Born: 1 July 1646
in Leipzig, Saxony (now Germany)

Died: 14 Nov 1716
in Hannover, Hanover (now Germany)

Gottfried Leibniz was the son of Friedrich Leibniz, a professor
of moral philosophy at Leipzig. Friedrich Leibniz:-

...was evidently a competent though not original scholar, who
devoted his time to his offices and to his family as a pious, Christian father.

Leibniz's mother was Catharina Schmuck, the daughter of a lawyer
and Friedrich Leibniz's third wife. However, Friedrich Leibniz died when
Leibniz was only six years old and he was brought up by his mother. 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. He seems to have been
motivated by wanting to read his father's books. As he progressed through school
he was taught Aristotle's logic and theory of categorising knowledge.
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. As well as his school work, Leibniz studied his father's
books. In particular he read metaphysics books and theology books from
both Catholic and Protestant writers.

In 1661, at the age of fourteen, Leibniz entered the University
of Leipzig. It may sound today as if this were a truly exceptionally early age
for anyone to enter university, but it is fair to say that by the standards of
the time he was quite young but there would be others of a similar age. He
studied philosophy, which was well taught at the University of Leipzig, and
mathematics which was very poorly taught. Among the other topics which were
included in this two year general degree course were rhetoric, Latin,
Greek and Hebrew. He graduated with a bachelors degree in 1663 with a thesis De
Principio Individui (On the Principle of the Individual) which:-

... emphasised the existential value of the individual, who is
not to be explained either by matter alone or by form alone but rather by his
whole being.

In this there is the beginning of his notion of
"monad". 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. He
was awarded his Master's Degree in philosophy for a dissertation which combined
aspects of philosophy and law studying relations in these subjects with
mathematical ideas that he had learnt from Weigel. A few days after Leibniz
presented his dissertation, his mother died.

After being awarded a bachelor's degree in law, Leibniz worked
on his habilitation in philosophy. His work was to be published in 1666
as Dissertatio de arte combinatoria (Dissertation on the combinatorial art). 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. It is a little unclear why
this happened. It is likely that, as one of the younger candidates and there
only being twelve law tutorships available, he would be expected to wait
another year. However, there is also a story that the Dean's wife persuaded the
Dean to argue against Leibniz, for some unexplained reason. 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. He served as secretary to the Nuremberg
alchemical society for a while (see [188]) then he met Baron Johann Christian
von Boineburg. By November 1667 Leibniz was living in Frankfurt, employed by
Boineburg. During the next few years Leibniz undertook a variety of different
projects, scientific, literary and political. He also continued his law career
taking up residence at the courts of Mainz before 1670. One of his tasks there,
undertaken for the Elector of Mainz, was to improve the Roman civil law code
for Mainz but:-

Leibniz was also occupied by turns as Boineburg's secretary,
assistant, librarian, lawyer and advisor, while at the same time a personal
friend of the Baron and his family.

Boineburg was a Catholic while Leibniz was a Lutheran but Leibniz
had as one of his lifelong aims the reunification of the Christian Churches and
:-

... with Boineburg's encouragement, he drafted a number of
monographs on religious topics, mostly to do with points at issue between the
churches...

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. In 1671 he published Hypothesis Physica Nova (New Physical
Hypothesis). In this work he claimed, as had Kepler, that movement
depends on the action of a spirit. He communicated with Oldenburg, the
secretary of the Royal Society of London, and dedicated some of his scientific
works to The Royal Society and the Paris Academy. Leibniz was also in contact
with Carcavi, the Royal Librarian in Paris. As Ross explains in :-

Although Leibniz'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.

Leibniz 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 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. Also in the autumn of 1672, Boineburg's
son was sent to Paris to study under Leibniz 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 Leibniz continued to be supported by the
Boineburg family.

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. 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 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. 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. 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. 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 Leibniz discovered the familiar d(xn) = nxn-1dx for
both integral and fractional n.

Newton wrote a letter to Leibniz, through Oldenburg, which
took some time to reach him. The letter listed many of Newton's results
but it did not describe his methods. 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.

Newton was to claim, with justification, that

..not a single previously unsolved problem was solved ...

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. This does not appear until the work of d'Alembert.

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. He left Paris in October 1676 making the journey to
Hanover via London and Holland. The rest of Leibniz'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 second-hand libraries, and
conventional cataloguing.

He undertook a whole collection of other projects however. For
example one major project begun in 1678-79 involved draining water from the
mines in the Harz mountains. His idea was to use wind power and water power to
operate pumps. He designed many different types of windmills, pumps, gears
but:-

... every one of these projects ended in failure. Leibniz
himself believed that this was because of deliberate obstruction by
administrators and technicians, and the worker's fear that technological
progress would cost them their jobs.

In 1680 Duke Johann Friedrich died and his brother Ernst August
became the new Duke. The Harz project had always been difficult and it failed
by 1684. 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. During this work he formed the hypothesis that the Earth was
at first molten.

Another of Leibniz's great achievements in mathematics was his
development of the binary system of arithmetic. He perfected his system by 1679
but he did not publish anything until 1701 when he sent the paper Essay d'une
nouvelle science des nombres to the Paris Academy to mark his election to the
Academy. Another major mathematical work by Leibniz was his work on
determinants which arose from his developing methods to solve systems of linear
equations. Although he never published this work in his lifetime, he developed
many different approaches to the topic with many different notations being
tried out to find the one which was most useful. An unpublished paper dated 22
January 1684 contains very satisfactory notation and results.

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).

Another major project which Leibniz undertook, this time for
Duke Ernst August, was writing the history of the Guelf family, of which the
House of Brunswick was a part. He made a lengthy trip to search archives for
material on which to base this history, visiting Bavaria, Austria and Italy
between November 1687 and June 1690. As always Leibniz took the opportunity to
meet with scholars of many different subjects on these journeys. In Florence,
for example, he discussed mathematics with Viviani who had been
Galileo's last pupil. Although Leibniz published nine large volumes of archival
material on the history of the Guelf family, he never wrote the work that was
commissioned.

In 1684 Leibniz published details of his differential calculus
in Nova Methodus pro Maximis et Minimis, itemque Tangentibus... in Acta
Eruditorum, a journal established in Leipzig two years earlier. The paper
contained the familiar d notation, the rules for computing the derivatives of
powers, products and quotients. However it contained no proofs and Jacob
Bernoulli called it an enigma rather than an explanation.

In 1686 Leibniz published, in Acta Eruditorum, a paper dealing
with the integral calculus with the first appearance in print of the notation.

Newton's Principia appeared the following year.
Newton's 'method of fluxions' was written in 1671 but Newton failed to
get it published and it did not appear in print until John Colson produced an
English translation in 1736. 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. He criticised Descartes' ideas of
mechanics and examined what are effectively kinetic energy, potential energy
and momentum. This work was begun in 1676 but he returned to it at various
times, in particular while he was in Rome in 1689. It is clear that while he
was in Rome, in addition to working in the Vatican library, Leibniz worked with
members of the Accademia. He was elected a member of the Accademia at this
time. Also while in Rome he read Newton's Principia. His two part
treatise Dynamica studied abstract dynamics and concrete dynamics and is
written in a somewhat similar style to Newton's Principia. Ross writes in
:-

... 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. ... It was only by simplifying the
issues... that Newton succeeded in reducing them to manageable
proportions.

Leibniz put much energy into promoting scientific societies. He
was involved in moves to set up academies in Berlin, Dresden, Vienna, and St
Petersburg. He began a campaign for an academy in Berlin in 1695, he visited
Berlin in 1698 as part of his efforts and on another visit in 1700 he finally
persuaded Friedrich to found the Brandenburg Society of Sciences on 11 July.
Leibniz was appointed its first president, this being an appointment for life.
However, the Academy was not particularly successful and only one volume of the
proceedings were ever published. It did lead to the creation of the Berlin
Academy some years later.

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. Similarly he did much of the
work to prompt the setting up of the St Petersburg Academy, but again it did
not come into existence until after his death.

It is no exaggeration to say that Leibniz corresponded with most
of the scholars in Europe. He had over 600 correspondents. Among the
mathematicians with whom he corresponded was Grandi. The correspondence
started in 1703, and later concerned the results obtained by putting x = 1 into
1/(1+x) = 1 - x + x2 - x3 + .... Leibniz also corresponded with Varignon
on this paradox. Leibniz discussed logarithms of negative numbers with
Johann Bernoulli, see [156].

In 1710 Leibniz published Théodicée 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. He then claims that the universe is the best possible
without being perfect. 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. His argument here is that the elimination
of natural disasters, for example, would involve such changes to the laws of
science that the world would be worse. In 1714 Leibniz wrote Monadologia which
synthesised the philosophy of his earlier work, the Théodicée.

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. Keill replied to Leibniz saying that the two
letters from Newton, sent through Oldenburg, had given:-

... pretty plain indications... 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. In response to this letter the
Royal Society set up a committee to pronounce on the priority dispute. 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. He learnt
of its contents in 1713 in
a letter from Johann Bernoulli, reporting on the copy of the work brought
from Paris by his nephew Nicolaus(I) Bernoulli. 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.

The argument continued with Keill who published a reply to
Charta volans. 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. From 1715 up until his death Leibniz
corresponded with Samuel Clarke, a supporter of Newton, on time,
space, freewill, gravitational attraction across a void and other topics, see,
, and .

In Leibniz is described as follows:-

Leibniz was a man of medium height with a stoop,
broad-shouldered but bandy-legged, as capable of thinking for several days
sitting in the same chair as of travelling the roads of Europe summer and winter.
He was an indefatigable worker, a universal letter writer (he had more than 600
correspondents), a patriot and cosmopolitan, a great scientist, and one of the
most powerful spirits of Western civilisation.

Ross, in , points out that Leibniz's legacy may have not been
quite what he had hoped for:-

It is ironical that one so devoted to the cause of mutual
understanding should have succeeded only in adding to intellectual chauvinism
and dogmatism. There is a similar irony in the fact that he was one of the last
great polymaths - not in the frivolous sense of having a wide general
knowledge, but in the deeper sense of one who is a citizen of the whole world
of intellectual inquiry. He deliberately ignored boundaries between
disciplines, and lack of qualifications never deterred him from contributing
fresh insights to established specialisms. Indeed, one of the reasons why he
was so hostile to universities as institutions was because their faculty
structure prevented the cross-fertilisation of ideas which he saw as essential
to the advance of knowledge and of wisdom. The irony is that he was himself
instrumental in bringing about an era of far greater intellectual and
scientific specialism, as technical advances pushed more and more disciplines
out of the reach of the intelligent layman and amateur.

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