**Gottfried Leibniz** was a German mathematician who developed the present day notation for the differential and integral calculus though he never thought of the derivative as a limit. His philosophy is also important and he invented an early calculating machine.

- Certainly Leibniz learnt his moral and religious values from her which would play an important role in his life and philosophy.
- At the age of seven, Leibniz entered the Nicolai School in Leipzig.
- Although he was taught Latin at school, Leibniz had taught himself far more advanced Latin and some Greek by the age of 12.
- Leibniz was clearly not satisfied with Aristotle's system and began to develop his own ideas on how to improve on it.
- In later life Leibniz recalled that at this time he was trying to find orderings on logical truths which, although he did not know it at the time, were the ideas behind rigorous mathematical proofs.
- In 1661, at the age of fourteen, Leibniz entered the University of Leipzig.
- Leibniz then went to Jena to spend the summer term of 1663.
- At Jena the professor of mathematics was Erhard Weigel but Weigel was also a philosopher and through him Leibniz began to understand the importance of the method of mathematical proof for subjects such as logic and philosophy.
- Weigel believed that number was the fundamental concept of the universe and his ideas were to have considerable influence of Leibniz.
- By October 1663 Leibniz was back in Leipzig starting his studies towards a doctorate in law.
- After being awarded a bachelor's degree in law, Leibniz worked on his habilitation in philosophy.
- In this work Leibniz aimed to reduce all reasoning and discovery to a combination of basic elements such as numbers, letters, sounds and colours.
- Despite his growing reputation and acknowledged scholarship, Leibniz was refused the doctorate in law at Leipzig.
- Leibniz was not prepared to accept any delay and he went immediately to the University of Altdorf where he received a doctorate in law in February 1667 for his dissertation De Casibus Perplexis Ⓣ(On Perplexing Cases).
- Leibniz declined the promise of a chair at Altdorf because he had very different things in view.
- By November 1667 Leibniz was living in Frankfurt, employed by Boineburg.
- Another of Leibniz's lifelong aims was to collate all human knowledge.
- Certainly he saw his work on Roman civil law as part of this scheme and as another part of this scheme, Leibniz tried to bring the work of the learned societies together to coordinate research.
- Leibniz began to study motion, and although he had in mind the problem of explaining the results of Wren and Huygens on elastic collisions, he began with abstract ideas of motion.
- Leibniz was also in contact with Carcavi, the Royal Librarian in Paris.
- Leibniz wished to visit Paris to make more scientific contacts.
- In 1672 Leibniz 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, Leibniz 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 Leibniz studied mathematics and physics under Christiaan Huygens beginning in the autumn of 1672.
- On Huygens' advice, Leibniz read Saint-Vincent's work on summing series and made some discoveries of his own in this area.
- In January 1673 Leibniz and Boineburg's nephew went to England to try the same peace mission, the French one having failed.
- Leibniz visited the Royal Society, and demonstrated his incomplete calculating machine.
- At the meeting of the Royal Society on 15 February, which Leibniz did not attend, Hooke made some unfavourable comments on Leibniz's calculating machine.
- Leibniz returned to Paris on hearing that the Elector of Mainz had died.
- Leibniz realised that his knowledge of mathematics was less than he would have liked so he redoubled his efforts on the subject.
- The Royal Society of London elected Leibniz a fellow on 19 April 1673.
- Leibniz met Ozanam and solved one of his problems.
- Leibniz was, however, not in the best of favours with the Royal Society since he had not kept his promise of finishing his mechanical calculating machine.
- Nor was Oldenburg to know that Leibniz 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 Leibniz which proved very mathematically profitable to both.
- It was during this period in Paris that Leibniz developed the basic features of his version of the calculus.
- Newton wrote a letter to Leibniz, through Oldenburg, which took some time to reach him.
- Leibniz replied immediately but Newton, not realising that his letter had taken a long time to reach Leibniz, thought he had had six weeks to work on his reply.
- Certainly one of the consequences of Newton's letter was that Leibniz realised he must quickly publish a fuller account of his own methods.
- Newton wrote a second letter to Leibniz on 24 October 1676 which did not reach Leibniz until June 1677 by which time Leibniz was in Hanover.
- This second letter, although polite in tone, was clearly written by Newton believing that Leibniz had stolen his methods.
- In his reply Leibniz gave some details of the principles of his differential calculus including the rule for differentiating a function of a function.
- by Leibniz's approach but the formalism was to prove vital in the latter development of the calculus.
- Leibniz never thought of the derivative as a limit.
- Leibniz would have liked to have remained in Paris in the Academy of Sciences, but it was considered that there were already enough foreigners there and so no invitation came.
- Reluctantly Leibniz accepted a position from the Duke of Hanover, Johann Friedrich, of librarian and of Court Councillor at Hanover.
- The rest of Leibniz's life, from December 1676 until his death, was spent at Hanover except for the many travels that he made.
- Leibniz himself believed that this was because of deliberate obstruction by administrators and technicians, and the workers' fear that technological progress would cost them their jobs.
- However Leibniz had achieved important scientific results becoming one of the first people to study geology through the observations he compiled for the Harz project.
- Another of Leibniz's great achievements in mathematics was his development of the binary system of arithmetic.
- Another major mathematical work by Leibniz was his work on determinants which arose from his developing methods to solve systems of linear equations.
- Leibniz continued to perfect his metaphysical system in the 1680s attempting to reduce reasoning to an algebra of thought.
- Leibniz published Meditationes de Cognitione, Veritate et Ideis Ⓣ(Reflections on Knowledge, Truth, and Ideas) which clarified his theory of knowledge.
- In February 1686, Leibniz wrote his Discours de métaphysique Ⓣ(Discourse on Metaphysics).
- As always Leibniz took the opportunity to meet with scholars of many different subjects on these journeys.
- In 1684 Leibniz published details of his differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus...
- In 1686 Leibniz published, in Acta Eruditorum, a paper dealing with the integral calculus with the first appearance in print of the ∫ notation.
- This time delay in the publication of Newton's work resulted in a dispute with Leibniz.
- Another important piece of mathematical work undertaken by Leibniz was his work on dynamics.
- It is clear that while he was in Rome, in addition to working in the Vatican library, Leibniz worked with members of the Accademia.
- Although Leibniz was ahead of his time in aiming at a genuine dynamics, it was this very ambition that prevented him from matching the achievement of his rival Newton.
- Leibniz put much energy into promoting scientific societies.
- Leibniz was appointed its first president, this being an appointment for life.
- Other attempts by Leibniz to found academies were less successful.
- He was appointed as Director of a proposed Vienna Academy in 1712 but Leibniz died before the Academy was created.
- It is no exaggeration to say that Leibniz corresponded with most of the scholars in Europe.
- Leibniz also corresponded with Varignon on this paradox.
- In 1710 Leibniz published Théodicée Ⓣ(Theodicy) a philosophical work intended to tackle the problem of evil in a world created by a good God.
- Leibniz claims that the universe had to be imperfect, otherwise it would not be distinct from God.
- Leibniz is aware that this argument looks unlikely - surely a universe in which nobody is killed by floods is better than the present one, but still not perfect.
- In 1714 Leibniz wrote Monadologia Ⓣ(Monadologia) which synthesised the philosophy of his earlier work, the Théodicée Ⓣ(Theodicy).
- Much of the mathematical activity of Leibniz's last years involved the priority dispute over the invention of the calculus.
- In 1711 he read the paper by Keill in the Transactions of the Royal Society of London which accused Leibniz of plagiarism.
- Leibniz demanded a retraction saying that he had never heard of the calculus of fluxions until he had read the works of Wallis.
- whence Leibniz derived the principles of that calculus or at least could have derived them.
- Leibniz wrote again to the Royal Society asking them to correct the wrong done to him by Keill's claims.
- It was totally biased, not asking Leibniz to give his version of the events.
- The report of the committee, finding in favour of Newton, was written by Newton himself and published as Commercium epistolicum near the beginning of 1713 but not seen by Leibniz until the autumn of 1714.
- Leibniz published an anonymous pamphlet Charta volans setting out his side in which a mistake by Newton in his understanding of second and higher derivatives, spotted by Johann Bernoulli, is used as evidence of Leibniz's case.
- Leibniz refused to carry on the argument with Keill, saying that he could not reply to an idiot.
- However, when Newton wrote to him directly, Leibniz did reply and gave a detailed description of his discovery of the differential calculus.
- When Leibniz died in Hannover, his only mourner was his secretary, and an eyewitness wrote: He was buried more like a robber than what he really was, the ornament of his century.

Born 1 July 1646, Leipzig, Saxony (now Germany). Died 14 November 1716, Hannover, Hanover (now Germany).

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Algebra, Analysis, Ancient Indian, Astronomy, Geography, Geometry, Origin Germany, Number Theory, Physics, Set Theory, Special Numbers And Numerals

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Propositions: 6

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive