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Archimedes: Spheres
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Other known works by Archimedes that are purely geometrical are On Conoids and Spheroids, On Spirals, and Quadrature of the Parabola. The first is concerned with volumes of segments of such figures as the hyperboloid of revolution. The second describes what is now known as Archimedes' spiral and contains area computations. The third is on finding areas of segments of the parabola.
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Archimedes made many contributions to geometry in his work on the areas of plane figures and on the areas of area and volumes of curved surfaces. His methods started the idea for calculus which was "invented" 2,000 years later by Sir Isaac Newton and Gottfried Wilhelm von Leibniz. Archimedes proved that the volume of an inscribed sphere is two-thirds the volume of a circumscribed cylinder. He requested that this formula/diagram be inscribed on his tomb.
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In the first part of this treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not
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In geometry Archimedes continued the work in Book XII of Euclid's Elements. In Book XII the method of exhaustion, discovered by Eudoxus, is used to prove theorems on areas of circles and volumes of spheres, pyramids, and cones. Two of the theorems are mentioned by Archimedes in the preface to On the Sphere and Cylinder. After stating the result concerning the ratio of the volumes of a cylinder and an inscribed sphere, he says that this result can be put side by side with his previous investigations and with those theorems of Eudoxus on solids, namely: the volume of a pyramid is one-third the volume of a prism with the same base and height; and the volume of a cone is one-third the volume of a cylinder with the same base and height.
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Archimedes continued Euclid's work more than anyone before him. One way he did this was to extend what is known as the "method of exhaustion." This method is used to determine the areas and volumes of figures with curved lines and surfaces, such as circles, spheres, pyramids, and cones. Archimedes's investigation of the method of exhaustion helped lead to the current form of mathematics called integral calculus. Although his method is now outdated, the advances that finally outdated it did not occur until about two thousand years after Archimedes lived.
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Archimedes considered his greatest achievement to be the discovery of how to calculate the volume of a sphere by comparing it with a similar-sized cylinder. Accordingly, a sphere inscribed in a cylinder was used to decorate his tombstone. More than a hundred years later the Roman Cicero was still able to locate and restore it, but the grave has subsequently been lost.
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