Gottfried Wilhelm Leibniz (1646-1716)



Wesley Lee

Ann Slate

July 3, 2008

MAT 5910 FINAL PROJECT

The Lives of Sir Isaac Newton and Gottfried Wilhelm Leibniz

Their Contributions to Calculus

Sir Isaac Newton

Christmas Day 1642 in Woolsthorpe England, a widow gave birth to one of the most remarkable mathematicians of all time. Little did his mother, Hanna, know when she placed her firstborn with his grandmother when he was merely three years old that she had given birth to a son that would later be known as the most influential and original theorist in science. As the keystone in the Scientific Revolution, Newton combined ideas from previous scholars such as Descartes and Copernicus into a new idea.

Newton’s life seemed to begin turmoil. At the time of his birth, England was in the midst of a civil war, which led to a commonwealth under Oliver Cromwell. In the universities the Aristotelian view of science was dominant, and new ideas were looked down upon. At an early age Newton was withdrawn from school to fulfill his duty as a farmer. To Newton’s comfort he failed miserably as a farmer and was soon re-enrolled to the King’s School at Grantham to prepare him for entrance into Cambridge. During his undergraduate studies at Cambridge Newton was likely bombarded with Aristotelian views, and those of other classical authors. Newton’s professors were highly disappointed with his lack of achievements in school especially those studies of Euclid. Newton, rather than studying Euclid, spent most of his time privately studying works from Rene Descartes, Pierre Gassendi, Thomas Hobbes, and other major figures of the scientific revolution. By the 1660’s Newton had begun to master Descartes’ works and other form of mathematics far advanced from the works of Euclid.

After graduation in 1665, Newton returned home to Woolsthorpe because of the plague at Cambridge. For 18 months Newton devoted his time to studying and research, and it was during this time that Newton conceived his idea of Calculus. Two years after returning to Cambridge where he became a fellow of Trinity in 1667, Newton was appointed second Lucasian Professor of Mathematics. It wasn’t until 1672 when Newton’s name began to ring throughout the scientific community after his invention of the reflecting telescope. With this, Newton was elected a member of the Royal Society. The Royal Society consisted of individuals mostly from Oxford University and the mathematician John Wallis was one of the leaders. In 1662, King Charles II granted a royal Charter of incorporation and presented a royal mace, a treasure still recognized to this day. After his election to the royal society Newton published his first paper, a controversial study on the nature of light. Newton seemingly could not cooperate with the Society’s curator of experiments, Robert Hooke, whom did not agree with Newton’s views. After being accused of plagiarizing from Hooke throughout his second paper, Newton later withdrew from the Society.

In 1678 Newton lost his mother and decided to cut off all contact from others, and focus on research. Once an embarrassment to Newton’s scholars, this research led to the Laws of motion that we now see are apparent everyday in nature. Planetary motion was also discovered by Newton. Through letters to Hooke and other members of the Royal Society, Newton stated that planets moved in an elliptical motion around the sun, an idea that Hooke and others could not comprehend. With promises to send them a copy of proof to how he knew this, Newton published de Motu in 1684. From this foundational work Newton produced the legendary Philosophiae Naturalis Principia Mathematica, one of the most important books in the history of science. After publication Hooke, with no justification, claimed that his letters back and forth to Newton earned him a role in Newton’s discovery. Newton was furious with Hooke’s presumptions and Newton deleted every mention of his Hooke’s name.

After publishing Principia Newton was elected to represent Cambridge in the Parliament in 1689. During his stay in London he became acquainted with philosopher John Locke. In 1693 Newton suffered a severe nervous disorder. Several controversies exist over the cause of this disorder, stress or even mercury poisoning from his many scientific experiments. Shortly after his recovery Newton was appointed Warden and then Master of the Mint and left Cambridge for London without regret. With this position Newton lived a comfortable economic and social lifestyle. Soon after the death of Hooke, Newton was elected president of the Royal Society. With the power that he held being president of the Royal Society Newton seemingly got away with acts that would have been easily disputed if it were not for his position. With this Newton was able to advance his ideas of Calculus shadowing the ideas of Liebniz about the subject. Until his death on March 20, 1727, Newton dominated the science world without competition.

Newton’s most capable adversary Leibniz began publishing papers nearly 20 years after Newton’s commencement. Their dispute over the subject delayed Newton’s publication of the matter. Rumors have it that Leibniz borrowed ideas from Newton and quickly published them, before Newton. For a while disputes that it was pure plagiarism and dishonesty floated throughout the science world. Now it is generally agreed that they each developed calculus independently.

Gottfried Wilhelm Leibniz (1646-1716)

History:

Gottfried Wilhelm Leibniz was born in Leipzig, Germany in 1646. His father, a professor of moral philosophy, died when Leibniz was 6. Leibniz taught himself Latin at age 8 and Greek and age 12 in order to gain access to his father’s extensive library. Many of these books were on topics such as metaphysics and theology.

Leibniz entered the University of Leipzig in the fall of 1661 when he was roughly 15 years old. He was young to be entering the university, but entering at this age was more common during the 17th century. One source identifies Leipzig as a “prodigy” (Burton, 2007.) He finished his bachelor’s degree in 1663, and earned a master’s degree in legal studies a year later. He qualified for a teaching position in philosophy at the university by defending his work on using mathematics (particularly permutations and combinations) for making logical deductions. This new language of reasoning hypothesized that all scientific concepts could be expressed using combinations of the basic alphabet of human thought which combines numbers, letters, sounds, and colors (Burton, 2007; O’Conner & Robertson, 1998.) This language is linked to George Boole and his Calculus of Logic developed in the mid-nineteenth century, and currently widely used in computer programming and technology.

Leibniz left Leipzig when he was not awarded a doctorate of law. He landed at the University of Altdorf where he was awarded a Doctorate in Law in February, 1667. He was offered a position teaching at the University, but declined to pursue other avenues. During his lifetime, Leibniz was involved in scientific discovery, literary research, and political endeavors. He held positions as librarian, legal counsel, historian, engineering consult, and diplomat, all while continuing to explore academic pursuits.

Two of Leibniz’s life goals were the reunification of the Christian churches and a collation of human knowledge. He was considered by many to be a Renaissance Humanist, defined as someone suspended between faith and reason (Kreis, 2004.) His projects included an attempt at creating an early calculating device (unfinished), development of wind and water powered pumps to help reduce water levels in underground mines (unsuccessful due to resistance and sabotage by mine workers), efforts in geology in which he hypothesized that the Earth was initially molten, and publications of Discourse in Metaphysics in 1686 and Theodicy in 1710 which examine the existence and role of God, and state that while our created world is not perfect, it is the least of all possible evils (O’Conner and Robertson, 1998; Burton, 2007.)

General Mathematical Background and Contributions:

Leibniz’s early mathematical instruction in Leipzig is described as elementary. He did not truly start to investigate mathematics until traveling to Paris in 1672. While in Paris for diplomatic reasons, he also made several scientific and mathematical contacts. The Dutch mathematician Christiaan Huygens became a friend and mentor to Leibniz. Huygens recognized Leibniz’s deficiencies in mathematical understanding but also realized his potential genius. Huygens encouraged Leibniz to study existing mathematical publications, including works by Blaise Pascal and translations of Descartes’ Géométrie (Burton, 2007.) Leibniz was motivated by a desire to expand his knowledge and level of understanding of mathematics, and quickly excelled in his comprehension of and research in mathematics.

In addition to his contributions to calculus (discussed in a separate section) Leibniz made other significant mathematical contributions. In an unpublished document from 1684 Leibniz worked with determinants in solving systems of linear equations. He illustrated satisfactory notation and results in his documentation that had not been used previously. Material published in 1701 discusses his ideas of binary arithmetic, which are used in computer languages today. Leibniz also studied dynamics and worked extensively with the areas of kinetic energy, potential energy, and momentum from 1676 through at least 1689 (O’Conner & Robertson, 1998.)

Contributions to Calculus:

During his time in Paris, Leibniz began developing his ideas for calculus. He attributes the writings of Pascal on infinite series for inspiring much of his work (Burton, 2007.) In November of 1675, Leibniz published a manuscript containing the notation ∫f(x) dx for integrals. In 1676 Leibniz develops the notation and understanding of d(xn)=nxn-1dx for derivatives. While Leibniz could successfully take higher order derivatives, he did not consider the derivative to be a limit. In 1684, the Nova Methodus pro Maximus et Minimus, Intemque Tangentibus was published. This document contained the familiar dx notation, and also rules for computing derivatives of powers, products and quotients but contained no proofs. One source credits Jacob Bernoulli with describing the work as an enigma rather than an explanation (O’Conner & Robertson, 1998.)

The work previously completed by Newton had focused primarily on the geometric properties of calculus. Leibniz’s method moved calculus away from the geometric perspective and allowed the achievement of results without as much effort (Kreis, 2004.)

The Scandal

While visiting London on diplomatic pursuits, Leibniz became aware of work Newton had done regarding fluxions and fluents. The results Newton achieved were similar to what Leibniz was working on for his calculus. Through connections, Leibniz requested information from Newton regarding his work. On June 13, 1676, Newton wrote to Leibniz listing his results, but not his methods. The letter took several weeks to reach Leibniz. Although Leibniz responded immediately, Newton mistakenly thought that Leibniz had taken roughly six weeks to formulate his response. It was at this point that Leibniz realized he needed to publish a “fuller” account of his methods to illustrate the differences between his work and Newton’s (Burton, 2007; O’Conner & Robertson, 1998.)

Newton responded with a second letter written on October 24, 1676. This letter did not reach Leibniz until June of 1677. In this letter, Newton implied that Leibniz had stolen his methods. Leibniz’s reply included details of his differential calculus including a rule for differentiating a composite function. In receiving Leibniz’s response, Newton stated that, “…not a previously unsolved problem was solved…” by Leibniz’s approach. It was not determined until later that the letters from Newton to Leibniz, which passed through the intermediary Henry Oldenburg, had been edited and condensed because of their size. This discovery contributed to the vindication of Leibniz because there was not enough material in the communications for him to gain an understanding of Newton’s work (O’Conner & Robertson, 1998.)

In 1687, Newton published his Principia exposing his method of fluxions and fluents. This information was originally written and shared with his friends and colleagues in 1671, but not published. Newton’s delay in publishing this work is part of what generated the controversy over priority.

In 1711, John Keill, a Newton supporter, published work accusing Leibniz of plagiarism. Upon learning of this charge, Leibniz approached the Royal Society about having Keill publish a retraction. Keill responded with another public accusation stating that Newton’s two letters to Leibniz provided enough information that Leibniz could have easily derived the principles of Newton’s calculus from them. At this time, Keill is referring to the letters in their entirety, not the edited versions received by Leibniz.

Leibniz, a longstanding foreign member of the Royal Society, again asked for retraction of the plagiarism charges. The Royal Society appointed a committee to investigate the issue. It is noteworthy to mention that at this time Newton was president of the Royal Society. The committee was comprised of Newton supporters and the final opinion awarding priority to Newton was penned by Newton himself. Leibniz was not informed of the decision, but found out the following year in communication from John Bernoulli.

In retaliation, Leibniz anonymously published the Charta Volans explaining his side. To support his work, he includes information regarding Newton’s incorrect method of taking higher order derivatives. This work is supported by John Bernoulli. John Keill tried to respond to the Charta Volans, but Leibniz refused to communicate with Keill.

Newton’s dislike of the public forum initially fueled his decision not to pursue priority. However, the pressure and innuendo of supporters like Keill and John Wallis forced him to denounce Leibniz’s work and claim priority over the development of calculus. Leibniz and Newton did eventually communicate regarding the matter following the public controversy. Today, both men are given priority when discussing the development of calculus.

Sources:

Leibniz:

Burton, D. M. (2007). The History Of Mathematics, (6th ed). Boston: McGraw Hill Higher Education.

Kreis, S. (2004). Renaissance Humanism. The History Guide: Lectures on Modern European Intellectual History. Retrieved June 23, 2008 from intellect/humanism.html.

O’Conner, J.J. & Peterson, E.F. (1998). Leibniz Biography. Retrieved June 23, 2008, from history.mcs.st-andrews.ac.uk/Biographies/Leibniz.html.

Newton:

Brabenec, R. L. (2004). Resources for the Study of Real Analysis. Massachusetts: Cambridge University Press.

Hatch, Robert. "Sir Isaac Newton." Professor Robert A. Hatch. 1998. University Of Florida. Retrieved June 30, 2008 from clas.ufl.edu/users/rhatch/pages/01-Courses/current-courses/08sr-newton.htm.

O'Connor, J.J. & Robertson, E. F. (2000). "Sir Isaac Newton." University of St Andrews, Scotland. University of St Andrews, Scotland. Retrieved June 30, 2008 from www-history.mcs.st-andrews.ac.uk/Biographies/Newton.html.

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