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Yet Archimedes’ results are no less impressive than theirs.The same freedom from conventional ways of thinking is apparent in the arithmetical field in , which shows a deep understanding of the nature of the numerical system.Its purpose is to determine the positions that various solids will assume when floating in a fluid, according to their form and the variation in their specific gravities.In the first book various general principles are established, notably what has come to be known as Archimedes’ principle: a solid denser than a fluid will, when immersed in that fluid, be lighter by the weight of the fluid it displaces.

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What Archimedes does, in effect, is to create a place-value system of notation, with a base of 100,000,000.Archimedes’ approach to determining π, which consists of inscribing and circumscribing regular polygons with a large number of sides, was followed by everyone until the development of infinite series expansions in India during the 15th century and in Europe during the 17th century.That work also contains accurate approximations (expressed as ratios of integers) to the square roots of 3 and several large numbers.His contribution was rather to extend those concepts to conic sections.written for the layman—it is addressed to Gelon, son of Hieron—that nevertheless contains some profoundly original mathematics.

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