Archimedes: Contributions to mathematics

On Floating Bodies was not Archimdedes' only contribution to applied mathematics. On the Equilibrium of Planes being the other treatise of only two on applied mathematics that have survived. It is concerned with the centres of gravity of the triangle, the trapezium, and that of a parabolic segment. Although these are the only surviving treatises on applied mathematics, Pappus of Alexandria mentions a work On Levers, and Theon of Alexandria quotes a theorem from another on the properties of mirrors.

Archimedes' contributions to physics and applied mathematics should not be underrated, since it was not until the 16th century that the science of statics and the theory of hydrostatics were appreciably advanced beyond the points reached by Archimedes.

Archimedes' contributions to mathematics were not limited to applied mathematics, however; for it was Archimedes who initiated work on the calculus. At least another eight of Archimedes' treatises have survived. Each work is a model of mathematical rigour, precision, and originality. In his work On Spirals, Archimedes computed the area swept out by a spiral. In his work On Conoids and Spheroids, he computed the area of an ellipse. In his treatise On the Sphere and the Cylinder, Archimedes proved that if a sphere is inscribed in a cylinder the height of which is equal to its diameter, then the ratio of volumes of the sphere and the cylinder is equal to the ratio of their surface areas (3 to 2). He was so pleased with this result that he had a sketch of a sphere inscribed in a cylinder carved on his tombstone. In his work Quadrature of the Parabola, Archimedes described a method for estimating the area under a parabola by computing the area under a sum of rectangles. This method, called the method of exhaustion, forms the the basis for the study of integral calculus.

Perhaps the most remarkable of all Archimedes' works is Method, which is written in the form of a letter to Eratosthenes. In it, Archimedes describes the process by which he discovered many of his theorems. It is remarkable because in all his other works only finished, polished results were presented, with no indication of how they were obtained. Method, perhaps more than any of his other works, demonstrates the genius of this most remarkable of mathematicians.

Other achievements that have been attributed to Archimedes include:

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