1.
Milliradian
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A milliradian, often called a mil or mrad, is an SI derived unit for angular measurement which is defined as a thousandth of a radian. Mils are used in adjustment of firearm sights by adjusting the angle of the sight compared to the barrel, Mils are also used for comparing shot groupings, or to compare the difficulty of hitting different sized targets at different distances. Using optics with mil markings in the one can make a range estimation of a known size target, or vice versa to determine a target size if the distance is known. In such applications it is useful to use a unit for target size that is a thousandth of the unit for range, for instance by using the metric units millimeters for target size and meters for range. This coincides with the definition of the milliradian where the arc length is defined as 1/1000 of the radius, a common adjustment value in firearm sights is 1 cm at 100 meters which equals 10 mm/100 m = 1/10 mil. The true definition of a milliradian is based on a circle with a radius of one. There are other definitions used for mapping and artillery which are rounded to more easily be divided into smaller parts. The milliradian was first used in the mid nineteenth century by Charles-Marc Dapples, degrees and minutes were the usual units of angular measurement but others were being proposed, with grads under various names having considerable popularity in much of northern Europe. However, Imperial Russia used a different approach, dividing a circle into equilateral triangles, around the time of the start of World War I, France was experimenting with the use of milliemes for use with artillery sights instead of decigrades. The United Kingdom was also trialing them to replace degrees and minutes and they were adopted by France although decigrades also remained in use throughout World War I. The United States, which copied many French artillery practices, adopted mils, before 2007 the Swedish defence forces used streck which is closer to the milliradian but then changed to NATO mils. After the Bolshevik Revolution and the adoption of the system of measurement the Red Army expanded the 600 unit circle into a 6000 mil one. Hence the Russian mil has a different origin than those derived from French artillery practices. In the 1950s, NATO adopted metric units of measurement for land, Mils, meters, and kilograms became standard, although degrees remained in use for naval and air purposes, reflecting civil practices. The approximation error by using the linear formula will increase as the angle increases. New shooters are often explained the principle of subtensions in order to understand that a milliradian is an angular measurement, subtension is the physical amount of space covered by an angle and varies with distance. Thus, the corresponding to a mil varies with range. Subtensions always change with distance, but a mil is always a mil regardless of distance, therefore ballistic tables and shot corrections are given in mils thereby avoiding the need of mathematical calculations
2.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles
3.
Pi
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The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
4.
International System of Units
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The International System of Units is the modern form of the metric system, and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, the system also establishes a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system was published in 1960 as the result of an initiative began in 1948. It is based on the system of units rather than any variant of the centimetre-gram-second system. The motivation for the development of the SI was the diversity of units that had sprung up within the CGS systems, the International System of Units has been adopted by most developed countries, however, the adoption has not been universal in all English-speaking countries. The metric system was first implemented during the French Revolution with just the metre and kilogram as standards of length, in the 1830s Carl Friedrich Gauss laid the foundations for a coherent system based on length, mass, and time. In the 1860s a group working under the auspices of the British Association for the Advancement of Science formulated the requirement for a coherent system of units with base units and derived units. Meanwhile, in 1875, the Treaty of the Metre passed responsibility for verification of the kilogram, in 1921, the Treaty was extended to include all physical quantities including electrical units originally defined in 1893. The units associated with these quantities were the metre, kilogram, second, ampere, kelvin, in 1971, a seventh base quantity, amount of substance represented by the mole, was added to the definition of SI. On 11 July 1792, the proposed the names metre, are, litre and grave for the units of length, area, capacity. The committee also proposed that multiples and submultiples of these units were to be denoted by decimal-based prefixes such as centi for a hundredth, on 10 December 1799, the law by which the metric system was to be definitively adopted in France was passed. Prior to this, the strength of the magnetic field had only been described in relative terms. The technique used by Gauss was to equate the torque induced on a magnet of known mass by the earth’s magnetic field with the torque induced on an equivalent system under gravity. The resultant calculations enabled him to assign dimensions based on mass, length, a French-inspired initiative for international cooperation in metrology led to the signing in 1875 of the Metre Convention. Initially the convention only covered standards for the metre and the kilogram, one of each was selected at random to become the International prototype metre and International prototype kilogram that replaced the mètre des Archives and kilogramme des Archives respectively. Each member state was entitled to one of each of the prototypes to serve as the national prototype for that country. Initially its prime purpose was a periodic recalibration of national prototype metres. The official language of the Metre Convention is French and the version of all official documents published by or on behalf of the CGPM is the French-language version
5.
William Thomson, 1st Baron Kelvin
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William Thomson, 1st Baron Kelvin, OM, GCVO, PC, FRS, FRSE was a Scots-Irish mathematical physicist and engineer who was born in Belfast in 1824. He worked closely with mathematics professor Hugh Blackburn in his work and he also had a career as an electric telegraph engineer and inventor, which propelled him into the public eye and ensured his wealth, fame and honour. For his work on the telegraph project he was knighted in 1866 by Queen Victoria. He had extensive maritime interests and was most noted for his work on the mariners compass, absolute temperatures are stated in units of kelvin in his honour. He was ennobled in 1892 in recognition of his achievements in thermodynamics and he was the first British scientist to be elevated to the House of Lords. The title refers to the River Kelvin, which close by his laboratory at the University of Glasgow. His home was the red sandstone mansion Netherhall, in Largs. William Thomsons father, James Thomson, was a teacher of mathematics and engineering at Royal Belfast Academical Institution, James Thomson married Margaret Gardner in 1817 and, of their children, four boys and two girls survived infancy. Margaret Thomson died in 1830 when William was six years old, William and his elder brother James were tutored at home by their father while the younger boys were tutored by their elder sisters. James was intended to benefit from the share of his fathers encouragement, affection. In 1832, his father was appointed professor of mathematics at Glasgow, the Thomson children were introduced to a broader cosmopolitan experience than their fathers rural upbringing, spending mid-1839 in London and the boys were tutored in French in Paris. Mid-1840 was spent in Germany and the Netherlands, language study was given a high priority. His sister, Anna Thomson, was the mother of James Thomson Bottomley FRSE, Thomson had heart problems and nearly died when he was 9 years old. In school, Thomson showed a keen interest in the classics along with his natural interest in the sciences, at the age of 12 he won a prize for translating Lucian of Samosatas Dialogues of the Gods from Latin to English. In the academic year 1839/1840, Thomson won the prize in astronomy for his Essay on the figure of the Earth which showed an early facility for mathematical analysis. Throughout his life, he would work on the problems raised in the essay as a strategy during times of personal stress. On the title page of this essay Thomson wrote the lines from Alexander Popes Essay on Man. These lines inspired Thomson to understand the world using the power and method of science, Go
6.
Queen's University Belfast
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Queens University Belfast is a public research university in Belfast, Northern Ireland. The university was chartered in 1845, and opened in 1849 as Queens College, Belfast, but has roots going back to 1810, the university forms the focal point of the Queens Quarter area of the city, one of Belfasts four cultural districts. It offers academic degrees at various levels and across a broad subject range, since 1 March 2014, Patrick Johnston has been the university’s 12th President and Vice-Chancellor. The university is associated with two Nobel laureates and one Turing Award laureate, Queens College, Belfast, opened in 1849. Its main building, the Lanyon Building, was designed by the English architect, at its opening, it had 23 professors and 343 students. Some early students at Queens University Belfast took University of London examinations, the university was one of only eight United Kingdom universities to hold a parliamentary seat in the House of Commons at Westminster until such representation was abolished in 1950. The university was represented in the Parliament of Northern Ireland from 1920–1968. On 20 June 2006, the university announced a £259 million investment programme focusing on facilities, recruitment, the building has been named in honour of Sir Allen McClay, a major benefactor of Queens University and of the Library. In June 2010, the university announced the launch of a £7. 5m Ansin international research hub with Seagate Technologies. In addition to the campus not far from the centre of Belfast. Although offering a range of courses, these colleges primarily provide training for those wishing to enter the teaching profession. The university has formal agreements with colleges in Northern Ireland. The current chancellor is the businessman Thomas Moran, academics at Queens are organised into fifteen schools across three faculties. The three faculties are the Faculty of Arts, Humanities & Social Sciences, the Faculty of Engineering & Physical Sciences, each of the 20 schools operates as a primary management unit of the university and the schools are the focus for education and research for their respective subject areas. On Feb 18th 2016 BBC Northern Ireland reported that this Institute was to be closed, Institute of Cognition and Culture - Founded in 2004, this is one of the world’s first centres for research in the cognitive science of culture. It has brought together a range of cutting-edge cognitive scientists via a series of visiting fellowships, Institute of Irish Studies - The Institute of Irish Studies celebrated its 50th birthday in March 2015. Institute of Professional Legal Studies - The Institute of Professional Legal Studies was established in 1977 at Queens University Belfast and it provides an internationally recognised and unique one-year postgraduate course for trainee barristers and trainee solicitors who study together. In all five colleges teach any programmes with a emphasis on behalf of the university
7.
Belfast
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Belfast is the capital and largest city of Northern Ireland, the second largest on the island of Ireland, and the heart of the tenth largest Primary Urban Area in the United Kingdom. On the River Lagan, it had a population of 286,000 at the 2011 census and 333,871 after the 2015 council reform, Belfast was granted city status in 1888. Belfast played a key role in the Industrial Revolution, and was an industrial centre until the latter half of the 20th century. It has sustained a major aerospace and missiles industry since the mid 1930s, industrialisation and the inward migration it brought made Belfast Irelands biggest city at the beginning of the 20th century. Today, Belfast remains a centre for industry, as well as the arts, higher education, business, and law, additionally, Belfast city centre has undergone considerable expansion and regeneration in recent years, notably around Victoria Square. Belfast is served by two airports, George Best Belfast City Airport in the city, and Belfast International Airport 15 miles west of the city. Although the county borough of Belfast was created when it was granted city status by Queen Victoria in 1888, the site of Belfast has been occupied since the Bronze Age. The Giants Ring, a 5, 000-year-old henge, is located near the city, Belfast remained a small settlement of little importance during the Middle Ages. The ONeill clan had a presence in the area, in the 14th century, Cloinne Aodha Buidhe, descendants of Aodh Buidhe ONeill built Grey Castle at Castlereagh, now in the east of the city. Conn ONeill of the Clannaboy ONeills owned vast lands in the area and was the last inhabitant of Grey Castle, evidence of this period of Belfasts growth can still be seen in the oldest areas of the city, known as the Entries. Belfast blossomed as a commercial and industrial centre in the 18th and 19th centuries, industries thrived, including linen, rope-making, tobacco, heavy engineering and shipbuilding, and at the end of the 19th century, Belfast briefly overtook Dublin as the largest city in Ireland. The Harland and Wolff shipyards became one of the largest shipbuilders in the world, in 1886 the city suffered intense riots over the issue of home rule, which had divided the city. In 1920–22, Belfast became the capital of the new entity of Northern Ireland as the island of Ireland was partitioned, the accompanying conflict cost up to 500 lives in Belfast, the bloodiest sectarian strife in the city until the Troubles of the late 1960s onwards. Belfast was heavily bombed during World War II, in one raid, in 1941, German bombers killed around one thousand people and left tens of thousands homeless. Apart from London, this was the greatest loss of life in a raid during the Blitz. Belfast has been the capital of Northern Ireland since its establishment in 1921 following the Government of Ireland Act 1920 and it had been the scene of various episodes of sectarian conflict between its Catholic and Protestant populations. These opposing groups in conflict are now often termed republican and loyalist respectively. The most recent example of conflict was known as the Troubles – a civil conflict that raged from around 1969 to 1998
8.
University of St Andrews
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The University of St Andrews is a British public research university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland, St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy. St Andrews is made up from a variety of institutions, including three constituent colleges and 18 academic schools organised into four faculties, the university occupies historic and modern buildings located throughout the town. The academic year is divided into two terms, Martinmas and Candlemas, in term time, over one-third of the towns population is either a staff member or student of the university. It is ranked as the third best university in the United Kingdom in national league tables, the Times Higher Education World Universities Ranking names St Andrews among the worlds Top 50 universities for Social Sciences, Arts and Humanities. St Andrews has the highest student satisfaction amongst all multi-faculty universities in the United Kingdom, St Andrews has many notable alumni and affiliated faculty, including eminent mathematicians, scientists, theologians, philosophers, and politicians. Six Nobel Laureates are among St Andrews alumni and former staff, a charter of privilege was bestowed upon the society of masters and scholars by the Bishop of St Andrews, Henry Wardlaw, on 28 February 1411. Wardlaw then successfully petitioned the Avignon Pope Benedict XIII to grant the university status by issuing a series of papal bulls. King James I of Scotland confirmed the charter of the university in 1432, subsequent kings supported the university with King James V confirming privileges of the university in 1532. A college of theology and arts called St Johns College was founded in 1418 by Robert of Montrose, St Salvators College was established in 1450, by Bishop James Kennedy. St Leonards College was founded in 1511 by Archbishop Alexander Stewart, St Johns College was refounded by Cardinal James Beaton under the name St Marys College in 1538 for the study of divinity and law. Some university buildings that date from this period are still in use today, such as St Salvators Chapel, St Leonards College Chapel, at this time, the majority of the teaching was of a religious nature and was conducted by clerics associated with the cathedral. During the 17th and 18th centuries, the university had mixed fortunes and was beset by civil. He described it as pining in decay and struggling for life, in the second half of the 19th century, pressure was building upon universities to open up higher education to women. In 1876, the University Senate decided to allow women to receive an education at St Andrews at a roughly equal to the Master of Arts degree that men were able to take at the time. The scheme came to be known as the L. L. A and it required women to pass five subjects at an ordinary level and one at honours level and entitled them to hold a degree from the university. In 1889 the Universities Act made it possible to admit women to St Andrews. Agnes Forbes Blackadder became the first woman to graduate from St Andrews on the level as men in October 1894
9.
Trigonometric functions
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
10.
Theta
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Theta is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth. In the system of Greek numerals it has a value of 9, in Ancient Greek, θ represented the aspirated voiceless dental plosive /t̪ʰ/, but in Modern Greek it represents the voiceless dental fricative /θ/. In its archaic form, θ was written as a cross within a circle, archaic crossed forms of theta are seen in the wheel letters of Linear A and Linear B. The cursive form ϑ was retained by Unicode as U+03D1 ϑ GREEK THETA SYMBOL, for the purpose of writing Greek text, the two can be font variants of a single character, but θ and ϑ are also used as distinct symbols in technical and mathematical contexts. In the Latin script used for the Gaulish language, theta developed into the tau gallicum, conventionally transliterated as Ð, the phonetic value of the tau gallicum is thought to have been. The early Cyrillic letter fita developed from θ and this letter existed in the Russian alphabet until the 1918 Russian orthography reform. In the International Phonetic Alphabet, represents the voiceless dental fricative and it does not represent the consonant in the, which is the voiced dental fricative. The lower-case letter θ is used as a symbol for, A plane angle in geometry, a Variable in trigonometry A special function of several complex variables. One of the Chebyshev functions in prime number theory, the score of a test taker in item response theory. Theta Type Replication, a type of bacterial DNA replication specific to circular chromosomes, threshold value of an artificial neuron. A Bayer designation letter applied to a star in a constellation, usually the star so labelled. The parameter frequently used in writing the likelihood function, the Watterson estimator for the population mutation rate in population genetics. Indicates a minimum optimum integration level determined by the intersection of GG, the GG-LL schedules are a tool used in analyzing the potential benefits of a country pegging their domestic currency to a foreign currency. The reserve ratio of banks in economic models, the ordinal collapsing function developed by Solomon Feferman The upper-case letter Θ is used as a symbol for, Quantity or temperature, by SI standard. An asymptotically tight bound in the analysis of algorithms, a certain ordinal number in set theory. Pentaquarks, exotic baryons in particle physics, a brain signal frequency ranging from 4–8 Hz. One of the known as Greeks in finance, representing time decay of options or the change in the intrinsic value of an option divided by the number of days until the option expires. A variable indicating temperature difference in heat transfer, according to Porphyry of Tyros, the Egyptians used an X within a circle as a symbol of the soul, having a value of nine, it was used as a symbol for Ennead