3000 Years of Analysis : Mathematics in History and Culture

What exactly is analysis? What are infinitely small or infinitely large quantities? What are indivisibles and infinitesimals? What are real numbers, continuity, the continuum, differentials, and integrals? You'll find the answers to these and other questions in this unique book! It explains in detail the origins and evolution of this important branch of mathematics, which Euler dubbed the "analysis of the infinite." A wealth of diagrams, tables, color images and figures serve to illustrate the fascinating history of analysis from Antiquity to the present. Further, the content is presented in connection with the historical and cultural events of the respective epochs, the lives of the scholars seeking knowledge, and insights into the subfields of analysis they created and shaped, as well as the applications in virtually every aspect of modern life that were made possible by analysis. From the reviews of the German edition: Three wishes granted at once: the book "3000 Jahre Analysis" is anything but a dry mixture of mathematical facts and formulae. Instead, it is a skillful blend of textbook, nonfiction, and history book rolled into one, which has been very vibrantly written. The author Thomas Sonar has managed to present the history of Analysis vividly, thrillingly, and full of intriguing details. Florian Modler, Spektrum der Wissenschaft The book [...] is simply wonderful. Everybody [...] picks up this book with a certain expectation, of course, and they won't be disappointed! ... It is truly fun browsing the book. Peter Littelmann, Mitteilungen der DMV About the Author Preface of the Author Preface of the Editors Advice to the reader Contents 1 Prologue: 3000 Years of Analysis 1.1 What is Analysis? 1.2 Precursors of 1.3 The of the Bible 1.4 Volume of a Frustum of a Pyramid 1.5 Babylonian Approximation of 2 2 The Continuum in Greek-Hellenistic Antiquity 2.1 The Greeks Shape Mathematics 2.1.1 The Very Beginning: Thales of Miletus and his Pupils 2.1.2 The Pythagoreans 2.1.3 The Proportion Theory of Eudoxus in Euclids Elements 2.1.4 The Method of Exhaustion Integration in the Greek Fashion 2.1.5 The Problem of Horn Angles 2.1.6 The Three Classical Problems of Antiquity Concerning the Quadrature of the Circle Concerning the Trisection of the Angle Concerning the Doubling of the Cube Remarks 2.2 Continuum versus Atoms Infinitesimals versus Indivisibles 2.2.1 The Eleatics 2.2.2 Atomism and the Theory of the Continuum 2.2.3 Indivisibles and Infinitesimals 2.2.4 The Paradoxes of Zeno 2.3 Archimedes 2.3.1 Life, Death, and Anecdotes 2.3.2 The Fate of Archimedess Writings 2.3.3 The Method: Access with Regard to Mechanical Theorems Weighing the Area Under a Parabola The Volume of a Paraboloid of Rotation 2.3.4 The Quadrature of the Parabola by means of Exhaustion 2.3.5 On Spirals 2.3.6 Archimedes traps 2.4 The Contributions of the Romans Approaches to Analysis in the Greek Antiquity 3 How Knowledge Migrates From Orient to Occident 3.1 The Decline of Mathematics and the Rescue by the Arabs 3.2 The Contributions of the Arabs Concerning Analysis 3.2.1 Avicenna (Ibn Sina): Polymath in the Orient 3.2.2 Alhazen (Ibn al-Haytham): Physicist and Mathematician 3.2.3 Averroes (Ibn Rushd): Islamic Aristotelian Contributions of Islamic Scholars to Analysis 4 Continuum and Atomism in Scholasticism 4.1 The Restart in Europe 4.2 The Great Time of the Translators 4.3 The Continuum in Scholasticism 4.3.1 Robert Grosseteste 4.3.2 Roger Bacon 4.3.3 Albertus Magnus 4.3.4 Thomas Bradwardine Life in the 14th Century: The Black Death Concerning Infinity Bradwardines Continuum Latitudes of Form: The Merton Rule as First Law of Motion 4.3.5 Nicole Oresme Summation of Infinite Serie Latitudes of Form and the Merton Rule The Doctrine of Proportions 4.4 Scholastic Dissenters 4.5 Nicholas of Cusa 4.5.1 The Mathematical Works Contributions to Analyis in the European Middle Ages 5 Indivisibles and Infinitesimals in the Renaissance 5.1 Renaissance: Rebirth of Antiquity 5.2 The Calculators of Barycentres 5.3 Johannes Kepler 5.3.1 New Stereometry of Wine Barrels 5.4 Galileo Galilei 5.4.1 Galileos Treatment of the Infinite Aristotles Wheel Galilei and Indivisibles The Cardinality of the Square Numbers 5.5 Cavalieri, Guldin, Torricelli, and the High Art of Indivisibles 5.5.1 Cavalieris Method of Indivisibles 5.5.2 The Criticism of Guldin 5.5.3 The Criticism of Galilei 5.5.4 Torricellis Apparent Paradox 5.5.5 De Saint-Vincent and the Area under the Hyperbola The Geometric Series of Saint-Vincent Horn Angles at Saint-Vincent The Area Under the Hyperbola Following Saint-Vincent Analysis and Astronomy during the Renaissance 6 At the Turn from the 16th to the 17th Century 6.1 Analysis in France before Leibniz 6.1.1 France at the turn of the 16th to the 17th Century 6.1.2 Ren Descartes The Circle Method of Descartes 6.1.3 Pierre de Fermat The Quadrature of Higher Parabolas Fermats Method of Pseudo-Equality 6.1.4 Blaise Pascal The Integration of xp The Characteristic Triangle Further Works Concerning Analysis 6.1.5 Gilles Personne de Roberval The Area Under the Cycloid The Quadrature of xp 6.2 Analysis Prior to Leibniz in the Netherlands 6.2.1 Frans van Schooten 6.2.2 Ren Franois Walther de Sluse 6.2.3 Johannes van Waveren Hudde 6.2.4 Christiaan Huygens 6.3 Analysis Before Newton in England 6.3.1 The Discovery of Logarithms 6.3.2 England at the Turn from the 16th to the 17th Century 6.3.3 John Napier and His Logarithms The Construction of Napiers Logarithms Napiers Kinematic Model The Early Meaning of Napiers Logarithms 6.3.4 Henry Briggs and His Logarithms The Construction Idea of Briggsian Logarithms The Successive Extraction of Roots Was Briggs Difference Calculus Stolen From Brgi? The Early Invention of the Binomial Theorem 6.3.5 England in the 17th Century 6.3.6 John Wallis and the Arithmetic of the Infinite Wallis and the Establishing of the Royal Society Wallis Mathematics at Oxford 6.3.7 Isaac Barrow and the Love of Geometry Barrows Mathematics 6.3.8 The Discovery of the Series of the Logarithms by Nicholas Mercator 6.3.9 The First Rectifications: Harriot and Neile Thomas Harriot William Neile 6.3.10 James Gregory 6.4 Analysis in India Development of Analysis in the 16th/17th Century 7 Newton and Leibniz Giants and Opponents 7.1 Isaac Newton 7.1.1 Childhood and Youth 7.1.2 Student in Cambridge 7.1.3 The Lucasian Professor 7.1.4 Alchemy, Religion, and the Great Crisis 7.1.5 Newton as President of the Royal Society 7.1.6 The Binomial Theorem 7.1.7 The Calculus of Fluxions 7.1.8 The Fundamental Theorem 7.1.9 Chain Rule and Substitutions 7.1.10 Computation with Series 7.1.11 Integration by Substitution 7.1.12 Newtons Last Works Concerning Analysis 7.1.13 Newton and Differential Equations 7.2 Gottfried Wilhelm Leibniz 7.2.1 Childhood, Youth, and Studies 7.2.2 Leibniz in the Service of the Elector of Mainz 7.2.3 Leibniz in Hanover 7.2.4 The Priority Dispute 7.2.5 First Achievements with Difference Sequences 7.2.6 Leibnizs Notation 7.2.7 The Characteristic Triangle 7.2.8 The Infinitely Small Quantities 7.2.9 The Transmutation Theorem 7.2.10 The Principle of Continuity 7.2.11 Differential Equations with Leibniz 7.3 First Critical Voice: George Berkeley Development of the Infinitesimal Calculus and the Priority Dispute 8 Absolutism, Enlightenment, Departure to New Shores 8.1 Historical Introduction 8.2 Jacob and John Bernoulli 8.2.1 The Calculus of Variations 8.3 Leonhard Euler 8.3.1 Eulers Notion of Function 8.3.2 The Infinitely Small in Eulers View 8.3.3 The Trigonometric Functions 8.4 Brook Taylor 8.4.1 The Taylor Series 8.4.2 Remarks Concerning the Calculus of Differences 8.5 Colin Maclaurin 8.6 The Beginnings of the Algebraic Interpretation 8.6.1 Lagranges Algebraic Analysis 8.7 Fourier Series and Multidimensional Analysis 8.7.1 Jean Baptiste Joseph Fourier 8.7.2 Early Discussions of the Wave Equation 8.7.3 Partial Differential Equations and Multidimensional Analysis 8.7.4 A Preview: The Importance of Fourier Series for Analysis Mathematicians and their Works Concerning the Analysis of the 18th Century 9 On the Way to Conceptual Rigour in the 19th Century From the French Revolution to the German Empire Science and Engineering in the Industrial Revolution 9.1 From the Congress of Vienna to the German Empire 9.2 Lines of Developments of Analysis in the 19th Century 9.3 Bernhard Bolzano and the Pradoxes of the Infinite 9.3.1 Bolzanos Contributions to Analysis 9.4 The Arithmetisation of Analysis: Cauchy 9.4.1 Limit and Continuity 9.4.2 The Convergence of Sequences and Series 9.4.3 Derivative and Integral 9.5 The Development of the Notion of Integral 9.6 The Final Arithmetisation of Analysis: Weierstra 9.6.1 The Real Numbers 9.6.2 Continuity, Differentiability, and Convergence 9.6.3 Uniformity 9.7 Richard Dedekind and his Companions 9.7.1 The Dedekind Cuts Substantial Results in Analysis 1800-1872 10 At the Turn to the 20th Century: Set Theory and the Search for the True Continuum General History 1871 to 1945 Technology and Natural sciences between 1871 and 1945 10.1 From the Establishment of the German Empire to the Global Catastrophes 10.2 Saint George Hunts Down the Dragon: Cantor and Set Theory 10.2.1 Cantors Construction of the Real Numbers 10.2.2 Cantor and Dedekind 10.2.3 The Transfinite Numbers 10.2.4 The Reception of Set Theory 10.2.5 Cantor and the Infinitely Small 10.3 Searching for the True Continuum: Paul Du Bois-Reymond 10.4 Searching for the True Continuum: The Intuitionists 10.5 Vector Analysis 10.6 Differential Geometry 10.7 Ordinary Differential Equantions 10.8 Partial Differential Equations 10.9 Analysis Becomes Even More Powerful: Functional Analysis 10.9.1 Basic Notions of Functional Analysis 10.9.2 A Historical Outline of Functional Analysis Development of Analysis in the 19th and 20th Century 11 Coming to full circle: Infinitesimals in Nonstandard Analysis General History From the End of WW II to Today Developments in Natural Sciences and Technology 11.1 From the Cold War up to today 11.1.1 Computer and Sputnik Shock 11.1.2 The Cold War and its End 11.1.3 Bologna Reform, Crises, Terrorism 11.2 The Rebirth of the Infinitely Small Numbers 11.2.1 Mathematics of Infinitesimals in the Black Book 11.2.2 The Nonstandard Analysis of Laugwitz and Schmieden 11.3 Robinson and the Nonstandard Analysis 11.4 Nonstandard Analysis by Axiomatisation: The Nelson Approach 11.5 Nonstandard Analysis and Smooth Worlds Development of Nonstandard Analysis 12 Analysis at Every Turn References List of Figures Index of persons Subject index