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Although Leibnitz’s interests were clearly developing in a scientific direction, he still hankered after a literary career. All his life he prided himself on his poetry (mostly Latin), and boasted that he could recite the bulk of Virgil’s Aeneid by heart. During this time with Boineburg he would have passed for a typical late Renaissance humanist.

Leibnitz wished to visit Paris to make more scientific contacts. He had begun construction of a calculating machine which he hoped would be of interest. He formed a political plan to try to persuade the French to attack Egypt and this proved the means of his visiting Paris. In 1672 Leibnitz went to Paris on behalf of Boineburg to try to use his plan to divert Louis XIV from attacking German areas. His first object in Paris was to make contact with the French government but, while waiting for such an opportunity, Leibnitz made contact with mathematicians and philosophers there, in particular Arnauld and Malebranche, discussing with Arnauld a variety of topics but particularly church reunification.

In Paris Leibnitz studied mathematics and physics under Christiaan Huygens beginning in the autumn of 1672. On Huygens’ advice, Leibnitz read Saint Vincent’s work on summing series and made some discoveries of his own in this area. Also in the autumn of 1672, Boineburg’s son was sent to Paris to study under Leibnitz, which meant that his financial support was secure. Accompanying Boineburg’s son was Boineburg’s nephew on a diplomatic mission to try to persuade Louis XIV to set up a peace congress. Boineburg died on 15 December but Leibnitz continued to be supported by the Boineburg family.

In January 1673 Leibnitz and Boineburg’s nephew went to England to try the same peace mission, the French one having failed. Leibnitz visited the Royal Society, and demonstrated his incomplete calculating machine. He also talked with Hooke, Boyle and Pell. While explaining his results on series to Pell, he was told that these were to be found in a book by Mouton. The next day he consulted Mouton’s book and found that Pell was correct. At a meeting of the Royal Society, which Leibnitz did not attend, Hooke made some unfavorable comments on Leibnitz’s calculating machine. Leibnitz returned to Paris on hearing that the Elector of Mainz had died. He realized that his knowledge of mathematics was less than he would have liked so he doubled his efforts on the subject.

The Royal Society of London elected Leibnitz a fellow on 19 April 1673. Leibnitz met Ozanam and solved one of his problems. He also met again with Huygens who gave him a reading list including works by Pascal, Fabri, Gregory, Saint Vincent, Descartes and Sluze. He began to study the geometry of infinitesimals and wrote to Oldenburg at the Royal Society in 1674. Oldenburg replied that Newton and Gregory had found general methods. Leibnitz was, however, not in the best of favors with the Royal Society since he had not kept his promise of finishing his mechanical calculating machine. Nor was Oldenburg to know that Leibnitz had changed from the rather ordinary mathematician who visited London, into a creative mathematical genius. In August 1675 Tschirnhaus arrived in Paris and he formed a close friendship with Leibnitz, which proved very mathematically profitable to both.

It was during this period in Paris that Leibnitz developed the basic features of his version of Calculus. In 1673 he was still struggling to develop a good notation for his calculus and his first calculations were clumsy. On 21 November 1675 he wrote a manuscript using the f(x) dx notation for the first time. In the same manuscript the product rule for differentiation is given. By autumn 1676 Leibnitz discovered the familiar d(xn) = nxn-1dx for both integral and fractional n.

Newton wrote a letter to Leibnitz, through Oldenburg, which took some time to reach him. The letter listed many of Newton’s results but it did not describe his methods. Leibnitz replied immediately but Newton, not realizing that his letter had taken a long time to reach Leibnitz, thought he had had six weeks to work on his reply. Certainly one of the consequences of Newton’s letter was that Leibnitz realized he must quickly publish a fuller account of his own methods.

Newton wrote a second letter to Leibnitz on 24 October 1676, which did not reach Leibnitz until June 1677 by which time Leibnitz was in Hanover. This second letter, although polite in tone, was clearly written by Newton believing that Leibnitz had stolen his methods. In his reply Leibnitz gave some details of the principles of his differential calculus including the rule for differentiating a function of a function.

Newton was to claim, with justification, that not a single previously unsolved problem was solved by Leibnitz’s approach but the formalism was to prove vital in the latter development of the calculus. Leibnitz never thought of the derivative as a limit. This does not appear until the work of d’Alembert. Leibnitz would have liked to remain in Paris in the Academy of Sciences, but it was considered that there were already enough foreigners there and so no invitation came. Reluctantly Leibnitz accepted a position from the Duke of Hanover, Johann Friedrich, of librarian and of Court Councilor at Hanover. He left Paris in October 1676 making the journey to Hanover via London and Holland. The rest of Leibnitz’s life, from December 1676 until his death, was spent at Hanover except for the many travels that he made.

His duties at Hanover as librarian were onerous, but fairly mundane: general administration, purchase of new books and

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