1.
1640s in architecture
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The Taj Mahal in Agra, India, is under construction. The mosque and jawab in the complex are completed in 1643,1640 Børsen in Copenhagen, designed by Lorentz and Hans van Steenwinckel the Younger and begun in 1619, is completed. 59-60 Lincolns Inn Fields, London, probably designed by Inigo Jones,1641 Tron Kirk in Edinburgh, Scotland, designed by John Mylne, is dedicated. The Mauritshuis at The Hague in the Dutch Republic, designed by Jacob van Campen, 1645–1648 - Main structure of Potala Palace in Lhasa, Tibet, is built. 1646 The St. Mary Magdalene Chapel, Dingli, Malta, is rebuilt after the chapel had collapsed. Chehel Sotoun in Isfahan, Persia, is completed,1647 - The Changdeokgung in Seoul, Korea, is reconstructed. 1648 - Jama Masjid, Agra, is built,1642 - Giovanni Barbara, Maltese architect and military engineer c

2.
Science
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Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. The formal sciences are often excluded as they do not depend on empirical observations, disciplines which use science, like engineering and medicine, may also be considered to be applied sciences. However, during the Islamic Golden Age foundations for the method were laid by Ibn al-Haytham in his Book of Optics. In the 17th and 18th centuries, scientists increasingly sought to formulate knowledge in terms of physical laws, over the course of the 19th century, the word science became increasingly associated with the scientific method itself as a disciplined way to study the natural world. It was during this time that scientific disciplines such as biology, chemistry, Science in a broad sense existed before the modern era and in many historical civilizations. Modern science is distinct in its approach and successful in its results, Science in its original sense was a word for a type of knowledge rather than a specialized word for the pursuit of such knowledge. In particular, it was the type of knowledge which people can communicate to each other, for example, knowledge about the working of natural things was gathered long before recorded history and led to the development of complex abstract thought. This is shown by the construction of calendars, techniques for making poisonous plants edible. For this reason, it is claimed these men were the first philosophers in the strict sense and they were mainly speculators or theorists, particularly interested in astronomy. In contrast, trying to use knowledge of nature to imitate nature was seen by scientists as a more appropriate interest for lower class artisans. A clear-cut distinction between formal and empirical science was made by the pre-Socratic philosopher Parmenides, although his work Peri Physeos is a poem, it may be viewed as an epistemological essay on method in natural science. Parmenides ἐὸν may refer to a system or calculus which can describe nature more precisely than natural languages. Physis may be identical to ἐὸν and he criticized the older type of study of physics as too purely speculative and lacking in self-criticism. He was particularly concerned that some of the early physicists treated nature as if it could be assumed that it had no intelligent order, explaining things merely in terms of motion and matter. The study of things had been the realm of mythology and tradition, however. Aristotle later created a less controversial systematic programme of Socratic philosophy which was teleological and he rejected many of the conclusions of earlier scientists. For example, in his physics, the sun goes around the earth, each thing has a formal cause and final cause and a role in the rational cosmic order. Motion and change is described as the actualization of potentials already in things, while the Socratics insisted that philosophy should be used to consider the practical question of the best way to live for a human being, they did not argue for any other types of applied science

3.
Technology
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Technology is the collection of techniques, skills, methods and processes used in the production of goods or services or in the accomplishment of objectives, such as scientific investigation. Technology can be the knowledge of techniques, processes, and the like, the human species use of technology began with the conversion of natural resources into simple tools. The steady progress of technology has brought weapons of ever-increasing destructive power. It has helped develop more advanced economies and has allowed the rise of a leisure class, many technological processes produce unwanted by-products known as pollution and deplete natural resources to the detriment of Earths environment. Various implementations of technology influence the values of a society and raise new questions of the ethics of technology, examples include the rise of the notion of efficiency in terms of human productivity, and the challenges of bioethics. Philosophical debates have arisen over the use of technology, with disagreements over whether technology improves the condition or worsens it. The use of the technology has changed significantly over the last 200 years. Before the 20th century, the term was uncommon in English, the term was often connected to technical education, as in the Massachusetts Institute of Technology. The term technology rose to prominence in the 20th century in connection with the Second Industrial Revolution, the terms meanings changed in the early 20th century when American social scientists, beginning with Thorstein Veblen, translated ideas from the German concept of Technik into technology. In German and other European languages, a distinction exists between technik and technologie that is absent in English, which translates both terms as technology. By the 1930s, technology referred not only to the study of the industrial arts, dictionaries and scholars have offered a variety of definitions. Ursula Franklin, in her 1989 Real World of Technology lecture, gave another definition of the concept, it is practice, the way we do things around here. The term is used to imply a specific field of technology, or to refer to high technology or just consumer electronics. Bernard Stiegler, in Technics and Time,1, defines technology in two ways, as the pursuit of life by other than life, and as organized inorganic matter. Technology can be most broadly defined as the entities, both material and immaterial, created by the application of mental and physical effort in order to some value. In this usage, technology refers to tools and machines that may be used to solve real-world problems and it is a far-reaching term that may include simple tools, such as a crowbar or wooden spoon, or more complex machines, such as a space station or particle accelerator. Tools and machines need not be material, virtual technology, such as software and business methods. W. Brian Arthur defines technology in a broad way as a means to fulfill a human purpose

4.
John Parkinson (botanist)
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John Parkinson was the last of the great English herbalists and one of the first of the great English botanists. He was apothecary to James I and a member of the Worshipful Society of Apothecaries in December 1617. Parkinson, born in 1567, spent his life in Yorkshire. He moved to London at the age of 14 years to become an apprentice apothecary, in addition, he assisted the Society in obtaining a grant of arms and in preparing a list of all medicines that should be stocked by an apothecary. He was on the committee published their Pharmacopœia Londinensis in 1618. Then, on the cusp of a new science, he became botanist to Charles I, jamess Palace, he took on the role of introducing the young queen to horticulturally sophisticated circles. Parkinson actively sought new varieties of plants through his contacts abroad and by financing William Boels plant-hunting expedition to Iberia and he introduced seven new plants into England and was the first gardener in England to grow the great double yellow Spanish daffodil. His piety as a Roman Catholic is evident from Paradisi in Sole, in his introduction, Parkinson saw the botanical world as an expression of divine creation, and believed that through gardens man could recapture something of Eden. However, struggles between Protestants and Catholics compelled Parkinson to keep a low profile and he did not attend any parish church. At the height of his success, the English Civil War tore his family apart, Parkinsons London house was in Ludgate Hill, but his botanical garden was in suburban Long Acre in Covent Garden, a district of market-gardens, today close to Trafalgar Square. Not much is known about the garden, but based on a study of the writings of Parkinson and others, John Riddell has suggested that it was at least 2 acres in size, four hundred and eighty-four types of plant are recorded as having been grown in the garden. Thomas Johnson and the Hampshire botanist, John Goodyer, both gathered seeds there, Parkinson has been called one of the most eminent gardeners of his day. Together, they belonged to the generation began to see extraordinary new plants coming from the Levant and from Virginia. In his writings, de Lobel frequently mentioned the Long Acre garden, Parkinson died in the summer of 1650, and was buried at St Martin-in-the-Fields, London, on 6 August. He is commemorated in the Central American genus of leguminous trees Parkinsonia, in the story, some children read Paradisi in Sole and are inspired to create their own garden. The magazine received much favourable correspondence about the story, and in July 1884 it was suggested that a Parkinson Society should be formed, to try to prevent the extermination of rare wild flowers, as well as of garden treasures. Paradisi in Sole Paradisus Terrestris describes the cultivation of plants in general. It contains illustrations of almost 800 plants in 108 full-page plates, in Paradisi in Sole Parkinson hinted that he hoped to add a fourth section, a garden of simples

5.
Cape Verde
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Located 570 kilometres off the coast of West Africa, the islands cover a combined area of slightly over 4,000 square kilometres. The Cape Verde archipelago was uninhabited until the 15th century, when Portuguese explorers discovered and colonized the islands, ideally located for the Atlantic slave trade, the islands grew prosperous throughout the 16th and 17th centuries, attracting merchants, privateers, and pirates. The end of slavery in the 19th century led to economic decline, Cape Verde gradually recovered as an important commercial center and stopover for shipping routes. Incorporated as a department of Portugal in 1951, the islands continued to agitate for independence. Since the early 1990s, Cape Verde has been a representative democracy. Lacking natural resources, its economy is mostly service-oriented, with a growing focus on tourism. Its population of around 512,000 is mostly of mixed European and sub-Saharan African heritage, a sizeable diaspora community exists across the world, slightly outnumbering inhabitants on the islands. Historically, the name Cape Verde has been used in English for the archipelago and, since independence in 1975, for the country. In 2013, the Cape Verdean government determined that the Portuguese designation Cabo Verde would henceforth be used for official purposes, such as at the United Nations, Cape Verde is a member of the African Union. The name of the stems from the nearby Cap-Vert, on the Senegalese coast. In 1444 Portuguese explorers had named that landmark as Cabo Verde, on 24 October 2013, the countrys delegation announced at the United Nations that the official name should no longer be translated into other languages. Instead of Cape Verde, the designation Republic of Cabo Verde is to be used, before the arrival of Europeans, the Cape Verde Islands were uninhabited. The islands of the Cape Verde archipelago were discovered by Genoese and Portuguese navigators around 1456, according to Portuguese official records, the first discoveries were made by Genoa-born António de Noli, who was afterwards appointed governor of Cape Verde by Portuguese King Afonso V. Other navigators mentioned as contributing to discoveries in the Cape Verde archipelago are Diogo Gomes, Diogo Dias, Diogo Afonso, in 1462, Portuguese settlers arrived at Santiago and founded a settlement they called Ribeira Grande. Ribeira Grande was the first permanent European settlement in the tropics, in the 16th century, the archipelago prospered from the Atlantic slave trade. Pirates occasionally attacked the Portuguese settlements, sir Francis Drake, an English corsair privateering under a letter of marque granted by the English crown, twice sacked the capital Ribeira Grande in 1585 when it was a part of the Iberian Union. After a French attack in 1712, the town declined in relative to nearby Praia. Decline in the trade in the 19th century resulted in an economic crisis

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.
Pascal's theorem
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The theorem is also valid in the Euclidean plane, but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel. The most natural setting for Pascals theorem is in a plane since all lines meet. However, with the interpretation of what happens when some opposite sides of the hexagon are parallel. If two pairs of sides are parallel, then all three pairs of opposite sides form pairs of parallel lines and there is no Pascal line in the Euclidean plane. This theorem is a generalization of Pappuss theorem – Pappuss theorem is the case of a degenerate conic of two lines. Pascals theorem is the reciprocal and projective dual of Brianchons theorem. It was formulated by Blaise Pascal in a written in 1639 when he was 16 years old. Par B. P. Pascals theorem is a case of the Cayley–Bacharach theorem. This can be proven independently using a property of pole-polar, six is the minimum number of points on a conic about which special statements can be made, as five points determine a conic. The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction, which is a construction of the conic defined by five points. Then if 2n of those points lie on a common line, if six unordered points are given on a conic section, they can be connected into a hexagon in 60 different ways, resulting in 60 different instances of Pascals theorem and 60 different Pascal lines. This configuration of 60 lines is called the Hexagrammum Mysticum, as Thomas Kirkman proved in 1849, these 60 lines can be associated with 60 points in such a way that each point is on three lines and each line contains three points. The 60 points formed in this way are now known as the Kirkman points, the Pascal lines also pass, three at a time, through 20 Steiner points. There are 20 Cayley lines which consist of a Steiner point, the Steiner points also lie, four at a time, on 15 Plücker lines. Furthermore, the 20 Cayley lines pass four at a time through 15 points known as the Salmon points, Pascals original note has no proof, but there are various modern proofs of the theorem. It is sufficient to prove the theorem when the conic is a circle and this was realised by Pascal, whose first lemma states the theorem for a circle. His second lemma states that what is true in one plane remains true upon projection to another plane, a short elementary proof of Pascals theorem in the case of a circle was found by van Yzeren, based on the proof in. This proof proves the theorem for circle and then generalizes it to conics, a short elementary computational proof in the case of the real projective plane was found by Stefanovic We can infer the proof from existence of isogonal conjugate too

8.
Marin Mersenne
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Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the father of acoustics. Mersenne, an ordained priest, had contacts in the scientific world and has been called the center of the world of science. Marin Mersenne was born of peasant parents near Oizé, Maine and he was educated at Le Mans and at the Jesuit College of La Flèche. On 17 July 1611, he joined the Minim Friars, and, after studying theology, between 1614 and 1618, he taught theology and philosophy at Nevers, but he returned to Paris and settled at the convent of LAnnonciade in 1620. There he studied mathematics and music and met with other kindred spirits such as René Descartes, Étienne Pascal, Pierre Petit, Gilles de Roberval and he corresponded with Giovanni Doni, Constantijn Huygens, Galileo Galilei, and other scholars in Italy, England and the Dutch Republic. He was a defender of Galileo, assisting him in translations of some of his mechanical works. For four years, Mersenne devoted himself entirely to philosophic and theological writing and it is sometimes incorrectly stated that he was a Jesuit. He was educated by Jesuits, but he never joined the Society of Jesus and he taught theology and philosophy at Nevers and Paris. He was not afraid to cause disputes among his friends in order to compare their views. In 1635 Mersenne met with Tommaso Campanella, but concluded that he could teach nothing in the sciences but still he has a good memory, Mersenne asked if René Descartes wanted Campanella to come to Holland to meet him, but Descartes declined. He visited Italy fifteen times, in 1640,1641 and 1645, in 1643–1644 Mersenne also corresponded with the German Socinian Marcin Ruar concerning the Copernican ideas of Pierre Gassendi, finding Ruar already a supporter of Gassendis position. Among his correspondents were Descartes, Galilei, Roberval, Pascal, Beeckman and he died September 1 through complications arising from a lung abscess. Some history scientists suggest he died for having drunk a huge quantity of water, along with Descartes. It was written as a commentary on the Book of Genesis, at first sight the book appears to be a collection of treatises on various miscellaneous topics. However Robert Lenoble has shown that the principle of unity in the work is a polemic against magical and divinatory arts, cabalism and he mentions Martin Del Rios Investigations into Magic and criticises Marsilio Ficino for claiming power for images and characters. He condemns astral magic and astrology and the anima mundi, a popular amongst Renaissance neo-platonists. Whilst allowing for an interpretation of the Cabala, he wholeheartedly condemned its magical application—particularly to angelology. He also criticises Pico della Mirandola, Cornelius Agrippa and Francesco Giorgio with Robert Fludd as his main target, Fludd responded with Sophia cum moria certamen, wherein Fludd admits his involvement with the Rosicrucians

9.
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

10.
Fermat's little theorem
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Fermats little theorem states that if p is a prime number, then for any integer a, the number a p − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as a p ≡ a, for example, if a =2 and p =7,27 =128, and 128 −2 =7 ×18 is an integer multiple of 7. If a is not divisible by p, Fermats little theorem is equivalent to the statement that a p −1 −1 is a multiple of p. For example, if a =2 and p =7 then 26 =64 and 64 −1 =63 is thus a multiple of 7, Fermats little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640 and it is called the little theorem to distinguish it from Fermats last theorem. Pierre de Fermat first stated the theorem in a letter dated October 18,1640, to his friend, an early use in English occurs in A. A. Albert, Modern Higher Algebra, which refers to the so-called little Fermat theorem on page 206, some mathematicians independently made the related hypothesis that 2p ≡2 if and only if p is a prime. Indeed, the if part is true, and is a case of Fermats little theorem. However, the if part of this hypothesis is false, for example,2341 ≡2. Several proofs of Fermats little theorem are known and it is frequently proved as a corollary of Eulers theorem. Fermats little theorem is a case of Eulers theorem, for any modulus n and any integer a coprime to n, we have a φ ≡1. Eulers theorem is indeed a generalization, because if n = p is a prime number, then φ = p −1. A slight generalization of Eulers theorem, which follows from it, is, if a, n, x, y are integers with n positive. This follows as x is of the form y + φk, in this form, the theorem finds many uses in cryptography and, in particular, underlies the computations used in the RSA public key encryption method. The special case with n a prime may be considered a consequence of Fermats little theorem, Fermats little theorem is also related to the Carmichael function and Carmichaels theorem, as well as to Lagranges theorem in group theory. The algebraic setting of Fermats little theorem can be generalized to finite fields, the converse of Fermats little theorem is not generally true, as it fails for Carmichael numbers. However, a stronger form of the theorem is true. The theorem is as follows, If there exists an a such that a p −1 ≡1 and this theorem forms the basis for the Lucas–Lehmer test, an important primality test

11.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n

12.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain

13.
Parity (mathematics)
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Parity is a mathematical term that describes the property of an integers inclusion in one of two categories, even or odd. An integer is even if it is divisible by two and odd if it is not even. For example,6 is even there is no remainder when dividing it by 2. By contrast,3,5,7,21 leave a remainder of 1 when divided by 2, examples of even numbers include −4,0,8, and 1738. In particular, zero is an even number, some examples of odd numbers are −5,3,9, and 73. Parity does not apply to non-integer numbers and this classification applies only to integers, i. e. non-integers like 1/2,4.201, or infinity are neither even nor odd. The sets of even and odd numbers can be defined as following and that is, if the last digit is 1,3,5,7, or 9, then it is odd, otherwise it is even. The same idea will work using any even base, in particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is even if. The following laws can be verified using the properties of divisibility and they are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, however, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. The structure is in fact a field with just two elements, the division of two whole numbers does not necessarily result in a whole number. For example,1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even, but when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor fully even. It is this, that two relatively different things or ideas there stands always a third, in a sort of balance. Thus, there is here between odd and even numbers one number which is neither of the two, similarly, in form, the right angle stands between the acute and obtuse angles, and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this, integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the cubic lattice and its higher-dimensional generalizations

14.
Micrometer
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Micrometers are usually, but not always, in the form of calipers, which is why micrometer caliper is another common name. The spindle is a very accurately machined screw and the object to be measured is placed between the spindle and the anvil, the spindle is moved by turning the ratchet knob or thimble until the object to be measured is lightly touched by both the spindle and the anvil. Micrometers are also used in telescopes or microscopes to measure the apparent diameter of celestial bodies or microscopic objects, the micrometer used with a telescope was invented about 1638 by William Gascoigne, an English astronomer. Colloquially the word micrometer is often shortened to mike or mic, the word micrometer is a neoclassical coinage from Greek micros, meaning small, and metron, meaning measure. The Merriam-Webster Collegiate Dictionary says that English got it from French, neither the metre nor the micrometre nor the micrometer as we know them today existed at that time. However, the people of that time did have much need for, and interest in, the word was no doubt coined in reference to this endeavor, even if it did not refer specifically to its present-day senses. In 1844 details of Whitworths workshop micrometer were published and this was described as having a strong frame of cast iron, the opposite ends of which were two highly finished steel cylinders, which traversed longitudinally by action of screws. The ends of the cylinders where they met was of hemispherical shape, one screw was fitted with a wheel graduated to measure to the ten thousandth of an inch. His object was to furnish ordinary mechanics with an instrument which, the micrometer caliper was introduced to the mass market in anglophone countries by Brown & Sharpe in 1867, allowing the penetration of the instruments use into the average machine shop. Brown & Sharpe were inspired by earlier devices, one of them being Palmers design. In 1888 Edward W. Morley added to the precision of micrometric measurements, each type of micrometer caliper can be fitted with specialized anvils and spindle tips for particular measuring tasks. For example, the anvil may be shaped in the form of a segment of screw thread, in the form of a v-block, universal micrometer sets come with interchangeable anvils, such as flat, spherical, spline, disk, blade, point, and knife-edge. The term universal micrometer may also refer to a type of micrometer whose frame has modular components, allowing one micrometer to function as outside mic, depth mic, step mic, blade micrometers have a matching set of narrow tips. They allow, for example, the measuring of a narrow o-ring groove, pitch-diameter micrometers have a matching set of thread-shaped tips for measuring the pitch diameter of screw threads. Limit mics have two anvils and two spindles, and are used like a snap gauge, the part being checked must pass through the first gap and must stop at the second gap in order to be within specification. The two gaps accurately reflect the top and bottom of the tolerance range, bore micrometer, typically a three-anvil head on a micrometer base used to accurately measure inside diameters. Tube micrometers have a cylindrical anvil positioned perpendicularly to a spindle and is used to measure the thickness of tubes. Micrometer stops are micrometer heads that are mounted on the table of a milling machine, bedways of a lathe, or other machine tool

15.
Bayonet
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In this regard, it is an ancillary close-quarter combat or last-resort weapon. Some modern bayonets, such as the one used on the British SA80 assault rifle, knife-shaped bayonets—when not fixed to a gun barrel—have long been utilized by soldiers in the field as general purpose cutting implements. The term bayonette dates back to the end of the 16th century, for example, Cotgraves 1611 Dictionarie describes the Bayonet as a kind of small flat pocket dagger, furnished with knives, or a great knife to hang at the girdle. Likewise, Pierre Borel wrote in 1655 that a kind of long-knife called a bayonette was made in Bayonne, the bayonet may have emerged to allow a hunter to fend off wild animals in the event of a missed shot. This idea was particularly persistent in Spain where hunting arms were equipped with bayonets from the 17th century until the advent of the cartridge era. The weapon was introduced into the French army by General Jean Martinet and was common in most European armies by the 1660s, the usefulness of such a dual-purpose arm soon became apparent. Early muskets fired at a rate, and could be both inaccurate and unreliable, depending on the quality of manufacture. A bayonet on a 5-foot tall musket achieved a similar to the infantry spear. The bayonet/musket combination was, however, considerably heavier than a polearm of the same length, early bayonets were of the plug type. This allowed light infantry to be converted to infantry and hold off cavalry charges. The bayonet had a handle that slid directly into the musket barrel. This naturally prevented the gun from being fired, in 1671, plug bayonets were issued to the French regiment of fusiliers then raised. They were issued to part of an English dragoon regiment raised in 1672 and disbanded in 1674, however, it was not widely adopted at the time. Soon socket bayonets would incorporate both ring mounts and a blade, keeping the bayonet well away from the muzzle blast of the musket barrel. In 1703, the French infantry adopted spring-loaded locking system that prevented the bayonet from accidentally separating from the musket, henceforward, the socket bayonet became, with the musket or other firearm, the typical weapon of the French infantry. The socket bayonet had by then adopted by most European armies. The British socket bayonet had a blade with a flat side towards the muzzle. However it had no lock to keep it fast to the muzzle and was well-documented for falling off in the heat of battle

16.
Pike (weapon)
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A pike is a pole weapon, a very long thrusting spear formerly used extensively by infantry. Unlike many similar weapons, the pike is not intended to be thrown, pikes were used regularly in European warfare from the early Middle Ages until around 1700, and were wielded by foot soldiers deployed in close quarters. The pike found extensive use with Landsknecht armies and Swiss mercenaries, a similar weapon, the sarissa, was also used by Alexander the Greats Macedonian phalanx infantry to great effect. Generally, a spear becomes a pike when it is too long to be wielded with one hand in combat, the pike was a long weapon, varying considerably in size, from 3 to 7.5 metres long. It was approximately 2. 5–6 kg in weight, with sixteenth-century military writer Sir John Smythe recommending lighter rather than heavier pikes and it had a wooden shaft with an iron or steel spearhead affixed. The shaft near the head was often reinforced with metal strips called cheeks or langets. When the troops of opposing armies both carried the pike, it grew in a sort of arms race, getting longer in both shaft and head length to give one sides pikemen an edge in combat. It is a mistake to refer to a bladed polearm as a pike, such weapons are more generally halberds. The great length of the pikes allowed a concentration of spearheads to be presented to the enemy, with their wielders at a greater distance. This meant that pikemen had to be equipped with an additional, shorter weapon such as a sword, mace, in general, however, pikemen attempted to avoid such disorganized combat, in which they were at a disadvantage. To compound their difficulties in a melee, the pikeman often did not have a shield, the pike, due to its unwieldy nature, was always intended to be used in a deliberate, defensive manner, often in conjunction with other missile and melee weapons. As long as it kept good order, such a formation could roll right over enemy infantry, the men were all moving forward facing in a single direction and could not turn quickly or efficiently to protect the vulnerable flanks or rear of the formation. The huge block of men carrying such unwieldy spears could be difficult to maneuver in any way other than straightforward movement, as a result, such mobile pike formations sought to have supporting troops protect their flanks or would maneuver to smash the enemy before they could be outflanked themselves. There was also the risk that the formation would become disordered, according to Sir John Smythe, there were two ways for two opposing pike formations to confront one another, cautious or aggressive. The cautious approach involved fencing at the length of the pike, while the approach involved quickly closing distance. In the aggressive approach, the first rank would then resort to swords. Smythe considered the cautious approach laughable, after the fall of the last successor of Macedon, the pike largely fell out of use for the next 1000 or so years. The one exception to this appears to have been in Germany and he consistently refers to the spears used by the Germans as being massive and very long suggesting that he is describing in essence a pike

17.
Sharpshooter
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A sharpshooter is one who is highly proficient at firing firearms or other projectile weapons accurately. Military units composed of sharpshooters were important factors in 19th century combat, along with marksman and expert, sharpshooter is one of the three marksmanship badges awarded by the U. S. Army. Another use of units of marksmen was during the Napoleonic Wars in the British Army, while most troops at that time used inaccurate smoothbore muskets, the British Green Jackets used the famous Baker rifle. Through the combination of a wad and tight grooves on the inside of the barrel. These Riflemen were the elite of the British Army, and served at the forefront of any engagement, most often in formation, scouting out. Another term, sharp shooter, was in use in British newspapers as early as 1801, the term appears even earlier, around 1781, in Continental Europe, translated from the German Scharfschütze. The sharpshooters used by both sides in the Civil War were less used as snipers, and more as skirmishers and these elite troops were well equipped and trained, and placed at the front of any column to first engage the enemy. Notable sharpshooter units of the Civil War included the 1st and 2nd United States Volunteer Sharpshooter Regiment, were organized by Colonel Hiram Berdan, a self-made millionaire who was reputed to be the best rifle marksman in the nation at that time. There was also an all-Native American company of sharpshooters in the Army of the Potomac and these men, primarily Odawa, Ojibwe, and Potawatomi from northern Michigan, comprised the members of Company K of the 1st Regiment Michigan Volunteer Sharpshooters. The regiment was raised by MG John C, fremont at St. Louis Benton Barracks as the Western Theater counterpart to Berdans sharpshooters. Members were recruited from most of the Western states, predominantly Ohio, Michigan, Illinois, competitive induction required candidates to place ten shots in a three-inch circle at 200 yards. They were initially armed with half-stock Plains Rifles built and procured by St. Louis custom gunmaker Horace Dimick and they were the only Federal unit completely armed with sporting rifles. Over 250 of the Western Sharpshooters purchased Henrys out of their own pocket, on the Confederate side, sharpshooter units functioned as light infantry. Their duties included skirmishing and reconnaissance, robert E. Rodes, a colonel and later major general of the 5th Alabama Infantry Regiment, was a leader in the development of sharpshooter units. Dedicated sharpshooter units included, 9th Battalion Missouri Sharpshooters, the 1st & 2nd Battalions Georgia Sharpshooters, in his memoirs, Louis Leon detailed his service as a sharpshooter in the Fifty-Third North Carolina Regiment during the Civil War. As a sharpshooter, he volunteered as a skirmisher, served on picket duty, of his companys original twelve sharpshooters, only he and one other were still alive after Gettysburg. As related by the commanding officer, Col. James Morehead. Leon killed a Union sharpshooter, whom the Confederates identified as a Native American from Canada

18.
Danish people
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Danes are the citizens of Denmark, most of whom speak Danish and consider themselves to be of Danish ethnicity. The first mention of Danes are from the 6th century in Jordanes Getica, by Procopius, the first mention of Danes within the Danish territory is on the Jelling Rune Stone which states how Harald Bluetooth converted the Danes to Christianity in the 10th century. Since the formulation of a Danish national identity in the 19th century, Danish national identity was built on a basis of peasant culture and Lutheran theology, theologian N. F. S. Grundtvig and his popular movement played a prominent part in the process. Today, the criterion for being considered a Dane is having Danish citizenship. Frankish annalists of the 8th century often refer to Danish kings, the Bobbio Orosius from the early 7th century, distinguishes between South Danes inhabiting Jutland and North Danes inhabiting the isles and the province of Scania. The first mention of Danes within the Danish territory is on the Jelling Rune Stone which mentions how Harald Bluetooth converted the Danes to Christianity in the 10th century. Between c.960 and the early 980s, Harald Bluetooth established a kingdom in the lands of the Danes, stretching from Jutland to Scania. Around the same time, he received a visit from a German missionary who, by surviving an ordeal by fire according to legend, the following years saw the Danish Viking expansion, which incorporated Norway and Northern England into the Danish kingdom. After the death of Canute the Great in 1035, England broke away from Danish control, canutes nephew Sweyn Estridson re-established strong royal Danish authority and built a good relationship with the archbishop of Bremen — at that time the Archbishop of all of Scandinavia. The Reformation, which originated in the German lands in the early 16th century from the ideas of Martin Luther, had a impact on Denmark. The Danish Reformation started in the mid-1520s, some Danes wanted access to the Bible in their own language. In 1524, Hans Mikkelsen and Christiern Pedersen translated the New Testament into Danish and those who had traveled to Wittenberg in Saxony and come under the influence of the teachings of Luther and his associates included Hans Tausen, a Danish monk in the Order of St John Hospitallers. The Dano-Norwegian Kingdom grew wealthy during the 16th century, largely because of the traffic through the Øresund. The Crown of Denmark could tax the traffic, because it controlled both sides of the Sound at the time, in the centuries after this loss of territory, the populations of the Scanian lands, who had previously been considered Danish, came to be fully integrated as Swedes. Later, in the early 19th century, Denmark suffered a defeat in the Napoleonic Wars, Denmark lost control over Norway, the political and economic defeat ironically sparked what is known as the Danish Golden Age during which a Danish national identity first came to be fully formed. The Danish liberal and national movements gained momentum in the 1830s, a new constitution emerged, separating the powers and granting the franchise to all adult males, as well as freedom of the press, religion, and association. The king became head of the executive branch, Danishness is the concept on which contemporary Danish national and ethnic identity is based. It is a set of values formed through the historic trajectory of the formation of the Danish nation, importantly, since its formulation, Danish identity has not been linked to a particular racial or biological heritage, as many other ethno-national identities have

19.
Mathematician
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A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and with whom the study of mathematics for its own sake begins, the first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked. Science and mathematics in the Islamic world during the Middle Ages followed various models and it was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences, an example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham, the Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking. ”Thus, seminars, overall, science became the focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems

20.
Robert Plot
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Robert Plot was an English naturalist, first Professor of Chemistry at the University of Oxford, and the first keeper of the Ashmolean Museum. Born in Borden, Kent to parents Robert Plot and Elisabeth Patenden and he entered Magdalen Hall, Oxford in 1658 where he graduated with a BA in 1661 and a MA in 1664. Plot subsequently taught and served as dean and vice principal at Magdalen Hall while preparing for his BCL and DCL, by this time, Plot had already developed an interest in the systematic study of natural history and antiquities. The favourable reception of his findings not only earned him the nickname of the learned Dr, in the field of chemistry he searched for a universal solvent that could be obtained from wine spirits, and believed that alchemy was necessary for medicine. In 1684, Plot published De origine fontium, a treatise on the source of springs, Plot shifted his focus towards archaeology in the 1686 publication of his second book, The Natural History of Staffordshire, but misinterpreted Roman remains as Saxon. He also describes a double sunset viewable from Leek and the Abbots Bromley Horn Dance, in 1687, Plot was made a notary public by the Archbishop of Canterbury as well as appointed the registrar to the Norfolk Court of Chivalry. The office of Mowbray Herald Extraordinary was created in January 1695 for Plot, although able to go on an archaeological tour of Anglia in September 1695, Plot was greatly suffering from urinary calculi, and succumbed to his illness on 30 April 1696. He was buried at Borden Church, where a plaque memorialises him, the Natural History of Oxford-shire, Being an Essay Towards the Natural History of England. The Correspondence of Robert Plot in EMLO Plot, Robert, Robert Plot from the Ashmolean Museum, Oxford Plot, Robert The natural history of Oxford-shire - digital facsimile from the Linda Hall Library

21.
English people
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The English are a nation and an ethnic group native to England, who speak the English language. The English identity is of medieval origin, when they were known in Old English as the Angelcynn. Their ethnonym is derived from the Angles, one of the Germanic peoples who migrated to Great Britain around the 5th century AD, England is one of the countries of the United Kingdom. Collectively known as the Anglo-Saxons, they founded what was to become England along with the later Danes, Normans, in the Acts of Union 1707, the Kingdom of England was succeeded by the Kingdom of Great Britain. Over the years, English customs and identity have become closely aligned with British customs. The English people are the source of the English language, the Westminster system and these and other English cultural characteristics have spread worldwide, in part as a result of the former British Empire. The concept of an English nation is far older than that of the British nation, many recent immigrants to England have assumed a solely British identity, while others have developed dual or mixed identities. Use of the word English to describe Britons from ethnic minorities in England is complicated by most non-white people in England identifying as British rather than English. In their 2004 Annual Population Survey, the Office for National Statistics compared the ethnic identities of British people with their national identity. They found that while 58% of white people in England described their nationality as English and it is unclear how many British people consider themselves English. Following complaints about this, the 2011 census was changed to allow respondents to record their English, Welsh, Scottish, another complication in defining the English is a common tendency for the words English and British to be used interchangeably, especially overseas. In his study of English identity, Krishan Kumar describes a common slip of the tongue in which people say English, I mean British. He notes that this slip is made only by the English themselves and by foreigners. Kumar suggests that although this blurring is a sign of Englands dominant position with the UK and it tells of the difficulty that most English people have of distinguishing themselves, in a collective way, from the other inhabitants of the British Isles. In 1965, the historian A. J. P. Taylor wrote, When the Oxford History of England was launched a generation ago and it meant indiscriminately England and Wales, Great Britain, the United Kingdom, and even the British Empire. Foreigners used it as the name of a Great Power and indeed continue to do so, bonar Law, by origin a Scotch Canadian, was not ashamed to describe himself as Prime Minister of England Now terms have become more rigorous. The use of England except for a geographic area brings protests and this version of history is now regarded by many historians as incorrect, on the basis of more recent genetic and archaeological research. The 2016 study authored by Stephan Schiffels et al, the remaining portion of English DNA is primarily French, introduced in a migration after the end of the Ice Age