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The Method Of Mechanical Theorems
''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is one of the major surviving works of the ancient Greek polymath Archimedes. ''The Method'' takes the form of a letter from Archimedes to Eratosthenes, the chief librarian at the Library of Alexandria, and contains the first attested explicit use of indivisibles (sometimes erroneously referred to as infinitesimals). The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated Archimedes Palimpsest. The palimpsest includes Archimedes' account of the "mechanical method", so called because it relies on the center of weights of figures (centroid) and the law of the lever, which were demonstrated by Archimedes in ''On the Equilibrium of Planes''. Archimedes did not admit the method of indivisibles as part of rigorous mathematics, and therefore did not publish his method in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean Sea, Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, classical antiquity ( AD 600), that comprised a loose collection of culturally and linguistically related polis, city-states and other territories. Most of these regions were officially unified only once, for 13 years, under Alexander the Great's Macedonian empire, empire from 336 to 323 BC (though this excludes a number of Greek city-states free from Alexander's jurisdiction in the western Mediterranean, around the Black Sea, Cyprus, and Cyrenaica). In Western history, the era of classical antiquity was immediately followed by the Early Middle Ages and the Byzantine period. Roughly three centuries after the Late Bronze Age collapse of Mycenaean Greece, Greek urban poleis began to form in the 8th century BC, ushering in the Archaic Greece, Archaic period and Greek colonis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a Point (geometry), point (the Focus (geometry), focus) and a Line (geometry), line (the Directrix (conic section), directrix). The focus does not lie on the directrix. The parabola is the locus (mathematics), locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane (geometry), plane Parallel (geometry), parallel to another plane that is tangential to the conical surface. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Exhaustion
The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the ''n''th polygon and the containing shape will become arbitrarily small as ''n'' becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members. The method of exhaustion typically required a form of proof by contradiction, known as ''reductio ad absurdum''. This amounts to finding an area of a region by first comparing it to the area of a second region, which can be "exhausted" so that its area becomes arbitrarily close to the true area. The proof involves assuming that the true area is greater than the second area, proving that assertion false, assuming it is less than the secon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Indivisibles
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. * 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes. Today Cavalieri's principle is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem, results using Cavalieri's principle can often be shown more directly via integration. In the other direction, Cavalieri's principle grew out of the ancient Greek method of exhaustion, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemisphere
Hemisphere refers to: * A half of a sphere As half of the Earth * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemisphere ** Land and water hemispheres * A half of the (geocentric) celestial sphere ** Northern celestial hemisphere ** Southern celestial hemisphere * A cultural hemisphere As half of the brain * A cerebral hemisphere, a division of the cerebrum * A half of the cerebellum, a smaller part of the brain Other * ''Hémisphère'' (Paradis), a 12-inch album by French artists Paradis * ''Hemispheres'' (magazine), an inflight publication * ''Hemispheres'' (TV series), Canadian and Australian news program * ''Hemispheres'' (Rush album), 1978 * ''Hemispheres'' (Lily Afshar album), 2006 * ''Hemispheres'' (Doseone album), 1998 * L'Hemisfèric at the Ciutat de les Arts i les Ciències, Valencia, Spain * Hemisphere Project, a counternarcotics program between United States federal and state drug officials and A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Hogendijk
Jan Pieter Hogendijk (born 21 July 1955) is a Dutch mathematician and historian of science. Since 2005, he is professor of history of mathematics at the University of Utrecht. Hogendijk became a member of the Royal Netherlands Academy of Arts and Sciences in 2010. Hogendijk has contributed to the study of Greek mathematics and mathematics in medieval Islam; he provides a list of Sources on his website (below). In 2012, he was awarded the inaugural Otto Neugebauer Prize for History of Mathematics, by the European Mathematical Society The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current ..., "for having illuminated how Greek mathematics was absorbed in the medieval Arabic world, how mathematics developed in medieval Islam, and how it was eventually transmitted to Europe." A bibliogr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicylinder
In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders is called a bicylinder. Topologically, it is equivalent to a square hosohedron. The intersection of three cylinders is called a tricylinder. A bisected bicylinder is called a vault, and a cloister vault in architecture has this shape. Steinmetz solids are named after mathematician Charles Proteus Steinmetz, who solved the problem of determining the volume of the intersection. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. Bicylinder A bicylinder generated by two cylinders with radius r has the ;volume :V=\frac r^3 and the ;surface area :A=16 r^2. The upper half of a bicylinder is the square ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On The Sphere And Cylinder
''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so. Contents The principal formulae derived in ''On the Sphere and Cylinder'' are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder. Let r be the radius of the sphere and cylinder, and h be the height of the cylinder, with the assumption that the cylinder is a right cylinder—the side is perpendicular to both caps. In his work, Archimedes showed that the surface area of a cylinder is equal to: :A_C = 2 \pi r^2 + 2 \pi r h = 2 \pi r ( r + h ).\, and that the volume of the same is: :V_C = \pi r^2 h. \, On the sphere, he showed that the surface area is four times the ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurement Of The Circle
''Measurement of a Circle'' or ''Dimension of the Circle'' (Greek: , ''Kuklou metrēsis'') is a treatise that consists of three propositions by Archimedes, ca. 250 BCE. The treatise is only a fraction of what was a longer work. Propositions Proposition one Proposition one states: The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle. Any circle with a circumference ''c'' and a radius ''r'' is equal in area with a right triangle with the two legs being ''c'' and ''r''. This proposition is proved by the method of exhaustion. Proposition two Proposition two states: The area of a circle is to the square on its diameter as 11 to 14. This proposition could not have been placed by Archimedes, for it relies on the outcome of the third proposition. Proposition three Proposition three states: The ratio of the circumference of any circle to its diameter is great ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spheroids
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry. If the ellipse is rotated about its major axis, the result is a ''prolate spheroid'', elongated like a rugby ball. The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis, the result is an ''oblate spheroid'', flattened like a lentil or a plain M&M. If the generating ellipse is a circle, the result is a sphere. Due to the combined effects of gravity and rotation, the figure of the Earth (and of all planets) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography and geodesy the Earth is often approximated by an oblate spheroid, known as the reference ellipsoid, instead of a sph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume Of A Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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