Kline M. Mathematical Thought from Ancient to Modern Times v.1-3

1990 paperback ed. — Oxford University Press USA — OSO, 1990. — 1252 p. — ISBN 9185961357.This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the discipline's origins in Babylon and Egypt, the creation of geometry and trigonometry by the Greeks, and the role of mathematics in the medieval and early modern periods. Volume 2 focuses on calculus, the rise of analysis in the 19th century, and the number theories of Dedekind and Dirichlet. The concluding volume covers the revival of projective geometry, the emergence of abstract algebra, the beginnings of topology, and the influence of Godel on recent mathematical study.Mathematics in Mesopotamia
Where Did Mathematics Begin?
Political History in Mesopotamia
The Number Symbols
Arithmetic Operations
Babylonian Algebra
Babylonian Geometry
The Uses of Mathematics in Babylonia
Evaluation of Babylonian MathematicsEgyptian Mathematics
Background
The Arithmetic
Algebra and Geometry
Egyptian Uses of MathematicsThe Creation of Classical Greek Mathematics
Background
The GeneralSources
The Major Schools of the Classical Period
The Ionian School
The Pythagoreans
The Eleatic School
The Sophist School
The PlatonicSchool
The School of Eudoxus
Aristotle and His SchoolEuclid and ApolloniusThe Background of Euclid's Elements
The Definitions and Axioms of the Elements
Books I to IV of the Elements
Book V The Theory of Proportion
Book VI. Similar Figures
Books VII, VIII, and IX: The Theory of Numbers
Book X: The Classification of Incommensurables
Books XI, XII, and XIII. Solid Geometry and the Method of Exhaustion
The Merits and Defects of the Elements
Other Mathematical Works by Euclid
The Mathematical Work of ApolloniusThe Alexandrian Greek Period: Geometry and Trigonometry
The Founding of Alexandria
The Character of Alexandrian Greek Mathematics
Areas and Volumes in the Work of Archimedes
Areas and Volumes in the Work of Heron
Some Exceptional Curves
The Creation of Trigonometry
Late Alexandrian Activity in GeometryThe Alexandrian Period: The Reemergence of Arithmetic and Algebra
The Symbols and Operations of Greek Arithmetic
Arithmetic and Algebra as an Independent DevelopmentThe Greek Rationalization of Nature
The Inspiration for Greek Mathematics
The Beginnings of a Rational View of Nature
The Development of the Belief in Mathematical Design
Greek Mathematical Astronomy
Geography
Mechanics
Optics
AstrologyThe Demise of the Greek World
A Review of the Greek Achievements
The Limitations of Greek Mathematics
The Problems Bequeathed by the Greeks
The Demise of the Greek CivilizationThe Mathematics of the Hindus and Arabs
Early Hindu Mathematics
Hindu Arithmetic and Algebra of the Period A.D. 200-1200
Hindu Geometry and Trigonometry of the Period A.D. 200-1200
The Arabs
Arabic Arithmetic and Algebra
Arabic Geometry and Trigonometry
Mathematics circa 1300The Medieval Period in Europe
The Beginnings of a European Civilization
The Materials Available for Learning
The Role of Mathematics in Early Medieval Europe
The Stagnation in Mathematics
The First Revival of the Greek Works
The Revival of Rationalism and Interest in Nature
Progress in Mathematics Proper
Progress in Physical ScienceThe Renaissance
Revolutionary Influences in Europe
The New Intellectual Outlook
The Spread of Learning
Humanistic Activity in Mathematics
The Clamor for the Reform of Science
The Rise of EmpiricismMathematical Contributions in the Renaissance
Perspective
Geometry Proper
Algebra
Trigonometry
The Major Scientific Progress in the Renaissance
Remarks on the RenaissanceArithmetic and Algebra in the Sixteenth and Seventeenth CenturiesThe Status of the Number System and Arithmetic
Symbolism
The Solution of Third and Fourth Degree Equations
The Theory of Equations
The Binomial Theorem and Allied Topics
The Theory of Numbers
The Relationship of Algebra to GeometryThe Beginnings of Projective Geometry
The Rebirth of Geometry
The Problems Raised by the Work on Perspective
The Work of Desargues
The Work of Pascal and La Hire
The Emergence of New Principles
Coordinate Geometry
The Motivation for Coordinate Geometry
The Coordinate Geometry of Fermat
Rene Descartes
Descartes's Work in Coordinate Geometry
Seventeenth-Century Extensions of Coordinate Geometry
The Importance of Coordinate GeometryThe Mathematization of ScienceDescartes's Concept of Science
Galileo's Approach to Science
The Function ConceptThe Creation of the Calculus
The Motivation for the Calculus
Early Seventeenth-Century Work on the Calculus
The Work of Newton
The Work of Leibniz
Comparison of the Work of Newton and Leibniz
The Controversy over Priority
Some Immediate Additions to the Calculus,
The Soundness of the CalculusMathematics as of 1700
The Transformation of Mathematics
Mathematics and Science
Communications Among Mathematicians
The Prospects for the Eighteenth CenturyCalculus in the Eighteenth CenturyThe Function Concept
The Technique of lntegration and Complex Quantities
Elliptic Integrals
Further Special Functions
The Calculus of Functions of Several Variables
The Attempts to Supply Rigor in the CalculusInfinite SeriesInitial Work on Infinite Series
The Expansion of Function
The Manipulation of Series
Trigonometric Series
Continued fractions
The Problem of Convergence and DivergenceOrdinary Differential Equations in the Eighteenth Century
Motivations
First Order Ordinary Differential Equations
Singular Solutions
Second Order Equations and the Riccati Equations
Higher Order Equations
The Method of Series
Systems of Differential EquationsPartial Differential Equations in the Eighteenth Century,
Introdurtion
The Wave Equation
Extensions of the Wave Equation
Potential Theory
First Order Partial Differential Equations
Monge and the Theory of Characteristics
Monge and Nonlinear Second Order Equations
Systems of First Order Partial Differential Equations
The Rise of the Mathematical SubjectAnalytic and Differential Geometry in the Eighteenth CenturyBasic Analytic Geometry
Higher Plane Curves
The Beginnings of Differential Geometry
Plane Curves
Space Curves
The Theory of Surfaces
The Mapping ProblemThe Calculus of Variations in the Eighteenth Century
The Initial Problems
The Early Work of Euler
The Principle of Least Action
The Methodology of Lagrange
Lagrange and Least Action
The Second VariationAlgebra in the Eighteenth Century
Status of the Number System
The Theory of Equations
Determinants and Elimination Theory
The Theory of NumbersMathematics as of 1800
The Rise of Analysis
The Motivation for the Eighteenth-Century Work
The Problem of Proof
The Metaphysical Basis
The Expansion of Mathematical Activity
A Glance AheadFunctions of a Complex VariableThe Beginnings of Complex Function Theory
The Geometrical Representation of Complex Numbers
The Foundation of Complex Function Theory
Weierstrass's Approach to Function Theory
Elliptic Functions
Hyperelliptic Integrals and Abel's Theorem
Riemann and Multiple-Valued Functions
Abelian Integrals and Functions
Conformal Mapping
The Representation of Functions and Excepional ValuesPartial Differential Equations in the Nineteenth CenturyThe Heat Equation and Fourier Series
Closed Solutions, the Fourier Integral
The Potential Equation and Green's Theorem
Curvilinear Coordinates
The Wave Equation and the Reduced Wave Equation
Systems of Partial Differential Equations
Existence TheoremsOrdinary Differential Equations in the Nineteenth CenturySeries Solutions and Special Functions
Sturm Liouville Theory
Existence Theorems
The Theory of Singularities
Automorphic Functions
Hill's Work on Periodic Solutions of Linear Equations
Nonlinear Differential Equations The Qualitative TheoryThe Calculus of Variations in the Nineteenth CenturyMathematical Physics and the Calculus of Variations
Mathematical Extensions of the Calculus of Variations Proper
Related Problems in the Calculus of VariationsGalois TheoryBinomial Equations
Abel's Work on the Solution of Equations by Radicals
Galois's Theory of Solvability
The Geometric Construction Problems
The Theory of Substitution GroupsQuaternions, Vectors, and Linear Associative Algebras
The Foundation of Algebra on Permanence of Form
The Search for a Three-Dimensional "Complex Number"
The Nature of Quaternions
Grassman's Calculus of Extension
From Quaternions to Vectors
Linear Associate AlgebrasDeterminants and MatricesSome New Uses of Determinants
Determinants and Quadratic Forms
MatricesThe Theory of Numbers in the Nineteenth CenturyThe Theory of Congruences
Algebraic Numbers
The Ideals of Dedekind
The Theory of Forms
Analytic Number TheoryThe Revival of Projective Geometry
The Renewal of I nterest in Geometry,
Synthetic Euclidean Geometry
The Revival of Synthetic Projective Geometry
Algebraic Projective Geometry
Higher Plane Curves and Surfaces36.Non-Euclidean GeometryThe Status of Euclidean Geometry About 1800
The Research on the Parallel Axiom
Foreshadowings of Non-Euclidean Geometry
The Creation of Non-Euclidean Geometry
The Technical Content of Non-Euclidian Geometry
The Claims of Lobatchevsky and Bolyai to Priority
The Implications of Non-Euclidean GeometryThe Differential Geometry of Gauss and RiemannGauss's Differential Geometry
Riemann's Approach to Geometry
The Successors of Riemann
Invariants of Differential FormsProjective and Metric GeometrySurfaces as Models ofNon-Euclidean Geometry
Projective and Metric Geometry
Models and the Consistency Problem
Geometry from the Transformation Viewpoint
The Reality of Non-Euclidean GeometryAlgebraic Geometry
Background
The Theory of Algebraic Invariants
The Concept of Birational Transformations
The Function-Theoretic Approach to Algebraic Geometry
The Uniformization Problem
The Algebraic-Geometric Approach
The Arithmetic Approach
The Algebraic Geometry of SurfacesThe Instillation of Rigor in AnalysisFunctions and Their Properties
The Derivative
The Integral
Infinite Series
Fourier Series
The Status of AnalysisThe Foundations of the Real and Transfinite NumbersAlgebraic and Transcendental Numbers
The Theory of Irrational Numbers
The Theory of Rational Numbers
Other Approaches to the Real Number System
The Concept of an Infinite Set
The Foundation of the Theory of Sets
Transfinite Cardinals and Ordinals
The Status of Set Theory by 1900The Foundations of Geometry
The Defects in Euclid
Contributions to the Foundations of Projective Geometry
The Foundations of Euclidean Geometry
Some Related Foundational Work
Some Open QuestionsMathematics as of 1900
The Chief Features ofthe Nineteenth-Century Developments
The Axiomatic Movement
Mathematics as Man`s Creation
The Loss of Truth
Mathematics as the Study of Arbitrary Structures
The Problem of Consistency
A Glance AheadThe Theory of Functions of Real Variables
The Origins
The Stieltjes Integral
Early Work on Content and Measure
The Lebesgue Integral
GeneralizationsIntegral EquationsThe Beginning of a General Theory
The Work of Hilbert
The Immediate Successors of Hilbert
Extensions of the TheoryFunctional Analysis
The Nature of Functional Analysis
The Theory of Functionals
Linear Functional Analysis
The Axiomatization of Hilbert SpaceDivergent SeriesThe Informal Uses of Divergent Series
The Formal Theory of Asymptotic Series
SummabilityTensor Analysis and Differential Geometry
The Origins of Tensor Analysis
The Notion of a Tensor
Covariant Differentiation
Parallel Displacement
Generalizations of Riemannian GeometryThe Emergence of Abstract Algebra
The Nineteenth-Century Background
Abstract Group Theory
The Abstract Theory of Fields
Rings
Non-Associative Algebras
The Range of Abstract AlgebraThe Beginnings of Topology
The Nature of Topology
Point Set Topology
The Beginnings of Combinational Topology
The Combinational Work of Poincare
Combinatorial Invariants
Fixed Point Theorems
Generalizations and ExtensionsThe Foundations of MathematicsThe Paradoxes of Set Theory
The Axiomatization of Set Theory
The Rise of Mathematical Logic
The Logistic School
The Intuitionist School
The Formalist School
Some Recent Developments
List of Abbreviations