Pierre De Fermat
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Differential Calculus and by Era Integral Calculus, Which Are Related by in Early Cultures in Classical Antiquity the Fundamental Theorem of Calculus
History of calculus - Wikipedia, the free encyclopedia 1/1/10 5:02 PM History of calculus From Wikipedia, the free encyclopedia History of science This is a sub-article to Calculus and History of mathematics. History of Calculus is part of the history of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The subject, known Background historically as infinitesimal calculus, Theories/sociology constitutes a major part of modern Historiography mathematics education. It has two major Pseudoscience branches, differential calculus and By era integral calculus, which are related by In early cultures in Classical Antiquity the fundamental theorem of calculus. In the Middle Ages Calculus is the study of change, in the In the Renaissance same way that geometry is the study of Scientific Revolution shape and algebra is the study of By topic operations and their application to Natural sciences solving equations. A course in calculus Astronomy is a gateway to other, more advanced Biology courses in mathematics devoted to the Botany study of functions and limits, broadly Chemistry Ecology called mathematical analysis. Calculus Geography has widespread applications in science, Geology economics, and engineering and can Paleontology solve many problems for which algebra Physics alone is insufficient. Mathematics Algebra Calculus Combinatorics Contents Geometry Logic Statistics 1 Development of calculus Trigonometry 1.1 Integral calculus Social sciences 1.2 Differential calculus Anthropology 1.3 Mathematical analysis -
Calculus? Can You Think of a Problem Calculus Could Be Used to Solve?
Mathematics Before you read Discuss these questions with your partner. What do you know about calculus? Can you think of a problem calculus could be used to solve? В A Vocabulary Complete the definitions below with words from the box. slope approximation embrace acceleration diverse indispensable sphere cube rectangle 1 If something is a(n) it В Reading 1 isn't exact. 2 An increase in speed is called Calculus 3 If something is you can't Calculus is the branch of mathematics that deals manage without it. with the rates of change of quantities as well as the length, area and volume of objects. It grew 4 If you an idea, you accept it. out of geometry and algebra. There are two 5 A is a three-dimensional, divisions of calculus - differential calculus and square shape. integral calculus. Differential calculus is the form 6 Something which is is concerned with the rate of change of quantities. different or of many kinds. This can be illustrated by slopes of curves. Integral calculus is used to study length, area 7 If you place two squares side by side, you and volume. form a(n) The earliest examples of a form of calculus date 8 A is a three-dimensional back to the ancient Greeks, with Eudoxus surface, all the points of which are the same developing a mathematical method to work out distance from a fixed point. area and volume. Other important contributions 9 A is also known as a fall. were made by the famous scientist and mathematician, Archimedes. In India, over the 98 Macmillan Guide to Science Unit 21 Mathematics 4 course of many years - from 500 AD to the 14th Pronunciation guide century - calculus was studied by a number of mathematicians. -
Historical Notes on Calculus
Historical notes on calculus Dr. Vladimir Dotsenko Dr. Vladimir Dotsenko Historical notes on calculus 1/9 Descartes: Describing geometric figures by algebraic formulas 1637: Ren´eDescartes publishes “La G´eom´etrie”, that is “Geometry”, a book which did not itself address calculus, but however changed once and forever the way we relate geometric shapes to algebraic equations. Later works of creators of calculus definitely relied on Descartes’ methodology and revolutionary system of notation in the most fundamental way. Ren´eDescartes (1596–1660), courtesy of Wikipedia Dr. Vladimir Dotsenko Historical notes on calculus 2/9 Fermat: “Pre-calculus” 1638: In a letter to Mersenne, Pierre de Fermat explains what later becomes the key point of his works “Methodus ad disquirendam maximam et minima” and “De tangentibus linearum curvarum”, that is “Method of discovery of maximums and minimums” and “Tangents of curved lines” (published posthumously in 1679). This is not differential calculus per se, but something equivalent. Pierre de Fermat (1601–1665), courtesy of Wikipedia Dr. Vladimir Dotsenko Historical notes on calculus 3/9 Pascal and Huygens: Implicit calculus In 1650s, slightly younger scientists like Blaise Pascal and Christiaan Huygens also used methods similar to those of Fermat to study maximal and minimal values of functions. They did not talk about anything like limits, but in fact were doing exactly the things we do in modern calculus for purposes of geometry and optics. Blaise Pascal (1623–1662), courtesy of Christiaan Huygens (1629–1695), Wikipedia courtesy of Wikipedia Dr. Vladimir Dotsenko Historical notes on calculus 4/9 Barrow: Tangents vs areas 1669-70: Isaac Barrow publishes “Lectiones Opticae” and “Lectiones Geometricae” (“Lectures on Optics” and “Lectures on Geometry”), based on his lectures in Cambridge. -
Evolution of Mathematics: a Brief Sketch
Open Access Journal of Mathematical and Theoretical Physics Mini Review Open Access Evolution of mathematics: a brief sketch Ancient beginnings: numbers and figures Volume 1 Issue 6 - 2018 It is no exaggeration that Mathematics is ubiquitously Sujit K Bose present in our everyday life, be it in our school, play ground, SN Bose National Center for Basic Sciences, Kolkata mobile number and so on. But was that the case in prehistoric times, before the dawn of civilization? Necessity is the mother Correspondence: Sujit K Bose, Professor of Mathematics (retired), SN Bose National Center for Basic Sciences, Kolkata, of invenp tion or discovery and evolution in this case. So India, Email mathematical concepts must have been seen through or created by those who could, and use that to derive benefit from the Received: October 12, 2018 | Published: December 18, 2018 discovery. The creative process of discovery and later putting that to use must have been arduous and slow. I try to look back from my limited Indian perspective, and reflect up on what might have been the course taken up by mathematics during 10, 11, 12, 13, etc., giving place value to each digit. The barrier the long journey, since ancient times. of writing very large numbers was thus broken. For instance, the numbers mentioned in Yajur Veda could easily be represented A very early method necessitated for counting of objects as Dasha 10, Shata (102), Sahsra (103), Ayuta (104), Niuta (enumeration) was the Tally Marks used in the late Stone Age. In (105), Prayuta (106), ArA bud (107), Nyarbud (108), Samudra some parts of Europe, Africa and Australia the system consisted (109), Madhya (1010), Anta (1011) and Parartha (1012). -
Pure Mathematics
good mathematics—to be without applications, Hardy ab- solved mathematics, and thus the mathematical community, from being an accomplice of those who waged wars and thrived on social injustice. The problem with this view is simply that it is not true. Mathematicians live in the real world, and their mathematics interacts with the real world in one way or another. I don’t want to say that there is no difference between pure and ap- plied math. Someone who uses mathematics to maximize Why I Don’t Like “Pure the time an airline’s fleet is actually in the air (thus making money) and not on the ground (thus costing money) is doing Mathematics” applied math, whereas someone who proves theorems on the Hochschild cohomology of Banach algebras (I do that, for in- † stance) is doing pure math. In general, pure mathematics has AVolker Runde A no immediate impact on the real world (and most of it prob- ably never will), but once we omit the adjective immediate, I am a pure mathematician, and I enjoy being one. I just the distinction begins to blur. don’t like the adjective “pure” in “pure mathematics.” Is mathematics that has applications somehow “impure”? The English mathematician Godfrey Harold Hardy thought so. In his book A Mathematician’s Apology, he writes: A science is said to be useful of its development tends to accentuate the existing inequalities in the distri- bution of wealth, or more directly promotes the de- struction of human life. His criterion for good mathematics was an entirely aesthetic one: The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colours or the words must fit together in a harmo- nious way. -
Dossier Pierre Duhem Pierre Duhem's Philosophy and History of Science
Transversal: International Journal for the Historiography of Science , 2 (201 7) 03 -06 ISSN 2526 -2270 www.historiographyofscience.org © The Author s 201 7 — This is an open access article Dossier Pierre Duhem Pierre Duhem’s Philos ophy and History of Science Introduction Fábio Rodrigo Leite 1 Jean-François Stoffel 2 DOI: http://dx.doi.org/10.24117/2526-2270.2017.i2.02 _____________________________________________________________________________ We are pleased to present in this issue a tribute to the thought of Pierre Duhem, on the occasion of the centenary of his death that occurred in 2016. Among articles and book reviews, the dossier contains 14 contributions of scholars from different places across the world, from Europe (Belgium, Greece, Italy, Portugal and Sweden) to the Americas (Brazil, Canada, Mexico and the United States). And this is something that attests to the increasing scope of influence exerted by the French physicist, philosopher and 3 historian. It is quite true that since his passing, Duhem has been remembered in the writings of many of those who knew him directly. However, with very few exceptions (Manville et al. 1927), the comments devoted to him exhibited clear biographical and hagiographic characteristics of a generalist nature (see Jordan 1917; Picard 1921; Mentré 1922a; 1922b; Humbert 1932; Pierre-Duhem 1936; Ocagne et al. 1937). From the 1950s onwards, when the studies on his philosophical work resumed, the thought of the Professor from Bordeaux acquired an irrevocable importance, so that references to La théorie physique: Son objet et sa structure became a common place in the literature of the area. As we know, this recovery was a consequence of the prominence attributed, firstly, to the notorious Duhem-Quine thesis in the English- speaking world, and secondly to the sparse and biased comments made by Popper that generated an avalanche of revaluations of the Popperian “instrumentalist interpretation”. -
Historiographie De Paul Tannery Et Receptions De Son Œuvre: Sur L'invention Du Metier D'historien Des Sciences
Historiographie de Paul Tannery et receptions de son œuvre : sur l’invention du metier d’historien des sciences François Pineau To cite this version: François Pineau. Historiographie de Paul Tannery et receptions de son œuvre : sur l’invention du metier d’historien des sciences. Histoire, Philosophie et Sociologie des sciences. Université de Nantes, 2010. Français. NNT : 2010NANT2076. tel-00726388 HAL Id: tel-00726388 https://tel.archives-ouvertes.fr/tel-00726388 Submitted on 29 Aug 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITÉ DE NANTES UFR SCIENCES ET TECHNIQUES _____ ECOLE DOCTORALE SOCIETES, CULTURES, ECHANGES Année 2010 N° attribué par la bibliothèque Historiographie de Paul Tannery et réceptions de son œuvre : sur l'invention du métier d'historien des sciences ___________ THÈSE DE DOCTORAT Discipline : Épistémologie et Histoire des sciences et des techniques Présentée et soutenue publiquement par François PINEAU Le 11 décembre 2010, devant le jury ci-dessous Rapporteurs : M. Eberhard KNOBLOCH, Professeur, Berlin Mme Jeanne PEIFFER, Directeur de recherche CNRS, Paris Examinateurs : M. Anastasios BRENNER, Professeur, Montpellier M. Bernard VITRAC, Directeur de recherche CNRS, Paris Co-directeurs de thèse : Mme Évelyne BARBIN, Professeur, Nantes M. -
Fermat's Technique of Finding Areas Under Curves
Fermat’s technique of finding areas under curves Ed Staples Erindale College, ACT <[email protected]> erhaps next time you head towards the fundamental theorem of calculus Pin your classroom, you may wish to consider Fermat’s technique of finding expressions for areas under curves, beautifully outlined in Boyer’s History of Mathematics. Pierre de Fermat (1601–1665) developed some impor- tant results in the journey toward the discovery of the calculus. One of these concerned finding areas under simple polynomial curves. It is an amazing fact that, despite deriving these results, he failed to document any connection between tangents and areas. The method is essentially an exercise in geometric series, and I think it is instructive and perhaps refreshing to introduce a little bit of variation from the standard introductions. The method is entirely accessible to senior students, and makes an interesting connection between the anti derivative result and the factorisation of (1 – rn+1). Fermat considered that the area between x = 0 and x = a below the curve y = xn and above the x-axis, to be approximated by circumscribed rectangles where the width of each rectangle formed a geometric sequence. The divi- sions into which the interval is divided, from the right hand side are made at x = a, x = ar, x = ar2, x = ar3 etc., where a and r are constants, with 0 < r < 1 as shown in the diagram. A u s t r a l i a n S e n i o r M a t h e m a t i c s J o u r n a l 1 8 ( 1 ) 53 s e 2 l p If we first consider the curve y = x , the area A contained within the rectan- a t S gles is given by: A = (a – ar)a2 + (ar – ar2)(ar)2 + (ar2 – ar3)(ar2)2 + … This is a geometric series with ratio r3 and first term (1 – r)a3. -
Leonhard Euler English Version
LEONHARD EULER (April 15, 1707 – September 18, 1783) by HEINZ KLAUS STRICK , Germany Without a doubt, LEONHARD EULER was the most productive mathematician of all time. He wrote numerous books and countless articles covering a vast range of topics in pure and applied mathematics, physics, astronomy, geodesy, cartography, music, and shipbuilding – in Latin, French, Russian, and German. It is not only that he produced an enormous body of work; with unbelievable creativity, he brought innovative ideas to every topic on which he wrote and indeed opened up several new areas of mathematics. Pictured on the Swiss postage stamp of 2007 next to the polyhedron from DÜRER ’s Melencolia and EULER ’s polyhedral formula is a portrait of EULER from the year 1753, in which one can see that he was already suffering from eye problems at a relatively young age. EULER was born in Basel, the son of a pastor in the Reformed Church. His mother also came from a family of pastors. Since the local school was unable to provide an education commensurate with his son’s abilities, EULER ’s father took over the boy’s education. At the age of 14, EULER entered the University of Basel, where he studied philosophy. He completed his studies with a thesis comparing the philosophies of DESCARTES and NEWTON . At 16, at his father’s wish, he took up theological studies, but he switched to mathematics after JOHANN BERNOULLI , a friend of his father’s, convinced the latter that LEONHARD possessed an extraordinary mathematical talent. At 19, EULER won second prize in a competition sponsored by the French Academy of Sciences with a contribution on the question of the optimal placement of a ship’s masts (first prize was awarded to PIERRE BOUGUER , participant in an expedition of LA CONDAMINE to South America). -
Historiographie De Paul Tannery Et Receptions De Son Œuvre : Sur L’Invention Du Metier D’Historien Des Sciences Fran¸Coispineau
Historiographie de Paul Tannery et receptions de son œuvre : sur l'invention du metier d'historien des sciences Fran¸coisPineau To cite this version: Fran¸coisPineau. Historiographie de Paul Tannery et receptions de son œuvre : sur l'invention du metier d'historien des sciences. Histoire, Philosophie et Sociologie des sciences. Universit´e de Nantes, 2010. Fran¸cais. <NNT : 2010NANT2076>. <tel-00726388> HAL Id: tel-00726388 https://tel.archives-ouvertes.fr/tel-00726388 Submitted on 29 Aug 2012 HAL is a multi-disciplinary open access L'archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destin´eeau d´ep^otet `ala diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publi´esou non, lished or not. The documents may come from ´emanant des ´etablissements d'enseignement et de teaching and research institutions in France or recherche fran¸caisou ´etrangers,des laboratoires abroad, or from public or private research centers. publics ou priv´es. UNIVERSITÉ DE NANTES UFR SCIENCES ET TECHNIQUES _____ ECOLE DOCTORALE SOCIETES, CULTURES, ECHANGES Année 2010 N° attribué par la bibliothèque Historiographie de Paul Tannery et réceptions de son œuvre : sur l'invention du métier d'historien des sciences ___________ THÈSE DE DOCTORAT Discipline : Épistémologie et Histoire des sciences et des techniques Présentée et soutenue publiquement par François PINEAU Le 11 décembre 2010, devant le jury ci-dessous Rapporteurs : M. Eberhard KNOBLOCH, Professeur, Berlin Mme Jeanne PEIFFER, Directeur de recherche CNRS, Paris Examinateurs : M. Anastasios BRENNER, Professeur, Montpellier M. Bernard VITRAC, Directeur de recherche CNRS, Paris Co-directeurs de thèse : Mme Évelyne BARBIN, Professeur, Nantes M. -
Beyond Riemann Sums: Fermat's Method of Integration
Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Calculus Mathematics via Primary Historical Sources (TRIUMPHS) Winter 2020 Beyond Riemann Sums: Fermat's Method of Integration Dominic Klyve Central Washington University, [email protected] Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_calculus Click here to let us know how access to this document benefits ou.y Recommended Citation Klyve, Dominic, "Beyond Riemann Sums: Fermat's Method of Integration" (2020). Calculus. 12. https://digitalcommons.ursinus.edu/triumphs_calculus/12 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Calculus by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. Beyond Riemann Sums: Fermat's method of integration Dominic Klyve∗ February 8, 2020 Introduction Finding areas of shapes, especially shapes with curved boundaries, has challenged mathematicians for thousands of years. Today calculus students learn to graph functions, approximate the area with rectangles of the same width, and use \Riemann Sums" to approximate the area under a curve (or, by taking limits, to find it exactly). Centuries before Riemann used this method, and a generation before Newton and Leibniz discovered calculus, seventeenth-century lawyer (and math fan) Pierre de Fermat (1607{1665) used a similar method, with an interesting twist that let the widths of the rectangles get larger as x grows. In this Primary Source Project, we will explore Fermat's method. Part 1: On \quadrature" It's worth noting that Fermat thought of area slightly differently than we usually do today. -
A Concise History of Mathematics the Beginnings 3
A CONCISE HISTORY OF A CONCISE HISTORY MATHEMATICS STRUIK OF MATHEMATICS DIRK J. STRUIK Professor Mathematics, BELL of Massachussetts Institute of Technology i Professor Struik has achieved the seemingly impossible task of compress- ing the history of mathematics into less than three hundred pages. By stressing the unfolding of a few main ideas and by minimizing references to other develop- ments, the author has been able to fol- low Egyptian, Babylonian, Chinese, Indian, Greek, Arabian, and Western mathematics from the earliest records to the beginning of the present century. He has based his account of nineteenth cen- tury advances on persons and schools rather than on subjects as the treatment by subjects has already been used in existing books. Important mathema- ticians whose work is analysed in detail are Euclid, Archimedes, Diophantos, Hammurabi, Bernoulli, Fermat, Euler, Newton, Leibniz, Laplace, Lagrange, Gauss, Jacobi, Riemann, Cremona, Betti, and others. Among the 47 illustra- tions arc portraits of many of these great figures. Each chapter is followed by a select bibliography. CHARLES WILSON A CONCISE HISTORY OF MATHEMATICS by DIRK. J. STRUIK Professor of Mathematics at the Massachusetts Institute of Technology LONDON G. BELL AND SONS LTD '954 Copyright by Dover Publications, Inc., U.S.A. FOR RUTH Printed in Great Britain by Butler & Tanner Ltd., Frame and London CONTENTS Introduction xi The Beginnings 1 The Ancient Orient 13 Greece 39 The Orient after the Decline of Greek Society 83 The Beginnings in Western Europe 98 The