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History of Calculus

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Caterina Urbano

on 28 February 2013

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Transcript of History of Calculus

By Caterina Urbano A History of Calculus What is Calculus? Sources: Archimede was the ancient Greece thinker that came closer to Calculus. He found a way to work with curved surface hundreds of years before Newton and Leibnitz. Following Aristotele's hypothesis on infinity, he never mentioned it in his work and, if he did use it as a concept, he always tought of it as a "potential" infinity. Calculus in Ancient Greece Cavalieri produced the law of indivisble in the 1630, which is used to calculate the area under a curve. Let's suppose that this area is made of lines (which can be seen as very tiny rectangles). To find the total area it's sufficient to sum all the rectangles together. This method demonstrated to be not completely reliable but it helped to solve many geometrical problems. Other precursors of Calculus They had different view regarding Calculus: Newton’s was based on limits and concrete reality, while Leibniz focused more on the infinite and the abstract.
Even if Newton reported to have discovered similar methods, Leibniz published "his" findings in 1684, the first account about differential calculus, and again in 1686, the second volume about integral calculus
Here the controversy started: Newton didn't publish until 1687.
Evidence shows that Newton was the first to establish what is called the "theory of fluxions" as well as 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.
No matter what, Leibniz had all the credit for the discover of Modern Calculus for many years and this lead to accuses of plagiarism.
Throughout the years, many hypothesis came up but many things are still to find out. Maybe Leibniz may have seen some of Newton's manuscripts before they were published, changed them using different symbols and published them under his name. It's also suspicious how Newton and Leibniz have been using corresponding letters, usually dealing with mathematical problems. Newton and Leibniz Davidson, Jon. "What Is Calculus?" MathAdvise. N.p., n.d. Web. 28 Feb. 2013. <http://www.sscc.edu/home/jdavidso/MathAdvising/AboutCalculus.html>.
"Archimedes Developed Calculus?" Kottke.org. N.p., n.d. Web. 28 Feb. 2013. <http://kottke.org/09/01/archimedes-developed-calculus>.
Otero, Daniel E. "Cavalieri: Indivisibles." Cerebro. N.p., n.d. Web. 28 Feb. 2013. <http://cerebro.xu.edu/math/math147/02f/cavalieri/cavintro.html>
"Newton vs. Leibniz; The Calculus Controversy." Angelfire. N.p., n.d. Web. 2013. <http://www.angelfire.com/md/byme/mathsample.html>.
Wallace, David F. Everything and More- A Compact History of Infinity. 2nd ed. New York: Atlas, 2010. N. pag. Print. Where does the word come from?
Calculus is a latin word that comes from the rocks used by the ancients to do arithmetic. What is its use?
Calculus has been fundamental to understand continuously changing qualities. Newton was interested in calculating gravity acceleration and the solution to it was the derivative. A type of mathematics that measures changes in one quantity in relation to another. Calculus is used for functions involving curves. Who and when invented Calculus?
Sir Isaac Newton and Goffried Leibnitz are both accredited for the discovery of Calculus during the 1670s. They accused each other for years but, eventually, Leibnitz symbols were adopted. Was there Calculus before Calculus?
Actually yes. The great thinkers, like Archimedes, in ancient times ran occasionally into calculus concepts and had many trouble dealing with them ( the way the concept of infinity has been treated in the past is a good example). The major issue was to find a way to compute the area under the curve using only straightedge and compass Greeks didn't like the concept of infinity. Aristotele himself decided that it was just an abstract concept without application in math, that for Greeks was, before everything, geometry.
This is one of the reasons why Greeks never arrived to invent calculus. Democritus was the first one to take into consideration the division of objects into infinite pieces. Although he couldn’t put those together rationalizing the cross sections into a smooth conical slope and that prevented him to consider infinity as an option. Edoxus invents the Exhaustion property, in which for the first time is introduced the concept of 'limit'. The exhaustion property basically allows you to consider an infinity number of polygons to measure the area of a circle. Fermat wrote "Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum", determining method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation That's it, we're at the point where the word " Calculus" assumed the meaning we give to it today, one of the principal fields of mathematics.
The birth of Calculus was also the moment of the biggest controversy in its history: two great scientists fought over the years accusing one another of plagiarism for being the actual inventor of calculus VS. Gottfried Leibnitz Isaac Newton The Indian mathematician-astronomer Aryabhata in 499 b.C. used a notion of infinitesimals and expressed an astronomical problem in the form of a basic differential equation.
In the late 12th century, the Persian mathematician Sharaf al-Din al-Tusi was the first to discover the derivative of cubic polynomials.
In the 15th century, an early version of the mean value theorem was first described by Parameshvara. Modern Calculus Many others contributed to the shape the Modern Calculus ad we know and among these it's important to name Isaac Barrow, Evangelista Torricelli, Micheal Rolle, Bernard Bolzano, and Christiaan Huygens What came after? Rolle proved the theorem that is named after him. 1691 L'hospital's rule Other scientists took the work done by Newton and Leibniz and made other important discoveries: What is the Rolle's Theorem?
It basically states that if a continuous curve passes through the same y-value twice and has a unique tangent line at every point of the interval, then somewhere between the endpoints it has a tangent parallel to the x-axis. Why is it useful?
It puts the basis of the Mean Value theorem, which is one of the most important tools in Calculus. 1694 Who invented it?
The swiss mathematician John Bernuolli Why that name then?
The explanation is that the rule was first published in a textbook by Marquis de l'Hospital but in fact de L'Hospital had bought the rights from Bernoulli in a curious businness arrangement. Why is it important?
What this rule does is to help of fining the limit in some indterminate forms. In a few words it can make your life easier when it comes to limits. Marquis de l'Hospital John Bernoulli 1797
Joseph-Louis Lagrange introduces the notations f'x and y' for the derivatives of f(x) and y, respectively.
Carl Gustav Jacob Jacobi adopts the modern notation for partial differentiation; Adrien-Marie Legendre originally introduced it in 1786, but immediately abandoned it.
Bernhard Riemann defines the integral in a way that does not require continuity.
Cantor comes up with the Cantor's set.
H. Eduard Heine, a student of Karl Weierstrass, presents the modern ``epsilon-delta'' definition of a limit in his Elements. In the last 200 years many discoveries have been made about calculus. Mathematics, that for greeks was something of extremely concrete (every theorem was proved by geometry) with Calculus and the progresses from there has begun to get more and more abstract. It's enough to think about concepts like infinity and irrational numbers, we know that even zero can create many problems.
The questions that comes natural after all those "inventions" and "discoveries" is: so mathematics was invented or discovered? Philosophers have been trying to give an answer for hundreds of years, but the question is still open.
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