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Archimedes: Spheres
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In this treatise, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter. He proves that the sphere will have exactly two thirds of the volume and area of the cylinder. A carving of this proof was used on his tomb.
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Archimedes requested that his tombstone be decorated with a sphere contained in the smallest possible cylinder and inscribed with the ratio of the cylinder's volume to that of the sphere. Archimedes considered the discovery of this ratio the greatest of all his accomplishments.
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The Method proved a revolution in the understanding of Archimedes’ thought. It provides a glimpse into the thinking which led Archimedes to many of his famous results, including the determination of the area of a parabola, the area and volume of a sphere, and the volume of an ellipsoid. Apparently, Archimedes, whose understanding of such matters as levers and centers of gravity was particularly insightful, was able to envision ways to "weigh" various geometric figures against one another so as to successfully compare their areas or volumes. In this sense, they are, perhaps, the original "extra-geometric" proofs. Notwithstanding the ingenious logic behind these demonstrations, Archimedes apparently considered them only informal ("back of the papyrus" ?) calculations, and preferred to publish these results with more formal double indirect proofs, now referred to as the "method of exhaustion."
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The written work of Archimedes has not survived as well as that of Euclid, and seven of his treatises are known to exist only through references made to them by other authors. Pappus of Alexandria mentions On Sphere-Making and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the
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The problem of determining the gold content of the royal crown was given to Archimedes, a noted Greek mathematician and natural philosopher. Needles to say, this was not a trivial problem! Archimedes knew that silver was less dense than gold, but did not know any way of determining the relative the density (mass/volume) of an irregularly shaped crown, The weight could be determined using a balance or scale, but the only way known to determine volume, using the geometry of the day, was to beat the crown into a solid sphere or cube. Since Hiero had specified that damage to the crown would be viewed with less than enthusiasm, Archimedes did not wish to risk the king�s wrath by pounding the crown into a cube and hoping that post-analysis it could be made all better again.
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