Rebecca Kratzer

Ellsworth High School

Calculus Class

December 5, 2000

Gottfried Wilhelm von Leibniz

Gottfried Wilhelm von Leibniz was considered a German philosopher, mathematician, universal genius, founder of modern science, and statesman. His work encompassed not only mathematics and philosophy, but also theology, law, diplomacy, politics, history, philology, and physics. Leibniz was born on July 1, 1646 in Leipzig, Saxony (now Germany). He died on November 14, 1716 in Hanover, Hanover (now Germany) at the age of 70. He was the son of Friedrich Leibniz and Catharina Schmuck. His father died when he was six years old and he was brought up primarily by his mother. Leibniz was educated at the universities of Leipzig, Jena, and Altdorf. He received his doctorate in law in February of 1667.

In 1672, Leibniz wished to go to Paris to make more scientific contacts. He had begun construction of a calculating machine. He set up a plan to try to persuade the French to attack Egypt for an excuse to go to Paris. It was during this duration in Paris that Leibniz developed the basic features for his version of the calculus. On November 21, 1675, he wrote a manuscript using the ƒf(x) dx notation for the first time. The product rule for differentiation was also given in this manuscript. By autumn of 1676 Leibniz had discovered the familiar d(xⁿ)=nx^n-1dx for both integral and fractional n.

Also during this time, Isaac Newton was working on his calculus discoveries. After Leibniz was published, and the work got to Newton, Newton wrote a letter to Leibniz. This letter took some time to reach Leibniz. The letter listed many of Newton’s results, but it did not describe his methods. Leibniz replied immediately, but not realizing that his letter had taken a long time to reach Leibniz, Newton thought that he had six weeks to reply. Unquestionably a consequence of Newton’s letter was that Leibniz recognized that he must quickly publish a fuller account of his own methods. Newton wrote Leibniz a second letter on October 24, 1676 which did not reach Leibniz until he was in Hanover in June of 1677. This letter was polite in tone, but it was clearly written by Newton believing that Leibniz had stolen his methods. Leibniz’s reply gave some details of the principles of his differential calculus including the rule for differentiating a function of a function.

One of Leibniz’s greatest achievements in mathematics was his development of the binary system of arithmetic. By 1679 this system was perfected, but Leibniz did not publish anything until 1701. Another of his major mathematical accomplishments was his work on determinants which arose from his developing methods to solve systems of linear equations. Leibniz developed many different approaches to the topic with many different notations being tried to find the one which was most useful, although he never published this work in his lifetime. An unpublished paper dated January 22, 1684 contains very sufficient notation and results. Leibniz continued to perfect his metaphysical system in the 1680s attempting to reduce reasoning to an algebra of thought.

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.

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. From 1715 until his death, Leibniz corresponded with Samuel Clarke, a supporter of Newton, on time, space, freewill, gravitational attraction across a void, and other topics.

Bibliography

http://encarta.msn.com/find/Concise.asp?ti=06023000

http://euler.ciens.ucv.ve/English/mathematics/leibniz.html

http://www-history.mcs.st-and.ac.uk/~history/mathematicians/Leibniz.html