1.
1797 in science
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The year 1797 in science and technology involved some significant events. Smithson Tennant demonstrates that diamond is a form of carbon. Joseph Proust proposes the law of definite proportions, which states that elements always combine in small, lagrange publishes his Théorie des fonctions analytiques. Giovanni Battista Venturi describes the Venturi effect, october 22 – André-Jacques Garnerin carries out the first descent using a frameless parachute, a 980 m drop from a balloon in Paris. Thomas Bewick publishes the first volume, Land Birds, of his History of British Birds
2.
1796 in architecture
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The year 1796 in architecture involved some significant events. October 8 - The Sans Souci Theatre in Westminster, London, opens to the public, built by dramatist, musician, somerset House in London, designed by William Chambers is completed. The parish church of Urtijëi in the Italian Tyrol, designed by Joseph Abenthung, is completed, hwaseong Fortress in Suwon, Korea, designed by Jeong Yak-yong, is completed. Work begins on Blaise Castle, commissioned by John Scandrett Harford from William Paty, the Mosque of Amr ibn al-As in Fustat, Egypt, is rebuilt by Mamluk leader Mourad Bey
3.
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
4.
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
5.
Pierre-Simon Laplace
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Pierre-Simon, marquis de Laplace was an influential French scholar whose work was important to the development of mathematics, statistics, physics and astronomy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste and this work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace, Laplace formulated Laplaces equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is named after him. Laplace is remembered as one of the greatest scientists of all time, sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries. Laplace became a count of the Empire in 1806 and was named a marquis in 1817, Laplace was born in Beaumont-en-Auge, Normandy on 23 March 1749, a village four miles west of Pont lEveque in Normandy. According to W. W. Rouse Ball, His father, Pierre de Laplace and his great-uncle, Maitre Oliver de Laplace, had held the title of Chirurgien Royal. It would seem that from a pupil he became an usher in the school at Beaumont, however, Karl Pearson is scathing about the inaccuracies in Rouse Balls account and states, Indeed Caen was probably in Laplaces day the most intellectually active of all the towns of Normandy. It was here that Laplace was educated and was provisionally a professor and it was here he wrote his first paper published in the Mélanges of the Royal Society of Turin, Tome iv. 1766–1769, at least two years before he went at 22 or 23 to Paris in 1771, thus before he was 20 he was in touch with Lagrange in Turin. He did not go to Paris a raw self-taught country lad with only a peasant background, the École Militaire of Beaumont did not replace the old school until 1776. His parents were from comfortable families and his father was Pierre Laplace, and his mother was Marie-Anne Sochon. The Laplace family was involved in agriculture until at least 1750, Pierre Simon Laplace attended a school in the village run at a Benedictine priory, his father intending that he be ordained in the Roman Catholic Church. At sixteen, to further his fathers intention, he was sent to the University of Caen to read theology, at the university, he was mentored by two enthusiastic teachers of mathematics, Christophe Gadbled and Pierre Le Canu, who awoke his zeal for the subject. Here Laplaces brilliance as a mathematician was recognised and while still at Caen he wrote a memoir Sur le Calcul integral aux differences infiniment petites et aux differences finies. About this time, recognizing that he had no vocation for the priesthood, in this connection reference may perhaps be made to the statement, which has appeared in some notices of him, that he broke altogether with the church and became an atheist. Laplace did not graduate in theology but left for Paris with a letter of introduction from Le Canu to Jean le Rond dAlembert who at time was supreme in scientific circles. According to his great-great-grandson, dAlembert received him rather poorly, and to get rid of him gave him a mathematics book
6.
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
7.
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
8.
Nebular hypothesis
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The nebular hypothesis is the most widely accepted model in the field of cosmogony to explain the formation and evolution of the Solar System. It suggests that the Solar System formed from nebulous material, the theory was developed by Immanuel Kant and published in his Allgemeine Naturgeschichte und Theorie des Himmels, published in 1755. Originally applied to the Solar System, the process of planetary formation is now thought to be at work throughout the Universe. The widely-accepted modern variant of the hypothesis is the solar nebular disk model or solar nebular model. It offered explanations for a variety of properties of the Solar System, including the circular and coplanar orbits of the planets. Some elements of the hypothesis are echoed in modern theories of planetary formation. According to the hypothesis, stars form in massive and dense clouds of molecular hydrogen—giant molecular clouds. These clouds are gravitationally unstable, and matter coalesces within them to smaller denser clumps, which rotate, collapse. Star formation is a process, which always produces a gaseous protoplanetary disk, proplyd. This may give birth to planets in certain circumstances, which are not well known, thus the formation of planetary systems is thought to be a natural result of star formation. A Sun-like star usually takes approximately 1 million years to form, the protoplanetary disk is an accretion disk that feeds the central star. Initially very hot, the disk later cools in what is known as the T tauri star stage, here, formation of small dust grains made of rocks, the grains eventually may coagulate into kilometer-sized planetesimals. If the disk is massive enough, the runaway accretions begin, near the star, the planetary embryos go through a stage of violent mergers, producing a few terrestrial planets. The last stage takes approximately 100 million to a billion years, the formation of giant planets is a more complicated process. It is thought to occur beyond the frost line, where planetary embryos mainly are made of various types of ice, as a result, they are several times more massive than in the inner part of the protoplanetary disk. What follows after the formation is not completely clear. Some embryos appear to continue to grow and eventually reach 5–10 Earth masses—the threshold value, jupiter- and Saturn-like planets are thought to accumulate the bulk of their mass during only 10,000 years. The accretion stops when the gas is exhausted, the formed planets can migrate over long distances during or after their formation
9.
Immanuel Kant
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Immanuel Kant was a German philosopher who is considered a central figure in modern philosophy. Kant took himself to have effected a Copernican revolution in philosophy and his beliefs continue to have a major influence on contemporary philosophy, especially the fields of metaphysics, epistemology, ethics, political theory, and aesthetics. Politically, Kant was one of the earliest exponents of the idea that peace could be secured through universal democracy. He believed that this will be the outcome of universal history. Kant wanted to put an end to an era of futile and speculative theories of human experience, Kant argued that our experiences are structured by necessary features of our minds. In his view, the shapes and structures experience so that, on an abstract level. Among other things, Kant believed that the concepts of space and time are integral to all human experience, as are our concepts of cause, Kant published other important works on ethics, religion, law, aesthetics, astronomy, and history. These included the Critique of Practical Reason, the Metaphysics of Morals, which dealt with ethics, and the Critique of Judgment, Immanuel Kant was born in 1724 in Königsberg, Prussia. His mother, Anna Regina Reuter, was born in Königsberg to a father from Nuremberg. His father, Johann Georg Kant, was a German harness maker from Memel, Immanuel Kant believed that his paternal grandfather Hans Kant was of Scottish origin. Kant was the fourth of nine children, baptized Emanuel, he changed his name to Immanuel after learning Hebrew. Young Kant was a solid, albeit unspectacular, student and he was brought up in a Pietist household that stressed religious devotion, humility, and a literal interpretation of the Bible. His education was strict, punitive and disciplinary, and focused on Latin and religious instruction over mathematics, despite his religious upbringing and maintaining a belief in God, Kant was skeptical of religion in later life, various commentators have labelled him agnostic. Common myths about Kants personal mannerisms are listed, explained, and refuted in Goldthwaits introduction to his translation of Observations on the Feeling of the Beautiful and Sublime. It is often held that Kant lived a strict and disciplined life. He never married, but seemed to have a social life — he was a popular teacher. He had a circle of friends whom he met, among them Joseph Green. A common myth is that Kant never traveled more than 16 kilometres from Königsberg his whole life, in fact, between 1750 and 1754 he worked as a tutor in Judtschen and in Groß-Arnsdorf
10.
Kingdom of Great Britain
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The Kingdom of Great Britain, officially Great Britain, was a sovereign state in western Europe from 1 May 1707 to 31 December 1800. It did not include Ireland, which remained a separate realm, the unitary state was governed by a single parliament and government that was based in Westminster. Also after the accession of George I to the throne of Great Britain in 1714, the early years of the unified kingdom were marked by Jacobite risings which ended in defeat for the Stuart cause at Culloden in 1746. On 1 January 1801, the kingdoms of Great Britain and Ireland were merged to form the United Kingdom of Great Britain and Ireland. In 1922, five-sixths of Ireland seceded from the United Kingdom, the name Britain descends from the Latin name for the island of Great Britain, Britannia or Brittānia, the land of the Britons via the Old French Bretaigne and Middle English Bretayne, Breteyne. The term Great Britain was first used officially in 1474, in the instrument drawing up the proposal for a marriage between Edward IV of Englands daughter Cecily and James III of Scotlands son James. The Treaty of Union and the subsequent Acts of Union state that England and Scotland were to be United into one Kingdom by the Name of Great Britain. However, both the Acts and the Treaty also refer numerous times to the United Kingdom and the longer form, other publications refer to the country as the United Kingdom after 1707 as well. The websites of the UK parliament, the Scottish Parliament, the BBC, additionally, the term United Kingdom was found in informal use during the 18th century to describe the state. The new state created in 1707 included the island of Great Britain, the kingdoms of England and Scotland, both in existence from the 9th century, were separate states until 1707. However, they had come into a union in 1603. Each of the three kingdoms maintained its own parliament and laws and this disposition changed dramatically when the Acts of Union 1707 came into force, with a single unified Crown of Great Britain and a single unified parliament. Ireland remained formally separate, with its own parliament, until the Acts of Union 1800, legislative power was vested in the Parliament of Great Britain, which replaced both the Parliament of England and the Parliament of Scotland. In practice it was a continuation of the English parliament, sitting at the location in Westminster. Newly created peers in the Peerage of Great Britain were given the right to sit in the Lords. Despite the end of a parliament for Scotland, it retained its own laws. As a result of Poynings Law of 1495, the Parliament of Ireland was subordinate to the Parliament of England, the Act was repealed by the Repeal of Act for Securing Dependence of Ireland Act 1782. The same year, the Irish constitution of 1782 produced a period of legislative freedom, the 18th century saw England, and after 1707 Great Britain, rise to become the worlds dominant colonial power, with France its main rival on the imperial stage
11.
Roman cement
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Roman cement is a substance developed by James Parker in the 1780s, being patented in 1796. The burnt nodules were ground to a fine powder and this product, made into a mortar with sand, set in 5–15 minutes. The success of Roman cement led other manufacturers to develop products by burning artificial mixtures of clay. There has been recent resurgence of interest in Natural cements and Roman cements due mainly to the need for repair of façades done in this material in the 19th century, the major confusion involved for many people in this subject is the terminology used. Roman cement was originally the name given, by Parker, to the cement he patented which is a Natural cement, in 1791, Parker was granted a patent Method of Burning bricks, Tiles, Chalk. His second patent in 1796 A certain Cement or Terras to be used in Aquatic and other Buildings and Stucco Work, covers Roman cement and he set up his manufacturing plant on Northfleetcreek, Kent. It was notably patented late on but James Parker is still the subject of all the credit. Later, in the 1800s various sources of the type of marl, known also as Cement Stone, were discovered across Europe. They do not slake in contact with water and must therefore be ground to a floury fineness, from around 1807 a number of people looked to make artificial versions of this cement. This was done by adding various materials together to make a version of natural cement. The name Portland cement is also recorded in a directory published in 1823 being associated with William Lockwood, Dave Stewart, there then followed a number of independently discovered or copied versions of this Portland cement. This cement is not, however, the same as the modern ordinary Portland cement, James Frost is reported to have erected a manufactory for making of an artificial cement in 1826. In 1843, Aspdins son William improved their cement, which was initially called Patent Portland cement, in 1848, William Aspdin further improved his cement and in 1853, he moved to Germany where he was involved in cement making. William Aspdin made what could be called meso-Portland cement, artificial cement, Development in the 1860s of rotating horizontal kiln technology brought dramatic changes in properties, arguably resulting in modern cement. Certainly it is difficult to define whether an old render was a cement or an artificial one. The names Natural cement or Roman cement then defines a cement coming from a source rock. Early or Proto Portland cement could be used for early cement that comes from a number of sourced and mixed materials, there is no widely used terminology for these 19th-century cements. There had been, in order to rediscover this technology, two carried out by the European Union ROCEM and subsequently ROCARE
12.
Mungo Park (explorer)
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Mungo Park was a Scottish explorer of West Africa. He was the first Westerner known to have travelled to the portion of the Niger River. Mungo Park was born in Selkirkshire, Scotland, at Foulshiels on the Yarrow Water, near Selkirk and he was the seventh in a family of thirteen. Although tenant farmers, the Parks were relatively well-off and they were able to pay for Park to receive a good education, and Parks father died leaving property valued at £3,000. His parents had intended him for the Church of Scotland. He was educated at home before attending Selkirk grammar school, at the age of fourteen, he was apprenticed to Thomas Anderson, a surgeon in Selkirk. During his apprenticeship, Park became friends with Andersons son Alexander and was introduced to Andersons daughter Allison, in October 1788, Park enrolled at the University of Edinburgh, attending for four sessions studying medicine and botany. Notably, during his time at university, he spent a year in the history course taught by Professor John Walker. After completing his studies, he spent a summer in the Scottish Highlands, engaged in fieldwork with his brother-in-law, James Dickson. In 1788 Dickson and Sir Joseph Banks had founded the London Linnean Society, in 1792 Park completed his medical studies at University of Edinburgh. Through a recommendation by Banks, he obtained the post of assistant surgeon on board the East India Companys ship Worcester, in February 1793 the Worcester sailed to Benkulen in Sumatra. Before departing, Park wrote his friend Alexander Anderson in terms that reflect his Calvinist upbringing, if I be deceived, may God alone put me right, for I would rather die in the delusion than wake to all the joys of earth. May the Holy Spirit dwell in your heart, my dear friend, on his return in 1794, Park gave a lecture to the Linnaean Society, describing eight new Sumatran fish. The paper was not published until three years later and he also presented Banks with various rare Sumatran plants. Supported by Sir Joseph Banks, Park was selected, on 22 May 1795, Park left Portsmouth, England, on the brig Endeavour, a vessel trading to the Gambia for beeswax and ivory. On 21 June 1795, he reached the Gambia River and ascended it 200 miles to a British trading station named Pisania, on 2 December, accompanied by two local guides, he started for the unknown interior. He chose the route crossing the upper Senegal basin and through the region of Kaarta. The journey was full of difficulties, and at Ludamar he was imprisoned by a Moorish chief for four months
13.
Niger River
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The Niger River is the principal river of western Africa, extending about 4,180 km. Its drainage basin is 2,117,700 km2 in area and its source is in the Guinea Highlands in southeastern Guinea. The Niger is the third-longest river in Africa, exceeded only by the Nile and its main tributary is the Benue River. The Scottish explorer Mungo Park was the first Westerner known to have travelled to the portion of the Niger River. The earliest use of the name Niger for the river is by Leo Africanus in his Della descrittione dell’Africa et delle cose notabili che iui sono published in Italian in 1550, the name may come from Berber phrase ger-n-ger meaning river of rivers. As Timbuktu was the end of the principal Trans-Saharan trade route to the western Mediterranean. When European colonial powers began to send ships along the West coast of Africa in the 16th and 17th centuries, the Niger Delta, pouring into the Atlantic through mangrove swamps and thousands of distributaries along more than 160 kilometres, was thought to be no more than coastal wetlands. The Niger River is a clear river, carrying only a tenth as much sediment as the Nile because the Nigers headwaters lie in ancient rocks that provide little silt. Like the Nile, the Niger floods yearly, this begins in September, peaks in November, an unusual feature of the river is the Inner Niger Delta, which forms where its gradient suddenly decreases. The result is a region of braided streams, marshes, and lakes the size of Belgium, the river loses nearly two-thirds of its potential flow in the Inner Delta between Ségou and Timbuktu to seepage and evaporation. All the water from the Bani River, which flows into the Delta at Mopti, the average loss is estimated at 31 km3/year, but varies considerably between years. The river is joined by various tributaries, but also loses more water to evaporation. The quantity of water entering Nigeria measured in Yola was estimated at 25 km3/year before the 1980s, the most important tributary of the Niger in Nigeria is the Benue River which merges with the river at Lokoja in Nigeria. The Niger takes one of the most unusual routes of any major river and this strange geography apparently came about because the Niger River is two ancient rivers joined together. Over time upstream erosion by the lower Niger resulted in capture of the upper Niger by the lower Niger. The northern part of the river, known as the Niger bend, is an important area because it is the major river and this made it the focal point of trade across the western Sahara, and the centre of the Sahelian kingdoms of Mali and Gao. The surrounding Niger River Basin is one of the distinct physiographic sections of the Sudan province, the origin of the rivers name remains unclear. What is clear is that Niger was an appellation applied in the Mediterranean world from at least the Classical era, when knowledge of the area by Europeans was slightly better than fable
14.
Germans
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Germans are a Germanic ethnic group native to Central Europe, who share a common German ancestry, culture and history. German is the mother tongue of a substantial majority of ethnic Germans. The English term Germans has historically referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages, before the collapse of communism and the reunification of Germany in 1990, Germans constituted the largest divided nation in Europe by far. Ever since the outbreak of the Protestant Reformation within the Holy Roman Empire, of approximately 100 million native speakers of German in the world, roughly 80 million consider themselves Germans. Thus, the number of Germans lies somewhere between 100 and more than 150 million, depending on the criteria applied. Today, people from countries with German-speaking majorities most often subscribe to their own national identities, the German term Deutsche originates from the Old High German word diutisc, referring to the Germanic language of the people. It is not clear how commonly, if at all, the word was used as an ethnonym in Old High German, used as a noun, ein diutscher in the sense of a German emerges in Middle High German, attested from the second half of the 12th century. The Old French term alemans is taken from the name of the Alamanni and it was loaned into Middle English as almains in the early 14th century. The word Dutch is attested in English from the 14th century, denoting continental West Germanic dialects, while in most Romance languages the Germans have been named from the Alamanni, the Old Norse, Finnish and Estonian names for the Germans were taken from that of the Saxons. In Slavic languages, the Germans were given the name of němьci, originally with a meaning foreigner, the English term Germans is only attested from the mid-16th century, based on the classical Latin term Germani used by Julius Caesar and later Tacitus. It gradually replaced Dutch and Almains, the latter becoming mostly obsolete by the early 18th century, the Germans are a Germanic people, who as an ethnicity emerged during the Middle Ages. Originally part of the Holy Roman Empire, around 300 independent German states emerged during its decline after the Peace of Westphalia in 1648 ending the Thirty Years War and these states eventually formed into modern Germany in the 19th century. The concept of a German ethnicity is linked to Germanic tribes of antiquity in central Europe, the early Germans originated on the North German Plain as well as southern Scandinavia. By the 2nd century BC, the number of Germans was significantly increasing and they began expanding into eastern Europe, during antiquity these Germanic tribes remained separate from each other and did not have writing systems at that time. In the European Iron Age the area that is now Germany was divided into the La Tène horizon in Southern Germany and the Jastorf culture in Northern Germany. By 55 BC, the Germans had reached the Danube river and had either assimilated or otherwise driven out the Celts who had lived there, and had spread west into what is now Belgium and France. Conflict between the Germanic tribes and the forces of Rome under Julius Caesar forced major Germanic tribes to retreat to the east bank of the Rhine, in Roman-held territories with Germanic populations, the Germanic and Roman peoples intermarried, and Roman, Germanic, and Christian traditions intermingled. The adoption of Christianity would later become an influence in the development of a common German identity
15.
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
16.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
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Polygon
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In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
18.
Heptadecagon
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In geometry, a heptadecagon or 17-gon is a seventeen-sided polygon. A regular heptadecagon is represented by the Schläfli symbol, as 17 is a Fermat prime, the regular heptadecagon is a constructible polygon, this was shown by Carl Friedrich Gauss in 1796 at the age of 19. This proof represented the first progress in regular polygon construction in over 2000 years, constructing a regular heptadecagon thus involves finding the cosine of 2 π /17 in terms of square roots, which involves an equation of degree 17—a Fermat prime. Gauss book Disquisitiones Arithmeticae gives this as,16 cos 2 π17 = −1 +17 +34 −217 +217 +317 −34 −217 −234 +217. The explicit construction of a heptadecagon was given by Herbert William Richmond in 1893, the following method of construction uses Carlyle circles, as shown below. Make OI one-fourth of OB, and the angle OIE one-fourth of OIA, another more recent construction is given by Callagy. The regular heptadecagon has Dih17 symmetry, order 34, since 17 is a prime number there is one subgroup with dihedral symmetry, Dih1, and 2 cyclic group symmetries, Z17, and Z1. These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon, john Conway labels these by a letter and group order. Full symmetry of the form is r34 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g17 subgroup has no degrees of freedom but can seen as directed edges. A heptadecagram is a 17-sided star polygon, there are seven regular forms given by Schläfli symbols, and. The regular heptadecagon is the Petrie polygon for one higher-dimensional regular convex polytope, projected in an orthogonal projection. – Describes the algebraic aspect, by Gauss, contains a description of the construction. Heptadecagon trigonometric functions heptadecagon building New R&D center for SolarUK BBC video of New R&D center for SolarUK Eisenbud, Heptadecagon Heptadecagon, a construction with only one point N, a variation of the design according to H. W. Richmond
19.
Quadratic reciprocity
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In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem, one version of the law states that for p and q odd prime numbers, = p −12 q −12 where denotes the Legendre symbol. This law, combined with the properties of the Legendre symbol and this makes it possible to determine, for any quadratic equation, x 2 ≡ a, where p is an odd prime, whether it has a solution. However, it does not provide any help at all for actually finding the solution, the solution can be found using quadratic residues. The theorem was conjectured by Euler and Legendre and first proved by Gauss and he refers to it as the fundamental theorem in the Disquisitiones Arithmeticae and his papers, writing The fundamental theorem must certainly be regarded as one of the most elegant of its type. Privately he referred to it as the golden theorem and he published six proofs, and two more were found in his posthumous papers. There are now over 200 published proofs, the first section of this article gives a special case of quadratic reciprocity that is representative of the general case. The second section gives the formulations of quadratic reciprocity found by Legendre, consider the polynomial f = n 2 −5 and its values for n ∈ N. The prime factorizations of these values are given as follows, The prime numbers that appear as factors are 2,5, no primes ending in 3 or 7 ever appear. Another way of phrasing this is that the primes p for which exists an n such that n2 ≡5 are precisely 2,5. Or in other words, when p is a prime that is neither 2 nor 5,5 is a residue modulo p iff p is 1 or 4 modulo 5. In other words,5 is a residue modulo p iff p is a quadratic residue modulo 5. The law of quadratic reciprocity gives a similar characterization of prime divisors of f = n2 − c for any integer c, a quadratic residue is any number congruent to a square. A quadratic nonresidue is any number that is not congruent to a square, the adjective quadratic can be dropped if the context makes it clear that it is implied. When working modulo primes, it is usual to treat zero as a special case, by doing so, the following statements become true, Modulo a prime, there are an equal number of quadratic residues and nonresidues. Modulo a prime, the product of two quadratic residues is a residue, the product of a residue and a nonresidue is a nonresidue, and this table is complete for odd primes less than 50. To check whether a number m is a quadratic residue mod one of these primes p, If a is in row p, then m is a residue, if a is not in row p of the table, then m is a nonresidue. The quadratic reciprocity law is the statement that certain patterns found in the table are true in general, in this article, p and q always refer to distinct positive odd prime numbers
20.
Quadratic equation
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If a =0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the coefficient, the linear coefficient. Because the quadratic equation involves only one unknown, it is called univariate, solutions to problems equivalent to the quadratic equation were known as early as 2000 BC. A quadratic equation with real or complex coefficients has two solutions, called roots and these two solutions may or may not be distinct, and they may or may not be real. It may be possible to express a quadratic equation ax2 + bx + c =0 as a product =0. In some cases, it is possible, by inspection, to determine values of p, q, r. If the quadratic equation is written in the form, then the Zero Factor Property states that the quadratic equation is satisfied if px + q =0 or rx + s =0. Solving these two linear equations provides the roots of the quadratic, for most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. As an example, x2 + 5x +6 factors as, the more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where b =0 or c =0 and this means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection. The process of completing the square makes use of the identity x 2 +2 h x + h 2 =2. Starting with an equation in standard form, ax2 + bx + c =0 Divide each side by a. Subtract the constant term c/a from both sides, add the square of one-half of b/a, the coefficient of x, to both sides. This completes the square, converting the left side into a perfect square, write the left side as a square and simplify the right side if necessary. Produce two linear equations by equating the square root of the side with the positive and negative square roots of the right side. Completing the square can be used to derive a formula for solving quadratic equations. The mathematical proof will now be briefly summarized and it can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation,2 = b 2 −4 a c 4 a 2. Taking the square root of both sides, and isolating x, gives, x = − b ± b 2 −4 a c 2 a and these result in slightly different forms for the solution, but are otherwise equivalent
21.
Modular arithmetic
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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, a familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7,00 now, then 8 hours later it will be 3,00. Usual addition would suggest that the time should be 7 +8 =15. Likewise, if the clock starts at 12,00 and 21 hours elapse, then the time will be 9,00 the next day, because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below,12 is congruent not only to 12 itself, Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations on integers, addition, subtraction, and multiplication. For a positive n, two integers a and b are said to be congruent modulo n, written, a ≡ b. The number n is called the modulus of the congruence, for example,38 ≡14 because 38 −14 =24, which is a multiple of 12. The same rule holds for negative values, −8 ≡72 ≡ −3 −3 ≡ −8. Equivalently, a ≡ b mod n can also be thought of as asserting that the remainders of the division of both a and b by n are the same, for instance,38 ≡14 because both 38 and 14 have the same remainder 2 when divided by 12. It is also the case that 38 −14 =24 is a multiple of 12. A remark on the notation, Because it is common to consider several congruence relations for different moduli at the same time, in spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been if the notation a ≡n b had been used. The properties that make this relation a congruence relation are the following, if a 1 ≡ b 1 and a 2 ≡ b 2, then, a 1 + a 2 ≡ b 1 + b 2 a 1 − a 2 ≡ b 1 − b 2. The above two properties would still hold if the theory were expanded to all real numbers, that is if a1, a2, b1, b2. The next property, however, would fail if these variables were not all integers, the notion of modular arithmetic is related to that of the remainder in Euclidean division. The operation of finding the remainder is referred to as the modulo operation. For example, the remainder of the division of 14 by 12 is denoted by 14 mod 12, as this remainder is 2, we have 14 mod 12 =2
22.
Prime number theorem
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In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896 using ideas introduced by Bernhard Riemann, the first such distribution found is π ~ N/log, where π is the prime-counting function and log is the natural logarithm of N. This means that for large enough N, the probability that an integer not greater than N is prime is very close to 1 / log. Consequently, an integer with at most 2n digits is about half as likely to be prime as a random integer with at most n digits. For example, among the integers of at most 1000 digits, about one in 2300 is prime, whereas among positive integers of at most 2000 digits. In other words, the gap between consecutive prime numbers among the first N integers is roughly log. Let π be the function that gives the number of primes less than or equal to x. For example, π =4 because there are four prime numbers less than or equal to 10, using asymptotic notation this result can be restated as π ∼ x log x. This notation does not say anything about the limit of the difference of the two functions as x increases without bound, instead, the theorem states that x / log x approximates π in the sense that the relative error of this approximation approaches 0 as x increases without bound. For example, the 7017200000000000000♠2×1017th prime number is 7018851267738604819♠8512677386048191063, and log rounds to 7018796741875229174♠7967418752291744388, a relative error of about 6. 4%. The prime number theorem is equivalent to lim x → ∞ ϑ x = lim x → ∞ ψ x =1, where ϑ and ψ are the first. Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre conjectured in 1797 or 1798 that π is approximated by the function a /, where A and B are unspecified constants. In the second edition of his book on number theory he made a more precise conjecture. Carl Friedrich Gauss considered the question at age 15 or 16 in the year 1792 or 1793. In 1838 Peter Gustav Lejeune Dirichlet came up with his own approximating function, in two papers from 1848 and 1850, the Russian mathematician Pafnuty Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. He was able to prove unconditionally that this ratio is bounded above, an important paper concerning the distribution of prime numbers was Riemanns 1859 memoir On the Number of Primes Less Than a Given Magnitude, the only paper he ever wrote on the subject. In particular, it is in paper of Riemann that the idea to apply methods of complex analysis to the study of the real function π originates
23.
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
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Triangular numbers
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A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side and it represents the number of distinct pairs that can be selected from n +1 objects, and it is read aloud as n plus one choose two. Carl Friedrich Gauss is said to have found this relationship in his early youth, however, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus, the triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with n +1 people shakes hands once with each person. In other words, the solution to the problem of n people is Tn−1. The function T is the analog of the factorial function. In the limit, the ratio between the two numbers, dots and line segments is lim n → ∞ T n L n =13, Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically, T n + T n −1 = + = + = n 2 =2, alternatively, the same fact can be demonstrated graphically, There are infinitely many triangular numbers that are also square numbers, e. g.1,36,1225. Some of them can be generated by a recursive formula. All square triangular numbers are found from the recursion S n =34 S n −1 − S n −2 +2 with S0 =0 and S1 =1. Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n and this can also be expressed as ∑ k =1 n k 3 =2. The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number, more generally, the difference between the nth m-gonal number and the nth -gonal number is the th triangular number. For example, the sixth heptagonal number minus the sixth hexagonal number equals the triangular number,15. Every other triangular number is a hexagonal number, knowing the triangular numbers, one can reckon any centered polygonal number, the nth centered k-gonal number is obtained by the formula C k n = k T n −1 +1 where T is a triangular number. The positive difference of two numbers is a trapezoidal number. Triangular numbers correspond to the case of Faulhabers formula. Alternating triangular numbers are also hexagonal numbers, every even perfect number is triangular, given by the formula M p 2 p −1 = M p 2 = T M p where Mp is a Mersenne prime