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.