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Carlyle Circle
In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation; it is named after Thomas Carlyle. The circle has the property that the equation solving, solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons. Definition Given the quadratic equation :''x''2 − ''sx'' + ''p'' = 0 the circle in the coordinate plane having the line segment joining the points ''A''(0, 1) and ''B''(''s'', ''p'') as a diameter is called the Carlyle circle of the quadratic equation.JSTOR Defining property The defining property of the Carlyle circle can be established thus: the equation of the circle having the line segment ''AB'' as diameter is :''x''(''x'' − ''s'') + (''y'' − 1)(''y'' − ''p'') =& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quartic Equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomial equation that can be solved by radicals in the general case. History Lodovico Ferrari is attributed with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it could not be published immediately. The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book '' Ars Magna'' (1545). The proof that this was the highest order general polynomial for which such solutions could be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by Évariste Galois before ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Polygons
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known. Conditions for constructibility Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969). This led to the question being posed: is it possible to construct ''all'' regular polygons with compass and straightedge? If not, which ''n''-gons (that is, polygons with ''n'' edges) are constructible and which are not? Carl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygons
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' or ''corners''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a ''polygonal region'' or ''polygonal area''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polygon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. One of those is the parallel postulate which relates to parallel lines on a Euclidean plane. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compass-and-straightedge Construction
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a compass. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so it may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formally, the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lill's Method
In mathematics, Lill's method is a visual method of finding the real number, real zero of a function, roots of a univariate polynomial of any degree of a polynomial, degree. It was developed by Austrian engineer Eduard Lill in 1867. A later paper by Lill dealt with the problem of complex numbers, complex roots. Lill's method involves drawing a path of straight line segments making Right angle, right angles, with lengths equal to the coefficients of the polynomial. The roots of the polynomial can then be found as the Slope, slopes of other right-angle paths, also connecting the start to the terminus, but with vertices on the lines of the first path. Description of the method To employ the method, a diagram is drawn starting at the origin. A line segment is drawn rightwards by the magnitude of the leading coefficient, so that with a negative coefficient, the segment will end left of the origin. From the end of the first segment, another segment is drawn upwards by the magnitude o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eduard Lill
Eduard Lill (1830–1900) was an Austrian engineer and army officer. Life Lill was born 20 October 1830 in Brüx (Bohemia). From 1848 to 1849 he studied mathematics at the Czech Technical University in Prague and in 1850 he joined the military engineering corps of the Austrian Empire. From 1852 to 1856 he continued his education at the military engineering academy at Klosterbruck near Znaim. Later he was stationed in Esseg, Kronstadt and Spalato until he retired from his military career in 1868 with the rank of captain (Hauptmann) of the engineering corps. In the same year he became an engineer for the Austrian Northwestern Railway and oversaw the railroad construction at Trautenau (Trutnov). A severe accident however restricted him soon to office work. From 1872-1875 he worked as a secretary for the director of construction of the railway company. Later he became a technical consultant for company's headquarters and in 1885 the head of its statistics department. He retired in 1894 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Leslie (physicist)
Sir John Leslie, FRSE Royal Guelphic Order, KH (10 April 1766 – 3 November 1832) was a Scottish mathematician and physicist best remembered for his research into heat. Leslie gave the first modern account of capillary action in 1802 and froze water using an air-pump in 1810, the first artificial production of ice. In 1804, he experimented with radiant heat using a cube, cubical vessel filled with boiling water. One side of the cube is composed of highly polished metal, two of dull metal (copper) and one side painted black. He showed that radiation was greatest from the black side and negligible from the polished side. The apparatus is known as a Leslie cube. Early life Leslie was born the son of Robert Leslie, a joiner and cabinetmaker, and his wife Anne Carstairs, in Lower Largo, Largo in Fife. He received his early education there and at Leven, Fife, Leven. In his thirteenth year, encouraged by friends who had even then remarked his aptitude for mathematical and physical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Howard Eves
Howard Whitley Eves (10 January 1911, 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics. Eves received his B.S. from the University of Virginia, an M.A. from Harvard University, and a Ph.D. in mathematics from Oregon State University in 1948, the last with a dissertation titled ''A Class of Projective Space Curves'' written under Ingomar Hostetter. He then spent most of his career at the University of Maine, 1954–1976. In later life, he occasionally taught at University of Central Florida. Eves was a strong spokesman for the Mathematical Association of America, which he joined in 1942, and whose Northeast Section he founded. For 25 years he edited the Elementary Problems section of the ''American Mathematical Monthly''. He solved over 300 problems proposed in various mathematical journals. His six volume ''Mathematical Circles'' series, collecting humorous and interesting anecdotes about mathematicians, was recently re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlyle Circle Original Problem
Carlyle may refer to: Places * Carlyle, Illinois, a US city * Carlyle, Kansas, an unincorporated place in the US * Carlyle, Montana, a ghost town in the US * Carlyle, Saskatchewan, a Canadian town, including: :: Carlyle Airport and :: Carlyle station * Carlyle Lake Resort, Saskatchewan, a Canadian hamlet * Carlyle Hotel, New York City * Carlyle Restaurant, New York City * The Carlyle, a residential condominium in Minneapolis, Minnesota * The Carlyle (Pittsburgh), a residential condominium in Pittsburgh, Pennsylvania Name * Carlyle (name) ** Carlyle (given name) ** Carlyle (surname) Other uses * The Carlyle Group, a private equity company based in the US * Carlyle Works, a former bus bodybuilder in the UK See also * Carlisle (other) * Carlile (other) * Carlyne Carlyne is both a given name that is a variant of Carly and Caroline (given name), Caroline. Notable people with the name include: *Arthur Carlyne Niven Dixey, full name of Arthur Dixey (1889 – 195 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
65537-gon
In geometry, a 65537-gon is a polygon with 65,537 (216 + 1) sides. The sum of the interior angles of any non– self-intersecting is 11796300°. Regular 65537-gon The area of a ''regular '' is (with ) :A = \frac t^2 \cot \frac A whole regular is not visually discernible from a circle, and its perimeter differs from that of the circumscribed circle by about 15 parts per billion. Construction The regular 65537-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 65,537 is a Fermat prime, being of the form 22''n'' + 1 (in this case ''n'' = 4). Thus, the values \cos \frac and \cos \frac are 32768- degree algebraic numbers, and like any constructible numbers, they can be written in terms of square roots and no higher-order roots. Although it was known to Carl Friedrich Gauss by 1801 that the regular 65537-gon was cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
