1.
Cardanus (crater)
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Cardanus is a lunar impact crater that is located in the western part of the Moon, in the western part of the Oceanus Procellarum. Due to its location the crater appears very oval because of foreshortening, Cardanus is distinctive for the chain of craters, designated Catena Krafft, that connect its northern rim with the crater Krafft to the north. The outer rim is sharp-edged and somewhat irregular, with an outer rampart. The crater floor has several small craterlets across its surface, the floor surface is somewhat irregular in the southwest, but nearly featureless elsewhere. To the southwest is the rille designated Rima Cardanus, a cleft in the mare that generally follows a northeasterly direction, to the southeast, beyond the rille, is the small crater Galilaei. By convention these features are identified on maps by placing the letter on the side of the crater midpoint that is closest to Cardanus
2.
Pavia
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Pavia is a town and comune of south-western Lombardy, northern Italy,35 kilometres south of Milan on the lower Ticino river near its confluence with the Po. It has a population of c, the city was the capital of the Kingdom of the Lombards from 572 to 774. Pavia is the capital of the province of Pavia, known for agricultural products including wine, rice, cereals. Although there are a number of industries located in the suburbs, Pavia is the episcopal seat of the Roman Catholic Bishop of Pavia. The city possesses many artistic and cultural treasures, including several important churches and museums, dating back to pre-Roman times, the town of Pavia, then known as Ticinum, was a municipality and an important military site under the Roman Empire. It was said by Pliny the Elder to have founded by the Laevi and Marici. It was at Pavia in 476 AD that the reign of Romulus Augustulus, ten months after Romulus Augustulus’s reign began, Orestes’s soldiers under the command of one of his officers named Odoacer, rebelled and killed Orestes in the city of Pavia in 476. Without his father Romulus Augustulus was powerless, instead of killing Romulus Augustulus, Odoacer pensioned him off at 6,000 solidi a year before declaring the end of the Western Roman Empire and himself king of the new Kingdom of Italy. Odoacer’s reign as king of Italy did not last long, because in 488 the Ostrogothic peoples led by their king Theoderic invaded Italy and waged war against Odoacer. After fighting for 5 years Theoderic defeated Odoacer and on March 15,493 assassinated Odoacer at a banquet meant to negotiate a peace between the two rulers, with the establishment of the Ostrogoth kingdom based in northern Italy, Theoderic began his vast program of public building. Pavia was among several cities that Theodoric chose to restore and expand and he began the construction of the vast palace complex that would eventually become the residence of Lombard monarchs several decades later. Near the end of Theoderic’s reign the Christian philosopher Boethius was imprisoned in one of Pavia’s churches from 522 to 525 before his execution for treason and it was during Boethius’s captivity in Pavia that he wrote his seminal work the Consolation of Philosophy. Pavia played an important role in the war between the Eastern Roman Empire and the Ostrogoths that began in 535, after the capitulation of the Ostrogothic leadership in 540 more than a thousand men remained garrisoned in Pavia and Verona dedicated to opposing Eastern Roman rule. The resilience of Ostrogoth strongholds like Pavia against invading forces allowed pockets of Ostrogothic rule to limp along until finally being defeated in 561, Pavia and the peninsula of Italy didn’t remain long under the rule of the Eastern Roman Empire for in 568 a new people invaded Italy. This new invading people in 568 were the Lombards, in their invasion of Italy in 568, the Lombards were led by their king Alboin, who would become the first Lombard king of Italy. Alboin captured much of northern Italy in 568 but his progress was halted in 569 by the city of Pavia. Meanwhile Alboin, after driving out the soldiers, took possession of everything as far as Tuscany except Rome and Ravenna and some other fortified places which were situated on the shore of the sea. ”The Siege of Ticinum finally ended with the Lombards capturing the city of Pavia in 572. Pavia’s strategic location and the Ostrogoth palaces located within it would make Pavia by the 620s the main capital of the Lombards’ Kingdom of Pavia, under Lombard rule many monasteries, nunneries, and churches were built at Pavia by the devout Christian Lombard monarchs
3.
University of Pavia
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The University of Pavia is a university located in Pavia, Lombardy, Italy. It was founded in 1361 and has thirteen faculties, an edict issued by the Frankish king of Italy Lothar I mentions the existence of a higher education institution at Pavia as early as AD825. This institution, mainly devoted to ecclesiastical and civil law as well as to divinity studies, was selected as the prime educational centre for northern Italy. In 1858, the University was the scene of student protests against Austrian rule in northern Italy. The authorities responded by ordering the universitys temporary closure, the incidents at Pavia were typical of the wave of nationalist demonstrations all over Italy that immediately preceded the Unification. During the following centuries, through periods of both adversity and prosperity, the fame of the University of Pavia grew over the last years due to the number of applicants. Three Nobel Prize winners also taught in Pavia, physician Camillo Golgi, chemist Giulio Natta, also critical to the universitys reputation was its distinguished record of public education, epitomised by the establishment of 5 private and public colleges. In 1997 the IUSS, was established, a Higher Learning Institution analogous to the Scuola Normale Superiore, the IUSS is the federal body that links the 5 colleges of Pavia which constitute the Pavia University System. Today, the University continues to offer a variety of disciplinary and inter-disciplinary teaching. Research is carried out in departments, institutes, clinics, centres and laboratories, in association with public and private institutions, enterprises. English - One single-cycle masters degree and seven masters degrees are offered in English
4.
Archimedes
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Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε
5.
Leonardo Fibonacci
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Fibonacci was an Italian mathematician, considered to be the most talented Western mathematician of the Middle Ages. The name he is called, Fibonacci, is short for figlio di Bonacci and he is also known as Leonardo Bonacci, Leonardo of Pisa, Leonardo Pisano Bigollo. Fibonacci popularized the Hindu–Arabic numeral system in the Western World primarily through his composition in 1202 of Liber Abaci and he also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci. Fibonacci was born around 1175 to Guglielmo Bonacci, a wealthy Italian merchant and, by some accounts, Guglielmo directed a trading post in Bugia, a port in the Almohad dynastys sultanate in North Africa. Fibonacci travelled with him as a boy, and it was in Bugia that he learned about the Hindu–Arabic numeral system. Fibonacci travelled extensively around the Mediterranean coast, meeting with many merchants and he soon realised the many advantages of the Hindu-Arabic system. In 1202, he completed the Liber Abaci which popularized Hindu–Arabic numerals in Europe, Fibonacci became a guest of Emperor Frederick II, who enjoyed mathematics and science. The date of Fibonaccis death is not known, but it has estimated to be between 1240 and 1250, most likely in Pisa. In the Liber Abaci, Fibonacci introduced the so-called modus Indorum, the book advocated numeration with the digits 0–9 and place value. The book was well-received throughout educated Europe and had a impact on European thought. No copies of the 1202 edition are known to exist, the book also discusses irrational numbers and prime numbers. Liber Abaci posed, and solved, a problem involving the growth of a population of rabbits based on idealized assumptions, the solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. Although Fibonaccis Liber Abaci contains the earliest known description of the sequence outside of India, in the Fibonacci sequence of numbers, each number is the sum of the previous two numbers. Fibonacci began the sequence not with 0,1,1,2, as modern mathematicians do but with 1,1,2, etc. He carried the calculation up to the place, that is 233. Fibonacci did not speak about the ratio as the limit of the ratio of consecutive numbers in this sequence. In the 19th century, a statue of Fibonacci was constructed and raised in Pisa, today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli. There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers, examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, and the Pisano period
6.
Blaise Pascal
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Blaise Pascal was a French mathematician, physicist, inventor, writer and Christian philosopher. He was a prodigy who was educated by his father. Pascal also wrote in defence of the scientific method, in 1642, while still a teenager, he started some pioneering work on calculating machines. After three years of effort and 50 prototypes, he built 20 finished machines over the following 10 years, following Galileo Galilei and Torricelli, in 1647, he rebutted Aristotles followers who insisted that nature abhors a vacuum. Pascals results caused many disputes before being accepted, in 1646, he and his sister Jacqueline identified with the religious movement within Catholicism known by its detractors as Jansenism. Following a religious experience in late 1654, he began writing works on philosophy. His two most famous works date from this period, the Lettres provinciales and the Pensées, the set in the conflict between Jansenists and Jesuits. In that year, he wrote an important treatise on the arithmetical triangle. Between 1658 and 1659 he wrote on the cycloid and its use in calculating the volume of solids, Pascal had poor health, especially after the age of 18, and he died just two months after his 39th birthday. Pascal was born in Clermont-Ferrand, which is in Frances Auvergne region and he lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal, who also had an interest in science and mathematics, was a local judge, Pascal had two sisters, the younger Jacqueline and the elder Gilberte. In 1631, five years after the death of his wife, the newly arrived family soon hired Louise Delfault, a maid who eventually became an instrumental member of the family. Étienne, who never remarried, decided that he alone would educate his children, for they all showed extraordinary intellectual ability, the young Pascal showed an amazing aptitude for mathematics and science. Particularly of interest to Pascal was a work of Desargues on conic sections and it states that if a hexagon is inscribed in a circle then the three intersection points of opposite sides lie on a line. Pascals work was so precocious that Descartes was convinced that Pascals father had written it, in France at that time offices and positions could be—and were—bought and sold. In 1631 Étienne sold his position as president of the Cour des Aides for 65,665 livres. The money was invested in a government bond which provided, if not a lavish, then certainly a comfortable income which allowed the Pascal family to move to, but in 1638 Richelieu, desperate for money to carry on the Thirty Years War, defaulted on the governments bonds. Suddenly Étienne Pascals worth had dropped from nearly 66,000 livres to less than 7,300 and it was only when Jacqueline performed well in a childrens play with Richelieu in attendance that Étienne was pardoned
7.
Pierre de Fermat
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He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermats principle for light propagation and his Fermats Last Theorem in number theory, Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a leather merchant. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth, there is little evidence concerning his school education, but it was probably at the Collège de Navarre in Montauban. He attended the University of Orléans from 1623 and received a bachelor in law in 1626. In Bordeaux he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apolloniuss De Locis Planis to one of the mathematicians there, there he became much influenced by the work of François Viète. In 1630, he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France and he held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat, fluent in six languages, Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often little or no proof of his theorems. In some of these letters to his friends he explored many of the ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession, nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus. Secrecy was common in European mathematical circles at the time and this naturally led to priority disputes with contemporaries such as Descartes and Wallis. Anders Hald writes that, The basis of Fermats mathematics was the classical Greek treatises combined with Vietas new algebraic methods, Fermats pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes famous La géométrie. This manuscript was published posthumously in 1679 in Varia opera mathematica, in these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of power functions. With his method, he was able to reduce this evaluation to the sum of geometric series, the resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus. In number theory, Fermat studied Pells equation, perfect numbers, amicable numbers and it was while researching perfect numbers that he discovered Fermats little theorem. Fermat developed the two-square theorem, and the polygonal number theorem, although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived
8.
Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus. Newtons Principia formulated the laws of motion and universal gravitation that dominated scientists view of the universe for the next three centuries. Newtons work on light was collected in his influential book Opticks. He also formulated a law of cooling, made the first theoretical calculation of the speed of sound. Newton was a fellow of Trinity College and the second Lucasian Professor of Mathematics at the University of Cambridge, politically and personally tied to the Whig party, Newton served two brief terms as Member of Parliament for the University of Cambridge, in 1689–90 and 1701–02. He was knighted by Queen Anne in 1705 and he spent the last three decades of his life in London, serving as Warden and Master of the Royal Mint and his father, also named Isaac Newton, had died three months before. Born prematurely, he was a child, his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug. When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabas Smith, leaving her son in the care of his maternal grandmother, Newtons mother had three children from her second marriage. From the age of twelve until he was seventeen, Newton was educated at The Kings School, Grantham which taught Latin and Greek. He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, Henry Stokes, master at the Kings School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a bully, he became the top-ranked student. In June 1661, he was admitted to Trinity College, Cambridge and he started as a subsizar—paying his way by performing valets duties—until he was awarded a scholarship in 1664, which guaranteed him four more years until he would get his M. A. He set down in his notebook a series of Quaestiones about mechanical philosophy as he found it, in 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became calculus. Soon after Newton had obtained his B. A. degree in August 1665, in April 1667, he returned to Cambridge and in October was elected as a fellow of Trinity. Fellows were required to become ordained priests, although this was not enforced in the restoration years, however, by 1675 the issue could not be avoided and by then his unconventional views stood in the way. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II. A and he was elected a Fellow of the Royal Society in 1672. Newtons work has been said to distinctly advance every branch of mathematics then studied and his work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newtons mathematical papers
9.
Gottfried Wilhelm von Leibniz
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Leibnizs notation has been widely used ever since it was published. It was only in the 20th century that his Law of Continuity and he became one of the most prolific inventors in the field of mechanical calculators. He also refined the number system, which is the foundation of virtually all digital computers. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th-century advocates of rationalism and he wrote works on philosophy, politics, law, ethics, theology, history, and philology. Leibnizs contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and he wrote in several languages, but primarily in Latin, French, and German. There is no complete gathering of the writings of Leibniz in English, Gottfried Leibniz was born on July 1,1646, toward the end of the Thirty Years War, in Leipzig, Saxony, to Friedrich Leibniz and Catharina Schmuck. Friedrich noted in his journal,21. Juny am Sontag 1646 Ist mein Sohn Gottfried Wilhelm, post sextam vespertinam 1/4 uff 7 uhr abents zur welt gebohren, in English, On Sunday 21 June 1646, my son Gottfried Wilhelm is born into the world a quarter after six in the evening, in Aquarius. Leibniz was baptized on July 3 of that year at St. Nicholas Church, Leipzig and his father died when he was six and a half years old, and from that point on he was raised by his mother. Her teachings influenced Leibnizs philosophical thoughts in his later life, Leibnizs father had been a Professor of Moral Philosophy at the University of Leipzig, and the boy later inherited his fathers personal library. He was given access to it from the age of seven. Access to his fathers library, largely written in Latin, also led to his proficiency in the Latin language and he also composed 300 hexameters of Latin verse, in a single morning, for a special event at school at the age of 13. In April 1661 he enrolled in his fathers former university at age 15 and he defended his Disputatio Metaphysica de Principio Individui, which addressed the principle of individuation, on June 9,1663. Leibniz earned his masters degree in Philosophy on February 7,1664, after one year of legal studies, he was awarded his bachelors degree in Law on September 28,1665. His dissertation was titled De conditionibus, in early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria, the first part of which was also his habilitation thesis in Philosophy, which he defended in March 1666. His next goal was to earn his license and Doctorate in Law, in 1666, the University of Leipzig turned down Leibnizs doctoral application and refused to grant him a Doctorate in Law, most likely due to his relative youth. Leibniz then enrolled in the University of Altdorf and quickly submitted a thesis, the title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure. Leibniz earned his license to practice law and his Doctorate in Law in November 1666 and he next declined the offer of an academic appointment at Altdorf, saying that my thoughts were turned in an entirely different direction
10.
Maria Gaetana Agnesi
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Maria Gaetana Agnesi was an Italian mathematician, philosopher, theologian and humanitarian. She was the first woman to write a handbook and the first woman appointed as a Mathematics Professor at a university. She is credited with writing the first book discussing both differential and integral calculus and was a member of the faculty at the University of Bologna and she devoted the last four decades of her life to studying theology and to charitable work and serving the poor. This extended to helping the sick by allowing them entrance into her home where she set up a hospital and she was a devout Catholic and wrote extensively on the marriage between intellectual pursuit and mystical contemplation, most notably in her essay Il cielo mistico. She saw the rational contemplation of God as a complement to prayer and contemplation of the life, death, Maria Teresa Agnesi Pinottini, clavicembalist and composer, was her sister. Maria Gaetana Agnesi was born in Milan, to a wealthy and her father Pietro Agnesi, a University of Bologna mathematics professor, wanted to elevate his family into the Milanese nobility. In order to achieve his goal, he had married Anna Fortunata Brivio in 1717 and her mothers death provided her the excuse to retire from public life. She took over management of the household and she was one of 21 children. Maria was recognized early on as a prodigy, she could speak both Italian and French at five years of age. By her eleventh birthday, she had also learned Greek, Hebrew, Spanish, German, and Latin and she even educated her younger brothers. When she was nine years old, she composed and delivered a speech in Latin to some of the most distinguished intellectuals of the day. The subject was womens right to be educated, Agnesi suffered a mysterious illness at the age of 12 that was attributed to her excessive studying and was prescribed vigorous dancing and horseback riding. This treatment did not work, she began to experience extreme convulsions, by age fourteen, she was studying ballistics and geometry. Maria was very shy in nature and did not like these meetings and her father remarried twice after Marias mother died, and Maria Agnesi ended up the eldest of 21 children, including her half-siblings. In addition to her performances and lessons, her responsibility was to teach her siblings and this task kept her from her own goal of entering a convent, as she had become strongly religious. During that time, Maria studied with him both differential and integral calculus and her family was recognized as one of the wealthiest in Milan. According to Dirk Jan Struik, Agnesi is the first important woman mathematician since Hypatia, the goal of this work was, according to Agnesi herself, to give a systematic illustration of the different results and theorems of infinitesimal calculus. The model for her treatise was Le calcul différentiel et intégral dans l’Analyse by Charles René Reyneau, in this treatise, she worked on integrating mathematical analysis with algebra
11.
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
12.
Carl Friedrich Gauss
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Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, as the son of poor working-class parents. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter and he was christened and confirmed in a church near the school he attended as a child. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100, there are many other anecdotes about his precocity while a toddler, and he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his opus, in 1798 at the age of 21. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day, while at university, Gauss independently rediscovered several important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone, the stonemason declined, stating that the difficult construction would essentially look like a circle. The year 1796 was most productive for both Gauss and number theory and he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory, on 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, the prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note, ΕΥΡΗΚΑ. On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, in 1831 Gauss developed a fruitful collaboration with the physics professor Wilhelm Weber, leading to new knowledge in magnetism and the discovery of Kirchhoffs circuit laws in electricity. It was during this time that he formulated his namesake law and they constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in Göttingen. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became associated member of the Royal Institute of the Netherlands, in 1854, Gauss selected the topic for Bernhard Riemanns Habilitationvortrag, Über die Hypothesen, welche der Geometrie zu Grunde liegen. On the way home from Riemanns lecture, Weber reported that Gauss was full of praise, Gauss died in Göttingen, on 23 February 1855 and is interred in the Albani Cemetery there. Two individuals gave eulogies at his funeral, Gausss son-in-law Heinrich Ewald and Wolfgang Sartorius von Waltershausen and his brain was preserved and was studied by Rudolf Wagner who found its mass to be 1,492 grams and the cerebral area equal to 219,588 square millimeters. Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius, Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen
13.
Latin language
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
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Polymath
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A polymath is a person whose expertise spans a significant number of different subject areas, such a person is known to draw on complex bodies of knowledge to solve specific problems. The term was first used in the 17th century, the related term, the term is often used to describe great thinkers of the Renaissance and the Enlightenment who excelled at several fields in science and the arts. In the Italian Renaissance, the idea of the polymath was expressed by Leon Battista Alberti and this term entered the lexicon during the twentieth century and has now been applied to great thinkers living before and after the Renaissance. Renaissance man was first recorded in written English in the early 20th century and it is now used to refer to great thinkers living before, during, or after the Renaissance. Leonardo da Vinci has often described as the archetype of the Renaissance man. These polymaths had an approach to education that reflected the ideals of the humanists of the time. A gentleman or courtier of that era was expected to speak several languages, play an instrument, write poetry. The idea of an education was essential to achieving polymath ability. At this time, universities did not specialize in specific areas but rather trained students in an array of science, philosophy. This universal education gave them a grounding from which they could continue into apprenticeship toward becoming a master of a specific field, aside from Renaissance man as mentioned above, similar terms in use are Homo Universalis and Uomo Universale, which translate to universal person or universal man. The related term generalist—contrasted with a used to describe a person with a general approach to knowledge. The term Universal Genius or Versatile Genius is also used, with Leonardo da Vinci as the prime example again. The term seems to be used especially when a person has made lasting contributions in at least one of the fields in which he was involved. When a person is described as having knowledge, they exhibit a vast scope of knowledge. This designation may be anachronistic, however, in the case of such as Eratosthenes whose reputation for having encyclopedic knowledge predates the existence of any encyclopedic object. One whose accomplishments are limited to athletics would not be considered a polymath in the sense of the word. An example is Howard Baker, who was called a sporting polymath by the Encyclopedia of British Football for winning high jump titles and playing cricket, football, many polymaths from across the centuries have their roots in medical applications. One of the well known polymaths, Leonardo da Vinci, was known for his immense interest in human anatomical structure
15.
Biologist
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A biologist, is a scientist who has specialized knowledge in the field of biology, the scientific study of life. Biologists involved in fundamental research attempt to explore and further explain the mechanisms that govern the functioning of living matter. Biologists involved in applied research attempt to develop or improve more specific processes and understanding, in such as medicine, industry. While biologist can apply to any scientist studying biology, most biologists research, in this way, biologists investigate large-scale organism interactions, whole multicellular organisms, organs, tissues, cells, and small-scale cellular and molecular processes. Other biologists study less direct aspects of life, such as phylogeny, Biologists conduct research based on the scientific method, to test the validity of a theory, with hypothesis formation, experimentation and documentation of methods and data. There are many types of biologists, some work on microorganisms, while others study multicellular organisms. Many jobs in biology as a field require an academic degree, a doctorate or its equivalent is generally required to direct independent research, and involves a specialization in a specific area of biology. Many biological scientists work in research and development, some conduct fundamental research to advance our knowledge of living organisms, including bacteria and other pathogens. This research enhances understanding and adds to the database of literature. Furthermore, it aids the development of solutions to problems in areas such as human health. These biological scientists work in government, university, and private industry laboratories. Many expand on specialized research that started in post-graduate qualifications. Biological scientists who work in applied research or product development often use knowledge gained by research to further knowledge in particular fields or applications. For example, this research may be used to develop new pharmaceutical drugs, treatments, and medical diagnostic tests, increase crop yields. These scientists must consider the effects of their work. Some biologists conduct laboratory experiments involving animals, plants or microorganisms, however, some biological research also occurs outside the laboratory and may involve natural observation rather than experimentation. For example, a botanist may investigate the plant species present in a particular environment, swift advances in knowledge of genetics and organic molecules spurred growth in the field of biotechnology, transforming the industries in which biological scientists work. Biological scientists can now manipulate the genetic material of animals and plants, basic and applied research on biotechnological processes, such as recombining DNA, has led to the production of important substances, including human insulin and growth hormone
16.
Physicist
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A physicist is a scientist who has specialized knowledge in the field of physics, the exploration of the interactions of matter and energy across the physical universe. A physicist is a scientist who specializes or works in the field of physics, physicists generally are interested in the root or ultimate causes of phenomena, and usually frame their understanding in mathematical terms. Physicists can also apply their knowledge towards solving real-world problems or developing new technologies, some physicists specialize in sectors outside the science of physics itself, such as engineering. The study and practice of physics is based on a ladder of discoveries. Many mathematical and physical ideas used today found their earliest expression in ancient Greek culture and Asian culture, the bulk of physics education can be said to flow from the scientific revolution in Europe, starting with the work of Galileo and Kepler in the early 1600s. New knowledge in the early 21st century includes an increase in understanding physical cosmology. The term physicist was coined by William Whewell in his 1840 book The Philosophy of the Inductive Sciences, many physicist positions require an undergraduate degree in applied physics or a related science or a Masters degree like MSc, MPhil, MPhys or MSci. In a research oriented level, students tend to specialize in a particular field, Physics students also need training in mathematics, and also in computer science and programming. For being employed as a physicist a doctoral background may be required for certain positions, undergraduate students like BSc Mechanical Engineering, BSc Electrical and Computer Engineering, BSc Applied Physics. etc. With physics orientation are chosen as research assistants with faculty members, the highest honor awarded to physicists is the Nobel Prize in Physics, awarded since 1901 by the Royal Swedish Academy of Sciences. The three major employers of career physicists are academic institutions, laboratories, and private industries, with the largest employer being the last, physicists in academia or government labs tend to have titles such as Assistants, Professors, Sr. /Jr. As per the American Institute for Physics, some 20% of new physics Ph. D. s holds jobs in engineering development programs, while 14% turn to computer software, a majority of physicists employed apply their skills and training to interdisciplinary sectors. For industry or self-employment. and also in science and programming. Hence a majority of Physics bachelors degree holders are employed in the private sector, other fields are academia, government and military service, nonprofit entities, labs and teaching
17.
Chemist
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A chemist is a scientist trained in the study of chemistry. Chemists study the composition of matter and its properties, chemists carefully describe the properties they study in terms of quantities, with detail on the level of molecules and their component atoms. Chemists carefully measure substance proportions, reaction rates, and other chemical properties, the word chemist is also used to address Pharmacists in Commonwealth English. Chemists may specialize in any number of subdisciplines of chemistry, materials scientists and metallurgists share much of the same education and skills with chemists. The roots of chemistry can be traced to the phenomenon of burning, fire was a mystical force that transformed one substance into another and thus was of primary interest to mankind. It was fire that led to the discovery of iron and glasses, after gold was discovered and became a precious metal, many people were interested to find a method that could convert other substances into gold. This led to the protoscience called alchemy, the word chemist is derived from the New Latin noun chimista, an abbreviation of alchimista. Alchemists discovered many chemical processes that led to the development of modern chemistry, Chemistry as we know it today, was invented by Antoine Lavoisier with his law of conservation of mass in 1783. The discoveries of the elements has a long history culminating in the creation of the periodic table by Dmitri Mendeleev. The Nobel Prize in Chemistry created in 1901 gives an excellent overview of chemical discovery since the start of the 20th century. Jobs for chemists usually require at least a degree, but many positions, especially those in research. At the Masters level and higher, students tend to specialize in a particular field, postdoctoral experience may be required for certain positions. Workers whose work involves chemistry, but not at a complexity requiring an education with a degree, are commonly referred to as chemical technicians. Such technicians commonly do such work as simpler, routine analyses for quality control or in clinical laboratories, there are also degrees specific to become a Chemical Technologist, which are somewhat distinct from those required when a student is interested in becoming a professional Chemist. A Chemical technologist is more involved in the management and operation of the equipment and they are part of the team of a chemical laboratory in which the quality of the raw material, intermediate products and finished products is analyzed. They also perform functions in the areas of quality control. The higher the level achieved in the field of Chemistry, the higher the responsibility given to that chemist. Chemistry, as a field, have so many applications that different tasks/objectives can be given to workers/scientists with these different levels of education and/or experience
18.
Astrologer
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Astrology is the study of the movements and relative positions of celestial objects as a means for divining information about human affairs and terrestrial events. Throughout most of its history astrology was considered a tradition and was common in academic circles, often in close relation with astronomy, alchemy, meteorology. It was present in political circles, and is mentioned in works of literature, from Dante Alighieri and Geoffrey Chaucer to William Shakespeare, Lope de Vega. Astrology thus lost its academic and theoretical standing, and common belief in it has largely declined, Astrology is now recognized to be pseudoscience. The word astrology comes from the early Latin word astrologia, which derives from the Greek ἀστρολογία—from ἄστρον astron, astrologia later passed into meaning star-divination with astronomia used for the scientific term. Many cultures have attached importance to astronomical events, and the Indians, Chinese, the majority of professional astrologers rely on such systems. Astrology has been dated to at least the 2nd millennium BCE, with roots in systems used to predict seasonal shifts. A form of astrology was practised in the first dynasty of Mesopotamia, Chinese astrology was elaborated in the Zhou dynasty. Hellenistic astrology after 332 BCE mixed Babylonian astrology with Egyptian Decanic astrology in Alexandria, Alexander the Greats conquest of Asia allowed astrology to spread to Ancient Greece and Rome. In Rome, astrology was associated with Chaldean wisdom, after the conquest of Alexandria in the 7th century, astrology was taken up by Islamic scholars, and Hellenistic texts were translated into Arabic and Persian. In the 12th century, Arabic texts were imported to Europe, major astronomers including Tycho Brahe, Johannes Kepler and Galileo practised as court astrologers. Astrological references appear in literature in the works of such as Dante Alighieri and Geoffrey Chaucer. Throughout most of its history, astrology was considered a scholarly tradition and it was accepted in political and academic contexts, and was connected with other studies, such as astronomy, alchemy, meteorology, and medicine. At the end of the 17th century, new concepts in astronomy. Astrology thus lost its academic and theoretical standing, and common belief in astrology has largely declined, Astrology, in its broadest sense, is the search for meaning in the sky. This was a first step towards recording the Moons influence upon tides and rivers, by the 3rd millennium BCE, civilisations had sophisticated awareness of celestial cycles, and may have oriented temples in alignment with heliacal risings of the stars. Scattered evidence suggests that the oldest known references are copies of texts made in the ancient world. The Venus tablet of Ammisaduqa thought to be compiled in Babylon around 1700 BCE, a scroll documenting an early use of electional astrology is doubtfully ascribed to the reign of the Sumerian ruler Gudea of Lagash
19.
Astronomer
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An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside of the scope of Earth. They look at stars, planets, moons, comets and galaxies, as well as other celestial objects — either in observational astronomy. Examples of topics or fields astronomers work on include, planetary science, solar astronomy, there are also related but distinct subjects like physical cosmology which studies the Universe as a whole. Astronomers usually fit into two types, Observational astronomers make direct observations of planets, stars and galaxies, and analyze the data, theoretical astronomers create and investigate models of things that cannot be observed. They use this data to create models or simulations to theorize how different celestial bodies work, there are further subcategories inside these two main branches of astronomy such as planetary astronomy, galactic astronomy or physical cosmology. Today, that distinction has disappeared and the terms astronomer. Professional astronomers are highly educated individuals who typically have a Ph. D. in physics or astronomy and are employed by research institutions or universities. They spend the majority of their time working on research, although quite often have other duties such as teaching, building instruments. The number of astronomers in the United States is actually quite small. The American Astronomical Society, which is the organization of professional astronomers in North America, has approximately 7,000 members. This number includes scientists from other such as physics, geology. The International Astronomical Union comprises almost 10,145 members from 70 different countries who are involved in research at the Ph. D. level. Before CCDs, photographic plates were a method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year, analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers time. Astronomers who serve as faculty spend much of their time teaching undergraduate and graduate classes, most universities also have outreach programs including public telescope time and sometimes planetariums as a public service to encourage interest in the field. Those who become astronomers usually have a background in maths, sciences. Taking courses that teach how to research, write and present papers are also invaluable, in college/university most astronomers get a Ph. D. in astronomy or physics. Keeping in mind how few there are it is understood that graduate schools in this field are very competitive
20.
Gambler
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Gambling is the wagering of money or something of value on an event with an uncertain outcome with the primary intent of winning money and/or material goods. Gambling thus requires three elements be present, consideration, chance and prize, the term gaming in this context typically refers to instances in which the activity has been specifically permitted by law. However, this distinction is not universally observed in the English-speaking world, for instance, in the United Kingdom, the regulator of gambling activities is called the Gambling Commission. Gambling is also an international commercial activity, with the legal gambling market totaling an estimated $335 billion in 2009. In other forms, gambling can be conducted with materials which have a value, many popular games played in modern casinos originate from Europe and China. Games such as craps, baccarat, roulette, and blackjack originate from different areas of Europe, a version of keno, an ancient Chinese lottery game, is played in casinos around the world. In addition, pai gow poker, a hybrid between pai gow and poker is also played, many jurisdictions, local as well as national, either ban gambling or heavily control it by licensing the vendors. Such regulation generally leads to gambling tourism and illegal gambling in the areas where it is not allowed, there is generally legislation requiring that the odds in gaming devices are statistically random, to prevent manufacturers from making some high-payoff results impossible. Since these high-payoffs have very low probability, a bias can quite easily be missed unless the odds are checked carefully. Most jurisdictions that allow gambling require participants to be above a certain age, in some jurisdictions, the gambling age differs depending on the type of gambling. For example, in many American states one must be over 21 to enter a casino, E. g. Nonetheless, both insurance and gambling contracts are typically considered aleatory contracts under most legal systems, though they are subject to different types of regulation. Under common law, particularly English Law, a contract may not give a casino bona fide purchaser status. For case law on recovery of gambling losses where the loser had stolen the funds see Rights of owner of money as against one who won it in gambling transaction from thief. This was a plot point in a Perry Mason novel, The Case of the Singing Skirt. Religious perspectives on gambling have been mixed, ancient Hindu poems like the Gamblers Lament and the Mahabharata testify to the popularity of gambling among ancient Indians. However, the text Arthashastra recommends taxation and control of gambling, ancient Jewish authorities frowned on gambling, even disqualifying professional gamblers from testifying in court. For these social and religious reasons, most legal jurisdictions limit gambling, in at least one case, the same bishop opposing a casino has sold land to be used for its construction. Although different interpretations of law exist in the Muslim world
21.
Renaissance
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The Renaissance was a period in European history, from the 14th to the 17th century, regarded as the cultural bridge between the Middle Ages and modern history. It started as a movement in Italy in the Late Medieval period and later spread to the rest of Europe. This new thinking became manifest in art, architecture, politics, science, Early examples were the development of perspective in oil painting and the recycled knowledge of how to make concrete. Although the invention of movable type sped the dissemination of ideas from the later 15th century. In politics, the Renaissance contributed to the development of the customs and conventions of diplomacy, the Renaissance began in Florence, in the 14th century. Other major centres were northern Italian city-states such as Venice, Genoa, Milan, Bologna, the word Renaissance, literally meaning Rebirth in French, first appeared in English in the 1830s. The word also occurs in Jules Michelets 1855 work, Histoire de France, the word Renaissance has also been extended to other historical and cultural movements, such as the Carolingian Renaissance and the Renaissance of the 12th century. The Renaissance was a movement that profoundly affected European intellectual life in the early modern period. Renaissance scholars employed the humanist method in study, and searched for realism, however, a subtle shift took place in the way that intellectuals approached religion that was reflected in many other areas of cultural life. In addition, many Greek Christian works, including the Greek New Testament, were back from Byzantium to Western Europe. Political philosophers, most famously Niccolò Machiavelli, sought to describe life as it really was. Others see more competition between artists and polymaths such as Brunelleschi, Ghiberti, Donatello, and Masaccio for artistic commissions as sparking the creativity of the Renaissance. Yet it remains much debated why the Renaissance began in Italy, accordingly, several theories have been put forward to explain its origins. During the Renaissance, money and art went hand in hand, Artists depended entirely on patrons while the patrons needed money to foster artistic talent. Wealth was brought to Italy in the 14th, 15th, and 16th centuries by expanding trade into Asia, silver mining in Tyrol increased the flow of money. Luxuries from the Eastern world, brought home during the Crusades, increased the prosperity of Genoa, unlike with Latin texts, which had been preserved and studied in Western Europe since late antiquity, the study of ancient Greek texts was very limited in medieval Western Europe. One of the greatest achievements of Renaissance scholars was to bring this entire class of Greek cultural works back into Western Europe for the first time since late antiquity, Arab logicians had inherited Greek ideas after they had invaded and conquered Egypt and the Levant. Their translations and commentaries on these ideas worked their way through the Arab West into Spain and Sicily and this work of translation from Islamic culture, though largely unplanned and disorganized, constituted one of the greatest transmissions of ideas in history
22.
Probability
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Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, the higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair coin, since the coin is unbiased, the two outcomes are both equally probable, the probability of head equals the probability of tail. Since no other outcomes are possible, the probability is 1/2 and this type of probability is also called a priori probability. Probability theory is used to describe the underlying mechanics and regularities of complex systems. For example, tossing a coin twice will yield head-head, head-tail, tail-head. The probability of getting an outcome of head-head is 1 out of 4 outcomes or 1/4 or 0.25 and this interpretation considers probability to be the relative frequency in the long run of outcomes. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, subjectivists assign numbers per subjective probability, i. e. as a degree of belief. The degree of belief has been interpreted as, the price at which you would buy or sell a bet that pays 1 unit of utility if E,0 if not E. The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as data to produce probabilities. The expert knowledge is represented by some prior probability distribution and these data are incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a probability distribution that incorporates all the information known to date. The scientific study of probability is a development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, there are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the study of probability. According to Richard Jeffrey, Before the middle of the century, the term probable meant approvable. A probable action or opinion was one such as people would undertake or hold. However, in legal contexts especially, probable could also apply to propositions for which there was good evidence, the sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes
23.
Binomial coefficients
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In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a coefficient is indexed by a pair of integers n ≥ k ≥0 and is written. It is the coefficient of the xk term in the expansion of the binomial power n. The value of the coefficient is given by the expression n. k, arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascals triangle. The properties of binomial coefficients have led to extending the definition to beyond the case of integers n ≥ k ≥0. Andreas von Ettingshausen introduced the notation in 1826, although the numbers were known centuries earlier, the earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingalas Chandaḥśāstra. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī, alternative notations include C, nCk, nCk, Ckn, Cnk, and Cn, k in all of which the C stands for combinations or choices. Many calculators use variants of the C notation because they can represent it on a single-line display, in this form the binomial coefficients are easily compared to k-permutations of n, written as P, etc. For natural numbers n and k, the binomial coefficient can be defined as the coefficient of the monomial Xk in the expansion of n, the same coefficient also occurs in the binomial formula, which explains the name binomial coefficient. This shows in particular that is a number for any natural numbers n and k. Most of these interpretations are easily seen to be equivalent to counting k-combinations, several methods exist to compute the value of without actually expanding a binomial power or counting k-combinations. It also follows from tracing the contributions to Xk in n−1, as there is zero Xn+1 or X−1 in n, one might extend the definition beyond the above boundaries to include =0 when either k > n or k <0. This recursive formula then allows the construction of Pascals triangle, surrounded by white spaces where the zeros, or the trivial coefficients, a more efficient method to compute individual binomial coefficients is given by the formula = n k _ k. = n ⋯ k ⋯1 = ∏ i =1 k n +1 − i i and this formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, the denominator counts the number of distinct sequences that define the same k-combination when order is disregarded. Due to the symmetry of the binomial coefficient with regard to k and n−k, calculation may be optimised by setting the limit of the product above to the smaller of k. This formula follows from the formula above by multiplying numerator and denominator by. As a consequence it involves many factors common to numerator and denominator and it is less practical for explicit computation unless common factors are first cancelled
24.
Binomial theorem
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In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. For example,4 = x 4 +4 x 3 y +6 x 2 y 2 +4 x y 3 + y 4, the coefficient a in the term of a xb yc is known as the binomial coefficient or. These coefficients for varying n and b can be arranged to form Pascals triangle and these numbers also arise in combinatorics, where gives the number of different combinations of b elements that can be chosen from an n-element set. Special cases of the theorem were known from ancient times. Greek mathematician Euclid mentioned the case of the binomial theorem for exponent 2. There is evidence that the theorem for cubes was known by the 6th century in India. Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting k objects out of n without replacement, were of interest to the ancient Hindus. The earliest known reference to this problem is the Chandaḥśāstra by the Hindu lyricist Pingala. The commentator Halayudha from the 10th century A. D. explains this method using what is now known as Pascals triangle. By the 6th century A. D. the Hindu mathematicians probably knew how to express this as a quotient n. k. the binomial theorem as such can be found in the work of 11th-century Persian mathematician Al-Karaji, who described the triangular pattern of the binomial coefficients. He also provided a proof of both the binomial theorem and Pascals triangle, using a primitive form of mathematical induction. The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, the binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui and also Chu Shih-Chieh. Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, in 1544, Michael Stifel introduced the term binomial coefficient and showed how to use them to express n in terms of n −1, via Pascals triangle. Blaise Pascal studied the eponymous triangle comprehensively in the treatise Traité du triangle arithmétique, however, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin. Isaac Newton is generally credited with the binomial theorem, valid for any rational exponent. This formula is also referred to as the formula or the binomial identity. Using summation notation, it can be written as n = ∑ k =0 n x n − k y k = ∑ k =0 n x k y n − k. A simple variant of the formula is obtained by substituting 1 for y
25.
Combination lock
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A combination lock is a type of locking device in which a sequence of numbers or symbols is used to open the lock. Types range from inexpensive three-digit luggage locks to high-security safes, unlike a regular padlock, combination locks do not use keys. The earliest known combination lock was excavated in a Roman period tomb on the Kerameikos, attached to a small box, it featured several dials instead of keyholes. In 1206, the Muslim engineer Al-Jazari documented a combination lock in his book al-Ilm Wal-Amal al-Nafi Fi Sinaat al-Hiyal, muhammad al-Astrulabi also made combination locks, two of which are kept in Copenhagen and Boston Museums. Gerolamo Cardano later described a combination lock in the 16th century, one of the simplest types of combination lock, often seen in low-security bicycle locks and in briefcases, uses several rotating discs with notches cut into them. The lock is secured by a pin with several teeth on it which hook into the rotating discs, when the notches in the discs align with the teeth on the pin, the lock can be opened. This lock is considered to be one of the least secure types of combination lock, opening one in this fashion depends on slight irregularities in the machining of the parts. Unless the lock is machined precisely, when the pin is pulled outward and this disc is then rotated until a slight click is heard, indicating that the tooth has settled into the notch. The procedure is repeated for the discs, resulting in an open lock. Combination locks found on padlocks, lockers, or safes may use a dial which interacts with several parallel discs or cams. Customarily, a lock of this type is opened by rotating the dial clockwise to the first numeral, counterclockwise to the second, and so on in an alternating fashion until the last numeral is reached. The cams typically have an indentation or notch, and when the combination is entered. Depending on the quality of the lock, some single-dial combination locks can also be defeated relatively easily, typical padlocks are manufactured with generous tolerances, allowing two, three or even more digits of play in the correct access sequence. Given a 60-number dial with three cams and three digits of play, the space is reduced from 60 ×60 ×60 to 20 ×20 ×20. Additionally, if testing the mechanism to open the lock does not modify the state of the lock, the first two digits are entered normally once, then, starting from the second digit, the dial is rotated sequentially through the digits, testing the lock on each. The reduced search time would be × 60², a reduction of nearly 82% from 360 hours to 65 hours and this strategy can be extended to the second digit as well, slightly reducing the search time further.78 hours. This is still significantly better security than multiple-dial locks and many keyed locks, inexpensive padlocks are often also susceptible to direct mechanical attacks, such as the use of a padlock shim which can release the shackle without entering a combination. This weakness reduces the number of possible combinations from 64,000 to a mere 100, in 1978 a combination lock which could be set by the user to a sequence of his own choosing was invented by Andrew Elliot Rae
26.
Gimbal
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A gimbal is a pivoted support that allows the rotation of an object about a single axis. A set of three gimbals, one mounted on the other with orthogonal axes, may be used to allow an object mounted on the innermost gimbal to remain independent of the rotation of its support. The gimbal suspension used for mounting compasses and the like is sometimes called a Cardan suspension after Italian mathematician, however, Cardano did not invent the gimbal, nor did he claim to. The device has been known since antiquity, first described in the 3rd c, BCE by Philo of Byzantium, although some modern authors support it may not have a single identifiable inventor. The gimbal was first described by the Greek inventor Philo of Byzantium and this was done by the suspension of the inkwell at the center, which was mounted on a series of concentric metal rings so that it remained stationary no matter which way the pot is turned. Thus, the sinologist Joseph Needham suspected Arab interpolation as late as 1965, however, Carra de Vaux, author of the French translation which still provides the basis for modern scholars, regards the Pneumatics as essentially genuine. The historian of technology George Sarton also asserts that it is safe to assume the Arabic version is a copying of Philos original. So does his colleague Michael Lewis, after antiquity, gimbals remained widely known in the Near East. In the Latin West, reference to the device appeared again in the 9th century recipe book called the Little Key of Painting Mappae clavicula, the French inventor Villard de Honnecourt depicts a set of gimbals in his famous sketchbook. In the early period, dry compasses were suspended in gimbals. In China, the Han Dynasty inventor Ding Huan created a gimbal incense burner around 180 CE, there is a hint in the writing of the earlier Sima Xiangru that the gimbal existed in China since the 2nd century BCE. There is mention during the Liang Dynasty that gimbals were used for hinges of doors and windows, extant specimens of Chinese gimbals used for incense burners date to the early Tang Dynasty, and were part of the silver-smithing tradition in China. In this application, the measurement unit is equipped with three orthogonally mounted gyros to sense rotation about all axes in three-dimensional space. The gyro outputs are kept to a null through drive motors on each gimbal axis, to accomplish this, the gyro error signals are passed through resolvers mounted on the three gimbals, roll, pitch and yaw. These resolvers perform an automatic matrix transformation according to each gimbal angle, the yaw torques must be resolved by roll and pitch transformations. The gimbal angle is never measured, similar sensing platforms are used on aircraft. In inertial navigation systems, gimbal lock may occur when vehicle rotation causes two of the three rings to align with their pivot axes in a single plane. When this occurs, it is no longer possible to maintain the sensing platforms orientation, to control roll, twin engines with differential pitch or yaw control signals are used to provide torque about the vehicles roll axis
27.
Compass
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A compass is an instrument used for navigation and orientation that shows direction relative to the geographic cardinal directions, or points. Usually, a called a compass rose shows the directions north, south, east. When the compass is used, the rose can be aligned with the geographic directions, so, for example. Frequently, in addition to the rose or sometimes instead of it, North corresponds to zero degrees, and the angles increase clockwise, so east is 90 degrees, south is 180, and west is 270. These numbers allow the compass to show azimuths or bearings, which are stated in this notation. The magnetic compass was first invented as a device for divination as early as the Chinese Han Dynasty, the first usage of a compass recorded in Western Europe and the Islamic world occurred around the early 13th century. The magnetic compass is the most familiar compass type and it functions as a pointer to magnetic north, the local magnetic meridian, because the magnetized needle at its heart aligns itself with the horizontal component of the Earths magnetic field. The needle is mounted on a pivot point, in better compasses a jewel bearing. When the compass is level, the needle turns until, after a few seconds to allow oscillations to die out. In navigation, directions on maps are usually expressed with reference to geographical or true north, the direction toward the Geographical North Pole, the rotation axis of the Earth. Depending on where the compass is located on the surface of the Earth the angle between north and magnetic north, called magnetic declination can vary widely with geographic location. The local magnetic declination is given on most maps, to allow the map to be oriented with a parallel to true north. The location of the Earths magnetic poles slowly change with time, the effect of this means a map with the latest declination information should be used. Some magnetic compasses include means to compensate for the magnetic declination. The first compasses in ancient Han dynasty China were made of lodestone, the compass was later used for navigation during the Song Dynasty of the 11th century. Later compasses were made of iron needles, magnetized by striking them with a lodestone, dry compasses began to appear around 1300 in Medieval Europe and the Islamic world. This was supplanted in the early 20th century by the magnetic compass. Modern compasses usually use a needle or dial inside a capsule completely filled with a liquid
28.
Gyroscope
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A gyroscope is a spinning wheel or disc in which the axis of rotation is free to assume any orientation by itself. When rotating, the orientation of this axis is unaffected by tilting or rotation of the mounting, according to the conservation of angular momentum, because of this, gyroscopes are useful for measuring or maintaining orientation. Due to their precision, gyroscopes are used in gyrotheodolites to maintain direction in tunnel mining. Gyroscopes can be used to construct gyrocompasses, which complement or replace magnetic compasses, a gyroscope is a wheel mounted in two or three gimbals, which are a pivoted supports that allow the rotation of the wheel about a single axis. In the case of a gyroscope with two gimbals, the gimbal, which is the gyroscope frame, is mounted so as to pivot about an axis in its own plane determined by the support. This outer gimbal possesses one degree of freedom and its axis possesses none. The inner gimbal is mounted in the frame so as to pivot about an axis in its own plane that is always perpendicular to the pivotal axis of the gyroscope frame. This inner gimbal has two degrees of rotational freedom, the axle of the spinning wheel defines the spin axis. The rotor is constrained to spin about an axis, which is perpendicular to the axis of the inner gimbal. So the rotor possesses three degrees of freedom and its axis possesses two. The wheel responds to a force applied to the axis by a reaction force to the output axis. The behaviour of a gyroscope can be most easily appreciated by consideration of the front wheel of a bicycle. If the wheel is leaned away from the vertical so that the top of the moves to the left. In other words, rotation on one axis of the turning wheel produces rotation of the third axis, a gyroscope flywheel will roll or resist about the output axis depending upon whether the output gimbals are of a free or fixed configuration. Examples of some free-output-gimbal devices would be the attitude reference gyroscopes used to sense or measure the pitch, roll, the centre of gravity of the rotor can be in a fixed position. Some gyroscopes have mechanical equivalents substituted for one or more of the elements, for example, the spinning rotor may be suspended in a fluid, instead of being pivotally mounted in gimbals. In some special cases, the outer gimbal may be omitted so that the rotor has two degrees of freedom. Essentially, a gyroscope is a top combined with a pair of gimbals, Tops were invented in many different civilizations, including classical Greece, Rome, and China
29.
Driveshaft
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Charlie Pace is a fictional character on ABCs Lost, a television series chronicling the lives of plane crash survivors on a mysterious tropical island. Played by Dominic Monaghan, Charlie was a character in the first three seasons, and continued to make occasional appearances until the final season. Charlie is introduced as one of the characters in the pilot episode. Flashbacks from the show that prior to the plane crash. Initially, Charlie battles with an addiction to heroin and he ultimately achieves sobriety, while establishing a romantic relationship with Claire Littleton, and acting as a father figure to Claires son Aaron. In addition, Charlie also develops friendships with Hugo Hurley Reyes. In season three of the show, Charlie begins to face his death when fellow castaway Desmond repeatedly foresees Charlies demise. In the season three finale Through the Looking Glass, Charlie drowns at the hands of Mikhail Bakunin, sacrificing himself in an effort to save the other survivors. Due to Losts non-linear presentation of events and supernatural elements, Charlie makes several appearances in seasons four, Charlie is prominent in the series finale, The End. Charlie was born in 1979 to Simon and Megan Pace, and lived in Manchester, when Charlie was young, he was given a piano as a gift on Christmas Day, thus beginning his career in music. Charlie honed his talents by playing on the street for money. Sometime later, Charlie and his brother Liam had formed a band called Drive Shaft, suddenly, Drive Shaft became extremely popular for their song You All Everybody, although they were a one hit wonder. Liam, amidst their popularity and success, turned to heroin, on Christmas Day, while touring Finland, Liam gave Charlie a family heirloom, saying because of his addiction he would never have a family so it should go to Charlie. Eventually, though, Charlie became addicted to heroin, as the bands fame began to decrease, Charlie sank further into his addiction. Liam sold his piano so he could enroll into rehabilitation, after dropping his newborn daughter, Liam left Charlie alone, in order to be with his wife, Karen, and daughter, Megan. With Drive Shaft disbanded, Charlie resorted to theft to support his heroin addiction and he charmed a wealthy woman, Lucy, in order to rob her, but she learned of his intentions and left him as he began to develop real feelings for her. Charlie then traveled to Australia to persuade a sober Liam to rejoin the band, Liam refused, offering instead to help him enter a rehabilitation program, but Charlie angrily refused and left to board a plane to Los Angeles the next day. The night before the flight, he took heroin with a woman named Lily, on the plane, he struggles without any heroin and goes to the bathroom to take some, when the plane begins to crash
30.
Universal joint
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A universal joint is a joint or coupling in a rigid rod that allows the rod to bend in any direction, and is commonly used in shafts that transmit rotary motion. It consists of a pair of hinges located close together, oriented at 90° to each other, the universal joint is not a constant-velocity joint. The main concept of the joint is based on the design of gimbals. One anticipation of the joint was its use by the ancient Greeks on ballistae. The mechanism was described in Technica curiosa sive mirabilia artis by Gaspar Schott. The first recorded use of the universal joint for this device was by Hooke in 1676. He published a description in 1678, resulting in the use of the term Hookes joint in the English-speaking world, christopher Polhem of Sweden later re-invented the universal joint, giving rise to the name Polhemsknut in Swedish. In 1841, the English scientist Robert Willis analyzed the motion of the universal joint, by 1845, the French engineer and mathematician Jean-Victor Poncelet had analyzed the movement of the universal joint using spherical trigonometry. The term universal joint was used in the 18th century and was in use in the 19th century. Edmund Morewoods 1844 patent for a metal coating machine called for a joint, by that name. Ephriam Shays locomotive patent of 1881, for example, used double universal joints in the drive shaft. Charles Amidon used a much smaller universal joint in his bit-brace patented 1884, beauchamp Towers spherical, rotary, high speed steam engine used an adaptation of the universal joint circa 1885. The term Cardan joint appears to be a latecomer to the English language, many early uses in the 19th century appear in translations from French or are strongly influenced by French usage. Examples include an 1868 report on the Exposition Universelle of 1867 and these variables are illustrated in the diagram on the right. Also shown are a set of fixed coordinate axes with unit vectors x ^ and y ^ and these planes of rotation are perpendicular to the axes of rotation and do not move as the axles rotate. The two axles are joined by a gimbal which is not shown, however, axle 1 attaches to the gimbal at the red points on the red plane of rotation in the diagram, and axle 2 attaches at the blue points on the blue plane. Coordinate systems fixed with respect to the axles are defined as having their x-axis unit vectors pointing from the origin towards one of the connection points. This configuration uses two U-joints joined by a shaft, with the second U-joint phased in relation to the first U-joint to cancel the changing angular velocity
31.
Hypocycloid
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In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid but instead of the circle rolling along a line, if k is an integer, then the curve is closed, and has k cusps. Specially for k=2 the curve is a line and the circles are called Cardano circles. Girolamo Cardano was the first to describe these hypocycloids and their applications to high-speed printing, if k is a rational number, say k = p/q expressed in simplest terms, then the curve has p cusps. If k is a number, then the curve never closes. A hypocycloid with three cusps is known as a deltoid, a hypocycloid curve with four cusps is known as an astroid. The hypocycloid with two cusps is a degenerate but still very interesting case, known as the Tusi couple and this motion looks like rolling, though it is not technically rolling in the sense of classical mechanics, since it involves slipping. Hypocycloid shapes can be related to special groups, denoted SU. For example, the values of the sum of diagonal entries for a matrix in SU, are precisely the points in the complex plane lying inside a hypocycloid of three cusps. Likewise, summing the entries of SU matrices give points inside an astroid. Thanks to this result, one can use the fact that SU fits inside SU as a subgroup to prove that an epicycloid with k cusps moves snugly inside one with k+1 cusps. The evolute of a hypocycloid is a version of the hypocycloid itself. The pedal of a hypocycloid with pole at the center of the hypocycloid is a rose curve, the isoptic of a hypocycloid is a hypocycloid. Curves similar to hypocyloids can be drawn with the Spirograph toy, specifically, the Spirograph can draw hypotrochoids and epitrochoids. The Pittsburgh Steelers logo, which is based on the Steelmark, in his weekly NFL. com column Tuesday Morning Quarterback, Gregg Easterbrook often refers to the Steelers as the Hypocycloids. Chilean soccer team CD Huachipato based their crest on the Steelers logo, the first Drew Carey season of The Price Is Rights set features astroids on the three main doors, giant price tag, and the turntable area. The astroids on the doors and turntable were removed when the show switched to high definition starting in 2008. Special cases, Astroid, Deltoid List of periodic functions Cyclogon Epicycloid Hypotrochoid Epitrochoid Spirograph Flag of Portland, Oregon, a catalog of special plane curves
32.
Printing press
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A printing press is a device for applying pressure to an inked surface resting upon a print medium, thereby transferring the ink. The printing press was invented in the Holy Roman Empire by the German Johannes Gutenberg around 1440, the printing press spread within several decades to over two hundred cities in a dozen European countries. By 1500, printing presses in operation throughout Western Europe had already produced more than twenty million volumes, in the 16th century, with presses spreading further afield, their output rose tenfold to an estimated 150 to 200 million copies. The operation of a press became so synonymous with the enterprise of printing that it lent its name to a new branch of media. The sharp rise of learning and literacy amongst the middle class led to an increased demand for books which the time-consuming hand-copying method fell far short of accommodating. Technologies preceding the press led to the presss invention included, manufacturing of paper, development of ink, woodblock printing. At the same time, a number of products and technological processes had reached a level of maturity which allowed their potential use for printing purposes. The device was used from very early on in urban contexts as a cloth press for printing patterns. Gutenberg may have also inspired by the paper presses which had spread through the German lands since the late 14th century. Gutenberg adopted the design, thereby mechanizing the printing process. Printing, however, put a demand on the quite different from pressing. Gutenberg adapted the construction so that the power exerted by the platen on the paper was now applied both evenly and with the required sudden elasticity. To speed up the process, he introduced a movable undertable with a plane surface on which the sheets could be swiftly changed. The known examples range from Germany to England to Italy, however, the various techniques employed did not have the refinement and efficiency needed to become widely accepted. Gutenberg greatly improved the process by treating typesetting and printing as two separate work steps, a goldsmith by profession, he created his type pieces from a lead-based alloy which suited printing purposes so well that it is still used today. The mass production of metal letters was achieved by his key invention of a hand mould. Another factor conducive to printing arose from the existing in the format of the codex. Considered the most important advance in the history of the prior to printing itself
33.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
34.
Cubic equation
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In algebra, a cubic function is a function of the form f = a x 3 + b x 2 + c x + d, where a is nonzero. Setting f =0 produces an equation of the form. The solutions of this equation are called roots of the polynomial f, If all of the coefficients a, b, c, and d of the cubic equation are real numbers then there will be at least one real root. All of the roots of the equation can be found algebraically. The roots can also be found trigonometrically, alternatively, numerical approximations of the roots can be found using root-finding algorithms like Newtons method. The coefficients do not need to be complex numbers, much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3. The solutions of the cubic equation do not necessarily belong to the field as the coefficients. For example, some cubic equations with rational coefficients have roots that are complex numbers. Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, the problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, in the 3rd century, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations. In the 11th century, the Persian poet-mathematician, Omar Khayyám, in an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution, in the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of an equation, x3 + 12x = 6x2 +35. He used what would later be known as the Ruffini-Horner method to approximate the root of a cubic equation. He also developed the concepts of a function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the equation to find algebraic solutions to certain types of cubic equations. Leonardo de Pisa, also known as Fibonacci, was able to approximate the positive solution to the cubic equation x3 + 2x2 + 10x =20
35.
Quartic equation
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Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function of a square, having the form f = a x 4 + c x 2 + e. The derivative of a function is a cubic function. Since a quartic function is defined by a polynomial of even degree, If a is positive, then the function increases to positive infinity at both ends, and thus the function has a global minimum. Likewise, if a is negative, it decreases to negative infinity and has a global maximum, in both cases it may or may not have another local maximum and another local minimum. The degree four is the highest degree such that every polynomial equation can be solved by radicals, the solution of the quartic was published together with that of the cubic by Ferraris mentor Gerolamo Cardano in the book Ars Magna. Depman claimed that even earlier, in 1486, Spanish mathematician Valmes was burned at the stake for claiming to have solved the quartic equation, inquisitor General Tomás de Torquemada allegedly told Valmes that it was the will of God that such a solution be inaccessible to human understanding. However Beckmann, who popularized this story of Depman in the West, said that it was unreliable, beckmanns version of this story has been widely copied in several books and internet sites, usually without his reservations and sometimes with fanciful embellishments. Several attempts to find corroborating evidence for this story, or even for the existence of Valmes, have failed. The notes left by Évariste Galois prior to dying in a duel in 1832 later led to an elegant complete theory of the roots of polynomials, each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus and it follows that quartic equations often arise in computational geometry and all related fields such as computer graphics, computer-aided design, computer-aided manufacturing and optics. Here are example of geometric problems whose solution amounts of solving a quartic equation. In computer-aided manufacturing, the torus is a shape that is associated with the endmill cutter. In optics, Alhazens problem is Given a light source and a spherical mirror and this leads to a quartic equation. Finding the distance of closest approach of two ellipses involves solving a quartic equation, the eigenvalues of a 4×4 matrix are the roots of a quartic polynomial which is the characteristic polynomial of the matrix. The characteristic equation of a linear difference equation or differential equation is a quartic equation. An example arises in the Timoshenko-Rayleigh theory of beam bending, intersections between spheres, cylinders, or other quadrics can be found using quartic equations. Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line, one of those regions is disjointed into sub-regions of equal area. The possible cases for the nature of the roots are as follows, If ∆ >0 then either the equations four roots are all real or none is
36.
Imaginary number
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An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2, for example, 5i is an imaginary number, and its square is −25. Zero is considered to be real and imaginary. Originally coined in the 17th century as a term and regarded as fictitious or useless. Some authors use the term pure imaginary number to denote what is called here an imaginary number, the concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time imaginary numbers, as well as numbers, were poorly understood and regarded by some as fictitious or useless. The use of numbers was not widely accepted until the work of Leonhard Euler. The geometric significance of numbers as points in a plane was first described by Caspar Wessel. This idea first surfaced with the articles by James Cockle beginning in 1848, geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. One way of viewing imaginary numbers is to consider a number line, positively increasing in magnitude to the right. This vertical axis is called the imaginary axis and is denoted iℝ, I. In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin, note that a 90-degree rotation in the negative direction also satisfies this interpretation. This reflects the fact that −i also solves the equation x2 = −1, in general, multiplying by a complex number is the same as rotating around the origin by the complex numbers argument, followed by a scaling by its magnitude. Care must be used when working with numbers expressed as the principal values of the square roots of negative numbers. For example,6 =36 = ≠ −4 −9 = =6 i 2 = −6, Imaginary unit de Moivres formula NaN Octonion Quaternion Nahin, Paul. An Imaginary Tale, the Story of the Square Root of −1, explains many applications of imaginary expressions. How can one show that imaginary numbers really do exist, – an article that discusses the existence of imaginary numbers. In our time, Imaginary numbers Discussion of imaginary numbers on BBC Radio 4, 5Numbers programme 4 BBC Radio 4 programme Why Use Imaginary Numbers
37.
Illegitimate
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Legitimacy, in traditional Western common law, is the status of a child born to parents who are legally married to each other, and of a child conceived before the parents obtain a legal divorce. Conversely, illegitimacy is the status of a child born outside marriage, depending on the cultural context, legitimacy can affect a childs rights of inheritance to the putative fathers estate and the childs right to bear the fathers surname or title. Illegitimacy has also had consequences for the mothers and childs right to support from the putative father, in medieval Wales, a bastard was defined simply as a child not acknowledged by its father. All children, whether born in or out of wedlock, that were acknowledged by the father enjoyed the legal rights. Englands Statute of Merton stated, regarding illegitimacy, He is a bastard that is born before the marriage of his parents and this definition also applied to situations when a childs parents could not marry, as when one or both were already married or when the relationship was incestuous. The Poor Law of 1576 formed the basis of English bastardy law and its purpose was to punish a bastard childs mother and putative father, and to relieve the parish from the cost of supporting mother and child. By an act of 1576, it was ordered that bastards should be supported by their putative fathers, if the genitor could be found, then he was put under very great pressure to accept responsibility and to maintain the child. Under English law, a bastard was unable to be an heir to real property, in contrast to the situation under civil law, a younger non-bastard brother would have no claim to the land. The Legitimacy Act 1926 of England and Wales legitimized the birth of a if the parents subsequently married each other. The Legitimacy Act 1959 extended the legitimization even if the parents had married others in the meantime, neither the 1926 nor 1959 Acts changed the laws of succession to the British throne and succession to peerage titles. The Family Law Reform Act 1969 allowed a bastard to inherit on the intestacy of his parents, in canon and in civil law, the offspring of putative marriages have also been considered legitimate. Since 2003 in England and Wales,2002 in Northern Ireland and 2006 in Scotland, still, children born out of wedlock may not be eligible for certain federal benefits unless the child has been legitimized in the appropriate jurisdiction. Many other countries have abolished by any legal disabilities of a child born out of wedlock. In France, legal reforms regarding illegitimacy began in the 1970s, the European Convention on the Legal Status of Children Born out of Wedlock came into force in 1978. Countries which ratify it must ensure that children born outside marriage are provided with legal rights as stipulated in the text of this Convention, the Convention was ratified by the UK in 1981 and by Ireland in 1988. Use of the illegitimate child is now rare, even in legal contexts. It has been stricken from passports and legal documents as needlessly insulting and stigmatizing to the child, terms such as extra-marital child, love child and child born out of wedlock are more commonly used. Also used in Britain and other English-speaking countries is bastard, though such as natural child are preferred in polite society
38.
Jurist
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A jurist, also known as legal scholar or legal theorist, is someone who researches and studies jurisprudence. Such a person can work as an academic, legal writer or law lecturer, thus a jurist, someone who studies, analyses and comments on law, stands in contrast with a lawyer, someone who applies law on behalf of clients and thinks about it in practical terms. Many legal scholars and authors have explained that a person may be both a lawyer and a jurist, but a jurist is not necessarily a lawyer, nor a lawyer necessarily a jurist, both must possess an acquaintance with the term law. The work of the jurist is the study, analysis and arrangement of the law — work which can be wholly in the seclusion of the library. Any highly civilized society requires both lawyers and jurists, both philosophers and doers and it is important however to note the fundamental difference between the work of the lawyer and that of the jurist. The term jurist has another sense, which is wider, synonymous with legal professional, i. e. anyone professionally involved with law, in some other European languages, a word resembling jurist is used in this major sense. This is a classification of some notable jurists. History of the legal profession Law professor Legal profession List of jurists Paralegal Media related to Jurists at Wikimedia Commons
39.
Leonardo da Vinci
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He has been variously called the father of palaeontology, ichnology, and architecture, and is widely considered one of the greatest painters of all time. Sometimes credited with the inventions of the parachute, helicopter and tank, many historians and scholars regard Leonardo as the prime exemplar of the Universal Genius or Renaissance Man, an individual of unquenchable curiosity and feverishly inventive imagination. Much of his working life was spent in the service of Ludovico il Moro in Milan. He later worked in Rome, Bologna and Venice, and he spent his last years in France at the home awarded to him by Francis I of France, Leonardo was, and is, renowned primarily as a painter. Among his works, the Mona Lisa is the most famous and most parodied portrait, Leonardos drawing of the Vitruvian Man is also regarded as a cultural icon, being reproduced on items as varied as the euro coin, textbooks, and T-shirts. Perhaps fifteen of his paintings have survived, Leonardo is revered for his technological ingenuity. He conceptualised flying machines, a type of armoured fighting vehicle, concentrated power, an adding machine. Some of his inventions, however, such as an automated bobbin winder. A number of Leonardos most practical inventions are nowadays displayed as working models at the Museum of Vinci. He made substantial discoveries in anatomy, civil engineering, geology, optics, and hydrodynamics, today, Leonardo is widely considered one of the most diversely talented individuals ever to have lived. Leonardo was born on 15 April 1452 at the hour of the night in the Tuscan hill town of Vinci. He was the son of the wealthy Messer Piero Fruosino di Antonio da Vinci, a Florentine legal notary, and Caterina. Leonardo had no surname in the modern sense – da Vinci simply meaning of Vinci, his birth name was Lionardo di ser Piero da Vinci, meaning Leonardo. The inclusion of the title ser indicated that Leonardos father was a gentleman, little is known about Leonardos early life. He spent his first five years in the hamlet of Anchiano in the home of his mother and his father had married a sixteen-year-old girl named Albiera Amadori, who loved Leonardo but died young in 1465 without children. When Leonardo was sixteen, his father married again to twenty-year-old Francesca Lanfredini, pieros legitimate heirs were born from his third wife Margherita di Guglielmo and his fourth and final wife, Lucrezia Cortigiani. Leonardo received an education in Latin, geometry and mathematics. In later life, Leonardo recorded only two childhood incidents, one, which he regarded as an omen, was when a kite dropped from the sky and hovered over his cradle, its tail feathers brushing his face
40.
Milan
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Milan is a city in Italy, capital of the Lombardy region, and the most populous metropolitan area and the second most populous comune in Italy. The population of the city proper is 1,351,000, Milan has a population of about 8,500,000 people. It is the industrial and financial centre of Italy and one of global significance. In terms of GDP, it has the largest economy among European non-capital cities, Milan is considered part of the Blue Banana and lies at the heart of one of the Four Motors for Europe. Milan is an Alpha leading global city, with strengths in the arts, commerce, design, education, entertainment, fashion, finance, healthcare, media, services, research, and tourism. Its business district hosts Italys Stock Exchange and the headquarters of the largest national and international banks, the city is a major world fashion and design capital, well known for several international events and fairs, including Milan Fashion Week and the Milan Furniture Fair. The city hosts numerous cultural institutions, academies and universities, with 11% of the national total enrolled students, Milans museums, theatres and landmarks attract over 9 million visitors annually. Milan – after Naples – is the second Italian city with the highest number of accredited stars from the Michelin Guide, the city hosted the Universal Exposition in 1906 and 2015. Milan is home to two of Europes major football teams, A. C. Milan and F. C. Internazionale, the etymology of Milan is uncertain. One theory holds that the Latin name Mediolanum comes from the Latin words medio, however, some scholars believe lanum comes from the Celtic root lan, meaning an enclosure or demarcated territory in which Celtic communities used to build shrines. Hence, Mediolanum could signify the central town or sanctuary of a Celtic tribe, indeed, the name Mediolanum is borne by about sixty Gallo-Roman sites in France, e. g. Saintes and Évreux. Alciato credits Ambrose for his account, around 400 BC, the Celtic Insubres settled Milan and the surrounding region. In 222 BC, the Romans conquered the settlement, renaming it Mediolanum, Milan was eventually declared the capital of the Western Roman Empire by Emperor Diocletian in 286 AD. Diocletian chose to stay in the Eastern Roman Empire and his colleague Maximianus ruled the Western one, immediately Maximian built several monuments, such as a large circus 470 m ×85 m, the Thermae Herculeae, a large complex of imperial palaces and several other buildings. With the Edict of Milan of 313, Emperor Constantine I guaranteed freedom of religion for Christians, after the city was besieged by the Visigoths in 402, the imperial residence was moved to Ravenna. In 452, the Huns overran the city, in 539, the Ostrogoths conquered and destroyed Milan during the Gothic War against Byzantine Emperor Justinian I. In the summer of 569, a Teutonic tribe, the Lombards, conquered Milan, some Roman structures remained in use in Milan under Lombard rule. Milan surrendered to the Franks in 774 when Charlemagne took the title of King of the Lombards, the Iron Crown of Lombardy dates from this period
41.
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
42.
Ars Magna (Gerolamo Cardano)
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The Ars Magna is an important book on algebra written by Girolamo Cardano. It was first published in 1545 under the title Artis Magnæ, there was a second edition in Cardanos lifetime, published in 1570. It is considered one of the three greatest scientific treatises of the early Renaissance, together with Copernicus De revolutionibus orbium coelestium, the first editions of these three books were published within a two-year span. In 1535 Niccolò Fontana Tartaglia became famous for having solved cubics of the form x3 + ax = b, however, he chose to keep his method secret. In 1539, Cardano, then a lecturer in mathematics at the Piatti Foundation in Milan, published his first mathematical book and that same year, he asked Tartaglia to explain to him his method for solving cubic equations. After some reluctance, Tartaglia did so, but he asked Cardano not to share the information until he published it, Cardano submerged himself in mathematics during the next several years working on how to extend Tartaglias formula to other types of cubics. Furthermore, his student Lodovico Ferrari found a way of solving quartic equations, then Cardano become aware of the fact that Scipione del Ferro had discovered Tartaglias formula before Tartaglia himself, a discovery that prompted him to publish these results. The book, which is divided into forty chapters, contains the first published solution to cubic and quartic equations, Cardano acknowledges that Tartaglia gave him the formula for solving a type of cubic equations and that the same formula had been discovered by Scipione del Ferro. He also acknowledges that it was Ferrari who found a way of solving quartic equations. Since at the negative numbers were not generally acknowledged, knowing how to solve cubics of the form x3 + ax = b did not mean knowing how to solve cubics of the form x3 = ax + b. Besides, Cardano, also explains how to reduce equations of the form x3 + ax2 + bx + c =0 to cubic equations without a quadratic term, in all, Cardano was driven to the study of thirteen different types of cubic equations. In Ars Magna the concept of multiple root appears for the first time, the first example that Cardano provides of a polynomial equation with multiple roots is x3 = 12x +16, of which −2 is a double root. Ars Magna also contains the first occurrence of complex numbers, the problem mentioned by Cardano which leads to square roots of negative numbers is, find two numbers whose sum is equal to 10 and whose product is equal to 40. The answer is 5 + √−15 and 5 − √−15, Cardano called this sophistic, because he saw no physical meaning to it, but boldly wrote nevertheless we will operate and formally calculated that their product does indeed equal 40. Cardano then says that this answer is “as subtle as it is useless” and it is a common misconception that Cardano introduced complex numbers in solving cubic equations. However, q2/4 + p3/27 never happens to be negative in the cases in which Cardano applies the formula. pdf of Ars Magna Cardanos biography
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Binomial coefficient
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In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a coefficient is indexed by a pair of integers n ≥ k ≥0 and is written. It is the coefficient of the xk term in the expansion of the binomial power n. The value of the coefficient is given by the expression n. k, arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascals triangle. The properties of binomial coefficients have led to extending the definition to beyond the case of integers n ≥ k ≥0. Andreas von Ettingshausen introduced the notation in 1826, although the numbers were known centuries earlier, the earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingalas Chandaḥśāstra. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī, alternative notations include C, nCk, nCk, Ckn, Cnk, and Cn, k in all of which the C stands for combinations or choices. Many calculators use variants of the C notation because they can represent it on a single-line display, in this form the binomial coefficients are easily compared to k-permutations of n, written as P, etc. For natural numbers n and k, the binomial coefficient can be defined as the coefficient of the monomial Xk in the expansion of n, the same coefficient also occurs in the binomial formula, which explains the name binomial coefficient. This shows in particular that is a number for any natural numbers n and k. Most of these interpretations are easily seen to be equivalent to counting k-combinations, several methods exist to compute the value of without actually expanding a binomial power or counting k-combinations. It also follows from tracing the contributions to Xk in n−1, as there is zero Xn+1 or X−1 in n, one might extend the definition beyond the above boundaries to include =0 when either k > n or k <0. This recursive formula then allows the construction of Pascals triangle, surrounded by white spaces where the zeros, or the trivial coefficients, a more efficient method to compute individual binomial coefficients is given by the formula = n k _ k. = n ⋯ k ⋯1 = ∏ i =1 k n +1 − i i and this formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, the denominator counts the number of distinct sequences that define the same k-combination when order is disregarded. Due to the symmetry of the binomial coefficient with regard to k and n−k, calculation may be optimised by setting the limit of the product above to the smaller of k. This formula follows from the formula above by multiplying numerator and denominator by. As a consequence it involves many factors common to numerator and denominator and it is less practical for explicit computation unless common factors are first cancelled
44.
Chess
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Chess is a two-player strategy board game played on a chessboard, a checkered gameboard with 64 squares arranged in an eight-by-eight grid. Chess is played by millions of people worldwide, both amateurs and professionals, each player begins the game with 16 pieces, one king, one queen, two rooks, two knights, two bishops, and eight pawns. Each of the six piece types moves differently, with the most powerful being the queen, the objective is to checkmate the opponents king by placing it under an inescapable threat of capture. To this end, a players pieces are used to attack and capture the opponents pieces, in addition to checkmate, the game can be won by voluntary resignation by the opponent, which typically occurs when too much material is lost, or if checkmate appears unavoidable. A game may result in a draw in several ways. Chess is believed to have originated in India, some time before the 7th century, chaturanga is also the likely ancestor of the Eastern strategy games xiangqi, janggi and shogi. The pieces took on their current powers in Spain in the late 15th century, the first generally recognized World Chess Champion, Wilhelm Steinitz, claimed his title in 1886. Since 1948, the World Championship has been controlled by FIDE, the international governing body. There is also a Correspondence Chess World Championship and a World Computer Chess Championship, online chess has opened amateur and professional competition to a wide and varied group of players. There are also many variants, with different rules, different pieces. FIDE awards titles to skilled players, the highest of which is grandmaster, many national chess organizations also have a title system. However, these are not recognised by FIDE, the term master may refer to a formal title or may be used more loosely for any skilled player. Until recently, chess was a sport of the International Olympic Committee. Chess was included in the 2006 and 2010 Asian Games, since the 1990s, computer analysis has contributed significantly to chess theory, particularly in the endgame. The computer IBM Deep Blue was the first machine to overcome a reigning World Chess Champion in a match when it defeated Garry Kasparov in 1997, the rise of strong computer programs that can be run on hand-held devices has led to increasing concerns about cheating during tournaments. The official rules of chess are maintained by FIDE, chesss international governing body, along with information on official chess tournaments, the rules are described in the FIDE Handbook, Laws of Chess section. Chess is played on a board of eight rows and eight columns. The colors of the 64 squares alternate and are referred to as light, the chessboard is placed with a light square at the right-hand end of the rank nearest to each player
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Chisholm, Hugh, ed. (1911). "Cardan, Girolamo". Encyclopædia Britannica (11th ed.). Cambridge University Press.
Chisholm, Hugh, ed. (1911). "
