1.
Turin
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Turin is a city and an important business and cultural centre in northern Italy, capital of the Piedmont region. The population of the city proper is 892,649 while the population of the urban area is estimated by Eurostat to be million inhabitants. The Turin metropolitan area is estimated by the OECD to have a population of million. Turin is well known for its renaissance, baroque, rococo, art nouveau architecture. The city currently hosts some of Italy's best universities, the Turin Polytechnic. Important museums, such as the Museo Egizio and the Mole Antonelliana are also found in the city. The tenth most visited city in Italy in 2008. Turin is ranked third after Milan and Rome, for economic strength. Turin is also home of the Italian automotive industry. The Taurini were an ancient Celto-Ligurian Alpine people, who occupied the upper valley of the Po River, in the centre of modern Piedmont. In 218 BC, they were attacked by Hannibal as he was allied with the Insubres. The Taurini chief town was captured after a three-day siege. As a people they are rarely mentioned in history. It is believed that a Roman colony was established under the name of Castra Taurinorum and afterwards Julia Augusta Taurinorum. In probably 28 BC, the Romans created a military camp, later dedicated to Augustus.
Turin
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From top to bottom, left to right: panorama of the Mole Antonelliana, Valentino Park with the medieval village, Piazza Castello with Palazzo Reale and Palazzo Madama, San Carlo Plaza with the Caval ëd Bronz, the Arco Olimpico and the Lingotto, the sarcophagus of Oki at the Egyptian Museum, a view of the hills, the Po, the Gran Madre, the Monte of Cappuccini and Palatine Towers.
Turin
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The Roman Palatine Towers.
Turin
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Siege of Turin
Turin
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Turin in the 17th century.
2.
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 today's Italy. When it was acquired in 1720, it was a small state with weak institutions. Its final capital was the centre of Savoyard power since the Middle Ages. Beginning in 1324, his successors conquered the island of Sardinia and established de facto their de jure authority. 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 Duke Victor Amadeus II of Savoy. There followed the annexation of Lombardy, the central Italian states and the Two Sicilies, the Papal States. In 238 BC Sardinia became, along with 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, who had also settled in north Africa. In 534 AD it was reconquered by the Romans, but now from Byzantium. It remained a Byzantine province in the 9th century. After that, powerful families of the island assumed control of the land. Starting from Saracens from north Africa harassed the population of the coastal cities.
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
3.
Paris
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Paris is the capital and the most populous city of France. It has a population in 2013 of 2,229,621 within the administrative limits. The agglomeration has grown well beyond the city's administrative limits. The Metropole of Grand Paris was created in 2016, combining its nearest suburbs into a single area for economic and environmental co-operation. Grand Paris has a population of 6.945 million persons. Paris was founded by a Celtic people called the Parisii, who gave the city its name. It retains that position still today. The city is also a major rail, highway, air-transport hub, served by the two international airports Paris-Charles de Gaulle and Paris-Orly. Opened in 1900, the Paris Métro, serves 5.23 million passengers daily. It is the second busiest system in Europe after Moscow Metro. Paris is surrounded by three orbital roads: the Périphérique, the A86 motorway, the Francilienne motorway. Most of France's major universities and écoles are located in Paris, as are France's major newspapers, including Le Monde, Le Figaro, Libération. The rugby union club Stade Français are based in Paris. The 80,000-seat Stade de France, built for the 1998 FIFA World Cup, is located just north in the neighbouring commune of Saint-Denis. Paris hosts the French Open Grand Slam tennis tournament on the red clay of Roland Garros.
Paris
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In the 1860s Paris streets and monuments were illuminated by 56,000 gas lamps, making it literally "The City of Light."
Paris
Paris
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Gold coins minted by the Parisii (1st century BC)
Paris
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The Palais de la Cité and Sainte-Chapelle, viewed from the Left Bank, from the Très Riches Heures du duc de Berry (month of June) (1410)
4.
Greater French Empire
–
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 into Poland. The plot included Bonaparte's Lucien, then serving as speaker of the Council of Five Hundred, Roger Ducos, another Director, Talleyrand. On the following day, troops led by Bonaparte seized control. They dispersed the legislative councils, leaving a legislature to name Bonaparte, Sieyès and Ducos as provisional Consuls to administer the government. The Battle of Marengo inaugurated the political idea, to continue its development until Napoleon's Moscow campaign. He was thought to prepare a new campaign in the East. The Peace of Amiens, which cost control of Egypt, was a temporary truce. Then Napoleon initiated the Concordat of 1801 to control the material claims of the pope. He would have ruling elites from a fusion of the old aristocracy. On 12 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 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 France's side.
Greater French Empire
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The Battle of Austerlitz
Greater French Empire
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Flag
Greater 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.
Greater French Empire
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Napoleon reviews the Imperial Guard before the Battle of Jena, 1806
5.
Piedmont
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It is one of the 20 regions of Italy. Piedmont has a population of about 4.6 million. The capital of Piedmont is Turin. The Piedmont comes from medieval Latin Pedemontium or Pedemontis, i.e. ad pedem montium, meaning "at the foot of the mountains". Piedmont borders for a very small fragment with Emilia Romagna. The geography of Piedmont is 43.3 % mountainous, along with extensive areas of plains. It is the second largest of Italy's 20 regions, after Sicily. 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, then to the upper, then to the lower great Padan Plain. 7.6% of the entire territory is considered protected area. There are 56 different regional parks, one of the most famous is the Gran Paradiso National Park located between Piedmont and the Aosta Valley. It was inhabited by Celtic-Ligurian tribes such as the Taurini and the Salassi. They were later subdued by the Romans, who founded several colonies there including Eporedia. After the fall of the Western Roman Empire, the region was repeatedly invaded by the Goths, Byzantines, Lombards, Franks. In the 9th–10th centuries there were further incursions by the Magyars and Saracens.
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.
6.
France
–
France, officially the French Republic, is a unitary sovereign state and transcontinental country consisting of territory in western Europe and several overseas regions and territories. Overseas France include several island territories in the Atlantic, Pacific and Indian oceans. France has a total population of 66.7 million. It is a semi-presidential republic with the capital in the country's largest city and main cultural and commercial centre. Other urban centres include Marseille, Lyon, Lille, Nice, Toulouse and Bordeaux. During the Iron Age, what is now metropolitan France was inhabited by a Celtic people. France emerged as a major European power with its victory in the Hundred Years' War strengthening state-building and political centralisation. During the Renaissance, a global colonial empire was established, which by the 20th century would be the second largest in the world. The 16th century was dominated by civil wars between Catholics and Protestants. France became Europe's dominant political, military power under Louis XIV. In the 19th century Napoleon established the First French Empire, whose subsequent Napoleonic Wars shaped the course of continental Europe. Following the collapse of the Empire, France endured a tumultuous succession of governments culminating in 1870. Following liberation in 1944, a Fourth Republic was later dissolved in the course of the Algerian War. The Fifth Republic, led by Charles de Gaulle, remains to this day. Algeria and nearly all the other colonies typically retained close economic and military connections with France.
France
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One of the Lascaux paintings: a horse – Dordogne, approximately 18,000 BC
France
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Flag
France
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The Maison Carrée was a temple of the Gallo-Roman city of Nemausus (present-day Nîmes) and is one of the best preserved vestiges of the Roman Empire.
France
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With Clovis ' conversion to Catholicism in 498, the Frankish monarchy, elective and secular until then, became hereditary and of divine right.
7.
Prussia
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Prussia was a historic state originating out of the Duchy of Prussia and the Margraviate of Brandenburg, centered on the region of Prussia. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually effective army. Prussia, 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 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, a state of Germany until 1933. Prussia existed de jure by the Allied Control Council Enactment No. 46 of 25 February 1947. The Prussia derives from the Old Prussians. In the 13th century, the Teutonic Knights -- an organized Catholic military order of German crusaders -- conquered the lands inhabited by them. In 1308, the Teutonic Knights conquered the region of Pomerelia with Gdańsk. In the south, it was Polonised by settlers from Masovia. The Duchy of Prussia in 1618 led to the proclamation of the Kingdom of Prussia in 1701. Prussia exercised most influence in the 18th and 19th centuries. During the 18th century it had a major say in 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.
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
Prussia
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Prussian King's Crown (Hohenzollern Castle Collection)
8.
First French Empire
–
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 into Poland. The plot included Bonaparte's Lucien, then serving as speaker of the Council of Five Hundred, Roger Ducos, another Director, Talleyrand. On the following day, troops led by Bonaparte seized control. They dispersed the legislative councils, leaving a legislature to name Bonaparte, Sieyès and Ducos as provisional Consuls to administer the government. The Battle of Marengo inaugurated the political idea, to continue its development until Napoleon's Moscow campaign. He was thought to prepare a new campaign in the East. The Peace of Amiens, which cost control of Egypt, was a temporary truce. Then Napoleon initiated the Concordat of 1801 to control the material claims of the pope. He would have ruling elites from a fusion of the old aristocracy. On 12 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 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 France's side.
First French Empire
–
The Battle of Austerlitz
First French Empire
–
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
9.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, change. There is a range of views among philosophers as to the exact scope and definition of mathematics. Mathematicians use them to formulate new conjectures. Mathematicians resolve the 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 logic, mathematics developed from counting, calculation, measurement, the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from 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 Euclid's Elements. Galileo Galilei said, "The universe can not become familiar with the characters in which it is written. Without these, one is wandering about in a dark labyrinth." Carl Friedrich Gauss referred 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.
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
10.
Mathematical physics
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Mathematical physics refers to development of mathematical methods for application to problems in physics. It is a branch of applied mathematics, but deals with physical problems. These roughly correspond to historical periods. The rigorous, abstract and advanced re-formulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics. Moreover, they have provided basic ideas in geometry. The theory of partial differential equations are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the eighteenth century until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, aerodynamics. It has connections to molecular physics. Quantum information theory is another subspecialty. The special and general theories of relativity require a rather different type of mathematics. This was theory, which played an important role in both quantum field theory and geometry. This was, however, gradually supplemented by functional analysis in the mathematical description of cosmological well as quantum field theory phenomena. In this area both homological theory are important nowadays.
Mathematical physics
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An example of mathematical physics: solutions of Schrödinger's equation for quantum harmonic oscillators (left) with their amplitudes (right).
11.
Alma mater
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Alma mater is an allegorical Latin phrase for a university or college. In modern usage, it is a university which an individual has attended. The phrase is variously translated as "nourishing mother", "fostering mother", suggesting that a school provides intellectual nourishment to its students. It is related to the alumnus, denoting a university graduate, literally meaning a "nursling" or "one, nourished". The phrase can also denote a hymn associated with a school. Although alma was a common epithet for other mother goddesses, it was not frequently used in conjunction with mater in classical Latin. "Alma Redemptoris Mater" is a 11th century antiphon devoted to Mary. Many European universities have adopted Alma Mater as part of the Latin translation of their official name. Alma Mater Studiorum, refers to its status as the oldest continuously operating university in the world. The ancient Roman world had many statues of the Alma Mater, some still extant. Modern sculptures are found in prominent locations on American university campuses. There is a well-known statue of Alma Mater by Daniel Chester French situated on the steps of Columbia University's Low Library. The University of Illinois at Urbana–Champaign also has an Alma Mater statue by Lorado Taft. 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. The statue was cast in 1919 with Feliciana Villalón Wilson as the inspiration for Alma Mater.
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
12.
University of Turin
–
The University of Turin is a university in the city of Turin in the Piedmont region of north-western Italy. It continues to play an important role in research and training. The University of Turin was founded under the initiative of Prince Ludovico di Savoia. From 1427 to 1436 the seat of the university was transferred to Chieri and Savigliano. It was reestablished by Duke Emmanuel Philibert thirty years later. With the reforms carried out by Victor Amadeus II, the University of Turin became a new model for many other universities. During the 18th century, the University faced an enormous growth in faculty and size, becoming a point of reference of the Italian Positivism. Notable scholars of this period include Cesare Lombroso, Arturo Graf. In the 20th century, the University of Turin was one of the centers of the Italian anti-fascism. By the end of the 1990s, the local campuses of Alessandria, Novara and Vercelli became autonomous units under the new University of Eastern Piedmont. The Bishop, as Rector of Studies, conferred the title on the new doctors. After a series of interruptions in its activities, the university was moved to Chieri and later, to Savigliano. The ducal licenses of 6 October 1436 chairs. The Study, closed with the French occupation, reopened in 1558 with lecturers at Mondovì; it was re-established in Turin in 1566. The opening of the new premises marked a major point in the history of the greatest Piedmontese educational institution.
University of Turin
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Hall of the Rectorate Palace of the University of Turin
University of Turin
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Seal of the University of Turin
University of Turin
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The Minerva Statue in front of the Rectorate Palace at the University of Turin.
University of Turin
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The revolt of the students of Turin University, 1821
13.
Leonhard Euler
–
He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. Euler is also known for his work in mechanics, music theory. Euler was one of the most eminent mathematicians of the 18th century, is held to be one of the greatest in history. He is also widely considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field. He spent most of his adult life in St. Petersburg, Russia, in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." He had two younger sisters: Anna Maria and Maria Magdalena, a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Euler later won this annual prize twelve times.
Leonhard Euler
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Portrait by Jakob Emanuel Handmann (1756)
Leonhard Euler
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1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Leonhard Euler
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Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as " v − e + f = 2".
Leonhard Euler
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Euler's grave at the Alexander Nevsky Monastery
14.
Joseph Fourier
–
The Fourier transform and Fourier's law are also named in his honour. Fourier is also generally credited with the discovery of the effect. Fourier was born at the son of a tailor. He was orphaned at age nine. Through this introduction, he was educated by the Benedictine Order of the Convent of St. Mark. He took a prominent part in promoting the French Revolution, serving on the local Revolutionary Committee. He was imprisoned briefly during the Terror but subsequently succeeded Joseph-Louis Lagrange at the École Polytechnique. Fourier was appointed secretary of the Institut d'Égypte. Cut off by the English fleet, he organized the workshops on which the French army had to rely for their munitions of war. He also contributed mathematical papers to the Egyptian Institute which Napoleon founded at Cairo, with a view of weakening English influence in the East. After the capitulation of the French under General Menou in 1801, Fourier returned to France. In 1801, Napoleon appointed Fourier Prefect of the Department of Isère in Grenoble, where he oversaw other projects. Hence being faithful to Napoleon, he took the office of Prefect. It was while at Grenoble that he began on the propagation of heat. He presented his paper to the Paris Institute on December 21, 1807.
Joseph Fourier
–
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
15.
Giovanni Antonio Amedeo Plana
–
Giovanni Antonio Amedeo Plana was an Italian astronomer and mathematician. Plana was born 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 was one of the students of Joseph-Louis Lagrange. In 1811 he was appointed to the chair of astronomy to the influence of Lagrange. He spent the remainder of his teaching at that institution. Plana's contributions included work on the motions of the Moon, well as integrals, elliptic functions, heat, electrostatics, geodesy. In 1834 he was awarded with the Copley Medal on lunar motion. He became royal, then in 1844 a Baron. At the age of 80 he was granted membership in the prestigious Académie des Sciences. He died in Turin. He is considered one of the Italian scientists of his age. The Plana on the Moon is named in his honor. Biography and a source for this page. O'Connor, 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.
16.
Analytical mechanics
–
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was onward, after Newtonian mechanics. A scalar is a quantity, whereas a vector is represented by direction. The equations of motion are derived by some underlying principle about the scalar's variation. Analytical mechanics takes advantage of a system's constraints to solve problems. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates. Two dominant branches of analytical mechanics are Hamiltonian mechanics. There are other formulations such as Hamilton -- Appell's equation of motion. All equations in any formalism, can be derived from the widely applicable result called the principle of least action. One result is a statement which connects conservation laws to their associated symmetries. Analytical mechanics is not more general than Newtonian mechanics. Rather it is a collection of equivalent formalisms which have broad application. Analytical mechanics is used widely, from fundamental physics to applied mathematics, particularly theory. The methods of analytical mechanics apply with a finite number of degrees of freedom. They can be modified to describe continuous fluids, which have infinite degrees of freedom.
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).
17.
Mathematical analysis
–
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, analytic functions. These theories are usually studied in the context of real and complex functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of Greek mathematics. For instance, an infinite sum is implicit in Zeno's paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes' The Method of a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. The Indian mathematician Bhāskara II used what is now known as Rolle's theorem in the 12th century. His followers at the Kerala school of mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. During this period, techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Instead, Cauchy formulated calculus in terms of geometric infinitesimals.
Mathematical analysis
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A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications to science and engineering.
18.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations. One may also study real numbers in relation to rational numbers, e.g. as approximated by the latter. The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". In particular, arithmetical is preferred as an adjective to number-theoretic. The triples are too many and too large to have been obtained by brute force. The heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly flowered only later. It has been suggested instead that the table was a source of numerical examples for school problems. While Babylonian theory -- or what survives of Babylonian mathematics that can be called thus -- consists of this striking fragment, Babylonian algebra was well developed. Late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt.
Number theory
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A Lehmer sieve, which is a primitive digital computer once used for finding primes and solving simple Diophantine equations.
Number theory
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The Plimpton 322 tablet
Number theory
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Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
Number theory
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Leonhard Euler
19.
Italians
–
Italians are a nation and ethnic group native to Italy who share a common Italian culture, ancestry and speak the Italian language as a mother tongue. Italians have greatly contributed to science, arts, technology, cuisine, sports, jurisprudence and banking both abroad and worldwide. Italian people are generally known to clothing and family values. The term Italian has a history that goes back to pre-Roman Italy. Greek historian Dionysius of Halicarnassus states this account together with the legend that Italy was named after Italus, mentioned also by Aristotle and Thucydides. This period of unification was followed by one of conquest beginning with the First Punic War against Carthage. In the course of the century-long struggle against Carthage, the Romans conquered Sicily, Sardinia and Corsica. The final victor, was accorded the title of Augustus by the Senate and thereby became the first Roman emperor. Emperor Diocletian's administrative division of the empire into two parts in 285 provided only temporary relief; it became permanent in 395. In 313, churches thereafter rose throughout the empire. However, he also moved his capital to Constantinople greatly reducing the importance of the former. Romulus Augustulus, was deposed in 476 by a Germanic foederati general in Italy, Odoacer. His defeat marked the end of the western part of the Roman Empire. Odoacer ruled well after gaining control of Italy in 476. Then he was defeated by Theodoric, the king of another Germanic tribe, the Ostrogoths.
Italians
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Amerigo Vespucci, the notable geographer and traveller from whose name the word America is derived.
Italians
Italians
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Christopher Columbus, the discoverer of the New World.
Italians
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Laura Bassi, the first chairwoman of a university in a scientific field of studies.
20.
Enlightenment Era
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The Enlightenment was an intellectual movement which dominated the world of ideas in Europe in the 18th century. French historians traditionally place the Enlightenment between the year that Louis XIV died, 1789, the beginning of the French Revolution. Some recent historians begin the period in the 1620s, with the start of the scientific revolution. Les philosophes of the period widely circulated their ideas at scientific academies, Masonic lodges, literary salons, coffee houses, through printed books and pamphlets. A variety including liberalism and neo-classicism, trace their intellectual heritage back to the Enlightenment. The Age of Enlightenment was closely associated with the scientific revolution. Earlier philosophers whose work influenced the Enlightenment included Francis Bacon, René Descartes, Baruch Spinoza. The major figures of the Enlightenment included Cesare Beccaria, Voltaire, Denis Diderot, Jean-Jacques Rousseau, David Hume, Immanuel Kant. Benjamin Franklin contributed actively to the scientific and political debates there and brought the newest ideas back to Philadelphia. Thomas Jefferson closely later incorporated some of the ideals of the Enlightenment into the Declaration of Independence. Others like James Madison incorporated them in 1787. The most influential publication of the Enlightenment was the Encyclopédie. The ideas of the Enlightenment played a major role in inspiring the French Revolution, which began in 1789. After the Revolution, the Enlightenment was followed by an intellectual movement known as Romanticism. René Descartes' philosophy laid the foundation for enlightenment thinking.
Enlightenment Era
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German philosopher Immanuel Kant
Enlightenment Era
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History of Western philosophy
Enlightenment Era
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Cesare Beccaria, father of classical criminal theory (1738–1794)
Enlightenment Era
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Like other Enlightenment philosophers, Rousseau was critical of the Atlantic slave trade.
21.
Mathematician
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A mathematician is someone who uses an extensive knowledge of mathematics in his/her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, change. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. It was the Pythagoreans who coined the term "mathematics", with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics. Mathematics in the Islamic world during the Middle Ages followed various modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham. The Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many mathematicians gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking.”
Mathematician
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Euclid (holding calipers), Greek mathematician, known as the "Father of Geometry"
Mathematician
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In 1938 in the United States, mathematicians were desired as teachers, calculating machine operators, mechanical engineers, accounting auditor bookkeepers, and actuary statisticians
Mathematician
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Archimedes, c. 287 – 212 BC
Mathematician
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Brahmagupta, c. 598 - 670
22.
Astronomer
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An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside of the scope of Earth. Examples of fields astronomers work on include: planetary science, solar astronomy, the formation of galaxies. There are also related but distinct subjects like cosmology which studies the Universe as a whole. Astronomers usually fit into two types: Observational astronomers make direct observations of planets, stars and galaxies, analyse the data. Theoretical astronomers create and investigate models of things that cannot be observed. They use this data to create simulations to theorize how celestial bodies work. There are further subcategories inside these two main branches of astronomy such as cosmology. That distinction the terms "astronomer" and "astrophysicist" are interchangeable. Professional astronomers are highly educated individuals who typically are employed by research universities. The number of professional astronomers in the United States is actually quite small. The American Astronomical Society, the major organization of professional astronomers in North America, has approximately 7,000 members. This number includes scientists from other fields such as engineering, whose research interests are closely related to astronomy. The International Astronomical Union comprises almost 10,145 members from 70 different countries who are involved in astronomical research at the Ph.D. level and beyond. Before CCDs, photographic plates were a common method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year.
Astronomer
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The Astronomer by Johannes Vermeer
Astronomer
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Galileo is often referred to as the Father of modern astronomy
Astronomer
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Guy Consolmagno (Vatikan observatory), analyzing a meteorite, 2014
Astronomer
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Emily Lakdawalla at the Planetary Conference 2013
23.
Classical mechanics
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In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the largest subjects in science, engineering and technology. It is also widely known as Newtonian mechanics. Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, well as astronomical objects, such as spacecraft, planets, stars, galaxies. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases and other specific sub-topics. When classical mechanics can not apply, such as at the quantum level with high speeds, quantum field theory becomes applicable. Since these aspects of physics were developed long before the emergence of quantum relativity, some sources exclude Einstein's theory of relativity from this category. However, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most accurate form. Later, more general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. They extend substantially beyond Newton's work, particularly through their use of analytical mechanics. The following introduces the basic concepts of classical mechanics. For simplicity, it often models real-world objects as point particles. The motion of a particle is characterized by a small number of parameters: its position, mass, the forces applied to it. Each of these parameters is discussed in turn.
Classical mechanics
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Sir Isaac Newton (1643–1727), an influential figure in the history of physics and whose three laws of motion form the basis of classical mechanics
Classical mechanics
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Diagram of orbital motion of a satellite around the earth, showing perpendicular velocity and acceleration (force) vectors.
Classical mechanics
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Hamilton 's greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics.
24.
Jean le Rond d'Alembert
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Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, music theorist. Until 1759 he was also co-editor with Denis Diderot of the Encyclopédie. D'Alembert's formula for obtaining solutions to the wave equation is named after him. The wave equation is sometimes referred to as d'Alembert's equation. Born in Paris, d'Alembert was the natural son of an artillery officer. Destouches was abroad at the time of d'Alembert's birth. Days after birth his mother left him on the steps of the Saint-Jean-le-Rond de Paris church. According to custom, he was named after the patron saint of the church. Destouches secretly paid for the education of Jean le Rond, but did not want his paternity officially recognized. D'Alembert first attended a private school. The chevalier Destouches left d'Alembert an annuity of 1200 livres on his death in 1726. Under the influence of the Destouches family, at the age of twelve d'Alembert entered the Jansenist Collège des Quatre-Nations. Here he studied the arts, graduating as baccalauréat en arts in 1735. In his later life, D'Alembert scorned the Cartesian principles he had been taught by the Jansenists: the vortices". The Jansenists steered D'Alembert toward an ecclesiastical career, attempting to deter him from pursuits such as poetry and mathematics.
Jean le Rond d'Alembert
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Jean-Baptiste le Rond d'Alembert, pastel by Maurice Quentin de La Tour
25.
Prussian Academy of Sciences
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In the 18th century it was a French-language institution, its most active members were Huguenots who had fled religious persecution in France. Unlike other academies, the Prussian Academy was not directly funded out of the state treasury. Frederick granted a suggestion by Leibniz. As Frederick was crowned "King in Prussia" in 1701, creating the Kingdom of Prussia, the academy was renamed Königlich Preußische Sozietät der Wissenschaften. While other academies focused on a few topics, the Prussian Academy was the first to teach both sciences and humanities. In 1710, the statute was set, dividing the academy in two humanities classes. This was not changed until 1830, when the physics-mathematics and the philosophy-history classes replaced the four old classes. The reign of King Frederick II saw major changes to the academy. In 1744, the Nouvelle Société Littéraire and the Society of Sciences were merged into the Königliche Akademie der Wissenschaften. An obligation from the new statute were public calls for ideas on unsolved scientific questions with a monetary reward for solutions. Frederick made French the official language and speculative philosophy the most important topic of study. The membership included Immanuel Kant, Jean D'Alembert, Etienne de Condillac. By d' Alembert emphasized the international Republic of Letters as the vehicle for scientific advance. By 1789, however, the academy had gained an international repute while making major contributions to German culture and thought. Frederick invited Joseph-Louis Lagrange to succeed Leonhard Euler as director; both were world-class mathematicians.
Prussian Academy of Sciences
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Entrance to the former Prussian Academy of Sciences on Unter Den Linden 8. Today it houses the Berlin State Library.
26.
Berlin
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Berlin is the capital and the largest city of Germany as well as one of its 16 states. Due to its location in the European Plain, Berlin is influenced by a seasonal climate. Around one-third of the city's area is composed of forests, parks, gardens, lakes. Berlin in the 1920s was the third largest municipality in the world. Following German reunification in 1990, Berlin again became the capital of a unified Germany. Berlin is a city of culture, politics, media and science. Its economy is based on the service sector, encompassing a diverse range of creative industries, research facilities, media corporations and convention venues. Berlin has a highly complex public transportation network. The metropolis is a popular destination. Significant industries also include IT, biomedical engineering, clean tech, biotechnology, construction and electronics. Modern Berlin is host to many sporting events. Its urban setting has made a sought-after location for international film productions. The city is well known for its festivals, diverse architecture, nightlife, a high quality of living. Over the last decade Berlin has seen the emergence of a entrepreneurial scene. All German place names ending on -ow, -itz and -in, of which there are many east of the River Elbe, are of Slavic origin.
Berlin
Berlin
Berlin
Berlin
27.
French Academy of Sciences
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It was at the forefront of scientific developments in Europe in the 17th and 18th centuries, is one of the earliest Academies of Sciences. Today, it is one of the five Académies of Institut de France. The Academy of Sciences owes its origin to Colbert's plan to create a general academy. He chose a small group of scholars who met on 22 December 1666 in the King's library, thereafter held twice-weekly working meetings there. The first 30 years of the Academy's existence were relatively informal, since no statutes had as yet been laid down for the institution. In contrast to its British counterpart, the Academy was founded as an organ of government. The Academy was expected to avoid discussion of social issues. On 20 Louis XIV gave its first rules. The Academy was installed in Paris. 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 and retook their ancient seats. 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; the head of State became its patron. In the Second Republic, the name returned to Académie des sciences.
French Academy of Sciences
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A heroic depiction of the activities of the Academy from 1698
French Academy of Sciences
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Colbert Presenting the Members of the Royal Academy of Sciences to Louis XIV in 1667
French Academy of Sciences
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The Institut de France in Paris where the Academy is housed
28.
Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations for classical mechanics. He shares credit with Gottfried Wilhelm Leibniz for the development of calculus. Newton's Principia formulated the laws of motion and universal gravitation, which dominated scientists' view of the physical universe for the next three centuries. This work also demonstrated that the motion of objects of celestial bodies could be described by the same principles. Newton formulated an empirical law of cooling, introduced the notion of a Newtonian fluid. He was the second Lucasian Professor of Mathematics at the University of Cambridge. In his later life, he became president of the Royal Society. He served the British government as Warden and Master of the Royal Mint. His father, also named Isaac Newton, had died three months before. Born prematurely, he was a small child; his mother Hannah Ayscough reportedly said that he could have fit inside a mug. Newton's mother had three children from her second marriage. Newton hated farming. Master at the King's 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 schoolyard bully, Newton became the top-ranked student, distinguishing himself mainly by building models of windmills. In June 1661, Newton was admitted on the recommendation of his uncle Rev William Ayscough who had studied there.
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
29.
History of the metric system
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Concepts similar to those behind the metric system had been discussed in the 16th and 17th centuries. The metric system was to be, in the words of philosopher and Condorcet, "for all people for all time". Reference copies for both units were placed in the custody of the French Academy of Sciences. Due to the unpopularity of the new metric system, France had reverted to units similar to those of their old system. In 1837 the metric system was re-adopted by France, also during the first half of the 19th century was adopted by the scientific community. Maxwell proposed three base units: length, time. This concept attempts to describe electromagnetic forces in terms of these units encountered difficulties. The mole was added as a seventh unit in 1971. The practical implementation of the metric system was the system implemented by French Revolutionaries towards the end of the 18th century. Its key features were that: It was decimal in nature. It derived its unit sizes from nature. Units that have different dimensions are related to each other in a rational manner. Prefixes are used to denote sub-multiples of its units. These features had already been explored and expounded by various scholars and academics prior to the French metric system being implemented. Simon Stevin is credited with introducing the decimal system into general use in Europe.
History of the metric system
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Frontispiece of the publication where John Wilkins proposed a metric system of units in which length, mass, volume and area would be related to each other
History of the metric system
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James Watt, British inventor and advocate of an international decimalized system of measure
History of the metric system
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A clock of the republican era showing both decimal and standard time
History of the metric system
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Repeating circle – the instrument used for triangulation when measuring the meridian
30.
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 under Napoleon although the form of the government changed several times. Under the Legislative Assembly, before the proclamation of the First Republic, France was engaged in war with Prussia and Austria. The foreign threat deepened the passion and sense of urgency among the various factions. This became known as the September Massacres. The resulting Convention was founded with the dual purpose of drafting a new constitution. The Convention's first act was to officially strip the king of all political powers. On 21 January, he was executed by guillotine. Throughout spring of 1793, Paris was plagued by food riots and mass hunger. The new Convention occupied instead with matters of war. Members of groups were executed including the Hébertists and the Dantonists. After the arrest and execution of Robespierre in July 1794, the surviving Girondins were reinstated. The National Convention adopted the Constitution of the Year III. They reestablished freedom of worship, most importantly, initiated elections for a new legislative body. On 3 the Directory was established.
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
31.
Bureau des Longitudes
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During the 19th century, it was responsible for synchronizing clocks across the world. It was headed by François Arago and Henri Poincaré. The Bureau now still meets monthly to discuss topics related to astronomy. As a result, the Bureau was established with all other astronomical establishments throughout France. By a decree of 30 January 1854, the Bureau's mission was extended to embrace geodesy, standardisation and astronomical measurements. This decree focused the efforts of the Bureau on time and astronomy. The Bureau was successful at setting a universal time in Paris via air pulses sent through pneumatic tubes. The French Bureau of Longitude established a commission in the year 1897 to extend the metric system to the measurement of time. Poincaré took its work very seriously, writing several of its reports. He was a fervent believer in a metric system. But he lost the battle. The French government was not prepared to go it alone. After three years of hard work, the commission was dissolved in 1900. Since 1970, the board has been constituted with 3 nominated by the Académie des Sciences. Since 1998, practical work has been carried out by the Institut de mécanique céleste et calcul des éphémérides.
Bureau des Longitudes
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ABBE GREGOIRE (1750-1831).
32.
Functional (mathematics)
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Its use originates in the calculus of variations, where one searches for a function that minimizes a given functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional. The mapping x 0 ↦ f is a function, where x0 is an argument of a function f. Integrals such as f ↦ I = ∫ Ω H μ form a special class of functionals. They map a f into a real number, provided that H is real-valued. Otherwise it is called non-local. This occurs commonly when integrals occur separately in the denominator of an equation such as in calculations of center of mass. Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals: i.e. they carry information on how a functional changes when the input function changes by a small amount. Richard Feynman used functional integrals over the histories formulation of quantum mechanics. This usage implies an integral taken over some space. Linear functional Optimization Tensor Hazewinkel, Michiel, ed. "Functional", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Rowland, Todd. "Functional". MathWorld. Lang, Serge, "III.
Functional (mathematics)
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The Riemann integral is a linear functional on the vector space of Riemann-integrable functions from to.
33.
Lagrange multipliers
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In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. For instance, consider the optimization problem maximize f subject to g = c. We g to have continuous partial derivatives. If f is a maximum of f for the original constrained problem, then there exists λ0 such, a stationary point for the Lagrange function. However, not all stationary points yield a solution of the original problem. Thus, the method of Lagrange multipliers yields a necessary condition for optimality in constrained problems. Sufficient conditions for a minimum or maximum also exist. Such difficulties often arise when one wishes to minimize a subject to fixed outside equality constraints. Consider the two-dimensional problem introduced above maximize f subject to g = 0. If it were, we could walk along g = 0 to get higher, meaning that the starting point wasn't actually the maximum. We can visualize contours of f given by f = d for various values of d, the contour of g given by g = 0. Suppose we walk along the contour line with g = 0. We are interested in finding points where f does not change as we walk, since these points might be maxima. This would mean that the contour lines of f and g are parallel here. The second possibility is that we have reached a "level" part of f, meaning that f does not change in any direction.
Lagrange multipliers
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Figure 1: Find x and y to maximize f (x, y) subject to a constraint (shown in red) g (x, y) = c.
34.
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, the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, biology. In pure mathematics, differential equations mostly concerned with their solutions -- the set of functions that satisfy the equation. 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 by Newton and Leibniz. Jacob Bernoulli proposed the Bernoulli equation in 1695. In 1746, 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. Lagrange sent the solution to Euler. Both further applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fourier's proposal of his equation for conductive diffusion of heat. This partial equation is now taught to every student of mathematical physics. In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one to express these variables dynamically as a equation for the unknown position of the body as a function of time.
Differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
35.
Differential calculus
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In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of the other being integral calculus. The primary objects of study in differential calculus are the derivative of a function, their applications. The derivative of a function at a chosen value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration. Differentiation has applications to nearly all quantitative disciplines. The rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. Derivatives are frequently used to find the minima of a function. Equations involving derivatives are fundamental in describing natural phenomena. Their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, abstract algebra. This relationship can be written as y = f. If f is the equation for a straight line, then there are b such that y = mx + b. It follows that Δy = m Δx.
Differential calculus
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The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line equals the derivative of the function at the marked point.
36.
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 mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature at atomic scales described in quantum mechanics. In the 19th century, Pierre Laplace completed what is today considered the classic interpretation. Initially, its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, presented his axiom system for probability theory in 1933. Most introductions to theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, more. Consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment.
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.
37.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of theory that have experienced advances and have become subject areas in their own right. Physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus the closely related representation theory have many important applications in physics, chemistry, materials science. Group theory is also central to key cryptography. The first class of groups to undergo a systematic study was permutation groups. An early construction due to Cayley exhibited any group as a 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. In this way one proves that for n ≥ 5, the alternating group An is simple, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a algebraic equation of degree n' ≥ 5 in radicals. The important class of groups is given by matrix groups, or linear groups. Here G is a set consisting of invertible matrices of given n over a field K, closed under the products and inverses. Such a group acts on the n-dimensional space Kn by linear transformations. In the case of permutation groups, X is a set; for matrix groups, X is a space.
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.
38.
Calculus
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It has two major branches, integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed by Isaac Newton and Gottfried Leibniz. Calculus has widespread uses in science, engineering and economics. Calculus is a part of modern education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". Calculus is also used for naming theories of computation, such as propositional calculus, calculus of variations, lambda calculus, process calculus. The method of exhaustion was later reinvented by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. Indian mathematicians gave a semi-rigorous method of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. The infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed 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 term.
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
39.
Lagrange interpolation
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In numerical analysis, Lagrange polynomials are used for polynomial interpolation. Although named after Joseph Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography. Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. And changing the points x j requires recalculating the entire interpolant, so it is often easier to use Newton polynomials instead. On the other hand, if also y i = y j, then those two points would actually be one single point. So: L = ∑ j = 0 k y j ℓ j = ∑ j = 0 k y j δ j i = y i. Thus the function L is a polynomial with degree at most k and where L = y i. Additionally, the interpolating polynomial is unique, as shown by the unisolvence theorem at the polynomial interpolation article. Solving an interpolation problem leads to a problem in linear algebra amounting to inversion of a matrix. This construction is analogous to the Chinese Remainder Theorem. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears. The Lagrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial. Therefore, it is preferred in proofs and theoretical arguments.
Lagrange interpolation
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Example of Lagrange polynomial interpolation divergence.
Lagrange interpolation
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This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L (x) (in black), which is the sum of the scaled basis polynomials y 0 ℓ 0 (x), y 1 ℓ 1 (x), y 2 ℓ 2 (x) and y 3 ℓ 3 (x). The interpolation polynomial passes through all four control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points.
40.
Taylor series
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The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715. A function can be approximated by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error introduced by the use of such an approximation. The polynomial formed by taking some initial terms of the Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of that function's Taylor polynomials as the degree increases, provided that the limit exists. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function, equal to its Taylor series in an open interval is known as an analytic function in that interval. + f ″ 2! 2 + f ‴ 3! 3 + ⋯. Which can be written as ∑ n = 0 ∞ f n! N where n! Denotes the factorial of f denotes the nth derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and 0 and 0! are both defined to be 1. When a = 0, the series is also called a Maclaurin series.
Taylor series
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As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin(x) and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.
41.
Three-body problem
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The three-body problem is a special case of the n-body problem. Historically, the first three-body problem to receive extended study was the one involving the Moon, the Earth and the Sun. In an 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". They submitted their competing first analyses in 1747. It was in connection with these researches, in the 1740s, that the name "three-body problem" began to be commonly used. An account published by Jean le Rond d'Alembert indicates that the name was first used in 1747. In 1887, mathematicians Heinrich Bruns and Henri Poincaré showed that there is no 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 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. The general statement for the three problem is as follows. 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.
Three-body problem
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Figure 1: Configuration of the Sitnikov Problem
Three-body problem
42.
Lagrangian point
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The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to orbit with them. There are five such points, labeled L1 to L5, all in the orbital plane of the two large bodies. The two latter points are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies. Several planets have satellites near their L4 and L5 points with respect with Jupiter in particular having more than a million of these. 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 three-body problem. In the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits. It is the most intuitively understood of the Lagrangian points: the one where the gravitational attraction of M2 partially cancels M1's gravitational attraction. The closer to Earth the object is, the greater this effect is. At the L1 point, the orbital period of the object becomes exactly equal to Earth's orbital period. L1 is about million kilometers from Earth. The L2 point lies on the line beyond the smaller of the two. Here, the gravitational 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 orbital period of an object would normally be greater than that of Earth.
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)
43.
Newtonian mechanics
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In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the largest subjects in science, engineering and technology. It is also widely known as Newtonian mechanics. Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, well as astronomical objects, such as spacecraft, planets, stars, galaxies. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases and other specific sub-topics. When classical mechanics can not apply, such as at the quantum level with high speeds, quantum field theory becomes applicable. Since these aspects of physics were developed long before the emergence of quantum relativity, some sources exclude Einstein's theory of relativity from this category. However, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most accurate form. Later, more general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. They extend substantially beyond Newton's work, particularly through their use of analytical mechanics. The following introduces the basic concepts of classical mechanics. For simplicity, it often models real-world objects as point particles. The motion of a particle is characterized by a small number of parameters: its position, mass, the forces applied to it. Each of these parameters is discussed in turn.
Newtonian mechanics
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Sir Isaac Newton (1643–1727), an influential figure in the history of physics and whose three laws of motion form the basis of classical mechanics
Newtonian mechanics
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Diagram of orbital motion of a satellite around the earth, showing perpendicular velocity and acceleration (force) vectors.
Newtonian mechanics
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Hamilton 's greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics.
44.
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. Newton's 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. Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, as a special case of Noether's theorem. Lagrangian mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of physics. It can also be applied to other systems by analogy, for instance to coupled electric circuits with inductances and capacitances. Lagrangian mechanics is widely used to solve mechanical problems in physics and engineering when Newton's formulation of classical mechanics is not convenient. Lagrangian mechanics applies to the dynamics of particles, fields are described using a Lagrangian density. Lagrange's equations are also used in optimisation problems of dynamic systems. In mechanics, Lagrange's equations of the second kind are used much more than those of the first kind. Suppose we have a bead sliding around on a wire, or a swinging simple pendulum, etc. 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 system of N point particles with masses m1, m2... mN, each particle has a position vector, denoted r1, r2... rN.
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)
45.
W.W. Rouse Ball
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Walter William Rouse Ball, known as W. W. Rouse Ball, was a British mathematician, lawyer, fellow at Trinity College, Cambridge from 1878 to 1905. He was also the founding president of the Cambridge Pentacle Club in 1919, one of the world's oldest magic societies. Ball was the heir of Walter Frederick Ball, of 3, St John's Park Villas, South Hampstead, London. He remained one for the rest of his life. He is buried in Cambridge. He is commemorated in the naming of the small pavilion, now used as changing 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:.
W.W. Rouse Ball
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W.W. Rouse Ball
46.
French people
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The French are an ethnic group and nation who are identified with the country of France. This connection may be legal, cultural. France was still regional differences in the late 19th century. According to the first article of the French Constitution, is to be a citizen of France, regardless of one's origin, race, or religion. The debate concerning the integration of this view with the principles underlying the European Community remains open. A large number of foreigners have traditionally been succeeded in doing so. Indeed, the country has long valued its openness, the quality of services available. Application for French citizenship is often interpreted as a renunciation of previous allegiance unless a dual citizenship agreement exists between the two countries. European citizens enjoy formal rights to employment in the state sector. Seeing itself as an inclusive nation with universal values, France strongly advocated assimilation. However, the success of such assimilation has recently been called into question. There is increasing dissatisfaction within, growing ethno-cultural enclaves. The 2005 French riots in some impoverished suburbs were an example of such tensions. However they should not be interpreted as ethnic conflicts but as social conflicts born out of socioeconomic problems endangering proper integration. The name "France" etymologically derives from the territory of the Franks.
French people
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Louis XIV of France "The Sun-King"
French people
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Row 1: Joan of Arc
French people
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Liberty Leading the People by Eugène Delacroix
French people
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French people in Paris, August 1944
47.
Kingdom of France
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The Kingdom of France was a medieval and early modern monarchy in Western Europe, the predecessor of the modern French Republic. It was one of the most powerful states in Europe, the Hundred Years' War. It was also an colonial power, with possessions around the world. France originated with the Treaty of Verdun. A branch of the Carolingian dynasty founded the Capetian dynasty. The territory remained known as Francia and its ruler well into the High Middle Ages. The first king calling Roi de France was Philip II, in 1190. France continued to be ruled by 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 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 yet a part of France. Subsequently France was defeated by Spain in the ensuing Italian Wars. Religiously France became divided between a Protestant minority, the Huguenots. After a series of the Wars of Religion, tolerance was granted to the Huguenots in the Edict of Nantes. France laid claim to large stretches of North America, known collectively as New France.
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.
48.
Roman Catholic
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The Catholic Church, also known as the Roman Catholic Church, is the largest Christian church, with more than 1.27 billion members worldwide. As one of the oldest religious institutions in the world, it has played a prominent role in the history of Western civilisation. Headed by the Bishop of Rome, known as the pope, its doctrines are summarised in the Nicene Creed. The Catholic Church is notable within Western Christianity for seven sacraments. The Catholic Church maintains that the doctrine on faith and morals that it declares as definitive is infallible. Of the seven sacraments, the principal one is the celebrated liturgically in the Mass.. The church teaches that through consecration by a priest wine become the body and blood of Christ. The Catholic Church practises closed communion, with only baptised members in a state of grace ordinarily permitted to receive the Eucharist. Mary is venerated as Queen of Heaven and honoured in numerous Marian devotions. The Catholic Church is the largest non-government provider in the world. The catholic is derived from the Greek word καθολικός, which means "universal". Katholikos is associated with a contraction of the phrase καθ' ὅλου, which means "according to the whole". Catholic was first used to describe the church in the 2nd century. The known use of the phrase "the catholic church" occurred in the letter from Saint Ignatius of Antioch to the Smyrnaeans, written about 110 AD. The name "Catholic Church" is only the website of the Holy See.
Roman Catholic
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Saint Peter's Basilica, Vatican City
Roman Catholic
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St. Peter's Basilica, Vatican City
Roman Catholic
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Pope Francis, elected in the papal conclave, 2013
Roman Catholic
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Traditional graphic representation of the Trinity: The earliest attested version of the diagram, from a manuscript of Peter of Poitiers ' writings, c. 1210
49.
Agnostic
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Agnosticism is the view that certain metaphysical claims – such as the existence of God or the supernatural – are unknown and perhaps unknowable. Agnosticism is a doctrine or set of tenets rather than a religion as such. Thomas Henry Huxley, an English biologist, coined the word "agnostic" in 1869. The Nasadiya Sukta in the Rigveda is agnostic about the origin of the universe. Agnosticism is of the essence of modern. It simply means that a man shall not say he believes that which he has no scientific grounds for professing to believe. Consequently, agnosticism puts aside not only the greater part of popular theology, but also the greater part of anti-theology. Agnosticism, in fact, is not a creed, but a method, the essence of which lies in the rigorous application of a single principle... And negatively: In matters of the intellect do not pretend that conclusions are certain which are not demonstrated or demonstrable. Being a scientist, above all else, Huxley presented agnosticism as a form of demarcation. A hypothesis with testable evidence is not an scientific claim. As such, there would be no way to test said hypotheses, leaving the results inconclusive. His agnosticism was not compatible with forming a belief as to the truth, or falsehood, of the claim at hand. Karl Popper would also describe himself as an agnostic. Others have redefined this concept, making it compatible with forming a belief, only incompatible with absolute certainty.
Agnostic
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Thomas Henry Huxley
Agnostic
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Robert G. Ingersoll
Agnostic
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Bertrand Russell
50.
Charles Emmanuel III of Sardinia
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Charles Emmanuel III was the Duke of Savoy and King of Sardinia from 1730 until his death. Charles Emmanuel was born a Prince of Savoy to his first wife the French Anne Marie d'Orléans. From his birth he was styled as the Duke of Aosta. Charles Emmanuel was the second of three males that would be born to his parents. His older brother died in 1715 and Charles Emmanuel then became heir apparent. 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, leaving the throne to Charles. He was not loved by Victor Amadeus, consequently received an incomplete education. He however acquired noteworthy knowledge in the military field along his father. In summer, 1731, after having recovered from a potentially fatal illness, Victor Amadeus returned to the throne. The old king was confined to the Castle of Rivoli, where he later 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 on 28 October 1733 Charles Emmanuel 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 Mantua in exchange.
Charles Emmanuel III of Sardinia
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Charles Emmanuel III
Charles Emmanuel III of Sardinia
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The young Charles Emmanuel as duke of Aosta, by an unknown artist.
Charles Emmanuel III of Sardinia
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A portrait of a young Charles Emmanuel
Charles Emmanuel III of Sardinia
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The children of Charles and his second wife; (L-R) Eleonora; Victor Amadeus; Maria Felicita and Maria Luisa Gabriella.
51.
Edmond Halley
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Edmond Halley, FRS was an English astronomer, geophysicist, mathematician, meteorologist, physicist, best known for computing the orbit of Halley's Comet. Halley was the second Astronomer Royal in Britain, succeeding John Flamsteed. He was born in east London. Edmond Halley Sr. came from a Derbyshire family and was a wealthy soap-maker in London. As a child, he was very interested in mathematics. Halley studied at The Queen's College, Oxford. While still an undergraduate, he published papers on sunspots. Halley returned in May 1678. In the following year Halley went 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. He observed and verified the quality of Hevelius' observations. In 1679 he published the results from his observations as Catalogus Stellarum Australium which included details of 341 southern stars. These additions to contemporary star maps earned comparison with Tycho Brahe: e.g. "the southern Tycho" as described by Flamsteed. He was elected as a Fellow of the Royal Society at the age of 22. In 1686, he published the second part of the results from his Helenian expedition, being a chart on trade winds and monsoons.
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
52.
Charles Emmanuel III
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Charles Emmanuel III was the Duke of Savoy and King of Sardinia from 1730 until his death. Charles Emmanuel was born a Prince of Savoy to Victor Amadeus II of Savoy and his first wife the French Anne Marie d'Orléans. From his birth Charles Emmanuel was styled as the Duke of Aosta. He was the second of three males that would be born to his parents. His older brother died in Charles Emmanuel then became heir apparent. Yet Charles Emmanuel retained his new title of King. However, Victor Amadeus in his late years was dominated by sadness, probably under the effect of some mental illness. In the end, on 3 September 1730, Charles Emmanuel abdicated, leaving the throne to Charles. Charles Emmanuel consequently received an incomplete education. Charles Emmanuel however acquired noteworthy knowledge along his father. In summer, 1731, after having recovered from a potentially fatal illness, Victor Amadeus returned to the throne. The old king was confined to the Castle of Rivoli, where he later died to Charles. In the War of the Polish Succession Charles Emmanuel sided with the French - backed I. After the treaty of alliance signed on 28 October 1733 Charles Emmanuel marched on Milan and occupied Lombardy without significant losses. However, when France tried to convince Philip V of Spain to join the coalition, Charles Emmanuel asked to receive Milan and Mantua in exchange.
Charles Emmanuel III
–
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.
53.
Tautochrone
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The time is equal to π times the square root of the radius over the acceleration of gravity. The tautochrone curve is the same as the brachistochrone curve for any given point. The attempt to identify this curve, was solved by Christiaan Huygens in 1659. He proved geometrically in his Horologium Oscillatorium, originally published in 1673, that the curve was a cycloid. This solution was later used to attack the problem of the brachistochrone curve. Jakob Bernoulli solved the problem using calculus in a paper that saw the first published use of the term integral. These attempts proved to not be useful for a number of reasons. First, the bending of the string causes friction, changing the timing. Second, there were much more significant sources of timing errors that overwhelmed any theoretical improvements that traveling on the tautochrone curve helps. Finally, the "circular error" of a pendulum decreases as length of the swing decreases, so better clock escapements could greatly reduce this source of inaccuracy. Later, the mathematicians Joseph Louis Lagrange and Leonhard Euler provided an analytical solution to the problem. If the particle's position is parametrized by the arclength s from the lowest point, the kinetic energy is proportional to s 2. The potential energy is proportional to the height y. Y = s 2 where the constant of proportionality has been set to 1 by changing units of length. 8 x = 2 sin θ cos θ + 2 θ = sin 2 θ + 2 θ.
Tautochrone
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Schematic of a cycloidal pendulum.
Tautochrone
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Four balls slide down a cycloid curve from different positions, but they arrive at the bottom at the same time. The blue arrows show the points' acceleration along the curve. On the top is the time-position diagram.
54.
Pierre Louis Maupertuis
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Pierre Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the first President of the Prussian Academy of Science, at the invitation of Frederick the Great. Maupertuis made an expedition to Lapland to determine the shape of the Earth. His work in natural history is interesting in relation to modern science, since he touched for life. Maupertuis was born to a moderately wealthy family of merchant-corsairs. Renė, had been involved in a number of enterprises that were central to the monarchy so that he thrived socially and politically. In 1723 he was admitted to the Académie des Sciences. In the 1730s, the shape of the Earth became a flashpoint among rival systems of mechanics. Maupertuis, based on his exposition of Newton predicted that the Earth should be oblate, while his rival Jacques Cassini measured it astronomically to be prolate. His results, which he published in a book detailing his procedures, essentially settled the controversy in his favor. The book included an account of the Käymäjärvi Inscriptions. On his home he became a member of almost all the scientific societies of Europe. He also expanded into the biological realm, anonymously publishing a book, part popular science, part erotica: Vénus physique. Finding his health declining, he retired to the south of France, but went in 1758 to Basel, where he died a year later. Maupertuis' difficult disposition involved him in constant quarrels, of which his controversies during the latter part of his life are examples.
Pierre Louis Maupertuis
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Maupertuis, wearing " lapmudes " from his Lapland expedition.
Pierre Louis Maupertuis
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Lettres
55.
Vibrating string
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A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with i.e. constant pitch. If the tension of the string is correctly adjusted, the sound produced is a musical note. Vibrating strings are the basis of string instruments such as guitars, pianos. Let Δ x be the length of a piece of string, μ its linear density. T 2 x = T 2 cos ≈ T. If both angles are small, the net horizontal force is zero.. . Once the speed of propagation is known, the frequency of the sound produced by the string can be calculated. A more controllable effect can be obtained using a stroboscope. This device allows matching the frequency of the xenon lamp to the frequency of vibration of the string. In a dark room, this clearly shows the waveform. In the case of a guitar, the 6th string pressed to the third fret gives a G at 97.999 Hz. Fretted instruments Musical acoustics Vibrations of a circular drum Melde's experiment 3rd bridge String resonance Reflection phase change Molteno, T. C. A.; N. B. Tufillaro.
Vibrating string
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This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (October 2010)
Vibrating string
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Vibration, standing waves in a string. The fundamental and the first 5 overtones in the harmonic series.
56.
Brook Taylor
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Brook Taylor FRS was an English mathematician, best known for Taylor's theorem and the Taylor series. Brook Taylor was born in Edmonton to John Taylor of Bifrons House in Patrixbourne, Kent, Olivia Tempest, daughter of Sir Nicholas Tempest, Bart. of Durham. He entered St John's College, Cambridge, as a fellow-commoner in 1701, took degrees of LL.B. and LL.D. in 1709 and 1714, respectively. Taylor's Methodus Incrementorum Directa et Inversa added a new branch to higher mathematics, now called the "calculus of finite differences". Among ingenious applications, he used it to determine the form of movement of a vibrating string, by him first successfully reduced to mechanical principles. From 1715 his studies took a religious bent. He corresponded on the subject of Nicolas Malebranche's tenets. By the date of his father's death in 1729 he had inherited the Bifrons estate. Taylor's fragile health gave way; he died aged 46, on 30 November 1731 at Somerset House, London. He was buried in London on 2 December 1731, in the churchyard of St Anne's, Soho. Short papers by Taylor were published in Phil. Trans. vols. xxvii to xxxii, including accounts of some interesting experiments in magnetism and capillary attraction. A French translation was published in 1757. In Methodus Incrementorum, Taylor gave the satisfactory investigation of astronomical refraction. Taylor, Brook, Methodus Incrementorum Directa et Inversa, London: William Innys.
Brook Taylor
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Brook Taylor (1685-1731)
Brook Taylor
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Methodus incrementorum directa et inversa, 1715
Brook Taylor
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Brook Taylor
57.
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 reflecting surface from the listener. A true echo is a single reflection of the sound source. The word echo derives from the Greek ἠχώ, itself from ἦχος, "sound". Some animals use echo for location navigation, such as cetaceans and bats. Acoustic waves are reflected by other hard surfaces, such as mountains and privacy fences. The reason of reflection may be explained as a discontinuity in the medium. This can delay to be perceived distinctly. When the echo itself, is reflected multiple times from multiple surfaces, the echo is characterized as a reverberation. The human ear can not distinguish echo from the 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 reflecting object is 343m away. In nature, canyon walls or rock cliffs facing water are the most common natural settings for hearing echoes. The strength of echo is frequently measured in sound pressure level relative to the directly transmitted wave.
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
58.
Beat (acoustics)
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When tuning instruments that can produce sustained tones, beats can readily be recognized. The volume varies like in a tremolo as the sounds alternately destructively. As the two tones gradually approach unison, the beating may become so slow as to be imperceptible. Instead, it is perceived above. It can be said that cosine term is an envelope for the higher frequency one, ie that its amplitude is modulated. The frequency of the modulation is 2, 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 versa. A physical interpretation is that they interfere constructively. When it is zero, they interfere destructively. Beats occur also in sounds of different volumes, though calculating them mathematically is not so easy. In the case of perfect fifth, the third harmonic of the bass note beats with the second harmonic of the other note. Musicians commonly use interference beats to objectively tuning at the unison, perfect fifth, or other simple harmonic intervals. 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 interference beats as their main focus.
Beat (acoustics)
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Diagram of beat frequency
59.
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 finite sequence has defined first and last terms, whereas a series continues indefinitely. Given an infinite sequence, a series is informally the result of adding all those terms together: a1 + a2 + a3 + ···. These can be written more compactly using the summation symbol ∑. A value may not always be given to such an infinite sum, and, in this case, the series is said to be divergent. The terms of the series are often produced according to a rule, such as by a formula, or by an algorithm. To emphasize that there are an infinite number of terms, a series is often called an infinite series. The study of infinite series is a major part of mathematical analysis. Series are used through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in quantitative disciplines such as physics, computer statistics and finance. This definition is usually written as L = ∑ n = 0 ∞ a n ⇔ L = lim k → ∞ S k. When the set is the natural numbers I = the function a: N ↦ G is a sequence denoted by a = a n. This definition is usually written as L = ∑ n = 0 ∞ a n ⇔ L = lim k → ∞ S k. A series ∑ an is said to ` be convergent' when the SN of partial sums has a finite limit. If the limit of SN is infinite or does not exist, the series is said to diverge.
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.
60.
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, of either "head" or "tail". This type of probability is also called a priori probability. Probability theory is also used to describe the underlying mechanics and regularities of complex systems. For example, tossing a fair coin twice will yield "head-head", "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 0.25. This interpretation considers probability to be the relative frequency "in the long run" of outcomes. Subjectivists assign numbers per subjective probability, i.e. as a degree of belief. The most popular version of subjective probability is Bayesian probability, which includes knowledge well as experimental data to produce probabilities. The knowledge is represented by some prior distribution. These data are incorporated in a likelihood function.
Probability
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Christiaan Huygens probably published the first book on probability
Probability
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Gerolamo Cardano
Probability
–
Carl Friedrich Gauss
61.
Principle of least action
–
This article discusses the history of the principle of least action. For the application, please refer to action. In relativity, a different action must be maximized. The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, even general relativity. It was historically called "least" because its solution requires finding the path that has the least change from nearby paths. The stationary action method helped in the development of quantum mechanics. Hamilton's principle exemplify the principle of stationary action. The principle is preceded by earlier ideas in surveying and optics. Rope stretchers in ancient Egypt stretched corded ropes to measure the distance between two points. Ptolemy, in his Geography, emphasized that one must correct for "deviations from a straight course". Hero of Alexandria later showed that this path was least time. Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote in 1744 and 1746. However, evidence shows that Gottfried Leibniz preceded both by 39 years. The starting point is the action, denoted S, of a physical system. Mathematically the principle is δ S = 0 where δ means a small change.
Principle of least action
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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).
62.
Integral calculus
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with differentiation, being the other. The area above the x-axis adds below the x-axis subtracts from the total. The operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is written: F = ∫ f d x. The integrals discussed in this article are those termed definite integrals. A mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a region by breaking the region into thin vertical slabs. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. A similar method was independently developed by Liu Hui, who used it to find the area of the circle. This method was later used by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere. The significant advances in integral calculus did not begin to appear until the 17th century. Further steps were made in the 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 fundamental theorem of calculus.
Integral calculus
–
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
63.
Pierre de Fermat
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Fermat optics. He is best known for Fermat's Last Theorem, which he described at the margin of a copy of Diophantus' Arithmetica. Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France -- the 15th-century mansion where Fermat was born is now a museum. His mother was either Claire de Long. Pierre was almost certainly brought up in the town of his birth. It was probably at the Collège de Navarre in Montauban. Fermat received a bachelor in civil law in 1626, before moving to Bordeaux. There Fermat became much influenced by the work of François Viète. Fermat held this office for the rest of his life. He thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fermat communicated most of his work in letters to friends, often with no proof of his theorems. In some of these letters to his friends Fermat explored many of the fundamental ideas of calculus before Newton or Leibniz. He was a trained lawyer making mathematics more of a hobby than a profession. Nevertheless, Fermat made important contributions to analytical geometry, probability, theory calculus. Secrecy was common in mathematical circles at the time.
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
–
Holographic will handwritten by Fermat on 4 March 1660 — kept at the Departmental Archives of Haute-Garonne, in Toulouse
64.
Square number
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For example, 9 is a square number, since it can be written as × 3. The equivalent exponentiation n2, usually pronounced as "n squared". The name square number comes from the name of the shape; see below. Square numbers are non-negative. Another way of saying that a integer is a square number, is that its square root is again an integer. For √ 9 = 3, so 9 is a square number. A positive integer that has no square divisors except 1 is called square-free. For a non-negative n, the nth square number is n2, with 02 = 0 being the zeroth one. The concept of square can be extended to some other number systems. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, where the expression ⌊x⌋ represents the floor of the number x. Hence, a square with side length n has area n2. The expression for the square number is n2. The formula follows: n 2 = ∑ k = 1 n. So for example, 52 25 = 1 + 3 + 5 + 7 + 9. There are several recursive methods for computing square numbers.
Square number
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m = 1 2 = 1
65.
N-body problem
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In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, the visible stars. In the 20th century, understanding the dynamics of globular star systems became an important n-body problem. The n-body problem in general relativity is considerably more difficult to solve. To this purpose the two-body problem is discussed below; as is the famous restricted 3-Body Problem. Newton realized it was amongst all the planets was affecting all their orbits. Thus came the awareness and rise of the n-body "problem" in the early 17th century. Ironically, this conformity led to the wrong approach. After Newton's time the n-body problem historically was not stated correctly because it did not include a reference to those interactive forces. Newton implies in his Principia the n-body problem is unsolvable because of those gravitational interactive forces. Newton said in paragraph 21: And hence it is that the attractive force is found in both bodies. The Sun attracts Jupiter and the other planets, similarly the satellites act on one another. Two bodies can be drawn by the contraction of rope between them. This last statement, which implies the existence of interactive forces, is key. The problem of finding the general solution of the n-body problem was considered very challenging.
N-body problem
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The Real Motion v.s. Kepler's Apparent Motion
N-body problem
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Restricted 3-Body Problem
66.
Libration
–
Lunar libration is distinct from the slight changes in the Moon's visual size as seen from Earth. The Moon generally has one hemisphere facing the Earth, due to tidal locking. Therefore, humans' first view of the far side of the Moon resulted on October 7, 1959. However, this simple picture is only approximately true: over time, slightly more than half of the Moon's surface is seen from Earth due to libration. Libration in latitude results from the normal to the plane of its orbit around Earth. Its origin is analogous to how the seasons arise from Earth's revolution about the Sun. Such ` asteroids' have been found co-orbiting with Earth, Jupiter, Mars, Neptune.
Libration
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Theoretical extent of visible lunar surface due to libration in Winkel tripel projection
Libration
67.
Moon
–
The Moon is Earth's only permanent natural satellite. It is the largest among planetary satellites relative to the size of the planet that it orbits. It is the second-densest satellite among those whose densities are known. The average distance of the Moon from the Earth is 1.28 light-seconds. The Moon is thought to have formed not long after Earth. It is the second-brightest regularly visible celestial object in Earth's sky, as measured by illuminance on Earth's surface. Its surface is actually dark, although compared to the sky it appears very bright, with a reflectance just slightly higher than that of worn asphalt. The Moon's gravitational influence produces the ocean tides, the slight lengthening of the day. This matching of visual size will not continue in the far future. This rate is not constant. Since the Apollo 17 mission in 1972, the Moon has been visited only by uncrewed spacecraft. The usual proper name for Earth's natural satellite is "the Moon". Occasionally, the name "Luna" is used. The principal English adjective pertaining to the Moon is lunar, derived from the Latin Luna. A less common adjective is selenic, derived from, derived the prefix "seleno -".
Moon
–
Full moon as seen from Earth's northern hemisphere
Moon
–
The Moon, tinted reddish, during a lunar eclipse
Moon
–
Near side of the Moon
Moon
–
Far side of the Moon
68.
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 according to the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work. The principle of virtual work had always been used in the study of statics. It was used by the Greeks, Renaissance Italians. Working with Leibnizian concepts, Johann Bernoulli made explicit the concept of infinitesimal displacement. He was able to solve problems for both rigid bodies well as fluids. His idea was to convert a dynamical problem by introducing inertial force. Consider a particle that moves along a path, described by a function r from point A, where r, to point B, where r. The δr satisfies the requirement δr = δr = 0. The components of the variation, δr1, δr3, are called virtual displacements. This can be generalized to an arbitrary mechanical system defined by the generalized coordinates qi, i = 1... n. In which case, the variation of the trajectory qi is defined by the virtual displacements δqi, i = 1... n.
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
69.
Euler
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Euler also introduced much of the modern mathematical terminology and notation, particularly such as the notion of a mathematical function. Euler is also known for his work in mechanics, fluid dynamics, optics, music theory. He is held to be one of the greatest in history. Euler is also widely considered to be the most prolific mathematician of all time. His collected works fill more than anybody in the field. Euler spent most of his adult life in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Euler had a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved to the town of Riehen where Euler spent most of his childhood. Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. During that time, Euler was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. In 1726, he completed a dissertation with the title De Sono. At that time, Euler was unsuccessfully attempting to obtain a position at the University of Basel. Euler took second place. He later won twelve times.
Euler
–
Portrait by Jakob Emanuel Handmann (1756)
Euler
–
1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Euler
–
Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as " v − e + f = 2".
Euler
–
Euler's grave at the Alexander Nevsky Monastery
70.
Maupertuis
–
Pierre Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, the first President of the Prussian Academy of Science, at the invitation of Frederick the Great. Maupertuis made an expedition to Lapland to determine the shape of the Earth. His work in natural history is interesting in relation to modern science, since he touched on aspects of heredity and the struggle for life. Maupertuis was born at Saint-Malo, France, to a moderately wealthy family of merchant-corsairs. His father, Renė, had been involved in a number of enterprises that were central to the monarchy so that he thrived socially and politically. In 1723 he was admitted to the Académie des Sciences. In the 1730s, the shape of the Earth became a flashpoint in the battle among rival systems of mechanics. Maupertuis, based on his exposition of Newton predicted that the Earth should be oblate, while his rival Jacques Cassini measured it astronomically to be prolate. His results, which he published in a book detailing his procedures, essentially settled the controversy in his favor. The book included an adventure narrative of the expedition, an account of the Käymäjärvi Inscriptions. On his return home he became a member of almost all the scientific societies of Europe. He also expanded into the biological realm, anonymously publishing a book, part erotica: Vénus physique. Finding his health declining, he retired to the south of France, but went to Basel, where he died a year later. Maupertuis' difficult disposition involved him in constant quarrels, of which his controversies with Samuel König and Voltaire during the latter part of his life are examples.
Maupertuis
–
Maupertuis, wearing " lapmudes " from his Lapland expedition.
Maupertuis
–
Lettres
71.
Frederick the Great
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Frederick II was King of Prussia from 1740 until 1786, the longest reign of any Hohenzollern king. He declared himself King of Prussia after achieving full sovereignty for all historical Prussian lands. Prussia became a leading military power in Europe under his rule. Frederick was affectionately nicknamed Der Alte Fritz by the Prussian people. In his youth, he was more interested in philosophy than the art of war. Frederick sought to run away with his close friend Hans Hermann von Katte. They were caught at King Frederick William I nearly executed his son for desertion. After being pardoned, Frederick was forced to watch the official beheading of Hans. Upon ascending to the Prussian throne, Frederick claimed Silesia during the Silesian Wars, winning military acclaim for himself and Prussia. Near the end of his life, Frederick physically connected most of his realm by conquering Polish territories in the First Partition of Poland. Frederick was an 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", he was a proponent of enlightened absolutism. Frederick pursued religious policies throughout his realm that ranged from tolerance to segregation. Frederick made it possible for men not of noble stock to become judges and senior bureaucrats. He also encouraged immigrants of various faiths to come to Prussia.
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
–
Frederick as Crown Prince (1739)
Frederick the Great
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Rheinsberg Palace, Frederick's residence 1736-1740
72.
Saint Petersburg
–
Saint Petersburg is Russia's second-largest city after Moscow, with five million inhabitants in 2012, an important Russian port on the Baltic Sea. It is politically incorporated as a federal subject. In 1914, the name was changed from Saint Petersburg to Petrograd, in 1924 to Leningrad, in 1991 back to Saint Petersburg. Between 1713–1728 and 1732–1918, Saint Petersburg was the imperial capital of Russia. In 1918, the central government bodies moved to Moscow. Saint Petersburg is the most Westernized city 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 foreign consulates, international corporations, banks, businesses have offices in Saint Petersburg. A small town called "Nyen" grew up around it. He needed a better seaport than Arkhangelsk, on the White Sea to the north and closed to shipping for months during the winter. On May 12 1703, during the Great Northern War, Peter the Great captured Nyenskans, soon replaced the fortress. Tens of thousands of serfs died building the city. Later, the city became the centre of the Saint Petersburg Governorate. During its first few years, the city developed around Trinity Square on the right bank of the Neva, near the Peter and Paul Fortress.
Saint Petersburg
–
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
–
The Bronze Horseman, monument to Peter the Great
Saint Petersburg
–
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
–
Map of Saint Petersburg, 1903
73.
Spain
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Along with France and Morocco, it is one of only three countries to have both Atlantic and Mediterranean coastlines. By population, Spain is the fifth in the European Union, after Italy. Largest city is Madrid, other major urban areas include Barcelona, Valencia, Seville, Bilbao and Málaga. Modern humans first arrived around 35,000 years ago. In the Middle Ages, the area was later by the Moors. Spain is a democracy organised under a constitutional monarchy. It is a developed country with the world's fourteenth largest economy by nominal GDP and sixteenth largest by purchasing power parity. Jesús Luis Cunchillos argues that the root of the span is the Phoenician word spy, meaning "to forge metals". Therefore, i-spn-ya would mean "the land where metals are forged". Don Isaac Abravanel and Solomon ibn Verga, gave an explanation now considered folkloric. This man was a Grecian by birth, but, given a kingdom in Spain. He became related by marriage to the nephew of king Heracles, who also ruled over a kingdom in Spain. Based upon their testimonies, this eponym would have already been by c. 350 BCE. Iberia enters written records as a land populated largely by Basques and Celts. After an arduous conquest, the peninsula came under the rule of the Roman Empire.
Spain
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Lady of Elche
Spain
–
Flag
Spain
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Altamira Cave paintings, in Cantabria.
Spain
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Celtic castro in A Guarda, Galicia.
74.
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 city's administrative limits. The Metropolitan City of Naples had a population of 3,115,320. Naples is the populous urban area in the European Union with a population of between 3 million and 3.7 million. About million people live in the Naples metropolitan area, one of the largest metropolises on the Mediterranean Sea. Naples is one of the oldest continuously inhabited cities in the world. Bronze Age Greek settlements were established in the second millennium BC. Naples remained influential after the fall of the Western Roman Empire, serving as the city of the Kingdom of Naples between 1282 and 1816. Thereafter, in union with Sicily, it became the capital of the Two Sicilies in 1861. Naples was the Italian city during World War II. Much of the city's 20th-century periphery was constructed after World War II. Unemployment levels in the city and surrounding Campania have decreased since 1999. However, unemployment levels remain high. Naples has the urban economy in Italy, after Milan, Rome and Turin. It is the world's 103rd-richest city with an estimated 2011 GDP of US$83.6 billion.
Naples
–
Naples Napoli
Naples
–
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
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The Gothic Battle of Mons Lactarius on Vesuvius, painted by Alexander Zick.
75.
Louis XVI of France
–
He was guillotined on 21 January 1793. The first part of Louis' reign was marked by attempts to reform France in accordance with Enlightenment ideas. These included efforts to increase tolerance toward non-Catholics. The French nobility reacted to the proposed reforms with hostility, successfully opposed their implementation. It resulted in bread prices. In periods of bad harvests, it would lead to food scarcity which would prompt the masses to revolt. The ensuing debt and financial crisis contributed to the unpopularity of the Ancien Régime which culminated at the Estates-General of 1789. In 1789, the storming of the Bastille during riots in Paris marked the beginning of the French Revolution. The abolition of the establishment of a republic became an ever increasing possibility. Louis-Auguste de France, given the title Duc de Berry at birth, was born in the Palace of Versailles. His mother was Prince-Elector of Saxony and King of Poland. He enjoyed physical activities such as rough-playing with his younger brothers, Charles-Philippe, comte d'Artois. From an early age, Louis-Auguste had been encouraged in another of his hobbies: locksmithing, seen as a'useful' pursuit for a child. Upon the death of his father, who died of tuberculosis on 20 December 1765, the eleven-year-old Louis-Auguste became the new Dauphin. His mother never recovered from the loss of her husband, died on 13 March 1767, also from tuberculosis.
Louis XVI of France
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King Louis XVI by Antoine-François Callet
Louis XVI of France
–
Marie Antoinette Queen of France with her three eldest children, Marie-Thérèse, Louis-Charles and Louis-Joseph. By Marie Louise Élisabeth Vigée-Lebrun
Louis XVI of France
–
Louis XVI at the age of 20
Louis XVI of France
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Louis-Charles, the dauphin of France and future Louis XVII. By Marie Louise Élisabeth Vigée-Lebrun.
76.
French Institute
–
The Institut de France is a French learned society, grouping five académies, the most famous of, the Académie française. The Institute, located in Paris, manages foundations, as well as museums and châteaux open for visit. It also awards subsidies, which amounted to a total of $5,028,190.55 for 2002. Most of these prizes are awarded 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 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
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Esplanade in front of the Institut, 1898.
77.
French revolution
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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. 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. The few years featured right-wing supporters of the monarchy 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 with estimates ranging from 16,000 to 40,000. 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, significant military conquests abroad. 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.
French revolution
–
The August Insurrection in 1792 precipitated the last days of the monarchy.
French revolution
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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.
78.
Pierre Charles Le Monnier
–
Pierre Charles Le Monnier was a French astronomer. His name is sometimes given as Lemonnier. Le Monnier was born in Paris, where his father Pierre, also an astronomer, was professor of philosophy at the college d'Harcourt. Shortly after his return, he explained, in a memoir read before the advantages of John Flamsteed's mode of determining right ascensions. Le Monnier's hasty speech resulted in many grudges. He fell out with Lalande "during an entire revolution of the moon's nodes". His career was ended by paralysis late in 1791, 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, was one of the 144 original members of the Institute. 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. "Lemonnier, Pierre Charles". Encyclopædia Britannica. 16.
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.
79.
Reign of Terror
–
The death toll ranged in the tens of thousands, with 16,594 executed by guillotine, another 25,000 in summary executions across France. During 1794, revolutionary France was beset by internal and foreign enemies. Within France, the revolution was opposed by the French nobility, which had lost its inherited privileges. They had the support of the Parisian population. Through the Revolutionary Tribunal, the Terror's leaders used them to eliminate the internal and external enemies of the republic. The Reign was a manifestation of the strong strain on centralized power. Many historians have debated the reasons the French Revolution took such a radical turn during the Reign of Terror of 1793–94. The public was frustrated that anti-poverty measures that the revolution originally promised were not materializing. The foundation of the Terror is centered on the April 1793 creation of the Committee of its militant Jacobin delegates. Those in power believed the Committee of Public Safety was an necessary and temporary, reaction to the pressures of foreign and civil war. Similar to Mathiez, Richard Cobb introduced competing circumstances of revolt and re-education as an explanation for the Terror. Marseille were threatening the revolution with royalist ideas. Terror was used in these rebellions both to provide a very visible example to those who might be considering rebellion. At the same time, Cobb rejects Mathiez's Marxist interpretation that elites controlled the Reign of Terror to the significant benefit of the bourgeoisie. Instead, Cobb argues that social struggles between the classes were seldom the reason for revolutionary sentiments.
Reign of Terror
–
Nine emigrants are executed by guillotine, 1793
Reign of Terror
–
Heads of aristocrats, on spikes (pikes)
Reign of Terror
–
Maximilien Robespierre had others executed via his role on the Revolutionary Tribunal and the Committee of Public Safety
Reign of Terror
–
A satirical engraving of Robespierre guillotining the executioner after having guillotined everyone else in France
80.
Lavoisier
–
Lavoisier is widely considered in popular literature as the "father of modern chemistry". It is generally accepted that Lavoisier's great accomplishments in chemistry largely stem from his changing the science from a qualitative to a quantitative one. He is most noted for his discovery of the oxygen plays in combustion. Lavoisier opposed the phlogiston theory. He helped construct the metric system, helped to reform chemical nomenclature. Lavoisier was also the first to establish that sulfur was an element rather than a compound. Lavoisier discovered that, although matter may change its shape, its mass always remains the same. He was an administrator of the Ferme générale. All of these economic activities enabled him to fund his scientific research. Antoine-Laurent Lavoisier was born on 26 August 1743. The son of an attorney at the Parliament of Paris, Lavoisier inherited a large fortune with the passing of his mother. He began his schooling at the Collège des Quatre-Nations, University of Paris at the age of 11. In his last two years at the school, he studied chemistry, botany, astronomy, mathematics. He entered the school of law, where he received a bachelor's degree in 1764. He was admitted to the bar, but never practiced as a lawyer.
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."
81.
Order of the Reunion
–
It was abolished in 1815. It was set up as an order of merit to replace Louis Bonaparte's Order of the Union. It Napoleon himself was its Grand Master. The knights of the order were authorised to exchange them for ones of the new order. Napoleon disliked the idea of a poor nobility and so assigned 500,000 francs annually to provide pensions to the order's members. This great event that truly characterises the Empire, could be called the Order of the Union.". Duc de Plaisance and Napoleon's representative in Amsterdam as ` Prins-stadhouder', oversaw the order and its membership numbers. Louis continued to wear ‘his’ Order of the Union throughout his life and old-established nobles did not receive the Order of the Reunion. The Dutch statesmen Godert van der Capellen, Vischer did not accept the Order of the Reunion, thinking it humiliating to the Netherlands. Knights of the new Order were appointed right up in 1814. On their initial restoration in 1814 the Bourbons neither awarded the Order of the Reunion and Napoleon awarded it during the Hundred Days. According to a statement by 11 great crosses, 36 commanders' crosses and 59 knights' crosses were handed in and melted down. The French state replaced them, though it was usually paid for by the recipient himself, honouring the awards of the Order of the Reunion. An official statement said that by its end the order had been awarded 1,622 times, with 1,364 knights, 131 Grand Crosses. 614 of these cases involved a foreigner, those who were not subjects of Napoleon.
Order of the Reunion
–
Insignia of the Order
82.
Astronomy
–
Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. The objects of interest include planets, moons, stars, comets; while the phenomena include supernovae explosions, gamma ray bursts, cosmic microwave background radiation. More generally all astronomical phenomena that originate outside Earth's atmosphere is within the perview of astronomy. 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 Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, Maya performed methodical observations of the night sky. During the 20th century, the field of professional astronomy split into theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, then analyzed using basic principles of physics. Theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the observational observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can still play an active role, especially in the observation of transient phenomena. Amateur astronomers have contributed to many important astronomical discoveries, such as finding new comets. Astronomy means "law of the stars". Astronomy should not be confused with the belief system which claims that human affairs are correlated with the positions of celestial objects.
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.
83.
Infinite series
–
In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence has defined last terms, whereas a series continues indefinitely. Given an infinite sequence, a series is informally the result of adding all those terms together: a1 + a2 + a3 + ···. These can be written more compactly using the summation ∑. A value may not always be given to such an infinite sum, and, in this case, the series is said to be divergent. The terms of the series are often produced according by an algorithm. To emphasize that there are an infinite number of terms, a series is often called an infinite series. The study of infinite series is a major part of mathematical analysis. Series are used in most areas of mathematics, through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in quantitative disciplines such as physics, computer science, statistics and finance. This definition is usually written as L = ∑ n = 0 ∞ a n ⇔ L = lim k → ∞ S k. When the set is the natural numbers I = N, the function a: N ↦ G is a sequence denoted by a = a n. This definition is usually written as L = ∑ n = 0 ∞ a n ⇔ L = lim k → ∞ S k. A series ∑ an is said to ` be convergent' when the sequence SN of partial sums has a finite limit. If the limit of SN does not exist, the series is said to diverge.
Infinite series
–
Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.
84.
Jupiter
–
Jupiter is the fifth planet from the Sun and the largest in the Solar System. Jupiter is a giant, along with Saturn, with the other two giant planets, Uranus and Neptune, being ice giants. Jupiter was known to astronomers of ancient times. The Romans named it after their Jupiter. Jupiter is primarily composed with a quarter of its mass being helium, though helium comprises only about a tenth of the number of molecules. Like the other giant planets, Jupiter lacks a well-defined solid surface. Because of its rapid rotation, the planet's shape is that of an oblate spheroid. The outer atmosphere is visibly segregated into several bands at different latitudes, resulting along their interacting boundaries. Surrounding Jupiter is a powerful magnetosphere. Jupiter has at least 67 moons, including the four large Galilean moons discovered by Galileo Galilei in 1610. The largest of these, has a diameter greater than that of the planet Mercury. Jupiter has been explored by robotic spacecraft, most notably during the early Pioneer and Voyager flyby missions and later by the Galileo orbiter. The latest probe to visit the planet is Juno, which entered on July 4, 2016. Future targets for exploration in the Jupiter system include the probable liquid ocean of its moon Europa. Its neighbor planets may have formed from fragments of planets after collisions with Jupiter destroyed those super-Earths near the Sun.
Jupiter
–
Jupiter in natural color, photographed by the Cassini spacecraft in 2001
Jupiter
Jupiter
–
Jupiter's diameter is one order of magnitude smaller (×0.10045) than the Sun, and one order of magnitude larger (×10.9733) than the Earth. The Great Red Spot is roughly the same size as the Earth.
Jupiter
–
This view of Jupiter's Great Red Spot and its surroundings was obtained by Voyager 1 on February 25, 1979, when the spacecraft was 9.2 million km (5.7 million mi) from Jupiter. The white oval storm directly below the Great Red Spot is approximately the same diameter as Earth.
85.
Second law of motion
–
Newton's laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between the forces acting upon it, its motion in response to those forces. They can be summarised as follows. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, first published in 1687. Newton used them to investigate the motion of many physical objects and systems. In this way, even a planet can be idealised around a star. In their original form, Newton's laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Euler's laws can, however, be taken as axioms describing the laws of any particle structure. Newton's laws hold only to a certain set of frames of reference called Newtonian or inertial reference frames. Other authors do treat the first law as a corollary of the second. The explicit concept of an inertial frame of reference was not developed after Newton's death. In the given mass, acceleration, momentum, force are assumed to be externally defined quantities. Not the only interpretation of the way one can consider the laws to be a definition of these quantities. The first law states that if the net force is zero, then the velocity of the object is constant. The first law can be stated mathematically when the mass is a constant, as, ∑ F = 0 ⇔ d v d t = 0.
Second law of motion
–
Newton's First and Second laws, in Latin, from the original 1687 Principia Mathematica.
Second law of motion
–
Isaac Newton (1643–1727), the physicist who formulated the laws
86.
Continuum mechanics
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The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century. Research in the area continues today. Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Continuum mechanics deals with physical properties of fluids which are independent of any particular coordinate system in which they are observed. These physical properties are then represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience. Materials, such as solids, gases, are composed of molecules separated by "empty" space. On a microscopic scale, materials have discontinuities. A continuum is a body that can be continually sub-divided with properties being those of the bulk material. More specifically, the hypothesis/assumption hinges on the concepts of a representative elementary volume and separation of scales based on the Hill -- Mandel condition. The latter then provide a micromechanics basis for finite elements. The levels of SVE and RVE link continuum mechanics to statistical mechanics. The RVE may be assessed only in a limited way via experimental testing: when the constitutive response becomes spatially homogeneous. Specifically for fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made. Consider traffic on a highway -- with just one lane for simplicity.
Continuum mechanics
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Figure 1. Configuration of a continuum body
87.
Kinematics
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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 and collections of such bodies. Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism. The term kinematic is the English version of A.M. Ampère's cinématique, which he constructed from the Greek κίνημα kinema, itself derived from κινεῖν kinein. Kinematic and cinématique are related to the French word cinéma, but neither are directly derived from it. Particle kinematics is the study of the trajectory of a particle. The position of a particle is defined to be the coordinate vector from the origin of a coordinate frame to the particle. If the tower is 50 m high, then the coordinate vector to the top of the tower is r=. In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move in a surface, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without those observations being described with respect to a reference frame.
Kinematics
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Each particle on the wheel travels in a planar circular trajectory (Kinematics of Machinery, 1876).
Kinematics
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Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.
Kinematics
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Illustration of a four-bar linkage from http://en.wikisource.org/wiki/The_Kinematics_of_Machinery Kinematics of Machinery, 1876
88.
Statics
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The application of Newton's 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 forces acting on the system, a is the acceleration of the system. The magnitude of the acceleration will be inversely proportional to the mass. The assumption of static equilibrium of a = 0 leads to: F = 0. 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 system leads to: M = I α = 0. The summation of moments, one of which might be unknown, allows that unknown to be found. These two equations together, can be applied to solve for as many as two loads acting on the system. From Newton's first law, this implies that the net force and torque on every part of the system is zero. See statically determinate. A scalar is a quantity which only has a magnitude, such as temperature. A vector has a direction. Vectors are added using the triangle law. Vectors contain components in orthogonal bases.
Statics
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Example of a beam in static equilibrium. The sum of force and moment is zero.
89.
Statistical mechanics
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A common use of statistical mechanics is in explaining the thermodynamic behaviour of large systems. This branch of statistical mechanics which extends classical thermodynamics is known as statistical thermodynamics or equilibrium statistical mechanics. Statistical mechanics also finds use outside equilibrium. An important subbranch known as statistical mechanics deals with the issue of microscopically modelling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical flows of particles and heat. In physics there are two types of mechanics usually examined: quantum mechanics. The statistical ensemble is a distribution over all possible states of the system. In statistical mechanics, the ensemble is a probability distribution over phase points, usually represented as a distribution in a phase space with canonical coordinates. In statistical mechanics, the ensemble is a probability distribution over pure states, can be compactly summarized as a density matrix. These two meanings will be used interchangeably in this article. However the probability is interpreted, each state in the ensemble evolves according to the equation of motion. Thus, the ensemble itself also evolves, as the virtual systems in the ensemble enter another. The evolution is given by the Liouville equation or the von Neumann equation. One special class of ensemble is those ensembles that do not evolve over time. Their condition is known as statistical equilibrium.
Statistical mechanics
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Statistical mechanics
90.
Acceleration
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Acceleration, in physics, is the rate of change of velocity of an object with respect to time. An object's acceleration is the net result of all forces acting on the object, as described by Newton's Second Law. The SI unit for acceleration is metre per second squared. Accelerations add according to the parallelogram law. As a vector, the net force is equal to the product of the object's mass and its acceleration. For example, when a car travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the car turns, there is an acceleration toward the new direction. When changing direction, we might call this "non-linear acceleration", which passengers might experience as a sideways force. If the speed of the car decreases, this is an acceleration in the opposite direction from the direction of the vehicle, sometimes called deceleration. Passengers may experience deceleration as a force lifting them forwards. Mathematically, there is no separate formula for deceleration: both are changes in velocity. Each of these accelerations might be felt by passengers until their velocity matches that of the car. An object's average acceleration over a period of time is its change in velocity divided by the duration of the period. Mathematically, a ¯ = Δ v Δ t. Instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimal interval of time.
Acceleration
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Components of acceleration for a curved motion. The tangential component a t is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector (or in the opposite direction). The normal component (also called centripetal component for circular motion) a c is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.
Acceleration
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Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as time interval Δt → 0 of Δ v / Δt
91.
Angular momentum
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In physics, angular momentum is the rotational analog of linear momentum. This definition can be applied to each point in continua like solids or fluids, or physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object via the moment of inertia I. Angular momentum is additive; the total angular momentum of a system is the vector sum of the angular momenta. For continua or fields one uses integration. Torque can be defined as the rate of change of angular momentum, analogous to force. Applications include the gyrocompass, control moment gyroscope, Earth's rotation to name a few. In general, conservation does limit the possible motion of a system, but does not uniquely determine what the exact motion is. In quantum mechanics, angular momentum is an operator with quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the "spin" of elementary particles does not correspond to literal spinning motion. Angular momentum is a vector quantity that represents the product of a body's rotational inertia and rotational velocity about a particular axis. Angular momentum can be considered a rotational analog of linear momentum. Unlike linear speed, which occurs in a straight line, angular speed occurs about a center of rotation.
Angular momentum
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This gyroscope remains upright while spinning due to the conservation of its angular momentum.
Angular momentum
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An ice skater conserves angular momentum – her rotational speed increases as her moment of inertia decreases by drawing in her arms and legs.
92.
Couple (mechanics)
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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. 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, where it is merely a synonym of moment. 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, as described below. Definition A couple is a pair of forces, 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. This is called a "simple couple". The forces have a moment called a torque about an axis, normal to the plane of the forces. The SI unit for the torque of the couple is newton metre. The moment of a force is only defined with respect to a certain point P, in general when P is changed, the moment changes. However, the moment of a couple is independent of the P: Any point will give the same moment.
Couple (mechanics)
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Classical mechanics
93.
D'Alembert's principle
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D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, mathematician Jean le Rond d'Alembert. A holonomic constraint depends only on the time. It does not depend on the velocities. More general specification of the irreversibility is required. D'Alembert's 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. It is equivalent to the somewhat more cumbersome Gauss's principle of least constraint. The general statement of d'Alembert's principle mentions "the time derivatives of the momenta of the system". To date, nobody has shown that D'Alembert's principle is equivalent to Newton's Second Law. This is true only for some very special cases e.g. rigid body constraints. However, an approximate solution to this problem does exist. Consider Newton's law for a system of i. If 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.
D'Alembert's principle
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Jean d'Alembert (1717—1783)
D'Alembert's principle
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Free body diagram of a wire pulling on a mass with weight W, showing the d’Alembert inertia “force” ma.
D'Alembert's principle
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Free body diagram depicting an inertia moment and an inertia force on a rigid body in free fall with an angular velocity.
94.
Energy
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In physics, energy is a property of objects which can be transferred to other objects or converted into different forms. It is misleading because energy is not necessarily available to do work. All of the many forms of energy are convertible to other kinds of energy. This means that it is impossible to destroy energy. This creates a limit to the amount of energy that can do work in a cyclic process, a limit called the available energy. Other forms of energy can be transformed in the other direction into thermal energy without such limitations. The total energy of a system can be calculated by adding up all forms of energy in the system. Lifting against gravity performs mechanical work on the object and stores gravitational potential energy in the object. Energy are closely related. With a sensitive enough scale, one could measure an increase in mass after heating an object. Living organisms require available energy to stay alive, such as the energy humans get from food. Civilisation gets the energy it needs from energy resources such as fossil fuels, renewable energy. The processes of Earth's ecosystem are driven by the radiant energy Earth receives from the sun and the geothermal energy contained within the earth. In biology, energy can be thought of as what's needed to keep entropy low. The total energy of a system can be classified in various ways.
Energy
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In a typical lightning strike, 500 megajoules of electric potential energy is converted into the same amount of energy in other forms, mostly light energy, sound energy and thermal energy.
Energy
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Thermal energy is energy of microscopic constituents of matter, which may include both kinetic and potential energy.
Energy
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Thomas Young – the first to use the term "energy" in the modern sense.
Energy
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A Turbo generator transforms the energy of pressurised steam into electrical energy
95.
Kinetic energy
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In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done in decelerating from its current speed to a state of rest. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a v is 1 2 m v 2. In relativistic mechanics, this is a good approximation only when v is much less than the speed of light. The standard unit of kinetic energy is the joule. The kinetic has its roots in the Greek word κίνησις kinesis, meaning "motion". The dichotomy between potential energy can be traced back to Aristotle's concepts of actuality and potentiality. Willem's Gravesande of the Netherlands provided experimental evidence of this relationship. Émilie du Châtelet published an explanation. Work in their present scientific meanings date back to the mid-19th century. William Thomson, later Lord Kelvin, is given the credit for coining 1849 -- 51. 1849–51. Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, rest energy.
Kinetic energy
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The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. When they start rising, the kinetic energy begins to be converted to gravitational potential energy. The sum of kinetic and potential energy in the system remains constant, ignoring losses to friction.
96.
Potential energy
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In physics, potential energy is energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, other factors. The unit for energy in the International System of Units is the joule, which has the symbol J. Potential energy is the stored energy of an object. It is the energy by virtue of an object's position relative to other objects. Potential energy is often associated with restoring forces such as the force of gravity. The action of lifting the mass is performed by an external force that works against the force field of the potential. This work is stored in the field, said to be stored as potential energy. Suppose a ball which it is in h position in height. If the acceleration of free fall is g, the weight of the ball is mg. There are various types of potential energy, each associated with a particular type of force. Thermal energy usually has the potential energy of their mutual positions. Forces derivable from a potential are also called conservative forces. The negative sign provides the convention while work done by the force field decreases potential energy. Common notations for potential energy are U, V, also Ep. Potential energy is closely linked with forces.
Potential energy
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In the case of a bow and arrow, when the archer does work on the bow, drawing the string back, some of the chemical energy of the archer's body is transformed into elastic potential-energy in the bent limbs of the bow. When the string is released, the force between the string and the arrow does work on the arrow. Thus, the potential energy in the bow limbs is transformed into the kinetic energy of the arrow as it takes flight.
Potential energy
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A trebuchet uses the gravitational potential energy of the counterweight to throw projectiles over two hundred meters
Potential energy
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Springs are used for storing elastic potential energy
Potential energy
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Archery is one of humankind's oldest applications of elastic potential energy
97.
Force
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In physics, a force is any interaction that, when unopposed, will change the motion of an object. In other words, a force can cause an object with mass i.e. to accelerate. Force can also be described by intuitive concepts such as a pull. A force has both direction, making it a vector quantity. It is represented by the symbol F. In an extended body, each part usually applies forces on the adjacent parts; the distribution of such forces through the body is the mechanical stress. Pressure is a simple type of stress. Stress usually causes flow in fluids. A fundamental error was the belief that a force is required to maintain motion, even at a constant velocity. Most of the previous misunderstandings about force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved-on for nearly hundred years. The Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong, weak, gravitational. High-energy physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Since antiquity the concept of force has been recognized to the functioning of each of the simple machines.
Force
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Aristotle famously described a force as anything that causes an object to undergo "unnatural motion"
Force
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Forces are also described as a push or pull on an object. They can be due to phenomena such as gravity, magnetism, or anything that might cause a mass to accelerate.
Force
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Though Sir Isaac Newton 's most famous equation is, he actually wrote down a different form for his second law of motion that did not use differential calculus.
Force
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Galileo Galilei was the first to point out the inherent contradictions contained in Aristotle's description of forces.
98.
Frame of reference
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In n dimensions, n+1 reference points are sufficient to fully define a reference frame. In Einsteinian relativity, reference frames are used to specify the relationship between the phenomenon or phenomena under observation. A relativistic 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 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 coordinate system is attached as a modifier, as in Cartesian frame of reference. Sometimes the state of motion is emphasized, in rotating frame of reference. Sometimes the way it transforms to frames considered as related is emphasized in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in microscopic frames of reference. On the other hand, a coordinate system may be employed for many purposes where the state of motion is not the primary concern. For example, a coordinate system may be adopted to take advantage of the symmetry of a system. It seems useful to divorce the various aspects of a frame for the discussion below. A coordinate system is a mathematical concept, amounting to a choice of language used to describe observations. Consequently, an observer in an observational frame of reference can choose to employ any coordinate system to describe observations made from that frame of reference. This viewpoint can be found elsewhere well.
Frame of reference
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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.
99.
Impulse (physics)
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In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Since force is a vector quantity, impulse is also a vector in the same direction. Impulse applied to an object produces an equivalent vector change in its linear momentum, also in the same direction. The SI unit of impulse is the newton second, the dimensionally equivalent unit of momentum is the kilogram meter per second. The corresponding English engineering units are the pound-second and the slug-foot per second. A resultant force causes acceleration and a change in the velocity of the body for as long as it acts. Conversely, a small force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly. This is often called the impulse-momentum theorem. As a result, an impulse may also be regarded as the change in momentum of an object to which a resultant force is applied. Impulse has the same units and dimensions as momentum. In the International System of Units, these are kg·m/s = N·s. In English engineering units, they are slug·ft/s = lbf·s. The term "impulse" is also used to refer to a fast-acting force or impact. This type of impulse is often idealized so that the change in momentum produced by the force happens with no change in time. This sort of change is a step change, is not physically possible.
Impulse (physics)
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A large force applied for a very short duration, such as a golf shot, is often described as the club giving the ball an impulse.
Impulse (physics)
100.
Inertia
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It is the tendency of objects to keep moving in a straight line at constant velocity. Inertia comes from iners, meaning idle, sluggish. Inertia is one of the primary manifestations of mass, a quantitative property of physical systems. Thus, an object will continue moving at its current velocity until some force causes its direction to change. Aristotle concluded that violent motion in a void was impossible. Despite its general acceptance, Aristotle's concept of motion was disputed by notable philosophers over nearly two millennia. For example, Lucretius stated that the "state" of matter was motion, not stasis. This view was strongly opposed by many scholastic philosophers who supported Aristotle. However, this view did not go unchallenged in the Islamic world, where Philoponus did have several supporters who further developed his ideas. In the 14th century, Jean Buridan rejected the notion that a motion-generating property, which he named impetus, dissipated spontaneously. Buridan also maintained that impetus increased with speed; thus, his initial idea of impetus was similar in many ways to the modern concept of momentum. Buridan also believed that impetus could be not only linear, but also circular in nature, causing objects to move in a circle. Buridan's thought was followed up by the Oxford Calculators, who performed various experiments that further undermined the classical, Aristotelian view. Their work in turn was elaborated by Nicole Oresme who pioneered the practice of demonstrating laws of motion in the form of graphs. Benedetti cites the motion of a rock in a sling as an example of the linear motion of objects, forced into circular motion.
Inertia
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Galileo Galilei
101.
Moment of inertia
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It depends on the axis chosen, with larger moments requiring more torque to change the body's rotation. It is an extensive property: the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems. For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia to the plane. When a body is free to rotate, around an axis, a torque must be applied to change its angular momentum. The amount of torque needed for any given rate of change in momentum is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of metre squared in SI units and pound-square feet in imperial or US units. The moment of inertia depends on how mass will vary depending on the chosen axis. In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum. There is an interesting difference in the moment of inertia appears in planar and spatial movement. The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. If the momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their outstretched divers curl their bodies into a tuck position during a dive, to spin faster. Moment of inertia can be measured using a simple pendulum, because it is the resistance to the rotation caused by gravity. Here r is the distance perpendicular to and from the force to the torque axis. Here F is the tangential component of the net force on the mass.
Moment of inertia
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Tightrope walker Samuel Dixon using the long rod's moment of inertia for balance while crossing the Niagara River in 1890.
Moment of inertia
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Flywheels have large moments of inertia to smooth out mechanical motion. This example is in a Russian museum.
Moment of inertia
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Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to conservation of angular momentum.
Moment of inertia
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Pendulums used in Mendenhall gravimeter apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth.
102.
Mass
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In physics, mass is a property of a physical body. It is the measure of an object's resistance to acceleration when a force is applied. It also determines the strength of its gravitational attraction to other bodies. In the theory of relativity a related concept is the mass -- content of a system. The SI unit of mass is the kilogram. It would still have the same mass. 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 in an object. However, at very high speeds, special relativity postulates that energy is an additional source of mass. Thus, all forms of energy resist acceleration by a force and have gravitational attraction. In addition, "matter" thus can not be precisely measured. There are distinct phenomena which can be used to measure mass. Gravitational mass measures the gravitational force exerted by an object. Passive gravitational mass measures the gravitational force exerted on an object in a gravitational field. Mass–energy measures the total amount of energy contained within a body, using E = mc2.
Mass
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Depiction of early balance scales in the Papyrus of Hunefer (dated to the 19th dynasty, ca. 1285 BC). The scene shows Anubis weighing the heart of Hunefer.
Mass
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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
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Galileo Galilei (1636)
Mass
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Distance traveled by a freely falling ball is proportional to the square of the elapsed time
103.
Power (physics)
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In physics, power is the rate of doing work. It is the amount of energy consumed per time. Having no direction, it is a quantity. Another traditional measure is horsepower. Being the rate of work, the equation for power can be written: P = W t The integral of power over time defines the work performed. As a physical concept, power requires both a change in a specified time in which the change occurs. This is distinct from the concept of work, only measured in terms of a net change in the state of the physical universe. The power of an electric motor is the product of the torque that the motor generates and the angular velocity of its output shaft. The power involved in moving a vehicle is the product of the velocity of the vehicle. The dimension of power is energy divided by time. The SI unit of power is the watt, equal to one joule per second. Other units of power include foot-pounds per minute. Other units include a relative logarithmic measure with 1 milliwatt as reference; food calories per hour; Btu per hour; and tons of refrigeration. This shows how power is an amount of energy consumed per time. It is the average amount of energy converted per unit of time.
Power (physics)
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Ansel Adams photograph of electrical wires of the Boulder Dam Power Units, 1941–1942
104.
Work (physics)
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The SI unit of work is the joule. Non-SI units of work include the erg, the foot-pound, the foot-poundal, the horsepower-hour. This is approximately the work done lifting a 1 weight from ground level over a person's head against the force of gravity. Notice that the work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Work is closely related to energy. Conversely, a decrease in kinetic energy is caused by an equal amount of negative work done by the resultant force. The work of forces generated by a potential function is known as potential energy and the forces are said to be conservative. These formulas demonstrate that work is the energy associated with the action of a force, so work subsequently possesses the physical dimensions, units, of energy. 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, which means the constraint forces do not perform work on the system. This only applies for a single particle system. In an Atwood machine, the rope does work on each body, but keeping always the virtual work null. There are, however, cases where this is not true. This force does zero work because it is perpendicular to the velocity of the ball.
Work (physics)
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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)
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A force of constant magnitude and perpendicular to the lever arm
Work (physics)
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Gravity F = mg does work W = mgh along any descending path
Work (physics)
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Lotus type 119B gravity racer at Lotus 60th celebration.
105.
Momentum
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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 the product of force and time, quantified in newton-seconds. Newton's 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 more slowly, then it would therefore require less impulse to start or stop. Linear momentum is also a conserved quantity, meaning that if a closed system is not affected by external forces, its linear momentum can not change. In classical mechanics, conservation of linear momentum is implied by Newton's laws. With appropriate definitions, a linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory, general relativity. It is ultimately an expression of the fundamental symmetries of space and time, that of translational symmetry. 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 provided the system is isolated. Momentum has a direction well as magnitude. Quantities that have both 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, well as their speeds. Below, the basic properties of momentum are described in one dimension.
Momentum
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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.
106.
Space
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Space is the boundless three-dimensional extent in which objects and events have relative position and direction. The concept of space is considered to be to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, part of a conceptual framework. Many of these philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, particularly during the early development of classical mechanics. In Isaac Newton's view, space was absolute -- in the sense that it existed independently of whether there was any matter in the space. Kant referred to the experience of "space" as being a subjective "pure a priori form of intuition". In the 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space. In the seventeenth century, the philosophy of time emerged as a central issue in epistemology and metaphysics. At its heart, the English physicist-mathematician, set out two opposing theories of what space is. Unoccupied regions are those that could have objects in them, thus spatial relations with other places. Space could be thought in a similar 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. According to the principle of sufficient reason, any theory of space that implied that there could be these two possible universes must therefore be wrong.
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
107.
Speed
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In everyday use and in kinematics, the speed of an object is the magnitude of its velocity; it is thus a scalar quantity. Speed has the dimensions of distance divided by time. For air and marine travel the knot is commonly used. Matter cannot quite reach the speed of light, as this would require an infinite amount of energy. In relativity physics, the concept of rapidity replaces the classical idea of speed. The time it takes. Galileo defined speed as the distance covered per unit of time. In equation form, this is v = d t, t is time. A cyclist who covers 30 metres in a time of 2 seconds, for example, has a speed of 15 metres per second. Objects in motion often have variations in speed. If s is the length of the path travelled until t, the speed equals the time derivative of s: v = d s d t. In the special case where the velocity is constant, this can be simplified to v = s / t. The average speed over a finite interval is the total distance travelled divided by the time duration. Assumed constant during a very short period of time, is called instantaneous speed. By looking at a speedometer, one can read the instantaneous speed of a car at any instant.
Speed
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Speed can be thought of as the rate at which an object covers distance. A fast-moving object has a high speed and covers a relatively large distance in a given amount of time, while a slow-moving object covers a relatively small amount of distance in the same amount of time.
108.
Time
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Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future. Time is often referred to as the fourth dimension, along with the three spatial dimensions. Nevertheless, diverse fields such as business, industry, sports, the performing arts all incorporate some notion of time into their respective measuring systems. Two contrasting viewpoints on time divide prominent philosophers. One view is that time is part of the fundamental structure of the universe -- a independent of events, in which events occur in sequence. Hence it is sometimes referred to as Newtonian time. Time in physics is unambiguously operationally defined as "what a clock reads". Time is one of International System of Quantities. Time is used to define other quantities—such as velocity—so defining time in terms of such quantities would result in circularity of definition. Temporal measurement was a prime motivation in navigation and astronomy. Periodic motion have long served as standards for units of time. Currently, the international unit of the second, is defined by measuring the electronic transition frequency of caesium atoms. In day-to-day life, the clock is consulted than a day whereas the calendar is consulted for periods longer than a day. Increasingly, electronic devices display both calendars and clocks simultaneously. The number that marks the occurrence of a specified event as to date is obtained by counting from a fiducial epoch -- a central reference point.
Time
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The flow of sand in an hourglass can be used to keep track of elapsed time. It also concretely represents the present as being between the past and the future.
Time
Time
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Horizontal sundial in Taganrog
Time
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A contemporary quartz watch
109.
Torque
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Torque, moment, or moment of force is the tendency of a force to rotate an object around an axis, fulcrum, or pivot. Just as a force is a pull, a torque can be thought of as a twist to an object. Loosely speaking, torque is a measure of the force on an object such as a bolt or a flywheel. The symbol for torque is typically the lowercase Greek letter tau. When it is called moment of force, it is commonly denoted by M. The SI unit for torque is the metre. For more on the units of torque, see Units. This article follows US physics terminology in its use of the torque. In the UK and in US mechanical engineering, this is called moment of force, usually shortened to moment. Torque is defined mathematically as the rate of change of momentum of an object. The definition of torque states that the moment of inertia of an object are changing. For a rotational force applied to a shaft causing acceleration, such as a drill bit accelerating from rest, results in a moment called a torque. Similarly with any couple on an object that has no change to its angular momentum, such moment is also not called a torque. The concept of torque, also called couple, originated with the studies of Archimedes on levers. The torque was apparently introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884.
Torque
110.
Velocity
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The velocity of an object is the rate of change of its position with respect to a frame of reference, is a function of time. Velocity is equivalent to a specification of its speed and direction of motion. Velocity is an important concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical quantity; both direction are needed to define it. For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change in both, then the object has a changing velocity and is said to be undergoing an acceleration. To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. Hence, the car is considered to be undergoing an acceleration. Speed describes only how fast an object is moving, whereas velocity gives both how fast and in what direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified. The big difference can be noticed when we consider movement around a circle. Average velocity can be calculated as: v ¯ = Δ x Δ t. The average velocity is always less than or equal to the average speed of an object.
Velocity
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As a change of direction occurs while the cars turn on the curved track, their velocity is not constant.
111.
Newton's laws of motion
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Newton's laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, its motion in response to those forces. They have been expressed in several different ways, over nearly three centuries, can be summarised as follows. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, first published in 1687. Newton used them to explain and investigate the motion of many physical objects and systems. In this way, even a planet can be idealised as a particle for analysis of its orbital motion around a star. In their original form, Newton's laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Euler's laws can, however, be taken as axioms describing the laws of motion for extended bodies, independently of any particle structure. Newton's laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Other authors do treat the first law as a corollary of the second. The explicit concept of an inertial frame of reference was not developed until long after Newton's death. In the given mass, force are assumed to be externally defined quantities. This is the most common, but not the only interpretation of the way one can consider the laws to be a definition of these quantities. The first law states that if the net force is zero, then the velocity of the object is constant. The first law can be stated mathematically when the mass is a non-zero constant, as, ∑ F = 0 ⇔ d v d t = 0.
Newton's laws of motion
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Newton's First and Second laws, in Latin, from the original 1687 Principia Mathematica.
Newton's laws of motion
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Isaac Newton (1643–1727), the physicist who formulated the laws
112.
Routhian mechanics
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In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions. The difference between the Lagrangian, Hamiltonian, Routhian functions are their variables. The Routhian differs from these functions in that some coordinates are chosen to have corresponding generalized velocities, the rest to have corresponding generalized momenta. This choice is arbitrary, can be done to simplify the problem. In each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian. The Lagrangian equations are powerful results, used frequently in theory and practice, since the equations of motion in the coordinates are easy to set up. However, if cyclic coordinates occur there will still be equations to solve for all the coordinates, including the cyclic coordinates despite their absence in the Lagrangian. Overall fewer equations need to be solved compared to the Lagrangian approach. As with the rest of analytical mechanics, Routhian mechanics is completely equivalent to Newtonian mechanics, all other formulations of classical mechanics, introduces no new physics. It offers an alternative way to solve mechanical problems. The velocities dqi/dt are expressed as functions of their corresponding momenta by inverting their defining relation. In this context, pi is said to be the momentum "canonically conjugate" to qi. The choice of which n coordinates are to have corresponding momenta, out of the n + s coordinates, is arbitrary. The above is used by Landau and Lifshitz, Goldstein.
Routhian mechanics
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Edward John Routh, 1831–1907.
113.
Damping
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If a frictional 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, an amplitude decreasing with time. Decay without oscillations. The solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called "critically damped." If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator. Mechanical examples include acoustical systems. Other analogous systems include harmonic oscillators such as RLC circuits. Harmonic oscillators are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal waves. A harmonic oscillator is an oscillator, neither driven nor damped. The motion is periodic, repeating itself in a sinusoidal fashion with A. The position at a given t also depends on the phase, φ, which determines the starting point on the sine wave. The 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 opposite direction as the displacement. The potential energy stored in a harmonic oscillator at position x is U = 1 2 k x 2.
Damping
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Mass attached to a spring and damper.
114.
Damping ratio
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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 released, bounce up and down. On each bounce, the system overshoots it. Sometimes losses 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. This case is called overdamped. Commonly, the mass tends overshooting again. With each overshoot, the oscillations die towards zero. This case is called underdamped. This case is called critical damping. The key difference between critical 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, that characterizes the response of a second order ordinary differential equation.
Damping ratio
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The effect of varying damping ratio on a second-order system.
115.
Displacement (vector)
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A displacement is a vector, 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 straight line from the initial position to the final position of the point. The velocity then is distinct from the instantaneous speed, the time rate of change of the distance traveled along a specific path. The velocity may be equivalently defined as the rate of change of the vector. For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity. In dealing with the motion of a rigid 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, while the rotation of the body is called angular displacement. For a vector s, a function of t, the derivatives can be computed with respect to t. These derivatives have common utility in the study of kinematics, other sciences and engineering disciplines. By extension, the higher order derivatives can be computed in a similar fashion. Study of these higher order derivatives can improve approximations of the original displacement function. The fourth derivative is called jounce, the sixth pop. Equipollence Position vector Affine space
Displacement (vector)
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Displacement versus distance traveled along a path
116.
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 replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions to the differential equations describing the motion of the dynamics. There are two main descriptions of motion: kinematics. Dynamics is general, since momenta, energy of the particles are taken into account. In this instance, sometimes the term refers to the differential equations that the system satisfies, sometimes to the solutions to those equations. However, kinematics is simpler as it concerns time. Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the main types of motion are translations, rotations, any combinations of these. Solving the 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, which fixes the values of the constants. Euclidean vectors in 3D are denoted throughout in bold. The initial conditions are given by the constant values at t r, r ˙. The solution r with specified initial values, describes the system for all times t after t = 0.
Equations of motion
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Kinematic quantities of a classical particle of mass m: position r, velocity v, acceleration a.
117.
Fictitious force
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The 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 nonuniform relative motion of two reference frames is called a pseudo-force. Assuming Newton's second law in the F = ma, fictitious forces are always proportional to the mass m. A fictitious force on an object arises when the frame of reference used to describe the object's motion is accelerating compared to a non-accelerating frame. As a frame can accelerate in any arbitrary way, so can fictitious forces be as arbitrary. Gravitational force would also be a fictitious force based upon a model in which particles distort spacetime due to their mass. The surface of the Earth is a rotating frame. The Euler force is typically ignored because the variations in the velocity of the rotating Earth surface are usually insignificant. They can be detected under careful conditions. For example, Léon Foucault was able to show that the Coriolis force results from the Earth's rotation using the Foucault pendulum. Other accelerations also give rise to fictitious forces, as described below. An example of the detection of a 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 fictitious force is necessary. Figure 1 shows an accelerating car.
Fictitious force
118.
Friction
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Friction is the force resisting the relative motion of solid surfaces, fluid layers, material elements sliding against each other. There are several types of friction: Dry friction resists lateral motion of two solid surfaces in contact. Dry friction is subdivided into kinetic friction between moving surfaces. Fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other. Lubricated friction is a case of fluid friction where a fluid separates two solid surfaces. Friction is a component of drag, the force resisting the motion of a fluid across the surface of a body. Internal friction is the force resisting motion between the elements making up a solid material while it undergoes deformation. When surfaces in contact move relative to each other, the friction between the two surfaces converts kinetic energy into thermal energy. This property can have dramatic consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to thermal energy whenever motion with friction occurs, for example when a viscous fluid is stirred. Another important consequence of many types of friction can be wear, which damage to components. Friction is a component of the science of tribology. Friction is not itself a fundamental force. Dry friction arises from a combination of inter-surface adhesion, surface roughness, surface contamination. Friction is a non-conservative force - work done against friction is path dependent.
Friction
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When the mass is not moving, the object experiences static friction. The friction increases as the applied force increases until the block moves. After the block moves, it experiences kinetic friction, which is less than the maximum static friction.
119.
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 lower in an amplitude decreasing with time. Decay to the equilibrium position, without oscillations. The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called "critically damped." If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator. Mechanical examples include pendulums, masses connected to springs, acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves. A simple harmonic oscillator is an oscillator, neither driven nor damped. The motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude, A. The position at a given time t also depends on the phase, φ, which determines the starting point on the sine wave. 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 opposite direction as the displacement. The potential energy stored in a simple harmonic oscillator at position x is U = 1 2 k x 2.
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
120.
Inertial frame of reference
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The physics of a system in an inertial frame have no causes external to the system. Measurements in one inertial frame can be converted to measurements in another by a simple transformation. Systems in non-inertial frames in general relativity don't have external causes because of the principle of geodesic motion. Physical laws take the same form in all inertial frames. For example, a ball dropped towards the ground does not go exactly straight down because the Earth is rotating. Someone rotating with the Earth must account for the Coriolis effect—in this case thought of as a force—to predict the horizontal motion. Another example of such a fictitious force associated with rotating reference frames is the centrifugal effect, or centrifugal force. The motion of a body can only be described relative to something else -- a set of space-time coordinates. These are called frames of reference. If the coordinates are chosen badly, the laws of motion may be more complex than necessary. For example, suppose a free body that has no external forces on it is at rest at some instant. In many coordinate systems, it would begin to move at the next instant, even though there are no forces on it. However, a frame of reference can always be chosen in which it remains stationary. Indeed, an intuitive summary of inertial frames can be given as: In an inertial reference frame, the laws of mechanics take their simplest form. In an inertial frame, the law of inertia, is satisfied: Any free motion has direction.
Inertial frame of reference
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Figure 1: Two frames of reference moving with relative velocity. Frame S' has an arbitrary but fixed rotation with respect to frame S. They are both inertial frames provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.
121.
Mechanics of planar particle motion
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This article describes a particle in planar motion when observed from non-inertial reference frames. See centrifugal force, two-body problem, Kepler's laws of planetary motion. Those problems fall from given laws of force. The Lagrangian approach to fictitious forces is introduced. Unlike real forces such as electromagnetic forces, fictitious forces do not originate from physical interactions between objects. This allows us to detect experimentally the non-inertial nature of a system. Pretend you are in an inertial frame. Elaboration of some citations on the subject follow. Examples are Cartesian coordinates, polar coordinates and curvilinear coordinates. The corresponding set of axes, sharing the rigid motion of the frame R, can be considered to give a physical realization of R. In traditional developments of general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply to events in spacetime neighborhoods. Or as seen from a rotating frame. A time-dependent description of observations does not change the frame of reference in which the observations are recorded. In discussion of a particle moving in an inertial frame of reference one can identify the centripetal and tangential forces.
Mechanics of planar particle motion
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The arc length s(t) measures distance along the skywriter's trail. Image from NASA ASRS
Mechanics of planar particle motion
Mechanics of planar particle motion
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Figure 2: Two coordinate systems differing by a displacement of origin. Radial motion with constant velocity v in one frame is not radial in the other frame. Angular rate, but
122.
Motion (physics)
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In physics, motion is a change in position of an object over time. Motion is typically described in terms of displacement, distance, velocity, acceleration, speed. An object's motion can not change unless it is acted by a force, as described. Momentum is a quantity, used for measuring motion of an object. As there is no absolute frame of reference, absolute motion cannot be determined. Thus, everything in the universe can be considered to be moving. One can also speak of motion of boundaries. So, the motion in general signifies a continuous change in the configuration of a physical system. In physics, motion is described through two sets of apparently contradictory laws of mechanics. Motions of familiar objects in the universe are described by classical mechanics. Whereas the motion of sub-atomic objects is described by quantum mechanics. It is one of the oldest and largest in science, engineering, technology. Classical mechanics is fundamentally based on Newton's laws of motion. These laws describe the relationship between the forces acting on the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, first published on July 1687.
Motion (physics)
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Motion involves a change in position, such as in this perspective of rapidly leaving Yongsan Station.
123.
Newton's law of universal gravitation
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This is a general physical law derived from empirical observations by what Isaac Newton called induction. It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica, first published on 5 July 1687. In modern language, the law states: Every mass attracts every other mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them. The first test of Newton's theory of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798. It took place 111 years after the publication of Newton's Principia and approximately 71 years after his death. Newton's law of gravitation resembles Coulomb's law of electrical forces, used to calculate the magnitude of the electrical force arising between two charged bodies. Both are inverse-square laws, where force is inversely proportional to the square of the distance between the bodies. Coulomb's law has the constant in place of the constant. At the same time Hooke agreed that "the Demonstration of the Curves generated thereby" was wholly Newton's. In this way, the question arose as to what, if anything, Newton owed to Hooke. This is a subject extensively discussed since that time and on which some points, outlined below, continue to excite controversy. Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses. He also did not provide accompanying evidence or mathematical demonstration.
Newton's law of universal gravitation
124.
Relative velocity
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We begin with relative motion in the classical, that all speeds are much less than the speed of light. This limit is associated with the Galilean transformation. The figure shows a man on top of a train, at the back edge. At 1:00 pm he begins to walk forward at a walking speed of 10 km/hr. The train is moving at 40 km/hr. The figure depicts the train at two different times: first, when the journey began, also one hour later at 2:00 pm. The figure suggests that the man is 50 km from the starting point after having traveled for one hour. This, by definition, is 50 km/hour, which suggests that the prescription for calculating relative velocity in this fashion is to add the two velocities. V → M | T is the velocity of the Man relative to the Train. V → T | E is the velocity of the Train relative to Earth. The figure shows two objects moving at constant velocity. The difference between the two displacement vectors, r → B − r → A, represents the location of B as seen from A. To construct a theory of relative motion consistent with the theory of special relativity, we must adopt a different convention. Recall that v is the motion of a stationary object in the primed frame, as seen from the unprimed frame. Hence relative speed is symmetrical.
Relative velocity
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Relative velocities between two particles in classical mechanics.
125.
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 given points of a rigid body remains constant in time regardless of external forces exerted on it. 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. 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, provided that their time-invariant position relative to the three selected particles is known. Typically a different, mathematically equivalent approach is used. Thus, the position of a rigid body has two components: linear and angular, respectively. This reference point may define the origin of a coordinate system fixed to the body. In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, respectively. Indeed, the position of a rigid body can be viewed starting from a hypothetic position. Velocity are measured with respect to a frame of reference. The linear velocity of a rigid body is a quantity, equal to the rate of change of its linear position. Thus, it is the velocity of a reference point fixed to the body.
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).
126.
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 plastic behavior. The solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems. If a system of particles moves parallel to a fixed plane, the system is said to be constrained to movement. In this case, Newton's laws for a rigid system of Pi, i = 1... N, simplify because there is no movement in the k direction. Several methods to describe orientations of a rigid 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 these three rotations are called Euler angles. These are three angles, also known as yaw, roll, Navigation angles and Cardan angles. In engineering they are usually referred to as Euler angles. Euler also realized that the composition of two rotations is equivalent to a single rotation about a fixed axis. Therefore, the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed. 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.
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
127.
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. In many cases, however, vibration is wasting energy and creating unwanted sound. For example, the vibrational motions of engines, any mechanical device in operation are typically unwanted. Such vibrations could be caused by imbalances in the rotating parts, the meshing of gear teeth. Careful designs usually minimize unwanted vibrations. The studies of vibration are closely related. Pressure waves, are generated by vibrating structures; these pressure waves can also induce the vibration of structures. Hence, attempts to reduce noise are often related to issues of vibration. Free vibration occurs when a mechanical system is allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and letting go, or letting it ring. The mechanical system damps down to motionlessness. Forced vibration is when a time-varying disturbance is applied to a mechanical system. The disturbance can be a periodic and steady-state input, a random input.
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).
128.
Circular motion
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In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body. Without this acceleration, the object would move according to Newton's laws of motion. In physics, circular motion describes the motion of a body traversing a circular path at constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times. Though the body's speed is constant, its velocity is not constant: a vector quantity, depends on both the body's speed and its direction of travel. This changing velocity indicates the presence of an acceleration; this centripetal acceleration is of constant magnitude and directed at all times towards the axis of rotation. This acceleration is, in turn, directed towards the axis of rotation. Note: The magnitude of the angular velocity is the angular speed. For motion in a circle of radius r, the circumference of the circle is C = 2π r. The axis of rotation is shown as a vector perpendicular to the plane of the orbit and with a magnitude ω = dθ / dt. The direction of ω is chosen using the right-hand rule. In the simplest case the mass and radius are constant.
Circular motion
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Figure 1: Velocity v and acceleration a in uniform circular motion at angular rate ω; the speed is constant, but the velocity is always tangent to the orbit; the acceleration has constant magnitude, but always points toward the center of rotation
129.
Centripetal force
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A centripetal force is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force responsible for astronomical orbits. One common example involving centripetal force is the case in which a body moves with uniform speed along a circular path. The centripetal force is directed at right angles to the motion and also along the radius towards the centre of the circular path. The mathematical description was derived in 1659 by the Dutch physicist Christiaan Huygens. The direction of the force is toward the center of the circle in which the object is moving, or the osculating circle. The speed in the formula is squared, so twice the speed needs four times the force. The inverse relationship with the radius of curvature shows that half the radial distance requires twice the force. Expressed using the orbital period T for one revolution of the circle, ω = 2 π T the equation becomes F = m r 2. The rope example is an example involving a'pull' force. Newton's idea of a centripetal force corresponds to what is nowadays referred to as a central force. In this case, the magnetic force is the centripetal force that acts towards the helix axis. Below are three examples of increasing complexity, with derivations of the formulas governing velocity and acceleration.
Centripetal force
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A body experiencing uniform circular motion requires a centripetal force, towards the axis as shown, to maintain its circular path.
130.
Centrifugal force
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The term has sometimes also been used for the force, a reaction to a centripetal force. All measurements of position and velocity must be made relative to some frame of reference. An inertial frame of reference is one, not accelerating. In terms of an inertial frame of reference, the centrifugal force does not exist. All calculations can be performed using only Newton's laws of motion and the real forces. In its current usage the term'centrifugal force' has no meaning in an inertial frame. In an inertial frame, an object that has no forces acting on it travels in a straight line, according to Newton's first law. If it is desired to apply Newton's laws in the rotating frame, it is necessary to introduce new, fictitious, forces to account for this curved motion. This is the centrifugal force. Consider a stone being whirled round on a string, in a horizontal plane. The only real force acting on the stone in the horizontal plane is the tension in the string. There are no other forces acting on the stone so there is a net force on the stone in the horizontal plane. In order to keep the stone moving in a circular path, this force, known as the centripetal force, must be continuously applied to the stone. As soon as it is removed the stone moves in a straight line. In a frame of reference rotating with the stone around the same axis as the stone, the stone is stationary.
Centrifugal force
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The interface of two immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.
131.
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 Newton's first law of motion, an object moves in a straight line in the absence of any external forces acting on the object. It is the reactive force, 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 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 model, the string is assumed the rotational motion frictionless, so no propelling force is needed to keep the ball in circular motion. The string transmits the centrifugal force from the ball to the fixed post, pulling upon the post. Again according to the post exerts a reaction upon the string, labeled the post reaction, pulling upon the string. The two forces upon the string are 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. Even though the reactive centrifugal is rarely used in analyses in the literature, the concept is applied within some mechanical engineering concepts. An example of this kind of concept is an analysis of the stresses within a rapidly rotating turbine blade.
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.
132.
Coriolis force
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In physics, the Coriolis force is an inertial force that acts on objects that are in motion relative to a rotating reference frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise rotation, the force 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'. Newton's laws of motion describe the motion of an object in an inertial frame of reference. When Newton's laws are transformed to a rotating frame of reference, the Coriolis force and centrifugal force appear. 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 centrifugal force acts outwards in the radial 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. They allow the application of Newton's 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 and the inertia of the mass experiencing the effect.
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.
133.
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. The motion does not lose energy to resistance. The gravitational field is uniform. The support does not move. The differential equation given above is not easily solved, there is no solution that can be written in terms of elementary functions. However adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained. The error due to the approximation is of order θ3. The period of the motion, the time for a complete oscillation is, known as Christiaan Huygens's law for the period. Note that under the small-angle approximation, the period is independent of the amplitude θ0; this is the property of isochronism that Galileo discovered. T 0 = 2 π ℓ g can be expressed as ℓ = g π 2 T 0 2 4. If assuming the measurement is taking place on the Earth's surface, then g / π2 ≈ 1. The linear approximation gives s. Less than 0.2 %, is much less than that caused with geographical location. From here there are many ways to proceed to calculate the elliptic integral.
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.
134.
Angular displacement
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When dealing with the rotation of an object, it becomes simpler to consider the body itself rigid. In a realistic sense, all things can be deformable, however this impact is negligible. 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 fixed r from the origin, O, rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates. In this particular example, the value of θ is changing, while the value of the radius remains the same. . If using radians, it provides a very simple relationship between distance traveled from the centre. Therefore 1 revolution is 2 π radians. Δ θ = θ − θ 1 which equals the Angular Displacement. In three dimensions, displacement is an entity with a direction and a magnitude. This entity is called an axis-angle. Despite having magnitude, angular displacement is not a vector because it does not obey the commutative law for addition. Nevertheless, in this case commutativity appears. Several ways to describe displacement exist, like rotation matrices or Euler angles.
Angular displacement
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Rotation of a rigid object P about a fixed object about a fixed axis O.
135.
Angular velocity
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This speed can be measured in terms of degrees per second, degrees per hour, etc.. Angular velocity is usually represented by the omega. The direction of the angular vector is perpendicular to the plane of rotation, in a direction, usually specified by the right-hand rule. The velocity of a particle is measured around or relative to a point, called the origin. If there is no radial component, then the particle moves in a circle. On the other hand, if there is no cross-radial component, then the particle moves along a straight line from the origin. Therefore, the angular velocity is completely determined by this component. The 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 from the x axis. If the parity is inverted, but the orientation of a rotation is not, then the sign of the velocity changes. There are three types of velocity involved in the movement on an ellipse corresponding to the three anomalies. In three dimensions, the velocity becomes a bit more complicated. The velocity in this case is generally thought of as a vector, or more precisely, a pseudovector. It now has not only a direction as well. The direction describes the axis of rotation that Euler's rotation theorem guarantees must exist.
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.
136.
Galileo Galilei
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Galileo Galilei was an Italian polymath: astronomer, physicist, engineer, philosopher, mathematician, he played a major role in the scientific revolution of the seventeenth century. Galileo has been called the "father of the "father of science". Galileo also worked in applied science and technology, inventing an improved military compass and other instruments. Galileo's championing of heliocentrism and Copernicanism was controversial during his lifetime, when most subscribed to either geocentrism or the Tychonic system. He met with opposition from astronomers, who doubted heliocentrism because of the absence of an observed stellar parallax. He was tried by the Inquisition, found "vehemently suspect of heresy", forced to recant. He spent the rest of his life under house arrest. Three of Galileo's five siblings survived infancy. Michelangelo, also became a noted composer although he contributed during Galileo's young adulthood. Michelangelo would occasionally have to support his musical excursions. These financial burdens may have contributed to Galileo's early fire to develop inventions that would bring him additional income. When Galileo Galilei was eight, his family moved to Florence, but he was left with Jacopo Borghini for two years. Galileo then was educated at 35 southeast of Florence. The Italian male given name "Galileo" derives from the Latin "Galilaeus", meaning "of Galilee", a biblically significant region in Northern Israel. The biblical roots of Galileo's name and surname were to become the subject of a famous pun.
Galileo Galilei
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Portrait of Galileo Galilei by Giusto Sustermans
Galileo Galilei
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Galileo's beloved elder daughter, Virginia (Sister Maria Celeste), was particularly devoted to her father. She is buried with him in his tomb in the Basilica of Santa Croce, Florence.
Galileo Galilei
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Galileo Galilei. Portrait by Leoni
Galileo Galilei
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Cristiano Banti 's 1857 painting Galileo facing the Roman Inquisition
137.
Johannes Kepler
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Johannes Kepler was a German mathematician, astronomer, astrologer. These works also provided one of the foundations for Isaac Newton's theory of universal gravitation. Kepler was a teacher at a seminary school in Graz, Austria, where he became an associate of Prince Hans Ulrich von Eggenberg. He was also an adviser to General Wallenstein. There was a strong division between astronomy and physics. Kepler was born on the feast day of St John the Evangelist, 1571, in the Free Imperial City of Weil der Stadt. Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, the Kepler family fortune was in decline. He left the family when Johannes was five years old. He was believed to have died in the Eighty Years' War in the Netherlands. An innkeeper's daughter, was a healer and herbalist. Born prematurely, Johannes claimed to have been sickly as a child. Nevertheless, he often impressed travelers with his phenomenal mathematical faculty. He was developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, writing that he "was taken to a high place to look at it."
Johannes Kepler
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A 1610 portrait of Johannes Kepler by an unknown artist
Johannes Kepler
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Birthplace of Johannes Kepler in Weil der Stadt
Johannes Kepler
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Portraits of Kepler and his wife in oval medallions
Johannes Kepler
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House of Johannes Kepler and Barbara Müller in Gössendorf near Graz (1597–1599)
138.
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 a former royal deer park near Liverpool, Lancashire. His father James had moved to Toxteth Park to be subsequently married his master's daughter Mary. Both families were well educated Puritans; the Horrocks sent their younger sons to the University of Cambridge and the Aspinwalls favoured Oxford. In 1632 Horrocks matriculated as a sizar. At Cambridge he associated with the platonist John Worthington. 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 seafaring town so navigational instruments such as the astrolabe and staff were easy to find. But there was no market for the specialised astronomical instruments he needed, so his only option was to make his own. He was well placed to do this; his father and uncles were watchmakers with expertise in creating precise instruments. According to local tradition in Much Hoole, he lived within the Bank Hall Estate, Bretherton. He posited that comets followed elliptical orbits. He anticipated Isaac Newton in suggesting the influence of the Sun well as the Earth on the moon's orbit. In the Principia Newton acknowledged Horrocks's work to his theory of lunar motion.
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
139.
Alexis Clairaut
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Alexis Claude Clairaut was a French mathematician, astronomer, geophysicist. Clairaut was one of the key figures in the expedition to Lapland that helped to confirm Newton's theory for the figure of the Earth. In that context, Clairaut worked out a mathematical result now known as "Clairaut's theorem". He also tackled the gravitational three-body problem, being the first to obtain a satisfactory result for the apsidal precession of the Moon's orbit. In mathematics he is also credited with Clairaut's equation and Clairaut's relation. He was born to Catherine Petit Clairaut. The couple had 20 children, however only a few of them survived childbirth. His father taught mathematics. Alexis was a prodigy — at the age of ten he began studying calculus. He known for leading an social life. Though he led a fulfilling 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. They sought to prove if Newton's theory and calculations were correct or not. Before the expedition team returned to Paris, Clairaut sent his calculations to the Royal Society of London.
Alexis Clairaut
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Alexis Claude Clairaut
140.
Pierre-Simon Laplace
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Pierre-Simon, marquis de Laplace was an influential French scholar whose work was important to the development of mathematics, statistics, physics and astronomy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste. This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace. The Laplacian differential operator, widely used in mathematics, is also named after him. Laplace is remembered as one of the greatest scientists of all time. Laplace was named a marquis after the Restoration. Laplace was born in Beaumont-en-Auge, Normandy on 23 March 1749 at Beaumont-en-Auge, a village four miles west of Pont l'Eveque in Normandy. According to W. W. Rouse Ball, His father, Pierre de Laplace, owned and farmed the small estates of Maarquis. His great-uncle, Maitre Oliver de Laplace, had held the title of Chirurgien Royal. It was here that Laplace was educated and was provisionally a professor. It was here he wrote his first paper published in the Mélanges of the Royal Society of Turin, Tome iv. 1766–1769, at least two years before he went at 22 or 23 to Paris in 1771. Thus before he was 20 he was in touch with Lagrange in Turin. He did not go to Paris a self-taught lad with only a background!
Pierre-Simon Laplace
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Pierre-Simon Laplace (1749–1827). Posthumous portrait by Jean-Baptiste Paulin Guérin, 1838.
Pierre-Simon Laplace
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Laplace's house at Arcueil.
Pierre-Simon Laplace
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Laplace.
Pierre-Simon Laplace
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Tomb of Pierre-Simon Laplace
141.
William Rowan Hamilton
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Sir William Rowan Hamilton PRIA FRSE was an Irish physicist, astronomer, mathematician, who made important contributions to classical mechanics, optics, algebra. His studies of optical systems led him to discover new mathematical concepts and techniques. His best known contribution to mathematical physics is the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the development of quantum mechanics. In pure mathematics, Hamilton is best known as the inventor of quaternions. He 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 is, the first mathematician of his age.' He also invented "icosian calculus", which he used to investigate closed edge paths on a dodecahedron that visit each vertex once. He was the fourth of nine children born to Archibald Hamilton, who lived in Dublin at 38 Dominick Street. Hamilton's father, from Dunboyne, worked as a solicitor. Meath. His uncle soon discovered that Hamilton had a remarkable ability to learn languages, from a young age, had displayed an uncanny ability to acquire them. These included Persian, Arabic, Hindustani, Sanskrit, even Marathi and Malay. In September 1813 the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, a year older than Hamilton.
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.
142.
Daniel Bernoulli
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Daniel Bernoulli FRS was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics for his pioneering work in probability and statistics. Daniel Bernoulli was born in Groningen, into a family of distinguished mathematicians. The Bernoulli family emigrated to escape the Spanish persecution of the Huguenots. After a brief period in Frankfurt the family moved in Switzerland. Daniel was the son of nephew of Jacob Bernoulli. He had two brothers, Johann II. Daniel Bernoulli was described by W. W. Rouse Ball as "by far the ablest of the younger Bernoullis". He is said to have had a bad relationship with his father. Johann Bernoulli also plagiarized some key ideas in his own book Hydraulica which he backdated to before Hydrodynamica. Despite Daniel's attempts at reconciliation, his father carried the grudge until his death. Around age, his father, Johann, encouraged him to study business, there being poor rewards awaiting a mathematician. However, Daniel refused, because he wanted to study mathematics. He later studied business. Daniel earned a PhD in anatomy and botany in 1721.
Daniel Bernoulli
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Daniel Bernoulli
143.
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 educating Leonhard Euler in the pupil's youth. Johann began studying medicine at Basel University. However, Johann Bernoulli 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. They were among the first mathematicians 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 at the University of Groningen. At the request of Johann Bernoulli's father-in-law, Johann Bernoulli began the voyage back in 1705. Just after setting out on the journey he learned to tuberculosis. As a student of Leibniz's calculus, Johann Bernoulli sided with him 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' theory over Newton's theory of gravitation. This ultimately delayed acceptance of Newton’s theory in continental Europe. In consequence he was disqualified for the prize, won by Maclaurin.
Johann Bernoulli
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Johann Bernoulli (portrait by Johann Rudolf Huber, circa 1740)
144.
Augustin-Louis Cauchy
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Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician reputed as a pioneer of analysis. Cauchy was one of the first to prove theorems of calculus rigorously, rejecting the heuristic principle of the generality of algebra of earlier authors. Cauchy singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. Cauchy had a great influence over his contemporaries and successors. His writings range widely in mathematical physics. "More theorems have been named for Cauchy than for any other mathematician." Cauchy was a prolific writer; he wrote five complete textbooks. He was the son of Louis François Cauchy and Marie-Madeleine Desestre. He married Aloise de Bure in 1818. She was a close relative of the publisher who published most of Cauchy's works. By her Cauchy had two daughters, Marie Mathilde. Cauchy's father was a high official in the Parisian Police of the New Régime. Cauchy lost his position because of the French Revolution that broke out month before Augustin-Louis was born. The Cauchy family survived 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.
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
145.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. The letter c is a constant, the speed of light in a vacuum. Algebra gives methods for expressing formulas that are much easier than the older method of writing everything out in words. The algebra is also used in certain specialized ways. A mathematician who does research in algebra is called an algebraist. The algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wa ` l-muḳābala by al-Khwarizmi. The word entered the English language from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting dislocated bones. The mathematical meaning was first recorded in the sixteenth century. The word "algebra" has related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, "algebra" names a broad part of mathematics.
Algebra
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A page from Al-Khwārizmī 's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Algebra
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Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
146.
Lagrange's theorem (group theory)
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The theorem is named after Joseph-Louis Lagrange. This can be shown using the concept of left cosets of H in G. If we can show that all cosets of H have the same number of elements, then each coset of H has precisely |H| elements. This map is bijective because its inverse is given by f − 1 = a b − 1 y. This proof also shows that the quotient of the orders |G| / |H| is equal to the index. If the group has n elements, it follows a n = e. This can be used to prove Euler's theorem. These special cases were known long before the general theorem was proved. The theorem also shows that any group of prime order is cyclic and simple. This in turn can be used to prove Wilson's theorem, that if p is prime then p is a factor of! + 1. Hence p < q, contradicting the assumption that p is the largest prime. Lagrange's theorem raises the converse question as to whether every divisor of the order of a group is the order of some subgroup. The smallest example is the alternating group G = A4, which has 12 elements but no subgroup of order 6. There are partial converses to Lagrange's theorem.
Lagrange's theorem (group theory)
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G is the group, the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to. There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is an additive group). Together they partition the entire group G into equal-size, non-overlapping sets. Thus the index [G: H] is 4.
147.
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, in some sense, simpler and better understood. 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, involves studying automorphisms of field extensions. Further abstraction of Galois theory is achieved by the theory of Galois connections. Further, it gives a conceptually clear, often practical, means of telling when some particular equation of higher degree can be solved in that manner. Galois theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterisation of the ratios of lengths that can be constructed with this method. This was first formalized by the 16th-century French mathematician François Viète, in Viète's formulas, for the case of positive real roots. 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, who shared it with Gerolamo Cardano, asking him to not publish it. Cardano then extended this to numerous other cases, using similar arguments; see more details at Cardano's method. After the discovery of Ferro's work, he felt that Tartaglia's method was no longer secret, thus he published his solution in his 1545 Ars Magna. His student Lodovico Ferrari solved the quartic polynomial; his solution was also included in Ars Magna.
Galois theory
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Évariste Galois (1811–1832)
148.
Functional determinant
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The corresponding det is called the functional determinant of S. There are several formulas for the functional determinant. They are all based on the fact that the determinant of a finite-dimensional matrix is equal to the product of the eigenvalues of the matrix. This integral is only well defined up to some divergent multiplicative constant. To give a rigorous meaning it must be divided by another functional determinant, thus effectively cancelling the problematic ` constants'. These are ostensibly, two different definitions for the functional determinant, one coming from quantum field theory and one coming from spectral theory. The problem is to find a way to make sense of the determinant of an operator S on an dimensional function space. In the basis of the functions fi, the functional integration reduces to an integration over all basisfunctions. 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 perform some kind of regularization. The most popular of which for computing functional determinants is the zeta function regularization. For instance, this allows on a Riemannian manifold using the Minakshisundaram -- Pleijel zeta function. Otherwise, it is also possible to consider the quotient of two determinants, making the divergent constants cancel. Let S be an elliptic operator with smooth coefficients, positive on functions of compact support.
Functional determinant
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The infinite potential well with A = 0.
149.
Jacobian matrix and determinant
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In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature. Suppose f: ℝn → ℝm is a function which takes as input the vector x ∈ ℝn and produces as output the vector f ∈ ℝm. This matrix, whose entries are functions of x, is also denoted by Df, Jf, ∂/∂. This linear map is thus the generalization of the usual notion of derivative, is called the derivative or the differential of f at x. If m = n, the Jacobian matrix is a square matrix, its determinant, a function of x1, …, xn, is the Jacobian determinant of f. It carries important information about the local behavior of f. The Jacobian determinant also appears when changing the variables in multiple integrals. These concepts are named after the mathematician Carl Gustav Jacob Jacobi. The Jacobian generalizes the gradient of a scalar-valued function of multiple variables, which itself generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian for a scalar-valued multivariate function is the gradient and that of a scalar-valued function of single variable is simply its derivative. The Jacobian can also be thought of as describing the amount of "stretching", "rotating" or "transforming" that a transformation imposes locally. For example, if = f is used to transform an image, the Jacobian Jf, describes how the image in the neighborhood of is transformed. If p is a point in ℝn and f is differentiable at p, then its derivative is given by Jf. Compare this to a Taylor series for a scalar function of a scalar argument, truncated to first order: f = f + f ′ + o.
Jacobian matrix and determinant
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A nonlinear map f: R 2 → R 2 sends a small square to a distorted parallelepiped close to the image of the square under the best linear approximation of f near the point.
150.
Volume
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Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the cubic metre. Three mathematical shapes are also assigned volumes. Circular shapes can be easily calculated using arithmetic formulas. Volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shape's boundary. Two-dimensional shapes are assigned zero volume in the three-dimensional space. The volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the combined volume is not additive. In geometry, volume is expressed by means of the volume form, is an important global Riemannian invariant. In thermodynamics, volume is a conjugate variable to pressure. Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length. In the International System of Units, the standard unit of volume is the cubic metre.
Volume
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A measuring cup can be used to measure volumes of liquids. This cup measures volume in units of cups, fluid ounces, and millilitres.
151.
Tetrahedron
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In geometry, a tetrahedron is a polyhedron composed of four triangular faces, six straight edges, four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle, so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere on which all four vertices lie, another sphere tangent to the tetrahedron's faces. A regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have been known since antiquity. In a regular tetrahedron, not only are all its faces the same shape but so are all its edges. If alternated with regular octahedra they form the cubic honeycomb, a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two dual tetrahedra form a stellated octahedron or octangula. This form has Schläfli h.
Tetrahedron
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(Click here for rotating model)
Tetrahedron
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4-sided die
152.
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 | 0 | = 0. For example, the absolute value of − 3 is also 3. The absolute value of a number may be thought as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, vector spaces. The absolute value is closely related to the notions of magnitude, norm in various mathematical and physical contexts. The term absolute value has been used 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 magnitude. The same notation is used when x is a set to denote cardinality; the meaning depends on context. As can be seen from the above definition, the absolute value of x is always never negative. Indeed, the notion of an abstract function in mathematics can be seen to be a generalisation of the absolute value of the difference. For example: Absolute value is used to define the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value can not 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.
153.
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 A is denoted | A |. It can be viewed as the scaling factor of the transformation described by the matrix. Each determinant of a 2 × 2 matrix in this equation is called a "minor" of the matrix A. Determinants occur throughout mathematics. The use of determinants in calculus includes the Jacobian determinant in the change of variables rule for integrals of functions of several variables. Determinants are also used to define the characteristic polynomial of a matrix, essential for eigenvalue problems in linear algebra. In analytical geometry, determinants express the signed n-dimensional volumes of n-dimensional parallelepipeds. Sometimes, determinants are used merely as a compact notation for expressions that would otherwise be unwieldy to write down. There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns. Another way to define the determinant is expressed in terms of the columns of the matrix. These properties mean that the determinant is an alternating function of the columns that maps the matrix to the underlying 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, it can be shown to be unique. Assume A is a square matrix with n rows and n columns, so that it can be written as A =.
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.
154.
Pell's equation
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Joseph Louis Lagrange proved that, long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the x/y. His Brahma Sphuta Siddhanta was subsequently translated into Latin in 1126. Bhaskara II in Narayana Pandit in the 14th century both found general solutions to Pell's equation and other quadratic indeterminate equations. The name of Pell's equation arose from Leonhard Euler's mistakenly attributing Lord Brouncker's solution of the equation to John Pell. Indeed, if y are positive integers satisfying this equation, then x/y is an approximation of √ 2. Later, Archimedes approximated the square root of 3 by the rational 1351/780. Although he did not explain his methods, this approximation may be obtained in the same way, as a solution to Pell's equation. Archimedes' cattle problem involves solving a Pellian equation. It is now generally accepted that this problem is due to Archimides. This equation is equivalent to it. Diophantus solved the equation to and. A 10th-century Persian mathematician, worked on similar problems to Diophantus. In Indian mathematics, Brahmagupta discovered that = − N 2 = 2 − N 2. For instance, for N = 92, Brahmagupta composed the triple with itself to get the new triple.
Pell's equation
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Pell's equation for n = 2 and six of its integer solutions
155.
Bachet
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Claude Gaspard Bachet de Méziriac was a French mathematician, linguist, poet and classics scholar born in Bourg-en-Bresse, at that time belonging to Duchy of Savoy. Bachet was a pupil of the Jesuit mathematician Jacques de Billy at the Jesuit College in Rheims. They became close friends. He also wrote Les éléments arithmétiques, which exists in manuscript; and a translation, from Greek to Latin, of the Arithmetica of Diophantus. It was this very translation in which Fermat wrote his famous note claiming that he had a proof of Fermat's last theorem. The same text renders Diophantus' παρισὀτης as adaequalitat, which became Fermat's technique of adequality, a pioneering method of infinitesimal calculus. Bachet was the earliest writer who discussed the solution of indeterminate equations by means of continued fractions. He also found a method of constructing magic squares. Some credible sources also name the founder of the Bézout's identity. For a year in 1601 Bachet was a member of the Jesuit Order. He married in 1612. He was elected member of the Académie française in 1635. Singh, Simon. 1997. Fermat's Last Theorem. Pg 61.
Bachet
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Claude-Gaspard Bachet
Bachet
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Title page of the 1621 edition of Diophantus ' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
156.
List of important publications in mathematics
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This is a list of important publications in mathematics, organized by field. Baudhayana Believed to have been written around the 8th BC, this is one of the oldest mathematical texts. It was influential in South Asia and its surrounding regions, perhaps even Greece. The Nine Chapters on the Mathematical Art from the 10th–2nd century BCE. Contains the earliest description of Gaussian elimination for solving system of linear equations, it also contains method for finding cubic root. Liu Hui Contains the application of distant objects. Sunzi Contains the earlist description of Chinese theorem. Aryabhata Aryabhata introduced the method of the Indians that has become our algebra today. This algebra then migrated to Europe. The text contains 33 verses covering mensuration, geometric progressions, gnomon / shadows, simple, quadratic, simultaneous, indeterminate equations. It also gave the standard algorithm for solving first-order diophantine equations. Jigu Suanjing This book by Tang mathematician Wang Xiaotong Contains the world's earliest third order equation. Brahmagupta Contained rules for manipulating general methods of solving linear and some quadratic equations. Muhammad ibn Mūsā al-Khwārizmī The first book by the Persian scholar Muhammad ibn Mūsā al-Khwārizmī. The book is considered to be the foundation of Islamic mathematics.
List of important publications in mathematics
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One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
List of important publications in mathematics
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Institutiones calculi differentialis
157.
Analytical geometry
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In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian system is applied to manipulate equations for planes, straight lines, squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean Euclidean space. The numerical output, however, might also be a shape. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. He further developed relations between the corresponding ordinates that are equivalent to rhetorical equations of curves. Curves were not determined by equations. Coordinates, 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. The alternative term used for analytic geometry, is named after Descartes. This work, written in its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, 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 Descartes's masterpiece receive due recognition.
Analytical geometry
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Cartesian coordinates
158.
Quadric
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In mathematics, a quadric or quadric surface, is a generalization of conic sections to any number of dimensions. It is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial. A quadric may be defined over any ring. In general, the locus of zeros of a set of polynomials is studied in the branch of algebraic geometry. A quadric is thus an example of an algebraic set. For the projective theory see Quadric. Quadrics in the Euclidean plane are those of dimension D = 1, to say that they are curves. Such quadrics are typically known as conics rather than quadrics. In Euclidean space, quadrics are known as quadric surfaces. In Euclidean space there are 16 such normal forms. Of these 16 forms, the remaining are degenerate forms. Degenerate forms include planes, lines, even no points at all. The quadrics can be treated by introducing homogeneous coordinates on a Euclidean space, thus effectively regarding it as a projective space. Regarding Q = 0 as an equation in projective space exhibits the quadric as a algebraic variety. The quadric is said to be non-degenerate if the quadratic form is non-singular; equivalently, if the matrix is invertible.
Quadric
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Ellipse (e = 1/2), parabola (e =1) and hyperbola (e = 2) with fixed focus F and directrix.
159.
Partial differential equation
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In mathematics, a partial differential equation is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in partial differential equations. Partial differential equations are equations that involve rates of change with respect to continuous variables. The dynamics for the rigid body take place in a finite-dimensional space; the dynamics for the ﬂuid occur in an infinite-dimensional conﬁguration space. Here again, there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid dynamics, heat transfer. A partial equation for the function u is an equation of the form f = 0. If f is a linear function of its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Poisson's equation. A relatively simple PDE is ∂ u ∂ x = 0. This relation implies that the u is independent of x.
Partial differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
160.
Colin Maclaurin
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Colin Maclaurin /məˈklɔːrən/ Scottish Gaelic: Cailean MacLabhruinn was a Scottish mathematician who made important contributions to geometry and algebra. A special case of the Taylor series, is named after him. Owing since that time, his surname is alternatively written MacLaurin. Maclaurin was born in Kilmodan, Argyll. His mother died before he reached nine years of age. He was then educated under the care of the Reverend Daniel Maclaurin, minister of Kilfinan. At eleven, Maclaurin entered the University of Glasgow. This record as the world's youngest professor endured until March 2008, when the record was officially given to Alia Sabur. He was admitted a member of the Royal Society. During their time in Lorraine, he wrote his essay on the percussion of bodies, which gained the prize of the Royal Academy of Sciences in 1724. Upon the death of his pupil at Montpellier, Maclaurin returned to Aberdeen. In 1725 Maclaurin was appointed deputy upon the recommendation of Isaac Newton. On 3 November of that year Maclaurin went on to raise the character of that university as a school of science. Newton was so impressed with Maclaurin that he had offered to pay himself. Maclaurin used Taylor series to characterize maxima, points of inflection for infinitely differentiable functions in his Treatise of Fluxions.
Colin Maclaurin
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Colin Maclaurin (1698–1746)
Colin Maclaurin
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Memorial, Greyfriars Kirkyard, Edinburgh
161.
Orbit
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Orbits of planets are typically elliptical, the central mass being orbited is at a focal point of the ellipse. For ease of calculation, relativity is commonly approximated based on Kepler's laws of planetary motion. Historically, the apparent motions of the planets were described by Arabic philosophers using the idea of celestial spheres. This model posited the existence of rings to which the stars and planets were attached. It was developed without any understanding of gravity. After the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycles were added. 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 modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. Third, Kepler found a universal 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 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 5.23 / 11.862, is practically equal to that for Venus, 0.7233 / 0.6152, in accord with the relationship. Lagrange made progress on the three body problem, discovering the Lagrangian points. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits.
Orbit
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The International Space Station orbits above Earth.
Orbit
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Planetary orbits
Orbit
Orbit
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Conic sections describe the possible orbits (yellow) of small objects around the earth. A projection of these orbits onto the gravitational potential (blue) of the earth makes it possible to determine the orbital energy at each point in space.
162.
Comet
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Community of Metros is a system of international railway benchmarking. CoMET consists from around the world. Each metro has a volume of at least million passengers annually. The four main objectives of CoMET are: To build measures to establish metro best practice. To provide comparative information both for the government. To introduce a system of measures for management. To prioritise areas for improvement. These objectives were discussed in detail at the CoMET Annual Meeting hosted by SMRT Trains of SMRT Corporation. The meeting was held in November 2016. The project was successful despite the fact that metros were very different in sizes, accounting practices. However, CoMET used the Key Performance Indicator innovatively to solve the problem. Thus, the metros can investigate best practice amongst similar heavy metros. These five metros are later known as the Group of Five. Over time, large transit systems joined the group. For example, Mexico City Metro, Tokyo Metro joined in 1996.
Comet
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"CoMET" redirects here. For the geoprofession, see Geoprofessions § Construction-materials engineering and testing (CoMET).
Comet
163.
Orbital elements
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Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used. A real orbit changes due to the effects of relativity. A Keplerian orbit is merely an idealized, mathematical approximation at a particular time. The orbital elements are the six Keplerian elements, after his laws of planetary motion. When viewed from an inertial frame, two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the common center of mass. An orbit has two sets of Keplerian elements depending on which body is used as the point of reference. The reference body is called the primary, the other body is called the secondary. Semimajor axis —the sum of the periapsis and apoapsis distances divided by two. For circular orbits, the semimajor axis is the distance between the centers of the bodies, not the distance of the bodies from the center of mass. Tilt angle is measured perpendicular to line of intersection between orbital plane. Any three points on an ellipse will define the ellipse orbital plane. The plane and the ellipse are both two-dimensional objects defined in three-dimensional space. Longitude of the ascending node —horizontally orients the ascending node of the ellipse with respect to the reference frame's vernal point.
Orbital elements
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In this diagram, the orbital plane (yellow) intersects a reference plane (gray). For earth-orbiting satellites, the reference plane is usually the Earth's equatorial plane, and for satellites in solar orbits it is the ecliptic plane. The intersection is called the line of nodes, as it connects the center of mass with the ascending and descending nodes. This plane, together with the Vernal Point (♈), establishes a reference frame.
164.
Urbain Le Verrier
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The calculations were made to explain discrepancies with Uranus's orbit and the laws of Kepler and Newton. Le Verrier sent the coordinates to Johann Gottfried Galle in Berlin, asking him to verify. Galle found Neptune in the same night he received Le Verrier's letter, within 1° of the predicted position. The discovery of Neptune is widely regarded as a dramatic validation of celestial mechanics, is one of the most remarkable moments of 19th century science. Le Verrier was born at Saint-Lô, Manche, France, studied at École Polytechnique. He briefly studied chemistry under Gay-Lussac, writing papers on the combinations of phosphorus and hydrogen, phosphorus and oxygen. He then switched to astronomy, particularly celestial mechanics, accepted a job at the Paris Observatory. He spent most of his professional life there, eventually became that institution's Director, from 1854 to 1870 and again from 1873 to 1877. Le Verrier's name is one of the 72 names inscribed on the Eiffel Tower. Le Verrier's first work in astronomy was presented to the Académie des Sciences in September 1839, entitled Sur les variations séculaires des orbites des planètes. This work addressed the then most-important question in astronomy: the stability of the Solar System, first investigated by Laplace. From 1844 to 1847, Le Verrier published a series of works on periodic comets, in particular those of Lexell, Faye and DeVico. 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, to an extent still is, controversy over the apportionment of credit for the discovery.
Urbain Le Verrier
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Urbain Le Verrier
Urbain Le Verrier
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Signature of M. LeVerrier
Urbain Le Verrier
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The grave of Urbain Le Verrier.
165.
Mechanics
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The scientific discipline has its origins in Ancient Greece with the writings of Aristotle and Archimedes. During the modern period, scientists such as Khayaam, Galileo, Kepler, Newton, laid the foundation for what is now known as classical mechanics. It can also be defined as a branch of science which deals with forces on objects. Historically, classical mechanics came first, while quantum mechanics is a comparatively recent invention. Classical mechanics originated in Principal Mathematical; Quantum Mechanics was discovered in the early 20th century. Both are commonly held to constitute the most certain knowledge that exists about physical nature. Classical mechanics has often been viewed as a model for other so-called exact sciences. Essential in this respect is the relentless use of mathematics in theories, well as the decisive role played by experiment in generating and testing them. Quantum mechanics is of a bigger scope, as it encompasses classical mechanics as a sub-discipline which applies under certain restricted circumstances. According to the principle, there is no contradiction or conflict between the two subjects, each simply pertains to specific situations. The principle states that the behavior of systems described by quantum theories reproduces classical physics in the limit of large quantum numbers. However, for macroscopic processes classical mechanics is well used. Modern descriptions of such behavior begin as displacement, time, velocity, acceleration, mass, force. Until about 400 years ago, however, motion was explained from a very different point of view. He showed that the speed of falling objects increases steadily during the time of their fall.
Mechanics
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Arabic Machine Manuscript. Unknown date (at a guess: 16th to 19th centuries).
166.
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 practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing the real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. 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, "logistic", now called arithmetic. The term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a separate discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between pure and applied. 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, strongly influenced by David Hilbert's example. 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 pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality.
Pure mathematics
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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.
167.
Adrien-Marie Legendre
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Adrien-Marie Legendre was a French mathematician. Legendre made numerous contributions to mathematics. 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. He defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Berlin Academy awarded a prize for his treatise on projectiles in resistant media. This treatise also brought him to the attention of Lagrange. The Académie des Sciences made an adjoint member in 1783 and an associé in 1785. In 1789 he was elected a Fellow of the Royal Society. He assisted with the Anglo-French Survey to calculate the precise distance between the Royal Greenwich Observatory by means of trigonometry. To this end in 1787 he visited Dover and London together with Dominique, comte de Cassini and Pierre Méchain. The three also visited the discoverer of the planet Uranus. Legendre lost his private fortune during the French Revolution.
Adrien-Marie Legendre
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1820 watercolor caricature of Adrien-Marie Legendre by French artist Julien-Leopold Boilly (see portrait debacle), the only existing portrait known
Adrien-Marie Legendre
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1820 watercolor caricatures of the French mathematicians Adrien-Marie Legendre (left) and Joseph Fourier (right) by French artist Julien-Leopold Boilly, watercolor portrait numbers 29 and 30 of Album de 73 portraits-charge aquarellés des membres de I’Institut.
Adrien-Marie Legendre
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Side view sketching of French politician Louis Legendre (1752–1797), whose portrait has been mistakenly used, for nearly 200 years, to represent French mathematician Adrien-Marie Legendre, i.e. up until 2005 when the mistake was discovered.
168.
Taylor's theorem
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In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. The exact content of "Taylor's theorem" is not universally agreed upon. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1712. Yet an explicit expression of the error was not provided until much later on by Joseph-Louis Lagrange. An earlier version of the result was already mentioned in 1671 by James Gregory. Taylor's theorem is taught in introductory level calculus courses and it is one of the central elementary tools in mathematical analysis. Taylor's theorem also generalizes to vector valued functions f: R n → m on any dimensions m. This generalization of Taylor's theorem is the basis for the definition of so-called jets which appear in differential geometry and partial differential equations. If a real-valued function f is differentiable at the point a then it has a linear approximation at the point a. Here P 1 = f + f ′ is the linear approximation of f at the point a. The graph of y = P1 is the tangent line to the graph of f at x = a. The error in the approximation is R 1 = f − P 1 = h 1. Note that this goes to zero a little bit faster than x − a as x tends to a, given the limiting behavior of h1. If we wanted a better approximation to f, we might instead try a quadratic polynomial instead of a linear function. The quadratic polynomial in question is P 2 = f + f ′ + f ″ 2 2.
Taylor's theorem
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The exponential function y = e x (solid red curve) and the corresponding Taylor polynomial of degree four (dashed green curve) around the origin.
169.
Augustin Louis Cauchy
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Baron Augustin-Louis Cauchy FRS FRSE was a French mathematician reputed as a pioneer of analysis. Cauchy was one of the first to prove theorems of calculus rigorously, rejecting the heuristic principle of the generality of algebra of earlier authors. Cauchy singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. Cauchy had a great influence over his contemporaries and successors. His writings range widely in mathematical physics. "More theorems have been named for Cauchy than for any other mathematician." Cauchy was a prolific writer; he wrote five complete textbooks. He was the son of Louis François Cauchy and Marie-Madeleine Desestre. He married Aloise de Bure in 1818. She was a close relative of the publisher who published most of Cauchy's works. By her Cauchy had two daughters, Marie Mathilde. Cauchy's father was a high official in the Parisian Police of the New Régime. Cauchy lost his position because of the French Revolution that broke out month before Augustin-Louis was born. The Cauchy family survived 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.
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
170.
Carl Gustav Jakob Jacobi
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Carl Gustav Jacob Jacobi was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, number theory. His name is occasionally written as Carolus Gustavus Iacobus Iacobi in his Latin books, his first name is sometimes given as Karl. Jacobi was the first Jewish mathematician to be appointed professor at a German university. Jacobi was born of Ashkenazi Jewish parentage in Potsdam on 10 December 1804. He was the second of the four children of banker Simon Jacobi. His elder brother Moritz von Jacobi would also become known later as an engineer and physicist. He was initially home schooled by his uncle Lehman, who instructed him in the classical languages and elements of mathematics. In 1816, the twelve-year-old Jacobi went to the Potsdam Gymnasium, where students were being taught classical languages, German history as well as mathematics. However, as the University was not accepting students younger than 16 years old, he had to remain in the senior class until 1821. Jacobi used this time to advance his knowledge, showing interest including Greek, history and mathematics. During this period he also made the first attempts at research trying to solve the quintic equation by radicals. In 1821 Jacobi went to study at the Berlin University, where initially he divided his attention between his passions for philology and mathematics. In philology he participated in the seminars of Böckh, drawing the professor's attention with his talent. However, he continued with his private study of the more advanced works of Euler, Lagrange and Laplace. By 1823 he understood that he needed to make a decision between his competing interests and he chose to devote all his attention to mathematics.
Carl Gustav Jakob Jacobi
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Carl Gustav Jacob Jacobi
171.
Karl Weierstrass
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Karl Theodor Wilhelm Weierstrass was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he trained as a teacher, eventually teaching mathematics, physics, botany and gymnastics. Weierstrass was born in part of Ennigerloh, Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, Theodora Vonderforst. His interest in mathematics began while he was a student at the Theodorianum in Paderborn. He was sent upon graduation to prepare for a government position. Because his studies were to be in the fields of law, finance, he was immediately in conflict with his hopes to study mathematics. He continued private study in mathematics. The outcome was to leave the university without a degree. Later he was certified as a teacher in that city. During this period of study, Weierstrass became interested in elliptic functions. Since 1848 he taught at the Lyceum Hosianum in Braunsberg. Besides mathematics he also taught physics, gymnastics. Weierstrass may have had an illegitimate child named Franz with the widow of his friend Carl Wilhelm Borchardt. After 1850 Weierstrass was able to publish papers that brought him fame and distinction.
Karl Weierstrass
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Karl Theodor Wilhelm Weierstrass (Weierstraß)
172.
Fermat's little theorem
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In the notation of modular arithmetic, this is expressed as a p ≡ a. For example, if a = 2 and p = 7 then 26 = 64 and 64 − 1 = 63 is thus a multiple of 7. Fermat's little theorem is one of the fundamental results of elementary theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's last theorem. Pierre de Fermat first stated the theorem to confidant Frénicle de Bessy. An early use in English occurs in A.A. Albert, Modern Higher Algebra, which refers to "the so-called "little" Fermat theorem" on page 206. Some mathematicians independently made the related hypothesis that 2p ≡ 2 if and only if p is a prime. The "if" part is a special case of Fermat's little theorem. However, the "only if" part of this hypothesis is false: for 341 = 11 × 31 is a pseudoprime. See below. Several proofs of Fermat's little theorem are known. It is frequently proved as a corollary of Euler's theorem. Euler's theorem is indeed a generalization, because if n = p is a prime number, then φ = p − 1.
Fermat's little theorem
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Pierre de Fermat
173.
Planetary motion
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The central mass being orbited is at a focal point of the ellipse. For ease of calculation, relativity is commonly approximated based on Kepler's laws of planetary motion. Historically, the apparent motions of the planets were described by Arabic philosophers using the idea of celestial spheres. This model posited the existence of rings to which the stars and planets were attached. It was developed without any understanding of gravity. After the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycles were added. 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 modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. Third, Kepler found a universal 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 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 5.23 / 11.862, is practically equal to that for Venus, 0.7233 / 0.6152, in accord with the relationship. Lagrange made progress on the three body problem, discovering the Lagrangian points. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits.
Planetary motion
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The International Space Station orbits above Earth.
Planetary motion
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Planetary orbits
Planetary motion
Planetary motion
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Conic sections describe the possible orbits (yellow) of small objects around the earth. A projection of these orbits onto the gravitational potential (blue) of the earth makes it possible to determine the orbital energy at each point in space.
174.
Royal Society of Edinburgh
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The Royal Society of Edinburgh is Scotland's national academy of science and letters. It is a registered charity, providing public benefit throughout Scotland. It was established in 1783. As of 2014 it has more than 1,500 Fellows. The Society covers a broader selection including literature and history. Fellowship includes people from a wide range of disciplines -- technology, arts, humanities, medicine, social science, business and public service. This breadth of expertise makes the Society unique in the UK. The Medals were instituted by Queen Elizabeth II, whose permission is required to make a presentation. Past winners include: The Lord Kelvin Medal is the Senior Prize for Physical, Engineering and Informatics Sciences. Winners are required to deliver a public lecture in Scotland. Senior Prize-winners can be based anywhere in the world. It is awarded alternately for papers on Environmental Sciences. The medal was founded in 1827 by Alexander Keith of Dunottar, the first Treasurer of the Society. The prize was founded by Sir Thomas Makdougall Brisbane, the long-serving fourth President of the Society. The cumbersome name was changed the following year to the Edinburgh Philosophical Society.
Royal Society of Edinburgh
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The Royal Society building, at the junction of George Street and Hanover Street in the New Town
Royal Society of Edinburgh
Royal Society of Edinburgh
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The cover of a 1788 volume of the journal Transactions of the Royal Society of Edinburgh. This is the issue where James Hutton published his Theory of the Earth.
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Royal Society
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Founded in November 1660, it was granted a royal charter by King Charles II as "The Royal Society". The society is governed by its Council, chaired according to a set of statutes and standing orders. As of 2016, there are about 1,600 fellows, allowed to use the postnominal FRS, with up to 52 new fellows appointed each year. There are also royal fellows, 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 November 2015. The Royal Society started from groups of natural philosophers, meeting at variety of locations, including Gresham College in London. They were influenced by the "new science", as promoted 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, at least, did help them, hinder us. Since then, every monarch has been the patron of the society. The society's early meetings included experiments performed first by Hooke and then by Denis Papin, appointed in 1684. These experiments were both important in some cases and trivial in others. The Society returned in 1673.
Royal Society
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The entrance to the Royal Society in Carlton House Terrace, London
Royal Society
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The President, Council, and Fellows of the Royal Society of London for Improving Natural Knowledge
Royal Society
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John Evelyn, who helped to found the Royal Society
Royal Society
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Mace granted by Charles II
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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. Etc. The Academy has elected about 1.700 Swedish and 1.200 foreign members since it was founded in 1739. 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, to publish in Swedish in order to widely disseminate the academy's findings. The academy was intended to be different from the Royal Society of Sciences in Uppsala, founded in 1719 and published in Latin. The location close to the commercial activities in Sweden's capital was also intentional. 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
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Napoleon
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Napoleon Bonaparte was a French military and political leader who rose to prominence during the French Revolution and led several successful campaigns during the Revolutionary Wars. As Napoleon I, he was Emperor of the French from 1804 until 1814, again in 1815. Napoleon dominated European and global affairs for more than a decade while leading France against a series of coalitions in the Napoleonic Wars. One of the greatest commanders in history, his wars and campaigns are studied at military schools worldwide. Napoleon's political and cultural legacy has ensured his status as one of the most celebrated and controversial leaders in human history. He was born in Corsica to a relatively modest family from the minor nobility. When the Revolution broke out in 1789, Napoleon was serving as an artillery officer in the French army. He attempted to capitalize quickly on the new political situation by returning to Corsica in hopes of starting a political career. In 1798, he led a military expedition to Egypt that served as a springboard to political power. He engineered a coup in November 1799 and became First Consul of the Republic. His rising ambition inspired him to go further, in 1804 he became the first Emperor of the French. Intractable differences with the British meant that the French were facing a Third Coalition by 1805. In 1806, the Fourth Coalition took up arms against him because Prussia became worried about growing French influence on the continent. France then forced the defeated nations of the Fourth Coalition to sign the Treaties of Tilsit in July 1807, bringing an uneasy peace to the continent. Tilsit signified the high watermark of the French Empire.
Napoleon
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The Emperor Napoleon in His Study at the Tuileries, by Jacques-Louis David, 1812
Napoleon
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Imperial coat of arms
Napoleon
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Napoleon's father Carlo Buonaparte was Corsica 's representative to the court of Louis XVI of France.
Napoleon
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Napoleon Bonaparte, aged 23, Lieutenant-Colonel of a battalion of Corsican Republican volunteers
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Legion of Honour
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The order is divided into five degrees of increasing distinction: Chevalier, Officier, Commandeur, Grand Officier and Grand-Croix. In the French Revolution, all French orders of chivalry were abolished, replaced with Weapons of Honour. The Légion however did use the organization of old French Orders of Chivalry, like the Ordre de Saint-Louis. 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, concessions. The Légion was loosely patterned with legionaries, a grand council. The highest rank was not a grand cross but a Grand Aigle, a rank that wore all the insignia common to grand crosses. 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? Never. That is good only for the scholar in his study. The soldier needs glory, distinctions, rewards." This has been often quoted as "It is with such baubles that men are led." The order was the first modern order of merit.
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)
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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. The Grand Dignitaries of the Empire ranked, regardless of noble title, immediately behind the Princes of France. Enoblement started with the creation of the princely title for members of Napoleon's imperial family. Others followed. In 1806 ducal titles were created and in 1808 those of count, knight. Napoleon founded the concept of nobility of Empire by an imperial decree on 1 March 1808. The purpose of this creation was to amalgamate the revolutionary middle-class in one peerage system. A council of the titles was also created and charged with establishing armorial bearings, had a monopoly of this new nobility. These creations are to be distinguished from an order such as the Order of the Bath. There were 239 remaining families belonging to the First Empire nobility in 1975. Of those, about 135 were titled. 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 characterised by a stronger sense of hierarchy. It employed a rigid system of additional marks in the shield to indicate official positions.
Count of the Empire
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Imperial coat of arms
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Satellites of Jupiter
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There are 67 known moons of Jupiter. This gives the largest number of moons with reasonably stable orbits of any planet in the Solar System. Of Jupiter's moons, eight are nearly circular orbits that are not greatly inclined with respect to Jupiter's equatorial plane. The other four regular satellites are closer to Jupiter; these serve as sources of the dust that makes up Jupiter's rings. The remainder of Jupiter's moons are irregular satellites whose retrograde orbits are much farther from Jupiter and have high inclinations and eccentricities. These moons were probably captured from solar orbits. Sixteen irregular satellites have not yet been named. The physical and orbital characteristics of the moons vary widely. All Jovian moons are less than 250 kilometres in diameter, with most barely exceeding 5 kilometres. Their orbital shapes range to Jupiter's spin. Orbital periods range to some three thousand times more. Jupiter's regular satellites are believed to have formed from a ring of accreting gas and solid debris analogous to a protoplanetary disk. They may be the remnants of a score of Galilean-mass satellites that formed early in Jupiter's history. 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 Jupiter's early history.
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
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List of the 72 names on the Eiffel Tower
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On the Eiffel Tower, seventy-two names of French scientists, engineers, 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 under the first balcony. The Tower is owned by the city of Paris. The letters are about 60 cm high. The repainting of 2010/2011 restored the letters to their original colour. The list is split in four parts. The list contains no women. In 1913 John Augustine Zahm suggested that Germain was excluded because she was a woman. Scholars are listed on the Eiffel Tower. Eiffel acknowledged most of the leading scientists in the field. Henri Philibert Gaspard Darcy is missing; some of his work did not come until the 20th century. Also missing are Antoine Chézy, less famous, Joseph Valentin Boussinesq, early in his career at the time. Also missing is the mathematician Evariste Galois. Barral, Georges.
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
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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 and built the tower. 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 same height as an 81-storey building, the tallest structure in Paris. Its base is square, measuring 125 metres on each side. 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 level's 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, as is the climb from the first level to the second. 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, 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. After some debate about the exact location of the tower, a contract was signed on 8 January 1887.
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
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Lunar crater
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Lunar craters are impact craters on Earth's Moon. The Moon's surface has many craters, almost all of which were formed by impacts. The crater was adopted by Galileo from the Greek word for vessel -. Galileo turned it to the Moon for the first time on November 30, 1609. Scientific opinion as to the origin of craters swung forth over the ensuing centuries. The formation of new craters is studied at NASA. The biggest recorded creation was caused by an impact recorded on March 2013. Visible to the naked eye, the impact is believed to be from an approximately 40 kg meteoroid striking the surface at a speed of 90,000 km/h. The age of large craters is determined by the number of smaller craters contained within older craters generally accumulating more small, contained craters. The smallest craters found have been microscopic in size, found in rocks returned from the Moon. The largest crater called such is about 290 kilometres across in diameter, located near the lunar South Pole. However, it is believed that many of the maria were formed by giant impacts, with the resulting depression filled by upwelling lava. In 1978, Leif Andersson of the Lunar & Planetary Lab devised a system of categorization of lunar impact craters. They used a sampling of craters that were relatively unmodified by subsequent impacts, then grouped the results into five broad categories. These successfully accounted for about 99% of all lunar impact craters.
Lunar crater
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Side view of the Moltke crater taken from Apollo 11.
Lunar crater
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Webb crater, as seen from Lunar Orbiter 1. Several smaller craters can be seen in and around Webb crater.
Lunar crater
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Lunar craters as captured through the backyard telescope of an amateur astronomer, partially illuminated by the sun on a waning crescent moon.
Lunar crater
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Albategnius
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Public domain
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Works in the public domain are those whose exclusive intellectual property rights have expired, have been forfeited, or are inapplicable. Examples for works actively dedicated by their authors are reference implementations of cryptographic algorithms, NIH's ImageJ, the CIA's World Factbook. As rights vary, a work may be subject to rights in one country and be in the public domain in another. The res nullius was defined as things not yet appropriated. The res communes was defined as "things that could be commonly enjoyed by mankind, such as air, sunlight and ocean." When the early copyright law was first established in Britain with the Statute of Anne in 1710, public domain did not appear. However, similar concepts were developed in the eighteenth century. Instead of "public domain" they used terms such as propriété publique to describe works that were not covered by copyright law. The phrase "fall in the public domain" can be traced to mid-nineteenth century France to describe the end of term. In this historical context Paul Torremans describes copyright as a "little reef of private right jutting up from the ocean of the public domain." Because law is different from country to country, Pamela Samuelson has described the public domain as being "different sizes at different times in different countries". However, the usage of the term domain can be more granular, including for example uses of works in copyright permitted by copyright exceptions. Such a definition regards work in copyright as limitation on ownership. The materials that compose our cultural heritage must be free for all living to use no less than matter necessary for biological survival." Edgar Huntly, Wieland and Sky-Walk by Charles Brockden Brown Camilla, Evelina and Cecilia by Frances Burney Jonathan Dickinson's Journal by Jonathan Dickinson.
Public domain
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Newton's own copy of his Principia, with hand-written corrections for the second edition
Public domain
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L.H.O.O.Q. (1919). Derivative work by the Dadaist Marcel Duchamp based on the Mona Lisa.
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Rouse History of Mathematics
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Walter William Rouse Ball, known as W. W. Rouse Ball, was a British mathematician, lawyer, fellow at Trinity College, Cambridge from 1878 to 1905. He was also the founding president of the Cambridge Pentacle Club in 1919, one of the world's oldest magic societies. Ball was the heir of Walter Frederick Ball, of 3, St John's Park Villas, South Hampstead, London. He remained one for the rest of his life. He is buried in Cambridge. He is commemorated in the naming of the small pavilion, now used as changing 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:.
Rouse History of Mathematics
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W.W. Rouse Ball
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W. W. Rouse Ball
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Walter William Rouse Ball, known as W. W. Rouse Ball, was a British mathematician, lawyer, fellow at Trinity College, Cambridge from 1878 to 1905. He was also the founding president of the Cambridge Pentacle Club in 1919, one of the world's oldest magic societies. Ball was the heir of Walter Frederick Ball, of 3, St John's Park Villas, South Hampstead, London. He remained one for the rest of his life. He is buried in Cambridge. He is commemorated in the naming of the small pavilion, now used as changing 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:.
W. W. Rouse Ball
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W.W. Rouse Ball
187.
Merriam-Webster Dictionary
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Merriam-Webster, Incorporated, is an American company that publishes reference books, especially dictionaries. In 1831, George and Charles Merriam founded the company as G & C Merriam Co. in Springfield, Massachusetts. In 1843, after Noah Webster died, the company bought the rights to An American Dictionary of the English Language from Webster's estate. All Merriam-Webster dictionaries trace their lineage to this source. In 1964, Encyclopædia Britannica, Inc. acquired Merriam-Webster, Inc. as a subsidiary. The company adopted its current name in 1982. In 1806, Webster published his first dictionary, A Compendious Dictionary of the English Language. In 1807 Webster started two decades of intensive work to expand his publication into a fully comprehensive dictionary, An American Dictionary of the English Language. To help him trace the etymology of words, Webster learned 26 languages. Webster hoped to standardize American speech, since Americans in different parts of the country used somewhat different vocabularies and spelled, pronounced, used words differently. Webster completed his dictionary during his year abroad in 1825 in Paris, at the University of Cambridge. His 1820s book contained 70,000 words, of which about 12,000 had never appeared in a dictionary before. He also added American words, including skunk and squash, that did not appear in British dictionaries. At the age of 70 in 1828, Webster published his dictionary; it sold poorly, with only 2,500 copies putting him in debt. However, in 1840, he published the second edition in two volumes with much greater success.
Merriam-Webster Dictionary
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Merriam-Webster's eleventh edition of the Collegiate Dictionary
Merriam-Webster Dictionary
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Merriam-Webster
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each variation of a book. For example, an e-book, a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned after 1 January 2007, 10 digits long if assigned before 2007. The method of assigning an ISBN varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated based upon the 9-digit Standard Book Numbering created in 1966. The 10-digit ISBN format was published in 1970 as international standard ISO 2108. The International Standard Serial Number, identifies periodical publications such as magazines; and the International Standard Music Number covers for musical scores. The ISBN configuration of recognition was generated in 1967 in the United Kingdom by Emery Koltay. The 10-digit ISBN format was published as international standard ISO 2108. The United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978. An SBN may be converted by prefixing the digit "0". This can be converted to ISBN 0-340-01381-8; the digit does not need to be re-calculated. Since 1 ISBNs have contained 13 digits, a format, compatible with "Bookland" European Article Number EAN-13s.
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
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Isoperimetric
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In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume. The equality holds when S is a ball in R n. Isoperimetric literally means "having the same perimeter". The isoperimetric problem is to determine a figure of the largest possible area whose boundary has a specified length. It is named after Dido, first queen of Carthage. The solution to the isoperimetric problem was known already in Ancient Greece. However, the first rigorous proof of this fact was obtained only in the 19th century. Since then, other proofs have been found. The isoperimetric problem has been extended, for example, to curves on surfaces and to regions in higher-dimensional spaces. Perhaps the most familiar physical manifestation of the isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric shape. The isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed perimeter, which curve maximizes the area of its enclosed region? Astrologer Johannes Kepler invoked the isoperimetric principle in discussing the morphology of the solar system, in Mysterium Cosmographicum. Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult.
Isoperimetric
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If a region is not convex, a "dent" in its boundary can be "flipped" to increase the area of the region while keeping the perimeter unchanged.
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Ivor Grattan-Guinness
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Ivor Owen Grattan-Guinness was a historian of mathematics and logic. Grattan-Guinness was born in Bakewell, England; his father was educational administrator. He gained both the doctorate in 1969, higher doctorate in 1978, at the University of London. He was a Visiting Research Associate at the London School of Economics. In 2010, he was elected an Honorary Member of the Bertrand Russell Society. Grattan-Guinness spent much of his career at Middlesex University. From 1974 to 1981, Grattan-Guinness was editor of the history of journal Annals of Science. In 1979 he edited it until 1992. He was an associate editor of Historia Mathematica for twenty years from its inception in 1974, again from 1996. He also acted to the editions of the writings of C.S. Peirce and Bertrand Russell, to several other journals and book series. He was a member of the Executive Committee of the International Commission on the History of Mathematics from 1977 to 1993. Grattan-Guinness gave over 570 invited lectures in over 20 countries around the world. These lectures include tours undertaken in Australia, New Zealand, Italy, Portugal. From 1986 to 1988, Grattan-Guinness was the President of the British Society for 1992 the Vice-President. In 1991, he was elected an effective member of the Académie Internationale d'Histoire des Sciences.
Ivor Grattan-Guinness
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Ivor Grattan-Guinness in 2003.
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Cambridge University Press
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Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent in 1534, Cambridge University is the world's oldest publishing house and the second-largest university press in the world. Cambridge University also holds letters patent as the Queen's Printer. Cambridge University Press is both an academic and educational publisher. With a global sales presence, offices in more than 40 countries, Cambridge University publishes over 50,000 titles by authors from over 100 countries. Its publishing includes academic journals, monographs, reference works, English-language teaching and learning publications. Cambridge University Press is a charitable enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest house in the world and the oldest university press. Cambridge is one of the two privileged presses. Authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Stephen Hawking. In 1591, John Legate, printed the first Cambridge Bible, an octavo edition of the popular Geneva Bible. The London Stationers objected strenuously, claiming that they had the monopoly on Bible printing. The university's response was to point out the provision in its charter to print'all manner of books'. Cambridge University was in 1698, that a body of senior scholars was appointed to be responsible to the university for the Press's affairs. Its role still includes the review and approval of the Press's planned output.
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
192.
Wikisource
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Wikisource is an online digital library of free content textual sources on a wiki, operated by the Wikimedia Foundation. The project's aims are to host all forms of free text, in many languages, translations. Originally conceived as an archive to store important historical texts, it has expanded to become a general-content library. The project officially began under the name Project Sourceberg. It received its own domain name seven months later. It is also cited by organisations such as the National Archives and Records Administration. Verification was initially made offline, or by trusting the reliability of digital libraries. Now works are supported by online scans via the ProofreadPage extension, which ensures the accuracy of the project's texts. Each representing a specific language, now only allow works backed up with scans. While the bulk of its collection are texts, Wikisource as a whole hosts other media, to audio books. Some Wikisources allow user-generated annotations, subject to the specific policies of the Wikisource in question. Wikisource's early history included the move to language subdomains in 2005. The original concept for Wikisource was as storage for important historical texts. These texts were intended to support Wikipedia articles, as an archive in its own right. The collection was initially focused on important cultural material, distinguishing it from other digital archives such as Project Gutenberg.
Wikisource
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The original Wikisource logo
Wikisource
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Screenshot of wikisource.org home page
Wikisource
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::: Original text
Wikisource
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::: Action of the modernizing tool
193.
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 and the third oldest university in the English-speaking world. St Andrews was founded between 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small group of Augustinian clergy. St Andrews is made up including 18 academic schools organised into four faculties. The university occupies historic and modern buildings located throughout the town. The academic year is divided into Candlemas. In time, over one-third of the town's population is either a staff student of the university. It is ranked behind Oxbridge. The Times Higher Education World Universities Ranking names St Andrews among the world's 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 affiliated faculty, including eminent mathematicians, scientists, theologians, politicians. Six Nobel Laureates are amongst St Andrews' alumni and former staff: two in Chemistry and Physiology or Medicine, one each in Peace and Literature. A charter of privilege was bestowed by the Bishop of Henry Wardlaw, on 28 February 1411. 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.
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
194.
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, open formats that can be used on almost any computer. As of 3 October 2015, Project Gutenberg reached 50,000 items in its collection. The releases are available in plain text but, wherever other formats are included, such as Plucker. Non-English works are also available. There are affiliated projects that are providing additional content, including language-specific works. 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. A student at the University of Illinois, obtained access in the university's Materials Research Lab. Hart has said he wanted to "give back" this gift by doing something that could be considered to be of great value. This particular computer was one of the 15 nodes on the network that would become the Internet. Hart decided to make works of literature available for free. He used a copy of the United States Declaration of Independence in his backpack, this became the first Project Gutenberg e-text.
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
195.
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 open Internet. The Wayback Machine, contains over 150 billion web captures. The Archive also oversees one of the world's largest digitization projects. Founded by Brewster Kahle in May 1996, the Archive is a 501 nonprofit operating in the United States. Its headquarters are in 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, Richmond. The Archive was officially designated as a library by the State of California in 2007. Brewster Kahle founded the Archive at around the same time that he began the for-profit web crawling company Alexa Internet. The archived content wasn't available to the general public until 2001, when it developed the Wayback Machine. In late 1999, the Archive expanded its collections beginning with the Prelinger Archives. Now the Internet Archive includes texts, audio, software. According to its site: Most societies place importance on preserving artifacts of their culture and heritage.
Internet Archive
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Since 2009, headquarters have been at 300 Funston Avenue in San Francisco, a former Christian Science Church
Internet Archive
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Internet Archive
Internet Archive
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Mirror of the Internet Archive in the Bibliotheca Alexandrina
Internet Archive
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From 1996 to 2009, headquarters were in the Presidio of San Francisco, a former U.S. military base
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Virtual International Authority File
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The Virtual International Authority File is an international authority file. It operated by the Online Computer Library Center. The project was initiated by the US Library of Congress. The aim is to link the national authority files to a virtual authority file. In this file, identical records from the different sets are linked together. The data are available for research and data exchange and sharing. Reciprocal updating uses the Open Archives Initiative protocol. The file numbers are incorporated into Wikidata. VIAF's clustering algorithm is run every month. Integrated Authority File International Standard Name Identifier Wikipedia's authority control template for articles Official website
Virtual International Authority File
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Screenshot 2012
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Integrated Authority File
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The Integrated Authority File or GND is an international authority file for the organisation of personal names, subject headings and corporate bodies from catalogues. It is used mainly increasingly also by archives and museums. The GND is managed with various regional library networks in German-speaking Europe and other partners. The GND falls under the Creative Commons Zero license. The GND specification provides a hierarchy of high-level sub-classes, useful in library classification, an approach to unambiguous identification of single elements. It also comprises an ontology intended for knowledge representation in the semantic web, available in the RDF format.
Integrated Authority File
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GND screenshot
198.
National Library of Australia
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In 2012 -- 2013, the National Library collection comprised 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 building was opened in 1968. The building was designed by the architectural firm of Bunning and Madden. The foyer is decorated in marble, with stained-glass windows by Mathieu Matégot. In 2012–2013 the Library collection comprised 6,496,772 items, with an estimated additional 2,325,900 items held in the manuscripts collection. The Library's collections of Australiana have developed into the nation's single most important resource of materials recording the cultural heritage. Australian writers, illustrators are actively sought and well represented -- whether published in Australia or overseas. Approximately 92.1 % of the Library's collection is discoverable through the online catalogue. The Library has digitized over 174,000 items from its collection and, where possible, delivers these directly across the Internet. The Library maintains an Internet-accessible archive of selected Australian websites called the Pandora Archive. The Library has particular collection strengths in the performing arts, including dance. The Library's considerable collections of general rare book materials, as well as world-class Asian and Pacific collections which augment the Australiana collections. The print collections are further supported by extensive microform holdings.
National Library of Australia
National Library of Australia
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National Library of Australia as viewed from Lake Burley Griffin, Canberra
National Library of Australia
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The original National Library building on Kings Avenue, Canberra, was designed by Edward Henderson. Originally intended to be several wings, only one wing was completed and was demolished in 1968. Now the site of the Edmund Barton Building.
National Library of Australia
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The library seen from Lake Burley Griffin in autumn.
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National Diet Library
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The National Diet Library is the only national library in Japan. It was established for the purpose of assisting members of the National Diet of Japan in researching matters of public policy. The library is similar in scope to the United States Library of Congress. The National Diet Library consists of several other branch libraries throughout Japan. Its need for information was "correspondingly small." The original Diet libraries "never developed either the services which might have made them vital adjuncts of genuinely responsible legislative activity." Until Japan's defeat, moreover, the executive had controlled all political documents, depriving the Diet of access to vital information. In 1946, each house of the Diet formed its own National Diet Library Standing Committee. 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 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 became the only national library in Japan. At this time the collection gained an additional million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, adjacent to the National Diet.
National Diet Library
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Tokyo Main Library of the National Diet Library
National Diet Library
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Kansai-kan of the National Diet Library
National Diet Library
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The National Diet Library
National Diet Library
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Main building in Tokyo
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National Library of the Czech Republic
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The National Library of the Czech Republic is the central library of the Czech Republic. It is directed by the Ministry of Culture. The library's main building is located in the historical Clementinum building in Prague, where approximately half of its books are kept. The other half of the collection is stored in the district of Hostivař. The National Library is the biggest library in its funds there are around 6 million documents. The library has around 60,000 registered readers. As well as Czech texts, the library also stores older material from Turkey, Iran and India. The library also houses books for Charles University in Prague. The project, which commenced in 1992, involved the digitisation of 1,700 documents in its first 13 years. The most medieval manuscripts preserved in the National Library are the Codex Vyssegradensis and the Passional of Abbes Kunigunde. Later in 2007 the project was delayed following objections regarding its proposed location from government officials including President Václav Klaus. The library was affected with some documents moved to upper levels to avoid the excess water. Over 4,000 books were removed following flooding in parts of the main building. Nobody was injured in the event. List of national and state libraries Official website
National Library of the Czech Republic
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Baroque library hall in the National Library of the Czech Republic
National Library of the Czech Republic
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General reading room (former refectory of the Jesuit residence in Clementinum)
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Joseph-Louis Lagrange
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Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia or Giuseppe Ludovico De la Grange Tournier, was an Italian Enlightenment Era mathematician and astronomer. Lagrange made significant contributions to the fields of both celestial mechanics. In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy. 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, arriving at the method of Lagrange multipliers. 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, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series. Born as Giuseppe Lodovico Lagrangia, Lagrange was of Italian and French descent. 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, certainly Lagrange seems to have accepted this willingly. He studied at the University of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.
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