1.
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
2.
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
3.
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
4.
Rotation
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A rotation is a circular movement of an object around a center of rotation. A three-dimensional object always rotates around a line called a rotation axis. If the axis passes through the center of mass, the body is said to rotate upon itself. A rotation about a point, e. g. the Earth about the Sun, is called a revolution or orbital revolution. The axis is called a pole, mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two, a rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion, the axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question then the body is said to orbit, there is no fundamental difference between a “rotation” and an “orbit” and or spin. The key distinction is simply where the axis of the rotation lies and this distinction can be demonstrated for both “rigid” and “non rigid” bodies. If a rotation around a point or axis is followed by a rotation around the same point/axis. The reverse of a rotation is also a rotation, thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis and that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the rotations are known as yaw, pitch. This terminology is used in computer graphics. In astronomy, rotation is an observed phenomenon. Stars, planets and similar bodies all spin around on their axes, the rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features and this rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the equator
5.
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
6.
Points of the compass
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The points of the compass, specifically on the compass rose, mark divisions of a compass, such divisions may be referred to as winds or directions. A compass point allows reference to a heading in a general or colloquial fashion. A compass is primarily divided into the four cardinal points—north, south, east and these are often further subdivided by the addition of the four intercardinal directions—northeast between north and east, southeast, southwest, and northwest —to indicate the eight principal winds. In meteorological usage, further intermediate points between cardinal and ordinal points, such as north-northeast between north and northeast, are added to give the sixteen points of a wind compass, for most applications, the fractional points have been superseded by degrees measured clockwise from North. In ancient China 24 points of the compass were used, measuring fifteen degrees between points. The names of the compass directions follow the 32-point wind compass rose follow these rules, The cardinal directions are north, east, south, west, the ordinal directions are northeast, southeast, southwest and northwest, formed by bisecting the angle of the cardinal winds. The name is merely a combination of the cardinals it bisects, the eight principal winds are the cardinals and ordinals considered together, that is N, NE, E, SE, S, SW, W, NW. Each principal wind is 45° from its neighbour, the principal winds form the basic eight-wind compass rose. The eight half-winds are the points obtained by bisecting the angles between the principal winds, the half-winds are north-northeast, east-northeast, east-southeast, south-southeast, south-southwest, west-southwest, west-northwest and north-northwest. Notice that the name is constructed simply by combining the names of the winds to either side, with the cardinal wind coming first. The eight principal winds and the eight half-winds together yield a 16-wind compass rose, all of the above named points plus the sixteen quarter winds listed in the next paragraph define the 32 points of the wind compass rose. The sixteen quarter winds are the points obtained by bisecting the angles between the points on a 16-wind compass rose. The name of a quarter-wind is X by Y, where X is a principal wind, so northeast by east means one quarter from NE towards E, southwest by south means one quarter from SW towards S. The eight principal winds, eight half-winds and sixteen quarter winds together yield a 32-wind compass rose, in the mariners exercise of boxing the compass, all thirty-two points of the compass are named in clockwise order. The title of the Alfred Hitchcock 1959 movie, North by Northwest, is not a direction point on the 32-wind compass. The traditional compass rose of eight winds was invented by seafarers in the Mediterranean Sea during the Middle Ages. This Italianate patois was used to designate the names of the winds on the compass rose found in mariner compasses. Tramutana, Gregale, Grecho, Sirocco, Xaloc, Lebeg, Libezo, Leveche, Mezzodi, Migjorn, Magistro, Mestre, etc
7.
Protractor
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A protractor is a measuring instrument, typically made of transparent plastic or glass, for measuring angles. Most protractors measure angles in degrees, radian-scale protractors measure angles in radians. Most protractors are divided into 180 equal parts and they are used for a variety of mechanical and engineering-related applications, but perhaps the most common use is in geometry lessons in schools. More advanced protractors, such as the bevel protractor, have one or two swinging arms, which can be used to measure the angle. A bevel protractor is a graduated circular protractor with one pivoted arm used for measuring or marking off angles, sometimes Vernier scales are attached to give more precise readings. It has wide application in architectural and mechanical drawing, although its use is decreasing with the availability of drawing software or CAD. Universal bevel protractors are also used by toolmakers, as they measure angles by mechanical contact they are classed as mechanical protractors, the bevel protractor is used to establish and test angles to very close tolerances. It reads to 5 minutes or 1/12° and can measure any angle from 0° to 360°, the bevel protractor consists of a beam, a graduated dial and a blade which is connected to a swivel plate by thumb nut and clamp. When the edges of the beam and blade are parallel, a mark on the swivel plate coincides with the zero line on the graduated dial. To measure an angle between the beam and the blade of 90° or less, the reading may be obtained direct from the number on the dial indicated by the mark on the swivel plate. To measure an angle of over 90°, subtract the number of degrees as indicated on the dial from 180°, as the dial is graduated from opposite zero marks to 90° each way. Since the spaces, both on the scale and the Vernier scale, are numbered both to the right and to the left from zero, any angle can be measured. The readings can be either to the right or to the left. The above picture illustrates a variety of uses of the bevel protractor, reading the Vernier scale, The bevel protractor Vernier scale may have graduations of 5′ or 1/12°. Each space on the Vernier scale is 5′ less than two spaces on the main scale, twenty four spaces on the Vernier scale equal in extreme length twenty three double degrees. Thus the difference between the occupied by 2° on a main scale and the space of the Vernier scale is equal to one twenty-fourth of 2°. Read off directly from the scale the number of whole degrees between 0 on this scale and the 0 of the Vernier scale. For example, Zero on the scale has moved 28 whole degrees to the right of the 0 on the main scale
8.
Percentage
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In mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, %, or the abbreviations pct. pct, a percentage is a dimensionless number. For example, 45% is equal to 45⁄100,45,100, percentages are often used to express a proportionate part of a total. If 50% of the number of students in the class are male. If there are 1000 students, then 500 of them are male, an increase of $0.15 on a price of $2.50 is an increase by a fraction of 0. 15/2.50 =0.06. Expressed as a percentage, this is a 6% increase, while many percentage values are between 0 and 100, there is no mathematical restriction and percentages may take on other values. For example, it is common to refer to 111% or −35%, especially for percent changes, in Ancient Rome, long before the existence of the decimal system, computations were often made in fractions which were multiples of 1⁄100. For example, Augustus levied a tax of 1⁄100 on goods sold at auction known as centesima rerum venalium, computation with these fractions was equivalent to computing percentages. Many of these texts applied these methods to profit and loss, interest rates, by the 17th century it was standard to quote interest rates in hundredths. The term per cent is derived from the Latin per centum, the sign for per cent evolved by gradual contraction of the Italian term per cento, meaning for a hundred. The per was often abbreviated as p. and eventually disappeared entirely, the cento was contracted to two circles separated by a horizontal line, from which the modern % symbol is derived. The percent value is computed by multiplying the value of the ratio by 100. For example, to find 50 apples as a percentage of 1250 apples, first compute the ratio 50⁄1250 =0.04, and then multiply by 100 to obtain 4%. The percent value can also be found by multiplying first, so in this example the 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1250 to give 4%. To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decimals, and multiply them, for example, 50% of 40% is, 50⁄100 × 40⁄100 =0.50 ×0.40 =0.20 = 20⁄100 = 20%. It is not correct to divide by 100 and use the percent sign at the same time, whenever we talk about a percentage, it is important to specify what it is relative to, i. e. what is the total that corresponds to 100%. The following problem illustrates this point, in a certain college 60% of all students are female, and 10% of all students are computer science majors. If 5% of female students are computer science majors, what percentage of computer science majors are female and we are asked to compute the ratio of female computer science majors to all computer science majors
9.
Binary scaling
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Binary scaling is a computer programming technique used mainly by embedded C, DSP and assembler programmers to perform a pseudo floating point using integer arithmetic. Binary scaling is both faster and more accurate than directly using floating point instructions, however, care must be not to cause an arithmetic overflow. A position for the binary point is taken, and then subsequent arithmetic operations determine the binary point. Binary points obey the laws of exponentiation. To give an example, a way to use integer arithmetic to simulate floating point is to multiply the coefficients by 65536. Using binary scientific notation, this place the binary point at 1B16. For instance, to represent 1.2 and 5.6 floating point real numbers as 1B16 one multiplies them by 216, multiplying these together gives 28862059643 To convert it back to 1B16, divide it by 216. This gives 440400B16, which converted back to a floating point number gives 6.71999. The correct floating point result is 6.72, the scaling range here is for any number between 65535.9999 and −65536.0 with 16 bits to hold fractional quantities. Note that some computer architectures may restrict arithmetic to 32 bit results, in this case extreme care must be taken not to overflow the 32 bit register. For other number ranges the binary scale can be adjusted for optimum accuracy, the example above for a B16 multiplication is a simplified example. Re-scaling depends on both the B scale value and the word size, B16 is often used in 32 bit systems because it works simply by multiplying and dividing by 65536. Placing the binary point at 0 gives a range of −1.0 to 0.999999,1 gives a range of −2.0 to 1.9999992 gives a range of −4.0 to 3.999999 and so on. When using different B scalings the complete B scaling formula must be used, consider a 32 bit word size, and two variables, one with a B scaling of 2 and the other with a scaling of 4. 1.4 @ B2 is 1.4 * ==1.4 *2 ^29 == 0x2CCCCCCD Note that here the 1.4 values is very well represented with 30 fraction bits, a 32 bit floating-point number has 23 bits to store the fraction in. This is why B scaling is always more accurate than floating point of the word size. This is especially useful in integrators or repeated summing of small quantities where rounding error can be a subtle but very dangerous problem, now a larger number 15.2 at B4. 15.2 @ B4 is 15.2 * ==15.2 *2 ^27 == 0x7999999A Again the number of bits to store the fraction is 28 bits
10.
Byte
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The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of used to encode a single character of text in a computer. The size of the byte has historically been hardware dependent and no standards existed that mandated the size. The de-facto standard of eight bits is a convenient power of two permitting the values 0 through 255 for one byte, the international standard IEC 80000-13 codified this common meaning. Many types of applications use information representable in eight or fewer bits, the popularity of major commercial computing architectures has aided in the ubiquitous acceptance of the 8-bit size. The unit symbol for the byte was designated as the upper-case letter B by the IEC and IEEE in contrast to the bit, internationally, the unit octet, symbol o, explicitly denotes a sequence of eight bits, eliminating the ambiguity of the byte. It is a respelling of bite to avoid accidental mutation to bit. Early computers used a variety of four-bit binary coded decimal representations and these representations included alphanumeric characters and special graphical symbols. S. Government and universities during the 1960s, the prominence of the System/360 led to the ubiquitous adoption of the eight-bit storage size, while in detail the EBCDIC and ASCII encoding schemes are different. In the early 1960s, AT&T introduced digital telephony first on long-distance trunk lines and these used the eight-bit µ-law encoding. This large investment promised to reduce costs for eight-bit data. The development of microprocessors in the 1970s popularized this storage size. A four-bit quantity is called a nibble, also nybble. The term octet is used to specify a size of eight bits. It is used extensively in protocol definitions, historically, the term octad or octade was used to denote eight bits as well at least in Western Europe, however, this usage is no longer common. The exact origin of the term is unclear, but it can be found in British, Dutch, and German sources of the 1960s and 1970s, and throughout the documentation of Philips mainframe computers. The unit symbol for the byte is specified in IEC 80000-13, IEEE1541, in the International System of Quantities, B is the symbol of the bel, a unit of logarithmic power ratios named after Alexander Graham Bell, creating a conflict with the IEC specification. However, little danger of confusion exists, because the bel is a used unit
11.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane
12.
Lathe
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Lathes are used in woodturning, metalworking, metal spinning, thermal spraying, parts reclamation, and glass-working. Lathes can be used to shape pottery, the design being the potters wheel. Most suitably equipped metalworking lathes can also be used to produce most solids of revolution, plane surfaces, ornamental lathes can produce three-dimensional solids of incredible complexity. The workpiece is held in place by either one or two centers, at least one of which can typically be moved horizontally to accommodate varying workpiece lengths. Other work-holding methods include clamping the work about the axis of rotation using a chuck or collet, or to a faceplate, using clamps or dogs. Examples of objects that can be produced on a lathe include candlestick holders, gun barrels, cue sticks, table legs, bowls, baseball bats, musical instruments, crankshafts, and camshafts. The lathe is an ancient tool, dating at least to ancient Egypt and known to be used in Assyria, the lathe was very important to the Industrial Revolution. It is known as the mother of machine tools, as it was the first machine tool that lead to the invention of other machine tools, the origin of turning dates to around 1300 BCE when the Ancient Egyptians first developed a two-person lathe. One person would turn the work piece with a rope while the other used a sharp tool to cut shapes in the wood. Ancient Rome improved the Egyptian design with the addition of a turning bow, in the Middle Ages a pedal replaced hand-operated turning, allowing a single person to rotate the piece while working with both hands. The pedal was usually connected to a pole, often a straight-grained sapling, the system today is called the spring pole lathe. Spring pole lathes were in use into the early 20th century. An important early lathe in the UK was the boring machine that was installed in 1772 in the Royal Arsenal in Woolwich. It was horse-powered and allowed for the production of more accurate. One of the key characteristics of this machine was that the workpiece was turning as opposed to the tool, henry Maudslay who later developed many improvements to the lathe worked at the Royal Arsenal from 1783 being exposed to this machine in the Verbruggen workshop. During the Industrial Revolution, mechanized power generated by water wheels or steam engines was transmitted to the lathe via line shafting, allowing faster and easier work, metalworking lathes evolved into heavier machines with thicker, more rigid parts. Between the late 19th and mid-20th centuries, individual electric motors at each lathe replaced line shafting as the power source, today manually controlled and CNC lathes coexist in the manufacturing industries. A lathe may or may not have legs, which sit on the floor, a lathe may be small and sit on a workbench or table, not requiring a stand
13.
David Gregory (mathematician)
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David Gregory FRS was a Scottish mathematician and astronomer. He was professor of mathematics at the University of Edinburgh, Savilian Professor of Astronomy at the University of Oxford, and a commentator on Isaac Newtons Principia. The fourth of the fifteen children of David Gregorie, a doctor from Kinnairdy, Banffshire, after his university studies, still only 16 years old, Gregory visited several countries on the continent, including the Netherlands and France, and did not return to Scotland until 1683. The same year he was elected to be a Fellow of the Royal Society, in 1692, he was elected a Fellow of Balliol College, Oxford. At the age of 24 he was appointed professor of mathematics at the University of Edinburgh, during 1694, he spent several days with Isaac Newton, discussing a second edition of Newtons Principia, but these plans came to nothing. At the Union of 1707, he was given the responsibility of reorganising the Scottish Mint and he was an uncle of philosopher Thomas Reid. Gregory and his wife, Elizabeth Oliphant, had nine children, on his death in Maidenhead, Berkshire he was buried in Maidenhead churchyard. OConnor, John J. Robertson, Edmund F. David Gregory, MacTutor History of Mathematics archive, significant Scots, David Gregory Papers of David Gregory David Gregroy Catoptricæ et dioptricæ sphæricæ elementa - digital facsimile from the Linda Hall Library
14.
Perimeter
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A perimeter is a path that surrounds a two-dimensional shape. The term may be used either for the path or its length—it can be thought of as the length of the outline of a shape, the perimeter of a circle or ellipse is called its circumference. Calculating the perimeter has several practical applications, a calculated perimeter is the length of fence required to surround a yard or garden. The perimeter of a wheel describes how far it will roll in one revolution, similarly, the amount of string wound around a spool is related to the spools perimeter. The perimeter is the distance around a shape, perimeters for more general shapes can be calculated, as any path, with ∫0 L d s, where L is the length of the path and d s is an infinitesimal line element. Both of these must be replaced with by algebraic forms in order to be practically calculated, the first mathematician known to have used this kind of reasoning is Archimedes, who approximated the perimeter of a circle by surrounding it with regular polygons. The perimeter of a polygon equals the sum of the lengths of its sides, in particular, the perimeter of a rectangle of width w and length ℓ equals 2 w +2 ℓ. An equilateral polygon is a polygon which has all sides of the same length, to calculate the perimeter of an equilateral polygon, one must multiply the common length of the sides by the number of sides. A regular polygon may be characterized by the number of its sides and by its circumradius, that is to say, the length of its sides can be calculated using trigonometry. If R is a regular polygons radius and n is the number of its sides, a splitter of a triangle is a cevian that divides the perimeter into two equal lengths, this common length being called the semiperimeter of the triangle. The three splitters of a triangle all intersect each other at the Nagel point of the triangle, a cleaver of a triangle is a segment from the midpoint of a side of a triangle to the opposite side such that the perimeter is divided into two equal lengths. The three cleavers of a triangle all intersect each other at the triangles Spieker center, the perimeter of a circle, often called the circumference, is proportional to its diameter and its radius. That is to say, there exists a constant number pi, π, such that if P is the perimeter and D its diameter then. In terms of the r of the circle, this formula becomes. To calculate a circles perimeter, knowledge of its radius or diameter, the problem is that π is not rational, nor is it algebraic. So, obtaining an approximation of π is important in the calculation. The computation of the digits of π is relevant to many fields, such as mathematical analysis, algorithmics, the perimeter and the area are two main measures of geometric figures. Confusing them is an error, as well as believing that the greater one of them is
15.
Circumference
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The circumference of a closed curve or circular object is the linear distance around its edge. The circumference of a circle is of importance in geometry and trigonometry. Informally circumference may also refer to the edge rather than to the length of the edge. The circumference of a circle is the distance around it, the term is used when measuring physical objects, as well as when considering abstract geometric forms. The circumference of a circle relates to one of the most important mathematical constants in all of mathematics and this constant, pi, is represented by the Greek letter π. The numerical value of π is 3.141592653589793, pi is defined as the ratio of a circles circumference C to its diameter d, π = C d Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference, the use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. The constant ratio of circumference to radius C / r =2 π also has uses in mathematics, engineering. These uses include but are not limited to radians, computer programming, the Greek letter τ is sometimes used to represent this constant, but is not generally accepted as proper notation. The circumference of an ellipse can be expressed in terms of the elliptic integral of the second kind. In graph theory the circumference of a graph refers to the longest cycle contained in that graph, arc length Area Caccioppoli set Isoperimetric inequality Pythagorean theorem Volume Numericana - Circumference of an ellipse Circumference of a circle With interactive applet and animation
16.
William Oughtred
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William Oughtred was an English mathematician and Anglican minister. Oughtred also introduced the × symbol for multiplication as well as the abbreviations sin, Oughtred was born at Eton in Buckinghamshire, and educated there and at Kings College, Cambridge, of which he became fellow. Being admitted to holy orders, he left the University of Cambridge about 1603, for a living at Shalford, he was presented in 1610 to the rectory of Albury, near Guildford in Surrey, where he settled. He married Christsgift Caryll, of the Caryll family of Tangley Hall at Wonersh, of which Lady Elizabeth Aungier, wife of Simon Caryll 1607-1619, was matriarch, about 1628 he was appointed by the Earl of Arundel to instruct his son in mathematics. He corresponded with some of the most eminent scholars of his time, including William Alabaster, Sir Charles Cavendish and he kept up regular contacts with Gresham College, where he knew Henry Briggs and Gunter. He offered free tuition to pupils, who included Richard Delamain. Seth Ward resided with Oughtred for six months to learn mathematics, and the physician Charles Scarburgh also stayed at Albury, John Wallis. Another Albury pupil was Robert Wood, who helped him get the Clavis through the press, the invention of the slide rule involved Oughtred in a priority dispute with Delamain. They also disagreed on pedagogy in mathematics, with Oughtred arguing that theory should precede practice and he remained rector until his death in 1660 at Albury, a month after the restoration of Charles II. Oughtred had an interest in alchemy and astrology, the testimony for his occult activities is quite slender, but there has been an accretion to his reputation based on his contemporaries. According to John Aubrey, he was not entirely sceptical about astrology, William Lilly, an eminent astrologer, claimed in his autobiography to have intervened on behalf of Oughtred to prevent his ejection by Parliament in 1646. In fact Oughtred was protected at this time by Bulstrode Whitelocke, Aubrey states that he was also defended by Sir Richard Onslow. It was used by George Wharton in publishing The Cabal of the Twelve Houses astrological by Morinus in 1659 and he expressed millenarian views to John Evelyn, as recorded in Evelyns Diary, entry for 28 August 1655. Oughtreds name is remembered in the Oughtred Society, a group formed in the United States in 1991 for collectors of slide rules and it produces the twice-yearly Journal of the Oughtred Society and holds meetings and auctions for its members. He published, among other works, Clavis Mathematicae, in 1631. It became a classic, reprinted in editions, and used by Wallis. It was not ambitious in scope, but an epitome aiming to represent current knowledge of algebra concisely, the book became popular around 15 years later, as mathematics took a greater role in higher education. Wallis wrote the introduction to his 1652 edition, and used it to publicise his skill as cryptographer, in another, Oughtred promoted the talents of Wren
17.
Wales
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Wales is a country that is part of the United Kingdom and the island of Great Britain. It is bordered by England to the east, the Irish Sea to the north and west, and it had a population in 2011 of 3,063,456 and has a total area of 20,779 km2. Wales has over 1,680 miles of coastline and is mountainous, with its higher peaks in the north and central areas, including Snowdon. The country lies within the temperate zone and has a changeable. Welsh national identity emerged among the Celtic Britons after the Roman withdrawal from Britain in the 5th century, Llywelyn ap Gruffudds death in 1282 marked the completion of Edward I of Englands conquest of Wales, though Owain Glyndŵr briefly restored independence to Wales in the early 15th century. The whole of Wales was annexed by England and incorporated within the English legal system under the Laws in Wales Acts 1535–1542, distinctive Welsh politics developed in the 19th century. Welsh Liberalism, exemplified in the early 20th century by Lloyd George, was displaced by the growth of socialism, Welsh national feeling grew over the century, Plaid Cymru was formed in 1925 and the Welsh Language Society in 1962. Established under the Government of Wales Act 1998, the National Assembly for Wales holds responsibility for a range of devolved policy matters, two-thirds of the population live in south Wales, mainly in and around Cardiff, Swansea and Newport, and in the nearby valleys. Now that the countrys traditional extractive and heavy industries have gone or are in decline, Wales economy depends on the sector, light and service industries. Wales 2010 gross value added was £45.5 billion, over 560,000 Welsh language speakers live in Wales, and the language is spoken by a majority of the population in parts of the north and west. From the late 19th century onwards, Wales acquired its popular image as the land of song, Rugby union is seen as a symbol of Welsh identity and an expression of national consciousness. The Old English-speaking Anglo-Saxons came to use the term Wælisc when referring to the Celtic Britons in particular, the modern names for some Continental European lands and peoples have a similar etymology. The modern Welsh name for themselves is Cymry, and Cymru is the Welsh name for Wales and these words are descended from the Brythonic word combrogi, meaning fellow-countrymen. The use of the word Cymry as a self-designation derives from the location in the post-Roman Era of the Welsh people in modern Wales as well as in northern England and southern Scotland. It emphasised that the Welsh in modern Wales and in the Hen Ogledd were one people, in particular, the term was not applied to the Cornish or the Breton peoples, who are of similar heritage, culture, and language to the Welsh. The word came into use as a self-description probably before the 7th century and it is attested in a praise poem to Cadwallon ap Cadfan c. 633. Thereafter Cymry prevailed as a reference to the Welsh, until c.1560 the word was spelt Kymry or Cymry, regardless of whether it referred to the people or their homeland. The Latinised forms of names, Cambrian, Cambric and Cambria, survive as lesser-used alternative names for Wales, Welsh
18.
William Jones (mathematician)
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William Jones, FRS was a Welsh mathematician, most noted for his proposal for the use of the symbol π to represent the ratio of the circumference of a circle to its diameter. He was a friend of Sir Isaac Newton and Sir Edmund Halley. In November,1711 he became a Fellow of the Royal Society, William Jones was born the son of Siôn Siôr and Elizabeth Rowland in the parish of Llanfihangel Trer Beirdd, about 4 miles west of Benllech on the Isle of Anglesey. He owed his career partly to the patronage of the distinguished Bulkeley family of north Wales. In this work he applied mathematics to navigation, studying methods of calculating position at sea. After his voyages were over he became a teacher in London, both in coffee houses and as a private tutor to the son of the future Earl of Macclesfield. He also held a number of undemanding posts in government offices with the help of his former pupils, Jones published Synopsis Palmariorum Matheseos in 1706, a work which was intended for beginners and which included theorems on differential calculus and infinite series. This used π as an abbreviation for perimeter and his 1711 work Analysis per quantitatum series, fluxiones ac differentias introduced the dot notation for differentiation in calculus. In 1731 he published Discourses of the Natural Philosophy of the Elements and he was noticed and befriended by two of Britain’s foremost mathematicians – Edmund Halley and Sir Isaac Newton – and was elected a fellow of the Royal Society in 1711. His son, also named William Jones and born in 1746, was a renowned philologist who established links between Latin, Greek and Sanskrit, leading to the concept of the Indo-European language group. William Jones and other important Welsh mathematicians William Jones and his Circle, The Man who invented Pi Pi Day 2015, meet the man who invented π
19.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
20.
Fred Hoyle
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He spent most of his working life at the Institute of Astronomy at Cambridge and served as its director for six years. He was a writer of fiction, and co-authored twelve books with his son. Hoyle was born near Bingley in Gilstead, West Riding of Yorkshire and his father, Ben Hoyle, worked in the wool trade in Bradford. His mother, Mabel Pickard, had studied music at the Royal College of Music in London, Hoyle was educated at Bingley Grammar School and read mathematics at Emmanuel College, Cambridge. In late 1940, Hoyle left Cambridge to go to Portsmouth to work for the Admiralty on radar research and he was also put in charge of countermeasures against the radar guided guns found on the Graf Spee. Britains radar project employed more personnel than the Manhattan project, and was probably the inspiration for the large British project in The Black Cloud, two key colleagues in this war work were Hermann Bondi and Thomas Gold, and the three had many and deep discussions on cosmology. The radar work paid for a couple of trips to North America and he had an intuition at the time I will make a name for myself if this works out. Eventually his prescient and ground breaking paper came out and he also formed a group at Cambridge exploring Stellar nucleosynthesis in ordinary stars and was bothered by the paucity of stellar carbon production in existing models. He noticed that one of the existing processes would be made a billion times more if the carbon-12 nucleus had a resonance at 7.7 MeV. After the war, in 1945, Hoyle returned to Cambridge University, Hoyles Cambridge years, 1945–1973, saw him rise to the top of world astrophysics theory, on the basis of a startling originality of ideas covering a very wide range of topics. In 1958, Hoyle was appointed to the illustrious Plumian Professor of Astronomy, in 1971 he was invited to deliver the MacMillan Memorial Lecture to the Institution of Engineers and Shipbuilders in Scotland. He chose the subject Astronomical Instruments and their Construction, after his leaving Cambridge, Hoyle wrote many popular science and science fiction books, as well as presenting lectures around the world. Part of the motivation for this was simply to provide a means of support, Hoyle was still a member of the joint policy committee, during the planning stage for the 150-inch Anglo-Australian Telescope at Siding Spring Observatory in New South Wales. He became chairman of the Anglo-Australian Telescope board in 1973, and presided at its inauguration in 1974 by Charles, on 24 November 1997, while hiking across moorlands in west Yorkshire, near his childhood home in Gilstead, Hoyle fell down into a steep ravine called Shipley Glen. Roughly twelve hours later, Hoyle was found by a search dog and he was hospitalized for two months with pneumonia, kidney problems as a result of hypothermia, and a smashed shoulder, while he ever afterwards suffered from memory and mental agility problems. In 2001, he suffered a series of strokes and died in Bournemouth on 20 August, Fred Hoyle authored the first two research papers ever published on the synthesis of the chemical elements heavier than helium by nuclear reactions in stars. This idea would later be called the e Process, Hoyles second foundational nucleosynthesis publication showed that the elements between carbon and iron cannot be synthesized by such equilibrium processes. Hoyle attributed those elements to specific nuclear fusion reactions between abundant constituents in concentric shells of evolved massive, pre-supernova stars and this startlingly modern picture is the accepted paradigm today for the supernova nucleosynthesis of these primary elements
21.
HP 39gII
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HP 39/40 series are graphing calculators from Hewlett-Packard, the successors of HP 38G. The series consists of six calculators, which all have algebraic entry modes, all calculators in this series are aimed at high school level students and are characterized by their ability to download APLETs or E-lessons. These are programs of varying complexity which are intended to be used in the classroom to enhance the learning of mathematics by the graphical and/or numerical exploration of concepts. The HP 39G was released in 2000, basic characteristics, CPU,4 MHz Yorke Communication, Proprietary infrared, Serial RS-232. Memory,256 KB Screen resolution, 131×64 pixels Includes a hard cover Limited symbolic equation functionality, HP 40G was released in 2000 in parallel with the HP 39G. The HP 40Gs operating system is identical to the HP 39G, differences detected in hardware during start-up trigger the differences in software functionality. The hardware is identical to the HP 49G/39G series, in contrast to the 39G, it integrates the same computer algebra system also found in the HP 49G, HP CAS. Unlike its bigger brothers, the HP 40G has no flags to set/mis-set resulting in a better behaved calculator for straightforward math analysis, additionally the HP 40G does not have infrared connectivity, and is limited to 27 variables. A list-based solver, and other handicaps make this simple-to-use calculator less adapted to end use. The HP 40G is not allowed for use in many standardized tests including the ACT, basic characteristics, Identical to HP 39G except, Communication, No infrared communication, Serial RS-232. Software, Includes an equation writer and advanced CAS, the hp 39g+ was released in September,2003. Basic characteristics, CPU,75 MHz Samsung S3C2410X Communication, USB port, memory,256 KB Power, 3×AAA as main power, CR2032 for memory backup Screen resolution, 131×64 pixels Does not come with a hard cover Limited symbolic equation functionality. The CAS component of the HP 40Gs operating system appears to have been totally removed, the HP 39gs was released in June 2006. Basic characteristics, CPU,75 MHz Samsung S3C2410A Communication, USB port, IrDA, Power, 4×AAA as main power, CR2032 for memory backup Screen resolution, 131×64 pixels Includes a hard cover Limited symbolic equation functionality. Flash memory to allow potential future upgrades/bug fixes, the HP 40gs was released in mid-2006. Basic characteristics, CPU,75 MHz Samsung S3C2410A Identical to HP 39gs except, Communication, software, Includes an equation writer and advanced CAS. This is necessary to accommodate the CAS software, the HP 39gII was released in October 2011. It is built around an 80 MHz Freescale STMP3770 processor with ARM926EJ-S core, the high-resolution monochrome gray-scale LCD provides 256x128 pixels
22.
HP Prime
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The HP Prime is a graphing calculator manufactured by Hewlett-Packard. It contains features common in smartphones, with a touchscreen and apps available to put onto it, there are two sides to the calculator, a numeric home screen and a computer algebra system homescreen. The calculator can quickly switch between the two, unlike its competitors, which either have a CAS model or a non-CAS model. The CAS is based on the free and open-source Xcas/Giac 1.1.2 engine by Bernard Parisse, the calculator has a 1500 mAh battery, which is expected to last up to 15 hours on a single charge. Prime emulator PC software is available as well and it has also, for now, taken the title for the worlds smallest CAS calculator at 18. 23×8.58 cm and is also the thinnest CAS calculator available currently, with a thickness of only 1.39 cm. The HP Prime has a feature called Exam Mode and this enables various features of the calculator to be selectively disabled for a specific time, from 15 minutes to 8 hours. This can be done manually within the menus, or by using a computer with HPs connectivity software. LEDs on the top of the calculator blink to let the instructor see that the calculator is in this mode, despite this feature, the Prime is still prohibited in many examinations, such as the USs ACT college-entry test. It is however starting to be accepted in other examinations, like those run by the Dutch CvTE, the HP Primes non-CAS home-screen supports textbook, algebraic and 128-level RPN entry logic. The calculator supports programming in a new, Pascal-like programming language now named HP PPL that also supports creating apps and this is based on a language introduced on the HP 38G and built on in subsequent models. The first production model in 2013 reports its hardware revision as A and this model does not support wireless connectivity, unit-to-unit USB communication, or data streaming. The second production model reports its hardware revision as C and it was introduced in May 2014. This model supports features lacking in the first production model, namely wireless connectivity, unit-to-unit USB communication, the wireless kit includes a base station connected to a PC and wireless modules to connect to up to 30 HP Prime calculators for use in a classroom. The third production model, which was introduced in August 2016, has a color scheme with darker blue. It still carries the model number G8X92AA and reports a hardware revision of C, but the package shows a 2016 copyright. org Reverse engineering the HP Prime USB protocol
23.
HP 49/50 series
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The HP 49/50 series are Hewlett-Packard manufactured graphing calculators. They are the successors of the popular HP48 series, there are five calculators in the 49/50 series of HP graphing calculators. Released in August 1999, the HP 49G calculator was the first HP unit to break from the more subdued coloration. In addition to having a blue color, the keyboard material was rubber. In addition, it lacked a large ↵ Enter key which was seen by many as the characteristic of an HP calculator. These changes were disliked by many traditional HP calculator users, the 49G was the first HP calculator to use flash memory and have an upgradable firmware. In addition, it had a hard sliding case as opposed to the soft pouches supplied with the HP48 series, almost the same hardware is also used by the HP 39G and HP 40G. The last officially supported firmware update for the 49G calculator was 1.18, the final firmware version was 1. 19-6. Several firmware versions for the successor hp 49g+ and HP 50g calculators have also released in builds intended for PC emulation software that lacked full utilization of the successors ARM CPU. Until at least firmware version 2.09, those emulator builds could be installed on the original HP 49G, in 2003, the CAS source code of the 49G firmware was released under the LGPL. In addition, this included an interactive geometry program and some commands to allow compatibility with certain programs written for the newer 49g+ calculator. Due to licensing restrictions, the recompiled firmware cannot be redistributed, in August 2003, Hewlett-Packard released the hp 49g+. This unit had metallic gold coloration and was compatible with the HP 49G. It was designed and manufactured by Kinpo Electronics for HP, the calculator system did not run directly on the new ARM processor, but rather on an emulation layer for the older Saturn processors found in previous HP calculators. Despite the emulation, the 49g+ was still faster than any older model of HP calculator. The speed increase over the HP 49G is around 3-7 times depending on the task and it is even possible to run programs written for the ARM processor thus bypassing the emulation layer completely. A port of the GNU C compiler is also available, the hp 48gII, which was announced on 2003-10-20, was not a replacement for the HP48 series as its name suggested. Rather it was a 49g+, also with an ARM processor, but with reduced memory, no expansion via an SD memory card, lower speed, a smaller screen
24.
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
25.
Tau
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Tau is the 19th letter of the Greek alphabet. In the system of Greek numerals it has a value of 300, the name in English is pronounced /taʊ/ or /tɔː/, but in modern Greek it is. This is because the pronunciation of the combination of Greek letters αυ has changed from ancient to modern times one of to either or. Tau was derived from the Phoenician letter taw, letters that arose from tau include Roman T and Cyrillic Te. The letter occupies the Unicode slots U+03C4 and U+03A4, in HTML, they can be produced with named entities, decimal references, or hexadecimal references. Kendall tau rank correlation coefficient in statistics Stopping time in stochastic processes Tau, in Biblical times, the taw was put on men to distinguish those who lamented sin, although newer versions of the Bible have replaced the ancient term taw with mark or signature. Tau in the Court of Seven Vowels, contains a reference to the cross attribution, stauros the vile engine is called, and it derives its vile name from him. Now, with all these crimes upon him, does he not deserve death, nay, francis love for it, symbol of the redemption and of the Cross. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style
26.
Electromagnetic coil
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An electromagnetic coil is an electrical conductor such as a wire in the shape of a coil, spiral or helix. A current through any conductor creates a magnetic field around the conductor due to Amperes law. The advantage of using the shape is that it increases the strength of magnetic field produced by a given current. The magnetic fields generated by the turns of wire all pass through the center of the coil. The more turns of wire, the stronger the field produced, conversely, a changing external magnetic flux induces a voltage in a conductor such as a wire, due to Faradays law of induction. The induced voltage can be increased by winding the wire into a coil, the direction of the magnetic field produced by a coil can be determined by the right hand grip rule. The end of a core from which the field lines emerge is defined to be the North pole. There are many different types of coils used in electric and electronic equipment, the wire or conductor which constitutes the coil is called the winding. The hole in the center of the coil is called the area or magnetic axis. Each loop of wire is called a turn, in windings in which the turns touch, the wire must be insulated with a coating of nonconductive insulation such as plastic or enamel to prevent the current from passing between the wire turns. The winding is often wrapped around a form made of plastic or other material to hold it in place. The ends of the wire are brought out and attached to an external circuit, windings may have additional electrical connections along their length, these are called taps. A winding which has a tap in the center of its length is called center-tapped. Coils can have more than one winding, insulated electrically from each other, when there are two or more windings around a common magnetic axis, the windings are said to be inductively coupled or magnetically coupled. A time-varying current through one winding will create a magnetic field which passes through the other winding. The winding to which current is applied, which creates the magnetic field, is called the primary winding, the other windings are called secondary windings. Many electromagnetic coils have a core, a piece of ferromagnetic material like iron in the center to increase the magnetic field. The current through the coil magnetizes the iron, and the field of the material adds to the field produced by the wire
27.
Winding number
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The term winding number may also refer to the rotation number of an iterated map. In mathematics, the number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point. The winding number depends on the orientation of the curve, and is if the curve travels around the point clockwise. Suppose we are given a closed, oriented curve in the xy plane and we can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the number of counterclockwise turns that the object makes around the origin. When counting the number of turns, counterclockwise motion counts as positive. For example, if the object first circles the four times counterclockwise. Using this scheme, a curve that does not travel around the origin at all has winding number zero, therefore, the winding number of a curve may be any integer. The following pictures show curves with winding numbers between −2 and 3, A curve in the xy plane can be defined by parametric equations, x = x and y = y for 0 ≤ t ≤1. If we think of the t as time, then these equations specify the motion of an object in the plane between t =0 and t =1. The path of motion is a curve as long as the functions x and y are continuous. This curve is closed as long as the position of the object is the same at t =0 and t =1 and we can define the winding number of such a curve using the polar coordinate system. Assuming the curve does not pass through the origin, we can rewrite the parametric equations in polar form, the functions r and θ are required to be continuous, with r >0. Because the initial and final positions are the same, θ and θ must differ by a multiple of 2π. This integer is the number, winding number = θ − θ2 π. This defines the number of a curve around the origin in the xy plane. By translating the coordinate system, we can extend this definition to include winding numbers around any point p. Winding number is defined in different ways in various parts of mathematics. Any curve partitions the plane into several connected regions, one of which is unbounded, the winding numbers of the curve around two points in the same region are equal
28.
External ray
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An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set. Although this curve is only rarely a half-line it is called a ray because it is an image of a ray, External rays are used in complex analysis, particularly in complex dynamics and geometric function theory. External rays were introduced in Douady and Hubbards study of the Mandelbrot set External rays of Julia sets on dynamical plane are called dynamic rays. External rays of the Mandelbrot set on parameter plane are called parameter rays, in other words the external rays define vertical foliation which is orthogonal to horizontal foliation defined by the level sets of potential. Let Ψ c be the mapping from the complement of the unit disk D ¯ to the complement of the filled Julia set K c. Eight parameter rays landing at this parameter are drawn in black.9, john Milnor, Dynamics in One Complex Variable, Third Edition, Princeton University Press,2006, ISBN 0-691-12488-4 Wolf Jung, Homeomorphisms on Edges of the Mandelbrot Set. Ph. D.1 videos by ImpoliteFruit Milan Va. Mandelbrot set drawing
29.
Pie chart
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A pie chart is a circular statistical graphic which is divided into slices to illustrate numerical proportion. In a pie chart, the arc length of each slice, is proportional to the quantity it represents, while it is named for its resemblance to a pie which has been sliced, there are variations on the way it can be presented. The earliest known pie chart is generally credited to William Playfairs Statistical Breviary of 1801, Pie charts are very widely used in the business world and the mass media. Pie charts can be replaced in most cases by other such as the bar chart. The earliest known pie chart is generally credited to William Playfairs Statistical Breviary of 1801, playfair presented an illustration, which contained a series of pie charts. One of those charts depicting the proportions of the Turkish Empire located in Asia, Europe and this invention was not widely used at first, The French engineer Charles Joseph Minard was one of the first to use pie charts in 1858, in particular in maps. Minards map,1858 used pie charts to represent the cattle sent from all around France for consumption in Paris, playfair thought that pie charts were in need of a third dimension to add additional information. It has been said that Florence Nightingale invented it, though in fact she just popularised it, a 3d pie cake, or perspective pie cake, is used to give the chart a 3D look. The use of superfluous dimensions not used to display the data of interest is discouraged for charts in general, a doughnut chart is a variant of the pie chart, with a blank center allowing for additional information about the data as a whole to be included. A chart with one or more separated from the rest of the disk is known as an exploded pie chart. This effect is used to highlight a sector, or to highlight smaller segments of the chart with small proportions. The polar area diagram is similar to a pie chart, except sectors are equal angles. The polar area diagram is used to plot cyclic phenomena, for example, if the count of deaths in each month for a year are to be plotted then there will be 12 sectors all with the same angle of 30 degrees each. The radius of each sector would be proportional to the root of the death count for the month. Léon Lalanne later used a diagram to show the frequency of wind directions around compass points in 1843. The wind rose is used by meteorologists. Nightingale published her rose diagram in 1858, the name coxcomb is sometimes used erroneously, this was the name Nightingale used to refer to a book containing the diagrams rather than the diagrams themselves. A ring chart, also known as a sunburst chart or a pie chart, is used to visualize hierarchical data
30.
Kinematics
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Kinematics as a field of study is often referred to as the geometry of motion and as such may be seen as a branch of mathematics. The study of the influence of forces acting on masses falls within the purview of kinetics, for further details, see analytical dynamics. Kinematics is used in astrophysics to describe the motion of celestial bodies, in mechanical engineering, robotics, and biomechanics kinematics is used to describe the motion of systems composed of joined parts such as an engine, a robotic arm or the human skeleton. Kinematic analysis is the process of measuring the quantities used to describe motion. In addition, kinematics applies geometry to the study of the mechanical advantage of a mechanical system or mechanism. The term kinematic is the English version of A. M, ampères cinématique, which he constructed from the Greek κίνημα kinema, itself derived from κινεῖν kinein. Kinematic and cinématique are related to the French word cinéma, particle kinematics is the study of the trajectory of a particle. The position of a particle is defined to be the vector from the origin of a coordinate frame to the particle. If the tower is 50 m high, then the vector to the top of the tower is r=. In the most general case, a coordinate system is used to define the position of a particle. However, if the particle is constrained to move in a surface, all observations in physics are incomplete without those observations being described with respect to a reference frame. The position vector of a particle is a vector drawn from the origin of the frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin, the magnitude of the position vector |P| gives the distance between the point P and the origin. | P | = x P2 + y P2 + z P2, the direction cosines of the position vector provide a quantitative measure of direction. It is important to note that the vector of a particle isnt unique. The position vector of a particle is different relative to different frames of reference. The velocity of a particle is a quantity that describes the direction of motion. More mathematically, the rate of change of the vector of a point
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Mathematical model
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A mathematical model is a description of a system using mathematical concepts and language. The process of developing a model is termed mathematical modeling. Mathematical models are used in the sciences and engineering disciplines. Physicists, engineers, statisticians, operations research analysts, and economists use mathematical models most extensively, a model may help to explain a system and to study the effects of different components, and to make predictions about behaviour. Mathematical models can take many forms, including systems, statistical models, differential equations. These and other types of models can overlap, with a model involving a variety of abstract structures. In general, mathematical models may include logical models, in many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed, in the physical sciences, the traditional mathematical model contains four major elements. These are Governing equations Defining equations Constitutive equations Constraints Mathematical models are composed of relationships. Relationships can be described by operators, such as operators, functions, differential operators. Variables are abstractions of system parameters of interest, that can be quantified, a model is considered to be nonlinear otherwise. The definition of linearity and nonlinearity is dependent on context, for example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, an equation is said to be linear if it can be written with linear differential operators. In a mathematical programming model, if the functions and constraints are represented entirely by linear equations. If one or more of the functions or constraints are represented with a nonlinear equation. Nonlinearity, even in simple systems, is often associated with phenomena such as chaos. Although there are exceptions, nonlinear systems and models tend to be difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be if one is trying to study aspects such as irreversibility
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Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
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Quaternion
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In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843, a feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a space or equivalently as the quotient of two vectors. Quaternions are generally represented in the form, a + bi + cj + dk where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units. In practical applications, they can be used other methods, such as Euler angles and rotation matrices, or as an alternative to them. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, in fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of quaternions is often denoted by H, or in blackboard bold by H and it can also be given by the Clifford algebra classifications Cℓ0,2 ≅ Cℓ03,0. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. The unit quaternions can be thought of as a choice of a structure on the 3-sphere S3 that gives the group Spin. Quaternion algebra was introduced by Hamilton in 1843, carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900. Hamilton knew that the numbers could be interpreted as points in a plane. Points in space can be represented by their coordinates, which are triples of numbers, however, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space. The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, as he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, i2 = j2 = k2 = ijk = −1, into the stone of Brougham Bridge as he paused on it. On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves and this letter was later published in the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. xxv, pp 489–95. In the letter, Hamilton states, And here there dawned on me the notion that we must admit, in some sense, an electric circuit seemed to close, and a spark flashed forth. Hamilton called a quadruple with these rules of multiplication a quaternion, Hamiltons treatment is more geometric than the modern approach, which emphasizes quaternions algebraic properties
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Complex plane
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In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the part of a complex number represented by a displacement along the x-axis. The concept of the plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors, in particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is known as the Argand plane. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In this customary notation the number z corresponds to the point in the Cartesian plane. In the Cartesian plane the point can also be represented in coordinates as = =. In the Cartesian plane it may be assumed that the arctangent takes values from −π/2 to π/2, and some care must be taken to define the real arctangent function for points when x ≤0. Here |z| is the value or modulus of the complex number z, θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π. Notice that without the constraint on the range of θ, the argument of z is multi-valued, because the exponential function is periodic. Thus, if θ is one value of arg, the values are given by arg = θ + 2nπ. The theory of contour integration comprises a part of complex analysis. In this context the direction of travel around a curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the direction is counterclockwise. Almost all of complex analysis is concerned with complex functions – that is, here it is customary to speak of the domain of f as lying in the z-plane, while referring to the range or image of f as a set of points in the w-plane. In symbols we write z = x + i y, f = w = u + i v and it can be useful to think of the complex plane as if it occupied the surface of a sphere. We can establish a correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows
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Polar coordinate
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The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, the concepts of angle and radius were already used by ancient peoples of the first millennium BC. In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle, the Greek work, however, did not extend to a full coordinate system. From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca —and its distance—from any location on the Earth, from the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. There are various accounts of the introduction of polar coordinates as part of a coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidges Origin of Polar Coordinates, grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs, in Method of Fluxions, Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the Seventh Manner, For Spirals, and nine other coordinate systems. In the journal Acta Eruditorum, Jacob Bernoulli used a system with a point on a line, called the pole, Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoullis work extended to finding the radius of curvature of curves expressed in these coordinates, the actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacocks 1816 translation of Lacroixs Differential and Integral Calculus, alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them. The radial coordinate is often denoted by r or ρ, the angular coordinate is specified as ϕ by ISO standard 31-11. Angles in polar notation are generally expressed in degrees or radians. Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics, in many contexts, a positive angular coordinate means that the angle ϕ is measured counterclockwise from the axis. In mathematical literature, the axis is often drawn horizontal. Adding any number of turns to the angular coordinate does not change the corresponding direction. Also, a radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the point can be expressed with an infinite number of different polar coordinates or
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Frank Morley
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Frank Morley was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry. Among his mathematical accomplishments was the discovery and proof of the celebrated Morleys trisector theorem in plane geometry. He led 50 Ph. D. s to their degrees, Morley was born in the town of Woodbridge in Suffolk, England. His parents were Elizabeth Muskett and Joseph Roberts Morley, Quakers who ran a china shop, after being educated at Woodbridge School, Morley went on to Kings College, Cambridge. In 1887 Morley moved to Pennsylvania and he taught at Haverford College until 1900, when he became chairman of the mathematics department at Johns Hopkins University. His publications include Elementary Treatise on the Theory of Functions, with James Harkness and he was President of the American Mathematical Society from 1919 to 1920 and was the editor of the American Journal of Mathematics from 1900 to 1921. He was a speaker at the International Congress of Mathematicians in 1912 at Cambridge, in 1924 at Toronto. In 1933 he and his son Frank Vigor published the stimulating volume, the book develops complex numbers as a tool for geometry and function theory. Some non-standard terminology is used such as base-circle for unit circle and he was a strong chess player and once beat world champion Emanuel Lasker in a game of chess. He died in Baltimore, Maryland at age 77 and his sons are novelist Christopher Morley, Pulitzer Prize winner Felix Morley, and his son, also mathematician, Frank Vigor Morley. Archibald, A Semicentennial History of the American Mathematical Society, Chapter 15, The Presidents, pp. 194–201, includes bibliography of Morleys papers. Works by or about Frank Morley at Internet Archive OConnor, John J. Robertson, Edmund F. Frank Morley, MacTutor History of Mathematics archive, Frank Morley at the Mathematics Genealogy Project Clark Kimberling, Frank Morley geometer
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Versor
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Versors are an algebraic parametrisation of rotations. In classical quaternion theory a versor is a quaternion of norm one. Each versor has the form q = exp = cos a + r sin a, r 2 = −1, a ∈, in case a = π/2, the versor is termed a right versor. The corresponding 3-dimensional rotation has the angle 2a about the axis r in axis–angle representation, the word is derived from Latin versare = to turn with the suffix -or forming a noun from the verb. It was introduced by William Rowan Hamilton in the context of his quaternion theory, for historical reasons, it sometimes is used synonymously with a unit quaternion without a reference to rotations. In the quaternion algebra a versor q = exp will rotate any quaternion v through the product map v ↦ q v q −1 such that the scalar part of v is preserved. If this scalar part is zero, i. e. v is a Euclidean vector in three dimensions, then the formula above defines the rotation through the angle 2a around the vector r. In other words, qvq−1 rotates the vector part of v around the vector r, see quaternions and spatial rotation for details. A quaternionic versor expressed in the complex 2×2 matrix representation is an element of the unitary group SU. Spin and SU are the same group, angles of rotation in this λ = 1/2 representation are equal to a, there is no 2 factor in angles unlike the λ =1 adjoint representation mentioned above, see representation theory of SU for details. For a fixed r, versors of the form exp where a ∈ (−π, π], in 2003 David W. Lyons wrote the fibers of the Hopf map are circles in S3. Lyons gives an introduction to quaternions to elucidate the Hopf fibration as a mapping on unit quaternions. Hamilton denoted the versor of a quaternion q by the symbol Uq and he was then able to display the general quaternion in polar coordinate form q = Tq Uq, where Tq is the norm of q. The norm of a versor is always equal to one, hence they occupy the unit 3-sphere in H, examples of versors include the eight elements of the quaternion group. Of particular importance are the right versors, which have angle π/2 and these versors have zero scalar part, and so are vectors of length one. The right versors form a sphere of square roots of −1 in the quaternion algebra, the generators i, j, and k are examples of right versors, as well as their additive inverses. Other versors include the twenty-four Hurwitz quaternions that have the norm 1, Hamilton defined a quaternion as the quotient of two vectors. A versor can be defined as the quotient of two unit vectors, for any fixed plane Π the quotient of two unit vectors lying in Π depends only on the angle between them, the same a as in the unit vector–angle representation of a versor explained above
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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
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Great circle
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A great circle, also known as an orthodrome or Riemannian circle, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. This partial case of a circle of a sphere is opposed to a circle, the intersection of the sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, a great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean 3-space is a circle of exactly one sphere. For most pairs of points on the surface of a sphere, there is a great circle through the two points. The exception is a pair of points, for which there are infinitely many great circles. The minor arc of a circle between two points is the shortest surface-path between them. In this sense, the arc is analogous to “straight lines” in Euclidean geometry. The length of the arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry. The great circles are the geodesics of the sphere, in higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn+1. To prove that the arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider the class of all paths from a point p to another point q. Introduce spherical coordinates so that p coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by θ = θ, ϕ = ϕ, a ≤ t ≤ b provided we allow φ to take on arbitrary real values. The infinitesimal arc length in these coordinates is d s = r θ ′2 + ϕ ′2 sin 2 θ d t. So the length of a curve γ from p to q is a functional of the curve given by S = r ∫ a b θ ′2 + ϕ ′2 sin 2 θ d t. Note that S is at least the length of the meridian from p to q, S ≥ r ∫ a b | θ ′ | d t ≥ r | θ − θ |. Since the starting point and ending point are fixed, S is minimized if and only if φ =0, so the curve must lie on a meridian of the sphere φ = φ0 = constant
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Spherical triangle
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Spherical trigonometry is of great importance for calculations in astronomy, geodesy and navigation. The origins of spherical trigonometry in Greek mathematics and the developments in Islamic mathematics are discussed fully in History of trigonometry. This book is now available on the web. The only significant developments since then have been the application of methods for the derivation of the theorems. A spherical polygon is a polygon on the surface of the sphere defined by a number of great-circle arcs, such polygons may have any number of sides. Two planes define a lune, also called a digon or bi-angle, the analogue of the triangle. Three planes define a triangle, the principal subject of this article. Four planes define a spherical quadrilateral, such a figure, and higher sided polygons, from this point the article will be restricted to spherical triangles, denoted simply as triangles. Both vertices and angles at the vertices are denoted by the upper case letters A, B and C. The angles of spherical triangles are less than π so that π < A + B + C < 3π. The sides are denoted by letters a, b, c. On the unit sphere their lengths are equal to the radian measure of the angles that the great circle arcs subtend at the centre. The sides of proper spherical triangles are less than π so that 0 < a + b + c < 3π, the radius of the sphere is taken as unity. For specific practical problems on a sphere of radius R the measured lengths of the sides must be divided by R before using the identities given below, likewise, after a calculation on the unit sphere the sides a, b, c must be multiplied by R. The polar triangle associated with a triangle ABC is defined as follows, consider the great circle that contains the side BC. This great circle is defined by the intersection of a plane with the surface. The points B and C are defined similarly, the triangle ABC is the polar triangle corresponding to triangle ABC. Therefore, if any identity is proved for the triangle ABC then we can derive a second identity by applying the first identity to the polar triangle by making the above substitutions