|
On Spirals
''On Spirals'' () is a treatise by 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 Hellenis ..., written around 225 BC. Notably, Archimedes employed the Archimedean spiral in this book to square the circle and trisect an angle. Contents Preface Archimedes begins ''On Spirals'' with a message to Dositheus of Pelusium mentioning the death of Conon as a loss to mathematics. He then goes on to summarize the results of '' On the Sphere and Cylinder'' (Περὶ σφαίρας καὶ κυλίνδρου) and ''On Conoids and Spheroids'' (Περὶ κωνοειδέων καὶ σφαιροειδέων). He continues to state his results of ''On Spirals''. Archimedean spiral The Archimedean spiral was first studied by Conon and was later studied by Archimedes in ''On Spirals''. Arc ... [...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] |
Square The Circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (\pi) is a transcendental number. That is, \pi is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if \pi were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found. Despite the proof that it is impossible, attempts to square the circle have been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trisect An Angle
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass. In 1837, Pierre Wantzel proved that the problem, as stated, is impossible to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a right angle. It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries. Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conon Of Samos
Conon of Samos (, ''Konōn ho Samios''; c. 280 – c. 220 BC) was a Greek astronomer and mathematician. He is primarily remembered for naming the constellation Coma Berenices. Life and work Conon was born on Samos, Ionia, and possibly died in Alexandria, Ptolemaic Egypt, where he was court astronomer to Ptolemy III Euergetes. He named the constellation Coma Berenices ("Berenice's Hair") after Ptolemy's wife Berenice II. She sacrificed her hair in exchange for her husband's safe return from the Third Syrian War, which began in 246 BC. When the lock of hair disappeared, Conon explained that the goddess had shown her favor by placing it in the sky. Not all Greek astronomers accepted the designation. In Ptolemy's ''Almagest'', Coma Berenices is not listed as a distinct constellation. However, Ptolemy does attribute several seasonal indications ('' parapegma'') to Conon. Conon was a friend of the mathematician Archimedes whom he probably met in Alexandria. Astronomical work In astro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On The Sphere And Cylinder
''On the Sphere and Cylinder'' () is a treatise that was published by Archimedes in two volumes . 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 area of its great circle. In modern terms, this means that the surf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Spiral
The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Ancient Greece, Greek mathematician Archimedes. The term ''Archimedean spiral'' is sometimes used to refer to the more general class of spirals of this type (see below), in contrast to ''Archimedes' spiral'' (the specific arithmetic spiral of Archimedes). It is the locus (mathematics), locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in Polar coordinate system, polar coordinates it can be described by the equation r = b\cdot\theta with real number . Changing the parameter controls the distance between loops. From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle as time elapses. Archimedes described such a spiral in his book ''On Spirals''. Conon of Samos was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangents
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is tangent to the curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. The point where the tangent line and the curve meet or intersect is called the ''point of tangency''. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a ''tangent line approximation'', the graph of the affine function that best approximates the original function at the given point. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes Trisect Spiral
Archimedes of Syracuse ( ; ) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in 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 analysis by applying the concept of the infinitesimals and the method of exhaustion to derive and rigorously prove many geometrical 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 mathematical achievements include deriving an approximation of pi (), defining and investigating the Archimedean spiral, and devising a system using exponentiation for expr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle Trisection
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a Compass (drawing tool), compass. In 1837, Pierre Wantzel proved that the problem, as stated, is Proof of impossibility, impossible to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a right angle. It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries. Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Spiral Circle Squaring Triangle
Archimedean means of or pertaining to or named in honor of the Greek mathematician Archimedes and may refer to: Mathematics *Archimedean absolute value * Archimedean circle * Archimedean constant * Archimedean copula *Archimedean field *Archimedean group * Archimedean point *Archimedean property *Archimedean solid *Archimedean spiral The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Ancient Greece, Greek mathematician Archimedes. The term ''Archimedean spiral'' is sometimes used to refer to the more gene ... * Archimedean tiling Other uses *Archimedean screw * Claw of Archimedes * The Archimedeans, the mathematical society of the University of Cambridge * Archimedean Dynasty * Archimedean Upper Conservatory See also * Archimedes (other) * {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurement Of A Circle
''Measurement of a Circle'' or ''Dimension of the Circle'' ( Greek: , ''Kuklou metrēsis'') is a treatise that consists of three propositions, probably made 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Works By Archimedes
Works may refer to: People * Caddy Works (1896–1982), American college sports coach * John D. Works (1847–1928), California senator and judge * Samuel Works (c. 1781–1868), New York politician Albums * ''Works'' (Pink Floyd album), a Pink Floyd album from 1983 * ''Works'', a Gary Burton album from 1972 * ''Works'', a Status Quo album from 1983 * ''Works'', a John Abercrombie album from 1991 * ''Works'', a Pat Metheny album from 1994 * ''Works'', an Alan Parson Project album from 2002 * ''Works Volume 1'', a 1977 Emerson, Lake & Palmer album * ''Works Volume 2'', a 1977 Emerson, Lake & Palmer album * '' The Works'', a 1984 Queen album Other uses *Good works, a topic in Christian theology * Microsoft Works, a collection of office productivity programs created by Microsoft * IBM Works, an office suite for the IBM OS/2 operating system * Mount Works, Victoria Land, Antarctica See also * The Works (other) * Work (other) Work may refer to: * Work ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
