3000 Years of Analysis: Mathematics in History and Culture

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 ˇ
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 Euclid’s 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 Archimedes’s 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 S¯in¯a): 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
Bradwardine’s 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 Galileo’s Treatment of the Infinite
Aristotle’s Wheel
Galilei and Indivisibles
The Cardinality of the Square Numbers
5.5 Cavalieri, Guldin, Torricelli, and the High Art of Indivisibles
5.5.1 Cavalieri’s Method of Indivisibles
5.5.2 The Criticism of Guldin
5.5.3 The Criticism of Galilei
5.5.4 Torricelli’s 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 René Descartes
The Circle Method of Descartes
6.1.3 Pierre de Fermat
The Quadrature of Higher Parabolas
Fermat’s 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 René François 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 Napier’s Logarithms
Napier’s Kinematic Model
The Early Meaning of Napier’s 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 Bürgi?
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 Leibniz’s 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 Euler’s Notion of Function
8.3.2 The Infinitely Small in Euler’s 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 Lagrange’s 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 Bolzano’s 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: Weierstraß
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 Cantor’s 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