Introduction

        The argument of who discovered calculus has been debated for over 300 years. Was it Sir Isaac Newton or was it Gottfried Wilhelm Leibniz? Even though the majority of people credited Sir Isaac Newton for its discovery, little is known about the mathematician and philosopher Gottfried Leibniz. It is important to examine both legends of the mathematics world before we devote all of our belief into one person.  Over the next couple blogs we will learn the history and discoveries of each mathematician and philosopher, and who the credit should be given too. Also we will see how the discovery has impacted us today and how we use it in everyday life without even knowing it.

Gottfried Wilhelm Leibniz

     
             Gottfried Leibniz was a mathematician, a philosopher, and an advisor to kings, that was born near the end of the thirty years war. He admired minds such as Aristotle’s and Socrates, and lived his life for the sophistication and the modernization of mathematics and science.  He was an influential politician of his time, but more importantly he is known for his inventions of integral and differential calculus. Even though history tells a different story of calculus’s invention we are going to dive into the life and inventions of Gottfried Leibniz.

             He was born in Germany, which lay in ruins because of the thirty years war, by prominent` Lutheran parents. Which was actually extremely unique because in that time because the  dominant religion was Catholicism, and most members of the Lutheran church were considered outcasts. Since his town and most of Germany was in ruins, he was mostly self-taught by his father, but most likely from his father’s extensive library. He decided in the spring of 1661 to enroll in the prestigious university of Liebzig in modern day Germany. This university was part of the same university that housed Galileo and other scientific minds like Francis Bacon.  At age 20 he applied for a doctor’s degree in law and was refused because of his young age. After the denial he fled his hometown, and would never return for the rest of his life. He ended up in the town of Altdolf and attended a university in that town and was automatically given his doctorates for his dissertation of Des Casibus Perplexis or in English On Perplexing Cases.  After a few fruitless jobs he meant young statesmen called Johann Christian of Boyenburg. During his employment under Johann Christian he worked on many pieces and works of literature that would reunite the Catholic Church, which was slightly ironic because he was born of Lutheran parents. After the completion of this work Johann Christian sent him to Paris, but shortly after the young statesman was killed and once again Liebniz was out of a job. But instead of rushing to find another job he used this time to study whatever he wanted. However since he was unemployed he needed a way to come up with some financial stability, so he could become free to study mathematics and other physical sciences. So after little trial and error he created a calculating machine and presented it to the royal family in London. He then had that financial stability he was seeking and was free to think and study.  

            Nobody knows how, but in later parts of 1675 Liebniz is said to have invented calculus, and had perfected his inventions and ideas in regards to integral and differential equations. With his creations he became obsessed with the idea of there is no such thing as time and space. He started to make predictions of phenomena’s based on his study of motions. He started to predict that motion was the fact of an invisible movement that could not only be explained in the idea of a force. To simply state what he was trying to say is that he was proposing the idea of gravity or more specifically a gravitational pull. With these conclusions stated with no factual information behind them he became discredited in his ideas of physics but the public also questioned his works in mathematics.

            In 1676 he took a job working for a duke in his native country of Germany where he became a very well-known philosopher. However it was a unique circumstance, he was the only well-known philosopher that had to still work for a living. Royalty as a result used his expertise and was actually very well known throughout all of Germany for a jack-of-all-trades.  Throughout the next five years however he felt underutilized so he started expanding his knowledge to the machine word. He started studying windmills and hydraulic pressures, but he also experimented with phosphorous and its reactions. In 1685 however while working on windmills he came up with the idea of a binary system. The same system is still used on modern computers and other devices that use basic coding for data. Throughout the next couple years he jumped between algebra and differential equations.

            After this discovery and publication of his book Nova Methodus pro Maximus et Minimus, or New Method for the Greatest and the Least, he really did not have any more mathematical discoveries for the rest of his life. He really focused more on theories of god and man and how we live together. He stated that the relations of God and his perspective and logic related to man and his logic. There was a change in Leibniz thinking, he was more influenced by the mysticism of God than the physical sciences of mathematics and physics. He even became a historian towards the end of his life.  

            Gottfried Leibniz was a man of many things whether it was inventing modern marvels or organizing theories of mathematics, he was a visionary but his history is overshadowed by Sir Isaac Newton.     

Sir Isaac Newton

   
           Isaac Newton was born on December 25, 1642 in the small English town of, Lincolnshire. He is without a doubt one of the most notable and influential figures of the 17^th century. He was a remarkable physicist and mathematician and with little argument one could say he was the face of the scientific revolution in the 17^th century. Isaac Newton discovered the three laws of motion which slowly turned into the building blocks of today’s physics. In our view and focus however he is the most commonly noted figure for the development and implication of calculus.

            In December of 1642 a local yeoman, or a man that owns his own farm, had a small and unhealthy baby named Isaac Newton and the doctors condemned young Isaac to not make it to his first birthday. Despite the doctors diagnosis Isaac ended up making it through his first birthday and 83 more of them. Isaac Newton lost his father three months before his birth but soon after his birth his mother abandoned him for a young minister named Barnabas Smith. Smith then sent young Isaac to live with his grandmother. It is unknown if Isaac would ever see his mother for at least nine years after that, but throughout his life he reports psychological and insecurity issues from the abandonment of his mother for the rest of his life.  However after paying no attention to Isaac Newton, she then willed all of her property to him after her second husband died. At a young age Newton was managing property and watching livestock. However when Isaac was out in the fields watching his cattle he would eventually curl up under tree and just read a book, which led to him being a sub-par cattle farmer. Some say this is where the idea for gravity came from when he was sitting under an apple tree and an apple fell to the ground. But where his ideas stemmed from nobody truly knows.

            He enrolled in a university where in 1665 he received his  bachelors, and surprisingly the most distinguished graduate of the college throughout their entire history, went relatively unnoticed.  After graduating he returned home and created his own innovative ways of theory, philosophy and solving mathematics. Around this time however the plague had shut down the university and most of England for about two years, but during this time he laid the foundations of calculus. Within these foundation of calculus he examined circular motion, or motion of the planets, but more importantly he solved the inverse square relation of the planets and the sun, that was based off his own philosophy of calculus. Not only did Newton invent calculus he applied it to something that was centuries before his time and literally light-years away.

            After simply creating calculus independently in his house, he did not publish it. He decided to work on other ideas that interested him such as optics. When the plague approached  it’s tail end and university’s started opening back up, Newton decided to become a professor.  Trinity College offered him a position as a professor and whether they simply hired him for mathematics or physics or his ideas on lenses nobody knows. But he was always passionate about lenses so he decided to pick optics. After teaching the course for two to three years he decided to developed the essay Of Colours which simply went into deeper understanding of Kepler’s, ideas of refraction and even some of the philosophy of Descartes. To say the least any impressions made from his essay’s were never noted in newspapers or talked about at a later date.

            Even though this was disappointing to Isaac Newton, he pressed on and perfected his work on mathematics which was called the Philosophe Naturalis Principia Mathematica, or in today’s modern English we would say the Principles of Planetary Motion. During the creation of the book Hooke and many other physicists contacted Newton but he disregarded there letters and never responded. Historians look at this in two different ways; one way being Newton used others ideas and simply rearranged thoughts and published them as his own, or the more commonly accepted idea of he disagreed with what others said most of the time but didn’t want to tell them why and jeopardize his autonomous claims of science. Why he did not respond to letters is not very important when you look at the whole picture however, because most of his claims where so far ahead of what any other physicist was even trying to prove. Even though in 1680 physicists were claiming some of his work was plagiarized, it is a common belief that some of the information may have been sent to him by others but that still does not account for the fact that he explained his theories using mathematics he created, therefore making some of their claims useless because they could not know the mathematics to back up most of their propositions or theories.

            The life of Isaac Newton can be spread over thousands of pages of any encyclopedia or webpage, but the purpose of this shorted biography was to show and examine Newton’s influence on the field of mathematics and how he led up to his independent discoveries calculus. 

                                   

           

            

The Legendary Debate

                There are few notable discoveries throughout man’s existence that the origins are concrete and the belief of the events that led up to and the discovery itself are without a doubt. Even the space race between the USSR and USA, has been debated since the minute Neil Armstrong took that small step off of  Apollo 11, there has been debate. Even though it is the common belief that the Americans landed on the moon first, there are still some skeptics and debate behind it. This is a little different from the debate between who invented calculus because unlike the facts of the 1960’s that is a little more concrete than the facts from the 17^th century. However there are historical references and books published by Leibniz and Newton that show when there works were published and how they invented calculus in their own ways.

            The invention of calculus was incredibly complex and believe it or not the two best mathematicians of the time created it two completely different ways. Leibniz created calculus by thinking of abstract and infinite series. While on the other hand Newton created calculus that was based on more concrete ideas such as limits and reality.

            Leibniz discovered his version of calculus between 1673 and 1676. He finally published his first book on differential calculus around 1684 and his explanation about integral calculus in 1686. During this time however Newton had already made the same claims in England, But due to Newton’s soft spookiness and resistance to publish when it came to his inventions he did not publish it till a few years after Leibniz published his works. Even though Newton did not publish any of his mathematical works in calculus there is documentation of him creating fluxional calculus in the mid-1660s.  Since he never published his works till after Leibniz, all the credit was given to Leibniz for several years. Eventually wind got around that Newton discovered the “theory of fluxions” before just never published it, which ended up leading to allegations about plagiarism that were always pointed at Leibniz.  

            Those ideas really only arose because there were letters from Newton to other physicist and mathematicians with calculus such notations that Leibniz had available to him. So in simpler terms the ideas could have stemmed from Newton’s discoveries. But like I mentioned earlier Newton was very uptight with who he talked to and what he talked to them about trying to avoid circumstances of plagiarism like this one. But what exactly the letters contained are unknown, whether they were as simple as one exact theorem or the whole idea nobody knows. But due to Newton’s past of being reluctant to publish or share I believe that it was most likely a theorem or simple idea that was exchanged. However there is evidence that Newton and Leibniz sent letters back and forth talking about mathematics and calculus. But due to the information Leibniz published was different type of thinking than that of Newton so any information that they exchanged may have been helpful to say the least but not enough to write a dissertation about the revolutionary idea of integral and differential calculus.

            However Newton was well respected in the world of physics and mathematics and had hundreds of educated followers while Leibniz only had one. Who actually wrote a letter to Newton and then when Newton spoke to him he denied ever writing the letter. In 1715 however only a couple months before his death the royal society claimed Sir Isaac Newton with the invention of calculus and Leibniz was convicted of plagiarism. However Liebniz name was tarnished for a very long time. To show how little support Liebniz actually did have, it was said by a reporter at his funeral that only one person attended his funeral, and it was his secretary. Eventually the controversy in the royal society was solved and Leibniz was no longer guilty of plagiarism and both gentlemen would go down in history as the co-inventers of calculus but this would come long after Leibniz’s death.

            What is very unique about this is that Leibniz’s version of calculus is still quite practical and usable in today’s modern day math while Newton’s has fallen by the wayside. England however reacted differently to the rulings of the royal society and decided that it was an international plea of egotism why they gave credit to Leibniz. So in England they refused to teach any other math than that of Newton till the year of 1820. This argument has been debated since the start of calculus. Even though the complete truth of who created calculus is unknown, it can stated simply that both inventors were visionaries and it was incredible discovery by both Leibniz and Newton.

Modern Day Uses of Calculus

            The roots of calculus can be found at the base of all geometry. Since the time of Euclid and Archimedes, man has wondered how to solve certain geometric and algebraic problems such as the area under the curve or the tangent to a curve. But as we know calculus was invented in the 17^th century and even in calculus in its most basic forms was extremely complicated for its times but could still solve these questions.  Calculus is a valuable tool that combines the fields of geometry and algebra, and is made up of two interconnected forms, one called differential calculus and the other integral calculus. Calculus is used in a lot of unknown aspects of life that we as a culture now take for granted. Certain items such as your cell-phone, to microwaves to motor vehicles, use calculus every time you turn them on.  

            An example of how we use calculus in everyday life can be as simple as a cardboard box or as complicated as tracking a space shuttle or satellite that is thousands of miles away.  But one simple example that most people do not think is calculus is a speed trap. A speed trap is a trap set by police officers between two set points to track a car or truck’s speed through two designated areas.  A speed trap can calculate your average speed, but more importantly increasing or decreasing speeds. It can tell by simply getting the initial speed, the final speed, the distance between the two trap points, and the time it takes the car to get through the trap points. The ending result would be a straight line if the speed remained the same but if the speed was changed then the result would be a concave or convex parabola and the vertex would be the top speed or the lowest speed depending on the time of the car through the two designated areas. Police officers can use this to administer tickets and citations. Even though in the reality of this knowledge your hate for calculus may have just gotten worse, but here are some different examples that may change your opinion of calculus.

            There are millions of uses for calculus throughout all spectrums of life whether you notice it or not, and the following are just a couple of professions or machines that use calculus. Biologist use calculus to see the exponential growth rate of bacteria on a certain specimen.  Credit card companies use calculus to calculate when and how much to charge the card holder by taking multiple variables into account, such as interest rates and amount of charged items. The government uses calculus in developing aircrafts such as the F35 and F22 along with using calculus to develop firewall programs to provide defense to national secrets and programs, whereas the graphic designer uses calculus to design three dimensional objects that appear real in video games or movies.

            The uses of calculus have been constantly evolving from its infant stages of determining gravity to today’s complex video games or computer programs. There is not an industry that doesn’t have its hand somewhere in the infamous calculus cookie jar. Whether it is a government agency, a credit card company, or a policeman using a speed trap, calculus will remain an intricate role in our lives and our kids’ lives. Whether it is protecting us against cyber warfare or something as simple as determining the average speed of an apple as it falls from a tree, it will remain a useful tool of describing and determining infinitely many problems    

               

                 

               

Conclusion

      The words above have credited both Gottfried Leibniz and Sir Isaac Newton with the invention of calculus and to say the least that is fair. Both legends solved the question within a couple years of each other and came up with their ideas in two different and unique ways. And I cannot think of any other invention that has stuck with the human race for over 300 years.  Calculus is used in every day in millions of different ways and only the future will tell how many more ways that humans will utilize this invention in the years to come. 

Sources

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The Cambridge companion to Leibniz / edited by N. The Cambridge companion to Leibniz. Cambridge, New York: Cambridge University Press, 1995. 475-493. Print.

Gale, Christianson E. In the presence of the Creator : Isaac Newton and his times. New York: Collier Macmillan, c1984, 1984. Print.

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