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Merila, Ubaldo Laya, Iligan City

S.Y. 2013- 2014

A Requirement in Mathematics IV:

Calculus Controversy: Leibniz vs. Newton

by

Noronsalih Ali, Jra

Submitted to

Ms. Moneerah A. Bint- Usman

Dedication

I would like to dedicate this research to my adviser, teacher Monie. And to all of the people who inspired me, especially my parents for their support and to God for giving me enough knowledge to make this study successful.

Acknowledgement

Abstract

This research explores more about the history of the two Mathematicians and how did they invent calculus with the same idea. This is a study about a controversy in Mathematics where Sir Isaac and Gottfried von Leibniz were involved. It tackles about who was the real father of calculus and who gets the credit of inventing it.

Inside this paper, the researcher will also discuss a brief summary about Calculus, and short biography of the Mathematicians that were involved in this matter. Many people debates about this matter and we will also tackle some of it in this study.

Introduction:

So who really invented calculus first? Was it Sir Isaac Newton or Gottfried von Leibniz? Well let's do some investigation. There is no doubt about it that Newton and Leibniz made great mathematical breakthroughs but even before they began studying Calculus there were other people such as Archimedes and Euclid who discovered the infinite and infinitesimal. Much of Newton and Leibniz's work was based off of previous discoveries of the subject.

In this controversy, inventors were almost always treated with suspicion, it is interesting to observe that from 1676, when Newton and Leibniz ﬁrst corresponded until about 1704 when the dispute was full-blown, neither Newton nor Leibniz considered the other to be guilty of plagiarism. Both seemed content in assuming that the other had independently arrived at his formulation of the calculus. However, by 1712, when the controversy was at its peak, each had accused the other of plagiarism. The relations between the two had soured to such an extent that Boyer views the resulting controversy to be “shamefully bitter”.

This study aims to answer the following questions: * What is Calculus? * Who invented Calculus? * Who are Leibniz and Newton? * Why we need Calculus?

I. Calculus:

According to Sherry (2007), Calculus is the mathematical study of change, in the same way that geometryis the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus, and integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally considered to have been founded in the 17th century by Isaac Newton and Gottfried Leibniz, today calculus has widespread uses in science, engineering and economics and can solve many problems that algebra alone cannot.

He added that Calculus was a part of modern mathematics education. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". The word "calculus" comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, calculus of variations, lambda calculus, and process calculus.

History of Calculus The history of Calculus does not begin with Newton and Leibniz’s findings. Their calculus was the culmination of centuries of work by other mathematicians rather than an instant epiphany that came individually to them. In 1600 B.C, the Rhind Papyrus showed that the ancient Egyptians knew that the volume of a rectangular pyramid volume is equal to 1/3 of the volume of a rectangular prism having the same base and height, but not how they reached this conclusion. In around 200 B.C, the ancient Greek also discovered that the diagonal of many geometric figures would never be a whole number and that determining this value involved an infinite process. Then many years after, many other nations also gave contribution to calculus even before this two mathematicians were born. By the early seventeenth century, everything was in place for Newton and Leibniz to take all the ideas of limited scope and compile them into methods of universal applicability.

II. The Great Controversy

The first ideas of calculus began with the Greek Mathematician Archimedes. He then found many formulas of how to solve a calculus, such as the formulas for the volume and surface area of a sphere. In the later 1500’s and 1600’s, many mathematicians worked on problems that called for Calculus.

Like most discoveries, calculus was the culmination of centuries of work rather than an instant epiphany. Mathematicians all over the world contributed to its development, but the two most recognized discoverers of calculus are Isaac Newton and Gottfried Wilhelm Leibniz. Although the credit is currently given to both men, there was a time when the debate over which of them truly deserved the recognition was both heated and widespread.

As the renowned author of Principia (1687) as well as a host of equally esteemed published works, it appears that Newton not only went much further in exploring the applications of calculus than Leibniz did, but he also ventured down a different road. Leibniz and Newton had very different views of calculus in that Newton’s was based on limits and concrete reality, while Leibniz focused more on the infinite and the abstract. However, regardless of the divergent paths these two scholars chose to venture down, the question of who took the first step remained the primary issue of debate.

According to Ball (1908), Newton was reported to have discovered similar methods, Leibniz discovered “his” calculus in Paris between 1673 and 1676. By 1676, Leibniz realized that he was onto something “big”; he just didn’t realize that Newton was on to the same big discovery because Newton was remaining somewhat tight lipped about his breakthroughs. Boyer (1968) added the fact that it was actually the delayed publication of Newton’s findings that caused the entire controversy. Leibniz published the first account of differential calculus in 1684 and then published the explanation of integral calculus in 1686. Struik (1948) says that Newton did not publish his findings until 1687. Yet evidence shows that Newton discovered his theories of fluxional calculus in 1665 and 1666, after having studied the work of other mathematicians such as Barrows and Wallis. Evidence also shows that Newton was the first to establish the general method called the "theory of fluxions" was the first to state the fundamental theorem of calculus and was also the first to explore applications of both integration and differentiation in a single work. However, since Leibniz was the first to publish a dissertation on calculus, he was given the total credit for the discovery for a number of years. This later led, of course, to accusations of plagiarism being hurled relentlessly in the direction of Leibniz.

Boyer (1968) also added that there was speculation that Leibniz may have gleaned some of his insights from two of Newton's manuscripts on fluxions, and that that is what sparked his understanding of calculus. Many believed that Leibniz used Newton's unpublished ideas, created a new notation and then published it as his own, which would obviously constitute plagiarism. The rumor that Leibniz may have seen some of Newton's manuscripts left little doubt in most people’s minds as to whether or not Leibniz arrived at his conclusions independently. The rumor was, after all, believable because Newton had admittedly bounced his ideas off a handful of colleagues, some of who were also in close contact with Leibniz.

National pride played a great role in the exacerbation of the dispute. Those involved realized that credit for the discovery of the whole new branch of mathematics was a stake, and each side wanted their country to get this credit. In 1711, the controversy was taken to court. A commission was appointed by the Royal Society, it is not all that surprising that Leibniz was found guilty of plagiarism.

Eventually, the mathematical community came to realize that Newton and Leibniz had made their own discoveries independently, but not until years after Leibniz’s death. During this time, continental Europe continued to use Leibniz’s easier notation and methods while England remained loyal to the more complicated methods and notations of their own Newton. For this reason, England was far behind the rest of the continent in mathematics for the entire eighteenth century.

III. Who is Leibniz?

-LEIBNIZ, Gottfried Wilhelm von (1646-1617)

Gottfried Wilhelm von Leibniz is one of the greatest thinkers of his era and perhaps in all history. He made important contributions to mathematics, philosophy law, politicians, theology, physics and history. Leibniz ranks with Descartes, Hobbes, Galileo and Spinoza as an example of the rational and scientific spirit that marked the change form medieval to modern thought. And by the way Leibniz was born on July 1, 1646 in Leipzig, Germany where his father was a Professor. Leibniz also had a large contribution in the discovery of calculus. Leibniz’s most important Philosophical writings are Theodicy (1710) and Monad logy (1714). But then he died in Hanover, Germany on November 14, 1716.

Calculus and Notation

While Newton thought of calculus in terms of motion, Leibniz viewed it in terms of sums and differences. Specifically, Leibniz used ordinates and sequences of the differences of these ordinates to calculate the area under curves. Following these discoveries, Leibniz introduced the notation

where was an elongated representation of the first letter of the Latin word summa, meaning summation and d was the first letter of the Latin word differentia, meaning differential (infinitesimal distance). Leibniz also used a differential triangle to discover the slope of a tangent line to a curve. He was thus able to derive the power, product, quotient, and chain rules.

IV. Who is Newton?

-NEWTON, Sir Isaac (1642-1727)

Newton, Sir Isaac is an English Scientist, Astronomer, and Mathematician invented a new kind of Mathematics; he discovered the secrets of light and color and showed how universe is held together. He is sometimes described as “one of the greatest names in the history of human thought” because of his great discoveries and contributions in Science and Mathematics. He discovered and invented a branch of Mathematics that is so called calculus. He discovered the calculus, the secrets of light and colors within 18 months and considered as his important discoveries and contributions in the history of Mathematics and Science.

Calculus and Notation

Using infinite series and the already established power rule for integrals, Newton was able to calculate the areas under curves that other previously could not. He was also able to calculate the tangent lines to these curves. He called his calculus the “method of fluxions” and he thought of everything in terms of motion. He considered the dependent variable x to be “fluent” and its velocity to be “fluxion”. He designated the fluxion , nowadays known as the derivative with respect time t, with the notation x.

V. Importance of Calculus

As Alexander (2013) said, Calculus is a very versatile and valuable tool. It is a form of mathematics which was developed from algebra and geometry. It is made up of two interconnected topics, differential calculus and integral calculus.

You can look at differential calculus as the mathematics of motion and change. Integral calculus covers the accumulation of quantities, such as areas under a curve. The two ideas work inversely together as defined by the Fundamental Theorem of Calculus.

Ted (2006) added that calculus is deeply integrated in every branch of the physical sciences, such as physics and biology. It is found in computer science, statistics, and engineering; in economics, business, and medicine. Modern developments such as architecture, aviation, and other technologies all make use of what calculus can offer. This page is designed to outline some of the applications of calculus and give you some idea of why calculus is so important and useful. * Finding the Slope of a Curve

Calculus can give us a generalized method of finding the slope of a curve. The slope of a line is fairly elementary, using some basic algebra it can be found. Although when we are dealing with a curve it is a different story. Calculus allows us to find out how steeply a curve will tilt at any given time. This can be very useful in any area of study. * Calculating the Area of Any Shape

Although we do have standard methods to calculate the area of some shapes, calculus allows us to do much more. Trying to find the area on a shape like this would be very difficult if it wasn’t for calculus * Calculate Complicated X-intercepts

Without an idea like the Intermediate Value Theorem it would be exceptionally hard to find or even know that a root existed in some functions. Using Newton’s Method you can also calculate an irrational root to any degree of accuracy, something your calculator would not be able to tell you if it wasn’t for calculus. * Visualizing Graphs

Using calculus you can practically graph any function or equation you would like. In fact you can find out the maximum and minimum values, where it increases and decreases and much more without even graphing a point, all using calculus. * Finding the Average of a Function

A function can represent many things. One example is the path of an airplane. Using calculus you can calculate its average cruising altitude, velocity and acceleration. Same goes for a car, bus, or anything else that moves along a path. Now what would you do without a speedometer on your car? * Calculating Optimal Values

By using the optimization of functions in just a few steps you can answer very practical and useful questions such as: “You have square piece of cardboard, with sides 1 meter in length. Using that piece of card board, you can make a box, what are the dimensions of a box containing the maximal volume?” These types of problems are a wonderful result of what calculus can do for us.

Calculus is one of the greatest inventions of modern science. There is just so much it can offer. The success of calculus has been extended over time into various other important topics in mathematics. Some are: differential equations, vector calculus, calculus of variations, complex analysis and differential topology. Studying calculus gives you the fundamentals you will need in all of your future studies.

Conclusion:

While making a research in this paper, I learned so much about the controversy and how it ended up to be both of them to get the credit of being entitled the ‘Father of Calculus’. Completing this study, I realized that calculus is also an important lesson that every learner must know because it can be used in reality. And about the history I studied, I agree with the statement that both of them should really get the credit of being the father of Calculus because they really made their own study individually. By this research, I conclude that we owe a great gratitude to the inventors of the Calculus because it is really one of the greatest inventions or formulas in the world of Mathematics. And I encourage you readers to study and practice this formula because of its importance and its help to the real world.

References:

http://www.angelfire.com/md/byme/mathsample.html http://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy https://www.google.com.ph/#q=leibniz+vs+newton http://ugrad-conf.fitchburgstate.edu/other/Sample-Math-Poster.pdf http://en.wikipedia.org/wiki/Leibniz's_notation http://www.angelfire.com/md/byme/mathsample.html http://historyofcalculus.weebly.com/calculus-in-the-real-world.html http://en.wikipedia.org/wiki/Calculus http://calculus.nipissingu.ca/calc_app.html

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...1&hid=117&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=tfh&AN=53475373 ANIME GOES MAINSTREAM. Princess Mononoke, Akira, and Cowboy Bebop may not be household names here. But in the world of anime, or Japanese animation, they're among the top 10 films ever made. You've seen it-colorful cartoon characters with big eyes, spiky hair, and small mouths. And you've probably heard about it-Hayao Miyazaki's latest film, Ponyo, was the ninth highest-grossing film in this country on its opening weekend last summer. With its complex plots and moral messages, anime is as intelligent as some of the best feature films. Take Grave of the Fireflies, written and directed by Isao Takahata. According to film critic Roger Ebert, it's one of the most powerful anti-war movies of all time. Or Paprika, directed by Satoshi Kon and animated by Madhouse Studios, a visual masterpiece where reality and dreams collide. From the epic fantasy Ninja Scroll and the cyberpunk Ghost in the Shell to the sci-fi romance The Girl Who Leapt Through Time, anime has something for everyone, whether it's romance, action, fantasy, adventure, or mystery. No wonder its gone mainstream, with thousands of young fans flocking to anime conventions across the country and teens from all socioeconomic and racial backgrounds trading cards, collecting art, and, of course, watching it on TV, DVDs, and the big screen every chance they can get. Anime dates back to the closing days of World War II, with fans here developing a cult...

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