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Born the son of a Lutheran professor of moral philosophy in Leipzig, Germany, Leibniz was educated at the universities of Leipzig, Jena, and Altdorf where he gained his doctorate in 1666. In 1667 he entered the service of the elector of Mainz for whom he spent the period 1673–76 on a diplomatic mission to Paris. Through meeting with such scholars as Christian Huygens in Paris and with members of the Royal Society, including Robert Boyle, during two trips to London in 1673 and 1676, Leibniz was introduced to the outstanding problems challenging the mathematicians and physicists of Europe. On leaving Paris he joined the staff of John Frederick the duke of Brunswick-Lüneburg, also Elector of Hannover, where he was given the commission to write the history of the House of Brunswick and the position of librarian. For the remaining 40 years of his life Leibniz dissipated his prodigious talents under three electors, including the future George I of Great Britain and Ireland, constructing genealogies of the numerous Brunswick progeny, both legitimate and, even more numerous, illegitimate. He also undertook a variety of administrative and diplomatic duties of which his attempt to unite the Protestant and Catholic churches in 1683 and the founding of the Berlin Academy of Sciences in 1700 are the most noteworthy. It was also to his Brunswick years that most of his philosophical writings belong although many of them remained unpublished until well after his death.

Leibniz's greatest achievement was undoubtedly his discovery of the differential and integral calculus, work which was to involve him in a bitter priority dispute with Isaac Newton. Newton's ideas on the calculus were developed first, as early as 1665, but remained unpublished until 1687; Leibniz, however, began work on problems of the calculus during his Paris years and published his results in 1684 in Nova methodus pro maximis et minimis(New Method for the Greatest and the Least). It was later suggested in 1699 that Leibniz's original inspiration may well have come from conversations in London in 1673 and in 1676 as well as letters of Newton to Henry Oldenburg shown to Leibniz. From this point the dispute became open to all and was conducted with considerable ferocity and not a little dishonesty. In fact, as became clear later on, the discoveries were made independently; the final triumph lay with Liebniz because it was his notation of differentiation and integration, rather than the fluxions of Newton, that have survived in modern textbooks.

In physics, Leibniz's metaphysical principles also led him to deny Newtonian gravity acting at a distance on the grounds that: “A body is never moved naturally, except by another body which touches it and pushes it; after that it continues until it is prevented by another body which touches it. Any other kind of operation on bodies is either miraculous or imaginary.” He also rejected Newtonian concepts of absolute space and time, arguing more plausibly that space was simply “the order of bodies among themselves” while time was their order of succession. Leibniz also, with Huygens, developed the concept of kinetic energy.

As well as his contributions to metaphysics and philosophy, Leibniz established the foundations of symbolic logic, probability theory, and combinatorial analysis, and was led to design a practical calculating machine. It was actually built and shown to the Royal Society in 1794.