1.
Minute and second 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″
2.
PSO-1
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The PSO-1 is a telescopic sight manufactured in Russia by the Novosibirsk instrument-making factory and issued with the Russian military Dragunov sniper rifle. The PSO-1 was, at the time of its introduction around 1964, the PSO-1 was specifically designed for the SVD as a telescopic sight for military designated marksman activities. The current version of the sight is the PSO-1M2 and this telescopic sight is different from the original PSO-1 only in that it lacks the now obsolete infrared detector. The metal body of the PSO-1 is made from a magnesium alloy, the PSO-1 features a battery-powered red illuminated reticle with light provided by a simple diode bulb. It features professionally ground, fully multi-coated optical elements, an enamel finish for scratch protection. The scope body is sealed and filled with nitrogen, which prevents fogging of optics and was designed to function within a -50 °C to 50 °C temperature range. For zeroing the sight the reticle can be adjusted by manipulating the elevation. Considered the higher end of Soviet military side-mount telescopic sights, the quality of the PSO-1 is higher than most other PSO-style telescopic sights, the PSO-1 has neither a focus adjustment nor a parallax compensation control. Most modern military tactical scopes with lower power fixed magnification such as the ACOG, modern fixed magnification military high-end-grade sniper telescopic sights scopes intended for long-range shooting usually offer one or both of these features. The positioning of the body to the left of the bore’s center line may not be comfortable to all shooters. The PSO-1 elevation turret features bullet drop compensation in 50 m or 100 m increments for engaging point, at longer distances the shooter must use the chevrons that would shift the trajectory by 100 m per each chevron. The BDC feature must be tuned at the factory for the ballistic trajectory of a particular combination of rifle. Inevitable BDC-induced errors will occur if the environmental and meteorological circumstances deviate from the circumstances the BDC was calibrated for, marksmen can be trained to compensate for these errors. The PSO-1 features a reticle with floating elements designed for use in range estimation and bullet drop, the top center chevron is used as the main aiming mark. The horizontal hash marks are for windage and lead corrections and can be used for ranging purposes as well, to the left is a stadiametric rangefinder that can be used to determine the distance from a 1.7 meters tall object/person from 200 m to 1000 m. For this the lowest part of the target is lined up on the horizontal line. Where the top of the target touches the top curved line the distance can be determined and this reticle lay out is also used in several other telescopic sights produced and used by other former Warsaw Pact member states. The three lower chevrons in the center are used as hold over points for engaging targets beyond 1,000 meters
3.
Dragunov sniper rifle
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The Dragunov sniper rifle is a semi-automatic sniper/designated marksman rifle chambered in 7. 62×54mmR and developed in the Soviet Union. For that reason, it was originally named Самозарядная Винтовка системы Драгунова образца1963 года Self-Loading Rifle, System of Dragunov and it was selected as the winner of a contest that included three competing designs, by Sergei Simonov, Aleksandr Konstantinov and Yevgeny Dragunov. Extensive field testing of the rifles conducted in a range of environmental conditions resulted in Dragunov’s proposal being accepted into service in 1963. An initial pre-production batch consisting of 200 rifles was assembled for evaluation purposes, since then, the Dragunov has become the standard squad support weapon of several countries, including those of the former Warsaw Pact. Licensed production of the rifle was established in China and Iran, the Dragunov is a semi-automatic, gas-operated rifle with a short-stroke gas-piston system. The barrel breech is locked through a rotating bolt and uses three locking lugs to engage corresponding locking recesses in the barrel extension, the rifle has a manual, two-position gas regulator. A gas regulator meters the portion of the gases fed into the action in order to cycle the weapon. The gas regulator can be set with the help of the rim of a cartridge, the normal position #1 leaves a gas escape port opened in the form of a hole that lets some combustion gas escape during cycling. Position #2 closes the gas port and directs extra combustion gas to the piston increasing the recoil velocity of the gas-piston system. It is used for when the rifle does not reliably cycle due to carbon fouling build-up in the gas port, when shooting in extreme cold or high altitude or using low powered ammunition. After discharging the last cartridge from the magazine, the bolt carrier, the rifle has a hammer-type striking mechanism and a manual lever safety selector. The firing pin is a type and, as a result. Thus, military-grade ammunition with primers confirmed to be seated is recommended for the Dragunov. This appears to have solved the slam fire issue, the rifles receiver is machined to provide additional accuracy and torsional strength. The Dragunovs receiver bears a number of similarities to the AK action, such as the large dust cover, iron sights and lever safety selector and these cosmetic similarities can lead to mis-categorization of the Dragunov as an AK variant. The barrel profile is relatively thin to save weight and is ended with a flash suppressor. The barrel’s bore is chrome-lined for increased resistance, and features 4 right-hand grooves. It is not rifled over its length but partly over a length of 547 mm
4.
Stadiametric rangefinding
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Stadiametric rangefinding, or the stadia method is a technique of measuring distances with a telescopic instrument. The term stadia comes from a Greek unit of length, stadiametric rangefinding is used for surveying and in the telescopic sights of firearms, artillery pieces, or tank guns, as well as some binoculars and other optics. It is still used in long-range military sniping, but in many professional applications it is being replaced with microwave, infrared. The stadia method is based upon the principle that in similar triangles homologous sides are proportional and this means that, for a right triangle with a given angle, the ratio of adjacent side length to opposite side length is constant. In either case, the parameter is used, in conjunction with the angular measurement. An object 5 meters high, for example, will cover 1 mrad at 5000 meters, or 5 mrad at 1000 meters, or 25 mrad at 200 meters. Since the radian expresses a ratio, it is independent of the units used, stadia readings used in surveying can be taken with modern instruments such as transits, theodolites, plane-table alidades and levels. When using the measuring method, a stadia rod is held so that it appears between two stadia marks visible on the instruments reticle. The stadia rod has measurements written on it that can be read through the telescope of the instrument, an instrument equipped for stadia work has two horizontal stadia marks spaced equidistant from the center crosshair of the reticle. The interval between stadia marks in most stadia instruments gives a stadia interval factor of 100, the distance between the instrument and a stadia rod can be determined simply by multiplying the measurement between the stadia hairs by 100. The instrument must be level for this method to work directly, if the instrument line of sight is inclined, the horizontal and vertical distance components must be determined. Some instruments have additional graduations on a circle to assist with these inclined measurements. These graduated circles, known as circles, provide the value of the horizontal. This system is sufficiently precise for locating topographic details such as rivers, bridges, buildings, the stadia method of distance measurement is primarily historical for surveying purposes, as distance nowadays is mostly measured by electronic or taping methods. Total station instruments do not have stadia lines marked on the reticle, traditional methods are still used in areas where modern instruments are not common or by aficionados to antique surveying methods
5.
System of measurement
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A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce, systems of measurement in modern use include the metric system, the imperial system, and United States customary units. The French Revolution gave rise to the system, and this has spread around the world. In most systems, length, mass, and time are base quantities, later science developments showed that either electric charge or electric current could be added to extend the set of base quantities by which many other metrological units could be easily defined. Other quantities, such as power and speed, are derived from the set, for example. Such arrangements were satisfactory in their own contexts, the preference for a more universal and consistent system only gradually spread with the growth of science. Changing a measurement system has substantial financial and cultural costs which must be offset against the advantages to be obtained using a more rational system. However pressure built up, including scientists and engineers for conversion to a more rational. The unifying characteristic is that there was some definition based on some standard, eventually cubits and strides gave way to customary units to met the needs of merchants and scientists. In the metric system and other recent systems, a basic unit is used for each base quantity. Often secondary units are derived from the units by multiplying by powers of ten. Thus the basic unit of length is the metre, a distance of 1.234 m is 1,234 millimetres. Metrication is complete or nearly complete in almost all countries, US customary units are heavily used in the United States and to some degree in Liberia. Traditional Burmese units of measurement are used in Burma, U. S. units are used in limited contexts in Canada due to the large volume of trade, there is also considerable use of Imperial weights and measures, despite de jure Canadian conversion to metric. In the United States, metric units are used almost universally in science, widely in the military, and partially in industry, but customary units predominate in household use. At retail stores, the liter is a used unit for volume, especially on bottles of beverages. Some other standard non-SI units are still in use, such as nautical miles and knots in aviation. Metric systems of units have evolved since the adoption of the first well-defined system in France in 1795, during this evolution the use of these systems has spread throughout the world, first to non-English-speaking countries, and then to English speaking countries
6.
SI derived unit
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The International System of Units specifies a set of seven base units from which all other SI units of measurement are derived. Each of these units is either dimensionless or can be expressed as a product of powers of one or more of the base units. For example, the SI derived unit of area is the metre. The degree Celsius has an unclear status, and is arguably an exception to this rule. The names of SI units are written in lowercase, the symbols for units named after persons, however, are always written with an uppercase initial letter. In addition to the two dimensionless derived units radian and steradian,20 other derived units have special names, some other units such as the hour, litre, tonne, bar and electronvolt are not SI units, but are widely used in conjunction with SI units. Until 1995, the SI classified the radian and the steradian as supplementary units, but this designation was abandoned, International System of Quantities International System of Units International Vocabulary of Metrology Metric prefix Metric system Non-SI units mentioned in the SI Planck units SI base unit I. Mills, Tomislav Cvitas, Klaus Homann, Nikola Kallay, IUPAC, Quantities, Units and Symbols in Physical Chemistry. CS1 maint, Multiple names, authors list
7.
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
8.
Metric prefix
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A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. While all metric prefixes in use today are decadic, historically there have been a number of binary metric prefixes as well. Each prefix has a symbol that is prepended to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand, the prefix milli-, likewise, may be added to metre to indicate division by one thousand, one millimetre is equal to one thousandth of a metre. Decimal multiplicative prefixes have been a feature of all forms of the system with six dating back to the systems introduction in the 1790s. Metric prefixes have even been prepended to non-metric units, the SI prefixes are standardized for use in the International System of Units by the International Bureau of Weights and Measures in resolutions dating from 1960 to 1991. Since 2009, they have formed part of the International System of Quantities, the BIPM specifies twenty prefixes for the International System of Units. Each prefix name has a symbol which is used in combination with the symbols for units of measure. For example, the symbol for kilo- is k, and is used to produce km, kg, and kW, which are the SI symbols for kilometre, kilogram, prefixes corresponding to an integer power of one thousand are generally preferred. Hence 100 m is preferred over 1 hm or 10 dam, the prefixes hecto, deca, deci, and centi are commonly used for everyday purposes, and the centimetre is especially common. However, some building codes require that the millimetre be used in preference to the centimetre, because use of centimetres leads to extensive usage of decimal points. Prefixes may not be used in combination and this also applies to mass, for which the SI base unit already contains a prefix. For example, milligram is used instead of microkilogram, in the arithmetic of measurements having units, the units are treated as multiplicative factors to values. If they have prefixes, all but one of the prefixes must be expanded to their numeric multiplier,1 km2 means one square kilometre, or the area of a square of 1000 m by 1000 m and not 1000 square metres. 2 Mm3 means two cubic megametres, or the volume of two cubes of 1000000 m by 1000000 m by 1000000 m or 2×1018 m3, and not 2000000 cubic metres, examples 5 cm = 5×10−2 m =5 ×0.01 m =0. The prefixes, including those introduced after 1960, are used with any metric unit, metric prefixes may also be used with non-metric units. The choice of prefixes with a unit is usually dictated by convenience of use. Unit prefixes for amounts that are larger or smaller than those actually encountered are seldom used
9.
Latin
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
10.
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
11.
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
12.
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
13.
Firearm
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A firearm is a portable gun - a barreled weapon that launches one or more projectiles, often driven by the action of an explosive force. The first primitive firearms originated in 13th-century China when the fire lance was combined with projectiles. The technology gradually spread through the rest of East Asia, South Asia, older firearms typically used black powder as a propellant, but modern firearms use smokeless powder or other propellants. Most modern firearms have rifled barrels to impart spin to the projectile for improved flight stability, modern firearms can be described by their caliber or in the case of shotguns their gauge, by the type of action employed together with the usual means of deportment. The word firearms usually is used in a sense restricted to small arms, shooters aim firearms at their targets with hand-eye co-ordination, using either iron sights or optical sights. The accurate range of pistols generally does not exceed 100 yards, while most rifles are accurate to 550 yards using iron sights, some purpose-built sniper rifles are accurate to ranges of more than 2,200 yards. The smallest of all firearms is the handgun, there are three common types of handguns, single-shot pistols, revolvers, and semi-automatic pistols. Revolvers have a number of firing chambers or charge holes in a revolving cylinder, semi-automatic pistols have a single fixed firing chamber machined into the rear of the barrel, and a magazine so they can be used to fire more than one round. Each press of the fires a cartridge, using the energy of the cartridge to activate the mechanism so that the next cartridge may be fired immediately. This is opposed to double-action revolvers which accomplish the end using a mechanical action linked to the trigger pull. Prior to the 19th century, virtually all handguns were single-shot muzzleloaders, with the invention of the revolver in 1818, handguns capable of holding multiple rounds became popular. Certain designs of auto-loading pistol appeared beginning in the 1870s and had largely supplanted revolvers in military applications by the end of World War I. By the end of the 20th century, most handguns carried regularly by military, police and civilians were semi-automatic, both designs are common among civilian gun owners, depending on the owners intention. A long gun is any firearm that is larger than a handgun and is designed to be held. Early long arms, from the Renaissance up to the century, were generally smoothbore firearms that fired one or more ball shot. Most modern long guns are either rifles or shotguns, both are the successors of the musket, diverging from their parent weapon in distinct ways. A rifle is so named for the spiral fluting machined into the surface of its barrel. Shotguns are predominantly smoothbore firearms designed to fire a number of shot, shotguns are also capable of firing single slugs, or specialty rounds such as bean bags, tear gas or breaching rounds
14.
Shot grouping
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In shooting sports, a shot grouping, or simply grouping, is the placement of multiple shots on a target, taken in one shooting session. The tightness of the grouping is a measure of the precision of a weapon, and a measure of the shooters consistency and skill. For firearms that shoot one round at a time, a shot grouping test can be used to measure the accuracy of the system, the weapons mechanical precision. The weapon is fixed into position on a test mount, multiple shots using rounds from the same type and batch are fired to observe how the weapon groups the shots. If a person holds the weapon and shoots it, the measures the combination of the persons skill. In shotgun shooting, the grouping is called the pattern. The pattern is the spread of shot from a shotgun shell. The barrel of a shotgun is designed to deliver a wide or narrow grouping, depending on the expected use. Shooting at close range indicates a cylinder bore barrel to deliver a wide grouping, while for hunting at longer distances such as 50 yards or meters, in archery, a shot grouping is the result of one person shooting multiple arrows at a target. A tight grouping indicates consistency in form, mean point of impact is the calculated center of the grouping, which is the average center of all the shots, and is not necessarily located at a hole in the target. The size of the grouping is described as the smallest circle containing all the shots, a flier is a shot from the same shooting session that is farther from the general shot group, considered to be outside of the grouping. Such shots may be the result of an unexpected gust of wind, rarely, it may indicate inconsistency in the ammunition. Single fliers may be discounted when evaluating a grouping, but if fliers occur often, then the problem should be traced to its origin. When using SI units, shooting accuracy is measured in mils by measuring the size in millimeters. The error of assuming that 0.1 mil equals 1 cm at 100 meters is about 1 × 10-7. When using imperial units, shooting accuracy is measured in minutes of arc by measuring the size in inches, multiplying by a factor of 100. The error of assuming that 1′ equals 1 at 100 yd is about 4.5 %, when shooting at a target 100 yards away, 1′ is a circle 1.047 inches in diameter, while at 100 meters, 1′ is a 29. 1-millimeter circle. The arcminute is a measurement, with 1′ equal to one minute of one degree
15.
Shooting target
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The center is often called the bullseye. Targets can for instance be made of paper, self healing rubber or steel, there are also electronic targets that electronically can provide the shooter with precise feedback of the shot placement. For instance, the well known circular bullseye target might originally have resembled a human torso or an animal being hunted, the origin of these targets are not usually given much attention, and have been kept around for variation and tradition. Some types of targets are, Steel targets Paper or cardboard Frangible Self healing rubber target Reactive targets are designed to move and/or bounce along the ground when hit, Targets are designed to explode when stuck with a bullet traveling at a suitable velocity to induce detonation. In the outdoor air gun discipline field target metal targets of various shape, the metal plates are often shaped to resemble small game animals, although there is currently a move towards simple geometric shapes. FITA targets are used in shooting competitions within the World Archery Federation. The targets have 10 evenly spaced concentric rings, generally with values from 1 through 10. In addition there is an inner 10 ring, sometimes called the X ring and this becomes the 10 ring at indoor compound competitions, while outdoors, it serves as a tiebreaker with the archer scoring the most Xs winning. The number of hits may also be taken into account as another tiebreaker, clay pigeon targets are usually used as flying targets for clay pigeon shooting, formally known as Inanimate Bird Shooting. The Popinjay is an ancient form of target for crossbow shooting, originally a bird tethered in a tree, it developed into a complex painted wood target atop a tall wooden pole. The popinjay would form the centrepiece of a shooting contest. Scoring was awarded for shooting off various parts of the target, dart targets are a special form of bullseye targets. Human silhouette targets are use for military, police and civilian firearms training, within the International Shooting Sport Federation mostly various bullseye targets are used, with variances depending on disciplines. For shotgun clay targets are used, in matches organized by the International Practical Shooting Confederation, both steel and paper targets are used. Currently the only paper targets used for handgun is the IPSC Target, additionally, for rifle and shotgun A3 and A4 paper targets and the Universal Target is used. For steel targets, standardized knock down targets called poppers are used, the two approved designs are the full size IPSC Popper and the 2/3 scaled down version IPSC Mini Popper, while the Pepper Popper and Mini Pepper Popper is now obsolete. In metallic silhouette shooting only knock down steel targets featuring animals are used, plinking refers to casual shooting practices aiming at informal target objects such as tin cans, glass bottles, steel barrels/plates, or anything else that draws the shooters attention. Electronic scoring system Grouping Steel target Targets on Wikimedia Commons
16.
Reticle
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Today, engraved lines or embedded fibers may be replaced by a computer-generated image superimposed on a screen or eyepiece. There are many variations of reticles, this article concerns itself mainly with a simple reticle, crosshairs. Crosshairs are most commonly represented as intersecting lines in the shape of a cross, +, though variations exist, including dots, posts, circles, scales, chevrons. The reticle is said to have been invented by Robert Hooke, another candidate as inventor is the amateur astronomer William Gascoigne, who predated Hooke. Telescopic sights for firearms, generally just called scopes, are probably the device most often associated with crosshairs, motion pictures and the media often use a view through crosshairs as a dramatic device, which has given crosshairs wide cultural exposure. Thicker bars are easier to discern against a complex background. The most popular types of cross-hair in modern scopes are variants on the duplex cross-hair, with bars that are thick on the perimeter, the thick bars allow the eye to quickly locate the center of the reticle, and the thin lines in the center allow for precision aiming. The thin bars in a duplex reticle may also be designed to be used as a measure, called a 30/30 reticle, the thin bars on such a reticle span 30 minutes of an arc, which happens to be close to 30 inches at 100 yards. This enables an experienced shooter to deduce the range within an acceptable error limit, originally crosshairs were constructed out of hair or spiderweb, these materials being sufficiently thin and strong. Many modern scopes use wire crosshairs, which can be flattened to various degrees to change the width and these wires are usually silver in color, but appear black when backlit by the image passing through the scopes optics. The advantage of wire crosshairs is that they are tough and durable. The first suggestion for etched glass reticles was made by Philippe de La Hire in 1700 and his method was based on engraving the lines on a glass plate with a diamond point. Many modern crosshairs are actually etched onto a plate of glass. Reticles may be illuminated, either by a plastic or fiber optic light pipe collecting ambient light or, in low light conditions, a graticule is another term for reticle, frequently encountered in British and British military technical manuals, and came into common use during World War One. The reticle may be located at the front or rear focal plane of the telescopic sight, american and European high end optics manufacturers often leave the customer the choice between a FFP or SFP mounted reticle. Collimated reticles are created using refractive or reflective optical collimators to generate an image of an illuminated or reflective reticle. These types of sights are used on surveying/triangulating equipment, to aid celestial telescope aiming, holographic weapon sights use a holographic image of a reticle at finite set range built into the viewing window and a collimated laser diode to illuminate it. The use of a hologram also eliminates the need for image dimming narrow band reflective coatings, a downside to the holographic weapon sight can be the weight and shorter battery life
17.
Metre
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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
18.
Unit circle
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In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1, the generalization to higher dimensions is the unit sphere, if is a point on the unit circles circumference, then | x | and | y | are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 =1. The interior of the circle is called the open unit disk. One may also use other notions of distance to define other unit circles, such as the Riemannian circle, see the article on mathematical norms for additional examples. The unit circle can be considered as the complex numbers. In quantum mechanics, this is referred to as phase factor, the equation x2 + y2 =1 gives the relation cos 2 + sin 2 =1. The unit circle also demonstrates that sine and cosine are periodic functions, triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P on the circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q and line segments PQ ⊥ OQ, the result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin = y1 and cos = x1. Having established these equivalences, take another radius OR from the origin to a point R on the circle such that the same angle t is formed with the arm of the x-axis. Now consider a point S and line segments RS ⊥ OS, the result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at in the way that P is at. The conclusion is that, since is the same as and is the same as, it is true that sin = sin and it may be inferred in a similar manner that tan = −tan, since tan = y1/x1 and tan = y1/−x1. A simple demonstration of the above can be seen in the equality sin = sin = 1/√2, when working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π
19.
Arc (geometry)
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In Euclidean geometry, an arc is a closed segment of a differentiable curve. A common example in the plane, is a segment of a circle called a circular arc, in space, if the arc is part of a great circle, it is called a great arc. Every pair of points on a circle determines two arcs. The length, L, of an arc of a circle with radius r and this is because L c i r c u m f e r e n c e = θ2 π. Substituting in the circumference L2 π r = θ2 π, and, with α being the angle measured in degrees, since θ = α/180π. For example, if the measure of the angle is 60 degrees and this is so because the circumference of a circle and the degrees of a circle, of which there are always 360, are directly proportional. The area of the sector formed by an arc and the center of a circle is A =12 r 2 θ. The area A has the proportion to the circle area as the angle θ to a full circle. We can cancel π on both sides, A r 2 = θ2, by multiplying both sides by r2, we get the final result, A =12 r 2 θ. Using the conversion described above, we find that the area of the sector for an angle measured in degrees is A = α360 π r 2. The area of the bounded by the arc and the straight line between its two end points is 12 r 2. To get the area of the arc segment, we need to subtract the area of the triangle, determined by the circles center and the two end points of the arc, from the area A. Using the intersecting chords theorem it is possible to calculate the radius r of a circle given the height H and its perpendicular bisector is another chord, which is a diameter of the circle. The length of the first chord is W, and it is divided by the bisector into two halves, each with length W/2. The total length of the diameter is 2r, and it is divided into two parts by the first chord, the length of one part is the sagitta of the arc, H, and the other part is the remainder of the diameter, with length 2r − H. Applying the intersecting chords theorem to these two chords produces H =2, whence 2 r − H = W24 H, so r = W28 H + H2
20.
Pi
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The number π is a mathematical constant, the ratio of a circles circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter π since the mid-18th century, being an irrational number, π cannot be expressed exactly as a fraction. Still, fractions such as 22/7 and other numbers are commonly used to approximate π. The digits appear to be randomly distributed, in particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a number, i. e. a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass, ancient civilizations required fairly accurate computed values for π for practical reasons. It was calculated to seven digits, using techniques, in Chinese mathematics. The extensive calculations involved have also used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. Because of its role as an eigenvalue, π appears in areas of mathematics. It is also found in cosmology, thermodynamics, mechanics, attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits. In English, π is pronounced as pie, in mathematical use, the lowercase letter π is distinguished from its capitalized and enlarged counterpart ∏, which denotes a product of a sequence, analogous to how ∑ denotes summation. The choice of the symbol π is discussed in the section Adoption of the symbol π, π is commonly defined as the ratio of a circles circumference C to its diameter d, π = C d The ratio C/d is constant, regardless of the circles size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of geometry, although the notion of a circle can be extended to any curved geometry. Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be defined independently of geometry using limits. An integral such as this was adopted as the definition of π by Karl Weierstrass, definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. One such definition, due to Richard Baltzer, and popularized by Edmund Landau, is the following, the cosine can be defined independently of geometry as a power series, or as the solution of a differential equation
21.
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
22.
Artillery
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Artillery is a class of large military weapons built to fire munitions far beyond the range and power of infantrys small arms. Early artillery development focused on the ability to breach fortifications, and led to heavy, as technology improved, lighter, more mobile field artillery developed for battlefield use. This development continues today, modern self-propelled artillery vehicles are highly mobile weapons of great versatility providing the largest share of an armys total firepower, in its earliest sense, the word artillery referred to any group of soldiers primarily armed with some form of manufactured weapon or armour. In common speech, the artillery is often used to refer to individual devices, along with their accessories and fittings. However, there is no generally recognised generic term for a gun, howitzer, mortar, and so forth, the United States uses artillery piece, the projectiles fired are typically either shot or shell. Shell is a widely used term for a projectile, which is a component of munitions. By association, artillery may also refer to the arm of service that customarily operates such engines, in the 20th Century technology based target acquisition devices, such as radar, and systems, such as sound ranging and flash spotting, emerged to acquire targets, primarily for artillery. These are usually operated by one or more of the artillery arms, Artillery originated for use against ground targets—against infantry, cavalry and other artillery. An early specialist development was coastal artillery for use against enemy ships, the early 20th Century saw the development of a new class of artillery for use against aircraft, anti-aircraft guns. Artillery is arguably the most lethal form of land-based armament currently employed, the majority of combat deaths in the Napoleonic Wars, World War I, and World War II were caused by artillery. In 1944, Joseph Stalin said in a speech that artillery was the God of War, although not called as such, machines performing the role recognizable as artillery have been employed in warfare since antiquity. The first references in the historical tradition begin at Syracuse in 399 BC. From the Middle Ages through most of the era, artillery pieces on land were moved by horse-drawn gun carriages. In the contemporary era, the artillery and crew rely on wheeled or tracked vehicles as transportation, Artillery used by naval forces has changed significantly also, with missiles replacing guns in surface warfare. The engineering designs of the means of delivery have likewise changed significantly over time, in some armies, the weapon of artillery is the projectile, not the equipment that fires it. The process of delivering fire onto the target is called gunnery, the actions involved in operating the piece are collectively called serving the gun by the detachment or gun crew, constituting either direct or indirect artillery fire. The term gunner is used in armed forces for the soldiers and sailors with the primary function of using artillery. The gunners and their guns are usually grouped in teams called either crews or detachments, several such crews and teams with other functions are combined into a unit of artillery, usually called a battery, although sometimes called a company
23.
Compass
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A compass is an instrument used for navigation and orientation that shows direction relative to the geographic cardinal directions, or points. Usually, a called a compass rose shows the directions north, south, east. When the compass is used, the rose can be aligned with the geographic directions, so, for example. Frequently, in addition to the rose or sometimes instead of it, North corresponds to zero degrees, and the angles increase clockwise, so east is 90 degrees, south is 180, and west is 270. These numbers allow the compass to show azimuths or bearings, which are stated in this notation. The magnetic compass was first invented as a device for divination as early as the Chinese Han Dynasty, the first usage of a compass recorded in Western Europe and the Islamic world occurred around the early 13th century. The magnetic compass is the most familiar compass type and it functions as a pointer to magnetic north, the local magnetic meridian, because the magnetized needle at its heart aligns itself with the horizontal component of the Earths magnetic field. The needle is mounted on a pivot point, in better compasses a jewel bearing. When the compass is level, the needle turns until, after a few seconds to allow oscillations to die out. In navigation, directions on maps are usually expressed with reference to geographical or true north, the direction toward the Geographical North Pole, the rotation axis of the Earth. Depending on where the compass is located on the surface of the Earth the angle between north and magnetic north, called magnetic declination can vary widely with geographic location. The local magnetic declination is given on most maps, to allow the map to be oriented with a parallel to true north. The location of the Earths magnetic poles slowly change with time, the effect of this means a map with the latest declination information should be used. Some magnetic compasses include means to compensate for the magnetic declination. The first compasses in ancient Han dynasty China were made of lodestone, the compass was later used for navigation during the Song Dynasty of the 11th century. Later compasses were made of iron needles, magnetized by striking them with a lodestone, dry compasses began to appear around 1300 in Medieval Europe and the Islamic world. This was supplanted in the early 20th century by the magnetic compass. Modern compasses usually use a needle or dial inside a capsule completely filled with a liquid
24.
Palais de Rumine
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The Palais de Rumine is a late 19th-century building in Florentine Renaissance style in Lausanne, Switzerland. On his death, Gabriel de Rumine, son of Russian nobility, building began in 1892 according to the design of the Lyonnais architect Gaspard André. The building was inaugurated on the 3 November 1902, although building work continued until 1904 and it housed facilities such as the library of the University of Lausanne, and scientific and artistic collections belonging to the Canton of Vaud. In the 1980s, the university moved to its current location by Lake Geneva due to lack of space, the building currently hosts one of the three sites of the Cantonal and University Library of Lausanne. Dossier des latinistes - Lausanne, quelques exemples darchitecture néo-classique, list of cultural property of national significance in Switzerland, Vaud University of Lausanne Page on the website of the City of Lausanne
25.
University of Lausanne
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The University of Lausanne in Lausanne, Switzerland was founded in 1537 as a school of theology, before being made a university in 1890. Today about 13,500 students and 2,200 researchers study, approximately 1,500 international students attend the university, which has a wide curriculum including exchange programs with world-renowned universities. Since 2005, the University follows the requirements of the Bologna process, the 2011 Times Higher Education World University Rankings ranked the University of Lausanne 116th globally. The CWTS Leiden Ranking 2015 ranks the University of Lausanne 11th in Europe and 41st globally, together with the École polytechnique fédérale de Lausanne the university forms a vast campus at the shores of Lake Geneva. Its vocation at that time was to train ministers for the church, the university enjoyed a certain renown due to the fact that it was the only French language Protestant school of theology. As the centuries passed, the number of faculties increased and diversified until, in 1890, in 1909 Rodolphe Archibald Reiss founded the first school of forensic science in the world, the Institut de police scientifique. From 1970, the university moved progressively from the old centre of Lausanne, around the Cathedral and Château, in 2003 two new faculties were founded, concentrating on the life and human sciences, the Faculty of Biology and Medicine and the Faculty of Geosciences and Environment. On 1 January 2014, the Swiss Graduate School of Public Administration was integrated into the University of Lausanne, the main campus is presently situated outside the city of Lausanne, on the shores of Lake Léman, in Dorigny. It is adjacent to the Swiss Federal Institute of Technology in Lausanne and is served by the Lausanne Metro Line 1, the two schools together welcome about 20,000 students. The UNIL and the EPFL share an active sports centre located on the campus, on the shores of Lake Geneva, the university campus is made up of individual buildings with a park and arboretum in between. The university library serves as eating hall and is centrally located. The view from the library across the fields to the lake of Geneva. On a clear day, Mont Blanc can be seen, the Swiss Institute of Comparative Law and the central administration of the Swiss Institute of Bioinformatics are also located on the main campus. In addition to its campus at the lakeside, the University of Lausanne also has other sites. The Faculty of Biology and Medicine is also located in two sites, around the University Hospital of Lausanne and in Épalinges. The Biopôle was built next to the Épalinges campus, the Faculty of Biology and Medicine also comprises a fourth site, the Psychiatric Hospital of Cery, in Prilly. The UNIL also publish Allez savoir, a magazine aimed at a larger audience, published three times a year and whose subscription is free. Besides these, Lauditoire is the newspapers from both UNIL and EPFL, of a circulation 19000 free copies
26.
Swiss people
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The Swiss are citizens of Switzerland. The demonym derives from the toponym of Schwyz and has been in use to refer to the Old Swiss Confederacy since the 16th century. The number of Swiss nationals has grown from 1.7 million in 1815 to 6.76 million in 2009 with 90% of them living in Switzerland, about 60% of those living abroad reside in the European Union. The largest groups of Swiss descendants and nationals outside of Europe are found in the United States, closely related German-speaking peoples are the Alsatians, the Swabians and the Vorarlbergians. The French-speaking Swiss, traditionally speaking Franco-Provençal dialects, today largely assimilated to the standard French language, amalgamated from the Gallo-Roman population and they are closely related to the French. The Italian-speaking Swiss, traditionally speakers of Lombard language today partly assimilated to the standard Italian language, amalgamated from Raetians and they are closely related to the Italians. The Romansh, speakers of the Romansh language, settling in parts of the Grisons, with worldwide human migration, there are an increasing number of Swiss not descended or only partially descended from the core ethnic groups listed above. Most naturalized Swiss citizens will be linguistically oriented according to their canton of residence, the Swiss populace historically derives from an amalgamation of Gaulish or Gallo-Roman, Alamannic and Rhaetic stock. Their cultural history is dominated by the Alps, and the environment is often cited as an important factor in the formation of the Swiss national character. Political allegiance and patriotism was directed towards the cantons, not the federal level, from the 19th century there were conscious attempts to foster a federal Pan-Swiss national identity that would replace or alleviate the cantonal patriotisms. An additional symbol of national identity at the federal level was introduced with the Swiss national holiday in 1889. The bonfires associated with the holiday have become so customary since then that they have displaced the Funken traditions of greater antiquity. These specifically include Grisons, Valais, Ticino, Vaud and Geneva, according to Hartley-Moore, Localized equivalents of nationalist symbols were also essential to the creation of Swiss civil society. In the Swiss model, pride in local identity is to some degree synonymous with loyalty to the larger state, as Gottfried Keller argued in the nineteenth century, Without cantons and without their differences and competition, no Swiss federation could exist. Facilitated naturalization for foreign spouses and children of Swiss citizens requires a total minimum residence of five years, with more than 20% of the population resident aliens, Switzerland has one of the highest ratios of non-naturalized inhabitants in Europe. In 2003,35,424 residents were naturalized, a number exceeding net population growth. Over the 25-year period of 1983 to 2007,479,264 resident foreigners were naturalized, compare the figure of 0. 2% in the United Kingdom. The genetic composition of the Swiss population is similar to that of Central Europe in general, Switzerland is on one hand at the crossroads of several prehistoric migrations, while on the other hand the Alps acted as a refuge in some cases
27.
Equilateral triangle
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In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular and they are regular polygons, and can therefore also be referred to as regular triangles. Thus these are properties that are unique to equilateral triangles, the three medians have equal lengths. The three angle bisectors have equal lengths, every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral, in particular, A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide. It is also equilateral if its circumcenter coincides with the Nagel point, for any triangle, the three medians partition the triangle into six smaller triangles. A triangle is equilateral if and only if any three of the triangles have either the same perimeter or the same inradius. A triangle is equilateral if and only if the circumcenters of any three of the triangles have the same distance from the centroid. Morleys trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, a version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, PA, PB, and PC satisfy the inequality that any two of them sum to at least as great as the third. By Eulers inequality, the triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle, specifically. The triangle of largest area of all those inscribed in a circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, the ratio of the area to the square of the perimeter of an equilateral triangle,1123, is larger than that for any other triangle. If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A1 and A2, then 79 ≤ A1 A2 ≤97, in no other triangle is there a point for which this ratio is as small as 2. For any point P in the plane, with p, q, and t from the vertices A, B. For any point P on the circle of an equilateral triangle, with distances p, q. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral, an equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the group of order 6 D3
28.
World War I
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World War I, also known as the First World War, the Great War, or the War to End All Wars, was a global war originating in Europe that lasted from 28 July 1914 to 11 November 1918. More than 70 million military personnel, including 60 million Europeans, were mobilised in one of the largest wars in history and it was one of the deadliest conflicts in history, and paved the way for major political changes, including revolutions in many of the nations involved. The war drew in all the worlds great powers, assembled in two opposing alliances, the Allies versus the Central Powers of Germany and Austria-Hungary. These alliances were reorganised and expanded as more nations entered the war, Italy, Japan, the trigger for the war was the assassination of Archduke Franz Ferdinand of Austria, heir to the throne of Austria-Hungary, by Yugoslav nationalist Gavrilo Princip in Sarajevo on 28 June 1914. This set off a crisis when Austria-Hungary delivered an ultimatum to the Kingdom of Serbia. Within weeks, the powers were at war and the conflict soon spread around the world. On 25 July Russia began mobilisation and on 28 July, the Austro-Hungarians declared war on Serbia, Germany presented an ultimatum to Russia to demobilise, and when this was refused, declared war on Russia on 1 August. Germany then invaded neutral Belgium and Luxembourg before moving towards France, after the German march on Paris was halted, what became known as the Western Front settled into a battle of attrition, with a trench line that changed little until 1917. On the Eastern Front, the Russian army was successful against the Austro-Hungarians, in November 1914, the Ottoman Empire joined the Central Powers, opening fronts in the Caucasus, Mesopotamia and the Sinai. In 1915, Italy joined the Allies and Bulgaria joined the Central Powers, Romania joined the Allies in 1916, after a stunning German offensive along the Western Front in the spring of 1918, the Allies rallied and drove back the Germans in a series of successful offensives. By the end of the war or soon after, the German Empire, Russian Empire, Austro-Hungarian Empire, national borders were redrawn, with several independent nations restored or created, and Germanys colonies were parceled out among the victors. During the Paris Peace Conference of 1919, the Big Four imposed their terms in a series of treaties, the League of Nations was formed with the aim of preventing any repetition of such a conflict. This effort failed, and economic depression, renewed nationalism, weakened successor states, and feelings of humiliation eventually contributed to World War II. From the time of its start until the approach of World War II, at the time, it was also sometimes called the war to end war or the war to end all wars due to its then-unparalleled scale and devastation. In Canada, Macleans magazine in October 1914 wrote, Some wars name themselves, during the interwar period, the war was most often called the World War and the Great War in English-speaking countries. Will become the first world war in the sense of the word. These began in 1815, with the Holy Alliance between Prussia, Russia, and Austria, when Germany was united in 1871, Prussia became part of the new German nation. Soon after, in October 1873, German Chancellor Otto von Bismarck negotiated the League of the Three Emperors between the monarchs of Austria-Hungary, Russia and Germany
29.
October Revolution
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It took place with an armed insurrection in Petrograd on 25 October 1917. During this time, urban workers began to organize into councils wherein revolutionaries criticized the provisional government and this immediately initiated the establishment of the Russian Socialist Federative Soviet Republic, the worlds first self-proclaimed socialist state. The revolution was led by the Bolsheviks, who used their influence in the Petrograd Soviet to organize the armed forces, Bolshevik Red Guards forces under the Military Revolutionary Committee began the takeover of government buildings on 24 October 1917. The following day, the Winter Palace, was captured, the long-awaited Constituent Assembly elections were held on 12 November 1917. The Bolsheviks only won 175 seats in the 715-seat legislative body, coming in second behind the Socialist Revolutionary party, the Constituent Assembly was to first meet on 28 November 1917, but its convocation was delayed until 5 January 1918 by the Bolsheviks. On its first and only day in session, the body rejected Soviet decrees on peace and land, as the revolution was not universally recognized, there followed the struggles of the Russian Civil War and the creation of the Soviet Union in 1922. At first, the event was referred to as the October coup or the Uprising of 25th, in Russian, however, переворот has a similar meaning to revolution and also means upheaval or overturn, so coup is not necessarily the correct translation. With time, the term October Revolution came into use and it is also known as the November Revolution having occurred in November according to the Gregorian Calendar. The Great October Socialist Revolution was the name for the October Revolution in the Soviet Union after the 10th anniversary of the Revolution in 1927. The February Revolution had toppled Tsar Nicolas II of Russia, however, the provisional government was weak and riven by internal dissension. It continued to wage World War I, which became increasingly unpopular, a nationwide crisis developed in Russia, affecting social, economic, and political relations. Disorder in industry and transport had intensified, and difficulties in obtaining provisions had increased, gross industrial production in 1917 had decreased by over 36% from what it had been in 1914. In the autumn, as much as 50% of all enterprises were closed down in the Urals, the Donbas, at the same time, the cost of living increased sharply. Real wages fell about 50% from what they had been in 1913, russias national debt in October 1917 had risen to 50 billion rubles. Of this, debts to foreign governments constituted more than 11 billion rubles, the country faced the threat of financial bankruptcy. In these months alone, more than a million took part in strikes. Workers established control over production and distribution in many factories and plants in a social revolution, by October 1917, there had been over 4,000 peasant uprisings against landowners. When the Provisional Government sent punitive detachments, it only enraged the peasants
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NATO
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The North Atlantic Treaty Organization, also called the North Atlantic Alliance, is an intergovernmental military alliance based on the North Atlantic Treaty which was signed on 4 April 1949. The organization constitutes a system of collective defence whereby its member states agree to mutual defence in response to an attack by any external party, three NATO members are permanent members of the United Nations Security Council with the power to veto and are officially nuclear-weapon states. NATOs headquarters are located in Haren, Brussels, Belgium, while the headquarters of Allied Command Operations is near Mons. NATO is an Alliance that consists of 28 independent member countries across North America and Europe, an additional 22 countries participate in NATOs Partnership for Peace program, with 15 other countries involved in institutionalized dialogue programmes. The combined military spending of all NATO members constitutes over 70% of the global total, Members defence spending is supposed to amount to 2% of GDP. The course of the Cold War led to a rivalry with nations of the Warsaw Pact, politically, the organization sought better relations with former Warsaw Pact countries, several of which joined the alliance in 1999 and 2004. N. The Treaty of Brussels, signed on 17 March 1948 by Belgium, the Netherlands, Luxembourg, France, the treaty and the Soviet Berlin Blockade led to the creation of the Western European Unions Defence Organization in September 1948. However, participation of the United States was thought necessary both to counter the power of the USSR and to prevent the revival of nationalist militarism. He got a hearing, especially considering American anxiety over Italy. In 1948 European leaders met with U. S. defense, military and diplomatic officials at the Pentagon, marshalls orders, exploring a framework for a new and unprecedented association. Talks for a new military alliance resulted in the North Atlantic Treaty and it included the five Treaty of Brussels states plus the United States, Canada, Portugal, Italy, Norway, Denmark and Iceland. The first NATO Secretary General, Lord Ismay, stated in 1949 that the goal was to keep the Russians out, the Americans in. Popular support for the Treaty was not unanimous, and some Icelanders participated in a pro-neutrality, the creation of NATO can be seen as the primary institutional consequence of a school of thought called Atlanticism which stressed the importance of trans-Atlantic cooperation. The members agreed that an attack against any one of them in Europe or North America would be considered an attack against them all. The treaty does not require members to respond with military action against an aggressor, although obliged to respond, they maintain the freedom to choose the method by which they do so. This differs from Article IV of the Treaty of Brussels, which states that the response will be military in nature. It is nonetheless assumed that NATO members will aid the attacked member militarily, the treaty was later clarified to include both the members territory and their vessels, forces or aircraft above the Tropic of Cancer, including some Overseas departments of France. The creation of NATO brought about some standardization of allied military terminology, procedures, and technology, the roughly 1300 Standardization Agreements codified many of the common practices that NATO has achieved
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Skinny triangle
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A skinny triangle in trigonometry is a triangle whose height is much greater than its base. The solution of triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to the angle in radians. The solution is simple for skinny triangles that are also isosceles or right triangles. The skinny triangle finds uses in surveying, astronomy and shooting, the proof of the skinny triangle solution follows from the small-angle approximation by applying the law of sines. Applying the small angle approximations to the law of sines above results in and this result is equivalent to assuming that the length of the base of the triangle is equal to the length of the arc of circle of radius r subtended by angle θ. This approximation becomes more accurate for smaller and smaller θ. The error is 10% or less for angles less than about 43°, the error of this approximation is less than 10% for angles 31° or less. Applications of the skinny triangle occur in any situation where the distance to a far object is to be determined and this can occur in surveying, astronomy, and also has military applications. The skinny triangle is used in astronomy to measure the distance to solar system objects. The base of the triangle is formed by the distance between two measuring stations and the angle θ is the angle formed by the object as seen by the two stations. This baseline is usually very long for best accuracy, in principle the stations could be on opposite sides of the Earth. However, this distance is short compared to the distance to the object being measured. The alternative method of measuring the angles is theoretically possible. The base angles are very nearly right angles and would need to be measured with much greater precision than the angle in order to get the same accuracy. The same method of measuring angles and applying the skinny triangle can be used to measure the distances to stars. In the case of stars however, a longer baseline than the diameter of the Earth is usually required, instead of using two stations on the baseline, two measurements are made from the same station at different times of year. During the intervening period, the orbit of the Earth around the Sun moves the station a great distance. This baseline can be as long as the axis of the Earths orbit or, equivalently
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Small-angle approximation
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The small-angle approximation is a useful simplification of the basic trigonometric functions which is approximately true in the limit where the angle approaches zero. They are truncations of the Taylor series for the trigonometric functions to a second-order approximation. This truncation gives, sin θ ≈ θ cos θ ≈1 − θ22 tan θ ≈ θ, where θ is the angle in radians. The small angle approximation is useful in areas of engineering and physics, including mechanics, electromagnetics, optics, cartography, astronomy. The accuracy of the approximations can be seen below in Figure 1, as the angle approaches zero, it is clear that the gap between the approximation and the original function quickly vanishes. The red section on the right, d, is the difference between the lengths of the hypotenuse, H, and the adjacent side, A. As is shown, H and A are almost the length, meaning cos θ is close to 1. Cos θ ≈1 − θ22 The opposite leg, O, is equal to the length of the blue arc. Simplifying leaves, sin θ ≈ tan θ ≈ θ, the Maclaurin expansion of the relevant trigonometric function is sin θ = ∑ n =0 ∞ n. + ⋯ where θ is the angle in radians.01, Figure 3 shows the relative errors of the small angle approximations. The angles at which the error exceeds 1% are as follows. Sin θ ≈ θ at about 0.244 radians, cos θ ≈1 − θ2/2 at about 0.664 radians. In astronomy, the angle subtended by the image of a distant object is only a few arcseconds. The linear size is related to the size and the distance from the observer by the simple formula D = X d 206265 where X is measured in arcseconds. The number 7005206265000000000♠206265 is approximately equal to the number of arcseconds in a circle, the exact formula is D = d tan and the above approximation follows when tan X is replaced by X. The second-order cosine approximation is useful in calculating the potential energy of a pendulum. The small-angle approximation also appears in structural mechanics, especially in stability and this leads to significant simplifications, though at a cost in accuracy and insight into the true behavior. The 1 in 60 rule used in air navigation has its basis in the small-angle approximation, skinny triangle Infinitesimal oscillations of a pendulum Versine and haversine Exsecant and excosecant
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Sine
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In mathematics, the sine is a trigonometric function of an angle. More generally, the definition of sine can be extended to any value in terms of the length of a certain line segment in a unit circle. The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy, via translation from Sanskrit to Arabic and then from Arabic to Latin. The word sine comes from a Latin mistranslation of the Arabic jiba, to define the trigonometric functions for an acute angle α, start with any right triangle that contains an angle of measure α, in the accompanying figure, angle A in triangle ABC has measure α. The three sides of the triangle are named as follows, The opposite side is the side opposite to the angle of interest, the hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle, the adjacent side is the remaining side, in this case side b. It forms a side of both the angle of interest and the right angle, once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse. As stated, the value sin appears to depend on the choice of right triangle containing an angle of measure α, however, this is not the case, all such triangles are similar, and so the ratio is the same for each of them. The trigonometric functions can be defined in terms of the rise, run, when the length of the line segment is 1, sine takes an angle and tells the rise. Sine takes an angle and tells the rise per unit length of the line segment, rise is equal to sin θ multiplied by the length of the line segment. In contrast, cosine is used for telling the run from the angle, arctan is used for telling the angle from the slope. The line segment is the equivalent of the hypotenuse in the right-triangle, in trigonometry, a unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. Let a line through the origin, making an angle of θ with the half of the x-axis. The x- and y-coordinates of this point of intersection are equal to cos θ and sin, the points distance from the origin is always 1. Unlike the definitions with the triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function. Exact identities, These apply for all values of θ. sin = cos =1 csc The reciprocal of sine is cosecant, i. e. the reciprocal of sin is csc, or cosec. Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side, the inverse function of sine is arcsine or inverse sine
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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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Theta
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Theta is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth. In the system of Greek numerals it has a value of 9, in Ancient Greek, θ represented the aspirated voiceless dental plosive /t̪ʰ/, but in Modern Greek it represents the voiceless dental fricative /θ/. In its archaic form, θ was written as a cross within a circle, archaic crossed forms of theta are seen in the wheel letters of Linear A and Linear B. The cursive form ϑ was retained by Unicode as U+03D1 ϑ GREEK THETA SYMBOL, for the purpose of writing Greek text, the two can be font variants of a single character, but θ and ϑ are also used as distinct symbols in technical and mathematical contexts. In the Latin script used for the Gaulish language, theta developed into the tau gallicum, conventionally transliterated as Ð, the phonetic value of the tau gallicum is thought to have been. The early Cyrillic letter fita developed from θ and this letter existed in the Russian alphabet until the 1918 Russian orthography reform. In the International Phonetic Alphabet, represents the voiceless dental fricative and it does not represent the consonant in the, which is the voiced dental fricative. The lower-case letter θ is used as a symbol for, A plane angle in geometry, a Variable in trigonometry A special function of several complex variables. One of the Chebyshev functions in prime number theory, the score of a test taker in item response theory. Theta Type Replication, a type of bacterial DNA replication specific to circular chromosomes, threshold value of an artificial neuron. A Bayer designation letter applied to a star in a constellation, usually the star so labelled. The parameter frequently used in writing the likelihood function, the Watterson estimator for the population mutation rate in population genetics. Indicates a minimum optimum integration level determined by the intersection of GG, the GG-LL schedules are a tool used in analyzing the potential benefits of a country pegging their domestic currency to a foreign currency. The reserve ratio of banks in economic models, the ordinal collapsing function developed by Solomon Feferman The upper-case letter Θ is used as a symbol for, Quantity or temperature, by SI standard. An asymptotically tight bound in the analysis of algorithms, a certain ordinal number in set theory. Pentaquarks, exotic baryons in particle physics, a brain signal frequency ranging from 4–8 Hz. One of the known as Greeks in finance, representing time decay of options or the change in the intrinsic value of an option divided by the number of days until the option expires. A variable indicating temperature difference in heat transfer, according to Porphyry of Tyros, the Egyptians used an X within a circle as a symbol of the soul, having a value of nine, it was used as a symbol for Ennead
36.
Trigonometric functions
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
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Taylor series
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In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the functions derivatives at a single point. The concept of a Taylor series was formulated by the Scottish mathematician James Gregory, a function can be approximated by using a finite number of terms of its Taylor series. Taylors theorem gives quantitative estimates on the error introduced by the use of such an approximation, the polynomial formed by taking some initial terms of the Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of that functions Taylor polynomials as the degree increases, a function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an interval is known as an analytic function in that interval. The Taylor series of a real or complex-valued function f that is differentiable at a real or complex number a is the power series f + f ′1. Which can be written in the more compact sigma notation as ∑ n =0 ∞ f n, N where n. denotes the factorial of n and f denotes the nth derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and 0 and 0. are both defined to be 1, when a =0, the series is also called a Maclaurin series. The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for 1/1 − x is the geometric series 1 + x + x 2 + x 3 + ⋯ so the Taylor series for 1/x at a =1 is 1 − +2 −3 + ⋯. The Taylor series for the exponential function ex at a =0 is x 00, + ⋯ =1 + x + x 22 + x 36 + x 424 + x 5120 + ⋯ = ∑ n =0 ∞ x n n. The above expansion holds because the derivative of ex with respect to x is also ex and this leaves the terms n in the numerator and n. in the denominator for each term in the infinite sum. The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a result, but rejected it as an impossibility. It was through Archimedess method of exhaustion that a number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later, in the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. The Kerala school of astronomy and mathematics further expanded his works with various series expansions, in the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a method for constructing these series for all functions for which they exist was finally provided by Brook Taylor. The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, if f is given by a convergent power series in an open disc centered at b in the complex plane, it is said to be analytic in this disc
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Approximation error
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The approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because the measurement of the data is not precise due to the instruments, or approximations are used instead of the real data. In the mathematical field of analysis, the numerical stability of an algorithm in numerical analysis indicates how the error is propagated by the algorithm. One commonly distinguishes between the error and the absolute error. Given some value v and its approximation vapprox, the error is ϵ = | v − v approx |. In words, the error is the magnitude of the difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value, the percent error is the relative error expressed in terms of per 100. These definitions can be extended to the case when v and v approx are n-dimensional vectors, by replacing the absolute value with an n-norm. As an example, if the value is 50 and the approximation is 49.9, then the absolute error is 0.1. Another example would be if, in measuring a 6mL beaker, the correct reading being 6mL, this means the percent error in that particular situation is, rounded,16. 7%.003 and in the second it is only 0.000003. There are two features of relative error that should be kept in mind, firstly, relative error is undefined when the true value is zero as it appears in the denominator. Secondly, relative error only makes sense when measured on a ratio scale, otherwise it would be sensitive to the measurement units. For example, when an error in a temperature measurement given in Celsius scale is 1 °C, and the true value is 2 °C, the relative error is 0.5. Celsius temperature is measured on a scale, whereas the Kelvin scale has a true zero. In most indicating instruments, the accuracy is guaranteed to a percentage of full-scale reading. The limits of deviations from the specified values are known as limiting errors or guarantee errors
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Iron sights
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Iron sights are typically composed of two component sights, formed by metal blades, a rear sight mounted perpendicular to the line of sight and a front sight that is a post, bead, or ring. Open sights use a notch of some sort as the rear sight, civilian, hunting, and police firearms usually feature open sights, while many military battle rifles employ aperture sights. The earliest and simplest iron sights are fixed and cannot be easily adjusted, many iron sights are designed to be adjustable, so that the sights can be adjusted for elevation or windage. On many firearms it is the sight that is adjustable. For precision applications such as hunting or sniping, the sights are usually replaced by a telescopic sight. Iron sights may still be fitted alongside other sighting devices for back-up usage, iron sights provide horizontal and vertical reference points that allow the shooter to train the weapon. Rear sights are mounted in a dovetail on the barrel or receiver, closer to the eye of the shooter. Front sights are mounted to the barrel by dovetailing, soldering, screwing, or staking close to the muzzle, frequently on a ramp. Some front sight assemblies include a detachable hood intended to reduce glare, with typical blade or post iron sights, the shooter would center the front post in the notch of the rear sight and the tops of both sights should be level. Since the eye is capable of focusing on one plane, and the rear sight, front sight. At 1,000 m, that same misalignment would be magnified 100 times, sights for shotguns used for shooting small, moving targets work quite differently. The rear sight is completely discarded, and the reference point is provided by the correct. A brightly colored round bead is placed at the end of the barrel, often, this bead will be placed along a raised, flat rib, which is usually ventilated to keep it cool and reduce mirage effects from a hot barrel. This method of aiming is not as precise as that of a front sight/rear sight combination, but it is faster. Some shotguns also provide a mid-bead, which is a smaller bead located halfway down the rib, open sights generally are used where the rear sight is at significant distance from the shooters eye. They provide minimum occlusion of the view, but at the expense of precision. Open sights generally use either a square post or a bead on a post for a front sight, to use the sight, the post or bead is positioned both vertically and horizontally in the center of the rear sight notch. For a center hold, the front sight is positioned on the center of the target, for a 6oclock hold, the front sight is positioned just below the target and centered horizontally
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Telescopic sight
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A telescopic sight, commonly called a scope, is a sighting device that is based on an optical refracting telescope. They are equipped with some form of graphic image pattern mounted in an appropriate position in their optical system to give an accurate aiming point. Telescopic sights are used all types of systems that require accurate aiming but are most commonly found on firearms. Other types of sights are iron sights, reflector sights, the optical components may be combined with optoelectronics to form a night scope. The first experiments directed to give shooters optical aiming aids go back to the early 17th century, for centuries different optical aiming aids and primitive predecessors of telescopic sights were created that had practical or performance limitations. The first documented telescopic rifle sight was invented between 1835 and 1840, in 1855, William Malcolm of Syracuse, NY began producing his own sight. Malcolm used a design incorporating achromatic lenses like those used in telescopes. They were between three and twenty magnification, malcolms and those made by L. M. Amidon of Vermont were the standard during the Civil War. Still other telescopic sights of the same period were the Davidson. An early practical refractor telescope based telescopic sight was built in 1880 by August Fiedler, later telescopic sights with extra long eye relief became available for handgun and scout rifle use. A historic example of a sight with a long eye relief is the German ZF41 which was used during World War II on Karabiner 98k rifles. An early example of a man portable telescopic sight for low visibility/night use is the Zielgerät 1229, the ZG1229 Vampir was a Generation 0 active infrared night vision device developed for the Wehrmacht for the StG44 assault rifle, intended primarily for night use. The issuing of the ZG1229 Vampir system to the military started in 1944, Telescopic sights are classified in terms of the optical magnification and the objective lens diameter, e. g. 10×50. This would denote 10 times magnification with a 50 mm objective lens, in general terms, larger objective lens diameters, due to their ability to gather larger amounts of light, provide a larger exit pupil and hence provide a brighter image at the eyepiece. On fixed magnification sights the magnification power and objective diameter should be chosen on the basis of the intended use, there are also telescopic sights with variable magnification. The magnification can be varied by manually operating a zoom mechanism, variable sights offer more flexibility regarding shooting at varying ranges, targets and light conditions and offer a relative wide field of view at lower magnification settings. The syntax for variable sights is the following, minimal magnification – maximum magnification × objective lens, for example, 3–9×40. Confusingly, some older telescopic sights, mainly of German or other European manufacture, have a different classification where the part of the designation refers to light gathering power