Preface
The purpose of this paper is to evaluate and analyze the scientific controversy of the creation and writing of “the” Calculus. The “priority dispute” over who wrote the scientific work that became Calculus was between Sir Isaac Newton and Gottfried Wilhelm Leibniz. This dispute is no longer a controversy, though it is still researched and debated by historians to this day. As documentation held by the repositories (Cambridge University, UK; University of Sussex, UK; The National Library of Israel, Israel; University of Kings College, Nova Scotia Canada) of Newton’s Papers has been transcribed and analyzed, the evidence has been accepted, and the verdict rendered by historians including Westfall, Manuel, Keynes, Illiffe, and Conduitt that Isaac Newton was the first to substantially discover and document the Calculus. But even Newton admitted, as did Leibniz, that they owed debts to “Isaac Barrow, John Wallis and others.” To paraphrase Newton himself, science itself stands on other giant shoulders, although Newton’s original quote was in fact a sarcastic comment written to Robert Hooke.
Introduction
The Newton versus Leibniz Calculus controversy is perhaps one of the most infamous priority disputes in the history of science. The Calculus controversy emerged largely due to the timing of the publications of both Newton and Leibniz’s work on Calculus. Historically it is said that Newton made his discoveries around 1672 but did not have them published until 1693. While Leibniz made his discoveries later in 1684, after Newton, but they were published in 1686 before Newton’s work. Due to the impact of the discovery and priority claims of the two claimants, the mathematical community began to question if Leibniz had stolen the ideas from Newton or if he had in fact developed them on his own. During the 17th century debates between scholars were common, but it was the issues between Newton and Leibniz that many found surprising. The controversy was not only intense but continued for a lengthy amount of time . Newton and Leibniz were not only arguing over the invention of Calculus but also did not agree on other topics including Newton’s action-at -a-distance theory. Leibniz viewed Newton as reverting to occultism and stifling science. Between the differing world philosophical view and other scientific disagreements the controversy over Calculus only grew.
Sir Isaac Newton
To understand the controversy between Newton and Leibniz the background of both scientists must also be explored. A great deal of national pride played a role in the controversy so analyzing the origins is imperative.
Sir Isaac Newton was born 25 December 1642 in Woolsthorpe, Lincolnshire, England. Newton’s life started abruptly as he was born prematurely and fatherless. The very beginning of Newton’s life was riddled with sadness and separation from his family. His mother left him early with his grandmother and he was born a posthumous son. Many claims have been made about Newton’s mental state due to his unfortunate childhood. Many believe that this is the reasoning for his complex character.
Newton attended school at Cambridge University and was later elected a Fellow of Trinity College in 1667 and Lucasian Professor of Mathematics in 1669. It was during these years that Newton was his most creative. It was this time that Newton wrote Philosaphiae Natrualia Principa Mathematica (Mathematical Principles of Natural Philosophy). Commonly known as Principia, this work of Newton was not published until 1687. Not long after the publication of Philosophiæ Naturalis Principia, Newton's fame exploded throughout England and the European continent and continued around the world. He is still arguably the most famous scientist who ever lived or may ever live. Some assert that it is Isaac Newton who would be deemed the father of the scientific method. His laws of motion alone changed the world and science forever. Principia is as brilliant a work that has had ever been written and only one of the amazing Newtonian papers; yet it was largely unintelligible to the overwhelming majority of those who would even attempt its' understanding.
Newton was elected a Memer of Parliament for the University of Cambridge to the Convention Parliament in 1689. He held some form of Parliamentary position all the way to Warden of the Royal Mint which he held until his death. Newtonian science was increasingly accepted following the end of the War of Spanish Succession. Newton quickly rose during a time of great discovery and philosophical thought as a highly esteemed philosopher .
Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz was born 21 June 1646 in Leipzig, Germany. Leibniz was a German philosopher, politician, theologist, physicist, and mathematician. Leibniz was born at the end of the Thirty Years’ War in Germany which had left the country in ruins. Leibniz younger years were consumed with learning and great tragedy, much like Isaac Newton, as Leibniz’s father died in 1652. Nine years later Leibniz started schooling at the University of Leipzig as a law student. This is where Leibniz found his passion for science and philosophy. Prominent figures such as Galileo, Bacon, Hobbes, and Descartes inspired the young scholar.
Leibniz completed his legal studies in 1666 and applied for his doctorate which he was denied entry for due to his youth. He left his home town to Altdorf where he was immediately given a doctorate for his dissertation De Casibus Perplexis (On Perplexing Cases). After acquiring his doctorate he began to work with the church to stabilize with Demonstrationes Catholicae.
History and Development of The Calculus
The first steps to discovering Calculus were taken by Greek Mathematicians. The Greeks viewed numbers as ratios of integers on a numberline. They saw this as having holes in it and in 450 BC Zeno of Elea argued:
“If a body moves from A to B then before it reaches B it passes through the mid-point, say B1 of AB. Now to move to B1 it must first reach the mid-point B2 of AB1 . Continue this argument to see that A must move through an infinite number of distances and so cannot move.”
It was Archimedes around the time of 225 BC that made one of the most significant contributions to Calculus during the time. His first important advance was to show that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex and 2/3 of the area of the circumscribed parallelogram.Archimedes constructed an infinite sequence of triangles starting with one of area A and continually adding further triangles between the existing ones and the parabola to get areas
A, A + A/4 , A + A/4 + A/16 , A + A/4 + A/16 + A/64 , …
The area of the segment of the parabola is therefore A(1 + 1/4 + 1/42 + 1/43 + ….) = (4/3)A.
This is the first known example of the summation of an infinite series. It was not until the 16th century that Calculus progressed any further. Torricelli and Barrow determined the next big steps to Calculus as they provided a method of tangents to come together as the differential triangle.
Mathematicians such as Valerio, Cavalieri, Roverval, Fermat, and Hudde all contributed to the furthering of Calculus in the early 1600s and influenced the work of Newton and Leibniz.
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Controversy of Discovery
The controversy over who discovered Calculus created an alarmingly heated dispute between the two scientists. Plagiarism in the 17th century took a different light than what it does in the modern day. In the 17th century works that were not officially published could still be under review for plagiarism. Private correspondence and other personal documents that were never meant to be made public were made open to the people. The integrity of documents was stifled by the suspicion of having stolen secrets from a colleague. The controversy over Calculus is also quite heated because of the length of time it spans and the fact that not one singular person created it. With the long and vast history of mathematicians providing breakthroughs it is hard to single out one singular source of discovery.
During the time the controversy erupted, both England and Germany could have used the discovery to their own political advantage, and both countries naturally wished to support their native sons. National pride played a significant role in accentuating the controversy. An entirely new branch of mathematical thinking was at stake and each country wanted to take claim. Due to the poor documentation there is not an exact accounting of what happened in the dispute of who first wrote the Calculus.
When Newton and Leibniz are mentioned in the same conversation it is usually with Calculus in mind. However, their debate goes back to many other disputes over math and philosophy. “The drama of this debate, and the way in which it poisoned numerous relationships between thinkers in England and their colleagues on the Continent, tends to suggest to contemporary readers that Leibniz and Newton were destined for disagreement.” Not all evidence does point to the two scientists becoming bitter enemies. Leibniz and Newton grew up in the same kind of environment fatherless and in the “heyday of Cartesianism” with each arguing in particular that Cartesian views failed to capture the full force of nature.
Newton had problems through his early career of publishing his mathematical works due to publishers being weary of mathematical publications. Newton wrote Tractatus de Quadratura Curvarum in 1693 but was not published until 1704 and was only an Appendix to his more famous work, Optiks. The book Newton is famous for his definition of limits.
“In the time in which x by flowing becomes x+o, the quantity xn becomes (x+o)n i.e. by the method of infinite series, xn + noxn-1 + (nn-n)/2 ooxn-2 + . . .” At the same time Newton was working on his papers; Leibniz was continuing his learning on his European tour meeting many mathematicians and having correspondence with Barrow, the scientist who had most recently published on the differential triangle. Newton and Leibniz were both working on Calculus but challenged the idea of variables in different ways. Newton considered variable as they changed in time while Leibniz thought of variable x and y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a defining property.
Newton was focused on finding fluents for a given fluxion so the fact that integration and differentiation were inverses was implied. Leibniz used integration as a sum, in a rather similar way to Cavalieri. He was also happy to use 'infinitesimals' dx and dy where Newton used x' and y' which were finite velocities. Of course neither Leibniz nor Newton thought in terms of functions, however, but both always thought in terms of graphs. For Newton the Calculus was geometrical while Leibniz took it towards analysis.
Leibniz was very conscious that finding a good notation was of fundamental importance and thought a lot about it. Newton, on the other hand, wrote more for himself and, as a consequence, tended to use whatever notation he thought of on the day. Leibniz's notation of d and ∫ highlighted the operator aspect which proved important in later developments. By 1675 Leibniz had settled on the notation ∫ y dy = y2/2 written exactly as it would be today. His results on the integral Calculus were published in 1684 and 1686 under the name 'Calculus summatorius', the name integral Calculus was suggested by Jacob Bernoulli in 1690.
After Newton and Leibniz the development of the Calculus was continued by Jacob Bernoulli and Johann Bernoulli. However when Berkeley published his Analyst in 1734 attacking the lack of rigour in the Calculus and disputing the logic on which it was based much effort was made to tighten the reasoning. Maclaurin attempted to put the Calculus on a rigorous geometrical basis but the really satisfactory basis for the Calculus had to wait for the work of Cauchy in the 19th Century.
The infinitesimal Calculus can be expressed either in the notation of fluxions or in that of differentials, or, as noted above, it was also expressed by Newton in geometrical form, as in the 'Principia' of 1687. Newton employed fluxions as early as 1666, but did not publish an account of his notation until 1693. The earliest use of differentials in Leibniz's notebooks may be traced to 1675. He employed this notation in a 1677 letter to Newton. The differential notation also appeared in Leibniz's memoir of 1684.
The claim that Leibniz invented the Calculus independently of Newton rests on the facts that Leibniz published a description of his method predating Newton, showed his own private papers of his invention of Calculus, and plea for good faith. The main claimants against him are that of the date differences of original work and the possibility of Leibniz having seen Newton's papers.
The evidence of the discovery is questionable which furthered the controversy. It is definite that Newton wrote his work predating Leibniz but the question is whether Leibniz’s notes changed foundational principles, dating, or if he had access to Newton’s notes before presenting his. However, in either case this does not take away from Leibniz's notes showing he was the first to come to integration, which he saw as a generalization of the summation of infinite series. Newton began from derivatives.
Conclusion
However, it has since been confirmed that it was Isaac Newton that wrote The Calculus first. After the discovery of the Newton Papers and the work of Dr. Sarah Dry in the chronicling of Newton’s life it was confirmed that Newton was the first to make the discovery. In the Newton Papers, Dr. Sarah Dry sheds light on the history of the writings of Isaac Newton. Ranging from alchemy, religion, to The Calculus; the Newton Papers outline the life of Sir Isaac Newton. Due to Newton's development of fluxions theory in 1665 Newton was able to set the standard algorithms that leibniz found two decades later. There was no way Newton could have plagiarised anything from Leibniz. Many historians believe that there is still not enough concrete evidence into whether Leibniz stole the ideas from Newton. However, it is very possible that Leibniz arrived at the idea of Calculus independently from Newton. This does not negate that Leibniz in his own discovered and wrote portions of Calculus that were not yet written by Newton. The development of new science or mathematics is not about