1.
Grade (slope)
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The grade of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a case of the slope, where zero indicates horizontality. A larger number indicates higher or steeper degree of tilt, often slope is calculated as a ratio of rise to run, or as a fraction in which run is the horizontal distance and rise is the vertical distance. The grades or slopes of existing physical features such as canyons and hillsides, stream and river banks, grades are typically specified for new linear constructions. The grade may refer to the slope or the perpendicular cross slope. There are several ways to express slope, as an angle of inclination to the horizontal, as a percentage, the formula for which is 100 rise run which could also be expressed as the tangent of the angle of inclination times 100. In the U. S. this percentage grade is the most commonly used unit for communicating slopes in transportation, surveying, construction, and civil engineering. As a per mille figure, the formula for which is 1000 rise run which could also be expressed as the tangent of the angle of inclination times 1000 and this is commonly used in Europe to denote the incline of a railway. As a ratio of one part rise to so many parts run, for example, a slope that has a rise of 5 feet for every 100 feet of run would have a slope ratio of 1 in 20. This is generally the method used to describe railway grades in Australia, any of these may be used. Grade is usually expressed as a percentage, but this is converted to the angle α from horizontal or the other expressions. Slope may still be expressed when the run is not known. This is not the way to specify slope, it follows the sine function rather than the tangent function. But in practice the way to calculate slope is to measure the distance along the slope and the vertical rise. When the angle of inclination is small, using the slope length rather than the horizontal displacement makes only an insignificant difference, Railway gradients are usually expressed in terms of the rise in relation to the distance along the track as a practical measure. In cases where the difference between sin and tan is significant, the tangent is used. In any case, the identity holds for all inclinations up to 90 degrees, tan α = sin α1 − sin 2 α In Europe. Grades are related using the equations with symbols from the figure at top
2.
Gradient
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In mathematics, the gradient is a multi-variable generalization of the derivative. While a derivative can be defined on functions of a variable, for functions of several variables. The gradient is a function, as opposed to a derivative. If f is a differentiable, real-valued function of several variables, like the derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, the components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. The Jacobian is the generalization of the gradient for vector-valued functions of several variables, a further generalization for a function between Banach spaces is the Fréchet derivative. Consider a room in which the temperature is given by a field, T. At each point in the room, the gradient of T at that point will show the direction in which the temperature rises most quickly, the magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level at point is H, the gradient of H at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, suppose that the steepest slope on a hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%, if, instead, the road goes around the hill at an angle, then it will have a shallower slope. This observation can be stated as follows. If the hill height function H is differentiable, then the gradient of H dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when H is differentiable, the dot product of the gradient of H with a unit vector is equal to the directional derivative of H in the direction of that unit vector. The gradient of a function f is denoted ∇f or ∇→f where ∇ denotes the vector differential operator. The notation grad f is commonly used for the gradient. The gradient of f is defined as the vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is
3.
Radian
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The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
4.
Regular polygons
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In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
5.
Angle
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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
6.
Degree (angle)
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A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of measure is the radian. Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians, the original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the path over the course of the year. Some ancient calendars, such as the Persian calendar, used 360 days for a year, the use of a calendar with 360 days may be related to the use of sexagesimal numbers. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle, a chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree, Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes, eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. Furthermore, it is divisible by every number from 1 to 10 except 7 and this property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention. Finally, it may be the case more than one of these factors has come into play. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates, degree measurements may be written using decimal degrees, with the symbol behind the decimals. Alternatively, the sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes, and one minute into 60 seconds, use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are represented by a single and double prime. For example,40. 1875° = 40° 11′ 15″, or, using quotation mark characters, additional precision can be provided using decimals for the arcseconds component. The older system of thirds, fourths, etc. which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today
7.
Minutes of arc
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A minute of arc, arcminute, arc minute, or minute arc is a unit of angular measurement equal to 1/60 of one degree. Since one degree is 1/360 of a turn, one minute of arc is 1/21600 of a turn, a second of arc, arcsecond, or arc second is 1/60 of an arcminute, 1/3600 of a degree, 1/1296000 of a turn, and π/648000 of a radian. To express even smaller angles, standard SI prefixes can be employed, the number of square arcminutes in a complete sphere is 4 π2 =466560000 π ≈148510660 square arcminutes. The standard symbol for marking the arcminute is the prime, though a single quote is used where only ASCII characters are permitted. One arcminute is thus written 1′ and it is also abbreviated as arcmin or amin or, less commonly, the prime with a circumflex over it. The standard symbol for the arcsecond is the prime, though a double quote is commonly used where only ASCII characters are permitted. One arcsecond is thus written 1″ and it is also abbreviated as arcsec or asec. In celestial navigation, seconds of arc are used in calculations. This notation has been carried over into marine GPS receivers, which normally display latitude and longitude in the format by default. An arcsecond is approximately the angle subtended by a U. S. dime coin at a distance of 4 kilometres, a milliarcsecond is about the size of a dime atop the Eiffel Tower as seen from New York City. A microarcsecond is about the size of a period at the end of a sentence in the Apollo mission manuals left on the Moon as seen from Earth, since antiquity the arcminute and arcsecond have been used in astronomy. The principal exception is Right ascension in equatorial coordinates, which is measured in units of hours, minutes. These small angles may also be written in milliarcseconds, or thousandths of an arcsecond, the unit of distance, the parsec, named from the parallax of one arcsecond, was developed for such parallax measurements. It is the distance at which the radius of the Earths orbit would subtend an angle of one arcsecond. The ESA astrometric space probe Gaia is hoped to measure star positions to 20 microarcseconds when it begins producing catalog positions sometime after 2016, there are about 1.3 trillion µas in a turn. Currently the best catalog positions of stars actually measured are in terms of milliarcseconds, apart from the Sun, the star with the largest angular diameter from Earth is R Doradus, a red supergiant with a diameter of 0.05 arcsecond. The dwarf planet Pluto has proven difficult to resolve because its angular diameter is about 0.1 arcsecond, space telescopes are not affected by the Earths atmosphere but are diffraction limited. For example, the Hubble space telescope can reach a size of stars down to about 0. 1″
8.
Unit of measurement
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A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same quantity. Any other value of quantity can be expressed as a simple multiple of the unit of measurement. For example, length is a physical quantity, the metre is a unit of length that represents a definite predetermined length. When we say 10 metres, we actually mean 10 times the definite predetermined length called metre, the definition, agreement, and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. Different systems of units used to be very common, now there is a global standard, the International System of Units, the modern form of the metric system. In trade, weights and measures is often a subject of regulation, to ensure fairness. The International Bureau of Weights and Measures is tasked with ensuring worldwide uniformity of measurements, metrology is the science for developing nationally and internationally accepted units of weights and measures. In physics and metrology, units are standards for measurement of quantities that need clear definitions to be useful. Reproducibility of experimental results is central to the scientific method, a standard system of units facilitates this. Scientific systems of units are a refinement of the concept of weights, science, medicine, and engineering often use larger and smaller units of measurement than those used in everyday life and indicate them more precisely. The judicious selection of the units of measurement can aid researchers in problem solving, in the social sciences, there are no standard units of measurement and the theory and practice of measurement is studied in psychometrics and the theory of conjoint measurement. A unit of measurement is a quantity of a physical property. Units of measurement were among the earliest tools invented by humans, primitive societies needed rudimentary measures for many tasks, constructing dwellings of an appropriate size and shape, fashioning clothing, or bartering food or raw materials. Weights and measures are mentioned in the Bible and it is a commandment to be honest and have fair measures. As of the 21st Century, multiple unit systems are used all over the world such as the United States Customary System, the British Customary System, however, the United States is the only industrialized country that has not yet completely converted to the Metric System. The systematic effort to develop an acceptable system of units dates back to 1790 when the French National Assembly charged the French Academy of Sciences to come up such a unit system. After this treaty was signed, a General Conference of Weights, the CGPM produced the current SI system which was adopted in 1954 at the 10th conference of weights and measures. Currently, the United States is a society which uses both the SI system and the US Customary system
9.
Turn (geometry)
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A turn is a unit of plane angle measurement equal to 2π radians, 360° or 400 gon. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot, a turn can be subdivided in many different ways, into half turns, quarter turns, centiturns, milliturns, binary angles, points etc. A turn can be divided in 100 centiturns or 1000 milliturns, with each corresponding to an angle of 0. 36°. A protractor divided in centiturns is normally called a percentage protractor, binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, the binary degree, also known as the binary radian, is 1⁄256 turn. The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte, other measures of angle used in computing may be based on dividing one whole turn into 2n equal parts for other values of n. The notion of turn is used for planar rotations. Two special rotations have acquired appellations of their own, a rotation through 180° is commonly referred to as a half-turn, the word turn originates via Latin and French from the Greek word τόρνος. In 1697, David Gregory used π/ρ to denote the perimeter of a divided by its radius. However, earlier in 1647, William Oughtred had used δ/π for the ratio of the diameter to perimeter, the first use of the symbol π on its own with its present meaning was in 1706 by the Welsh mathematician William Jones. Euler adopted the symbol with that meaning in 1737, leading to its widespread use, percentage protractors have existed since 1922, but the terms centiturns and milliturns were introduced much later by Sir Fred Hoyle. The German standard DIN1315 proposed the unit symbol pla for turns, since 2011, the HP 39gII and HP Prime support the unit symbol tr for turns. In 2016, support for turns was also added to newRPL for the HP 50g, one turn is equal to 2π radians. In 1958, Albert Eagle proposed the Greek letter tau τ as a symbol for 1/2π and his proposal used a pi with three legs symbol to denote the constant. In 2010, Michael Hartl proposed to use tau to represent Palais circle constant, τ=2π. First, τ is the number of radians in one turn, which allows fractions of a turn to be expressed directly, for instance. Second, τ visually resembles π, whose association with the constant is unavoidable. Hartls Tau Manifesto gives many examples of formulas that are simpler if tau is used instead of pi, however, a rebuttal was given in The Pi Manifesto, stating a variety of reasons tau should not supplant pi
10.
Europe
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Europe is a continent that comprises the westernmost part of Eurasia. Europe is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, yet the non-oceanic borders of Europe—a concept dating back to classical antiquity—are arbitrary. Europe covers about 10,180,000 square kilometres, or 2% of the Earths surface, politically, Europe is divided into about fifty sovereign states of which the Russian Federation is the largest and most populous, spanning 39% of the continent and comprising 15% of its population. Europe had a population of about 740 million as of 2015. Further from the sea, seasonal differences are more noticeable than close to the coast, Europe, in particular ancient Greece, was the birthplace of Western civilization. The fall of the Western Roman Empire, during the period, marked the end of ancient history. Renaissance humanism, exploration, art, and science led to the modern era, from the Age of Discovery onwards, Europe played a predominant role in global affairs. Between the 16th and 20th centuries, European powers controlled at times the Americas, most of Africa, Oceania. The Industrial Revolution, which began in Great Britain at the end of the 18th century, gave rise to economic, cultural, and social change in Western Europe. During the Cold War, Europe was divided along the Iron Curtain between NATO in the west and the Warsaw Pact in the east, until the revolutions of 1989 and fall of the Berlin Wall. In 1955, the Council of Europe was formed following a speech by Sir Winston Churchill and it includes all states except for Belarus, Kazakhstan and Vatican City. Further European integration by some states led to the formation of the European Union, the EU originated in Western Europe but has been expanding eastward since the fall of the Soviet Union in 1991. The European Anthem is Ode to Joy and states celebrate peace, in classical Greek mythology, Europa is the name of either a Phoenician princess or of a queen of Crete. The name contains the elements εὐρύς, wide, broad and ὤψ eye, broad has been an epithet of Earth herself in the reconstructed Proto-Indo-European religion and the poetry devoted to it. For the second part also the divine attributes of grey-eyed Athena or ox-eyed Hera. The same naming motive according to cartographic convention appears in Greek Ανατολή, Martin Litchfield West stated that phonologically, the match between Europas name and any form of the Semitic word is very poor. Next to these there is also a Proto-Indo-European root *h1regʷos, meaning darkness. Most major world languages use words derived from Eurṓpē or Europa to refer to the continent, in some Turkic languages the originally Persian name Frangistan is used casually in referring to much of Europe, besides official names such as Avrupa or Evropa
11.
Celsius
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Celsius, also known as centigrade, is a metric scale and unit of measurement for temperature. As an SI derived unit, it is used by most countries in the world and it is named after the Swedish astronomer Anders Celsius, who developed a similar temperature scale. The degree Celsius can refer to a temperature on the Celsius scale as well as a unit to indicate a temperature interval. Before being renamed to honour Anders Celsius in 1948, the unit was called centigrade, from the Latin centum, which means 100, and gradus, which means steps. The scale is based on 0° for the point of water. This scale is widely taught in schools today, by international agreement the unit degree Celsius and the Celsius scale are currently defined by two different temperatures, absolute zero, and the triple point of VSMOW. This definition also precisely relates the Celsius scale to the Kelvin scale, absolute zero, the lowest temperature possible, is defined as being precisely 0 K and −273.15 °C. The temperature of the point of water is defined as precisely 273.16 K at 611.657 pascals pressure. This definition fixes the magnitude of both the degree Celsius and the kelvin as precisely 1 part in 273.16 of the difference between absolute zero and the point of water. Thus, it sets the magnitude of one degree Celsius and that of one kelvin as exactly the same, additionally, it establishes the difference between the two scales null points as being precisely 273.15 degrees. In his paper Observations of two persistent degrees on a thermometer, he recounted his experiments showing that the point of ice is essentially unaffected by pressure. He also determined with precision how the boiling point of water varied as a function of atmospheric pressure. He proposed that the point of his temperature scale, being the boiling point. This pressure is known as one standard atmosphere, the BIPMs 10th General Conference on Weights and Measures later defined one standard atmosphere to equal precisely 1013250dynes per square centimetre. On 19 May 1743 he published the design of a mercury thermometer, in 1744, coincident with the death of Anders Celsius, the Swedish botanist Carolus Linnaeus reversed Celsiuss scale. In it, Linnaeus recounted the temperatures inside the orangery at the University of Uppsala Botanical Garden, since the 19th century, the scientific and thermometry communities worldwide referred to this scale as the centigrade scale. Temperatures on the scale were often reported simply as degrees or. More properly, what was defined as centigrade then would now be hectograde.2 gradians, for scientific use, Celsius is the term usually used, with centigrade otherwise continuing to be in common but decreasing use, especially in informal contexts in English-speaking countries
12.
French Revolution
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Through the Revolutionary Wars, it unleashed a wave of global conflicts that extended from the Caribbean to the Middle East. Historians widely regard the Revolution as one of the most important events in human history, the causes of the French Revolution are complex and are still debated among historians. Following the Seven Years War and the American Revolutionary War, the French government was deeply in debt, Years of bad harvests leading up to the Revolution also inflamed popular resentment of the privileges enjoyed by the clergy and the aristocracy. Demands for change were formulated in terms of Enlightenment ideals and contributed to the convocation of the Estates-General in May 1789, a central event of the first stage, in August 1789, was the abolition of feudalism and the old rules and privileges left over from the Ancien Régime. The next few years featured political struggles between various liberal assemblies and right-wing supporters of the intent on thwarting major reforms. The Republic was proclaimed in September 1792 after the French victory at Valmy, in a momentous event that led to international condemnation, Louis XVI was executed in January 1793. External threats closely shaped the course of the Revolution, internally, popular agitation radicalised the Revolution significantly, culminating in the rise of Maximilien Robespierre and the Jacobins. Large numbers of civilians were executed by revolutionary tribunals during the Terror, after the Thermidorian Reaction, an executive council known as the Directory assumed control of the French state in 1795. The rule of the Directory was characterised by suspended elections, debt repudiations, financial instability, persecutions against the Catholic clergy, dogged by charges of corruption, the Directory collapsed in a coup led by Napoleon Bonaparte in 1799. The modern era has unfolded in the shadow of the French Revolution, almost all future revolutionary movements looked back to the Revolution as their predecessor. The values and institutions of the Revolution dominate French politics to this day, the French Revolution differed from other revolutions in being not merely national, for it aimed at benefiting all humanity. Globally, the Revolution accelerated the rise of republics and democracies and it became the focal point for the development of all modern political ideologies, leading to the spread of liberalism, radicalism, nationalism, socialism, feminism, and secularism, among many others. The Revolution also witnessed the birth of total war by organising the resources of France, historians have pointed to many events and factors within the Ancien Régime that led to the Revolution. Over the course of the 18th century, there emerged what the philosopher Jürgen Habermas called the idea of the sphere in France. A perfect example would be the Palace of Versailles which was meant to overwhelm the senses of the visitor and convince one of the greatness of the French state and Louis XIV. Starting in the early 18th century saw the appearance of the sphere which was critical in that both sides were active. In France, the emergence of the public sphere outside of the control of the saw the shift from Versailles to Paris as the cultural capital of France. In the 1750s, during the querelle des bouffons over the question of the quality of Italian vs, in 1782, Louis-Sébastien Mercier wrote, The word court no longer inspires awe amongst us as in the time of Louis XIV
13.
France
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France, officially the French Republic, is a country with territory in western Europe and several overseas regions and territories. The European, or metropolitan, area of France extends from the Mediterranean Sea to the English Channel and the North Sea, Overseas France include French Guiana on the South American continent and several island territories in the Atlantic, Pacific and Indian oceans. France spans 643,801 square kilometres and had a population of almost 67 million people as of January 2017. It is a unitary republic with the capital in Paris. Other major urban centres include Marseille, Lyon, Lille, Nice, Toulouse, during the Iron Age, what is now metropolitan France was inhabited by the Gauls, a Celtic people. The area was annexed in 51 BC by Rome, which held Gaul until 486, France emerged as a major European power in the Late Middle Ages, with its victory in the Hundred Years War strengthening state-building and political centralisation. During the Renaissance, French culture flourished and a colonial empire was established. The 16th century was dominated by civil wars between Catholics and Protestants. France became Europes dominant cultural, political, and military power under Louis XIV, in the 19th century Napoleon took power and established the First French Empire, whose subsequent Napoleonic Wars shaped the course of continental Europe. Following the collapse of the Empire, France endured a succession of governments culminating with the establishment of the French Third Republic in 1870. Following liberation in 1944, a Fourth Republic was established and later dissolved in the course of the Algerian War, the Fifth Republic, led by Charles de Gaulle, was formed in 1958 and remains to this day. Algeria and nearly all the colonies became independent in the 1960s with minimal controversy and typically retained close economic. France has long been a centre of art, science. It hosts Europes fourth-largest number of cultural UNESCO World Heritage Sites and receives around 83 million foreign tourists annually, France is a developed country with the worlds sixth-largest economy by nominal GDP and ninth-largest by purchasing power parity. In terms of household wealth, it ranks fourth in the world. France performs well in international rankings of education, health care, life expectancy, France remains a great power in the world, being one of the five permanent members of the United Nations Security Council with the power to veto and an official nuclear-weapon state. It is a member state of the European Union and the Eurozone. It is also a member of the Group of 7, North Atlantic Treaty Organization, Organisation for Economic Co-operation and Development, the World Trade Organization, originally applied to the whole Frankish Empire, the name France comes from the Latin Francia, or country of the Franks
14.
Metric system
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The metric system is an internationally agreed decimal system of measurement. Many sources also cite Liberia and Myanmar as the other countries not to have done so. Although the originators intended to devise a system that was accessible to all. Control of the units of measure was maintained by the French government until 1875, when it was passed to an intergovernmental organisation. From its beginning, the features of the metric system were the standard set of interrelated base units. These base units are used to larger and smaller units that could replace a huge number of other units of measure in existence. Although the system was first developed for use, the development of coherent units of measure made it particularly suitable for science. Although the metric system has changed and developed since its inception, designed for transnational use, it consisted of a basic set of units of measurement, now known as base units. At the outbreak of the French Revolution in 1789, most countries, the metric system was designed to be universal—in the words of the French philosopher Marquis de Condorcet it was to be for all people for all time. However, these overtures failed and the custody of the metric system remained in the hands of the French government until 1875. In languages where the distinction is made, unit names are common nouns, the concept of using consistent classical names for the prefixes was first proposed in a report by the Commission on Weights and Measures in May 1793. The prefix kilo, for example, is used to multiply the unit by 1000, thus the kilogram and kilometre are a thousand grams and metres respectively, and a milligram and millimetre are one thousandth of a gram and metre respectively. These relations can be written symbolically as,1 mg =0, however,1935 extensions to the prefix system did not follow this convention, the prefixes nano- and micro-, for example have Greek roots. During the 19th century the prefix myria-, derived from the Greek word μύριοι, was used as a multiplier for 10000, prefixes are not usually used to indicate multiples of a second greater than 1, the non-SI units of minute, hour and day are used instead. On the other hand, prefixes are used for multiples of the unit of volume. The base units used in the system must be realisable. Each of the units in SI is accompanied by a mise en pratique published by the BIPM that describes in detail at least one way in which the base unit can be measured. In practice, such realisation is done under the auspices of a mutual acceptance arrangement, in the original version of the metric system the base units could be derived from a specified length and the weight of a specified volume of pure water
15.
German language
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German is a West Germanic language that is mainly spoken in Central Europe. It is the most widely spoken and official language in Germany, Austria, Switzerland, South Tyrol, the German-speaking Community of Belgium and it is also one of the three official languages of Luxembourg. Major languages which are most similar to German include other members of the West Germanic language branch, such as Afrikaans, Dutch, English, Luxembourgish and it is the second most widely spoken Germanic language, after English. One of the languages of the world, German is the first language of about 95 million people worldwide. The German speaking countries are ranked fifth in terms of publication of new books. German derives most of its vocabulary from the Germanic branch of the Indo-European language family, a portion of German words are derived from Latin and Greek, and fewer are borrowed from French and English. With slightly different standardized variants, German is a pluricentric language, like English, German is also notable for its broad spectrum of dialects, with many unique varieties existing in Europe and also other parts of the world. The history of the German language begins with the High German consonant shift during the migration period, when Martin Luther translated the Bible, he based his translation primarily on the standard bureaucratic language used in Saxony, also known as Meißner Deutsch. Copies of Luthers Bible featured a long list of glosses for each region that translated words which were unknown in the region into the regional dialect. Roman Catholics initially rejected Luthers translation, and tried to create their own Catholic standard of the German language – the difference in relation to Protestant German was minimal. It was not until the middle of the 18th century that a widely accepted standard was created, until about 1800, standard German was mainly a written language, in urban northern Germany, the local Low German dialects were spoken. Standard German, which was different, was often learned as a foreign language with uncertain pronunciation. Northern German pronunciation was considered the standard in prescriptive pronunciation guides though, however, German was the language of commerce and government in the Habsburg Empire, which encompassed a large area of Central and Eastern Europe. Until the mid-19th century, it was essentially the language of townspeople throughout most of the Empire and its use indicated that the speaker was a merchant or someone from an urban area, regardless of nationality. Some cities, such as Prague and Budapest, were gradually Germanized in the years after their incorporation into the Habsburg domain, others, such as Pozsony, were originally settled during the Habsburg period, and were primarily German at that time. Prague, Budapest and Bratislava as well as cities like Zagreb, the most comprehensive guide to the vocabulary of the German language is found within the Deutsches Wörterbuch. This dictionary was created by the Brothers Grimm and is composed of 16 parts which were issued between 1852 and 1860, in 1872, grammatical and orthographic rules first appeared in the Duden Handbook. In 1901, the 2nd Orthographical Conference ended with a standardization of the German language in its written form
16.
Swedish language
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Swedish is a North Germanic language, spoken natively by more than 9 million people predominantly in Sweden and parts of Finland, where it has equal legal standing with Finnish. It is largely mutually intelligible with Norwegian and Danish, along with the other North Germanic languages, Swedish is a descendant of Old Norse, the common language of the Germanic peoples living in Scandinavia during the Viking Era. It is currently the largest of the North Germanic languages by number of speakers, Standard Swedish, spoken by most Swedes, is the national language that evolved from the Central Swedish dialects in the 19th century and was well established by the beginning of the 20th century. While distinct regional varieties descended from the rural dialects still exist. The standard word order is, as in most Germanic languages, V2, Swedish morphology is similar to English, that is, words have comparatively few inflections. There are two genders, no cases, and a distinction between plural and singular. Older analyses posit the cases nominative and genitive and there are remains of distinct accusative and dative forms as well. Adjectives are compared as in English, and are inflected according to gender, number. The definiteness of nouns is marked primarily through suffixes, complemented with separate definite and indefinite articles, the prosody features both stress and in most dialects tonal qualities. The language has a large vowel inventory. Swedish is also notable for the voiceless velar fricative, a highly variable consonant phoneme. Swedish is an Indo-European language belonging to the North Germanic branch of the Germanic languages, by many general criteria of mutual intelligibility, the Continental Scandinavian languages could very well be considered dialects of a common Scandinavian language. In the 8th century, the common Germanic language of Scandinavia, Proto-Norse, had some changes. This language began to undergo new changes that did not spread to all of Scandinavia, the dialects of Old East Norse that were spoken in Sweden are called Runic Swedish while the dialects of Denmark are referred to as Runic Danish. The dialects are described as runic because the body of text appears in the runic alphabet. Unlike Proto-Norse, which was written with the Elder Futhark alphabet, Old Norse was written with the Younger Futhark alphabet, from 1200 onwards, the dialects in Denmark began to diverge from those of Sweden. An early change that separated Runic Danish from the dialects of Old East Norse was the change of the diphthong æi to the monophthong é. This is reflected in runic inscriptions where the older read stain, there was also a change of au as in dauðr into a long open ø as in døðr dead
17.
Danish language
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There are also minor Danish-speaking communities in Norway, Sweden, Spain, the United States, Canada, Brazil and Argentina. Due to immigration and language shift in urban areas, around 15–20% of the population of Greenland speak Danish as their home language. Along with the other North Germanic languages, Danish is a descendant of Old Norse, until the 16th century, Danish was a continuum of dialects spoken from Schleswig to Scania with no standard variety or spelling conventions. With the Protestant Reformation and the introduction of printing, a language was developed which was based on the educated Copenhagen dialect. It spread through use in the system and administration though German. Today, traditional Danish dialects have all but disappeared, though there are variants of the standard language. The main differences in language are between generations, with youth language being particularly innovative, Danish has a very large vowel inventory comprising 27 phonemically distinctive vowels, and its prosody is characterized by the distinctive phenomenon stød, a kind of laryngeal phonation type. The grammar is moderately inflective with strong and weak conjugations and inflections, nouns and demonstrative pronouns distinguish common and neutral gender. As in English, Danish only has remnants of a case system, particularly in the pronouns. Its syntax is V2, with the verb always occupying the second slot in the sentence. Danish is a Germanic language of the North Germanic branch, other names for this group are the Nordic or Scandinavian languages. Along with Swedish, Danish descends from the Eastern dialects of the Old Norse language, Scandinavian languages are often considered a dialect continuum, where there are no sharp dividing lines between the different vernacular languages. Like Norwegian and Swedish, Danish was significantly influenced by Low German in the Middle Ages, Danish itself can be divided into three main dialect areas, West Danish, Insular Danish, and East Danish. Danish is largely mutually intelligible with Norwegian and Swedish, both Swedes and Danes also understand Norwegian better than they understand each others languages. By the 8th century, the common Germanic language of Scandinavia, Proto-Norse, had some changes. This language was called the Danish tongue, or Norse language. Norse was written in the alphabet, first with the elder futhark. From the 7th century the common Norse language began to undergo changes that did not spread to all of Scandinavia, most of the changes separating East Norse from West Norse started as innovations in Denmark, that spread through Scania into Sweden and by maritime contact to southern Norway
18.
Norwegian language
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Norwegian is a North Germanic language spoken mainly in Norway, where it is the official language. Along with Swedish and Danish, Norwegian forms a continuum of more or less mutually intelligible local and regional variants. These Scandinavian languages, together with Faroese and Icelandic as well as extinct languages. Faroese and Icelandic are hardly mutually intelligible with Norwegian in their spoken form because continental Scandinavian has diverged from them, as established by law and governmental policy, the two official forms of written Norwegian are Bokmål and Nynorsk. The official Norwegian Language Council is responsible for regulating the two forms, and recommends the terms Norwegian Bokmål and Norwegian Nynorsk in English. Two other written forms without official status also exist, one and it is regulated by the unofficial Norwegian Academy, which translates the name as Standard Norwegian. Nynorsk and Bokmål provide standards for how to write Norwegian, no standard of spoken Norwegian is officially sanctioned, and most Norwegians speak their own dialects in all circumstances. Thus, unlike in other countries, the use of any Norwegian dialect. Outside Eastern Norway, this variation is not used. From the 16th to the 19th centuries, Danish was the written language of Norway. As a result, the development of modern written Norwegian has been subject to strong controversy related to nationalism, rural versus urban discourse, historically, Bokmål is a Norwegianised variety of Danish, while Nynorsk is a language form based on Norwegian dialects and puristic opposition to Danish. The unofficial form known as Riksmål is considered more conservative than Bokmål, Norwegians are educated in both Bokmål and Nynorsk. A2005 poll indicates that 86. 3% use primarily Bokmål as their written language,5. 5% use both Bokmål and Nynorsk, and 7. 5% use primarily Nynorsk. Thus, 13% are frequently writing Nynorsk, though the majority speak dialects that resemble Nynorsk more closely than Bokmål. Broadly speaking, Nynorsk writing is widespread in western Norway, though not in major urban areas, examples are Setesdal, the western part of Telemark county and several municipalities in Hallingdal, Valdres, and Gudbrandsdalen. It is little used elsewhere, but 30–40 years ago, it also had strongholds in rural parts of Trøndelag. Today, not only is Nynorsk the official language of four of the 19 Norwegian counties, NRK, the Norwegian broadcasting corporation, broadcasts in both Bokmål and Nynorsk, and all governmental agencies are required to support both written languages. Bokmål is used in 92% of all publications, and Nynorsk in 8%
19.
Icelandic language
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Icelandic /aɪsˈlændɪk/ is a North Germanic language, the language of Iceland. It is an Indo-European language belonging to the North Germanic or Nordic branch of the Germanic languages, historically, it was the westernmost of the Indo-European languages prior to the colonisation of the Americas. Icelandic, Faroese, Norn, and Western Norwegian formerly constituted West Nordic, Danish, Eastern Norwegian, modern Norwegian Bokmål is influenced by both groups, leading the Nordic languages to be divided into mainland Scandinavian languages and Insular Nordic. Most Western European languages have reduced levels of inflection, particularly noun declension. In contrast, Icelandic retains a four-case synthetic grammar comparable to, Icelandic is distinguished by a wide assortment of irregular declensions. Icelandic also has many instances of oblique cases without any governing word, for example, many of the various Latin ablatives have a corresponding Icelandic dative. The vast majority of Icelandic speakers—about 320, 000—live in Iceland, more than 8,000 Icelandic speakers live in Denmark, of whom approximately 3,000 are students. The language is spoken by some 5,000 people in the United States. Notably in the province of Manitoba, while 97% of the population of Iceland consider Icelandic their mother tongue, the language is in decline in some communities outside Iceland, particularly in Canada. Icelandic speakers outside Iceland represent recent emigration in almost all cases except Gimli, Manitoba, the state-funded Árni Magnússon Institute for Icelandic Studies serves as a centre for preserving the medieval Icelandic manuscripts and studying the language and its literature. Since 1995, on 16 November each year, the birthday of 19th-century poet Jónas Hallgrímsson is celebrated as Icelandic Language Day, the oldest preserved texts in Icelandic were written around 1100 AD. Much of the texts are based on poetry and laws traditionally preserved orally, the most famous of the texts, which were written in Iceland from the 12th century onward, are the Icelandic Sagas. They comprise the historical works and the eddaic poems, the language of the sagas is Old Icelandic, a western dialect of Old Norse. Danish rule of Iceland from 1380 to 1918 had little effect on the evolution of Icelandic, though more archaic than the other living Germanic languages, Icelandic changed markedly in pronunciation from the 12th to the 16th century, especially in vowels. The modern Icelandic alphabet has developed from an established in the 19th century. The later Rasmus Rask standard was a re-creation of the old treatise, with changes to fit concurrent Germanic conventions. Various archaic features, as the letter ð, had not been used much in later centuries, rasks standard constituted a major change in practice. Later 20th-century changes include the use of é instead of je, apart from the addition of new vocabulary, written Icelandic has not changed substantially since the 11th century, when the first texts were written on vellum
20.
Surveying
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Surveying or land surveying is the technique, profession, and science of determining the terrestrial or three-dimensional position of points and the distances and angles between them. A land surveying professional is called a land surveyor, Surveyors work with elements of geometry, trigonometry, regression analysis, physics, engineering, metrology, programming languages and the law. Surveying has been an element in the development of the environment since the beginning of recorded history. The planning and execution of most forms of construction require it and it is also used in transport, communications, mapping, and the definition of legal boundaries for land ownership. It is an important tool for research in other scientific disciplines. Basic surveyance has occurred since humans built the first large structures, the prehistoric monument at Stonehenge was set out by prehistoric surveyors using peg and rope geometry. In ancient Egypt, a rope stretcher would use simple geometry to re-establish boundaries after the floods of the Nile River. The almost perfect squareness and north-south orientation of the Great Pyramid of Giza, built c.2700 BC, the Groma instrument originated in Mesopotamia. The mathematician Liu Hui described ways of measuring distant objects in his work Haidao Suanjing or The Sea Island Mathematical Manual, the Romans recognized land surveyors as a profession. They established the basic measurements under which the Roman Empire was divided, Roman surveyors were known as Gromatici. In medieval Europe, beating the bounds maintained the boundaries of a village or parish and this was the practice of gathering a group of residents and walking around the parish or village to establish a communal memory of the boundaries. Young boys were included to ensure the memory lasted as long as possible, in England, William the Conqueror commissioned the Domesday Book in 1086. It recorded the names of all the owners, the area of land they owned, the quality of the land. It did not include maps showing exact locations, abel Foullon described a plane table in 1551, but it is thought that the instrument was in use earlier as his description is of a developed instrument. Gunters chain was introduced in 1620 by English mathematician Edmund Gunter and it enabled plots of land to be accurately surveyed and plotted for legal and commercial purposes. Leonard Digges described a Theodolite that measured horizontal angles in his book A geometric practice named Pantometria, joshua Habermel created a theodolite with a compass and tripod in 1576. Johnathon Sission was the first to incorporate a telescope on a theodolite in 1725, in the 18th century, modern techniques and instruments for surveying began to be used. Jesse Ramsden introduced the first precision theodolite in 1787 and it was an instrument for measuring angles in the horizontal and vertical planes
21.
Mining
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Mining is extraction of valuable minerals or other geological materials from the earth usually from an orebody, lode, vein, seam, reef or placer deposits. These deposits form a mineralized package that is of economic interest to the miner, ores recovered by mining include metals, coal, oil shale, gemstones, limestone, chalk, dimension stone, rock salt, potash, gravel, and clay. Mining is required to obtain any material that cannot be grown through agricultural processes, Mining in a wider sense includes extraction of any non-renewable resource such as petroleum, natural gas, or even water. Mining of stones and metal has been a human activity since pre-historic times, Mining operations usually create a negative environmental impact, both during the mining activity and after the mine has closed. Hence, most of the nations have passed regulations to decrease the impact. Work safety has long been a concern as well, and modern practices have significantly improved safety in mines, levels of metals recycling are generally low. Unless future end-of-life recycling rates are stepped up, some rare metals may become unavailable for use in a variety of consumer products, due to the low recycling rates, some landfills now contain higher concentrations of metal than mines themselves. Since the beginning of civilization, people have used stone, ceramics and, later and these were used to make early tools and weapons, for example, high quality flint found in northern France, southern England and Poland was used to create flint tools. Flint mines have been found in areas where seams of the stone were followed underground by shafts. The mines at Grimes Graves and Krzemionki are especially famous, other hard rocks mined or collected for axes included the greenstone of the Langdale axe industry based in the English Lake District. The oldest-known mine on archaeological record is the Lion Cave in Swaziland, at this site Paleolithic humans mined hematite to make the red pigment ochre. Mines of an age in Hungary are believed to be sites where Neanderthals may have mined flint for weapons. Ancient Egyptians mined malachite at Maadi, at first, Egyptians used the bright green malachite stones for ornamentations and pottery. Later, between 2613 and 2494 BC, large building projects required expeditions abroad to the area of Wadi Maghareh in order to secure minerals and other resources not available in Egypt itself. Quarries for turquoise and copper were found at Wadi Hammamat, Tura, Aswan and various other Nubian sites on the Sinai Peninsula. Mining in Egypt occurred in the earliest dynasties, the gold mines of Nubia were among the largest and most extensive of any in Ancient Egypt. These mines are described by the Greek author Diodorus Siculus, who mentions fire-setting as one used to break down the hard rock holding the gold. One of the complexes is shown in one of the earliest known maps, the miners crushed the ore and ground it to a fine powder before washing the powder for the gold dust
22.
Geology
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Geology is an earth science concerned with the solid Earth, the rocks of which it is composed, and the processes by which they change over time. Geology can also refer generally to the study of the features of any terrestrial planet. Geology gives insight into the history of the Earth by providing the evidence for plate tectonics, the evolutionary history of life. Geology also plays a role in engineering and is a major academic discipline. The majority of data comes from research on solid Earth materials. These typically fall into one of two categories, rock and unconsolidated material, the majority of research in geology is associated with the study of rock, as rock provides the primary record of the majority of the geologic history of the Earth. There are three types of rock, igneous, sedimentary, and metamorphic. The rock cycle is an important concept in geology which illustrates the relationships between three types of rock, and magma. When a rock crystallizes from melt, it is an igneous rock, the sedimentary rock can then be subsequently turned into a metamorphic rock due to heat and pressure and is then weathered, eroded, deposited, and lithified, ultimately becoming a sedimentary rock. Sedimentary rock may also be re-eroded and redeposited, and metamorphic rock may also undergo additional metamorphism, all three types of rocks may be re-melted, when this happens, a new magma is formed, from which an igneous rock may once again crystallize. Geologists also study unlithified material which typically comes from more recent deposits and these materials are superficial deposits which lie above the bedrock. Because of this, the study of material is often known as Quaternary geology. This includes the study of sediment and soils, including studies in geomorphology, sedimentology and this theory is supported by several types of observations, including seafloor spreading, and the global distribution of mountain terrain and seismicity. This coupling between rigid plates moving on the surface of the Earth and the mantle is called plate tectonics. The development of plate tectonics provided a basis for many observations of the solid Earth. Long linear regions of geologic features could be explained as plate boundaries, mid-ocean ridges, high regions on the seafloor where hydrothermal vents and volcanoes exist, were explained as divergent boundaries, where two plates move apart. Arcs of volcanoes and earthquakes were explained as convergent boundaries, where one plate subducts under another, transform boundaries, such as the San Andreas Fault system, resulted in widespread powerful earthquakes. Plate tectonics also provided a mechanism for Alfred Wegeners theory of continental drift and they also provided a driving force for crustal deformation, and a new setting for the observations of structural geology
23.
Scientific calculator
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A scientific calculator is a type of electronic calculator, usually but not always handheld, designed to calculate problems in science, engineering, and mathematics. They have almost completely replaced slide rules in almost all traditional applications, there is also some overlap with the financial calculator market. A few have multi-line displays, with recent models from Hewlett-Packard, Texas Instruments, Casio, Sharp. By providing a method to enter an entire problem in as it is written on the page using simple formatting tools, the HP-35, introduced on February 1,1972, was Hewlett-Packards first pocket calculator and the worlds first handheld scientific calculator. Like some of HPs desktop calculators it used RPN, introduced at US$395, the HP-35 was available from 1972 to 1975. Texas Instruments, after the introduction of units with scientific notation, came out with a handheld scientific calculator on January 15,1974. TI continues to be a player in the calculator market. Casio and Sharp have also been major players, with Casios fx series being a common brand. Casio is also a player in the graphing calculator market
24.
Trigonometric functions
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
25.
Meter
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The metre or meter, is the base unit of length in the International System of Units. The metre is defined as the length of the path travelled by light in a vacuum in 1/299792458 seconds, the metre was originally defined in 1793 as one ten-millionth of the distance from the equator to the North Pole. In 1799, it was redefined in terms of a metre bar. In 1960, the metre was redefined in terms of a number of wavelengths of a certain emission line of krypton-86. In 1983, the current definition was adopted, the imperial inch is defined as 0.0254 metres. One metre is about 3 3⁄8 inches longer than a yard, Metre is the standard spelling of the metric unit for length in nearly all English-speaking nations except the United States and the Philippines, which use meter. Measuring devices are spelled -meter in all variants of English, the suffix -meter has the same Greek origin as the unit of length. This range of uses is found in Latin, French, English. Thus calls for measurement and moderation. In 1668 the English cleric and philosopher John Wilkins proposed in an essay a decimal-based unit of length, as a result of the French Revolution, the French Academy of Sciences charged a commission with determining a single scale for all measures. In 1668, Wilkins proposed using Christopher Wrens suggestion of defining the metre using a pendulum with a length which produced a half-period of one second, christiaan Huygens had observed that length to be 38 Rijnland inches or 39.26 English inches. This is the equivalent of what is now known to be 997 mm, no official action was taken regarding this suggestion. In the 18th century, there were two approaches to the definition of the unit of length. One favoured Wilkins approach, to define the metre in terms of the length of a pendulum which produced a half-period of one second. The other approach was to define the metre as one ten-millionth of the length of a quadrant along the Earths meridian, that is, the distance from the Equator to the North Pole. This means that the quadrant would have defined as exactly 10000000 metres at that time. To establish a universally accepted foundation for the definition of the metre, more measurements of this meridian were needed. This portion of the meridian, assumed to be the length as the Paris meridian, was to serve as the basis for the length of the half meridian connecting the North Pole with the Equator
26.
Equator
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The Equator usually refers to an imaginary line on the Earths surface equidistant from the North Pole and South Pole, dividing the Earth into the Northern Hemisphere and Southern Hemisphere. The Equator is about 40,075 kilometres long, some 78. 7% lies across water and 21. 3% over land, other planets and astronomical bodies have equators similarly defined. Generally, an equator is the intersection of the surface of a sphere with the plane that is perpendicular to the spheres axis of rotation. The latitude of the Earths equator is by definition 0° of arc, the equator is the only line of latitude which is also a great circle — that is, one whose plane passes through the center of the globe. The plane of Earths equator when projected outwards to the celestial sphere defines the celestial equator, in the cycle of Earths seasons, the plane of the equator passes through the Sun twice per year, at the March and September equinoxes. To an observer on the Earth, the Sun appears to travel North or South over the equator at these times, light rays from the center of the Sun are perpendicular to the surface of the Earth at the point of solar noon on the Equator. Locations on the Equator experience the quickest sunrises and sunsets because the sun moves nearly perpendicular to the horizon for most of the year. The Earth bulges slightly at the Equator, the diameter of the Earth is 12,750 kilometres. Because the Earth spins to the east, spacecraft must also launch to the east to take advantage of this Earth-boost of speed, seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earths axis relative to the plane of revolution. During the year the northern and southern hemispheres are inclined toward or away from the sun according to Earths position in its orbit, the hemisphere inclined toward the sun receives more sunlight and is in summer, while the other hemisphere receives less sun and is in winter. At the equinoxes, the Earths axis is not tilted toward the sun, instead it is perpendicular to the sun meaning that the day is about 12 hours long, as is the night, across the whole of the Earth. Near the Equator there is distinction between summer, winter, autumn, or spring. The temperatures are usually high year-round—with the exception of high mountains in South America, the temperature at the Equator can plummet during rainstorms. In many tropical regions people identify two seasons, the wet season and the dry season, but many places close to the Equator are on the oceans or rainy throughout the year, the seasons can vary depending on elevation and proximity to an ocean. The Equator lies mostly on the three largest oceans, the Pacific Ocean, the Atlantic Ocean, and the Indian Ocean. The highest point on the Equator is at the elevation of 4,690 metres, at 0°0′0″N 77°59′31″W and this is slightly above the snow line, and is the only place on the Equator where snow lies on the ground. At the Equator the snow line is around 1,000 metres lower than on Mount Everest, the Equator traverses the land of 11 countries, it also passes through two island nations, though without making a landfall in either. Starting at the Prime Meridian and heading eastwards, the Equator passes through, Despite its name, however, its island of Annobón is 155 km south of the Equator, and the rest of the country lies to the north
27.
Complex plane
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In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the part of a complex number represented by a displacement along the x-axis. The concept of the plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors, in particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is known as the Argand plane. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In this customary notation the number z corresponds to the point in the Cartesian plane. In the Cartesian plane the point can also be represented in coordinates as = =. In the Cartesian plane it may be assumed that the arctangent takes values from −π/2 to π/2, and some care must be taken to define the real arctangent function for points when x ≤0. Here |z| is the value or modulus of the complex number z, θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π. Notice that without the constraint on the range of θ, the argument of z is multi-valued, because the exponential function is periodic. Thus, if θ is one value of arg, the values are given by arg = θ + 2nπ. The theory of contour integration comprises a part of complex analysis. In this context the direction of travel around a curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the direction is counterclockwise. Almost all of complex analysis is concerned with complex functions – that is, here it is customary to speak of the domain of f as lying in the z-plane, while referring to the range or image of f as a set of points in the w-plane. In symbols we write z = x + i y, f = w = u + i v and it can be useful to think of the complex plane as if it occupied the surface of a sphere. We can establish a correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows
28.
Imaginary unit
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The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 +1 =0. The term imaginary is used there is no real number having a negative square. There are two square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι is sometimes used, in the disciplines of electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current. For the history of the unit, see Complex number § History. The imaginary number i is defined solely by the property that its square is −1, with i defined this way, it follows directly from algebra that i and −i are both square roots of −1. In polar form, i is represented as 1eiπ/2, having a value of 1. In the complex plane, i is the point located one unit from the origin along the imaginary axis, more precisely, once a solution i of the equation has been fixed, the value −i, which is distinct from i, is also a solution. Since the equation is the definition of i, it appears that the definition is ambiguous. However, no ambiguity results as long as one or other of the solutions is chosen and labelled as i and this is because, although −i and i are not quantitatively equivalent, there is no algebraic difference between i and −i. Both imaginary numbers have equal claim to being the number whose square is −1, the issue can be a subtle one. See also Complex conjugate and Galois group, a more precise explanation is to say that the automorphism group of the special orthogonal group SO has exactly two elements — the identity and the automorphism which exchanges CW and CCW rotations. All these ambiguities can be solved by adopting a rigorous definition of complex number. For example, the pair, in the usual construction of the complex numbers with two-dimensional vectors. The imaginary unit is sometimes written √−1 in advanced mathematics contexts, however, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation is reserved either for the square root function. Similarly,1 i =1 −1 =1 −1 = −11 = −1 = i, the calculation rules a ⋅ b = a ⋅ b and a b = a b are only valid for real, non-negative values of a and b. These problems are avoided by writing and manipulating expressions like i√7, for a more thorough discussion, see Square root and Branch point
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Argument (complex analysis)
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In mathematics, arg is a function operating on complex numbers. It gives the angle between the real axis to the line joining the point to the origin, shown as φ in figure 1. The numeric value is given by the angle in radians and is positive if measured counterclockwise, algebraically, as any real quantity φ such that z = r = r e i φ for some positive real r. The quantity r is the modulus of z, denoted |z|, the names magnitude for the modulus and phase for the argument are sometimes used equivalently. Similarly, from the periodicity of sin and cos, the definition also has this property. Because a complete rotation around the leaves a complex number unchanged. This is shown in figure 3, a representation of the multi-valued function and this represents an angle of up to half a complete circle from the positive real axis in either direction. Some authors define the range of the value as being in the closed-open interval, Arg. The principal value Arg of a number given as x + iy is normally available in math libraries of many programming languages using the function atan2 or some language-specific variant. The value of atan2 is the value in the range. If z ≠0 and n is any integer, then Arg ≡ n Arg ( mod, Arg = Arg − Arg = −3 π4 − π2 = −5 π4 =3 π4 ( mod
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Angular mil
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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
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Harmonic analysis
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In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, tidal analysis and neuroscience. The term harmonics originated as the ancient Greek word, harmonikos, the classical Fourier transform on Rn is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. For instance, if we impose some requirements on a distribution f, the Paley–Wiener theorem is an example of this. The Paley–Wiener theorem immediately implies that if f is a distribution of compact support. This is an elementary form of an uncertainty principle in a harmonic analysis setting. See also, Convergence of Fourier series, Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis. Many applications of analysis in science and engineering begin with the idea or hypothesis that a phenomenon or signal is composed of a sum of individual oscillatory components. Ocean tides and vibrating strings are common and simple examples, the specific equations depend on the field, but theories generally try to select equations that represent major principles that are applicable. The experimental approach is usually to acquire data that accurately quantifies the phenomenon, for example, the top signal at the right is a sound waveform of a bass guitar playing an open string corresponding to an A note with a fundamental frequency of 55 Hz. The waveform appears oscillatory, but it is more complex than a simple sine wave, the different wave components contributing to the sound can be revealed by applying a mathematical analysis technique known as the Fourier transform, which is shown in the lower figure. Note that there is a prominent peak at 55 Hz, but that there are other peaks at 110 Hz,165 Hz, in this case,55 Hz is identified as the fundamental frequency of the string vibration, and the integer multiples are known as harmonics. One of the most modern branches of analysis, having its roots in the mid-twentieth century, is analysis on topological groups. The core motivating ideas are the various Fourier transforms, which can be generalized to a transform of functions defined on Hausdorff locally compact topological groups, the theory for abelian locally compact groups is called Pontryagin duality. Harmonic analysis studies the properties of that duality and Fourier transform, for general non-abelian locally compact groups, harmonic analysis is closely related to the theory of unitary group representations. For compact groups, the Peter–Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each class of representations. If the group is neither abelian nor compact, no satisfactory theory is currently known. However, many cases have been analyzed, for example SLn. In this case, representations in infinite dimensions play a crucial role, study of the eigenvalues and eigenvectors of the Laplacian on domains, manifolds, and graphs is also considered a branch of harmonic analysis
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Steradian
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The steradian or square radian is the SI unit of solid angle. It is used in geometry, and is analogous to the radian which quantifies planar angles. The name is derived from the Greek stereos for solid and the Latin radius for ray and it is useful, however, to distinguish between dimensionless quantities of a different nature, so the symbol sr is used to indicate a solid angle. For example, radiant intensity can be measured in watts per steradian, the steradian was formerly an SI supplementary unit, but this category was abolished in 1995 and the steradian is now considered an SI derived unit. A steradian can be defined as the angle subtended at the center of a unit sphere by a unit area on its surface. For a general sphere of radius r, any portion of its surface with area A = r2 subtends one steradian, because the surface area A of a sphere is 4πr2, the definition implies that a sphere measures 4π steradians. By the same argument, the solid angle that can be subtended at any point is 4π sr. Since A = r2, it corresponds to the area of a cap. Therefore one steradian corresponds to the angle of the cross-section of a simple cone subtending the plane angle 2θ, with θ given by, θ = arccos = arccos = arccos ≈0.572 rad. This angle corresponds to the plane angle of 2θ ≈1.144 rad or 65. 54°. A steradian is also equal to the area of a polygon having an angle excess of 1 radian, to 1/4π of a complete sphere. The solid angle of a cone whose cross-section subtends the angle 2θ is, Ω =2 π s r. In two dimensions, an angle is related to the length of the arc that it spans, θ = l r r a d where l is arc length, r is the radius of the circle. For example, a measurement of the width of an object would be given in radians. At the same time its visible area over ones visible field would be given in steradians. Just as the area of a circle is related to its diameter or radius. One-dimensional circular measure has units of radians or degrees, while two-dimensional spherical measure is expressed in steradians, in higher dimensional mathematical spaces, units for analogous solid angles have not been explicitly named. When they are used, they are dealt with by analogy with the circular or spherical cases and that is, as a proportion of the relevant unit hypersphere taken up by the generalized angle, or point set expressed in spherical coordinates
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Trigonometry
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Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies, Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles, thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees, the ancient Nubians used a similar method. In 140 BC, Hipparchus gave the first tables of chords, analogous to modern tables of sine values, in the 2nd century AD, the Greco-Egyptian astronomer Ptolemy printed detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta and these Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, at about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond, Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for maps of large geographic areas. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595, gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry, the works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series. Also in the 18th century, Brook Taylor defined the general Taylor series, if one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees, they are complementary angles, the shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, if the length of one of the sides is known, the other two are determined. Sin A = opposite hypotenuse = a c, Cosine function, defined as the ratio of the adjacent leg to the hypotenuse
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Jean-Charles de Borda
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Jean-Charles, chevalier de Borda was a French mathematician, physicist, political scientist, and sailor. Borda was born in the city of Dax to Jean‐Antoine de Borda, in 1756, Borda wrote Mémoire sur le mouvement des projectiles, a product of his work as a military engineer. For that, he was elected to the French Academy of Sciences in 1764, Borda was a mariner and a scientist, spending time in the Caribbean testing out advances in chronometers. Between 1777 and 1778, he participated in the American Revolutionary War, in 1781, he was put in charge of several vessels in the French Navy. In 1782, he was captured by the English, and was returned to France shortly after and he returned as an engineer in the French Navy, making improvements to waterwheels and pumps. In 1770, Borda formulated a ranked voting system that is referred to as the Borda count. The Borda count is in use today in some institutions, competitions. The Borda count has also served as a basis for other such as the Quota Borda system. Another of his contributions is his construction of the standard metre, as an instrument maker, he improved the reflecting circle and the repeating circle, the latter used to measure the meridian arc from Dunkirk to Barcelona by Delambre and Méchain. This required the calculation of trigonometric tables and logarithms corresponding to the new size of the degree, the tables of logarithms of sines, secants, and tangents were also required for the purposes of navigation. The division of the degree into hundredths was accompanied by the division of the day into 10 hours of 100 minutes, the Republican Calendar was abolished by Napoleon in 1806, but the 400-degree circle lived on as the Gradian. Five French ships were named Borda in his honour, the crater Borda on the Moon is named after him. Asteroide 175726 has been called Borda in his honour and his name is one of the 72 names inscribed on the Eiffel Tower. Cape Borda on the northwest coast of Kangaroo Island in South Australia is named in his honour, Île Borda was the name given to Kangaroo Island in his honour by Nicholas Baudin. Borda–Carnot equation OConnor, John J. Robertson, Edmund F, jean Charles de Borda, MacTutor History of Mathematics archive, University of St Andrews
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Repeating circle
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The repeating circle is an instrument for geodetic surveying, invented by Etienne Lenoir in 1784, while an assistant of Jean-Charles de Borda, who later improved the instrument. It was notable as being the equal of the great theodolite created by the instrument maker. It was used to measure the arc from Dunkirk to Barcelona by Jean Baptiste Delambre and Pierre Méchain. The repeating circle is made of two mounted on a shared axis with scales to measure the angle between the two. The instrument combines multiple measurements to increase accuracy with the procedure, At this stage. Repeating the procedure causes the instrument to show 4x the angle of interest, with further iterations increasing it to 6x, 8x, in this way, many measurements can be added together, allowing some of the random measurement errors to cancel out
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
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Wayback Machine
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The Internet Archive launched the Wayback Machine in October 2001. It was set up by Brewster Kahle and Bruce Gilliat, and is maintained with content from Alexa Internet, the service enables users to see archived versions of web pages across time, which the archive calls a three dimensional index. Since 1996, the Wayback Machine has been archiving cached pages of websites onto its large cluster of Linux nodes and it revisits sites every few weeks or months and archives a new version. Sites can also be captured on the fly by visitors who enter the sites URL into a search box, the intent is to capture and archive content that otherwise would be lost whenever a site is changed or closed down. The overall vision of the machines creators is to archive the entire Internet, the name Wayback Machine was chosen as a reference to the WABAC machine, a time-traveling device used by the characters Mr. Peabody and Sherman in The Rocky and Bullwinkle Show, an animated cartoon. These crawlers also respect the robots exclusion standard for websites whose owners opt for them not to appear in search results or be cached, to overcome inconsistencies in partially cached websites, Archive-It. Information had been kept on digital tape for five years, with Kahle occasionally allowing researchers, when the archive reached its fifth anniversary, it was unveiled and opened to the public in a ceremony at the University of California, Berkeley. Snapshots usually become more than six months after they are archived or, in some cases, even later. The frequency of snapshots is variable, so not all tracked website updates are recorded, Sometimes there are intervals of several weeks or years between snapshots. After August 2008 sites had to be listed on the Open Directory in order to be included. As of 2009, the Wayback Machine contained approximately three petabytes of data and was growing at a rate of 100 terabytes each month, the growth rate reported in 2003 was 12 terabytes/month, the data is stored on PetaBox rack systems manufactured by Capricorn Technologies. In 2009, the Internet Archive migrated its customized storage architecture to Sun Open Storage, in 2011 a new, improved version of the Wayback Machine, with an updated interface and fresher index of archived content, was made available for public testing. The index driving the classic Wayback Machine only has a bit of material past 2008. In January 2013, the company announced a ground-breaking milestone of 240 billion URLs, in October 2013, the company announced the Save a Page feature which allows any Internet user to archive the contents of a URL. This became a threat of abuse by the service for hosting malicious binaries, as of December 2014, the Wayback Machine contained almost nine petabytes of data and was growing at a rate of about 20 terabytes each week. Between October 2013 and March 2015 the websites global Alexa rank changed from 162 to 208, in a 2009 case, Netbula, LLC v. Chordiant Software Inc. defendant Chordiant filed a motion to compel Netbula to disable the robots. Netbula objected to the motion on the ground that defendants were asking to alter Netbulas website, in an October 2004 case, Telewizja Polska USA, Inc. v. Echostar Satellite, No.02 C3293,65 Fed. 673, a litigant attempted to use the Wayback Machine archives as a source of admissible evidence, Telewizja Polska is the provider of TVP Polonia and EchoStar operates the Dish Network