Leibniz summed sequences of reciprocal polygonal numbers and, following the work of St. Vincent, summed and analyzed geometric sequences.  He studied trigonometric functions from the works of Huygens. In 1671 Leibniz developed a machine that could not only add and subtract but also multiply, divide, and extract square roots.  The  Leibniz calculator was gear-operated, and it provided a carry from one order to the next.  His machine was the first general-purpose calculator whose principles are still used in mechanical calculating machines. 

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