested aid from the republics of Pisa. Even the title of Judices was a Byzantine reminder of the Greek church and state, of these sovereigns only two names are known, Turcoturiu and Salusiu, who probably ruled in the 10th century. The Archons still wrote in Greek or Latin, but one of the first documents of the Judex of Cagliari, their successor, was written in romance Sardinian language. The realm was divided into four kingdoms, the Judicati, perfectly organized as was the previous realm, but was now under the influence of the Pope. That was the cause of leading to a long war between the Judices, who regarded themselves as kings fighting against rebellious nobles
Kingdom of Sardinia
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A map of the Kingdom of Sardinia in 1856, after the fusion of all its provinces into a single jurisdiction
Kingdom of Sardinia
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The final flag used by the kingdom under the " Perfect Fusion " (1848–1861)
Kingdom of Sardinia
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Flag of the Kingdom of Sardinia in the middle of the 16th century
Kingdom of Sardinia
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Kingdom of Sardinia 16th-century map
2.
Piedmont
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Piedmont is one of the 20 regions of Italy. It has an area of 25,402 square kilometres and a population of about 4.6 million, the capital of Piedmont is Turin. The name Piedmont comes from medieval Latin Pedemontium or Pedemontis, i. e. ad pedem montium, meaning “at the foot of the mountains”. Other towns of Piedmont with more than 20,000 inhabitants sorted by population and it borders with France, Switzerland and the Italian regions of Lombardy, Liguria, Aosta Valley and for a very small fragment with Emilia Romagna. The geography of Piedmont is 43. 3% mountainous, along with areas of hills. Piedmont is the second largest of Italys 20 regions, after Sicily and it is broadly coincident with the upper part of the drainage basin of the river Po, which rises from the slopes of Monviso in the west of the region and is Italy’s largest river. The Po collects all the waters provided within the semicircle of mountains which surround the region on three sides, from the highest peaks the land slopes down to hilly areas, and then to the upper, and then to the lower great Padan Plain. 7. 6% of the territory is considered protected area. There are 56 different national or regional parks, one of the most famous is the Gran Paradiso National Park located between Piedmont and the Aosta Valley, Piedmont was inhabited in early historic times by Celtic-Ligurian tribes such as the Taurini and the Salassi. They were later subdued by the Romans, who founded several colonies there including Augusta Taurinorum, after the fall of the Western Roman Empire, the region was repeatedly invaded by the Burgundians, the Goths, Byzantines, Lombards, Franks. In the 9th–10th centuries there were incursions by the Magyars. At the time Piedmont, as part of the Kingdom of Italy within the Holy Roman Empire, was subdivided into several marks, in 1046, Oddo of Savoy added Piedmont to their main territory of Savoy, with a capital at Chambéry. Other areas remained independent, such as the powerful comuni of Asti and Alessandria, the County of Savoy was elevated to a duchy in 1416, and Duke Emanuele Filiberto moved the seat to Turin in 1563. In 1720, the Duke of Savoy became King of Sardinia, founding what evolved into the Kingdom of Sardinia, the Republic of Alba was created in 1796 as a French client republic in Piedmont. A new client republic, the Piedmontese Republic, existed between 1798 and 1799 before it was reoccupied by Austrian and Russian troops, in June 1800 a third client republic, the Subalpine Republic, was established in Piedmont. It fell under full French control in 1801 and it was annexed by France in September 1802, in the congress of Vienna, the Kingdom of Sardinia was restored, and furthermore received the Republic of Genoa to strengthen it as a barrier against France. Piedmont was a springboard for Italys unification in 1859–1861, following earlier unsuccessful wars against the Austrian Empire in 1820–1821 and this process is sometimes referred to as Piedmontisation. However, the efforts were countered by the efforts of rural farmers
Piedmont
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A Montferrat landscape, with the distant Alps in the background.
Piedmont
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Piedmont Piemonte
Piedmont
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The Palazzina di caccia of Stupinigi, in Nichelino, is a UNESCO World Heritage Site.
Piedmont
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The Kingdom of Sardinia in 1856.
3.
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)
Prussia
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... according to the design of 1702
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Prussian King's Crown (Hohenzollern Castle Collection)
4.
First 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
First French Empire
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The Battle of Austerlitz
First French Empire
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Flag
First French Empire
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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.
First French Empire
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Napoleon reviews the Imperial Guard before the Battle of Jena, 1806
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
Alma mater
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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.
Joseph Fourier
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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
Giovanni Antonio Amedeo Plana
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Giovanni Antonio Amedeo Plana.
9.
Analytical mechanics
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In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by scientists and mathematicians during the 18th century and onward. A scalar is a quantity, whereas a vector is represented by quantity, the equations of motion are derived from the scalar quantity by some underlying principle about the scalars variation. Analytical mechanics takes advantage of a systems constraints to solve problems, the constraints limit the degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates and it does not always work for non-conservative forces or dissipative forces like friction, in which case one may revert to Newtonian mechanics or use the Udwadia–Kalaba equation. Two dominant branches of mechanics are Lagrangian mechanics and Hamiltonian mechanics. There are other such as Hamilton–Jacobi theory, Routhian mechanics. All equations of motion for particles and fields, in any formalism, one result is Noethers theorem, a statement which connects conservation laws to their associated symmetries. Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics, rather it is a collection of equivalent formalisms which have broad application. In fact the principles and formalisms can be used in relativistic mechanics and general relativity. Analytical mechanics is used widely, from physics to applied mathematics. The methods of analytical mechanics apply to particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom, the definitions and equations have a close analogy with those of mechanics. Generalized coordinates and constraints In Newtonian mechanics, one customarily uses all three Cartesian coordinates, or other 3D coordinate system, to refer to a position during its motion. In physical systems, however, some structure or other system usually constrains the motion from taking certain directions. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motions geometry and these are known as generalized coordinates, denoted qi. Difference between curvillinear and generalized coordinates Generalized coordinates incorporate constraints on the system, there is one generalized coordinate qi for each degree of freedom, i. e. each way the system can change its configuration, as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates, DAlemberts principle The foundation which the subject is built on is DAlemberts principle
Analytical mechanics
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As the system evolves, q traces a path through configuration space (only some are shown). The path taken by the system (red) has a stationary action (δ S = 0) under small changes in the configuration of the system (δ q).
10.
Berlin
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Berlin is the capital and the largest city of Germany as well as one of its constituent 16 states. With a population of approximately 3.5 million, Berlin is the second most populous city proper, due to its location in the European Plain, Berlin is influenced by a temperate seasonal climate. Around one-third of the area is composed of forests, parks, gardens, rivers. Berlin in the 1920s was the third largest municipality in the world, following German reunification in 1990, Berlin once again became the capital of all-Germany. Berlin is a city of culture, politics, media. Its economy is based on high-tech firms and the sector, encompassing a diverse range of creative industries, research facilities, media corporations. Berlin serves as a hub for air and rail traffic and has a highly complex public transportation network. The metropolis is a popular tourist destination, significant industries also include IT, pharmaceuticals, biomedical engineering, clean tech, biotechnology, construction and electronics. Modern Berlin is home to world renowned universities, orchestras, museums and its urban setting has made it a sought-after location for international film productions. The city is known for its festivals, diverse architecture, nightlife, contemporary arts. Since 2000 Berlin has seen the emergence of a cosmopolitan entrepreneurial scene, the name Berlin has its roots in the language of West Slavic inhabitants of the area of todays Berlin, and may be related to the Old Polabian stem berl-/birl-. All German place names ending on -ow, -itz and -in, since the Ber- at the beginning sounds like the German word Bär, a bear appears in the coat of arms of the city. It is therefore a canting arm, the first written records of towns in the area of present-day Berlin date from the late 12th century. Spandau is first mentioned in 1197 and Köpenick in 1209, although these areas did not join Berlin until 1920, the central part of Berlin can be traced back to two towns. Cölln on the Fischerinsel is first mentioned in a 1237 document,1237 is considered the founding date of the city. The two towns over time formed close economic and social ties, and profited from the right on the two important trade routes Via Imperii and from Bruges to Novgorod. In 1307, they formed an alliance with a common external policy, in 1415 Frederick I became the elector of the Margraviate of Brandenburg, which he ruled until 1440. In 1443 Frederick II Irontooth started the construction of a new palace in the twin city Berlin-Cölln
Berlin
Berlin
Berlin
Berlin
11.
French Academy of Sciences
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The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the forefront of developments in Europe in the 17th and 18th centuries. Currently headed by Sébastien Candel, it is one of the five Academies of the Institut de France, the Academy of Sciences makes its origin to Colberts plan to create a general academy. He chose a group of scholars who met on 22 December 1666 in the Kings library. The first 30 years of the Academys existence were relatively informal, in contrast to its British counterpart, the Academy was founded as an organ of government. The Academy was expected to remain apolitical, and to avoid discussion of religious, on 20 January 1699, Louis XIV gave the Company its first rules. The Academy received the name of Royal Academy of Sciences and was installed in the Louvre in Paris, following this reform, the Academy began publishing a volume each year with information on all the work done by its members and obituaries for members who had died. This reform also codified the method by which members of the Academy could receive pensions for their work, on 8 August 1793, the National Convention abolished all the academies. Almost all the old members of the previously abolished Académie were formally re-elected, among the exceptions was Dominique, comte de Cassini, who refused to take his seat. In 1816, the again renamed Royal Academy of Sciences became autonomous, while forming part of the Institute of France, in the Second Republic, the name returned to Académie des sciences. During this period, the Academy was funded by and accountable to the Ministry of Public Instruction, the Academy came to control French patent laws in the course of the eighteenth century, acting as the liaison of artisans knowledge to the public domain. As a result, academicians dominated technological activities in France, the Academy proceedings were published under the name Comptes rendus de lAcadémie des sciences. The Comptes rendus is now a series with seven titles. The publications can be found on site of the French National Library, in 1818 the French Academy of Sciences launched a competition to explain the properties of light. The civil engineer Augustin-Jean Fresnel entered this competition by submitting a new theory of light. Siméon Denis Poisson, one of the members of the judging committee, being a supporter of the particle-theory of light, he looked for a way to disprove it. The Poisson spot is not easily observed in every-day situations, so it was natural for Poisson to interpret it as an absurd result. However, the head of the committee, Dominique-François-Jean Arago, and he molded a 2-mm metallic disk to a glass plate with wax
French Academy of Sciences
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A heroic depiction of the activities of the Academy from 1698
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Colbert Presenting the Members of the Royal Academy of Sciences to Louis XIV in 1667
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The Institut de France in Paris where the Academy is housed
12.
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
13.
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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Flag
14.
Bureau des Longitudes
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During the 19th century, it was responsible for synchronizing clocks across the world. It was headed during this time by François Arago and Henri Poincaré, the Bureau now functions as an academy and still meets monthly to discuss topics related to astronomy. The Bureau was founded by the National Convention after it heard a report drawn up jointly by the Committee of Navy, the Committee of Finances, as a result, the Bureau was established with authority over the Paris Observatory and all other astronomical establishments throughout France. The Bureau was charged with taking control of the seas away from the English and improving accuracy when tracking the longitudes of ships through astronomical observations, by a decree of 30 January 1854, the Bureaus mission was extended to embrace geodesy, time standardisation and astronomical measurements. This decree granted independence to the Paris Observatory, separating it from the Bureau, the Bureau was successful at setting a universal time in Paris via air pulses sent through pneumatic tubes. It later worked to synchronize time across the French colonial empire by determining the length of time for a signal to make a trip to. The French Bureau of Longitude established a commission in the year 1897 to extend the system to the measurement of time. They planned to abolish the antiquated division of the day hours, minutes, and seconds, and replace it by a division into tenths, thousandths. This was a revival of a dream that was in the minds of the creators of the system at the time of the French Revolution a hundred years earlier. Some members of the Bureau of Longitude commission introduced a proposal, retaining the old-fashioned hour as the basic unit of time. Poincaré served as secretary of the commission and took its work very seriously and he was a fervent believer in a universal metric system. The rest of the world outside France gave no support to the proposals. After three years of work, the commission was dissolved in 1900. Since 1970, the board has been constituted with 13 members,3 nominated by the Académie des Sciences, since 1998, practical work has been carried out by the Institut de mécanique céleste et de calcul des éphémérides. Institut de mécanique céleste et de calcul des éphémérides Bureau Des Longitudes Galison, einsteins Clocks, Poincarés Maps, Empires of Time
Bureau des Longitudes
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ABBE GREGOIRE (1750-1831).
15.
Differential equation
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A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, if a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli. In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
Differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
16.
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.
17.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
Group theory
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Water molecule with symmetry axis
Group theory
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The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation groups.
18.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
Calculus
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Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Calculus
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Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
Calculus
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Maria Gaetana Agnesi
Calculus
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The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
19.
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
20.
Lagrangian point
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The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centrifugal force required to orbit with them. There are five points, labeled L1 to L5, all in the orbital plane of the two large bodies. The first three are on the line connecting the two bodies, the last two, L4 and L5, each form an equilateral triangle with the two large bodies. The two latter points are stable, which implies that objects can orbit around them in a coordinate system tied to the two large bodies. Several planets have satellites near their L4 and L5 points with respect to the Sun, the three collinear Lagrange points were discovered by Leonhard Euler a few years before Lagrange discovered the remaining two. In 1772, Joseph-Louis Lagrange published an Essay on the three-body problem, in the first chapter he considered the general three-body problem. From that, in the chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits. The five Lagrangian points are labeled and defined as follows, The L1 point lies on the line defined by the two large masses M1 and M2, and between them. It is the most intuitively understood of the Lagrangian points, the one where the attraction of M2 partially cancels M1s gravitational attraction. Explanation An object that orbits the Sun more closely than Earth would normally have an orbital period than Earth. If the object is directly between Earth and the Sun, then Earths gravity counteracts some of the Suns pull on the object, the closer to Earth the object is, the greater this effect is. At the L1 point, the period of the object becomes exactly equal to Earths orbital period. L1 is about 1.5 million kilometers from Earth, the L2 point lies on the line through the two large masses, beyond the smaller of the two. Here, the forces of the two large masses balance the centrifugal effect on a body at L2. Explanation On the opposite side of Earth from the Sun, the period of an object would normally be greater than that of Earth. The extra pull of Earths gravity decreases the orbital period of the object, like L1, L2 is about 1.5 million kilometers from Earth. The L3 point lies on the line defined by the two masses, beyond the larger of the two. Explanation L3 in the Sun–Earth system exists on the side of the Sun
Lagrangian point
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Visualisation of the relationship between the Lagrangian points (red) of a planet (blue) orbiting a star (yellow) anticlockwise, and the effective potential in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential). Click for animation.
Lagrangian point
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Lagrange points in the Sun–Earth system (not to scale)
21.
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)
22.
W.W. Rouse Ball
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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
W.W. Rouse Ball
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W.W. Rouse Ball
23.
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.
24.
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
25.
Charles Emmanuel III
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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
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Charles Emmanuel III
Charles Emmanuel III
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The young Charles Emmanuel as duke of Aosta, by an unknown artist.
Charles Emmanuel III
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A portrait of a young Charles Emmanuel
Charles Emmanuel III
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The children of Charles and his second wife; (L-R) Eleonora; Victor Amadeus; Maria Felicita and Maria Luisa Gabriella.
26.
Echo (phenomenon)
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In audio signal processing and acoustics, echo is a reflection of sound that arrives at the listener with a delay after the direct sound. The delay is proportional to the distance of the surface from the source. Typical examples are the echo produced by the bottom of a well, by a building, or by the walls of an enclosed room, a true echo is a single reflection of the sound source. The word echo derives from the Greek ἠχώ, itself from ἦχος, echo in the folk story of Greek is a mountain nymph whose ability to speak was cursed, only able to repeat the last words anyone spoke to her. Some animals use echo for location sensing and navigation, such as cetaceans, acoustic waves are reflected by walls or other hard surfaces, such as mountains and privacy fences. The reason of reflection may be explained as a discontinuity in the propagation medium and this can be heard when the reflection returns with sufficient magnitude and delay to be perceived distinctly. When sound, or the echo itself, is reflected multiple times from multiple surfaces, the human ear cannot distinguish echo from the original direct sound if the delay is less than 1/15 of a second. The velocity of sound in dry air is approximately 343 m/s at a temperature of 25 °C, therefore, the reflecting object must be more than 17. 2m from the sound source for echo to be perceived by a person located at the source. When a sound produces an echo in two seconds, the object is 343m away. In nature, canyon walls or rock cliffs facing water are the most common settings for hearing echoes. The strength of echo is frequently measured in dB sound pressure level relative to the transmitted wave. Echoes may be desirable or undesirable, in music performance and recording, electric echo effects have been used since the 1950s. The Echoplex is a delay effect, first made in 1959 that recreates the sound of an acoustic echo. Designed by Mike Battle, the Echoplex set a standard for the effect in the 1960s and was used by most of the guitar players of the era. While Echoplexes were used heavily by guitar players, many recording studios used the Echoplex. Beginning in the 1970s, Market built the solid-state Echoplex for Maestro, in the 2000s, most echo effects units use electronic or digital circuitry to recreate the echo effect. Hamilton Mausoleum, Hamilton, South Lanarkshire, Scotland, Its high stone holds the record for the longest echo in the world, gol Gumbaz of Bijapur, India, Any whisper, clap or sound gets echoed repeatedly. The gazebo of Napier Museum in Trivandrum, Kerala, India, listen to Duck echoes and an animated demonstration of how an echo is formed
Echo (phenomenon)
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This illustration depicts the principle of sediment echo sounding, which uses a narrow beam of high energy and low frequency
27.
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
28.
Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
Series (mathematics)
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Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.
29.
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
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Gerolamo Cardano
Probability
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Carl Friedrich Gauss
30.
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.
31.
Pierre de Fermat
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He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermats principle for light propagation and his Fermats Last Theorem in number theory, Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a leather merchant. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth, there is little evidence concerning his school education, but it was probably at the Collège de Navarre in Montauban. He attended the University of Orléans from 1623 and received a bachelor in law in 1626. In Bordeaux he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apolloniuss De Locis Planis to one of the mathematicians there, there he became much influenced by the work of François Viète. In 1630, he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France and he held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat, fluent in six languages, Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often little or no proof of his theorems. In some of these letters to his friends he explored many of the ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession, nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus. Secrecy was common in European mathematical circles at the time and this naturally led to priority disputes with contemporaries such as Descartes and Wallis. Anders Hald writes that, The basis of Fermats mathematics was the classical Greek treatises combined with Vietas new algebraic methods, Fermats pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes famous La géométrie. This manuscript was published posthumously in 1679 in Varia opera mathematica, in these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of power functions. With his method, he was able to reduce this evaluation to the sum of geometric series, the resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pells equation, perfect numbers, amicable numbers and it was while researching perfect numbers that he discovered Fermats little theorem. Fermat developed the two-square theorem, and the polygonal number theorem, although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived
Pierre de Fermat
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Pierre de Fermat
Pierre de Fermat
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Bust in the Salle des Illustres in Capitole de Toulouse
Pierre de Fermat
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Place of burial of Pierre de Fermat in Place Jean Jaurés, Castres. Translation of the plaque: in this place was buried on January 13, 1665, Pierre de Fermat, councilor of the chamber of Edit [Parlement of Toulouse] and mathematician of great renown, celebrated for his theorem, a n + b n ≠ c n for n>2
Pierre de Fermat
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Holographic will handwritten by Fermat on 4 March 1660 — kept at the Departmental Archives of Haute-Garonne, in Toulouse
32.
Virtual work
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Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements, among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the according to the principle of least action. The work of a force on a particle along a displacement is known as the virtual work. The principle of work had always been used in some form since antiquity in the study of statics. It was used by the Greeks, medieval Arabs and Latins, working with Leibnizian concepts, Johann Bernoulli systematized the virtual work principle and made explicit the concept of infinitesimal displacement. He was able to solve problems for both bodies as well as fluids. Bernoullis version of virtual work law appeared in his letter to Pierre Varignon in 1715 and this formulation of the principle is today known as the principle of virtual velocities and is commonly considered as the prototype of the contemporary virtual work principles. In 1743 DAlembert published his Traite de Dynamique where he applied the principle of work, based on the Bernoullis work. His idea was to convert a dynamical problem into static problem by introducing inertial force, consider a point particle that moves along a path which is described by a function r from point A, where r, to point B, where r. It is possible that the moves from A to B along a nearby path described by r + δr. The variation δr satisfies the requirement δr = δr =0, the components of the variation, δr1, δr2 and δr3, are called virtual displacements. This can be generalized to a mechanical system defined by the generalized coordinates qi. In which case, the variation of the qi is defined by the virtual displacements δqi. Virtual work is the work done by the applied forces. When considering forces applied to a body in equilibrium, the principle of least action requires the virtual work of these forces to be zero. Consider a particle P that moves from a point A to a point B along a trajectory r and it is important to notice that the value of the work W depends on the trajectory r. Suppose the force F is the same as F, the variation of the work δW associated with this nearby path, known as the virtual work, can be computed to be δ W = W ¯ − W = ∫ t 0 t 1 d t
Virtual work
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This is an engraving from Mechanics Magazine published in London in 1824.
Virtual work
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Illustration from Army Service Corps Training on Mechanical Transport, (1911), Fig. 112 Transmission of motion and force by gear wheels, compound train
33.
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
34.
Saint Petersburg
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Saint Petersburg is Russias second-largest city after Moscow, with five million inhabitants in 2012, and an important Russian port on the Baltic Sea. It is politically incorporated as a federal subject, situated on the Neva River, at the head of the Gulf of Finland on the Baltic Sea, it was founded by Tsar Peter the Great on May 271703. In 1914, the name was changed from Saint Petersburg to Petrograd, in 1924 to Leningrad, between 1713 and 1728 and 1732–1918, Saint Petersburg was the capital of imperial Russia. In 1918, the government bodies moved to Moscow. Saint Petersburg is one of the cities of Russia, as well as its cultural capital. The Historic Centre of Saint Petersburg and Related Groups of Monuments constitute a UNESCO World Heritage Site, Saint Petersburg is home to The Hermitage, one of the largest art museums in the world. A large number of consulates, international corporations, banks. Swedish colonists built Nyenskans, a fortress, at the mouth of the Neva River in 1611, in a then called Ingermanland. A small town called Nyen grew up around it, Peter the Great was interested in seafaring and maritime affairs, and he intended to have Russia gain a seaport in order to be able to trade with other maritime nations. He needed a better seaport than Arkhangelsk, which was on the White Sea to the north, on May 1703121703, during the Great Northern War, Peter the Great captured Nyenskans, and soon replaced the fortress. On May 271703, closer to the estuary 5 km inland from the gulf), on Zayachy Island, he laid down the Peter and Paul Fortress, which became the first brick and stone building of the new city. The city was built by conscripted peasants from all over Russia, tens of thousands of serfs died building the city. Later, the city became the centre of the Saint Petersburg Governorate, Peter moved the capital from Moscow to Saint Petersburg in 1712,9 years before the Treaty of Nystad of 1721 ended the war, he referred to Saint Petersburg as the capital as early as 1704. During its first few years, the city developed around Trinity Square on the bank of the Neva, near the Peter. However, Saint Petersburg soon started to be built out according to a plan, by 1716 the Swiss Italian Domenico Trezzini had elaborated a project whereby the city centre would be located on Vasilyevsky Island and shaped by a rectangular grid of canals. The project was not completed, but is evident in the layout of the streets, in 1716, Peter the Great appointed French Jean-Baptiste Alexandre Le Blond as the chief architect of Saint Petersburg. In 1724 the Academy of Sciences, University and Academic Gymnasium were established in Saint Petersburg by Peter the Great, in 1725, Peter died at the age of fifty-two. His endeavours to modernize Russia had met opposition from the Russian nobility—resulting in several attempts on his life
Saint Petersburg
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Top left to bottom right: Peter and Paul Fortress on Zayachy Island, Smolny Cathedral, Moyka river with the General Staff Building, Trinity Cathedral, Bronze Horseman on Senate Square, and the Winter Palace.
Saint Petersburg
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The Bronze Horseman, monument to Peter the Great
Saint Petersburg
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Palace Square backed by the General Staff arch and building, as the main square of the Russian Empire it was the setting of many events of historic significance
Saint Petersburg
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Map of Saint Petersburg, 1903
35.
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.
36.
Naples
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Naples is the capital of the Italian region Campania and the third-largest municipality in Italy, after Rome and Milan. In 2015, around 975,260 people lived within the administrative limits. The Metropolitan City of Naples had a population of 3,115,320, Naples is the 9th-most populous urban area in the European Union with a population of between 3 million and 3.7 million. About 4.4 million people live in the Naples metropolitan area, Naples is one of the oldest continuously inhabited cities in the world. Bronze Age Greek settlements were established in the Naples area in the second millennium BC, a larger colony – initially known as Parthenope, Παρθενόπη – developed on the Island of Megaride around the ninth century BC, at the end of the Greek Dark Ages. Naples remained influential after the fall of the Western Roman Empire, thereafter, in union with Sicily, it became the capital of the Two Sicilies until the unification of Italy in 1861. Naples was the most-bombed Italian city during World War II, much of the citys 20th-century periphery was constructed under Benito Mussolinis fascist government, and during reconstruction efforts after World War II. The city has experienced significant economic growth in recent decades, and unemployment levels in the city, however, Naples still suffers from political and economic corruption, and unemployment levels remain high. Naples has the fourth-largest urban economy in Italy, after Milan, Rome and it is the worlds 103rd-richest city by purchasing power, with an estimated 2011 GDP of US$83.6 billion. The port of Naples is one of the most important in Europe, numerous major Italian companies, such as MSC Cruises Italy S. p. A, are headquartered in Naples. The city also hosts NATOs Allied Joint Force Command Naples, the SRM Institution for Economic Research, Naples is a full member of the Eurocities network of European cities. The city was selected to become the headquarters of the European institution ACP/UE and was named a City of Literature by UNESCOs Creative Cities Network, the Villa Rosebery, one of the three official residences of the President of Italy, is located in the citys Posillipo district. Naples historic city centre is the largest in Europe, covering 1,700 hectares and enclosing 27 centuries of history, Naples has long been a major cultural centre with a global sphere of influence, particularly during the Renaissance and Enlightenment eras. In the immediate vicinity of Naples are numerous culturally and historically significant sites, including the Palace of Caserta, culinarily, Naples is synonymous with pizza, which originated in the city. Neapolitan music has furthermore been highly influential, credited with the invention of the romantic guitar, according to CNN, the metro stop Toledo is the most beautiful in Europe and it won also the LEAF Award 2013 as Public building of the year. Naples is the Italian city with the highest number of accredited stars from the Michelin Guide, Naples sports scene is dominated by football and Serie A club S. S. C. Napoli, two-time Italian champions and winner of European trophies, who play at the San Paolo Stadium in the south-west of the city, the Phlegraean Fields around Naples has been inhabited since the Neolithic period. The earliest Greek settlements were established in the Naples area in the second millennium BC, sailors from the Greek island of Rhodes established a small commercial port called Parthenope on the island of Megaride in the ninth century BC
Naples
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Naples Napoli
Naples
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Ancient map of the Bay of Naples area from Vatican Museum
Naples
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A scene featuring the siren Parthenope, the mythological founder of Naples.
Naples
–
The Gothic Battle of Mons Lactarius on Vesuvius, painted by Alexander Zick.
37.
French Institute
–
The Institut de France is a French learned society, grouping five académies, the most famous of which is the Académie française. The Institute, located in Paris, manages approximately 1,000 foundations, as well as museums and it also awards prizes and subsidies, which amounted to a total of €5,028,190.55 for 2002. Most of these prizes are awarded by the Institute on the recommendation of the académies, the Institut de France was established on 25 October 1795, by the French government. Académie française – initiated 1635, suppressed 1793, restored 1803 as a division of the institute, Académie des inscriptions et belles-lettres – initiated 1663. Académie des sciences – initiated 1666, the Royal Society of Canada, initiated 1882, was modeled after the Institut de France and the Royal Society of London
French Institute
–
Institut de France, from the pont des Arts
French Institute
–
Cupola of the Institut de France
French Institute
–
Henri Grégoire was a founding member of the Institut de France.
French Institute
–
Esplanade in front of the Institut, 1898.
38.
French revolution
–
Through the Revolutionary Wars, it unleashed a wave of global conflicts that extended from the Caribbean to the Middle East. Historians widely regard the Revolution as one of the most important events in human history, the causes of the French Revolution are complex and are still debated among historians. Following the Seven Years War and the American Revolutionary War, the French government was deeply in debt, Years of bad harvests leading up to the Revolution also inflamed popular resentment of the privileges enjoyed by the clergy and the aristocracy. Demands for change were formulated in terms of Enlightenment ideals and contributed to the convocation of the Estates-General in May 1789, a central event of the first stage, in August 1789, was the abolition of feudalism and the old rules and privileges left over from the Ancien Régime. The next few years featured political struggles between various liberal assemblies and right-wing supporters of the intent on thwarting major reforms. The Republic was proclaimed in September 1792 after the French victory at Valmy, in a momentous event that led to international condemnation, Louis XVI was executed in January 1793. External threats closely shaped the course of the Revolution, internally, popular agitation radicalised the Revolution significantly, culminating in the rise of Maximilien Robespierre and the Jacobins. Large numbers of civilians were executed by revolutionary tribunals during the Terror, after the Thermidorian Reaction, an executive council known as the Directory assumed control of the French state in 1795. The rule of the Directory was characterised by suspended elections, debt repudiations, financial instability, persecutions against the Catholic clergy, dogged by charges of corruption, the Directory collapsed in a coup led by Napoleon Bonaparte in 1799. The modern era has unfolded in the shadow of the French Revolution, almost all future revolutionary movements looked back to the Revolution as their predecessor. The values and institutions of the Revolution dominate French politics to this day, the French Revolution differed from other revolutions in being not merely national, for it aimed at benefiting all humanity. Globally, the Revolution accelerated the rise of republics and democracies and it became the focal point for the development of all modern political ideologies, leading to the spread of liberalism, radicalism, nationalism, socialism, feminism, and secularism, among many others. The Revolution also witnessed the birth of total war by organising the resources of France, historians have pointed to many events and factors within the Ancien Régime that led to the Revolution. Over the course of the 18th century, there emerged what the philosopher Jürgen Habermas called the idea of the sphere in France. A perfect example would be the Palace of Versailles which was meant to overwhelm the senses of the visitor and convince one of the greatness of the French state and Louis XIV. Starting in the early 18th century saw the appearance of the sphere which was critical in that both sides were active. In France, the emergence of the public sphere outside of the control of the saw the shift from Versailles to Paris as the cultural capital of France. In the 1750s, during the querelle des bouffons over the question of the quality of Italian vs, in 1782, Louis-Sébastien Mercier wrote, The word court no longer inspires awe amongst us as in the time of Louis XIV
French revolution
–
The August Insurrection in 1792 precipitated the last days of the monarchy.
French revolution
–
The French government faced a fiscal crisis in the 1780s, and King Louis XVI was blamed for mishandling these affairs.
French revolution
–
Caricature of the Third Estate carrying the First Estate (clergy) and the Second Estate (nobility) on its back.
French revolution
–
The meeting of the Estates General on 5 May 1789 at Versailles.
39.
Pierre Charles Le Monnier
–
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
–
Latin and French inscriptions at the base of the obelisk of the Gnomon of Saint-Sulpice, mentioning Pierre Charles Claude Le Monnier.
40.
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
–
Line engraving by Louis Jean Desire Delaistre, after a design by Julien Leopold Boilly
Lavoisier
–
Antoine-Laurent Lavoisier by Jules Dalou 1866
Lavoisier
–
Portrait of Antoine-Laurent Lavoisier and his wife by Jacques-Louis David, ca. 1788
Lavoisier
–
Joseph Priestley, an English chemist known for isolating oxygen, which he termed "dephlogisticated air."
41.
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
–
A giant Hubble mosaic of the Crab Nebula, a supernova remnant
Astronomy
–
19th century Sydney Observatory, Australia (1873)
Astronomy
–
19th century Quito Astronomical Observatory is located 12 minutes south of the Equator in Quito, Ecuador.
42.
Kinematics
–
Kinematics as a field of study is often referred to as the geometry of motion and as such may be seen as a branch of mathematics. The study of the influence of forces acting on masses falls within the purview of kinetics, for further details, see analytical dynamics. Kinematics is used in astrophysics to describe the motion of celestial bodies, in mechanical engineering, robotics, and biomechanics kinematics is used to describe the motion of systems composed of joined parts such as an engine, a robotic arm or the human skeleton. Kinematic analysis is the process of measuring the quantities used to describe motion. In addition, kinematics applies geometry to the study of the mechanical advantage of a mechanical system or mechanism. The term kinematic is the English version of A. M, ampères cinématique, which he constructed from the Greek κίνημα kinema, itself derived from κινεῖν kinein. Kinematic and cinématique are related to the French word cinéma, particle kinematics is the study of the trajectory of a particle. The position of a particle is defined to be the vector from the origin of a coordinate frame to the particle. If the tower is 50 m high, then the vector to the top of the tower is r=. In the most general case, a coordinate system is used to define the position of a particle. However, if the particle is constrained to move in a surface, all observations in physics are incomplete without those observations being described with respect to a reference frame. The position vector of a particle is a vector drawn from the origin of the frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin, the magnitude of the position vector |P| gives the distance between the point P and the origin. | P | = x P2 + y P2 + z P2, the direction cosines of the position vector provide a quantitative measure of direction. It is important to note that the vector of a particle isnt unique. The position vector of a particle is different relative to different frames of reference. The velocity of a particle is a quantity that describes the direction of motion. More mathematically, the rate of change of the vector of a point
Kinematics
–
Each particle on the wheel travels in a planar circular trajectory (Kinematics of Machinery, 1876).
Kinematics
–
Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.
Kinematics
–
Illustration of a four-bar linkage from http://en.wikisource.org/wiki/The_Kinematics_of_Machinery Kinematics of Machinery, 1876
43.
Statics
–
When in static equilibrium, the acceleration of the system is zero and the system is either at rest, or its center of mass moves at constant velocity. The application of Newtons second law to a system gives, F = m a, where bold font indicates a vector that has magnitude and direction. F is the total of the acting on the system, m is the mass of the system. The summation of forces will give the direction and the magnitude of the acceleration will be proportional to the mass. The assumption of static equilibrium of a =0 leads to, the summation of forces, one of which might be unknown, allows that unknown to be found. Likewise the application of the assumption of zero acceleration to the summation of moments acting on the leads to. The summation of moments, one of which might be unknown and these two equations together, can be applied to solve for as many as two loads acting on the system. From Newtons first law, this implies that the net force, the net forces equaling zero is known as the first condition for equilibrium, and the net torque equaling zero is known as the second condition for equilibrium. A scalar is a quantity which only has a magnitude, such as mass or temperature, a vector has a magnitude and a direction. There are several notations to identify a vector, including, A bold faced character V An underlined character V A character with an arrow over it V →, vectors are added using the parallelogram law or the triangle law. Vectors contain components in orthogonal bases, unit vectors i, j, and k are, by convention, along the x, y, and z axes, respectively. Force is the action of one body on another, a force is either a push or a pull. A force tends to move a body in the direction of its action, the action of a force is characterized by its magnitude, by the direction of its action, and by its point of application. Thus, force is a quantity, because its effect depends on the direction as well as on the magnitude of the action. Forces are classified as either contact or body forces, a contact force is produced by direct physical contact, an example is the force exerted on a body by a supporting surface. A body force is generated by virtue of the position of a body within a field such as a gravitational, electric. An example of a force is the weight of a body in the Earths gravitational field. In addition to the tendency to move a body in the direction of its application, the axis may be any line which neither intersects nor is parallel to the line of action of the force
Statics
–
Example of a beam in static equilibrium. The sum of force and moment is zero.
44.
Couple (mechanics)
–
In mechanics, a couple is a system of forces with a resultant moment but no resultant force. A better term is force couple or pure moment and its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass. In rigid body mechanics, force couples are free vectors, meaning their effects on a body are independent of the point of application, the resultant moment of a couple is called a torque. This is not to be confused with the term torque as it is used in physics, instead, torque is a special case of moment. Torque has special properties that moment does not have, in particular the property of being independent of reference point, definition A couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance or moment. The simplest kind of couple consists of two equal and opposite forces whose lines of action do not coincide and this is called a simple couple. The forces have an effect or moment called a torque about an axis which is normal to the plane of the forces. The SI unit for the torque of the couple is newton metre. When d is taken as a vector between the points of action of the forces, then the couple is the product of d and F, i. e. τ = | d × F |. The moment of a force is defined with respect to a certain point P, and in general when P is changed. However, the moment of a couple is independent of the reference point P, in other words, a torque vector, unlike any other moment vector, is a free vector. The proof of claim is as follows, Suppose there are a set of force vectors F1, F2, etc. that form a couple, with position vectors r1, r2. The moment about P is M = r 1 × F1 + r 2 × F2 + ⋯ Now we pick a new reference point P that differs from P by the vector r. The new moment is M ′ = × F1 + × F2 + ⋯ Now the distributive property of the cross product implies M ′ = + r ×, however, the definition of a force couple means that F1 + F2 + ⋯ =0. Therefore, M ′ = r 1 × F1 + r 2 × F2 + ⋯ = M This proves that the moment is independent of reference point, which is proof that a couple is a free vector. A force F applied to a body at a distance d from the center of mass has the same effect as the same force applied directly to the center of mass. The couple produces an acceleration of the rigid body at right angles to the plane of the couple. The force at the center of mass accelerates the body in the direction of the force without change in orientation, conversely, a couple and a force in the plane of the couple can be replaced by a single force, appropriately located
Couple (mechanics)
–
Classical mechanics
45.
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
–
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.
46.
Frame of reference
–
In physics, a frame of reference consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements. In n dimensions, n+1 reference points are sufficient to define a reference frame. Using rectangular coordinates, a frame may be defined with a reference point at the origin. In Einsteinian relativity, reference frames are used to specify the relationship between an observer and the phenomenon or phenomena under observation. In this context, the phrase often becomes observational frame of reference, a relativistic reference frame includes the coordinate time, which does not correspond across different frames moving relatively to each other. The situation thus differs from Galilean relativity, where all possible coordinate times are essentially equivalent, the need to distinguish between the various meanings of frame of reference has led to a variety of terms. For example, sometimes the type of system is attached as a modifier. Sometimes the state of motion is emphasized, as in rotating frame of reference, sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference, in this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a system may be employed for many purposes where the state of motion is not the primary concern. For example, a system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors and it seems useful to divorce the various aspects of a reference frame for the discussion below. A coordinate system is a concept, amounting to a choice of language used to describe observations. Consequently, an observer in a frame of reference can choose to employ any coordinate system to describe observations made from that frame of reference. A change in the choice of coordinate system does not change an observers state of motion. This viewpoint can be found elsewhere as well, which is not to dispute that some coordinate systems may be a better choice for some observations than are others. Choice of what to measure and with what observational apparatus is a separate from the observers state of motion. D. Norton, The discussion is taken beyond simple space-time coordinate systems by Brading, extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations of quantum field theory, classical relativistic mechanics, and quantum gravity
Frame of reference
–
An observer O, situated at the origin of a local set of coordinates – a frame of reference F. The observer in this frame uses the coordinates (x, y, z, t) to describe a spacetime event, shown as a star.
47.
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
–
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
–
Distance traveled by a freely falling ball is proportional to the square of the elapsed time
48.
Work (physics)
–
In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and then dropped, the SI unit of work is the joule. The SI unit of work is the joule, which is defined as the work expended by a force of one newton through a distance of one metre. The dimensionally equivalent newton-metre is sometimes used as the unit for work, but this can be confused with the unit newton-metre. Usage of N⋅m is discouraged by the SI authority, since it can lead to confusion as to whether the quantity expressed in newton metres is a torque measurement, or a measurement of energy. Non-SI units of work include the erg, the foot-pound, the foot-poundal, the hour, the litre-atmosphere. Due to work having the physical dimension as heat, occasionally measurement units typically reserved for heat or energy content, such as therm, BTU. The work done by a constant force of magnitude F on a point that moves a distance s in a line in the direction of the force is the product W = F s. For example, if a force of 10 newtons acts along a point that travels 2 meters and this is approximately the work done lifting a 1 kg weight from ground level to over a persons head against the force of gravity. Notice that the work is doubled either by lifting twice the weight the distance or by lifting the same weight twice the distance. Work is closely related to energy, the work-energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in energy is caused by an equal amount of negative work done by the resultant force. From Newtons second law, it can be shown that work on a free, rigid body, is equal to the change in energy of the velocity and rotation of that body. The work of forces generated by a function is known as potential energy. These formulas demonstrate that work is the associated with the action of a force, so work subsequently possesses the physical dimensions. The work/energy principles discussed here are identical to Electric work/energy principles, constraint forces determine the movement of components in a system, constraining the object within a boundary. Constraint forces ensure the velocity in the direction of the constraint is zero and this only applies for a single particle system. For example, in an Atwood machine, the rope does work on each body, there are, however, cases where this is not true
Work (physics)
–
A baseball pitcher does positive work on the ball by applying a force to it over the distance it moves while in his grip.
Work (physics)
–
A force of constant magnitude and perpendicular to the lever arm
Work (physics)
–
Gravity F = mg does work W = mgh along any descending path
Work (physics)
–
Lotus type 119B gravity racer at Lotus 60th celebration.
49.
Momentum
–
In classical mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object, quantified in kilogram-meters per second. It is dimensionally equivalent to impulse, the product of force and time, Newtons second law of motion states that the change in linear momentum of a body is equal to the net impulse acting on it. If the truck were lighter, or moving slowly, then it would have less momentum. Linear momentum is also a quantity, meaning that if a closed system is not affected by external forces. In classical mechanics, conservation of momentum is implied by Newtons laws. It also holds in special relativity and, with definitions, a linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory. It is ultimately an expression of one of the symmetries of space and time. Linear momentum depends on frame of reference, observers in different frames would find different values of linear momentum of a system. But each would observe that the value of linear momentum does not change with time, momentum has a direction as well as magnitude. Quantities that have both a magnitude and a direction are known as vector quantities, because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds. Below, the properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations, the momentum of a particle is traditionally represented by the letter p. It is the product of two quantities, the mass and velocity, p = m v, the units of momentum are the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity in meters per second then the momentum is in kilogram meters/second, in cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters/second. Being a vector, momentum has magnitude and direction, for example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg m/s due north measured from the ground. The momentum of a system of particles is the sum of their momenta, if two particles have masses m1 and m2, and velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2. If all the particles are moving, the center of mass will generally be moving as well, if the center of mass is moving at velocity vcm, the momentum is, p = m v cm. This is known as Eulers first law, if a force F is applied to a particle for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t
Momentum
–
In a game of pool, momentum is conserved; that is, if one ball stops dead after the collision, the other ball will continue away with all the momentum. If the moving ball continues or is deflected then both balls will carry a portion of the momentum from the collision.
50.
Space
–
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, in Isaac Newtons view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the visibility of spatial depth in his Essay Towards a New Theory of Vision. Kant referred to the experience of space in his Critique of Pure Reason as being a pure a priori form of intuition. In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in space is conceived as curved. According to Albert Einsteins theory of relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a model for the shape of space. In the seventeenth century, the philosophy of space and time emerged as an issue in epistemology. At its heart, Gottfried Leibniz, the German philosopher-mathematician, and Isaac Newton, unoccupied regions are those that could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete. Space could be thought of in a way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people, but since there would be no observational way of telling these universes apart then, according to the identity of indiscernibles, there would be no real difference between them. According to the principle of sufficient reason, any theory of space that implied that there could be two possible universes must therefore be wrong. Newton took space to be more than relations between objects and based his position on observation and experimentation
Space
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Gottfried Leibniz
Space
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A right-handed three-dimensional Cartesian coordinate system used to indicate positions in space.
Space
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Isaac Newton
Space
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Immanuel Kant
51.
Torque
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Torque, moment, or moment of force is rotational force. Just as a force is a push or a pull. Loosely speaking, torque is a measure of the force on an object such as a bolt or a flywheel. For example, pushing or pulling the handle of a wrench connected to a nut or bolt produces a torque that loosens or tightens the nut or bolt, the symbol for torque is typically τ, the lowercase Greek letter tau. When it is called moment of force, it is denoted by M. The SI unit for torque is the newton metre, for more on the units of torque, see Units. This article follows US physics terminology in its use of the word torque, in the UK and in US mechanical engineering, this is called moment of force, usually shortened to moment. In US physics and UK physics terminology these terms are interchangeable, unlike in US mechanical engineering, Torque is defined mathematically as the rate of change of angular momentum of an object. The definition of states that one or both of the angular velocity or the moment of inertia of an object are changing. Moment is the term used for the tendency of one or more applied forces to rotate an object about an axis. For example, a force applied to a shaft causing acceleration, such as a drill bit accelerating from rest. By contrast, a force on a beam produces a moment, but since the angular momentum of the beam is not changing. Similarly with any force couple on an object that has no change to its angular momentum and this article follows the US physics terminology by calling all moments by the term torque, whether or not they cause the angular momentum of an object to change. The concept of torque, also called moment or couple, originated with the studies of Archimedes on levers, the term torque was apparently introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884. A force applied at an angle to a lever multiplied by its distance from the levers fulcrum is its torque. A force of three newtons applied two metres from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum. More generally, the torque on a particle can be defined as the product, τ = r × F, where r is the particles position vector relative to the fulcrum. Alternatively, τ = r F ⊥, where F⊥ is the amount of force directed perpendicularly to the position of the particle, any force directed parallel to the particles position vector does not produce a torque
Torque
52.
Damping
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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
Damping
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Mass attached to a spring and damper.
53.
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
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The effect of varying damping ratio on a second-order system.
54.
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
55.
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.
56.
Fictitious force
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The force F does not arise from any physical interaction between two objects, but rather from the acceleration a of the non-inertial reference frame itself. As stated by Iro, Such an additional force due to relative motion of two reference frames is called a pseudo-force. Assuming Newtons second law in the form F = ma, fictitious forces are proportional to the mass m. A fictitious force on an object arises when the frame of reference used to describe the motion is accelerating compared to a non-accelerating frame. As a frame can accelerate in any way, so can fictitious forces be as arbitrary. Gravitational force would also be a force based upon a field model in which particles distort spacetime due to their mass. The role of forces in Newtonian mechanics is described by Tonnelat. To solve classical mechanics problems exactly in an Earth-bound reference frame, the Euler force is typically ignored because the variations in the angular velocity of the rotating Earth surface are usually insignificant. Both of the fictitious forces are weak compared to most typical forces in everyday life. For example, Léon Foucault was able to show that the Coriolis force results from the Earths rotation using the Foucault pendulum. If the Earth were to rotate a thousand times faster, people could easily get the impression that such forces are pulling on them. Other accelerations also give rise to forces, as described mathematically below. An example of the detection of a non-inertial, rotating reference frame is the precession of a Foucault pendulum, in the non-inertial frame of the Earth, the fictitious Coriolis force is necessary to explain observations. In an inertial frame outside the Earth, no such force is necessary. Figure 1 shows an accelerating car, when a car accelerates, a passenger feels like theyre being pushed back into the seat. In an inertial frame of reference attached to the road, there is no physical force moving the rider backward, however, in the riders non-inertial reference frame attached to the accelerating car, there is a backward fictitious force. We mention two possible reasons for the force to clarify its existence, Figure 1, to an observer at rest on an inertial reference frame, the car will seem to accelerate. In order for the passenger to stay inside the car, a force must be exerted on the passenger
Fictitious force
57.
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
58.
Motion (physics)
–
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.
59.
Rigid body
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In physics, a rigid body is an idealization of a solid body in which deformation is neglected. In other words, the distance between any two points of a rigid body remains constant in time regardless of external forces exerted on it. Even though such an object cannot physically exist due to relativity, in classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechanics a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics molecules are often seen as rigid bodies, the position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, if the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, however, typically a different, mathematically more convenient, but equivalent approach is used. Thus, the position of a body has two components, linear and angular, respectively. The same is true for other kinematic and kinetic quantities describing the motion of a body, such as linear and angular velocity, acceleration, momentum, impulse. This reference point may define the origin of a coordinate system fixed to the body, there are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix. In general, when a body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation of the body starting from a hypothetic reference position. Velocity and angular velocity are measured with respect to a frame of reference, the linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a point fixed to the body. During purely translational motion, all points on a body move with the same velocity. However, when motion involves rotation, the velocity of any two points on the body will generally not be the same. Two points of a body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation. Angular velocity is a quantity that describes the angular speed at which the orientation of the rigid body is changing. All points on a rigid body experience the same velocity at all times
Rigid body
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The position of a rigid body is determined by the position of its center of mass and by its attitude (at least six parameters in total).
60.
Rigid body dynamics
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Rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. This excludes bodies that display fluid highly elastic, and plastic behavior, the dynamics of a rigid body system is described by the laws of kinematics and by the application of Newtons second law or their derivative form Lagrangian mechanics. The formulation and solution of rigid body dynamics is an important tool in the simulation of mechanical systems. If a system of particles moves parallel to a fixed plane, in this case, Newtons laws for a rigid system of N particles, Pi, i=1. N, simplify because there is no movement in the k direction. Determine the resultant force and torque at a reference point R, to obtain F = ∑ i =1 N m i A i, T = ∑ i =1 N ×, where ri denotes the planar trajectory of each particle. In this case, the vectors can be simplified by introducing the unit vectors ei from the reference point R to a point ri. Several methods to describe orientations of a body in three dimensions have been developed. They are summarized in the following sections, the first attempt to represent an orientation is attributed to Leonhard Euler. The values of three rotations are called Euler angles. These are three angles, also known as yaw, pitch and roll, Navigation angles and Cardan angles, in aerospace engineering they are usually referred to as Euler angles. Euler also realized that the composition of two rotations is equivalent to a rotation about a different fixed axis. Therefore, the composition of the three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed. Based on this fact he introduced a way to describe any rotation, with a vector on the rotation axis. Therefore, any orientation can be represented by a vector that leads to it from the reference frame. When used to represent an orientation, the vector is commonly called orientation vector, or attitude vector. A similar method, called axis-angle representation, describes a rotation or orientation using a unit vector aligned with the axis. With the introduction of matrices the Euler theorems were rewritten, the rotations were described by orthogonal matrices referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a matrix is commonly called orientation matrix
Rigid body dynamics
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Human body modelled as a system of rigid bodies of geometrical solids. Representative bones were added for better visualization of the walking person.
Rigid body dynamics
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Movement of each of the components of the Boulton & Watt Steam Engine (1784) is modeled by a continuous set of rigid displacements
61.
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).
62.
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.
63.
Coriolis force
–
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.
64.
Pendulum (mathematics)
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The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allows the equations of motion to be solved analytically for small-angle oscillations. e, the bob does not trace an ellipse but an arc. The motion does not lose energy to friction or air resistance, the differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However adding a restriction to the size of the oscillations amplitude gives a form whose solution can be easily obtained, the error due to the approximation is of order θ3. Given the initial conditions θ = θ0 and dθ/dt =0, the period of the motion, the time for a complete oscillation is which is known as Christiaan Huygenss law for the period. Note that under the small-angle approximation, the period is independent of the amplitude θ0, T0 =2 π ℓ g can be expressed as ℓ = g π2 T024. If SI units are used, and assuming the measurement is taking place on the Earths surface, then g ≈9.81 m/s2, and g/π2 ≈1. Therefore, a reasonable approximation for the length and period are. Note that this integral diverges as θ0 approaches the vertical lim θ0 → π T = ∞, so that a pendulum with just the right energy to go vertical will never actually get there. For comparison of the approximation to the solution, consider the period of a pendulum of length 1 m on Earth at initial angle 10 degrees is 41 m g K ≈2.0102 s. The linear approximation gives 2 π1 m g ≈2.0064 s, the difference between the two values, less than 0. 2%, is much less than that caused by the variation of g with geographical location. From here there are ways to proceed to calculate the elliptic integral. Figure 4 shows the relative errors using the power series, T0 is the linear approximation, and T2 to T10 include respectively the terms up to the 2nd to the 10th powers. The resulting power series is, T =2 π ℓ g, given Eq.3 and the arithmetic–geometric mean solution of the elliptic integral, K = π2 M, where M is the arithmetic-geometric mean of x and y. This yields an alternative and faster-converging formula for the period, T =2 π M ℓ g, the animations below depict the motion of a simple pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity. The small graph above each pendulum is the phase plane diagram, the horizontal axis is displacement. With a large enough initial velocity the pendulum does not oscillate back and forth, a compound pendulum is one where the rod is not massless, and may have extended size, that is, an arbitrarily shaped rigid body swinging by a pivot. In this case the period depends on its moment of inertia I around the pivot point
Pendulum (mathematics)
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Figure 1. Force diagram of a simple gravity pendulum.
Pendulum (mathematics)
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Animation of a pendulum showing the velocity and acceleration vectors.
65.
Angular displacement
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Angular displacement of a body is the angle in radians through which a point or line has been rotated in a specified sense about a specified axis. When an object rotates about its axis, the motion cannot simply be analyzed as a particle, since in circular motion it undergoes a changing velocity, when dealing with the rotation of an object, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal, Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion. In the example illustrated to the right, a particle on object P is at a distance r from the origin, O. It becomes important to represent the position of particle P in terms of its polar coordinates. In this particular example, the value of θ is changing, if using radians, it provides a very simple relationship between distance traveled around the circle and the distance r from the centre. Therefore,1 revolution is 2 π radians, when object travels from point P to point Q, as it does in the illustration to the left, over δ t the radius of the circle goes around a change in angle. Δ θ = θ2 − θ1 which equals the Angular Displacement, in three dimensions, angular displacement is an entity with a direction and a magnitude. The direction specifies the axis of rotation, which exists by virtue of the Eulers rotation theorem. This entity is called an axis-angle, despite having direction and magnitude, angular displacement is not a vector because it does not obey the commutative law for addition. Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded, several ways to describe angular displacement exist, like rotation matrices or Euler angles. See charts on SO for others, given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being A0 and A f two matrices, the angular displacement matrix between them can be obtained as Δ A = A f, when this product is performed having a very small difference between both frames we will obtain a matrix close to the identity. In the limit, we will have a rotation matrix. An infinitesimal angular displacement is a rotation matrix, As any rotation matrix has a single real eigenvalue, which is +1. Its module can be deduced from the value of the infinitesimal rotation, when it is divided by the time, this will yield the angular velocity vector. Suppose we specify an axis of rotation by a unit vector, expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix ΔR is represented as, Δ R = + Δ θ = I + A Δ θ
Angular displacement
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Rotation of a rigid object P about a fixed object about a fixed axis O.
66.
Angular velocity
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This speed can be measured in the SI unit of angular velocity, radians per second, or in terms of degrees per second, degrees per hour, etc. Angular velocity is usually represented by the symbol omega, the direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction that is usually specified by the right-hand rule. The angular velocity of a particle is measured around or relative to a point, called the origin. As shown in the diagram, if a line is drawn from the origin to the particle, then the velocity of the particle has a component along the radius, if there is no radial component, then the particle moves in a circle. On the other hand, if there is no cross-radial component, a radial motion produces no change in the direction of the particle relative to the origin, so, for the purpose of finding the angular velocity, the radial component can be ignored. Therefore, the rotation is completely produced by the perpendicular motion around the origin, the angular velocity in two dimensions is a pseudoscalar, a quantity that changes its sign under a parity inversion. The positive direction of rotation is taken, by convention, to be in the direction towards the y axis from the x axis, if the parity is inverted, but the orientation of a rotation is not, then the sign of the angular velocity changes. There are three types of angular velocity involved in the movement on an ellipse corresponding to the three anomalies, in three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in case is generally thought of as a vector, or more precisely. It now has not only a magnitude, but a direction as well, the magnitude is the angular speed, and the direction describes the axis of rotation that Eulers rotation theorem guarantees must exist. The right-hand rule indicates the direction of the angular velocity pseudovector. Let u be a vector along the instantaneous rotation axis. This is the definition of a vector space, the only property that presents difficulties to prove is the commutativity of the addition. This can be proven from the fact that the velocity tensor W is skew-symmetric, therefore, R = e W t is a rotation matrix and in a time dt is an infinitesimal rotation matrix. Therefore, it can be expanded as R = I + W ⋅ d t +122 +, in such a frame, each vector is a particular case of the previous case, in which the module of the vector is constant. Though it just a case of a moving particle, this is a very important one for its relationship with the rigid body study. There are two ways to describe the angular velocity of a rotating frame, the angular velocity vector. Both entities are related and they can be calculated from each other, in a consistent way with the general definition, the angular velocity of a frame is defined as the angular velocity of each of the three vectors of the frame
Angular velocity
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The angular velocity of the particle at P with respect to the origin O is determined by the perpendicular component of the velocity vector v.
67.
Jeremiah Horrocks
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Jeremiah Horrocks, sometimes given as Jeremiah Horrox, was an English astronomer. Jeremiah Horrocks was born at Lower Lodge Farm in Toxteth Park and his father James had moved to Toxteth Park to be apprenticed to Thomas Aspinwall, a watchmaker, and subsequently married his masters daughter Mary. Both families were well educated Puritans, the Horrocks sent their sons to the University of Cambridge. For their unorthodox beliefs the Puritans were excluded from public office, in 1632 Horrocks matriculated at Emmanuel College at the University of Cambridge as a sizar. At Cambridge he associated with the mathematician John Wallis and the platonist John Worthington, at that time he was one of only a few at Cambridge to accept Copernicuss revolutionary heliocentric theory, and he studied the works of Johannes Kepler, Tycho Brahe and others. In 1635 for reasons not clear Horrocks left Cambridge without graduating, now committed to the study of astronomy, Horrocks began to collect astronomical books and equipment, by 1638 he owned the best telescope he could find. Liverpool was a town so navigational instruments such as the astrolabe. But there was no market for the very specialised astronomical instruments he needed and he was well placed to do this, his father and uncles were watchmakers with expertise in creating precise instruments. While a youth he read most of the treatises of his day and marked their weaknesses. Tradition has it that after he left home he supported himself by holding a curacy in Much Hoole, near Preston in Lancashire, according to local tradition in Much Hoole, he lived at Carr House, within the Bank Hall Estate, Bretherton. Carr House was a property owned by the Stones family who were prosperous farmers and merchants. Horrocks was the first to demonstrate that the Moon moved in a path around the Earth. He anticipated Isaac Newton in suggesting the influence of the Sun as well as the Earth on the moons orbit, in the Principia Newton acknowledged Horrockss work in relation to his theory of lunar motion. In the final months of his life Horrocks made detailed studies of tides in attempting to explain the nature of causation of tidal movements. Keplers tables had predicted a near-miss of a transit of Venus in 1639 but, having made his own observations of Venus for years, Horrocks predicted a transit would indeed occur. Horrocks made a simple helioscope by focusing the image of the Sun through a telescope onto a plane surface, from his location in Much Hoole he calculated the transit would begin at approximately 3,00 pm on 24 November 1639, Julian calendar. The weather was cloudy but he first observed the tiny black shadow of Venus crossing the Sun at about 3,15 pm, the 1639 transit was also observed by William Crabtree from his home in Broughton near Manchester. His figure of 95 million kilometres was far from the 150 million kilometres known today and it presented Horrocks enthusiastic and romantic nature, including humorous comments and passages of original poetry
Jeremiah Horrocks
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Making the first observation of the transit of Venus in 1639
Jeremiah Horrocks
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A representation of Horrocks' recording of the transit published in 1662 by Johannes Hevelius
Jeremiah Horrocks
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The title page of Jeremiah Horrocks' Opera Posthuma, published by the Royal Society in 1672.
Jeremiah Horrocks
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Jeremiah Horrocks Observatory on Moor Park, Preston
68.
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
69.
William Rowan Hamilton
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Sir William Rowan Hamilton PRIA FRSE was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and his best known contribution to mathematical physics is the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the study of classical field theories such as electromagnetism. In pure mathematics, he is best known as the inventor of quaternions, Hamilton is said to have shown immense talent at a very early age. Astronomer Bishop Dr. John Brinkley remarked of the 18-year-old Hamilton, This young man, I do not say will be, but is, Hamilton also invented icosian calculus, which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once. Hamilton was the fourth of nine born to Sarah Hutton and Archibald Hamilton. Hamiltons father, who was from Dunboyne, worked as a solicitor, by the age of three, Hamilton had been sent to live with his uncle James Hamilton, a graduate of Trinity College who ran a school in Talbots Castle in Trim, Co. His uncle soon discovered that Hamilton had an ability to learn languages. At the age of seven he had made very considerable progress in Hebrew. These included the classical and modern European languages, and Persian, Arabic, Hindustani, Sanskrit, in September 1813 the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, an older than Hamilton. The two were pitted against each other in a mental arithmetic contest with Colburn emerging the clear victor, in reaction to his defeat, Hamilton dedicated less time to studying languages and more time to studying mathematics. Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College, Dublin, which he entered at age 18. He studied both classics and mathematics, and was appointed Professor of Astronomy in 1827, prior to his taking up residence at Dunsink Observatory where he spent the rest of his life. Hamilton made important contributions to optics and to classical mechanics and his first discovery was in an early paper that he communicated in 1823 to Dr. Brinkley, who presented it under the title of Caustics in 1824 to the Royal Irish Academy. It was referred as usual to a committee, while their report acknowledged its novelty and value, they recommended further development and simplification before publication. Between 1825 and 1828 the paper grew to an immense size, but it also became more intelligible, and the features of the new method were now easily to be seen. Until this period Hamilton himself seems not to have fully understood either the nature or importance of optics and he proposed for it when he first predicted its existence in the third supplement to his Systems of Rays, read in 1832
William Rowan Hamilton
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Quaternion Plaque on Broom Bridge
William Rowan Hamilton
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William Rowan Hamilton (1805–1865)
William Rowan Hamilton
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Irish commemorative coin celebrating the 200th Anniversary of his birth.
70.
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)
71.
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
72.
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)
73.
Functional determinant
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The corresponding quantity det is called the functional determinant of S. There are several formulas for the functional determinant and they are all based on the fact that the determinant of a diagonalizable finite-dimensional matrix is equal to the product of the eigenvalues of the matrix. Another possible generalization, often used by physicists when using the Feynman path integral formalism in quantum theory, uses a functional integration. This path integral is only defined up to some divergent multiplicative constant. To give it a meaning it must be divided by another functional determinant. These are now, ostensibly, two different definitions for the determinant, one coming from quantum field theory and one coming from spectral theory. Each involves some kind of regularization, in the popular in physics. Osgood, Phillips & Sarnak have shown that the results obtained by comparing two functional determinants in the QFT formalism agree with the results obtained by the functional determinant. For a positive selfadjoint operator S on a finite-dimensional Euclidean space V, the problem is to find a way to make sense of the determinant of an operator S on an infinite dimensional function space. The basic assumption on S is that it should be selfadjoint and this roughly means all functions φ can be written as linear combinations of the functions fi, | ϕ ⟩ = ∑ i c i | f i ⟩ with c i = ⟨ f i | ϕ ⟩. In the basis of the fi, the functional integration reduces to an integration over all basisfunctions. Formally, assuming our intuition from the finite dimensional case carries over into the infinite dimensional setting, the product of all eigenvalues is equal to the determinant for finite-dimensional spaces, and we formally define this to be the case in our infinite-dimensional case also. This results in the formula ∫ V D ϕ e − ⟨ ϕ | S | ϕ ⟩ ∝1 det S, if all quantities converge in an appropriate sense, then the functional determinant can be described as a classical limit. Otherwise, it is necessary to some kind of regularization. The most popular of which for computing functional determinants is the zeta function regularization, for instance, this allows for the computation of the determinant of the Laplace and Dirac operators on a Riemannian manifold, using the Minakshisundaram–Pleijel zeta function. Otherwise, it is possible to consider the quotient of two determinants, making the divergent constants cancel. Let S be a differential operator with smooth coefficients which is positive on functions of compact support. That is, there exists a constant c >0 such that ⟨ ϕ, S ϕ ⟩ ≥ c ⟨ ϕ, ϕ ⟩ for all compactly supported smooth functions φ, then S has a self-adjoint extension to an operator on L2 with lower bound c
Functional determinant
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The infinite potential well with A = 0.
74.
Absolute value
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In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a x, |x| = −x for a negative x. For example, the value of 3 is 3. The absolute value of a number may be thought of as its distance from zero, generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, a value is also defined for the complex numbers. The absolute value is related to the notions of magnitude, distance. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English, the notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude, in programming languages and computational software packages, the absolute value of x is generally represented by abs, or a similar expression. Thus, care must be taken to interpret vertical bars as an absolute value sign only when the argument is an object for which the notion of an absolute value is defined. For any real number x the value or modulus of x is denoted by |x| and is defined as | x | = { x, if x ≥0 − x. As can be seen from the definition, the absolute value of x is always either positive or zero. Indeed, the notion of a distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Since the square root notation without sign represents the square root. This identity is used as a definition of absolute value of real numbers. The absolute value has the four fundamental properties, The properties given by equations - are readily apparent from the definition. To see that equation holds, choose ε from so that ε ≥0, some additional useful properties are given below. These properties are either implied by or equivalent to the properties given by equations -, for example, Absolute value is used to define the absolute difference, the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value cannot be directly generalised for a complex number
Absolute value
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The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complex conjugate z have the same absolute value.
75.
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.
76.
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
–
Pell's equation for n = 2 and six of its integer solutions
77.
Analytical geometry
–
In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete, usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane and Euclidean space, the numerical output, however, might also be a vector or a shape. That the algebra of the numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is thought to have anticipated the work of Descartes by some 1800 years. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves and that is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation, analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. Cartesian geometry, the term used for analytic geometry, is named after Descartes. This work, written in his native French tongue, and its philosophical principles, initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 did Descartess masterpiece receive due recognition, Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a form of Ad locos planos et solidos isagoge was circulating in Paris in 1637. Clearly written and well received, the Introduction also laid the groundwork for analytical geometry, as a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonard Euler who first applied the method in a systematic study of space curves and surfaces. In analytic geometry, the plane is given a coordinate system, similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the point of origin. These are typically written as an ordered pair and this system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates. In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ from the polar axis
Analytical geometry
–
Cartesian coordinates
78.
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.
79.
Orbit
–
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
–
The International Space Station orbits above Earth.
Orbit
–
Planetary orbits
Orbit
Orbit
–
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.
80.
Urbain Le Verrier
–
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
–
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.
Pure mathematics
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Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. Even though the pure and applied viewpoints are distinct philosophical positions, in there is much overlap in the activity of pure. To develop accurate models for describing the world, many applied mathematicians draw on tools. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research, ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between arithmetic, now called number theory, and logistic, now called arithmetic. Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, the term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, in the following years, specialisation and professionalisation started to make a rift more apparent. At the start of the twentieth century mathematicians took up the axiomatic method, in fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved, Pure mathematician became a recognized vocation, achievable through training. One central concept in mathematics is the idea of generality. One can use generality to avoid duplication of effort, proving a general instead of having to prove separate cases independently. Generality can facilitate connections between different branches of mathematics, category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math. Generalitys impact on intuition is both dependent on the subject and a matter of preference or learning style. Often generality is seen as a hindrance to intuition, although it can function as an aid to it. Each of these branches of abstract mathematics have many sub-specialties. A steep rise in abstraction was seen mid 20th century, in practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, the point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central
Pure mathematics
–
An illustration of the Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.
82.
Adrien-Marie Legendre
–
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
–
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
–
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
–
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.
Planetary motion
–
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
Planetary motion
–
The International Space Station orbits above Earth.
Planetary motion
–
Planetary orbits
Planetary motion
Planetary motion
–
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.
85.
Royal Society
–
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
–
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
86.
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
87.
Legion of Honour
–
The Legion of Honour, full name National Order of the Legion of Honour, is the highest French order of merit for military and civil merits, established 1802 by Napoléon Bonaparte. The order is divided into five degrees of increasing distinction, Chevalier, Officier, Commandeur, Grand Officier and Grand-Croix. The orders motto is Honneur et Patrie and its seat is the Palais de la Légion dHonneur next to the Musée dOrsay, in the French Revolution, all French orders of chivalry were abolished, and replaced with Weapons of Honour. The Légion however did use the organization of old French orders of chivalry, the badges of the legion also bear a resemblance to the Ordre de Saint-Louis, which also used a red ribbon. Napoleon originally created this to ensure political loyalty, the organization would be used as a facade to give political favours, gifts, and concessions. The Légion was loosely patterned after a Roman legion, with legionaries, officers, commanders, regional cohorts, the highest rank was not a grand cross but a Grand Aigle, a rank that wore all the insignia common to grand crosses. The members were paid, the highest of them extremely generously,5,000 francs to an officier,2,000 francs to a commandeur,1,000 francs to an officier,250 francs to a légionnaire. Napoleon famously declared, You call these baubles, well, it is with baubles that men are led, do you think that you would be able to make men fight by reasoning. That is good only for the scholar in his study, the soldier needs glory, distinctions, rewards. This has been quoted as It is with such baubles that men are led. The order was the first modern order of merit, under the monarchy, such orders were often limited to Roman Catholics, and all knights had to be noblemen. The military decorations were the perks of the officers, the Légion, however, was open to men of all ranks and professions—only merit or bravery counted. The new legionnaire had to be sworn in the Légion and it is noteworthy that all previous orders were crosses or shared a clear Christian background, whereas the Légion is a secular institution. The jewel of the Légion has five arms, in a decree issued on the 10 Pluviôse XIII, a grand decoration was instituted. This decoration, a cross on a sash and a silver star with an eagle, symbol of the Napoleonic Empire, became known as the Grand Aigle. After Napoleon crowned himself Emperor of the French in 1804 and established the Napoleonic nobility in 1808, the title was made hereditary after three generations of grantees. Napoleon had dispensed 15 golden collars of the legion among his family and this collar was abolished in 1815. The Légion dhonneur was prominent and visible in the French Empire, the Emperor always wore it and the fashion of the time allowed for decorations to be worn most of the time
Legion of Honour
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Order's streamer
Legion of Honour
Legion of Honour
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A depiction of Napoleon making some of the first awards of the Légion d'honneur, at a camp near Boulogne on 16 August 1804
Legion of Honour
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First Légion d'Honneur investiture, 15 July 1804, at Saint-Louis des Invalides by Jean-Baptiste Debret (1812)
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.
List of the 72 names on the Eiffel Tower
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On the Eiffel Tower, seventy-two names of French scientists, engineers, and mathematicians are engraved in recognition of their contributions. Gustave Eiffel chose this invocation of science because of his concern over the protests against the tower, the engravings are found on the sides of the tower under the first balcony. The Tower is owned by the city of Paris, the letters were originally painted in gold and are about 60 cm high. The repainting of 2010/2011 restored the letters to their gold colour. There are also names of engineers who helped build the tower and design its architecture on the top of the tower on a plaque. The list is split in four parts, the list has been criticized for excluding the name of Sophie Germain, a noted French mathematician whose work on the theory of elasticity was used in the construction of the tower itself. In 1913 John Augustine Zahm suggested that Germain was excluded because she was a woman,14 hydraulic engineers and scholars are listed on the Eiffel Tower. Eiffel acknowledged most of the scientists in the field. Henri Philibert Gaspard Darcy is missing, some of his work did not come into use until the 20th century. Also missing are Antoine Chézy, who was famous, Joseph Valentin Boussinesq. Also missing is the mathematician Evariste Galois, le Panthéon scientifique de la tour Eiffel, histoire des origines de la construction de la Tour. Le Panthéon scientifique de la tour Eiffel, histoire des origines de la de la Tour. Media related to 72 names on the Eiffel Tower at Wikimedia Commons Paris streets named for the 72 scientists
List of the 72 names on the Eiffel Tower
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The location of the names on the tower
List of the 72 names on the Eiffel Tower
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Petiet, Daguerre, Wurtz, Le Verrier, Perdonnet, Delambre, Malus, Breguet, Polonceau, Dumas, Clapeyron, Borda, Fourier, Bichat, Sauvage, Pelouze, Carnot, Lamé
List of the 72 names on the Eiffel Tower
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Cauchy, Belgrand, Regnault, Fresnel, De Prony, Vicat, Ebelmen, Coulomb, Poinsot, Foucault, Delaunay, Morin, Haüy, Combes, Thénard, Arago, Poisson, Monge
List of the 72 names on the Eiffel Tower
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Jamin, Gay-Lussac, Fizeau, Schneider, Le Chatelier, Berthier, Barral, De Dion, Goüin, Jousselin, Broca, Becquerel, Coriolis, Cail, Triger, Giffard, Perrier, Sturm
91.
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
92.
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
93.
W. W. Rouse Ball
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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
W. W. Rouse Ball
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W.W. Rouse Ball
94.
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
95.
University of St Andrews
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The University of St Andrews is a British public research university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland, St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy. St Andrews is made up from a variety of institutions, including three constituent colleges and 18 academic schools organised into four faculties, the university occupies historic and modern buildings located throughout the town. The academic year is divided into two terms, Martinmas and Candlemas, in term time, over one-third of the towns population is either a staff member or student of the university. It is ranked as the third best university in the United Kingdom in national league tables, the Times Higher Education World Universities Ranking names St Andrews among the worlds Top 50 universities for Social Sciences, Arts and Humanities. St Andrews has the highest student satisfaction amongst all multi-faculty universities in the United Kingdom, St Andrews has many notable alumni and affiliated faculty, including eminent mathematicians, scientists, theologians, philosophers, and politicians. Six Nobel Laureates are among St Andrews alumni and former staff, a charter of privilege was bestowed upon the society of masters and scholars by the Bishop of St Andrews, Henry Wardlaw, on 28 February 1411. Wardlaw then successfully petitioned the Avignon Pope Benedict XIII to grant the university status by issuing a series of papal bulls. King James I of Scotland confirmed the charter of the university in 1432, subsequent kings supported the university with King James V confirming privileges of the university in 1532. A college of theology and arts called St Johns College was founded in 1418 by Robert of Montrose, St Salvators College was established in 1450, by Bishop James Kennedy. St Leonards College was founded in 1511 by Archbishop Alexander Stewart, St Johns College was refounded by Cardinal James Beaton under the name St Marys College in 1538 for the study of divinity and law. Some university buildings that date from this period are still in use today, such as St Salvators Chapel, St Leonards College Chapel, at this time, the majority of the teaching was of a religious nature and was conducted by clerics associated with the cathedral. During the 17th and 18th centuries, the university had mixed fortunes and was beset by civil. He described it as pining in decay and struggling for life, in the second half of the 19th century, pressure was building upon universities to open up higher education to women. In 1876, the University Senate decided to allow women to receive an education at St Andrews at a roughly equal to the Master of Arts degree that men were able to take at the time. The scheme came to be known as the L. L. A and it required women to pass five subjects at an ordinary level and one at honours level and entitled them to hold a degree from the university. In 1889 the Universities Act made it possible to admit women to St Andrews. Agnes Forbes Blackadder became the first woman to graduate from St Andrews on the level as men in October 1894
University of St Andrews
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College Hall, within the 16th century St Mary's College building
University of St Andrews
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University of St Andrews shield
University of St Andrews
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St Salvator's Chapel in 1843
University of St Andrews
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The "Gateway" building, built in 2000 and now used for the university's management department
96.
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
97.
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
98.
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
99.
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.
100.
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
101.
Joseph-Louis Lagrange
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Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia or Giuseppe Ludovico De la Grange Tournier, was an Italian and French Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, in 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life, Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He also extended the method to take into account possible constraints and he proved that every natural number is a sum of four squares. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, in calculus, Lagrange developed a novel approach to interpolation and Taylor series. Born as Giuseppe Lodovico Lagrangia, Lagrange was of Italian and French descent and his mother was from the countryside of Turin. He was raised as a Roman Catholic, a career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the University of Turin and his subject was classical Latin. At first he had no enthusiasm for mathematics, finding Greek geometry rather dull. It was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident. Alone and unaided he threw himself into mathematical studies, at the end of a years incessant toil he was already an accomplished mathematician, in that capacity, Lagrange was the first to teach calculus in an engineering school. In this Academy one of his students was François Daviet de Foncenex, Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of tautochrone, Lagrange wrote several letters to Leonhard Euler between 1754 and 1756 describing his results. He outlined his δ-algorithm, leading to the Euler–Lagrange equations of variational calculus, Lagrange also applied his ideas to problems of classical mechanics, generalizing the results of Euler and Maupertuis. Euler was very impressed with Lagranges results, Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773. Many of these are elaborate papers, the article concludes with a masterly discussion of echoes, beats, and compound sounds. Other articles in volume are on recurring series, probabilities. The next work he produced was in 1764 on the libration of the Moon, and an explanation as to why the face was always turned to the earth
Joseph-Louis Lagrange
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Joseph-Louis (Giuseppe Luigi), comte de Lagrange
Joseph-Louis Lagrange
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Lagrange's tomb in the crypt of the Panthéon