1.
Greater French Empire
–
The First French Empire, Note 1 was the empire of Napoleon Bonaparte of France and the dominant power in much of continental Europe at the beginning of the 19th century. Its name was a misnomer, as France already had colonies overseas and was short lived compared to the Colonial Empire, a series of wars, known collectively as the Napoleonic Wars, extended French influence over much of Western Europe and into Poland. The plot included Bonapartes brother Lucien, then serving as speaker of the Council of Five Hundred, Roger Ducos, another Director, on 9 November 1799 and the following day, troops led by Bonaparte seized control. They dispersed the legislative councils, leaving a rump legislature to name Bonaparte, Sieyès, although Sieyès expected to dominate the new regime, the Consulate, he was outmaneuvered by Bonaparte, who drafted the Constitution of the Year VIII and secured his own election as First Consul. He thus became the most powerful person in France, a power that was increased by the Constitution of the Year X, the Battle of Marengo inaugurated the political idea that was to continue its development until Napoleons Moscow campaign. Napoleon planned only to keep the Duchy of Milan for France, setting aside Austria, the Peace of Amiens, which cost him control of Egypt, was a temporary truce. He gradually extended his authority in Italy by annexing the Piedmont and by acquiring Genoa, Parma, Tuscany and Naples, then he laid siege to the Roman state and initiated the Concordat of 1801 to control the material claims of the pope. Napoleon would have ruling elites from a fusion of the new bourgeoisie, on 12 May 1802, the French Tribunat voted unanimously, with exception of Carnot, in favour of the Life Consulship for the leader of France. This action was confirmed by the Corps Législatif, a general plebiscite followed thereafter resulting in 3,653,600 votes aye and 8,272 votes nay. On 2 August 1802, Napoleon Bonaparte was proclaimed Consul for life, pro-revolutionary sentiment swept through Germany aided by the Recess of 1803, which brought Bavaria, Württemberg and Baden to Frances side. The memories of imperial Rome were for a time, after Julius Caesar and Charlemagne. The Treaty of Pressburg, signed on 26 December 1805, did little other than create a more unified Germany to threaten France. On the other hand, Napoleons creation of the Kingdom of Italy, the occupation of Ancona, to create satellite states, Napoleon installed his relatives as rulers of many European states. The Bonapartes began to marry into old European monarchies, gaining sovereignty over many nations, in addition to the vassal titles, Napoleons closest relatives were also granted the title of French Prince and formed the Imperial House of France. Met with opposition, Napoleon would not tolerate any neutral power, Prussia had been offered the territory of Hanover to stay out of the Third Coalition. With the diplomatic situation changing, Napoleon offered Great Britain the province as part of a peace proposal and this, combined with growing tensions in Germany over French hegemony, Prussia responded by forming an alliance with Russia and sending troops into Bavaria on 1 October 1806. In this War of the Fourth Coalition, Napoleon destroyed the armies of Frederick William at Jena-Auerstedt, the Eylau and the Friedland against the Russians finally ruined Frederick the Greats formerly mighty kingdom, obliging Russia and Prussia to make peace with France at Tilsit. The Treaties of Tilsit ended the war between Russia and the French Empire and began an alliance between the two empires that held power of much of the rest of Europe, the two empires secretly agreed to aid each other in disputes
Greater French Empire
–
The Battle of Austerlitz
Greater French Empire
–
Flag
Greater French Empire
–
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
–
Napoleon reviews the
Imperial Guard before the
Battle of Jena, 1806
2.
Piedmont
–
Piedmont is one of the 20 regions of Italy. It has an area of 25,402 square kilometres and a population of about 4.6 million, the capital of Piedmont is Turin. The name Piedmont comes from medieval Latin Pedemontium or Pedemontis, i. e. ad pedem montium, meaning “at the foot of the mountains”. Other towns of Piedmont with more than 20,000 inhabitants sorted by population and it borders with France, Switzerland and the Italian regions of Lombardy, Liguria, Aosta Valley and for a very small fragment with Emilia Romagna. The geography of Piedmont is 43. 3% mountainous, along with areas of hills. Piedmont is the second largest of Italys 20 regions, after Sicily and it is broadly coincident with the upper part of the drainage basin of the river Po, which rises from the slopes of Monviso in the west of the region and is Italy’s largest river. The Po collects all the waters provided within the semicircle of mountains which surround the region on three sides, from the highest peaks the land slopes down to hilly areas, and then to the upper, and then to the lower great Padan Plain. 7. 6% of the territory is considered protected area. There are 56 different national or regional parks, one of the most famous is the Gran Paradiso National Park located between Piedmont and the Aosta Valley, Piedmont was inhabited in early historic times by Celtic-Ligurian tribes such as the Taurini and the Salassi. They were later subdued by the Romans, who founded several colonies there including Augusta Taurinorum, after the fall of the Western Roman Empire, the region was repeatedly invaded by the Burgundians, the Goths, Byzantines, Lombards, Franks. In the 9th–10th centuries there were incursions by the Magyars. At the time Piedmont, as part of the Kingdom of Italy within the Holy Roman Empire, was subdivided into several marks, in 1046, Oddo of Savoy added Piedmont to their main territory of Savoy, with a capital at Chambéry. Other areas remained independent, such as the powerful comuni of Asti and Alessandria, the County of Savoy was elevated to a duchy in 1416, and Duke Emanuele Filiberto moved the seat to Turin in 1563. In 1720, the Duke of Savoy became King of Sardinia, founding what evolved into the Kingdom of Sardinia, the Republic of Alba was created in 1796 as a French client republic in Piedmont. A new client republic, the Piedmontese Republic, existed between 1798 and 1799 before it was reoccupied by Austrian and Russian troops, in June 1800 a third client republic, the Subalpine Republic, was established in Piedmont. It fell under full French control in 1801 and it was annexed by France in September 1802, in the congress of Vienna, the Kingdom of Sardinia was restored, and furthermore received the Republic of Genoa to strengthen it as a barrier against France. Piedmont was a springboard for Italys unification in 1859–1861, following earlier unsuccessful wars against the Austrian Empire in 1820–1821 and this process is sometimes referred to as Piedmontisation. However, the efforts were countered by the efforts of rural farmers
Piedmont
–
A
Montferrat landscape, with the distant
Alps in the background.
Piedmont
–
Piedmont Piemonte
Piedmont
–
The
Palazzina di caccia of Stupinigi, in
Nichelino, is a
UNESCO World Heritage Site.
Piedmont
–
The
Kingdom of Sardinia in 1856.
3.
First French Empire
–
The First French Empire, Note 1 was the empire of Napoleon Bonaparte of France and the dominant power in much of continental Europe at the beginning of the 19th century. Its name was a misnomer, as France already had colonies overseas and was short lived compared to the Colonial Empire, a series of wars, known collectively as the Napoleonic Wars, extended French influence over much of Western Europe and into Poland. The plot included Bonapartes brother Lucien, then serving as speaker of the Council of Five Hundred, Roger Ducos, another Director, on 9 November 1799 and the following day, troops led by Bonaparte seized control. They dispersed the legislative councils, leaving a rump legislature to name Bonaparte, Sieyès, although Sieyès expected to dominate the new regime, the Consulate, he was outmaneuvered by Bonaparte, who drafted the Constitution of the Year VIII and secured his own election as First Consul. He thus became the most powerful person in France, a power that was increased by the Constitution of the Year X, the Battle of Marengo inaugurated the political idea that was to continue its development until Napoleons Moscow campaign. Napoleon planned only to keep the Duchy of Milan for France, setting aside Austria, the Peace of Amiens, which cost him control of Egypt, was a temporary truce. He gradually extended his authority in Italy by annexing the Piedmont and by acquiring Genoa, Parma, Tuscany and Naples, then he laid siege to the Roman state and initiated the Concordat of 1801 to control the material claims of the pope. Napoleon would have ruling elites from a fusion of the new bourgeoisie, on 12 May 1802, the French Tribunat voted unanimously, with exception of Carnot, in favour of the Life Consulship for the leader of France. This action was confirmed by the Corps Législatif, a general plebiscite followed thereafter resulting in 3,653,600 votes aye and 8,272 votes nay. On 2 August 1802, Napoleon Bonaparte was proclaimed Consul for life, pro-revolutionary sentiment swept through Germany aided by the Recess of 1803, which brought Bavaria, Württemberg and Baden to Frances side. The memories of imperial Rome were for a time, after Julius Caesar and Charlemagne. The Treaty of Pressburg, signed on 26 December 1805, did little other than create a more unified Germany to threaten France. On the other hand, Napoleons creation of the Kingdom of Italy, the occupation of Ancona, to create satellite states, Napoleon installed his relatives as rulers of many European states. The Bonapartes began to marry into old European monarchies, gaining sovereignty over many nations, in addition to the vassal titles, Napoleons closest relatives were also granted the title of French Prince and formed the Imperial House of France. Met with opposition, Napoleon would not tolerate any neutral power, Prussia had been offered the territory of Hanover to stay out of the Third Coalition. With the diplomatic situation changing, Napoleon offered Great Britain the province as part of a peace proposal and this, combined with growing tensions in Germany over French hegemony, Prussia responded by forming an alliance with Russia and sending troops into Bavaria on 1 October 1806. In this War of the Fourth Coalition, Napoleon destroyed the armies of Frederick William at Jena-Auerstedt, the Eylau and the Friedland against the Russians finally ruined Frederick the Greats formerly mighty kingdom, obliging Russia and Prussia to make peace with France at Tilsit. The Treaties of Tilsit ended the war between Russia and the French Empire and began an alliance between the two empires that held power of much of the rest of Europe, the two empires secretly agreed to aid each other in disputes
First French Empire
–
The Battle of Austerlitz
First French Empire
–
Flag
First French Empire
–
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
–
Napoleon reviews the
Imperial Guard before the
Battle of Jena, 1806
4.
Alma mater
–
Alma mater is an allegorical Latin phrase for a university or college. In modern usage, it is a school or university which an individual has attended, the phrase is variously translated as nourishing mother, nursing mother, or fostering mother, suggesting that a school provides intellectual nourishment to its students. Before its modern usage, Alma mater was a title in Latin for various mother goddesses, especially Ceres or Cybele. The source of its current use is the motto, Alma Mater Studiorum, of the oldest university in continuous operation in the Western world and it is related to the term alumnus, denoting a university graduate, which literally means a nursling or one who is nourished. The phrase can also denote a song or hymn associated with a school, although alma was a common epithet for Ceres, Cybele, Venus, and other mother goddesses, it was not frequently used in conjunction with mater in classical Latin. Alma Redemptoris Mater is a well-known 11th century antiphon devoted to Mary, the earliest documented English use of the term to refer to a university is in 1600, when University of Cambridge printer John Legate began using an emblem for the universitys press. In English etymological reference works, the first university-related usage is often cited in 1710, many historic European universities have adopted Alma Mater as part of the Latin translation of their official name. The University of Bologna Latin name, Alma Mater Studiorum, refers to its status as the oldest continuously operating university in the world. At least one, the Alma Mater Europaea in Salzburg, Austria, the College of William & Mary in Williamsburg, Virginia, has been called the Alma Mater of the Nation because of its ties to the founding of the United States. At Queens University in Kingston, Ontario, and the University of British Columbia in Vancouver, British Columbia, the ancient Roman world had many statues of the Alma Mater, some still extant. Modern sculptures are found in prominent locations on several American university campuses, outside the United States, there is an Alma Mater sculpture on the steps of the monumental entrance to the Universidad de La Habana, in Havana, Cuba. Media related to Alma mater at Wikimedia Commons The dictionary definition of alma mater at Wiktionary Alma Mater Europaea website
Alma mater
–
The Alma Mater statue by
Mario Korbel, at the entrance of the
University of Havana in
Cuba.
Alma mater
–
John Legate's Alma Mater for Cambridge in 1600
Alma mater
–
Alma Mater (1929, Lorado Taft),
University of Illinois at Urbana–Champaign
5.
University of Turin
–
The University of Turin is a university in the city of Turin in the Piedmont region of north-western Italy. It is one of the oldest universities in Europe, and continues to play an important role in research, the University of Turin was founded as a studium in 1404, under the initiative of Prince Ludovico di Savoia. From 1427 to 1436 the seat of the university was transferred to Chieri and it was closed in 1536, and 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 endowment size, notable scholars of this period include Cesare Lombroso, Carlo Forlanini and Arturo Graf. In the 20th century, the University of Turin was one of the centers of the Italian anti-fascism, the new impulse was performed in collaboration with other national and international research centers, as well as with local organizations and the Italian Minister of Public Instruction. By the end of the 1990s, the campuses of Alessandria, Novara. The new institution, which only held courses in civil. The Bishop, as Rector of Studies, proclaimed and conferred the title on the new doctors, after a series of interruptions in its activities, the university was moved to Chieri and later, in 1434, to Savigliano. In 1436, when the returned to Turin, Ludovico di Savoia. The ducal licenses of 6 October 1436 set up the three faculties of Theology, Arts and Medicine, and Law, and twenty-five lectureships or chairs. The growth and development of the role of Turin as the capital led to the consolidation of the University. The Study, closed at the beginning of 1536 with the French occupation, reopened in 1558 with lecturers at Mondovì, it was re-established in Turin in 1566. With Emmanuel Philibert and Charles Emmanuel I, the University enjoyed a season of great prosperity due to the presence of illustrious teachers, the opening of the new premises marked a major turning point in the history of the greatest Piedmontese educational institution. This had an effect on the cultural linguistic models of the Duchy. At the time, the Piedmontese Studium became a point of reference for university reforms at Parma and Modena, Charles Emmanuel III continued the policy of innovation and consolidation commenced by Victor Amadeus II and created a University Museum in 1739. However, in the last decades of the 18th century, the course of events at the University, closely connected to international developments, led to urban unrest. The revolt of university students in 1791 joined by artisans who stormed the Collegio delle Province in 1792 causing numerous victims, was an instance of this conflict
University of Turin
–
Hall of the Rectorate Palace of the University of Turin
University of Turin
–
Seal of the University of Turin
University of Turin
–
The
Minerva Statue in front of the Rectorate Palace at the University of Turin.
University of Turin
–
The revolt of the students of Turin University, 1821
6.
Joseph Fourier
–
The Fourier transform and Fouriers law are also named in his honour. Fourier is also credited with the discovery of the greenhouse effect. Fourier was born at Auxerre, the son of a tailor and he was orphaned at age nine. Fourier was recommended to the Bishop of Auxerre, and through this introduction, the commissions in the scientific corps of the army were reserved for those of good birth, and being thus ineligible, he accepted a military lectureship on mathematics. He took a prominent part in his own district in promoting the French Revolution and he was imprisoned briefly during the Terror but in 1795 was appointed to the École Normale, and subsequently succeeded Joseph-Louis Lagrange at the École Polytechnique. Fourier accompanied Napoleon Bonaparte on his Egyptian expedition in 1798, as scientific adviser, cut off from France by the English fleet, he organized the workshops on which the French army had to rely for their munitions of war. He also contributed several papers to the Egyptian Institute which Napoleon founded at Cairo. After the British victories and the capitulation of the French under General Menou in 1801, in 1801, Napoleon appointed Fourier Prefect of the Department of Isère in Grenoble, where he oversaw road construction and other projects. However, Fourier had previously returned home from the Napoleon expedition to Egypt to resume his academic post as professor at École Polytechnique when Napoleon decided otherwise in his remark. The Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place, hence being faithful to Napoleon, he took the office of Prefect. It was while at Grenoble that he began to experiment on the propagation of heat and he presented his paper On the Propagation of Heat in Solid Bodies to the Paris Institute on December 21,1807. He also contributed to the monumental Description de lÉgypte, Fourier moved to England in 1816. Later, he returned to France, and in 1822 succeeded Jean Baptiste Joseph Delambre as Permanent Secretary of the French Academy of Sciences, in 1830, he was elected a foreign member of the Royal Swedish Academy of Sciences. In 1830, his health began to take its toll, Fourier had already experienced, in Egypt and Grenoble. At Paris, it was impossible to be mistaken with respect to the cause of the frequent suffocations which he experienced. A fall, however, which he sustained on the 4th of May 1830, while descending a flight of stairs, shortly after this event, he died in his bed on 16 May 1830. His name is one of the 72 names inscribed on the Eiffel Tower, a bronze statue was erected in Auxerre in 1849, but it was melted down for armaments during World War II. Joseph Fourier University in Grenoble is named after him and this book was translated, with editorial corrections, into English 56 years later by Freeman
Joseph Fourier
–
Jean-Baptiste Joseph Fourier
Joseph Fourier
–
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
–
Sketch of Fourier, circa 1820.
Joseph Fourier
–
Bust of Fourier in Grenoble
7.
Giovanni Antonio Amedeo Plana
–
Giovanni Antonio Amedeo Plana was an Italian astronomer and mathematician. Plana was born in Voghera, Italy to Antonio Maria Plana and Giovanna Giacoboni, at the age of 15 he was sent to live with his uncles in Grenoble to complete his education. In 1800 he entered the École Polytechnique, and was one of the students of Joseph-Louis Lagrange, in 1811 he was appointed to the chair of astronomy at the University of Turin thanks to the influence of Lagrange. He spent the remainder of his teaching at that institution. Planas contributions included work on the motions of the Moon, as well as integrals, elliptic functions, heat, electrostatics, and geodesy. In 1820 he was one of the winners of an awarded by the Académie des Sciences in Paris based on the construction of lunar tables using the law of gravity. In 1832 he published the Théorie du mouvement de la lune, in 1834 he was awarded with the Copley Medal by the Royal Society for his studies on lunar motion. He became astronomer royal, and then in 1844 a Baron, at the age of 80 he was granted membership in the prestigious Académie des Sciences. He is considered one of the premiere Italian scientists of his age, the crater Plana on the Moon is named in his honor. Biography and a source for this page, oConnor, John J. Robertson, Edmund F. Giovanni Antonio Amedeo Plana, MacTutor History of Mathematics archive, University of St Andrews
Giovanni Antonio Amedeo Plana
–
Giovanni Antonio Amedeo Plana.
8.
Mathematical analysis
–
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are studied in the context of real and complex numbers. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis, analysis may be distinguished from geometry, however, it can be applied to any space of mathematical objects that has a definition of nearness or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, in Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieris principle to find the volume of a sphere in the 5th century, the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of analysis were established in 17th century Europe. During this period, calculus techniques were applied to approximate discrete problems by continuous ones, in the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the definition of continuity in 1816. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations, the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Riemann introduced his theory of integration, the last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the size of the set of discontinuities of real functions, also, monsters began to be investigated
Mathematical analysis
–
A
strange attractor arising from a
differential equation. Differential equations are an important area of mathematical analysis with many applications to
science and
engineering.
9.
Enlightenment Era
–
The Enlightenment was an intellectual movement which dominated the world of ideas in Europe during the 18th century, The Century of Philosophy. In France, the doctrines of les Lumières were individual liberty and religious tolerance in opposition to an absolute monarchy. French historians traditionally place the Enlightenment between 1715, the year that Louis XIV died, and 1789, the beginning of the French Revolution, some recent historians begin the period in the 1620s, with the start of the scientific revolution. Les philosophes of the widely circulated their ideas through meetings at scientific academies, Masonic lodges, literary salons, coffee houses. The ideas of the Enlightenment undermined the authority of the monarchy and the Church, a variety of 19th-century movements, including liberalism and neo-classicism, trace their intellectual heritage back to the Enlightenment. The Age of Enlightenment was preceded by and closely associated with the scientific revolution, earlier philosophers whose work influenced the Enlightenment included Francis Bacon, René Descartes, John Locke, and Baruch Spinoza. The major figures of the Enlightenment included Cesare Beccaria, Voltaire, Denis Diderot, Jean-Jacques Rousseau, David Hume, Adam Smith, Benjamin Franklin visited Europe repeatedly and contributed actively to the scientific and political debates there and brought the newest ideas back to Philadelphia. Thomas Jefferson closely followed European ideas and later incorporated some of the ideals of the Enlightenment into the Declaration of Independence, others like James Madison incorporated them into the Constitution in 1787. The most influential publication of the Enlightenment was the Encyclopédie, the ideas of the Enlightenment played a major role in inspiring the French Revolution, which began in 1789. After the Revolution, the Enlightenment was followed by an intellectual movement known as Romanticism. René Descartes rationalist philosophy laid the foundation for enlightenment thinking and his attempt to construct the sciences on a secure metaphysical foundation was not as successful as his method of doubt applied in philosophic areas leading to a dualistic doctrine of mind and matter. His skepticism was refined by John Lockes 1690 Essay Concerning Human Understanding and his dualism was challenged by Spinozas uncompromising assertion of the unity of matter in his Tractatus and Ethics. Both lines of thought were opposed by a conservative Counter-Enlightenment. In the mid-18th century, Paris became the center of an explosion of philosophic and scientific activity challenging traditional doctrines, the political philosopher Montesquieu introduced the idea of a separation of powers in a government, a concept which was enthusiastically adopted by the authors of the United States Constitution. Francis Hutcheson, a philosopher, described the utilitarian and consequentialist principle that virtue is that which provides, in his words. Much of what is incorporated in the method and some modern attitudes towards the relationship between science and religion were developed by his protégés David Hume and Adam Smith. Hume became a figure in the skeptical philosophical and empiricist traditions of philosophy. Immanuel Kant tried to reconcile rationalism and religious belief, individual freedom and political authority, as well as map out a view of the sphere through private
Enlightenment Era
–
German philosopher
Immanuel Kant
Enlightenment Era
–
History of
Western philosophy
Enlightenment Era
–
Cesare Beccaria, father of classical criminal theory (1738–1794)
Enlightenment Era
–
Like other Enlightenment philosophers,
Rousseau was critical of the
Atlantic slave trade.
10.
French Academy of Sciences
–
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the forefront of developments in Europe in the 17th and 18th centuries. Currently headed by Sébastien Candel, it is one of the five Academies of the Institut de France, the Academy of Sciences makes its origin to Colberts plan to create a general academy. He chose a group of scholars who met on 22 December 1666 in the Kings library. The first 30 years of the Academys existence were relatively informal, in contrast to its British counterpart, the Academy was founded as an organ of government. The Academy was expected to remain apolitical, and to avoid discussion of religious, on 20 January 1699, Louis XIV gave the Company its first rules. The Academy received the name of Royal Academy of Sciences and was installed in the Louvre in Paris, following this reform, the Academy began publishing a volume each year with information on all the work done by its members and obituaries for members who had died. This reform also codified the method by which members of the Academy could receive pensions for their work, on 8 August 1793, the National Convention abolished all the academies. Almost all the old members of the previously abolished Académie were formally re-elected, among the exceptions was Dominique, comte de Cassini, who refused to take his seat. In 1816, the again renamed Royal Academy of Sciences became autonomous, while forming part of the Institute of France, in the Second Republic, the name returned to Académie des sciences. During this period, the Academy was funded by and accountable to the Ministry of Public Instruction, the Academy came to control French patent laws in the course of the eighteenth century, acting as the liaison of artisans knowledge to the public domain. As a result, academicians dominated technological activities in France, the Academy proceedings were published under the name Comptes rendus de lAcadémie des sciences. The Comptes rendus is now a series with seven titles. The publications can be found on site of the French National Library, in 1818 the French Academy of Sciences launched a competition to explain the properties of light. The civil engineer Augustin-Jean Fresnel entered this competition by submitting a new theory of light. Siméon Denis Poisson, one of the members of the judging committee, being a supporter of the particle-theory of light, he looked for a way to disprove it. The Poisson spot is not easily observed in every-day situations, so it was natural for Poisson to interpret it as an absurd result. However, the head of the committee, Dominique-François-Jean Arago, and he molded a 2-mm metallic disk to a glass plate with wax
French Academy of Sciences
–
A heroic depiction of the activities of the Academy from 1698
French Academy of Sciences
–
Colbert Presenting the Members of the Royal Academy of Sciences to Louis XIV in 1667
French Academy of Sciences
–
The
Institut de France in Paris where the Academy is housed
11.
Bureau des Longitudes
–
During the 19th century, it was responsible for synchronizing clocks across the world. It was headed during this time by François Arago and Henri Poincaré, the Bureau now functions as an academy and still meets monthly to discuss topics related to astronomy. The Bureau was founded by the National Convention after it heard a report drawn up jointly by the Committee of Navy, the Committee of Finances, as a result, the Bureau was established with authority over the Paris Observatory and all other astronomical establishments throughout France. The Bureau was charged with taking control of the seas away from the English and improving accuracy when tracking the longitudes of ships through astronomical observations, by a decree of 30 January 1854, the Bureaus mission was extended to embrace geodesy, time standardisation and astronomical measurements. This decree granted independence to the Paris Observatory, separating it from the Bureau, the Bureau was successful at setting a universal time in Paris via air pulses sent through pneumatic tubes. It later worked to synchronize time across the French colonial empire by determining the length of time for a signal to make a trip to. The French Bureau of Longitude established a commission in the year 1897 to extend the system to the measurement of time. They planned to abolish the antiquated division of the day hours, minutes, and seconds, and replace it by a division into tenths, thousandths. This was a revival of a dream that was in the minds of the creators of the system at the time of the French Revolution a hundred years earlier. Some members of the Bureau of Longitude commission introduced a proposal, retaining the old-fashioned hour as the basic unit of time. Poincaré served as secretary of the commission and took its work very seriously and he was a fervent believer in a universal metric system. The rest of the world outside France gave no support to the proposals. After three years of work, the commission was dissolved in 1900. Since 1970, the board has been constituted with 13 members,3 nominated by the Académie des Sciences, since 1998, practical work has been carried out by the Institut de mécanique céleste et de calcul des éphémérides. Institut de mécanique céleste et de calcul des éphémérides Bureau Des Longitudes Galison, einsteins Clocks, Poincarés Maps, Empires of Time
Bureau des Longitudes
–
ABBE GREGOIRE (1750-1831).
12.
Lagrange multipliers
–
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 and we need both f and g to have continuous first partial derivatives. We introduce a new variable called a Lagrange multiplier and study the Lagrange function defined by L = f − λ ⋅, if f is a maximum of f for the original constrained problem, then there exists λ0 such that is 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, one of the most common problems in calculus is that of finding maxima or minima of a function, but it is often difficult to find a closed form for the function being extremized. Such difficulties often arise when one wishes to maximize or minimize a function subject to fixed outside equality constraints, the method of Lagrange multipliers is a powerful tool for solving this class of problems without the need to explicitly solve the conditions and use them to eliminate extra variables. Consider the two-dimensional problem introduced above maximize f subject to g =0, the method of Lagrange multipliers relies on the intuition that at a maximum, f cannot be increasing in the direction of any neighboring point where g =0. If it were, we could walk along g =0 to get higher and we can visualize contours of f given by f = d for various values of d, and the contour of g given by g =0. Suppose we walk along the line with g =0. We are interested in finding points where f does not change as we walk, there are two ways this could happen, First, we could be following a contour line of f, since by definition f does not change as we walk along its contour lines. This would mean that the lines of f and g are parallel here. The second possibility is that we have reached a part of f. Thus we want points where g =0 and ∇ x, y f = λ ∇ x, y g, the constant λ is required because although the two gradient vectors are parallel, the magnitudes of the gradient vectors are generally not equal. This constant is called the Lagrange multiplier, notice that this method also solves the second possibility, if f is level, then its gradient is zero, and setting λ =0 is a solution regardless of g. To incorporate these conditions into one equation, we introduce an auxiliary function L = f − λ ⋅ g, note that this amounts to solving three equations in three unknowns. This is the method of Lagrange multipliers, note that ∇ λ L =0 implies g =0. The constrained extrema of f are points of the Lagrangian L. One may reformulate the Lagrangian as a Hamiltonian, in case the solutions are local minima for the Hamiltonian
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.
13.
Taylor series
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In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the functions derivatives at a single point. The concept of a Taylor series was formulated by the Scottish mathematician James Gregory, a function can be approximated by using a finite number of terms of its Taylor series. Taylors 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 functions Taylor polynomials as the degree increases, a function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an interval is known as an analytic function in that interval. The Taylor series of a real or complex-valued function f that is differentiable at a real or complex number a is the power series f + f ′1. Which can be written in the more compact sigma notation as ∑ n =0 ∞ f n, N where n. denotes the factorial of n and 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. The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for 1/1 − x is the geometric series 1 + x + x 2 + x 3 + ⋯ so the Taylor series for 1/x at a =1 is 1 − +2 −3 + ⋯. The Taylor series for the exponential function ex at a =0 is x 00, + ⋯ =1 + x + x 22 + x 36 + x 424 + x 5120 + ⋯ = ∑ n =0 ∞ x n n. The above expansion holds because the derivative of ex with respect to x is also ex and this leaves the terms n in the numerator and n. in the denominator for each term in the infinite sum. The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a result, but rejected it as an impossibility. It was through Archimedess method of exhaustion that a number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later, in the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. The Kerala school of astronomy and mathematics further expanded his works with various series expansions, in the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a method for constructing these series for all functions for which they exist was finally provided by Brook Taylor. The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, if f is given by a convergent power series in an open disc centered at b in the complex plane, it is said to be analytic in this disc
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.
14.
Three-body problem
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The three-body problem is a special case of the n-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, in an extended modern sense, a three-body problem is a class of problems in classical or quantum mechanics that model the motion of three particles. The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia. The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei, however the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth. They submitted their competing first analyses to the Académie Royale des Sciences in 1747 and it was in connection with these researches, in Paris, in the 1740s, that the name three-body problem began to be commonly used. An account published in 1761 by Jean le Rond dAlembert indicates that the name was first used in 1747, in 1887, mathematicians Heinrich Bruns and Henri Poincaré showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases, a prominent example of the classical three-body problem is the movement of a planet with a satellite around a star. In this case, the problem is simplified to two instances of the two-body problem, however, the effect of the star on the movement of the satellite around the planet can be considered as a perturbation. While a spaceflight involving a gravity assist tends to be at least a problem, once far away from the Earth when Earths gravity becomes negligible. The general statement for the three body problem is as follows, in the circular restricted three-body problem, two massive bodies move in circular orbits around their common center of mass, and the third mass is negligible with respect to the other two. It can be useful to consider the effective potential, in 1767 Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. In 1772 Lagrange found a family of solutions in which the three form an equilateral triangle at each instant. Together, these form the central configurations for the three-body problem. These solutions are valid for any mass ratios, and the move on Keplerian ellipses. These five families are the only solutions for which there are explicit analytic formulae. In 1893 Meissel stated what is called the Pythagorean three-body problem. Burrau further investigated this problem in 1913, in 1967 Victor Szebehely and coworkers established eventual escape for this problem using numerical integration, while at the same time finding a nearby periodic solution. In 1911, United States scientist William Duncan MacMillan found one special solution, in 1961, Russian mathematician Sitnikov improved this solution
Three-body problem
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Figure 1: Configuration of the Sitnikov Problem
Three-body problem
15.
Lagrangian point
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The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centrifugal force required to orbit with them. There are five points, labeled L1 to L5, all in the orbital plane of the two large bodies. The first three are on the line connecting the two bodies, the last two, L4 and L5, each form an equilateral triangle with the two large bodies. The two latter points are stable, which implies that objects can orbit around them in a coordinate system tied to the two large bodies. Several planets have satellites near their L4 and L5 points with respect to the Sun, the three collinear Lagrange points were discovered by Leonhard Euler a few years before Lagrange discovered the remaining two. In 1772, Joseph-Louis Lagrange published an Essay on the three-body problem, in the first chapter he considered the general three-body problem. From that, in the chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits. The five Lagrangian points are labeled and defined as follows, The L1 point lies on the line defined by the two large masses M1 and M2, and between them. It is the most intuitively understood of the Lagrangian points, the one where the attraction of M2 partially cancels M1s gravitational attraction. Explanation An object that orbits the Sun more closely than Earth would normally have an orbital period than Earth. If the object is directly between Earth and the Sun, then Earths gravity counteracts some of the Suns pull on the object, the closer to Earth the object is, the greater this effect is. At the L1 point, the period of the object becomes exactly equal to Earths orbital period. L1 is about 1.5 million kilometers from Earth, the L2 point lies on the line through the two large masses, beyond the smaller of the two. Here, the forces of the two large masses balance the centrifugal effect on a body at L2. Explanation On the opposite side of Earth from the Sun, the period of an object would normally be greater than that of Earth. The extra pull of Earths gravity decreases the orbital period of the object, like L1, L2 is about 1.5 million kilometers from Earth. The L3 point lies on the line defined by the two masses, beyond the larger of the two. Explanation L3 in the Sun–Earth system exists on the side of the Sun
Lagrangian point
–
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)
16.
W.W. Rouse Ball
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Walter William Rouse Ball, known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge from 1878 to 1905. He was also an amateur magician, and the founding president of the Cambridge Pentacle Club in 1919. Ball was the son and heir of Walter Frederick Ball, of 3, St Johns Park Villas, South Hampstead, London. Educated at University College School, he entered Trinity College, Cambridge in 1870, became a scholar and first Smiths Prizeman and he became a Fellow of Trinity in 1875, and remained one for the rest of his life. He is buried at the Parish of the Ascension Burial Ground in Cambridge and he is commemorated in the naming of the small pavilion, now used as changing rooms and toilets, on Jesus Green in Cambridge. A History of the Study of Mathematics at Cambridge, Cambridge University Press,1889 A Short Account of the History of Mathematics at Project Gutenberg, dover 1960 republication of fourth edition. Mathematical Recreations and Essays at Project Gutenberg A History of the First Trinity Boat Club Cambridge Papers at Project Gutenberg, string Figures, Cambridge, W. Heffer & Sons Rouse Ball Professor of Mathematics Rouse Ball Professor of English Law Martin Gardner, another author of recreational mathematics. Singmaster, David,1892 Walter William Rouse Ball, Mathematical recreations and problems of past and present times, in Grattan-Guinness, W. W. Rouse Ball at the Mathematics Genealogy Project W. W. Rouse Ball at Find a Grave
W.W. Rouse Ball
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W.W. Rouse Ball
17.
Kingdom of France
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The Kingdom of France was a medieval and early modern monarchy in Western Europe. It was one of the most powerful states in Europe and a great power since the Late Middle Ages and it was also an early colonial power, with possessions around the world. France originated as West Francia, the half of the Carolingian Empire. A branch of the Carolingian dynasty continued to rule until 987, the territory remained known as Francia and its ruler as rex Francorum well into the High Middle Ages. The first king calling himself Roi de France was Philip II, France continued to be ruled by the Capetians and their cadet lines—the Valois and Bourbon—until the monarchy was overthrown in 1792 during the French Revolution. France in the Middle Ages was a de-centralised, feudal monarchy, in Brittany and Catalonia the authority of the French king was barely felt. Lorraine and Provence were states of the Holy Roman Empire and not yet a part of France, during the Late Middle Ages, the Kings of England laid claim to the French throne, resulting in a series of conflicts known as the Hundred Years War. Subsequently, France sought to extend its influence into Italy, but was defeated by Spain in the ensuing Italian Wars, religiously France became divided between the Catholic majority and a Protestant minority, the Huguenots, which led to a series of civil wars, the Wars of Religion. France laid claim to large stretches of North America, known collectively as New France, Wars with Great Britain led to the loss of much of this territory by 1763. French intervention in the American Revolutionary War helped secure the independence of the new United States of America, the Kingdom of France adopted a written constitution in 1791, but the Kingdom was abolished a year later and replaced with the First French Republic. The monarchy was restored by the great powers in 1814. During the later years of the elderly Charlemagnes rule, the Vikings made advances along the northern and western perimeters of the Kingdom of the Franks, after Charlemagnes death in 814 his heirs were incapable of maintaining political unity and the empire began to crumble. The Treaty of Verdun of 843 divided the Carolingian Empire into three parts, with Charles the Bald ruling over West Francia, the nucleus of what would develop into the kingdom of France. Viking advances were allowed to increase, and their dreaded longboats were sailing up the Loire and Seine rivers and other waterways, wreaking havoc. During the reign of Charles the Simple, Normans under Rollo from Norway, were settled in an area on either side of the River Seine, downstream from Paris, that was to become Normandy. With its offshoots, the houses of Valois and Bourbon, it was to rule France for more than 800 years. Henry II inherited the Duchy of Normandy and the County of Anjou, and married Frances newly divorced ex-queen, Eleanor of Aquitaine, after the French victory at the Battle of Bouvines in 1214, the English monarchs maintained power only in southwestern Duchy of Guyenne. The death of Charles IV of France in 1328 without male heirs ended the main Capetian line, under Salic law the crown could not pass through a woman, so the throne passed to Philip VI, son of Charles of Valois
Kingdom of France
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The Kingdom of France in 1789.
Ancien Régime provinces in
1789.
Kingdom of France
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Royal Standard^{a}
Kingdom of France
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Henry IV, by
Frans Pourbus the younger, 1610.
Kingdom of France
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Louis XIII, by
Philippe de Champaigne, 1647.
18.
Roman Catholic
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The Catholic Church, also known as the Roman Catholic Church or the Universal Church, is the largest Christian church, with more than 1.28 billion members worldwide. As one of the oldest religious institutions in the world, it has played a prominent role in the history, headed by the Bishop of Rome, known as the Pope, the churchs doctrines are summarised in the Nicene Creed and the Apostles Creed. Its central administration is located in Vatican City, enclaved within Rome, the Catholic Church is notable within Western Christianity for its sacred tradition and seven sacraments. It teaches that it is the one church founded by Jesus Christ, that its bishops are the successors of Christs apostles. The Catholic Church maintains that the doctrine on faith and morals that it declares as definitive is infallible. The Latin Church, the Eastern Catholic Churches, as well as such as mendicant orders and enclosed monastic orders. Among the sacraments, the one is the Eucharist, celebrated liturgically in the Mass. The church teaches that through consecration by a priest the sacrificial bread and wine become the body, the Catholic Church practises closed communion, with only baptised members in a state of grace ordinarily permitted to receive the Eucharist. The Virgin Mary is venerated in the Catholic Church as Queen of Heaven and is honoured in numerous Marian devotions. The Catholic Church has influenced Western philosophy, science, art and culture, Catholic spiritual teaching includes spreading the Gospel while Catholic social teaching emphasises support for the sick, the poor and the afflicted through the corporal and spiritual works of mercy. The Catholic Church is the largest non-government provider of education and medical services in the world, from the late 20th century, the Catholic Church has been criticised for its doctrines on sexuality, its refusal to ordain women and its handling of sexual abuse cases. Catholic was first used to describe the church in the early 2nd century, the first known use of the phrase the catholic church occurred in the letter from Saint Ignatius of Antioch to the Smyrnaeans, written about 110 AD. In the Catechetical Discourses of Saint Cyril of Jerusalem, the name Catholic Church was used to distinguish it from other groups that call themselves the church. The use of the adjective Roman to describe the Church as governed especially by the Bishop of Rome became more widespread after the Fall of the Western Roman Empire and into the Early Middle Ages. Catholic Church is the name used in the Catechism of the Catholic Church. The Catholic Church follows an episcopal polity, led by bishops who have received the sacrament of Holy Orders who are given formal jurisdictions of governance within the church. Ultimately leading the entire Catholic Church is the Bishop of Rome, commonly called the pope, in parallel to the diocesan structure are a variety of religious institutes that function autonomously, often subject only to the authority of the pope, though sometimes subject to the local bishop. Most religious institutes only have male or female members but some have both, additionally, lay members aid many liturgical functions during worship services
Roman Catholic
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Saint Peter's Basilica,
Vatican City
Roman Catholic
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St. Peter's Basilica,
Vatican City
Roman Catholic
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Pope Francis, elected in the
papal conclave, 2013
Roman Catholic
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Traditional graphic representation of the Trinity: The earliest attested version of the diagram, from a manuscript of
Peter of Poitiers ' writings, c. 1210
19.
Charles Emmanuel III of Sardinia
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Charles Emmanuel III was the Duke of Savoy and King of Sardinia from 1730 until his death. He was born a Prince of Savoy in Turin to Victor Amadeus II of Savoy and his maternal grandparents were Prince Philippe of France and his first wife Princess Henrietta Anne, the youngest daughter of King Charles I of England and Henrietta Maria of France. From his birth he was styled as the Duke of Aosta, at the time of his birth, Charles Emmanuel was not the heir to the Duchy of Savoy, his older brother Prince Victor Amadeus John Philip, Prince of Piedmont, was the heir apparent. Charles Emmanuel was the second of three males that would be born to his parents and his older brother died in 1715 and Charles Emmanuel then became heir apparent. As a result of his aid in the War of the Spanish Succession, Victor Amadeus was forced to exchange Sicily for the less important kingdom of Sardinia in 1720 after objections from an alliance of four nations, including several of his former allies. Yet he retained his new title of King, however, Victor Amadeus in his late years was dominated by shyness and sadness, probably under the effect of some mental illness. In the end, on 3 September 1730, he abdicated and he was not loved by Victor Amadeus, and consequently received an incomplete education. He however acquired noteworthy knowledge in the field along his father. In summer,1731, after having recovered from a fatal illness. The old king was confined to the Castle of Rivoli, where he died without any further harm to Charles. In the War of the Polish Succession Charles Emmanuel sided with the French- backed king Stanislaw I, after the treaty of alliance signed in Turin, on 28 October 1733 he marched on Milan and occupied Lombardy without significant losses. However, when France tried to convince Philip V of Spain to join the coalition, he asked to receive Milan and this was not acceptable for Charles Emmanuel, as it would recreate a Spanish domination in Italy as it had been in the previous centuries. While negotiations continued about the matter, the Savoy-French-Spanish troops attacked Mantua under the command of Charles Emmanuel himself. Sure that in the end Mantua would be assigned to Spain, the Franco-Piedmontese army was victorious in two battles at Crocetta and Guastalla. In the end, when Austria and France signed a peace, in exchange, he was given some territories, including Langhe, Tortona and Novara. Charles Emmanuel was involved in the War of the Austrian Succession, in which he sided with Maria Theresa of Austria, with financial and naval support from England. After noteworthy but inconclusive initial successes, he had to face the French-Spanish invasion of Savoy and, after a failed allied attempt to conquer the Kingdom of Naples, when the enemy army invaded Piedmont, in 1744 he personally defended Cuneo against the Spanish-French besiegers. The following year, with some 20,000 men, he was facing an invasion of two armies with a total of some 60,000 troops, the important strongholds of Alessandria, Asti and Casale fell
Charles Emmanuel III of Sardinia
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Charles Emmanuel III
Charles Emmanuel III of Sardinia
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The young Charles Emmanuel as duke of Aosta, by an unknown artist.
Charles Emmanuel III of Sardinia
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A portrait of a young Charles Emmanuel
Charles Emmanuel III of Sardinia
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The children of Charles and his second wife; (L-R)
Eleonora;
Victor Amadeus;
Maria Felicita and
Maria Luisa Gabriella.
20.
Charles Emmanuel III
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Charles Emmanuel III was the Duke of Savoy and King of Sardinia from 1730 until his death. He was born a Prince of Savoy in Turin to Victor Amadeus II of Savoy and his maternal grandparents were Prince Philippe of France and his first wife Princess Henrietta Anne, the youngest daughter of King Charles I of England and Henrietta Maria of France. From his birth he was styled as the Duke of Aosta, at the time of his birth, Charles Emmanuel was not the heir to the Duchy of Savoy, his older brother Prince Victor Amadeus John Philip, Prince of Piedmont, was the heir apparent. Charles Emmanuel was the second of three males that would be born to his parents and his older brother died in 1715 and Charles Emmanuel then became heir apparent. As a result of his aid in the War of the Spanish Succession, Victor Amadeus was forced to exchange Sicily for the less important kingdom of Sardinia in 1720 after objections from an alliance of four nations, including several of his former allies. Yet he retained his new title of King, however, Victor Amadeus in his late years was dominated by shyness and sadness, probably under the effect of some mental illness. In the end, on 3 September 1730, he abdicated and he was not loved by Victor Amadeus, and consequently received an incomplete education. He however acquired noteworthy knowledge in the field along his father. In summer,1731, after having recovered from a fatal illness. The old king was confined to the Castle of Rivoli, where he died without any further harm to Charles. In the War of the Polish Succession Charles Emmanuel sided with the French- backed king Stanislaw I, after the treaty of alliance signed in Turin, on 28 October 1733 he marched on Milan and occupied Lombardy without significant losses. However, when France tried to convince Philip V of Spain to join the coalition, he asked to receive Milan and this was not acceptable for Charles Emmanuel, as it would recreate a Spanish domination in Italy as it had been in the previous centuries. While negotiations continued about the matter, the Savoy-French-Spanish troops attacked Mantua under the command of Charles Emmanuel himself. Sure that in the end Mantua would be assigned to Spain, the Franco-Piedmontese army was victorious in two battles at Crocetta and Guastalla. In the end, when Austria and France signed a peace, in exchange, he was given some territories, including Langhe, Tortona and Novara. Charles Emmanuel was involved in the War of the Austrian Succession, in which he sided with Maria Theresa of Austria, with financial and naval support from England. After noteworthy but inconclusive initial successes, he had to face the French-Spanish invasion of Savoy and, after a failed allied attempt to conquer the Kingdom of Naples, when the enemy army invaded Piedmont, in 1744 he personally defended Cuneo against the Spanish-French besiegers. The following year, with some 20,000 men, he was facing an invasion of two armies with a total of some 60,000 troops, the important strongholds of Alessandria, Asti and Casale fell
Charles Emmanuel III
–
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
–
The children of Charles and his second wife; (L-R)
Eleonora;
Victor Amadeus;
Maria Felicita and
Maria Luisa Gabriella.
21.
Pierre Louis Maupertuis
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Pierre Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, Maupertuis made an expedition to Lapland to determine the shape of the Earth. He is often credited with having invented the principle of least action and his work in natural history is interesting in relation to modern science, since he touched on aspects of heredity and the struggle for life. Maupertuis was born at Saint-Malo, France, to a wealthy family of merchant-corsairs. His father, Renė, had involved in a number of enterprises that were central to the monarchy so that he thrived socially and politically. The son was educated in mathematics by a tutor, Nicolas Guisnée. In 1723 he was admitted to the Académie des Sciences and his early mathematical work revolved around the vis viva controversy, for which Maupertuis developed and extended the work of Isaac Newton and argued against the waning Cartesian mechanics. In the 1730s, the shape of the Earth became a flashpoint in the battle among rival systems of mechanics, Maupertuis, based on his exposition of Newton predicted that the Earth should be oblate, while his rival Jacques Cassini measured it astronomically to be prolate. In 1736 Maupertuis acted as chief of the French Geodesic Mission sent by King Louis XV to Lapland to measure the length of a degree of arc of the meridian and his results, which he published in a book detailing his procedures, essentially settled the controversy in his favor. The book included a narrative of the expedition, and an account of the Käymäjärvi Inscriptions. On his return home he became a member of almost all the societies of Europe. He also expanded into the realm, anonymously publishing a book that was part popular science, part philosophy. In 1740 Maupertuis went to Berlin at the invitation of Frederick II of Prussia, and took part in the Battle of Mollwitz, where he was taken prisoner by the Austrians. On his release he returned to Berlin, and thence to Paris, where he was elected director of the Academy of Sciences in 1742, and in the following year was admitted into the Académie française. His position became extremely awkward with the outbreak of the Seven Years War between his country and his patrons, and his reputation suffered in both Paris and Berlin. Finding his health declining, he retired in 1757 to the south of France, but went in 1758 to Basel, Maupertuis difficult disposition involved him in constant quarrels, of which his controversies with Samuel König and Voltaire during the latter part of his life are examples. The brilliance of much of what he did was undermined by his tendency to leave work unfinished and it was the insight of genius that led him to least-action principle, but a lack of intellectual energy or rigour that prevented his giving it the mathematical foundation that Lagrange would provide. He reveals remarkable powers of perception in heredity, in understanding the mechanism by which developed, even in immunology
Pierre Louis Maupertuis
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Maupertuis, wearing " lapmudes " from his Lapland expedition.
Pierre Louis Maupertuis
–
Lettres
22.
Echo (phenomenon)
–
In audio signal processing and acoustics, echo is a reflection of sound that arrives at the listener with a delay after the direct sound. The delay is proportional to the distance of the surface from the source. Typical examples are the echo produced by the bottom of a well, by a building, or by the walls of an enclosed room, a true echo is a single reflection of the sound source. The word echo derives from the Greek ἠχώ, itself from ἦχος, echo in the folk story of Greek is a mountain nymph whose ability to speak was cursed, only able to repeat the last words anyone spoke to her. Some animals use echo for location sensing and navigation, such as cetaceans, acoustic waves are reflected by walls or other hard surfaces, such as mountains and privacy fences. The reason of reflection may be explained as a discontinuity in the propagation medium and this can be heard when the reflection returns with sufficient magnitude and delay to be perceived distinctly. When sound, or the echo itself, is reflected multiple times from multiple surfaces, the human ear cannot distinguish echo from the original direct sound if the delay is less than 1/15 of a second. The velocity of sound in dry air is approximately 343 m/s at a temperature of 25 °C, therefore, the reflecting object must be more than 17. 2m from the sound source for echo to be perceived by a person located at the source. When a sound produces an echo in two seconds, the object is 343m away. In nature, canyon walls or rock cliffs facing water are the most common settings for hearing echoes. The strength of echo is frequently measured in dB sound pressure level relative to the transmitted wave. Echoes may be desirable or undesirable, in music performance and recording, electric echo effects have been used since the 1950s. The Echoplex is a delay effect, first made in 1959 that recreates the sound of an acoustic echo. Designed by Mike Battle, the Echoplex set a standard for the effect in the 1960s and was used by most of the guitar players of the era. While Echoplexes were used heavily by guitar players, many recording studios used the Echoplex. Beginning in the 1970s, Market built the solid-state Echoplex for Maestro, in the 2000s, most echo effects units use electronic or digital circuitry to recreate the echo effect. Hamilton Mausoleum, Hamilton, South Lanarkshire, Scotland, Its high stone holds the record for the longest echo in the world, gol Gumbaz of Bijapur, India, Any whisper, clap or sound gets echoed repeatedly. The gazebo of Napier Museum in Trivandrum, Kerala, India, listen to Duck echoes and an animated demonstration of how an echo is formed
Echo (phenomenon)
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This illustration depicts the principle of sediment echo sounding, which uses a narrow beam of high energy and low frequency
23.
Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
Series (mathematics)
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Illustration of 3
geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.
24.
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 minimized or maximized. The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, the physicist Richard Feynman demonstrated how this principle can also be used in quantum calculations. It was historically called least because its solution requires finding the path that has the least value and its classical mechanics and electromagnetic expressions are a consequence of quantum mechanics, but the stationary action method helped in the development of quantum mechanics. Maupertuis principle and Hamiltons principle exemplify the principle of stationary action, the action 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. In ancient Greece, Euclid wrote in his Catoptrica that, for the path of light reflecting from a mirror, hero of Alexandria later showed that this path was the shortest length and least time. Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744 and 1746, however, Leonhard Euler discussed the principle in 1744, and 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. In words this reads, The path taken by the system between times t1 and t2 is the one for which the action is stationary to first order. In applications the statement and definition of action are taken together, δ ∫ t 1 t 2 L d t =0 The action and Lagrangian both contain the dynamics of the system for all times. The term path simply refers to a curve traced out by the system in terms of the coordinates in the space, i. e. the curve q. In the 1600s, Pierre de Fermat postulated that light travels between two points along the path of shortest time, which is known as the principle of least time or Fermats principle. The movement of animals, the growth of plants. This notion of Maupertuis, although somewhat deterministic today, does much of the essence of mechanics. Leonhard Euler gave a formulation of the principle in 1744, in very recognizable terms. Curiously, Euler did not claim any priority, as the episode shows. Maupertuis priority was disputed in 1751 by the mathematician Samuel König, although similar to many of Leibnizs arguments, the principle itself has not been documented in Leibnizs works
Principle of least action
–
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).
25.
Integral calculus
–
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
Integral calculus
–
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
26.
Pierre de Fermat
–
He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermats principle for light propagation and his Fermats Last Theorem in number theory, Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a leather merchant. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth, there is little evidence concerning his school education, but it was probably at the Collège de Navarre in Montauban. He attended the University of Orléans from 1623 and received a bachelor in law in 1626. In Bordeaux he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apolloniuss De Locis Planis to one of the mathematicians there, there he became much influenced by the work of François Viète. In 1630, he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France and he held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat, fluent in six languages, Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often little or no proof of his theorems. In some of these letters to his friends he explored many of the ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession, nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus. Secrecy was common in European mathematical circles at the time and this naturally led to priority disputes with contemporaries such as Descartes and Wallis. Anders Hald writes that, The basis of Fermats mathematics was the classical Greek treatises combined with Vietas new algebraic methods, Fermats pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes famous La géométrie. This manuscript was published posthumously in 1679 in Varia opera mathematica, in these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of power functions. With his method, he was able to reduce this evaluation to the sum of geometric series, the resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pells equation, perfect numbers, amicable numbers and it was while researching perfect numbers that he discovered Fermats little theorem. Fermat developed the two-square theorem, and the polygonal number theorem, although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived
Pierre de Fermat
–
Pierre de Fermat
Pierre de Fermat
–
Bust in the Salle des Illustres in
Capitole de Toulouse
Pierre de Fermat
–
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
27.
N-body problem
–
In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, in the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is more difficult to solve. Having done so, he and others soon discovered over the course of a few years, Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits. Thus came the awareness and rise of the problem in the early 17th century. Ironically, this conformity led to the wrong approach, after Newtons time the n-body problem historically was not stated correctly because it did not include a reference to those gravitational interactive forces. Newton does not say it directly but implies in his Principia the n-body problem is unsolvable because of gravitational interactive forces. Newton said in his Principia, paragraph 21, And hence it is that the force is found in both bodies. The Sun attracts Jupiter and the planets, Jupiter attracts its satellites. Two bodies can be drawn to other by the contraction of rope between them. Newton concluded via his third law of motion according to this Law all bodies must attract each other. This last statement, which implies the existence of gravitational forces, is key. The problem of finding the solution of the n-body problem was considered very important. Indeed, in the late 19th century King Oscar II of Sweden, advised by Gösta Mittag-Leffler, in case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was awarded to Poincaré, even though he did not solve the original problem, the version finally printed contained many important ideas which led to the development of chaos theory. The problem as stated originally was finally solved by Karl Fritiof Sundman for n =3. The n-body problem considers n point masses mi, i =1,2, …, n in a reference frame in three dimensional space ℝ3 moving under the influence of mutual gravitational attraction. Each mass mi has a position vector qi, Newtons second law says that mass times acceleration mi d2qi/dt2 is equal to the sum of the forces on the mass
N-body problem
–
The Real Motion v.s. Kepler's Apparent Motion
N-body problem
–
Restricted 3-Body Problem
28.
Libration
–
Lunar libration is distinct from the slight changes in the Moons visual size as seen from Earth. Although this appearance can also be described as a motion, libration is caused by actual changes in the physical distance of the Moon. The Moon generally has one hemisphere facing the Earth, due to tidal locking, therefore, humans first view of the far side of the Moon resulted from lunar exploration on October 7,1959. However, this picture is only approximately true, over time. Libration is manifested as a slow rocking back and forth of the Moon as viewed from Earth, libration in latitude results from a slight inclination between the Moons axis of rotation and the normal to the plane of its orbit around Earth. Its origin is analogous to how the seasons arise from Earths revolution about the Sun. In 1772 Lagranges analyses determined that small bodies can stably share the orbit as a planet if they remain near Lagrange points. Such ‘trojan asteroids’ have been found co-orbiting with Earth, Jupiter, Mars, trojan asteroids associated with Earth are difficult to observe in the visible spectrum, as their libration paths are such that they would be visible primarily in the daylight sky. In 2010, however, using infrared observation techniques, the asteroid 2010 TK7 was found to be a companion of the Earth, it librates around the leading Lagrange point, L4
Libration
–
Theoretical extent of visible lunar surface due to libration in
Winkel tripel projection
Libration
29.
Moon
–
The Moon is an astronomical body that orbits planet Earth, being Earths only permanent natural satellite. It is the fifth-largest natural satellite in the Solar System, following Jupiters satellite Io, the Moon is second-densest satellite among those whose densities are known. The average distance of the Moon from the Earth is 384,400 km, the Moon is thought to have formed about 4.51 billion years ago, not long after Earth. It is the second-brightest regularly visible celestial object in Earths sky, after the Sun and its surface is actually dark, although compared to the night sky it appears very bright, with a reflectance just slightly higher than that of worn asphalt. Its prominence in the sky and its cycle of phases have made the Moon an important cultural influence since ancient times on language, calendars, art. The Moons gravitational influence produces the ocean tides, body tides, and this matching of apparent visual size will not continue in the far future. The Moons linear distance from Earth is currently increasing at a rate of 3.82 ±0.07 centimetres per year, since the Apollo 17 mission in 1972, the Moon has been visited only by uncrewed spacecraft. The usual English proper name for Earths natural satellite is the Moon, the noun moon is derived from moone, which developed from mone, which is derived from Old English mōna, which ultimately stems from Proto-Germanic *mǣnōn, like all Germanic language cognates. Occasionally, the name Luna is used, in literature, especially science fiction, Luna is used to distinguish it from other moons, while in poetry, the name has been used to denote personification of our moon. The principal modern English adjective pertaining to the Moon is lunar, a less common adjective is selenic, derived from the Ancient Greek Selene, from which is derived the prefix seleno-. Both the Greek Selene and the Roman goddess Diana were alternatively called Cynthia, the names Luna, Cynthia, and Selene are reflected in terminology for lunar orbits in words such as apolune, pericynthion, and selenocentric. The name Diana is connected to dies meaning day, several mechanisms have been proposed for the Moons formation 4.51 billion years ago, and some 60 million years after the origin of the Solar System. These hypotheses also cannot account for the angular momentum of the Earth–Moon system. This hypothesis, although not perfect, perhaps best explains the evidence, eighteen months prior to an October 1984 conference on lunar origins, Bill Hartmann, Roger Phillips, and Jeff Taylor challenged fellow lunar scientists, You have eighteen months. Go back to your Apollo data, go back to computer, do whatever you have to. Dont come to our conference unless you have something to say about the Moons birth, at the 1984 conference at Kona, Hawaii, the giant impact hypothesis emerged as the most popular. Afterward there were only two groups, the giant impact camp and the agnostics. Giant impacts are thought to have been common in the early Solar System, computer simulations of a giant impact have produced results that are consistent with the mass of the lunar core and the present angular momentum of the Earth–Moon system
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
30.
Virtual work
–
Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements, among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the according to the principle of least action. The work of a force on a particle along a displacement is known as the virtual work. The principle of work had always been used in some form since antiquity in the study of statics. It was used by the Greeks, medieval Arabs and Latins, working with Leibnizian concepts, Johann Bernoulli systematized the virtual work principle and made explicit the concept of infinitesimal displacement. He was able to solve problems for both bodies as well as fluids. Bernoullis version of virtual work law appeared in his letter to Pierre Varignon in 1715 and this formulation of the principle is today known as the principle of virtual velocities and is commonly considered as the prototype of the contemporary virtual work principles. In 1743 DAlembert published his Traite de Dynamique where he applied the principle of work, based on the Bernoullis work. His idea was to convert a dynamical problem into static problem by introducing inertial force, consider a point particle that moves along a path which is described by a function r from point A, where r, to point B, where r. It is possible that the moves from A to B along a nearby path described by r + δr. The variation δr satisfies the requirement δr = δr =0, the components of the variation, δr1, δr2 and δr3, are called virtual displacements. This can be generalized to a mechanical system defined by the generalized coordinates qi. In which case, the variation of the qi is defined by the virtual displacements δqi. Virtual work is the work done by the applied forces. When considering forces applied to a body in equilibrium, the principle of least action requires the virtual work of these forces to be zero. Consider a particle P that moves from a point A to a point B along a trajectory r and it is important to notice that the value of the work W depends on the trajectory r. Suppose the force F is the same as F, the variation of the work δW associated with this nearby path, known as the virtual work, can be computed to be δ W = W ¯ − W = ∫ t 0 t 1 d t
Virtual work
–
This is an engraving from Mechanics Magazine published in London in 1824.
Virtual work
–
Illustration from Army Service Corps Training on Mechanical Transport, (1911), Fig. 112 Transmission of motion and force by gear wheels, compound train
31.
Frederick the Great
–
Frederick II was King of Prussia from 1740 until 1786, the longest reign of any Hohenzollern king. Frederick was the last titled King in Prussia and declared himself King of Prussia after achieving full sovereignty for all historical Prussian lands, Prussia had greatly increased its territories and became a leading military power in Europe under his rule. He became known as Frederick the Great and was affectionately nicknamed Der Alte Fritz by the Prussian, in his youth, Frederick was more interested in music and philosophy than the art of war. Upon ascending to the Prussian throne, he attacked Austria and claimed Silesia during the Silesian Wars, winning acclaim for himself. Near the end of his life, Frederick physically connected most of his realm by conquering Polish territories in the First Partition of Poland and he was an influential military theorist whose analysis emerged from his extensive personal battlefield experience and covered issues of strategy, tactics, mobility and logistics. Considering himself the first servant of the state, Frederick was a proponent of enlightened absolutism and he modernized the Prussian bureaucracy and civil service and pursued religious policies throughout his realm that ranged from tolerance to segregation. He reformed the system and made it possible for men not of noble stock to become judges. Frederick also encouraged immigrants of various nationalities and faiths to come to Prussia, some critics, however, point out his oppressive measures against conquered Polish subjects during the First Partition. Frederick supported arts and philosophers he favored, as well as allowing complete freedom of the press, Frederick is buried at his favorite residence, Sanssouci in Potsdam. Because he died childless, Frederick was succeeded by his nephew, Frederick William II, son of his brother, historian Leopold von Ranke was unstinting in his praise of Fredericks Heroic life, inspired by great ideas, filled with feats of arms. Immortalized by the raising of the Prussian state to the rank of a power, Johann Gustav Droysen was even more extolling. However, by the 21st century, a re-evaluation of his legacy as a great warrior, Frederick, the son of Frederick William I and his wife, Sophia Dorothea of Hanover, was born in Berlin on 24 January 1712. The birth of Frederick was welcomed by his grandfather, Frederick I, with more than usual pleasure, with the death of his father in 1713, Frederick William became King of Prussia, thus making young Frederick the crown prince. The new king wished for his sons and daughters to be educated not as royalty and he had been educated by a Frenchwoman, Madame de Montbail, who later became Madame de Rocoulle, and he wished that she educate his children. However, he possessed a violent temper and ruled Brandenburg-Prussia with absolute authority. As Frederick grew, his preference for music, literature and French culture clashed with his fathers militarism, in contrast, Fredericks mother Sophia was polite, charismatic and learned. Her father, George Louis of Brunswick-Lüneburg, succeeded to the British throne as King George I in 1714, Frederick was brought up by Huguenot governesses and tutors and learned French and German simultaneously. Although Frederick William I was raised a Calvinist, he feared he was not of the elect, to avoid the possibility of Frederick being motivated by the same concerns, the king ordered that his heir not be taught about predestination
Frederick the Great
–
Portrait of Frederick the Great; By
Anton Graff, 1781
Frederick the Great
–
Baptism of Frederick, 1712 (Harper's Magazine, 1870)
Frederick the Great
–
Frederick as Crown Prince (1739)
Frederick the Great
–
Rheinsberg Palace, Frederick's residence 1736-1740
32.
Saint Petersburg
–
Saint Petersburg is Russias second-largest city after Moscow, with five million inhabitants in 2012, and an important Russian port on the Baltic Sea. It is politically incorporated as a federal subject, situated on the Neva River, at the head of the Gulf of Finland on the Baltic Sea, it was founded by Tsar Peter the Great on May 271703. In 1914, the name was changed from Saint Petersburg to Petrograd, in 1924 to Leningrad, between 1713 and 1728 and 1732–1918, Saint Petersburg was the capital of imperial Russia. In 1918, the government bodies moved to Moscow. Saint Petersburg is one of the cities of Russia, as well as its cultural capital. The Historic Centre of Saint Petersburg and Related Groups of Monuments constitute a UNESCO World Heritage Site, Saint Petersburg is home to The Hermitage, one of the largest art museums in the world. A large number of consulates, international corporations, banks. Swedish colonists built Nyenskans, a fortress, at the mouth of the Neva River in 1611, in a then called Ingermanland. A small town called Nyen grew up around it, Peter the Great was interested in seafaring and maritime affairs, and he intended to have Russia gain a seaport in order to be able to trade with other maritime nations. He needed a better seaport than Arkhangelsk, which was on the White Sea to the north, on May 1703121703, during the Great Northern War, Peter the Great captured Nyenskans, and soon replaced the fortress. On May 271703, closer to the estuary 5 km inland from the gulf), on Zayachy Island, he laid down the Peter and Paul Fortress, which became the first brick and stone building of the new city. The city was built by conscripted peasants from all over Russia, tens of thousands of serfs died building the city. Later, the city became the centre of the Saint Petersburg Governorate, Peter moved the capital from Moscow to Saint Petersburg in 1712,9 years before the Treaty of Nystad of 1721 ended the war, he referred to Saint Petersburg as the capital as early as 1704. During its first few years, the city developed around Trinity Square on the bank of the Neva, near the Peter. However, Saint Petersburg soon started to be built out according to a plan, by 1716 the Swiss Italian Domenico Trezzini had elaborated a project whereby the city centre would be located on Vasilyevsky Island and shaped by a rectangular grid of canals. The project was not completed, but is evident in the layout of the streets, in 1716, Peter the Great appointed French Jean-Baptiste Alexandre Le Blond as the chief architect of Saint Petersburg. In 1724 the Academy of Sciences, University and Academic Gymnasium were established in Saint Petersburg by Peter the Great, in 1725, Peter died at the age of fifty-two. His endeavours to modernize Russia had met opposition from the Russian nobility—resulting in several attempts on his life
Saint Petersburg
–
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
33.
Louis XVI of France
–
Louis XVI, born Louis-Auguste, was the last King of France and Navarre before the French Revolution, during which he was also known as Louis Capet. In 1765, at the death of his father, Louis, Dauphin of France, son and heir apparent of Louis XV of France, Louis XVI was guillotined on 21 January 1793. The first part of his reign was marked by attempts to reform France in accordance with Enlightenment ideas and these included efforts to abolish serfdom, remove the taille, and increase tolerance toward non-Catholics. The French nobility reacted to the reforms with hostility. Louis implemented deregulation of the market, advocated by his liberal minister Turgot. In periods of bad harvests, it would lead to food scarcity which would prompt the masses to revolt, from 1776, Louis XVI actively supported the North American colonists, who were seeking their independence from Great Britain, which was realized in the 1783 Treaty of Paris. The ensuing debt and financial crisis contributed to the unpopularity of the Ancien Régime and this led to the convening of the Estates-General of 1789. In 1789, the storming of the Bastille during riots in Paris marked the beginning of the French Revolution. Louiss indecisiveness and conservatism led some elements of the people of France to view him as a symbol of the tyranny of the Ancien Régime. The credibility of the king was deeply undermined, and the abolition of the monarchy, Louis XVI was the only King of France ever to be executed, and his death brought an end to more than a thousand years of continuous French monarchy. Louis-Auguste de France, who was given the title Duc de Berry at birth, was born in the Palace of Versailles. Out of seven children, he was the son of Louis, the Dauphin of France. His mother was Marie-Josèphe of Saxony, the daughter of Frederick Augustus II of Saxony, Prince-Elector of Saxony and King of Poland. A strong and healthy boy, but very shy, Louis-Auguste excelled in his studies and had a taste for Latin, history, geography, and astronomy. He enjoyed physical activities such as hunting with his grandfather, and rough-playing with his brothers, Louis-Stanislas, comte de Provence. From an early age, Louis-Auguste had been encouraged in another of his hobbies, locksmithing, upon the death of his father, who died of tuberculosis on 20 December 1765, the eleven-year-old Louis-Auguste became the new Dauphin. His mother never recovered from the loss of her husband, and died on 13 March 1767, throughout his education, Louis-Auguste received a mixture of studies particular to religion, morality, and humanities. His instructors may have also had a hand in shaping Louis-Auguste into the indecisive king that he became
Louis XVI of France
–
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
–
Louis-Charles, the dauphin of France and future Louis XVII. By
Marie Louise Élisabeth Vigée-Lebrun.
34.
Reign of Terror
–
The Reign of Terror or The Terror, is the label given by some historians to a period of violence during the French Revolution. Different historians place the date at either 5 September 1793 or June 1793 or March 1793 or September 1792 or July 1789. Between June 1793 and the end of July 1794, there were 16,594 official death sentences in France, but the total number of deaths in France in 1793–96 in only the civil war in the Vendée is estimated at 250,000 counter-revolutionaries and 200,000 republicans. During 1794, revolutionary France was beset with conspiracies by internal, within France, the revolution was opposed by the French nobility, which had lost its inherited privileges. The Catholic Church opposed the revolution, which had turned the clergy into employees of the state, in addition, the French First Republic was engaged in a series of wars with neighboring powers, and parts of France were engaging in civil war against the loyalist regime. The latter were grouped in the parliamentary faction called the Mountain. Through the Revolutionary Tribunal, the Terrors leaders exercised broad powers, the Reign was a manifestation of the strong strain on centralized power. Many historians have debated the reasons the French Revolution took such a turn during the Reign of Terror of 1793–94. The public was frustrated that the equality and anti-poverty measures that the revolution originally promised were not materializing. Jacques Rouxs Manifesto of the Enraged on 25 June 1793, describes the extent to which, four years into the revolution, the foundation of the Terror is centered on the April 1793 creation of the Committee of Public Safety and its militant Jacobin delegates. Those in power believed the Committee of Public Safety was an unfortunate, according to Mathiez, they touched only with trepidation and reluctance the regime established by the Constituent Assembly so as not to interfere with the early accomplishments of the revolution. Similar to Mathiez, Richard Cobb introduced competing circumstances of revolt, counter-revolutionary rebellions taking place in Lyon, Brittany, Vendée, Nantes, and Marseille were threatening the revolution with royalist ideas. Cobb writes, the revolutionaries themselves, living as if in combat… were easily persuaded that only terror, Terror was used in these rebellions both to execute inciters and to provide a very visible example to those who might be considering rebellion. Cobb agrees with Mathiez that the Terror was simply a response to circumstances, at the same time, Cobb rejects Mathiezs Marxist interpretation that elites controlled the Reign of Terror to the significant benefit of the bourgeoisie. Instead, Cobb argues that social struggles between the classes were seldom the reason for actions and sentiments. Widespread terror and a consequent rise in executions came after external and internal threats were vastly reduced, with the backing of the national guard, they persuaded the convention to arrest 29 Girondist leaders, including Jacques Pierre Brissot. On 13 July the assassination of Jean-Paul Marat – a Jacobin leader, georges Danton, the leader of the August 1792 uprising against the king, was removed from the committee. The Jacobins identified themselves with the movement and the sans-culottes
Reign of Terror
–
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
35.
Lavoisier
–
Antoine-Laurent de Lavoisier was a French nobleman and chemist central to the 18th-century chemical revolution and had a large influence on both the history of chemistry and the history of biology. He is widely considered in popular literature as the father of modern chemistry and it is generally accepted that Lavoisiers great accomplishments in chemistry largely stem from his changing the science from a qualitative to a quantitative one. Lavoisier is most noted for his discovery of the role oxygen plays in combustion and he recognized and named oxygen and hydrogen and opposed the phlogiston theory. Lavoisier helped construct the system, wrote the first extensive list of elements. He predicted the existence of silicon and was also the first to establish that sulfur was an element rather than a compound and he discovered that, although matter may change its form or shape, its mass always remains the same. Lavoisier was a member of a number of aristocratic councils. All of these political and economic activities enabled him to fund his scientific research, at the height of the French Revolution, he was accused by Jean-Paul Marat of selling adulterated tobaccoand of other crimes, and was eventually guillotined a year after Marats death. Antoine-Laurent Lavoisier was born to a family of the nobility in Paris on 26 August 1743. The son of an attorney at the Parliament of Paris, he inherited a fortune at the age of five with the passing of his mother. Lavoisier began his schooling at the Collège des Quatre-Nations, University of Paris in Paris in 1754 at the age of 11, in his last two years at the school, his scientific interests were aroused, and he studied chemistry, botany, astronomy, and mathematics. Lavoisier entered the school of law, where he received a degree in 1763. Lavoisier received a law degree and was admitted to the bar, however, he continued his scientific education in his spare time. Lavoisiers education was filled with the ideals of the French Enlightenment of the time and he attended lectures in the natural sciences. Lavoisiers devotion and passion for chemistry were largely influenced by Étienne Condillac and his first chemical publication appeared in 1764. From 1763 to 1767, he studied geology under Jean-Étienne Guettard, in collaboration with Guettard, Lavoisier worked on a geological survey of Alsace-Lorraine in June 1767. In 1768 Lavoisier received an appointment to the Academy of Sciences. In 1769, he worked on the first geological map of France, on behalf of the Ferme générale Lavoisier commissioned the building of a wall around Paris so that customs duties could be collected from those transporting goods into and out of the city. Lavoisier attempted to introduce reforms in the French monetary and taxation system to help the peasants, Lavoisier consolidated his social and economic position when, in 1771 at age 28, he married Marie-Anne Pierrette Paulze, the 13-year-old daughter of a senior member of the Ferme générale
Lavoisier
–
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."
36.
Astronomy
–
Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, physical cosmology, is concerned with the study of the Universe as a whole, Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
Astronomy
–
A
star -forming region in the
Large Magellanic Cloud, an
irregular galaxy.
Astronomy
–
A giant
Hubble mosaic of the
Crab Nebula, a
supernova remnant
Astronomy
–
19th century
Sydney Observatory,
Australia (1873)
Astronomy
–
19th century
Quito Astronomical Observatory is located 12 minutes south of the
Equator in
Quito,
Ecuador.
37.
Infinite series
–
In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
Infinite series
–
Illustration of 3
geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.
38.
Second law of motion
–
Newtons laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a measure of the force. These three laws have been expressed in different ways, over nearly three centuries, and can be summarised as follows. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation. Newtons laws are applied to objects which are idealised as single point masses, in the sense that the size and this can be done when the object is small compared to the distances involved in its analysis, or the deformation and rotation of the body are of no importance. In this way, even a planet can be idealised as a particle for analysis of its orbital motion around a star, in their original form, Newtons laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newtons laws of motion for rigid bodies called Eulers laws of motion, if a body is represented as an assemblage of discrete particles, each governed by Newtons laws of motion, then Eulers laws can be derived from Newtons laws. Eulers laws can, however, be taken as axioms describing the laws of motion for extended bodies, Newtons laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Other authors do treat the first law as a corollary of the second, the explicit concept of an inertial frame of reference was not developed until long after Newtons death. In the given mass, acceleration, momentum, and force are assumed to be externally defined quantities. This is the most common, but not the interpretation of the way one can consider the laws to be a definition of these quantities. Newtonian mechanics has been superseded by special relativity, but it is useful as an approximation when the speeds involved are much slower than the speed of light. The first law states that if the net force is zero, the first law can be stated mathematically when the mass is a non-zero constant, as, ∑ F =0 ⇔ d v d t =0. Consequently, An object that is at rest will stay at rest unless a force acts upon it, an object that is in motion will not change its velocity unless a force acts upon it. This is known as uniform motion, an object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest, if an object is moving, it continues to move without turning or changing its speed
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
39.
Statistical mechanics
–
Statistical mechanics is a branch of theoretical physics using probability theory to study the average behaviour of a mechanical system, where the state of the system is uncertain. A common use of mechanics is in explaining the thermodynamic behaviour of large systems. This branch of mechanics, which treats and extends classical thermodynamics, is known as statistical thermodynamics or equilibrium statistical mechanics. Statistical mechanics also finds use outside equilibrium, an important subbranch known as non-equilibrium 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 reactions or flows of particles, in physics there are two types of mechanics usually examined, classical mechanics and quantum mechanics. Statistical mechanics fills this disconnection between the laws of mechanics and the experience of incomplete knowledge, by adding some uncertainty about which state the system is in. The statistical ensemble is a probability distribution over all states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points, in quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a density matrix. These two meanings are equivalent for many purposes, and will be used interchangeably in this article, however the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself also evolves, as the systems in the ensemble continually leave one state. The ensemble 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. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium, Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics, non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems. The primary goal of thermodynamics is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles. Whereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium, Statistical equilibrium does not mean that the particles have stopped moving, rather, only that the ensemble is not evolving. A sufficient condition for statistical equilibrium with a system is that the probability distribution is a function only of conserved properties. There are many different equilibrium ensembles that can be considered, additional postulates are necessary to motivate why the ensemble for a given system should have one form or another. A common approach found in textbooks is to take the equal a priori probability postulate
Statistical mechanics
–
Statistical mechanics
40.
Acceleration
–
Acceleration, in physics, is the rate of change of velocity of an object with respect to time. An objects acceleration is the net result of any and all forces acting on the object, the SI unit for acceleration is metre per second squared. Accelerations are vector quantities and add according to the parallelogram law, as a vector, the calculated net force is equal to the product of the objects mass and its acceleration. For example, when a car starts from a standstill and travels in a line at increasing speeds. If the car turns, there is an acceleration toward the new direction, in this example, we can call the forward acceleration of the car a linear acceleration, which passengers in the car might experience as a force pushing them back into their seats. When changing direction, we call this non-linear acceleration, which passengers might experience as a sideways force. If the speed of the car decreases, this is an acceleration in the direction from the direction of the vehicle. Passengers may experience deceleration as a force lifting them forwards, mathematically, there is no separate formula for deceleration, both are changes in velocity. Each of these accelerations might be felt by passengers until their velocity matches that of the car, an objects average acceleration over a period of time is its change in velocity divided by the duration of the period. Mathematically, a ¯ = Δ v Δ t, instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimal interval of time. The SI unit of acceleration is the metre per second squared, or metre per second per second, as the velocity in metres per second changes by the acceleration value, every second. An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, in this case it is said to be undergoing centripetal acceleration. Proper acceleration, the acceleration of a relative to a free-fall condition, is measured by an instrument called an accelerometer. As speeds approach the speed of light, relativistic effects become increasingly large and these components are called the tangential acceleration and the normal or radial acceleration. Geometrical analysis of space curves, which explains tangent, normal and binormal, is described by the Frenet–Serret formulas. Uniform or constant acceleration is a type of motion in which the velocity of an object changes by an amount in every equal time period. A frequently cited example of uniform acceleration is that of an object in free fall in a gravitational field. The acceleration of a body in the absence of resistances to motion is dependent only on the gravitational field strength g
Acceleration
–
Components of acceleration for a curved motion. The tangential component a t is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector (or in the opposite direction). The normal component (also called centripetal component for circular motion) a c is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.
Acceleration
–
Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as
time interval Δt → 0 of Δ v / Δt
41.
Angular momentum
–
In physics, angular momentum is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque. The definition of momentum for a point particle is a pseudovector r×p. This definition can be applied to each point in continua like solids or fluids, unlike momentum, angular momentum does depend on where the origin is chosen, since the particles position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object via the moment of inertia I. However, while ω always points in the direction of the rotation axis, Angular momentum is additive, the total angular momentum of a system is the vector sum of the angular momenta. For continua or fields one uses integration, torque can be defined as the rate of change of angular momentum, analogous to force. Applications include the gyrocompass, control moment gyroscope, inertial systems, reaction wheels, flying discs or Frisbees. In general, conservation does limit the motion of a system. In quantum mechanics, angular momentum is an operator with quantized eigenvalues, Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the spin of elementary particles does not correspond to literal spinning motion, Angular momentum is a vector quantity that represents the product of a bodys rotational inertia and rotational velocity about a particular axis. Angular momentum can be considered an analog of linear momentum. Thus, where momentum is proportional to mass m and linear speed v, p = m v, angular momentum is proportional to moment of inertia I. Unlike mass, which only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation. Unlike linear speed, which occurs in a line, angular speed occurs about a center of rotation. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center and this simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case, L = r m v ⊥, where v ⊥ = v sin θ is the component of the motion. It is this definition, × to which the moment of momentum refers
Angular momentum
–
This
gyroscope remains upright while spinning due to the conservation of its angular momentum.
Angular momentum
–
An
ice skater conserves angular momentum – her
rotational speed increases as her
moment of inertia decreases by drawing in her arms and legs.
42.
Energy
–
In physics, energy is the property that must be transferred to an object in order to perform work on – or to heat – the object, and can be converted in form, but not created or destroyed. The SI unit of energy is the joule, which is the transferred to an object by the mechanical work of moving it a distance of 1 metre against a force of 1 newton. Mass and energy are closely related, for example, 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 humans get from food. Civilisation gets the energy it needs from energy resources such as fuels, nuclear fuel. The processes of Earths climate and ecosystem are driven by the radiant energy Earth receives from the sun, the total energy of a system can be subdivided and classified in various ways. It may also be convenient to distinguish gravitational energy, thermal energy, several types of energy, electric energy. Many of these overlap, for instance, thermal energy usually consists partly of kinetic. Some types of energy are a mix of both potential and kinetic energy. An example is energy which is the sum of kinetic. Whenever physical scientists discover that a phenomenon appears to violate the law of energy conservation. Heat and work are special cases in that they are not properties of systems, in general we cannot measure how much heat or work are present in an object, but rather only how much energy is transferred among objects in certain ways during the occurrence of a given process. Heat and work are measured as positive or negative depending on which side of the transfer we view them from, the distinctions between different kinds of energy is not always clear-cut. In contrast to the definition, energeia was a qualitative philosophical concept, broad enough to include ideas such as happiness. The modern analog of this property, kinetic energy, differs from vis viva only by a factor of two, in 1807, Thomas Young was possibly the first to use the term energy instead of vis viva, in its modern sense. Gustave-Gaspard Coriolis described kinetic energy in 1829 in its modern sense, the law of conservation of energy was also first postulated in the early 19th century, and applies to any isolated system. It was argued for years whether heat was a physical substance, dubbed the caloric, or merely a physical quantity. In 1845 James Prescott Joule discovered the link between mechanical work and the generation of heat and these developments led to the theory of conservation of energy, formalized largely by William Thomson as the field of thermodynamics
Energy
–
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
–
Thermal energy is energy of microscopic constituents of matter, which may include both
kinetic and
potential energy.
Energy
–
Thomas Young – the first to use the term "energy" in the modern sense.
Energy
–
A
Turbo generator transforms the energy of pressurised steam into electrical energy
43.
Force
–
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 to change its velocity, force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity and it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newtons second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. In an extended body, each part usually applies forces on the adjacent parts, such internal mechanical stresses cause no accelation of that body as the forces balance one another. Pressure, the distribution of small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate. Stress usually causes deformation of materials, or flow in fluids. In part this was due to an understanding of the sometimes non-obvious force of friction. A fundamental error was the belief that a force is required to maintain motion, most of the previous misunderstandings about motion and 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 three hundred years, the Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known, in order of decreasing strength, they are, strong, electromagnetic, weak, high-energy particle 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 as integral to the functioning of each of the simple machines. The mechanical advantage given by a machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was famous for formulating a treatment of buoyant forces inherent in fluids. Aristotle provided a discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotles view, the sphere contained four elements that come to rest at different natural places therein. Aristotle believed that objects on Earth, those composed mostly of the elements earth and water, to be in their natural place on the ground. He distinguished between the tendency of objects to find their natural place, which led to natural motion, and unnatural or forced motion
Force
–
Aristotle famously described a force as anything that causes an object to undergo "unnatural motion"
Force
–
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
–
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
–
Galileo Galilei was the first to point out the inherent contradictions contained in Aristotle's description of forces.
44.
Frame of reference
–
In physics, a frame of reference consists of an abstract coordinate system and the set of physical reference points that uniquely fix the coordinate system and standardize measurements. In n dimensions, n+1 reference points are sufficient to define a reference frame. Using rectangular coordinates, a frame may be defined with a reference point at the origin. In Einsteinian relativity, reference frames are used to specify the relationship between an observer and the phenomenon or phenomena under observation. In this context, the phrase often becomes observational frame of reference, a relativistic reference frame includes the coordinate time, which does not correspond across different frames moving relatively to each other. The situation thus differs from Galilean relativity, where all possible coordinate times are essentially equivalent, the need to distinguish between the various meanings of frame of reference has led to a variety of terms. For example, sometimes the type of system is attached as a modifier. Sometimes the state of motion is emphasized, as in rotating frame of reference, sometimes the way it transforms to frames considered as related is emphasized as in Galilean frame of reference. Sometimes frames are distinguished by the scale of their observations, as in macroscopic and microscopic frames of reference, in this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a system may be employed for many purposes where the state of motion is not the primary concern. For example, a system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors and it seems useful to divorce the various aspects of a reference frame for the discussion below. A coordinate system is a concept, amounting to a choice of language used to describe observations. Consequently, an observer in a frame of reference can choose to employ any coordinate system to describe observations made from that frame of reference. A change in the choice of coordinate system does not change an observers state of motion. This viewpoint can be found elsewhere as well, which is not to dispute that some coordinate systems may be a better choice for some observations than are others. Choice of what to measure and with what observational apparatus is a separate from the observers state of motion. D. Norton, The discussion is taken beyond simple space-time coordinate systems by Brading, extension to coordinate systems using generalized coordinates underlies the Hamiltonian and Lagrangian formulations of quantum field theory, classical relativistic mechanics, and quantum gravity
Frame of reference
–
An observer O, situated at the origin of a local set of coordinates – a frame of reference F. The observer in this frame uses the coordinates (x, y, z, t) to describe a spacetime event, shown as a star.
45.
Impulse (physics)
–
In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Since force is a quantity, impulse is also a vector in the same direction. Impulse applied to an object produces an equivalent vector change in its linear momentum, the SI unit of impulse is the newton second, and 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 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 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 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 and this sort of change is a step change, and is not physically possible. However, this is a model for computing the effects of ideal collisions. The application of Newtons second law for variable mass allows impulse, in the case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse. This fact can be used to derive the Tsiolkovsky rocket equation, which relates the vehicles propulsive change in velocity to the specific impulse. Wave–particle duality defines the impulse of a wave collision, the preservation of momentum in the collision is then called phase matching. Applications include, Compton effect nonlinear optics Acousto-optic modulator Electron phonon scattering Serway, Raymond A. Jewett, John W. Physics for Scientists, Physics for Scientists and Engineers, Mechanics, Oscillations and Waves, Thermodynamics
Impulse (physics)
–
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)
46.
Moment of inertia
–
It depends on the bodys mass distribution and the axis chosen, with larger moments requiring more torque to change the bodys rotation. It is a property, the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems. One of its definitions is the moment of mass with respect to distance from an axis r, I = ∫ Q r 2 d m. For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about a perpendicular to the plane. When a body is rotating, or free to rotate, around an axis, the amount of torque needed to cause any given angular acceleration is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram metre squared in SI units, moment of inertia plays the role in rotational kinetics that mass plays in linear kinetics - both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, for a point-like mass, the moment of inertia about some axis is given by mr2, where r is the distance to the axis, and m is the mass. For an extended body, the moment of inertia is just the sum of all the pieces of mass multiplied by the square of their distances from the axis in question. For an extended body of a shape and uniform density. In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, the term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Eulers second law. Comparison of this frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. Moment of inertia appears in momentum, kinetic energy, and in Newtons laws of motion for a rigid body as a physical parameter that combines its shape. There is a difference in the way moment of inertia appears in planar. The moment of inertia of a flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis, if the angular 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 arms or divers curl their bodies into a tuck position during a dive. For a simple pendulum, this yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as. Thus, moment of inertia depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation
Moment of inertia
–
Tightrope walker Samuel Dixon using the long rod's moment of inertia for balance while crossing the
Niagara River in 1890.
Moment of inertia
–
Flywheels have large moments of inertia to smooth out mechanical motion. This example is in a Russian museum.
Moment of inertia
–
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
–
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.
47.
Power (physics)
–
In physics, power is the rate of doing work. It is the amount of energy consumed per unit time, having no direction, it is a scalar quantity. In the SI system, the unit of power is the joule per second, known as the watt in honour of James Watt, another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written, because this integral depends on the trajectory of the point of application of the force and torque, this calculation of work is said to be path dependent. As a physical concept, power requires both a change in the universe and a specified time in which the change occurs. This is distinct from the concept of work, which is measured in terms of a net change in the state of the physical universe. The output power of a motor is the product of the torque that the motor generates. The power involved in moving a vehicle is the product of the force of the wheels. The dimension of power is divided by time. The SI unit of power is the watt, which is equal to one joule per second, other units of power include ergs per second, horsepower, metric horsepower, and foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the required to lift 550 pounds by one foot in one second. Other units include dBm, a logarithmic measure with 1 milliwatt as reference, food calories per hour, Btu per hour. This shows how power is an amount of energy consumed per unit time. If ΔW is the amount of work performed during a period of time of duration Δt and it is the average amount of work done or energy converted per unit of time. The average power is simply called power when the context makes it clear. The instantaneous power is then the value of the average power as the time interval Δt approaches zero. P = lim Δ t →0 P a v g = lim Δ t →0 Δ W Δ t = d W d t. In the case of constant power P, the amount of work performed during a period of duration T is given by, W = P t
Power (physics)
–
Ansel Adams photograph of electrical wires of the Boulder Dam Power Units, 1941–1942
48.
Momentum
–
In classical mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object, quantified in kilogram-meters per second. It is dimensionally equivalent to impulse, the product of force and time, Newtons second law of motion states that the change in linear momentum of a body is equal to the net impulse acting on it. If the truck were lighter, or moving slowly, then it would have less momentum. Linear momentum is also a quantity, meaning that if a closed system is not affected by external forces. In classical mechanics, conservation of momentum is implied by Newtons laws. It also holds in special relativity and, with definitions, a linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory. It is ultimately an expression of one of the symmetries of space and time. Linear momentum depends on frame of reference, observers in different frames would find different values of linear momentum of a system. But each would observe that the value of linear momentum does not change with time, momentum has a direction as well as magnitude. Quantities that have both a magnitude and a direction are known as vector quantities, because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds. Below, the properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations, the momentum of a particle is traditionally represented by the letter p. It is the product of two quantities, the mass and velocity, p = m v, the units of momentum are the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity in meters per second then the momentum is in kilogram meters/second, in cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters/second. Being a vector, momentum has magnitude and direction, for example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg m/s due north measured from the ground. The momentum of a system of particles is the sum of their momenta, if two particles have masses m1 and m2, and velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2. If all the particles are moving, the center of mass will generally be moving as well, if the center of mass is moving at velocity vcm, the momentum is, p = m v cm. This is known as Eulers first law, if a force F is applied to a particle for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t
Momentum
–
In a game of
pool, momentum is conserved; that is, if one ball stops dead after the collision, the other ball will continue away with all the momentum. If the moving ball continues or is deflected then both balls will carry a portion of the momentum from the collision.
49.
Space
–
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Many of these classical philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, in Isaac Newtons view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another. In the 18th century, the philosopher and theologian George Berkeley attempted to refute the visibility of spatial depth in his Essay Towards a New Theory of Vision. Kant referred to the experience of space in his Critique of Pure Reason as being a pure a priori form of intuition. In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in space is conceived as curved. According to Albert Einsteins theory of relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a model for the shape of space. In the seventeenth century, the philosophy of space and time emerged as an issue in epistemology. At its heart, Gottfried Leibniz, the German philosopher-mathematician, and Isaac Newton, unoccupied regions are those that could have objects in them, and thus spatial relations with other places. For Leibniz, then, space was an abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete. Space could be thought of in a way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people, but since there would be no observational way of telling these universes apart then, according to the identity of indiscernibles, there would be no real difference between them. According to the principle of sufficient reason, any theory of space that implied that there could be two possible universes must therefore be wrong. Newton took space to be more than relations between objects and based his position on observation and experimentation
Space
–
Gottfried Leibniz
Space
–
A right-handed three-dimensional
Cartesian coordinate system used to indicate positions in space.
Space
–
Isaac Newton
Space
–
Immanuel Kant
50.
Speed
–
In everyday use and in kinematics, the speed of an object is the magnitude of its velocity, it is thus a scalar quantity. Speed has the dimensions of distance divided by time, the SI unit of speed is the metre per second, but the most common unit of speed in everyday usage is the kilometre per hour or, in the US and the UK, miles per hour. For air and marine travel the knot is commonly used, the fastest possible speed at which energy or information can travel, according to special relativity, is the speed of light in a vacuum c =299792458 metres per second. Matter cannot quite reach the speed of light, as this would require an amount of energy. In relativity physics, the concept of rapidity replaces the classical idea of speed, italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time, in equation form, this is v = d t, where v is speed, d is distance, and t is time. A cyclist who covers 30 metres in a time of 2 seconds, objects in motion often have variations in speed. If s is the length of the path travelled until time t, in the special case where the velocity is constant, this can be simplified to v = s / t. The average speed over a time interval is the total distance travelled divided by the time duration. Speed at some instant, or assumed constant during a short period of time, is called instantaneous speed. By looking at a speedometer, one can read the speed of a car at any instant. A car travelling at 50 km/h generally goes for less than one hour at a constant speed, if the vehicle continued at that speed for half an hour, it would cover half that distance. If it continued for one minute, it would cover about 833 m. Different from instantaneous speed, average speed is defined as the distance covered divided by the time interval. For example, if a distance of 80 kilometres is driven in 1 hour, likewise, if 320 kilometres are travelled in 4 hours, the average speed is also 80 kilometres per hour. When a distance in kilometres is divided by a time in hours, average speed does not describe the speed variations that may have taken place during shorter time intervals, and so average speed is often quite different from a value of instantaneous speed. If the average speed and the time of travel are known, using this equation for an average speed of 80 kilometres per hour on a 4-hour trip, the distance covered is found to be 320 kilometres. Linear speed is the distance travelled per unit of time, while speed is the linear speed of something moving along a circular path
Speed
–
Speed can be thought of as the rate at which an object covers
distance. A fast-moving object has a high speed and covers a relatively large distance in a given amount of time, while a slow-moving object covers a relatively small amount of distance in the same amount of time.
51.
Torque
–
Torque, moment, or moment of force is rotational force. Just as a force is a push or a pull. Loosely speaking, torque is a measure of the force on an object such as a bolt or a flywheel. For example, pushing or pulling the handle of a wrench connected to a nut or bolt produces a torque that loosens or tightens the nut or bolt, the symbol for torque is typically τ, the lowercase Greek letter tau. When it is called moment of force, it is denoted by M. The SI unit for torque is the newton metre, for more on the units of torque, see Units. This article follows US physics terminology in its use of the word torque, in the UK and in US mechanical engineering, this is called moment of force, usually shortened to moment. In US physics and UK physics terminology these terms are interchangeable, unlike in US mechanical engineering, Torque is defined mathematically as the rate of change of angular momentum of an object. The definition of states that one or both of the angular velocity or the moment of inertia of an object are changing. Moment is the term used for the tendency of one or more applied forces to rotate an object about an axis. For example, a force applied to a shaft causing acceleration, such as a drill bit accelerating from rest. By contrast, a force on a beam produces a moment, but since the angular momentum of the beam is not changing. Similarly with any force couple on an object that has no change to its angular momentum and this article follows the US physics terminology by calling all moments by the term torque, whether or not they cause the angular momentum of an object to change. The concept of torque, also called moment or couple, originated with the studies of Archimedes on levers, the term torque was apparently introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884. A force applied at an angle to a lever multiplied by its distance from the levers fulcrum is its torque. A force of three newtons applied two metres from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum. More generally, the torque on a particle can be defined as the product, τ = r × F, where r is the particles position vector relative to the fulcrum. Alternatively, τ = r F ⊥, where F⊥ is the amount of force directed perpendicularly to the position of the particle, any force directed parallel to the particles position vector does not produce a torque
Torque
52.
Velocity
–
The velocity of an object is the rate of change of its position with respect to a frame of reference, and 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 vector quantity, both magnitude and direction are needed to define it. The scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI system as metres per second or as the SI base unit of. For example,5 metres per second is a scalar, whereas 5 metres per second east is a vector, if there is a change in speed, direction or 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. For example, a car moving at a constant 20 kilometres per hour in a path has 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 and this is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Velocity is defined as the rate of change of position with respect to time, average velocity can be calculated as, v ¯ = Δ x Δ t. The average velocity is less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, from this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time is the displacement, x. In calculus terms, the integral of the velocity v is the displacement function x. In the figure, this corresponds to the area under the curve labeled s. Since the derivative of the position with respect to time gives the change in position divided by the change in time, although velocity is defined as the rate of change of position, it is often common to start with an expression for an objects acceleration. As seen by the three green tangent lines in the figure, an objects instantaneous acceleration at a point in time is the slope of the tangent to the curve of a v graph at that point. In other words, acceleration is defined as the derivative of velocity with respect to time, from there, we can obtain an expression for velocity as the area under an a acceleration vs. time graph
Velocity
–
As a change of direction occurs while the cars turn on the curved track, their velocity is not constant.
53.
Routhian mechanics
–
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 Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, and has a similar mathematical form to the Hamiltonian, but is not exactly the same. The difference between the Lagrangian, Hamiltonian, and 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, and can be done to simplify the problem, in each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian. The full set thus has the advantages of both sets of equations, with the convenience of splitting one set of coordinates to the Hamilton equations, and the rest to the Lagrangian equations. The Lagrangian equations are powerful results, used frequently in theory, 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, and introduces no new physics. It offers a 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 Routhian is intermediate between L and H, some coordinates q1, q2, qn are chosen to have corresponding generalized momenta p1, p2. Pn, the rest of the coordinates ζ1, ζ2, ζs to have generalized velocities dζ1/dt, dζ2/dt. Dζs/dt, and time may appear explicitly, where again the generalized velocity dqi/dt is to be expressed as a function of generalized momentum pi via its defining relation. The choice of n coordinates are to have corresponding momenta. The above is used by Landau and Lifshitz, and Goldstein, some authors may define the Routhian to be the negative of the above definition. Below, the Routhian equations of motion are obtained in two ways, in the other useful derivatives are found that can be used elsewhere. Consider the case of a system with two degrees of freedom, q and ζ, with generalized velocities dq/dt and dζ/dt, now change variables, from the set to, simply switching the velocity dq/dt to the momentum p. This change of variables in the differentials is the Legendre transformation, the differential of the new function to replace L will be a sum of differentials in dq, dζ, dp, d, and dt. Notice the Routhian replaces the Hamiltonian and Lagrangian functions in all the equations of motion, the remaining equation states the partial time derivatives of L and R are negatives ∂ L ∂ t = − ∂ R ∂ t. n, and j =1,2
Routhian mechanics
–
Edward John Routh, 1831–1907.
54.
Damping
–
If a frictional force proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the coefficient, the system can, Oscillate with a frequency lower than in the non-damped case. Decay to the position, without oscillations. The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a value of the friction coefficient and is called critically damped. If an external time dependent force is present, the oscillator is described as a driven oscillator. Mechanical examples include pendulums, masses connected to springs, and acoustical systems, other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many devices, such as clocks. They are the source of virtually all sinusoidal vibrations and waves, a simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a m, which experiences a single force, F, which pulls the mass in the direction of the point x=0 and depends only on the masss position x. Balance of forces for the system is F = m a = m d 2 x d t 2 = m x ¨ = − k x. Solving this differential equation, we find that the motion is described by the function x = A cos , the motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude, A. The position at a time t also depends on the phase, φ. The period and frequency are determined by the size of the mass m, the velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the direction as the displacement. The potential energy stored in a harmonic oscillator at position x is U =12 k x 2. In real oscillators, friction, or damping, slows the motion of the system, due to frictional force, the velocity decreases in proportion to the acting frictional force. While simple harmonic motion oscillates with only the force acting on the system
Damping
–
Mass attached to a spring and damper.
55.
Equations of motion
–
In mathematical physics, equations of motion are equations that describe the behaviour of a physical system in terms of its motion as a function of time. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system, the functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions to the equations describing the motion of the dynamics. There are two descriptions of motion, dynamics and kinematics. Dynamics is general, since momenta, forces and energy of the particles are taken into account, in this instance, sometimes the term refers to the differential equations that the system satisfies, and sometimes to the solutions to those equations. However, kinematics is simpler as it concerns only variables derived from the positions of objects, equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the types of motion are translations, rotations, oscillations. A differential equation of motion, usually identified as some physical law, solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the initial values, to state this formally, in general an equation of motion M is a function of the position r of the object, its velocity, and its acceleration, and time t. Euclidean vectors in 3D are denoted throughout in bold and this is equivalent to saying an equation of motion in r is a second order ordinary differential equation in r, M =0, where t is time, and each overdot denotes one time derivative. The initial conditions are given by the constant values at t =0, r, r ˙, the solution r to the equation of motion, with specified initial values, describes the system for all times t after t =0. Sometimes, the equation will be linear and is likely to be exactly solvable. In general, the equation will be non-linear, and cannot be solved exactly so a variety of approximations must be used, the solutions to nonlinear equations may show chaotic behavior depending on how sensitive the system is to the initial conditions. Despite the great strides made in the development of geometry made by Ancient Greeks and surveys in Rome, the exposure of Europe to Arabic numerals and their ease in computations encouraged first the scholars to learn them and then the merchants and invigorated the spread of knowledge throughout Europe. These studies led to a new body of knowledge that is now known as physics, thomas Bradwardine, one of those scholars, extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested a law involving force, resistance, distance, velocity. Nicholas Oresme further extended Bradwardines arguments, for writers on kinematics before Galileo, since small time intervals could not be measured, the affinity between time and motion was obscure. They used time as a function of distance, and in free fall, de Sotos comments are shockingly correct regarding the definitions of acceleration and the observation that during the violent motion of ascent acceleration would be negative
Equations of motion
–
Kinematic quantities of a classical particle of mass m: position r, velocity v, acceleration a.
56.
Friction
–
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction, Dry friction resists relative lateral motion of two surfaces in contact. Dry friction is subdivided into static friction between non-moving surfaces, and kinetic friction between moving surfaces, fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other. Lubricated friction is a case of fluid friction where a lubricant fluid separates two solid surfaces, skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body. Internal friction is the force resisting motion between the making up a solid material while it undergoes deformation. When surfaces in contact move relative to other, the friction between the two surfaces converts kinetic energy into thermal energy. This property can have consequences, as illustrated by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is converted to thermal energy whenever motion with friction occurs, another important consequence of many types of friction can be wear, which may lead to performance degradation and/or damage to components. Friction is a component of the science of tribology, Friction is not itself a fundamental force. Dry friction arises from a combination of adhesion, surface roughness, surface deformation. The complexity of interactions makes the calculation of friction from first principles impractical and necessitates the use of empirical methods for analysis. Friction is a non-conservative force - work done against friction is path dependent, in the presence of friction, some energy is always lost in the form of heat. Thus mechanical energy is not conserved, the Greeks, including Aristotle, Vitruvius, and Pliny the Elder, were interested in the cause and mitigation of friction. They were aware of differences between static and kinetic friction with Themistius stating in 350 A. D. that it is easier to further the motion of a moving body than to move a body at rest. The classic laws of sliding friction were discovered by Leonardo da Vinci in 1493, a pioneer in tribology and these laws were rediscovered by Guillaume Amontons in 1699. Amontons presented the nature of friction in terms of surface irregularities, the understanding of friction was further developed by Charles-Augustin de Coulomb. Coulomb further considered the influence of sliding velocity, temperature and humidity, the distinction between static and dynamic friction is made in Coulombs friction law, although this distinction was already drawn by Johann Andreas von Segner in 1758. Leslie was equally skeptical about the role of adhesion proposed by Desaguliers, in Leslies view, friction should be seen as a time-dependent process of flattening, pressing down asperities, which creates new obstacles in what were cavities before
Friction
–
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.
57.
Harmonic oscillator
–
If a frictional force proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the coefficient, the system can, Oscillate with a frequency lower than in the non-damped case. Decay to the position, without oscillations. The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a value of the friction coefficient and is called critically damped. If an external time dependent force is present, the oscillator is described as a driven oscillator. Mechanical examples include pendulums, masses connected to springs, and acoustical systems, other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many devices, such as clocks. They are the source of virtually all sinusoidal vibrations and waves, a simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a m, which experiences a single force, F, which pulls the mass in the direction of the point x=0 and depends only on the masss position x. Balance of forces for the system is F = m a = m d 2 x d t 2 = m x ¨ = − k x. Solving this differential equation, we find that the motion is described by the function x = A cos , the motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude, A. The position at a time t also depends on the phase, φ. The period and frequency are determined by the size of the mass m, the velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the direction as the displacement. The potential energy stored in a harmonic oscillator at position x is U =12 k x 2. In real oscillators, friction, or damping, slows the motion of the system, due to frictional force, the velocity decreases in proportion to the acting frictional force. While simple harmonic motion oscillates with only the force acting on the system
Harmonic oscillator
–
Another damped harmonic oscillator
Harmonic oscillator
–
Dependence of the system behavior on the value of the damping ratio ζ
58.
Inertial frame of reference
–
In classical physics and special relativity, an inertial frame of reference is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner. The physics of a system in an inertial frame have no causes external to the system, all inertial frames are in a state of constant, rectilinear motion with respect to one another, an accelerometer moving with any of them would detect zero acceleration. Measurements in one frame can be converted to measurements in another by a simple transformation. In general relativity, in any region small enough for the curvature of spacetime and tidal forces to be negligible, systems in non-inertial frames in general relativity dont have external causes because of the principle of geodesic motion. Physical laws take the form in all inertial frames. For example, a ball dropped towards the ground does not go 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—other bodies, observers and 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, however, a frame of reference can always be chosen in which it remains stationary. Similarly, if space is not described uniformly or time independently, 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, Newtons first law, the law of inertia, is satisfied, Any free motion has a constant magnitude, the force F is the vector sum of all real forces on the particle, such as electromagnetic, gravitational, nuclear and so forth. The extra terms in the force F′ are the forces for this frame. The first extra term is the Coriolis force, the second the centrifugal force, also, fictitious forces do not drop off with distance. For example, the force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis. All observers agree on the forces, F, only non-inertial observers need fictitious forces. The laws of physics in the frame are simpler because unnecessary forces are not present. In Newtons time the stars were invoked as a reference frame
Inertial frame of reference
–
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.
59.
Motion (physics)
–
In physics, motion is a change in position of an object over time. Motion is described in terms of displacement, distance, velocity, acceleration, time, motion of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame. If the position of a body is not changing with respect to a frame of reference. An objects motion cannot change unless it is acted upon by a force, momentum is a quantity which is used for measuring motion of an object. As there is no frame of reference, absolute motion cannot be determined. Thus, everything in the universe can be considered to be moving, more generally, motion is a concept that applies to objects, bodies, and matter particles, to radiation, radiation fields and radiation particles, and to space, its curvature and space-time. One can also speak of motion of shapes and boundaries, so, the term motion in general signifies a continuous change in the configuration of a physical system. For example, one can talk about motion of a wave or about motion of a quantum particle, in physics, motion is described through two sets of apparently contradictory laws of mechanics. Motions of all large scale and familiar objects in the universe are described by classical mechanics, whereas the motion of very small atomic and sub-atomic objects is described by quantum mechanics. It produces very accurate results within these domains, and is one of the oldest and largest in science, engineering, classical mechanics is fundamentally based on Newtons laws of motion. These laws describe the relationship between the acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica and his three laws are, A body either is at rest or moves with constant velocity, until and unless an outer force is applied to it. An object will travel in one direction only until an outer force changes its direction, whenever one body exerts a force F onto a second body, the second body exerts the force −F on the first body. F and −F are equal in magnitude and opposite in sense, so, the body which exerts F will go backwards. Newtons three laws of motion, along with his Newtons law of motion, which were the first to provide a mathematical model for understanding orbiting bodies in outer space. This explanation unified the motion of bodies and motion of objects on earth. Classical mechanics was later enhanced by Albert Einsteins special relativity. Motion of objects with a velocity, approaching the speed of light
Motion (physics)
–
Motion involves a change in position, such as in this perspective of rapidly leaving
Yongsan Station.
60.
Rigid body
–
In physics, a rigid body is an idealization of a solid body in which deformation is neglected. In other words, the distance between any two points of a rigid body remains constant in time regardless of external forces exerted on it. Even though such an object cannot physically exist due to relativity, in classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechanics a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics molecules are often seen as rigid bodies, the position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, if the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, however, typically a different, mathematically more convenient, but equivalent approach is used. Thus, the position of a body has two components, linear and angular, respectively. The same is true for other kinematic and kinetic quantities describing the motion of a body, such as linear and angular velocity, acceleration, momentum, impulse. This reference point may define the origin of a coordinate system fixed to the body, there are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix. In general, when a body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation of the body starting from a hypothetic reference position. Velocity and angular velocity are measured with respect to a frame of reference, the linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a point fixed to the body. During purely translational motion, all points on a body move with the same velocity. However, when motion involves rotation, the velocity of any two points on the body will generally not be the same. Two points of a body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation. Angular velocity is a quantity that describes the angular speed at which the orientation of the rigid body is changing. All points on a rigid body experience the same velocity at all times
Rigid body
–
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).
61.
Vibration
–
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem, the oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road. Vibration can be desirable, for example, the motion of a fork, the reed in a woodwind instrument or harmonica. In many cases, however, vibration is undesirable, wasting energy, for example, the vibrational motions of engines, electric motors, or any mechanical device in operation are typically unwanted. Such vibrations could be caused by imbalances in the parts, uneven friction. Careful designs usually minimize unwanted vibrations, the studies of sound and vibration are closely related. Sound, or pressure waves, are generated by vibrating structures, hence, attempts to reduce noise are often related to issues of vibration. Free vibration occurs when a system is set in motion with an initial input. Examples of this type of vibration are pulling a child back on a swing and letting go, or hitting a tuning fork, the mechanical system vibrates at one or more of its natural frequencies and damps down to motionlessness. Forced vibration is when a disturbance is applied to a mechanical system. The disturbance can be a periodic and steady-state input, a transient input, the periodic input can be a harmonic or a non-harmonic disturbance. Damped vibration, When the energy of a system is gradually dissipated by friction and other resistances. The vibrations gradually reduce or change in frequency or intensity or cease, Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. Alternately, a DUT is attached to the table of a shaker, Vibration testing is performed to examine the response of a device under test to a defined vibration environment. The measured response may be life, resonant frequencies or squeak. Squeak and rattle testing is performed with a type of quiet shaker that produces very low sound levels while under operation. For relatively low frequency forcing, servohydraulic shakers are used, for higher frequencies, electrodynamic shakers are used. Generally, one or more input or control points located on the DUT-side of a fixture is kept at a specified acceleration, other response points experience maximum vibration level or minimum vibration level
Vibration
–
Car Suspension: designing vibration control is undertaken as part of
acoustic,
automotive or
mechanical engineering.
Vibration
–
One of the possible modes of
vibration of a circular drum (see other modes).
62.
Circular motion
–
In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, the rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body, since the objects velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, in physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times, though the bodys speed is constant, its velocity is not constant, velocity, a vector quantity, depends on both the bodys speed and its direction of travel. This changing velocity indicates the presence of an acceleration, this acceleration is of constant magnitude. This acceleration is, in turn, produced by a force which is also constant in magnitude. For motion in a circle of radius r, the circumference of the circle is C = 2π r, the axis of rotation is shown as a vector ω perpendicular to the plane of the orbit and with a magnitude ω = dθ / dt. The direction of ω is chosen using the right-hand rule, likewise, the acceleration is given by a = ω × v = ω ×, which is a vector perpendicular to both ω and v of magnitude ω |v| = ω2 r and directed exactly opposite to r. In the simplest case the speed, mass and radius are constant, consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second. The speed is one metre per second, the inward acceleration is one metre per square second, v2/r. It is subject to a force of one kilogram metre per square second. The momentum of the body is one kg·m·s−1, the moment of inertia is one kg·m2. The angular momentum is one kg·m2·s−1, the kinetic energy is 1/2 joule. The circumference of the orbit is 2π metres, the period of the motion is 2π seconds per turn. It is convenient to introduce the unit vector orthogonal to u ^ R as well and it is customary to orient u ^ θ to point in the direction of travel along the orbit. The velocity is the derivative of the displacement, v → = d d t r → = d R d t u ^ R + R d u ^ R d t. Because the radius of the circle is constant, the component of the velocity is zero
Circular motion
–
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
63.
Centripetal force
–
A centripetal force is a force that makes a body follow a curved path. Its direction is orthogonal to the motion of the body. Isaac Newton described it as a force by which bodies are drawn or impelled, or in any way tend, in Newtonian mechanics, gravity provides the centripetal force responsible for astronomical orbits. One common example involving centripetal force is the case in which a body moves with uniform speed along a circular path, the centripetal force is directed at right angles to the motion and also along the radius towards the centre of the circular path. The mathematical description was derived in 1659 by the Dutch physicist Christiaan Huygens, the direction of the force is toward the center of the circle in which the object is moving, or the osculating circle. The speed in the formula is squared, so twice the speed needs four times the force, the inverse relationship with the radius of curvature shows that half the radial distance requires twice the force. Expressed using the orbital period T for one revolution of the circle, the rope example is an example involving a pull force. The centripetal force can also be supplied as a push force, newtons idea of a centripetal force corresponds to what is nowadays referred to as a central force. Another example of centripetal force arises in the helix that is traced out when a particle moves in a uniform magnetic field in the absence of other external forces. In this case, the force is the centripetal force that acts towards the helix axis. Below are three examples of increasing complexity, with derivations of the formulas governing velocity and acceleration, uniform circular motion refers to the case of constant rate of rotation. Here are two approaches to describing this case, assume uniform circular motion, which requires three things. The object moves only on a circle, the radius of the circle r does not change in time. The object moves with constant angular velocity ω around the circle, therefore, θ = ω t where t is time. Now find the velocity v and acceleration a of the motion by taking derivatives of position with respect to time, consequently, a = − ω2 r. negative shows that the acceleration is pointed towards the center of the circle, hence it is called centripetal. While objects naturally follow a path, this centripetal acceleration describes the circular motion path caused by a centripetal force. The image at right shows the relationships for uniform circular motion. In this subsection, dθ/dt is assumed constant, independent of time, consequently, d r d t = lim Δ t →0 r − r Δ t = d ℓ d t
Centripetal force
–
A body experiencing
uniform circular motion requires a centripetal force, towards the axis as shown, to maintain its circular path.
64.
Centrifugal force
–
In Newtonian mechanics, the centrifugal force is an inertial force directed away from the axis of rotation that appears to act on all objects when viewed in a rotating reference frame. When they are analyzed in a coordinate system. The term has also been used for the force that is a reaction to a centripetal force. The centrifugal force is an outward force apparent in a reference frame. All measurements of position and velocity must be relative to some frame of reference. An inertial frame of reference is one that is not accelerating, the use of an inertial frame of reference, which will be the case for all elementary calculations, is often not explicitly stated but may generally be assumed unless stated otherwise. In terms of a frame of reference, the centrifugal force does not exist. All calculations can be performed using only Newtons laws of motion, in its current usage the term centrifugal force has no meaning in an inertial frame. In an inertial frame, an object that has no acting on it travels in a straight line. When measurements are made with respect to a reference frame, however. If it is desired to apply Newtons laws in the frame, it is necessary to introduce new, fictitious. Consider a stone being whirled round on a string, in a horizontal plane, the only real force acting on the stone in the horizontal plane is the tension in the string. There are no forces acting on the stone so there is a net force on the stone in the horizontal plane. In an inertial frame of reference, were it not for this net force acting on the stone, in order to keep the stone moving in a circular path, this force, known as the centripetal force, must be continuously applied to the stone. As soon as it is removed the stone moves in a straight line, in this inertial frame, the concept of centrifugal force is not required as all motion can be properly described using only real forces and Newtons laws of motion. In a frame of reference rotating with the stone around the axis as the stone. However, the tension in the string is still acting on the stone, if Newtons laws were applied in their usual form, the stone would accelerate in the direction of the net applied force, towards the axis of rotation, which it does not do. With this new the net force on the stone is zero, with the addition of this extra inertial or fictitious force Newtons laws can be applied in the rotating frame as if it were an inertial frame
Centrifugal force
–
The interface of two
immiscible liquids rotating around a vertical axis is an upward-opening circular paraboloid.
65.
William Rowan Hamilton
–
Sir William Rowan Hamilton PRIA FRSE was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and his best known contribution to mathematical physics is the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the study of classical field theories such as electromagnetism. In pure mathematics, he is best known as the inventor of quaternions, Hamilton is said to have shown immense talent at a very early age. Astronomer Bishop Dr. John Brinkley remarked of the 18-year-old Hamilton, This young man, I do not say will be, but is, Hamilton also invented icosian calculus, which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once. Hamilton was the fourth of nine born to Sarah Hutton and Archibald Hamilton. Hamiltons father, who was from Dunboyne, worked as a solicitor, by the age of three, Hamilton had been sent to live with his uncle James Hamilton, a graduate of Trinity College who ran a school in Talbots Castle in Trim, Co. His uncle soon discovered that Hamilton had an ability to learn languages. At the age of seven he had made very considerable progress in Hebrew. These included the classical and modern European languages, and Persian, Arabic, Hindustani, Sanskrit, in September 1813 the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, an older than Hamilton. The two were pitted against each other in a mental arithmetic contest with Colburn emerging the clear victor, in reaction to his defeat, Hamilton dedicated less time to studying languages and more time to studying mathematics. Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College, Dublin, which he entered at age 18. He studied both classics and mathematics, and was appointed Professor of Astronomy in 1827, prior to his taking up residence at Dunsink Observatory where he spent the rest of his life. Hamilton made important contributions to optics and to classical mechanics and his first discovery was in an early paper that he communicated in 1823 to Dr. Brinkley, who presented it under the title of Caustics in 1824 to the Royal Irish Academy. It was referred as usual to a committee, while their report acknowledged its novelty and value, they recommended further development and simplification before publication. Between 1825 and 1828 the paper grew to an immense size, but it also became more intelligible, and the features of the new method were now easily to be seen. Until this period Hamilton himself seems not to have fully understood either the nature or importance of optics and he proposed for it when he first predicted its existence in the third supplement to his Systems of Rays, read in 1832
William Rowan Hamilton
–
Quaternion Plaque on
Broom Bridge
William Rowan Hamilton
–
William Rowan Hamilton (1805–1865)
William Rowan Hamilton
–
Irish commemorative coin celebrating the 200th Anniversary of his birth.
66.
Volume
–
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
Volume
–
A
measuring cup can be used to measure volumes of
liquids. This cup measures volume in units of
cups,
fluid ounces, and
millilitres.
67.
Absolute value
–
In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a x, |x| = −x for a negative x. For example, the value of 3 is 3. The absolute value of a number may be thought of as its distance from zero, generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, a value is also defined for the complex numbers. The absolute value is related to the notions of magnitude, distance. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English, the notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude, in programming languages and computational software packages, the absolute value of x is generally represented by abs, or a similar expression. Thus, care must be taken to interpret vertical bars as an absolute value sign only when the argument is an object for which the notion of an absolute value is defined. For any real number x the value or modulus of x is denoted by |x| and is defined as | x | = { x, if x ≥0 − x. As can be seen from the definition, the absolute value of x is always either positive or zero. Indeed, the notion of a distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Since the square root notation without sign represents the square root. This identity is used as a definition of absolute value of real numbers. The absolute value has the four fundamental properties, The properties given by equations - are readily apparent from the definition. To see that equation holds, choose ε from so that ε ≥0, some additional useful properties are given below. These properties are either implied by or equivalent to the properties given by equations -, for example, Absolute value is used to define the absolute difference, the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value cannot be directly generalised for a complex number
Absolute value
–
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.
68.
Determinant
–
In linear algebra, the determinant is a useful value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det, detA and it can be viewed as the scaling factor of the transformation described by the matrix. In the case of a 2 ×2 matrix, the formula for the determinant. Each determinant of a 2 ×2 matrix in this equation is called a minor of the matrix A, the same sort of procedure can be used to find the determinant of a 4 ×4 matrix, the determinant of a 5 ×5 matrix, and so forth. The use of determinants in calculus includes the Jacobian determinant in the change of rule for integrals of functions of several variables. Determinants are also used to define the characteristic polynomial of a matrix, in analytical geometry, determinants express the signed n-dimensional volumes of n-dimensional parallelepipeds. Sometimes, determinants are used merely as a notation for expressions that would otherwise be unwieldy to write down. When the entries of the matrix are taken from a field, it can be proven that any matrix has an inverse if. There are various equivalent ways to define the determinant of a square matrix A, i. e. one with the number of rows. Another way to define the determinant is expressed in terms of the columns of the matrix and these properties mean that the determinant is an alternating multilinear function of the columns that maps the identity matrix to the underlying unit scalar. These suffice to uniquely calculate the determinant of any square matrix, provided the underlying scalars form a field, the definition below shows that such a function exists, and it can be shown to be unique. Assume A is a matrix with n rows and n columns. The entries can be numbers or expressions, the definition of the determinant depends only on the fact that they can be added and multiplied together in a commutative manner. The determinant of a 2 ×2 matrix is defined by | a b c d | = a d − b c. If the matrix entries are numbers, the matrix A can be used to represent two linear maps, one that maps the standard basis vectors to the rows of A. In either case, the images of the vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the rows of the matrix is the one with vertices at. The absolute value of ad − bc is the area of the parallelogram, the absolute value of the determinant together with the sign becomes the oriented area of the parallelogram
Determinant
–
The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.
69.
Bachet
–
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, W. Rouse Balls Mathematical Recreations and Essays. He also wrote Les éléments arithmétiques, which exists in manuscript, and it was this very translation in which Fermat wrote his famous margin note claiming that he had a proof of Fermats last theorem. The same text renders Diophantus term παρισὀτης as adaequalitat, which became Fermats technique of adequality, Bachet was the earliest writer who discussed the solution of indeterminate equations by means of continued fractions. He also did work in theory and found a method of constructing magic squares. Some credible sources also name him the founder of the Bézouts identity, for a year in 1601 Bachet was a member of the Jesuit Order. He lived a life in Bourg-en-Bresse and married in 1612. He was elected member of the Académie française in 1635, Diophantus Alexandrinus, Pierre de Fermat, Claude Gaspard Bachet de Meziriac, Diophanti Alexandrini Arithmeticorum libri 6, et De numeris multangulis liber unus. C G Bacheti et observationibus P de Fermat, acc. doctrinae analyticae inventum novum, coll
Bachet
–
Claude-Gaspard Bachet
Bachet
–
Title page of the 1621 edition of
Diophantus ' Arithmetica, translated into
Latin by Claude Gaspard Bachet de Méziriac.
70.
List of important publications in mathematics
–
This is a list of important publications in mathematics, organized by field. Baudhayana Believed to have written around the 8th century BC. It laid the foundations of Indian mathematics and was influential in South Asia and its surrounding regions, the Nine Chapters on the Mathematical Art from the 10th–2nd century BCE. Contains the earliest description of Gaussian elimination for solving system of equations, it also contains method for finding square root. Liu Hui Contains the application of right triangles for survey of depth or height of distant objects. Sunzi Contains the earlist description of Chinese remainder theorem, Aryabhata Aryabhata introduced the method known as Modus Indorum or the method of the Indians that has become our algebra today. This algebra came along with the Hindu Number system to Arabia, the text contains 33 verses covering mensuration, arithmetic and geometric progressions, gnomon / shadows, simple, quadratic, simultaneous, and indeterminate equations. It also gave the standard algorithm for solving first-order diophantine equations. Jigu Suanjing This book by Tang dynasty mathematician Wang Xiaotong Contains the worlds earliest third order equation, Brahmagupta Contained rules for manipulating both negative and positive numbers, a method for computing square roots, and general methods of solving linear and some quadratic equations. Muhammad ibn Mūsā al-Khwārizmī The first book on the algebraic solutions of linear. The book is considered to be the foundation of modern algebra, the word algebra itself is derived from the al-Jabr in the title of the book. One of the treatises on mathematics by Bhāskara II provides the solution for indeterminate equations of 1st. Liu Yi Contains the earliest invention of 4th order polynomial equation, qin Jiushao This 13th century book contains the earliest complete solution of 19th century Horners method of solving high order polynomial equations. It also contains a complete solution of Chinese remainder theorem, which predates Euler, li Zhi Contains the application of high order polynomial equation in solving complex geometry problems. Zhu Shijie Contains the method of establishing system of high order polynomial equations of up to four unknowns, leonhard Euler Also known as Elements of Algebra, Eulers textbook on elementary algebra is one of the first to set out algebra in the modern form we would recognize today. The first volume deals with equations, while the second part deals with Diophantine equations. The last section contains a proof of Fermats Last Theorem for the case n =3, Carl Friedrich Gauss Gauss doctoral dissertation, which contained a widely accepted but incomplete proof of the fundamental theorem of algebra. Joseph Louis Lagrange The title means Reflections on the solutions of equations
List of important publications in mathematics
–
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
–
Institutiones calculi differentialis
71.
Colin Maclaurin
–
Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a case of the Taylor series, is named after him. Owing to changes in orthography since that time, his surname is alternatively written MacLaurin, Maclaurin was born in Kilmodan, Argyll. His father, Reverend and Minister of Glendaruel John Maclaurin, died when Maclaurin was in infancy and he was then educated under the care of his uncle, the Reverend Daniel Maclaurin, minister of Kilfinan. At eleven, Maclaurin entered the University of Glasgow and this record as the worlds youngest professor endured until March 2008, when the record was officially given to Alia Sabur. In the vacations of 1719 and 1721, Maclaurin went to London, where he acquainted with Sir Isaac Newton, Dr Benjamin Hoadly, Samuel Clarke, Martin Folkes. He was admitted a member of the Royal Society, in 1722, having provided a substitute for his class at Aberdeen, he traveled on the Continent as tutor to George Hume, the son of Alexander Hume, 2nd Earl of Marchmont. During their time in Lorraine, he wrote his essay on the percussion of bodies, upon the death of his pupil at Montpellier, Maclaurin returned to Aberdeen. In 1725 Maclaurin was appointed deputy to the professor at Edinburgh, James Gregory. On 3 November of that year Maclaurin succeeded Gregory, and 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 his salary himself. Maclaurin used Taylor series to characterize maxima, minima, and points of inflection for infinitely differentiable functions in his Treatise of Fluxions. Maclaurin attributed the series to Taylor, though the series was known before to Newton and Gregory, nevertheless, Maclaurin received credit for his use of the series, and the Taylor series expanded around 0 is sometimes known as the Maclaurin series. Maclaurin also made significant contributions to the attraction of ellipsoids. Clairaut, Euler, Laplace, Legendre, Poisson and Gauss, Maclaurin showed that an oblate spheroid was a possible equilibrium in Newtons theory of gravity. The subject continues to be of scientific interest, and Nobel Laureate Subramanyan Chandrasekhar dedicated a chapter of his book Ellipsoidal Figures of Equilibrium to Maclaurin spheroids, independently from Euler and using the same methods, Maclaurin discovered the Euler–Maclaurin formula. He used it to sum powers of arithmetic progressions, derive Stirlings formula, Maclaurin contributed to the study of elliptic integrals, reducing many intractable integrals to problems of finding arcs for hyperbolas. His work was continued by dAlembert and Euler, who gave a more concise approach and this publication preceded by two years Cramers publication of a generalization of the rule to n unknowns, now commonly known as Cramers rule. In 1733, Maclaurin married Anne Stewart, the daughter of Walter Stewart, Maclaurin actively opposed the Jacobite Rebellion of 1745 and superintended the operations necessary for the defence of Edinburgh against the Highland army
Colin Maclaurin
–
Colin Maclaurin (1698–1746)
Colin Maclaurin
–
Memorial, Greyfriars Kirkyard, Edinburgh
72.
Urbain Le Verrier
–
Urbain Jean Joseph Le Verrier was a French mathematician who specialized in celestial mechanics and is best known for predicting the existence and position of Neptune using only mathematics. The calculations were made to explain discrepancies with Uranuss orbit and the laws of Kepler, Le Verrier sent the coordinates to Johann Gottfried Galle in Berlin, asking him to verify. Galle found Neptune in the night he received Le Verriers letter. The discovery of Neptune is widely regarded as a validation of celestial mechanics. Le Verrier was born at Saint-Lô, Manche, France, and he briefly studied chemistry under Gay-Lussac, writing papers on the combinations of phosphorus and hydrogen, and phosphorus and oxygen. He then switched to astronomy, particularly celestial mechanics, and accepted a job at the Paris Observatory and he spent most of his professional life there, and eventually became that institutions Director, from 1854 to 1870 and again from 1873 to 1877. In 1846, Le Verrier became a member of the French Academy of Sciences, Le Verriers name is one of the 72 names inscribed on the Eiffel Tower. Le Verriers first work in astronomy was presented to the Académie des Sciences in September 1839 and this work addressed the then most-important question in astronomy, the stability of the Solar System, first investigated by Laplace. He was able to some important limits on the motions of the system. From 1844 to 1847, Le Verrier published a series of works on periodic comets, in particular those of Lexell, Faye and DeVico. He was able to some interesting interactions with the planet Jupiter. Le Verriers most famous achievement is his prediction of the existence of the unknown planet Neptune, using only mathematics. At the same time, but unknown to Le Verrier, similar calculations were made by John Couch Adams in England, Le Verrier transmitted his own prediction by 18 September in a letter to Johann Galle of the Berlin Observatory. There was, and to an extent still is, controversy over the apportionment of credit for the discovery, there is no ambiguity to the discovery claims of Le Verrier, Galle, and dArrest. Adamss work was earlier than Le Verriers but was finished later and was unrelated to the actual discovery. Not even the briefest account of Adamss predicted orbital elements was published more than a month after Berlins visual confirmation. Galle, so that the facts stated above cannot detract, in the slightest degree, early in the 19th century, the methods of predicting the motions of the planets were somewhat scattered, having been developed over decades by many different researchers. In 1847, Le Verrier took on the task to, embrace in a single work the entire planetary system, put everything in harmony if possible, otherwise, declare with certainty that there are as yet unknown causes of perturbations
Urbain Le Verrier
–
Urbain Le Verrier
Urbain Le Verrier
–
Signature of M. LeVerrier
Urbain Le Verrier
–
The grave of Urbain Le Verrier.
73.
Mechanics
–
Mechanics is an area of science concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. The scientific discipline has its origins in Ancient Greece with the writings of Aristotle, during the early modern period, scientists such as Khayaam, Galileo, Kepler, and Newton, laid the foundation for what is now known as classical mechanics. It is a branch of physics that deals with particles that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as a branch of science which deals with the motion of, historically, classical mechanics came first, while quantum mechanics is a comparatively recent invention. Classical mechanics originated with Isaac Newtons laws of motion in Philosophiæ Naturalis Principia Mathematica, both are commonly held to constitute the most certain knowledge that exists about physical nature. Classical mechanics has especially often been viewed as a model for other so-called exact sciences, essential in this respect is the relentless use of mathematics in theories, as well as the decisive role played by experiment in generating and testing them. Quantum mechanics is of a scope, as it encompasses classical mechanics as a sub-discipline which applies under certain restricted circumstances. According to the principle, there is no contradiction or conflict between the two subjects, each simply pertains to specific situations. The correspondence principle states that the behavior of systems described by quantum theories reproduces classical physics in the limit of quantum numbers. Quantum mechanics has superseded classical mechanics at the level and is indispensable for the explanation and prediction of processes at the molecular, atomic. However, for macroscopic processes classical mechanics is able to solve problems which are difficult in quantum mechanics and hence remains useful. Modern descriptions of such behavior begin with a definition of such quantities as displacement, time, velocity, acceleration, mass. Until about 400 years ago, however, motion was explained from a different point of view. He showed that the speed of falling objects increases steadily during the time of their fall and this acceleration is the same for heavy objects as for light ones, provided air friction is discounted. The English mathematician and physicist Isaac Newton improved this analysis by defining force and mass, for objects traveling at speeds close to the speed of light, Newton’s laws were superseded by Albert Einstein’s theory of relativity. For atomic and subatomic particles, Newton’s laws were superseded by quantum theory, for everyday phenomena, however, Newton’s three laws of motion remain the cornerstone of dynamics, which is the study of what causes motion. In analogy to the distinction between quantum and classical mechanics, Einsteins general and special theories of relativity have expanded the scope of Newton, the differences between relativistic and Newtonian mechanics become significant and even dominant as the velocity of a massive body approaches the speed of light. Relativistic corrections are also needed for quantum mechanics, although general relativity has not been integrated, the two theories remain incompatible, a hurdle which must be overcome in developing a theory of everything
Mechanics
–
Arabic Machine Manuscript. Unknown date (at a guess: 16th to 19th centuries).
74.
Pure mathematics
–
Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. Even though the pure and applied viewpoints are distinct philosophical positions, in there is much overlap in the activity of pure. To develop accurate models for describing the world, many applied mathematicians draw on tools. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research, ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between arithmetic, now called number theory, and logistic, now called arithmetic. Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, the term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, in the following years, specialisation and professionalisation started to make a rift more apparent. At the start of the twentieth century mathematicians took up the axiomatic method, in fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved, Pure mathematician became a recognized vocation, achievable through training. One central concept in mathematics is the idea of generality. One can use generality to avoid duplication of effort, proving a general instead of having to prove separate cases independently. Generality can facilitate connections between different branches of mathematics, category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math. Generalitys impact on intuition is both dependent on the subject and a matter of preference or learning style. Often generality is seen as a hindrance to intuition, although it can function as an aid to it. Each of these branches of abstract mathematics have many sub-specialties. A steep rise in abstraction was seen mid 20th century, in practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, the point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central
Pure mathematics
–
An illustration of the
Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.
75.
Carl Gustav Jakob Jacobi
–
Carl Gustav Jacob Jacobi was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory. His name is written as Carolus Gustavus Iacobus Iacobi in his Latin books. 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 four children of banker Simon Jacobi and 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, 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. As a result of the education received from his uncle, as well as his own remarkable abilities. However, as the University was not accepting students younger than 16 years old and he used this time to advance his knowledge, showing interest in all subjects, including Latin and Greek, philology, history and mathematics. During this period he 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 professors attention with his talent. Jacobi did not follow a lot of mathematics classes at the University, 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 interests and he chose to devote all his attention to mathematics. In the same year he qualified to teach secondary school and was offered a position at the Joachimsthal Gymnasium in Berlin. Jacobi decided instead to continue to work towards a University position, in 1825 he obtained the degree of Doctor of Philosophy with a dissertation on the partial fraction decomposition of rational fractions defended before a commission led by Enno Dirksen. He followed immediately with his Habilitation and at the time converted to Christianity. Now qualifying for teaching University classes, the 21-year-old Jacobi lectured in 1825/26 on the theory of curves and surfaces at the University of Berlin, in 1827 he became a professor and in 1829, a tenured professor of mathematics at Königsberg University, and held the chair until 1842. Jacobi suffered a breakdown from overwork in 1843 and he then visited Italy for a few months to regain his health. On his return he moved to Berlin, where he lived as a royal pensioner until his death, during the Revolution of 1848 Jacobi was politically involved and unsuccessfully presented his parliamentary candidature on behalf of a Liberal club. This led, after the suppression of the revolution, to his royal grant being cut off – but his fame, in 1836, he had been elected a foreign member of the Royal Swedish Academy of Sciences
Carl Gustav Jakob Jacobi
–
Carl Gustav Jacob Jacobi
76.
Planetary motion
–
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet about a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating path around a body, to a close approximation, planets and satellites follow elliptical orbits, with the central mass being orbited at a focal point of the ellipse, as described by Keplers laws of planetary motion. For ease of calculation, in most situations orbital motion is adequately approximated by Newtonian Mechanics, historically, the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres. This model posited the existence of perfect moving spheres or rings to which the stars and it assumed the heavens were fixed apart from the motion of the spheres, and was developed without any understanding of gravity. After the planets motions were accurately measured, theoretical mechanisms such as deferent. Originally geocentric it was modified by Copernicus to place the sun at the centre to help simplify the model, the model was further challenged during the 16th century, as comets were observed traversing the spheres. The basis for the understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. Second, he found that the speed of each planet is not constant, as had previously been thought. Third, Kepler found a relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 AU distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter,5. 23/11.862, is equal to that for Venus,0. 7233/0.6152. Idealised orbits meeting these rules are known as Kepler orbits, isaac Newton demonstrated that Keplers laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections. Newton showed that, for a pair of bodies, the sizes are in inverse proportion to their masses. Where one body is more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, in a dramatic vindication of classical mechanics, in 1846 le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits, in relativity theory, orbits follow geodesic trajectories which are usually approximated very well by the Newtonian predictions but the differences are measurable. Essentially all the evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy
Planetary motion
–
The
International Space Station orbits above
Earth.
Planetary motion
–
Planetary orbits
Planetary motion
Planetary motion
–
Conic sections describe the possible orbits (yellow) of small objects around the earth. A projection of these orbits onto the gravitational potential (blue) of the earth makes it possible to determine the orbital energy at each point in space.
77.
Royal Swedish Academy of Sciences
–
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. Every year the Academy awards the Nobel Prizes in Physics and Chemistry, the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, the Crafoord Prize, the Academy has elected about 1.700 Swedish and 1.200 foreign members since it was founded in 1739. Hansson, appointed from 1 July 2015 The transactions of the Academy were published as its main series between 1739 and 1974, in parallel, other major series have appeared and gone, Öfversigt af Kungl. These lasted into the 1860s, when they were replaced by the single Bihang series, further restructuring of their topics occurred in 1949 and 1974. The purpose of the academy was to focus on practically useful knowledge, the academy was intended to be different from the Royal Society of Sciences in Uppsala, which had been founded in 1719 and published in Latin. The location close to the activities in Swedens capital was also intentional. The academy was modeled after the Royal Society of London and Academie Royale des Sciences in Paris, France, members of the Royal Swedish Academy of Sciences Official website Royal Swedish Academy of Sciences video site
Royal Swedish Academy of Sciences
–
Main building of the Royal Swedish Academy of Sciences in
Stockholm.
Royal Swedish Academy of Sciences
–
Kongl. Svenska Vetenskaps-Academiens handlingar, volume XI (1750).
Royal Swedish Academy of Sciences
–
The Royal Swedish Academy of Sciences
78.
Legion of Honour
–
The Legion of Honour, full name National Order of the Legion of Honour, is the highest French order of merit for military and civil merits, established 1802 by Napoléon Bonaparte. The order is divided into five degrees of increasing distinction, Chevalier, Officier, Commandeur, Grand Officier and Grand-Croix. The orders motto is Honneur et Patrie and its seat is the Palais de la Légion dHonneur next to the Musée dOrsay, in the French Revolution, all French orders of chivalry were abolished, and replaced with Weapons of Honour. The Légion however did use the organization of old French orders of chivalry, the badges of the legion also bear a resemblance to the Ordre de Saint-Louis, which also used a red ribbon. Napoleon originally created this to ensure political loyalty, the organization would be used as a facade to give political favours, gifts, and concessions. The Légion was loosely patterned after a Roman legion, with legionaries, officers, commanders, regional cohorts, the highest rank was not a grand cross but a Grand Aigle, a rank that wore all the insignia common to grand crosses. The members were paid, the highest of them extremely generously,5,000 francs to an officier,2,000 francs to a commandeur,1,000 francs to an officier,250 francs to a légionnaire. Napoleon famously declared, You call these baubles, well, it is with baubles that men are led, do you think that you would be able to make men fight by reasoning. That is good only for the scholar in his study, the soldier needs glory, distinctions, rewards. This has been quoted as It is with such baubles that men are led. The order was the first modern order of merit, under the monarchy, such orders were often limited to Roman Catholics, and all knights had to be noblemen. The military decorations were the perks of the officers, the Légion, however, was open to men of all ranks and professions—only merit or bravery counted. The new legionnaire had to be sworn in the Légion and it is noteworthy that all previous orders were crosses or shared a clear Christian background, whereas the Légion is a secular institution. The jewel of the Légion has five arms, in a decree issued on the 10 Pluviôse XIII, a grand decoration was instituted. This decoration, a cross on a sash and a silver star with an eagle, symbol of the Napoleonic Empire, became known as the Grand Aigle. After Napoleon crowned himself Emperor of the French in 1804 and established the Napoleonic nobility in 1808, the title was made hereditary after three generations of grantees. Napoleon had dispensed 15 golden collars of the legion among his family and this collar was abolished in 1815. The Légion dhonneur was prominent and visible in the French Empire, the Emperor always wore it and the fashion of the time allowed for decorations to be worn most of the time
Legion of Honour
–
Order's
streamer
Legion of Honour
Legion of Honour
–
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
–
First Légion d'Honneur investiture, 15 July 1804, at
Saint-Louis des Invalides by
Jean-Baptiste Debret (1812)
79.
Count of the Empire
–
Napoleon I created titles of nobility to institute a stable elite in the First French Empire, after the instability resulting from the French Revolution. Like many others, both before and since, Napoleon found that the ability to confer titles was also a tool of patronage which cost the state little treasure. The Grand Dignitaries of the Empire ranked, regardless of noble title, enoblement started in 1804 with the creation of the princely title for members of Napoleons imperial family. In 1806 ducal titles were created and in 1808 those of count, baron, Napoleon founded the concept of nobility of Empire by an imperial decree on 1 March 1808. The purpose of creation was to amalgamate the old nobility. This step, which aimed at the introduction of a elite, is fully in line with the creation of the Legion of Honour. A council of the seals and the titles was created and charged with establishing armorial bearings. These creations are to be distinguished from an order such as the Order of the Bath. These titles of nobility did not have any true privileges, with two exceptions, right of bearing, the lands granted with the title were held in a majorat. This nobility is essentially a nobility of service, to a large extent made up of soldiers, some civil servants, there were 239 remaining families belonging to the First Empire nobility in 1975. Of those, perhaps about 135 were titled, only one princely title and seven ducal titles remain today. Along with a new system of titles of nobility, the First French Empire also introduced a new system of heraldry, Napoleonic heraldry was based on traditional heraldry but was characterised by a stronger sense of hierarchy. It employed a system of additional marks in the shield to indicate official functions and positions. Another notable difference from traditional heraldry was the toques, which replaced coronets, the toques were surmounted by ostrich feathers, dukes had 7, counts had 5, barons 3, knights 1. The number of lambrequins was also regulated,3,2,1, as many grantees were new men, and the arms often alluded to their life or specific actions, many new or unusual charges were also introduced. The most characteristic mark of Napoleonic heraldry was the additional marks in the shield to indicate official functions and positions and these came in the form of quarters in various colours, and would be differenced further by marks of the specific rank or function. The said marks of the rank or function as used by Barons. A decree of 3 March 1810 states, The name, arms and this provision applied only to the bearers of Napoleonic titles
Count of the Empire
–
Imperial coat of arms
80.
Satellites of Jupiter
–
There are 67 known moons of Jupiter. This gives Jupiter the largest number of moons with reasonably stable orbits of any planet in the Solar System. The Galilean moons are by far the largest and most massive objects to orbit Jupiter, with the remaining 63 moons, of Jupiters moons, eight are regular satellites with prograde and nearly circular orbits that are not greatly inclined with respect to Jupiters equatorial plane. The Galilean satellites are nearly spherical in shape due to their planetary mass, the other four regular satellites are much smaller and closer to Jupiter, these serve as sources of the dust that makes up Jupiters rings. The remainder of Jupiters moons are irregular satellites whose prograde and retrograde orbits are farther from Jupiter and have high inclinations. These moons were probably captured by Jupiter from solar orbits, sixteen irregular satellites have been discovered since 2003 and have not yet been named. The physical and orbital characteristics of the moons vary widely, all other Jovian moons are less than 250 kilometres in diameter, with most barely exceeding 5 kilometres. Their orbital shapes range from perfectly circular to highly eccentric and inclined. Orbital periods range from seven hours, to three thousand times more. Jupiters regular satellites are believed to have formed from a circumplanetary disk and they may be the remnants of a score of Galilean-mass satellites that formed early in Jupiters history. Simulations suggest that, while the disk had a high mass at any given moment. However, only 2% the proto-disk mass of Jupiter is required to explain the existing satellites, thus there may have been several generations of Galilean-mass satellites in Jupiters early history. Each generation of moons might have spiraled into Jupiter, due to drag from the disk, by the time the present generation formed, the disk had thinned to the point that it no longer greatly interfered with the moons orbits. The current Galilean moons were still affected, falling into and being protected by an orbital resonance with each other, which still exists for Io, Europa. Ganymedes larger mass means that it would have migrated inward at a faster rate than Europa or Io, many broke up due to the mechanical stresses of capture, or afterward by collisions with other small bodies, producing the moons we see today. The first claimed observation of one of Jupiters moons is that of Chinese astronomer Gan De around 364 BC, however, the first certain observations of Jupiters satellites were those of Galileo Galilei in 1609. By January 1610, he had sighted the four massive Galilean moons with his 30× magnification telescope, no additional satellites were discovered until E. E. Barnard observed Amalthea in 1892. With the aid of photography, further discoveries followed quickly over the course of the twentieth century
Satellites of Jupiter
–
A montage of Jupiter and its four largest moons (distance and sizes not to scale)
Satellites of Jupiter
–
Jupiter and the
Galilean moons through a 10" (25 cm)
Meade LX200 telescope
Satellites of Jupiter
–
The Galilean moons. From left to right, in order of increasing distance from Jupiter:
Io,
Europa,
Ganymede,
Callisto
Satellites of Jupiter
81.
List of the 72 names on the Eiffel Tower
–
On the Eiffel Tower, seventy-two names of French scientists, engineers, and mathematicians are engraved in recognition of their contributions. Gustave Eiffel chose this invocation of science because of his concern over the protests against the tower, the engravings are found on the sides of the tower under the first balcony. The Tower is owned by the city of Paris, the letters were originally painted in gold and are about 60 cm high. The repainting of 2010/2011 restored the letters to their gold colour. There are also names of engineers who helped build the tower and design its architecture on the top of the tower on a plaque. The list is split in four parts, the list has been criticized for excluding the name of Sophie Germain, a noted French mathematician whose work on the theory of elasticity was used in the construction of the tower itself. In 1913 John Augustine Zahm suggested that Germain was excluded because she was a woman,14 hydraulic engineers and scholars are listed on the Eiffel Tower. Eiffel acknowledged most of the scientists in the field. Henri Philibert Gaspard Darcy is missing, some of his work did not come into use until the 20th century. Also missing are Antoine Chézy, who was famous, Joseph Valentin Boussinesq. Also missing is the mathematician Evariste Galois, le Panthéon scientifique de la tour Eiffel, histoire des origines de la construction de la Tour. Le Panthéon scientifique de la tour Eiffel, histoire des origines de la de la Tour. Media related to 72 names on the Eiffel Tower at Wikimedia Commons Paris streets named for the 72 scientists
List of the 72 names on the Eiffel Tower
–
The location of the names on the tower
List of the 72 names on the Eiffel Tower
–
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
–
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
–
Jamin,
Gay-Lussac,
Fizeau,
Schneider,
Le Chatelier,
Berthier,
Barral,
De Dion,
Goüin,
Jousselin,
Broca,
Becquerel,
Coriolis,
Cail,
Triger,
Giffard,
Perrier,
Sturm
82.
Eiffel Tower
–
The Eiffel Tower is a wrought iron lattice tower on the Champ de Mars in Paris, France. It is named after the engineer Gustave Eiffel, whose company designed, the Eiffel Tower is the most-visited paid monument in the world,6.91 million people ascended it in 2015. The tower is 324 metres tall, about the height as an 81-storey building. Its base is square, measuring 125 metres on each side, due to the addition of a broadcasting aerial at the top of the tower in 1957, it is now taller than the Chrysler Building by 5.2 metres. Excluding transmitters, the Eiffel Tower is the second-tallest structure in France after the Millau Viaduct, the tower has three levels for visitors, with restaurants on the first and second levels. The top levels upper platform is 276 m above the ground – the highest observation deck accessible to the public in the European Union, tickets can be purchased to ascend by stairs or lift to the first and second levels. The climb from ground level to the first level is over 300 steps, although there is a staircase to the top level, it is usually only accessible by lift. Eiffel openly acknowledged that inspiration for a tower came from the Latting Observatory built in New York City in 1853, sauvestre added decorative arches to the base of the tower, a glass pavilion to the first level, and other embellishments. Little progress was made until 1886, when Jules Grévy was re-elected as president of France and Édouard Lockroy was appointed as minister for trade. On 12 May, a commission was set up to examine Eiffels scheme and its rivals, which, after some debate about the exact location of the tower, a contract was signed on 8 January 1887. Eiffel was to all income from the commercial exploitation of the tower during the exhibition. He later established a company to manage the tower, putting up half the necessary capital himself. The proposed tower had been a subject of controversy, drawing criticism from those who did not believe it was feasible and these objections were an expression of a long-standing debate in France about the relationship between architecture and engineering. And for twenty years … we shall see stretching like a blot of ink the hateful shadow of the column of bolted sheet metal. Gustave Eiffel responded to criticisms by comparing his tower to the Egyptian pyramids. Will it not also be grandiose in its way, and why would something admirable in Egypt become hideous and ridiculous in Paris. Indeed, Garnier was a member of the Tower Commission that had examined the various proposals, some of the protesters changed their minds when the tower was built, others remained unconvinced. Guy de Maupassant supposedly ate lunch in the restaurant every day because it was the one place in Paris where the tower was not visible
Eiffel Tower
–
The Eiffel Tower as seen from the
Champ de Mars
Eiffel Tower
–
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
–
A
calligram by
Guillaume Apollinaire
83.
Lunar crater
–
Lunar craters are impact craters on Earths Moon. The Moons surface has many craters, almost all of which were formed by impacts, the word crater was adopted by Galileo from the Greek word for vessel -. Galileo built his first telescope in late 1609, and turned it to the Moon for the first time on November 30,1609. He discovered that, contrary to general opinion at that time, the Moon was not a perfect sphere, scientific opinion as to the origin of craters swung back and forth over the ensuing centuries. The formation of new craters is studied in the lunar impact monitoring program at NASA, the biggest recorded creation was caused by an impact recorded on March 17,2013. Visible to the eye, the impact is believed to be from an approximately 40 kg meteoroid striking the surface at a speed of 90,000 km/h. Because of the Moons lack of water, and atmosphere, or tectonic plates, there is little erosion, the age of large craters is determined by the number of smaller craters contained within it, older craters generally accumulating more small, contained craters. The smallest craters found have been microscopic in size, found in rocks returned to Earth from the Moon, the largest crater called such is about 290 kilometres across in diameter, located near the lunar South Pole. However, it is believed many of the lunar maria were formed by giant impacts. In 1978, Chuck Wood and Leif Andersson of the Lunar & Planetary Lab devised a system of categorization of lunar impact craters and they used a sampling of craters that were relatively unmodified by subsequent impacts, then grouped the results into five broad categories. These successfully accounted for about 99% of all lunar impact craters, the LPC Crater Types were as follows, ALC — small, cup-shaped craters with a diameter of about 10 km or less, and no central floor. The archetype for this category is Albategnius C, BIO — similar to an ALC, but with small, flat floors. Typical diameter is about 15 km, the lunar crater archetype is Biot. SOS — the interior floor is wide and flat, with no central peak, the inner walls are not terraced. The diameter is normally in the range of 15–25 km, TRI — these complex craters are large enough so that their inner walls have slumped to the floor. They can range in size from 15–50 km in diameter, TYC — these are larger than 50 km, with terraced inner walls and relatively flat floors. They frequently have large central peak formations, tycho is the archetype for this class. Beyond a couple of hundred kilometers diameter, the peak of the TYC class disappear
Lunar crater
–
Side view of the
Moltke crater taken from
Apollo 11.
Lunar crater
–
Webb crater, as seen from
Lunar Orbiter 1. Several smaller craters can be seen in and around Webb crater.
Lunar crater
–
Lunar craters as captured through the backyard telescope of an amateur astronomer, partially illuminated by the sun on a waning crescent moon.
Lunar crater
–
Albategnius
84.
Public domain
–
The term public domain has two senses of meaning. Anything published is out in the domain in the sense that it is available to the public. Once published, news and information in books is in the public domain, in the sense of intellectual property, works in the public domain are those whose exclusive intellectual property rights have expired, have been forfeited, or are inapplicable. Examples for works not covered by copyright which are therefore in the domain, are the formulae of Newtonian physics, cooking recipes. Examples for works actively dedicated into public domain by their authors are reference implementations of algorithms, NIHs ImageJ. The term is not normally applied to situations where the creator of a work retains residual rights, as rights are country-based and vary, a work may be subject to rights in one country and be in the public domain in another. Some rights depend on registrations on a basis, and the absence of registration in a particular country, if required. Although the term public domain did not come into use until the mid-18th century, the Romans had a large proprietary rights system where they defined many things that cannot be privately owned as res nullius, res communes, res publicae and res universitatis. The term res nullius was defined as not yet appropriated. The term res communes was defined as things that could be enjoyed by mankind, such as air, sunlight. The term res publicae referred to things that were shared by all citizens, when the first early copyright law was first established in Britain with the Statute of Anne in 1710, public domain did not appear. However, similar concepts were developed by British and French jurists in the eighteenth century, instead of public domain they used terms such as publici juris or propriété publique to describe works that were not covered by copyright law. The phrase fall in the domain can be traced to mid-nineteenth century France to describe the end of copyright term. In this historical context Paul Torremans describes copyright as a coral reef of private right jutting up from the ocean of the public domain. Because copyright law is different from country to country, Pamela Samuelson has described the public domain as being different sizes at different times in different countries. According to James Boyle this definition underlines common usage of the public domain and equates the public domain to public property. However, the usage of the public domain can be more granular. Such a definition regards work in copyright as private property subject to fair use rights, the materials that compose our cultural heritage must be free for all living to use no less than matter necessary for biological survival
Public domain
–
Newton's own copy of his
Principia, with hand-written corrections for the second edition
Public domain
–
L.H.O.O.Q. (1919). Derivative work by the
Dadaist Marcel Duchamp based on the
Mona Lisa.
85.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and 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 in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, 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 to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
International Standard Book Number
–
A 13-digit ISBN, 978-3-16-148410-0, as represented by an
EAN-13 bar code
86.
Isoperimetric
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In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume. The equality holds when S is a ball in R n, on a plane, i. e. when n =2, the isoperimetric inequality relates square of the circumference of a closed curve and the area of a plane region it encloses. Isoperimetric literally means having the same perimeter, the isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length. The closely related Didos problem asks for a region of the area bounded by a straight line. It is named after Dido, the founder and first queen of Carthage. The solution to the problem is given by a circle and was known already in Ancient Greece. However, the first mathematically rigorous proof of this fact was obtained only in the 19th century, since then, many other proofs have been found. The isoperimetric problem has been extended in multiple ways, for example, to curves on surfaces, perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will assume a symmetric round shape. Since the amount of water in a drop is fixed, surface forces the drop into a shape which minimizes the surface area of the drop. The classical isoperimetric problem dates back to antiquity, the problem can be stated as follows, Among all closed curves in the plane of fixed perimeter, which curve maximizes the area of its enclosed region. This question can be shown to be equivalent to the problem, Among all closed curves in the plane enclosing a fixed area. German astronomer and astrologer Johannes Kepler invoked the principle in discussing the morphology of the solar system. Although the circle appears to be a solution to the problem. The first progress toward the solution was made by Swiss geometer Jakob Steiner in 1838, Steiner showed that if a solution existed, then it must be the circle. Steiners proof was completed later by other mathematicians. It can further be shown that any closed curve which is not fully symmetrical can be tilted so that it encloses more area. The one shape that is convex and symmetrical is the circle, although this, in itself
Isoperimetric
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If a region is not convex, a "dent" in its boundary can be "flipped" to increase the area of the region while keeping the perimeter unchanged.
87.
University of St Andrews
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The University of St Andrews is a British public research university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland, St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy. St Andrews is made up from a variety of institutions, including three constituent colleges and 18 academic schools organised into four faculties, the university occupies historic and modern buildings located throughout the town. The academic year is divided into two terms, Martinmas and Candlemas, in term time, over one-third of the towns population is either a staff member or student of the university. It is ranked as the third best university in the United Kingdom in national league tables, the Times Higher Education World Universities Ranking names St Andrews among the worlds Top 50 universities for Social Sciences, Arts and Humanities. St Andrews has the highest student satisfaction amongst all multi-faculty universities in the United Kingdom, St Andrews has many notable alumni and affiliated faculty, including eminent mathematicians, scientists, theologians, philosophers, and politicians. Six Nobel Laureates are among St Andrews alumni and former staff, a charter of privilege was bestowed upon the society of masters and scholars by the Bishop of St Andrews, Henry Wardlaw, on 28 February 1411. Wardlaw then successfully petitioned the Avignon Pope Benedict XIII to grant the university status by issuing a series of papal bulls. King James I of Scotland confirmed the charter of the university in 1432, subsequent kings supported the university with King James V confirming privileges of the university in 1532. A college of theology and arts called St Johns College was founded in 1418 by Robert of Montrose, St Salvators College was established in 1450, by Bishop James Kennedy. St Leonards College was founded in 1511 by Archbishop Alexander Stewart, St Johns College was refounded by Cardinal James Beaton under the name St Marys College in 1538 for the study of divinity and law. Some university buildings that date from this period are still in use today, such as St Salvators Chapel, St Leonards College Chapel, at this time, the majority of the teaching was of a religious nature and was conducted by clerics associated with the cathedral. During the 17th and 18th centuries, the university had mixed fortunes and was beset by civil. He described it as pining in decay and struggling for life, in the second half of the 19th century, pressure was building upon universities to open up higher education to women. In 1876, the University Senate decided to allow women to receive an education at St Andrews at a roughly equal to the Master of Arts degree that men were able to take at the time. The scheme came to be known as the L. L. A and it required women to pass five subjects at an ordinary level and one at honours level and entitled them to hold a degree from the university. In 1889 the Universities Act made it possible to admit women to St Andrews. Agnes Forbes Blackadder became the first woman to graduate from St Andrews on the level as men in October 1894
University of St Andrews
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College Hall, within the 16th century St Mary's College building
University of St Andrews
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University of St Andrews
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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
88.
Internet Archive
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The Internet Archive is a San Francisco–based nonprofit digital library with the stated mission of universal access to all knowledge. As of October 2016, its collection topped 15 petabytes, in addition to its archiving function, the Archive is an activist organization, advocating for a free and open Internet. Its web archive, the Wayback Machine, contains over 150 billion web captures, the Archive also oversees one of the worlds largest book digitization projects. Founded by Brewster Kahle in May 1996, the Archive is a 501 nonprofit operating in the United States. It has a budget of $10 million, derived from a variety of sources, revenue from its Web crawling services, various partnerships, grants, donations. Its headquarters are in San Francisco, California, where about 30 of its 200 employees work, Most of its staff work in its book-scanning centers. The Archive has data centers in three Californian cities, San Francisco, Redwood City, and Richmond, the Archive is a member of the International Internet Preservation Consortium and was officially designated as a library by the State of California in 2007. Brewster Kahle founded the Archive in 1996 at around the time that he began the for-profit web crawling company Alexa Internet. In October 1996, the Internet Archive had begun to archive and preserve the World Wide Web in large quantities, the archived content wasnt available to the general public until 2001, when it developed the Wayback Machine. In late 1999, the Archive expanded its collections beyond the Web archive, Now the Internet Archive includes texts, audio, moving images, and software. It hosts a number of projects, the NASA Images Archive, the contract crawling service Archive-It. According to its web site, Most societies place importance on preserving artifacts of their culture, without such artifacts, civilization has no memory and no mechanism to learn from its successes and failures. Our culture now produces more and more artifacts in digital form, the Archives mission is to help preserve those artifacts and create an Internet library for researchers, historians, and scholars. In August 2012, the Archive announced that it has added BitTorrent to its file download options for over 1.3 million existing files, on November 6,2013, the Internet Archives headquarters in San Franciscos Richmond District caught fire, destroying equipment and damaging some nearby apartments. The nonprofit Archive sought donations to cover the estimated $600,000 in damage, in November 2016, Kahle announced that the Internet Archive was building the Internet Archive of Canada, a copy of the archive to be based somewhere in the country of Canada. The announcement received widespread coverage due to the implication that the decision to build an archive in a foreign country was because of the upcoming presidency of Donald Trump. Kahle was quoted as saying that on November 9th in America and it was a firm reminder that institutions like ours, built for the long-term, need to design for change. For us, it means keeping our cultural materials safe, private and it means preparing for a Web that may face greater restrictions
Internet Archive
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Since 2009, headquarters have been at 300 Funston Avenue in
San Francisco, a former
Christian Science Church
Internet Archive
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Internet Archive
Internet Archive
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Mirror of the Internet Archive in the
Bibliotheca Alexandrina
Internet Archive
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From 1996 to 2009, headquarters were in the
Presidio of San Francisco, a former U.S. military base
89.
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 for documentation in libraries and increasingly also by archives, the GND is managed by the German National Library in cooperation 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 entities and sub-classes, useful in library classification, and 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
90.
National Library of Australia
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In 2012–2013, the National Library collection comprised 6,496,772 items, and an additional 15,506 metres of manuscript material. In 1901, a Commonwealth Parliamentary Library was established to serve the newly formed Federal Parliament of Australia, from its inception the Commonwealth Parliamentary Library was driven to development of a truly national collection. The present library building was opened in 1968, the building was designed by the architectural firm of Bunning and Madden. The foyer is decorated in marble, with windows by Leonard French. In 2012–2013 the Library collection comprised 6,496,772 items, the Librarys collections of Australiana have developed into the nations single most important resource of materials recording the Australian cultural heritage. Australian writers, editors and illustrators are actively sought and well represented—whether published in Australia or overseas, approximately 92. 1% of the Librarys collection has been catalogued and is discoverable through the online catalogue. The Library has digitized over 174,000 items from its collection and, the Library is a world leader in digital preservation techniques, and maintains an Internet-accessible archive of selected Australian websites called the Pandora Archive. A core Australiana collection is that of John A. Ferguson, the Library has particular collection strengths in the performing arts, including dance. The Librarys considerable collections of general overseas and rare materials, as well as world-class Asian. The print collections are further supported by extensive microform holdings, the Library also maintains the National Reserve Braille Collection. The Library has acquired a number of important Western and Asian language scholarly collections from researchers, williams Collection The Asian Collections are searchable via the National Librarys catalogue. The National Library holds a collection of pictures and manuscripts. The manuscript collection contains about 26 million separate items, covering in excess of 10,492 meters of shelf space, the collection relates predominantly to Australia, but there are also important holdings relating to Papua New Guinea, New Zealand and the Pacific. The collection also holds a number of European and Asian manuscript collections or single items have received as part of formed book collections. Examples are the papers of Alfred Deakin, Sir John Latham, Sir Keith Murdoch, Sir Hans Heysen, Sir John Monash, Vance Palmer and Nettie Palmer, A. D. Hope, Manning Clark, David Williamson, W. M. The Library has also acquired the records of many national non-governmental organisations and they include the records of the Federal Secretariats of the Liberal party, the A. L. P, the Democrats, the R. S. L. Finally, the Library holds about 37,000 reels of microfilm of manuscripts and archival records, mostly acquired overseas and predominantly of Australian, the National Librarys Pictures collection focuses on Australian people, places and events, from European exploration of the South Pacific to contemporary events. Art works and photographs are acquired primarily for their informational value, media represented in the collection include photographs, drawings, watercolours, oils, lithographs, engravings, etchings and sculpture/busts
National Library of Australia
National Library of Australia
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National Library of Australia as viewed from
Lake Burley Griffin,
Canberra
National Library of Australia
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The original National Library building on Kings Avenue, Canberra, was designed by Edward Henderson. Originally intended to be several wings, only one wing was completed and was demolished in 1968. Now the site of the Edmund Barton Building.
National Library of Australia
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The library seen from Lake Burley Griffin in autumn.
91.
National Diet Library
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The National Diet Library is the only national library in Japan. It was established in 1948 for the purpose of assisting members of the National Diet of Japan in researching matters of public policy, the library is similar in purpose and scope to the United States Library of Congress. The National Diet Library consists of two facilities in Tokyo and Kyoto, and several other branch libraries throughout Japan. The Diets power in prewar Japan was limited, and its need for information was correspondingly small, the original Diet libraries never developed either the collections or the services which might have made them vital adjuncts of genuinely responsible legislative activity. Until Japans defeat, moreover, the executive had controlled all political documents, depriving the people and the Diet of access to vital information. The U. S. occupation forces under General Douglas MacArthur deemed reform of the Diet library system to be an important part of the democratization of Japan after its defeat in World War II. In 1946, each house of the Diet formed its own National Diet Library Standing Committee, hani Gorō, a Marxist historian who had been imprisoned during the war for thought crimes and had been elected to the House of Councillors after the war, spearheaded the reform efforts. Hani envisioned the new body as both a citadel of popular sovereignty, and the means of realizing a peaceful revolution, the National Diet Library opened in June 1948 in the present-day State Guest-House with an initial collection of 100,000 volumes. The first Librarian of the Diet Library was the politician Tokujirō Kanamori, the philosopher Masakazu Nakai served as the first Vice Librarian. In 1949, the NDL merged with the National Library and became the national library in Japan. At this time the collection gained a million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, in 1986, the NDLs Annex was completed to accommodate a combined total of 12 million books and periodicals. The Kansai-kan, which opened in October 2002 in the Kansai Science City, has a collection of 6 million items, in May 2002, the NDL opened a new branch, the International Library of Childrens Literature, in the former building of the Imperial Library in Ueno. This branch contains some 400,000 items of literature from around the world. Though the NDLs original mandate was to be a library for the National Diet. In the fiscal year ending March 2004, for example, the library reported more than 250,000 reference inquiries, in contrast, as Japans national library, the NDL collects copies of all publications published in Japan. The NDL has an extensive collection of some 30 million pages of documents relating to the Occupation of Japan after World War II. This collection include the documents prepared by General Headquarters and the Supreme Commander of the Allied Powers, the Far Eastern Commission, the NDL maintains a collection of some 530,000 books and booklets and 2 million microform titles relating to the sciences
National Diet Library
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Tokyo Main Library of the National Diet Library
National Diet Library
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Kansai-kan of the National Diet Library
National Diet Library
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The National Diet Library
National Diet Library
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Main building in Tokyo
92.
Joseph-Louis Lagrange
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Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia or Giuseppe Ludovico De la Grange Tournier, was an Italian and French Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, in 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life, Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He also extended the method to take into account possible constraints and he proved that every natural number is a sum of four squares. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, in calculus, Lagrange developed a novel approach to interpolation and Taylor series. Born as Giuseppe Lodovico Lagrangia, Lagrange was of Italian and French descent and his mother was from the countryside of Turin. He was raised as a Roman Catholic, a career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the University of Turin and his subject was classical Latin. At first he had no enthusiasm for mathematics, finding Greek geometry rather dull. It was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident. Alone and unaided he threw himself into mathematical studies, at the end of a years incessant toil he was already an accomplished mathematician, in that capacity, Lagrange was the first to teach calculus in an engineering school. In this Academy one of his students was François Daviet de Foncenex, Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of tautochrone, Lagrange wrote several letters to Leonhard Euler between 1754 and 1756 describing his results. He outlined his δ-algorithm, leading to the Euler–Lagrange equations of variational calculus, Lagrange also applied his ideas to problems of classical mechanics, generalizing the results of Euler and Maupertuis. Euler was very impressed with Lagranges results, Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773. Many of these are elaborate papers, the article concludes with a masterly discussion of echoes, beats, and compound sounds. Other articles in volume are on recurring series, probabilities. The next work he produced was in 1764 on the libration of the Moon, and an explanation as to why the face was always turned to the earth
Joseph-Louis Lagrange
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Joseph-Louis (Giuseppe Luigi), comte de Lagrange
Joseph-Louis Lagrange
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Lagrange's tomb in the crypt of the
Panthéon