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His approximation of between 3-1/2 and 3-10/71 was the most accurate of his time and he devised a new way to approximate square roots. Unhappy with the unwieldy Greek number system, he devised his own that could accommodate larger numbers more easily. He invented the entire field of hydrostatics with the discovery of the Archimedes' Principle...
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However, his greatest invention was integral calculus. To determine the area of sections bounded by geometric figures such as parabolas and ellipses, Archimedes broke the sections into an infinite number of rectangles and added the areas together. This is known as integration. He also anticipated the invention of differential calculus as he devised ways to approximate the slope of the tangent lines to his figures...
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