Gottfried Wilhelm Leibniz (1646 - 1716)

Leibniz was a contemporary of Newton and the story of their simultaneous work on the calculus, delays in receiving letters and replying to them, the intervention of other people as supporters, makes a real mystery ‘who dunnit’ that divided mathematicians then and now.

Without doubt they each made fundamental contributions to the establishment of the powerful subject we call the calculus, but it is mainly Leibniz’s notation that is commonly used. Despite this, Newton’s contribution was more profound and more fundamental.

Leibniz lost his father at a young age and that may have motivated him to immerse himself in self-study - covering subjects such as Latin, Greek and the available mathematics of that period.

Leibniz went to college at 15 to study Law, and by the age of 20 went to the University of Leipzig - but the authorities thought he was too young to receive his Law doctorate, so he moved, again, to the University of Altdorf in Nuremberg.

This move was a successful one as the college awarded him his doctorate and even offered him a position within the Law faculty. Strangely enough Leibniz turned down this job - preferring do to some legal work in the local community!

Leibniz’s life took an alternative route at this point. When he dedicated a paper on law to the Elector of Mainz, the Elector recruited Leibniz into the diplomatic world. In 1672, while in Paris on a diplomatic mission, Leibniz met Christian Huygens, a prominent physicist, who was persuaded to teach the young Leibniz the latest in mathematics. After Paris came a posting in London where again he was fortunate enough to be well received by the Royal Society and its members.

During the following years Leibniz’s new found interest in mathematics blossomed and eventually paved the way for his discoveries within the field of calculus. Initially, the starting point of calculus was to find a mathematical method of describing change - change that happened continuously.

Newton’s book on the calculus, describing what he called his ‘method of fluxions’ (Newton used the word ‘flux’ for change) was written in 1671 but did not appear in print until John Colson produced an English translation in 1736. This time delay in the publication of Newton’s work was at the heart of the dispute with Leibniz. What does seem certain, now, is that they had both worked independently and had equal claim to the subject.

Leibniz was still a diplomat at heart, and his endeavours had a more serious note. At a conference in 1683, he attempted to reunite the Catholic and Protestant churches which had split apart, but an agreement could not be reached and the two remain separate. Another attempt at reconciliation took place not long after, when Leibniz tried to bring together the two main factions within the Protestant church - again failure followed.

It is a sobering fact that despite all his endeavours and achievements, diplomatic and mathematical, his funeral was allegedly attended by only his secretary.

Leibniz’s mathematics

Lots of physical applications required Leibniz’s idea - the motion of a body that changed its position, the temperature of a hot liquid as it cooled. Giving the means to describe such change would lead to fundamental discoveries about the way the world behaves. Suppose that a function f describes something that changes - position, speed or temperature for example. It helps to picture the graph of such a function:

The rate of change of the function is measured by the tangent at any point - the steeper this is, the faster the function is growing. The tangent can be calculated as the limit of chords - one, shown dotted. The calculus seeks another function - sometimes called the derived function, or derivative - that measures this tangent at any point. Leibniz wrote this as {d\over{dx}}f(x); Newton preferred f^\prime(x). This led to the great controversy of Leibniz’s life - the question of who had originally ‘invented’ calculus. It was during the 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

\int f(x)dx

notation for the first time. This examined the inverse to finding the tangent, that is a function for which the given function was the tangent or derived function. In geometric terms this surprisingly turns out to be the area ‘under the graph’ of the given function. In the same manuscript the product rule for differentiation is given. By autumn 1676 Leibniz discovered the familiar

{d\over{dx}}\left(x^n\right)=nx^{n-1}

for both whole and fractional n. Letters written by Newton to Leibniz were claimed by the Royal Society and British mathematicians to be evidence that Leibniz had merely duplicated Newton’s work. This claim was undermined by the fact that a Royal Society committee set up to investigate had Newton write its final report.

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.

Such an arithmetical system now lies at the heart of computers. The system uses only the digits 0 and 1, and writes numbers as powers of two rather that the decimal equivalent of powers of ten. For example,

37=3.10+7; \mbox{ in binary notation is, } 100101=1.32+0.16+0.8+1.4+0.2+1.1

You might like to try some binary division and multiplication:

10100 divided by 11 and 10100 multiplied by 11.

The digit 1 in a computer is marked by the flow of current; 0 is marked by the absence of current. Digital computers using this idea provide many of the things we now regard as essential parts of our world - mobile telephones; digital TV, the internet, programmable video recorders…