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Salinon Shaded
The salinon (meaning 'salt-cellar' in Greek) is a geometrical figure that consists of four semicircles. It was first introduced in the ''Book of Lemmas'', a work attributed to Archimedes. Construction Let ''A'', ''D'', ''E'', and ''B'' be four points on a line in the plane, in that order, with ''AD'' = ''EB''. Let ''O'' be the bisector of segment ''AB'' (and of ''DE''). Draw semicircles above line ''AB'' with diameters ''AB'', ''AD'', and ''EB'', and another semicircle below with diameter ''DE''. A salinon is the figure bounded by these four semicircles. Properties Area Archimedes introduced the salinon in his ''Book of Lemmas'' by applying Book II, Proposition 10 of Euclid's ''Elements''. Archimedes noted that "the area of the figure bounded by the circumferences of all the semicircles sequal to the area of the circle on CF as diameter." Namely, if r_1 is the radius of large enclosing semicircle, and r_2 is the radius of the small central semicircle, then the area of the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Salinon
The salinon (meaning 'salt-cellar' in Greek) is a geometrical figure that consists of four semicircles. It was first introduced in the '' Book of Lemmas'', a work attributed to Archimedes. Construction Let ''A'', ''D'', ''E'', and ''B'' be four points on a line in the plane, in that order, with ''AD'' = ''EB''. Let ''O'' be the bisector of segment ''AB'' (and of ''DE''). Draw semicircles above line ''AB'' with diameters ''AB'', ''AD'', and ''EB'', and another semicircle below with diameter ''DE''. A salinon is the figure bounded by these four semicircles. Properties Area Archimedes introduced the salinon in his ''Book of Lemmas'' by applying Book II, Proposition 10 of Euclid's ''Elements''. Archimedes noted that "the area of the figure bounded by the circumferences of all the semicircles sequal to the area of the circle on CF as diameter." Namely, if r_1 is the radius of large enclosing semicircle, and r_2 is the radius of the small central semicircle, then the area of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrical Figure
A shape is a graphical representation of an object's form or its external boundary, outline, or external surface. It is distinct from other object properties, such as color, texture, or material type. In geometry, ''shape'' excludes information about the object's position, size, orientation and chirality. A ''figure'' is a representation including both shape and size (as in, e.g., figure of the Earth). A plane shape or plane figure is constrained to lie on a ''plane'', in contrast to ''solid'' 3D shapes. A two-dimensional shape or two-dimensional figure (also: 2D shape or 2D figure) may lie on a more general curved ''surface'' (a two-dimensional space). Classification of simple shapes Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, acute, scalene, etc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, radians, or a half-turn). It only has one line of symmetry (reflection symmetry). In non-technical usage, the term "semicircle" is sometimes used to refer to either a closed curve that also includes the diameter segment from one end of the arc to the other or to the half- disk, which is a two-dimensional geometric region that further includes all the interior points. By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a right triangle, with a right angle at the third vertex. All lines intersecting the semicircle perpendicularly are concurrent at the center of the circle containing the given semicircle. Arithmetic and geometric means A semicircle can be used to construct th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Book Of Lemmas
The ''Book of Lemmas'' or ''Book of Assumptions'' (Arabic ''Maʾkhūdhāt Mansūba ilā Arshimīdis'') is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositions ( lemmas) on circles. History Translations The ''Book of Lemmas'' was first introduced in Arabic by Thābit ibn Qurra; he attributed the work to Archimedes. A translation from Arabic into Latin by John Greaves and revised by Samuel Foster (c. 1650) was published in 1659 a''Lemmata Archimedis'' Another Latin translation by Abraham Ecchellensis and edited by Giovanni A. Borelli was published in 1661 under the name ''Liber Assumptorum''. T. L. Heath translated Heiburg's Latin work into English in his ''The Works of Archimedes''. A more recently discovered manuscript copy of Thābit ibn Qurra's Arabic translation was translated into English by Emre Coşkun in 2018. Authorship The original authorship of the ''Book of Lemmas'' has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes
Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenistic Sicily, Sicily. Although few details of his life are known, based on his surviving work, he is considered one of the leading scientists in classical antiquity, and one of the greatest mathematicians of all time. Archimedes anticipated modern calculus and mathematical analysis, analysis by applying the concept of the Cavalieri's principle, infinitesimals and the method of exhaustion to derive and rigorously prove many geometry, geometrical theorem, theorems, including the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. Archimedes' other math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length d of a diameter is also called the diameter. In this sense one speaks of diameter rather than diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the radius r. :d = 2r \qquad\text\qquad r = \frac. The word "diameter" is derived from (), "diameter of a circle", from (), "across, through" and (), "measure". It is often abbreviated \text, \text, d, or \varnothing. Constructions With straightedge and compass, a diameter of a given circle can be constructed as the perpendicular bisector of an arbitrary chord. Drawing two diameters in this way can be used to locate the center of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( ) is a mathematics, mathematical treatise written 300 BC by the Ancient Greek mathematics, Ancient Greek mathematician Euclid. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus and Theaetetus (mathematician), Theaetetus, the ''Elements'' is a collection in 13 books of definitions, postulates, propositions and mathematical proofs that covers plane and solid Euclidean geometry, elementary number theory, and Commensurability (mathematics), incommensurable lines. These include Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the Compass-and-straightedge construction, construction of regular polygons and Regular polyhedra, polyhedra. Often referred to as the most successful textbook ever written, the ''Elements'' has continued to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Mathematics, senior instructor at Hebrew University and software consultant at Ben Gurion University. He wrote extensively about arithmetic, probability, algebra, geometry, trigonometry and mathematical games. He was known for his contribution to heuristics and mathematics education, creating and maintaining the mathematically themed educational website ''Cut-the-Knot'' for the Mathematical Association of America (MAA) Online. He was a pioneer in mathematical education on the internet, having started ''Cut-the-Knot'' in October 1996. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arbelos
In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that contains their diameters. The earliest known reference to this figure is in Archimedes's ''Book of Lemmas'', where some of its mathematical properties are stated as Propositions 4 through 8. The word ''arbelos'' is Greek for 'shoemaker's knife'. The figure is closely related to the Pappus chain. Properties Two of the semicircles are necessarily concave, with arbitrary diameters and ; the third semicircle is Convex curve, convex, with diameter Let the diameters of the smaller semicircles be and ; then the diameter of the larger semircle is . Area Let be the intersection of the larger semicircle with the line perpendicular to at . Then the area (geometry), area of the arbelos is equal to the area of a circle with diameter . Proof: For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant under some Transformation (function), transformations, such as Translation (geometry), translation, Reflection (mathematics), reflection, Rotation (mathematics), rotation, or Scaling (geometry), scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a space, spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including scientific model, theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Y-axis
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular oriented lines, called '' coordinate lines'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the '' origin'' and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame. Similarly, the position of any point in three-dimensional space can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension . These coordinates are the signed distances from the point to mutually perpendicular fixed hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lune Of Hippocrates
In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex plane region bounded by one 180-degree circular arc and one 90-degree circular arc. It was the first curved figure to have its exact area calculated mathematically.. Translated from Postnikov's 1963 Russian book on Galois theory. History Hippocrates wanted to solve the classic problem of squaring the circle, i.e. constructing a square by means of straightedge and compass, having the same area as a given circle. He proved that the lune bounded by the arcs labeled ''E'' and ''F'' in the figure has the same area as triangle ''ABO''. This afforded some hope of solving the circle-squaring problem, since the lune is bounded only by arcs of circles. Heath concludes that, in proving his result, Hippocrates was also the first to prove that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
