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Playfair's Postulate
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): ''In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.'' It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry and was named after the Scottish mathematician John Playfair. The "at most" clause is all that is needed since it can be proved from the first four axioms that at least one parallel line exists given a line ''L'' and a point ''P'' not on ''L'', as follows: # ''Construct a perpendicular'': Using the axioms and previously established theorems, you can construct a line perpendicular to line ''L'' that passes through ''P''. # ''Construct another perpendicular'': A second perpendicular line is drawn to the first one, starting from point ''P''. # ''Parallel Line'': This second perpendicular line will be parallel to ''L'' by the definition of parallel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point And Line
A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topological space * Point, or Element (category theory), generalizes the set-theoretic concept of an element of a set to an object of any category * Critical point (mathematics), a stationary point of a function of an arbitrary number of variables * Decimal point * Point-free geometry * Stationary point, a point in the domain of a single-valued function where the value of the function ceases to change Places * Point, Cornwall, England, a settlement in Feock parish * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points, West Virginia, an unincorporated community in the United States Business an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Addison-Wesley
Addison–Wesley is an American publisher of textbooks and computer literature. It is an imprint of Pearson plc, a global publishing and education company. In addition to publishing books, Addison–Wesley also distributes its technical titles through the O'Reilly Online Learning e-reference service. Addison–Wesley's majority of sales derive from the United States (55%) and Europe (22%). The Addison–Wesley Professional Imprint produces content including books, eBooks, and video for the professional IT worker including developers, programmers, managers, system administrators. Classic titles include '' The Art of Computer Programming'', '' The C++ Programming Language'', '' The Mythical Man-Month'', and '' Design Patterns''. History Lew Addison Cummings and Melbourne Wesley Cummings founded Addison–Wesley in 1942, with the first book published by Addison–Wesley being Massachusetts Institute of Technology professor Francis Weston Sears' ''Mechanics''. Its first comput ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Geometry
300px, A sphere with a spherical triangle on it. Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres. Long studied for its practical applications to astronomy, navigation, and geodesy, spherical geometry and the metrical tools of spherical trigonometry are in many respects analogous to Euclidean plane geometry and trigonometry, but also have some important differences. The sphere can be studied either ''extrinsically'' as a surface embedded in 3-dimensional Euclidean space (part of the study of solid geometry), or ''intrinsically'' using methods that only involve the surface itself without reference to any surrounding space. Principles In plane (Euclidean) geometry, the basic concepts are points and (straight) lines. In spherical geometry, the basic concepts are points and great circles. However, two great circles on a plane intersect in two antipodal points, unlike coplan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logically Equivalent
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related. Logical equivalences In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these. General logical equivalences Logical equivalences involving conditional statements :#p \rightarrow q \equiv \neg p \vee q :#p \rightarrow q \equiv \neg q \rightarrow \neg p :#p \vee q \equiv \neg p \rightarrow q :#p \wedge q \equiv \neg (p \rightarrow \neg q) :#\neg (p \rightarrow q) \equiv p \wedge \neg q :#(p \rightarro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin ''angulus rectus''; here ''rectus'' means "upright", referring to the vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to Euclidean vector, vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry. Etymology The meaning of ''right'' in ''right angle'' possibly refers to the Classical Latin, Latin adjective ''rectus'' 'erect, straight, upright, perp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (mathematics)
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its ''endpoints''). Euclid's ''Elements'' defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. ''Euclidean line'' and ''Euclidean geometry'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry. Properties In the Greek deductive geometry of Euclid's ''Elements'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special case of an ''arc (geometry), arc'', with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline (vinculum (symbol), vinculum) above the symbols for the two endpoints, such as in . Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Postulate En
Parallel may refer to: Mathematics * Parallel (geometry), two lines in the Euclidean plane which never intersect * Parallel (operator), mathematical operation named after the composition of electrical resistance in parallel circuits Science and engineering * Parallel (latitude), an imaginary east–west line circling a globe * Parallel of declination, used in astronomy * Parallel, a geometric term of location meaning "in the same direction" * Parallel electrical circuits Computing * Parallel (software), a UNIX utility for running programs in parallel Language * Parallelism (grammar), a balance of two or more similar words, phrases, or clauses * Parallelism (rhetoric) Entertainment * ''Parallel'' (manga) * ''Parallel'' (2018 film), a Canadian science fiction thriller film * ''Parallel'' (2024 film) an American science fiction thriller film * ''Parallel'' (video), a compilation of music videos by R.E.M. * ''Parallel'' (The Black Dog album), 1995 * ''Parallel'' ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book ''Grundlagen der Geometrie'' (tr. ''The Foundations of Geometry'') as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Tarski's axioms, Alfred Tarski and of Birkhoff's axioms, George Birkhoff. The axioms Hilbert's axiom system is constructed with six primitive notions: three primitive terms: * Point (geometry), point; * Line (geometry), line; * Plane (mathematics), plane; and three primitive finitary relation, relations: * ''Betweenness'', a ternary relation linking points; * ''Lies on (Containment)'', three binary relations, one linking points and straight lines, one linking points and planes, and one linking straight lines and planes; * ''Congruence'', two binary relations, one linking line segments and one linking angles, each denoted by an infix ≅. Line segments, angles, and triangles may each be d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a cofounder of proof theory and mathematical logic. Life Early life and education Hilbert, the first of two children and only son of O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
British Association
The British Science Association (BSA) is a charity and learned society founded in 1831 to aid in the promotion and development of science. Until 2009 it was known as the British Association for the Advancement of Science (BA). The current Chief Executive is Hannah Russell. The BSA's mission is to get more people engaged in the field of science by coordinating, delivering, and overseeing different projects that are suited to achieve these goals. The BSA "envisions a society in which a diverse group of people can learn and apply the sciences in which they learn." and is managed by a professional staff located at their Head Office in the Wellcome Wolfson Building. The BSA offers a wide variety of activities and events that both recognise and encourage people to be involved in science. These include the British Science Festival, British Science Week, the CREST Awards, For Thought, The Ideas Fund, along with regional and local events. History Foundation The Association was founded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
