什么是数学：对思想和方法的基本研究（英文版?第2版）
PREFACE TO SECOND EDITION
PREFACE TO REVISED EDITIONS
PREFACE TO FIRST EDITION
HOW TO USE THE BOOK
WHAT IS MATHEMATICS?
CHARTER I. THE NATURAL NUMBERS
Introduction
1. Calculation with Integers
1. Laws of Arithmetic. 2. The Representation of Integers. 3. Computation in Sys…
查看完整
PREFACE TO REVISED EDITIONS
PREFACE TO FIRST EDITION
HOW TO USE THE BOOK
WHAT IS MATHEMATICS?
CHARTER I. THE NATURAL NUMBERS
Introduction
1. Calculation with Integers
1. Laws of Arithmetic. 2. The Representation of Integers. 3. Computation in Sys…
查看完整
Richard Courant（1888-1972）20世纪杰出的数学家，哥廷根学派重要成员。曾担任纽约大学数学系主任和数学科学研究院院长，为了纪念他，纽约大学数学科学研究院1964年改名为柯朗数学科学研究院！成为世界上最大的应用数学研究中心。他写的书《数学物理方程》为每一个物理学家所熟知，而他的《微积分学》也被认为是该学科的代表作。
Herbert Robbins（1915－2001） 美国著名数学家和统计学家.他的研究涉及拓扑学. 测度论. 统计学等诸多领域.经验贝叶斯方法中的 Robbins引理，图论中的Robbins定理，还有Robbins 代数和Robbins问题都以他的名字命名。
Herbert Robbins（1915－2001） 美国著名数学家和统计学家.他的研究涉及拓扑学. 测度论. 统计学等诸多领域.经验贝叶斯方法中的 Robbins引理，图论中的Robbins定理，还有Robbins 代数和Robbins问题都以他的名字命名。
《什么是数学：对思想和方法的基本研究(英文版·第2版)》是世界著名的数学科普读物。它荟萃了许多数学的奇珍异宝，对数学世界做了生动而易懂的描述。内容涵盖代数、几何、微积分、拓扑等领域，其中还穿插了许多相关的历史和哲学知识。《什么是数学：对思想和方法的基本研究(英文版·第2版)》不仅是数学专业人员的必读之物，也是任何愿意做科学思考者的优秀读物。对于中学数学教师、高中生和大学生来说，这都是一本极好的参考书。
PREFACE TO SECOND EDITION
PREFACE TO REVISED EDITIONS
PREFACE TO FIRST EDITION
HOW TO USE THE BOOK
WHAT IS MATHEMATICS?
CHARTER I. THE NATURAL NUMBERS
Introduction
1. Calculation with Integers
1. Laws of Arithmetic. 2. The Representation of Integers. 3. Computation in Systems Other than the Decimal
2. The Infinitude of the Number System. Mathematical Induction
1. The Principle of Mathematical .Induction. 2. The Arithmetical Progression. 3. The Geometrical Progression. 4. The Sum of the First n Squares. 5. An Important Inequality. 6. The Binomial Theorem. 7. Further Remarks on Mathematical Induction
SUPPLEMENT TO CHAPTER I. THE THEORY OF NUMBERS
Introduction
1. The Prime Numbers
I. Fundamental Facts. 2. The Distribution of the Primes. a. Formulas Producing Primes. b. Primes in Arithmetical Progressions. c. The Prime Number Theorem. d. Two Unsolved Problems Concerning Prime Numbers
2. Congruences
1. General Concepts. 2. Fermats Theorem. 3. Quadratic Residues
3. Pythagorean Numbers and Fermats Last Theorem
4. The Euclidean Algorithm
1. General Theory. 2. Application to the Fundamental Theorem of Arithmetic. 3. Eulers Function. Fermats Theorem Again. 4. Continued Fractions. Diophantine Equations
CHAPTER II. THE NUMBER SYSTEM OF MATHEMATICS
Introduction
1. The Rational Numbers
1. Rational Numbers as a Device for Measuring. 2. Intrinsic Need for the Rational Numbers. Principal of Generalization. 3. Geometrical Interpretation of Rational Numbers
2. Incommensurable Segments， Irrational Numbers， and the Concept of Limit
1. Introduction. 2. Decimal Fractions. Infinite Decimals. 3. Limits. Infinite Geometrical Series. 4. Rational Numbers and Periodic Decimals. 5. General Definition of Irrational Numbers by Nested Intervals. 6. Alternative Methods of Defining Irrational Numbers. Dedekind Cuts
3. Remarks on Analytic Geometry
1. The Basic Principle. 2. Equations of Lines and Curves
4. The Mathematical Analysis of Infinity
1. Fundamental Concepts. 2. The Denumerability of the Rational Numbers and the Non-Denumerability of the Continuum. 3. Cantors "Cardinal Numbers." 4. The Indirect Method of Proof， 5. The Paradoxes of the Infinite. 6. The Foundations of Mathematics
5. Complex Numbers
1. The Origin of Complex Numbers. 2. The Geometrical Interpretation of Complex Numbers. 3. De Moivres Formula and the Roots of Unity. 4. The Fundamental Theorem of Algebra
6. Algebraic and Transcendental Numbers
1. Definition and Existence. 2. Liouvilles Theorem and the Construction of Transcendental Numbers
SUPPLEMENT TO CHAPTER II. THE ALGEBRA OF SETS
1. General Theory. 2. Application to Mathematical Logic. 3. An Application to the Theory of Probability
CHAPTER III. GEOMETRICAL CONSTRUCTIONS. THE ALGEBRA OF NUMBER FIELDS
Introduction
Part I. Impossibility Proofs and Algebra
1. Fundamental Geometrical Constructions
1. Construction of Fields and Square Root Extraction. 2. Regular Polygons. 3. Apollonius Problem
2. Constructible Numbers and Number Fields
1. General Theory. 2. All Constructible Numbers are Algebraic
3. The Unsolvability of the Three Greek Problems
1. Doubling the Cube. 2. A Theorem on Cubic Equations. 3. Trisecting the Angle. 4. The Regular Heptagon. 5. Remarks on the Problem of Squaring the Circle
Part II. Various Methods for Performing Constructions
4. Geometrical Transformations. Inversion
1. General Remarks. 2. Properties of Inversion. 3. Geometrical Construction of Inverse Points. 4. How to Bisect a Segment and Find the Center of a Circle with the Compass Alone
5. Constructions with Other Tools. Mascheroni Constructions with Compass Alone
1. A Classical Construction for Doubling the Cube. 2. Restriction to the Use of the Compass Alone. 3. Drawing with Mechanical Instruments. Mechanical Curves. Cycloids. 4. Linkages. Peaucelliers and Harts Inversors
6. More About Inversions and its Applications
1. Invariance of Angles. Families of Circles. 2. Application to the Problem of Apoilonius. 3. Repeated Reflections
CHAPTER IV. PROJECTIVE GEOMETRY. AXIOMATICS. NoN-EUCLIDEAN GEOMETRIES
I. Introduction
1. Classification of Geometrical Properties. lnvariance under Transformations. 2. Projective Transformations
2. Fundamental Concepts
1. The Group of Projective Transformations. 2. Desarguess Theorem
3.Cross-Ratio
1. Definition and Proof of Invariance. 2. Application to the Complete Quadrilateral
4. Parallelism and Infinity
1. Points at Infinity as "Ideal Points." 2. Ideal Elements and Projection. 3. Cross-Ratio with Elements at Infinity
5. Applications
1. Preliminary Remarks. 2. Proof of Desarguess Theorem in the Plane. 3. Pascals Theorem. 4. Brianchons Theorem. 5. Remark on Duality
6. Analytic Representation
1. Introductory Remarks. 2. Homogeneous Co6rdinates. The Algebraic Basis of Duality
7. Problems on Constructions with the Straightedge Alone
8. Conics and Quadric Surfaces
1. Elementary Metric Geometry of Conics. 2. Projective Properties of Conics. 3. Conics as Line Curves. 4. Pascals and Brianchons General Theorems for Conics. 5. The Hyperboloid
9. Axiomatics and Non-Euclidean Geometry
1. The Axiomatic Method. 2. Hyperbolic Non-Euchdean Geometry. 3. Geometry and Reality. 4. Poincares Model. 5. Elliptic or Riemannian Geometry
APPENDIX. GEOMETRY IN MORE THAN THREE DIMENSIONS
1. Introduction. 2. Analytic Approach. 3. Geometrical or Combinatorial Approach
CHAPTER V. TOPOLOGY
Introduction
1. Eulers Formula for Polyhedra
2. Topological Properties of Figures
1. Topological Properties. 2. Connectivity
3. Other Examples of Topological Theorems
1. The Jordan Curve Theorem. 2. The Four Color Problem. 3. The Concept of Dimension. 4. A Fixed Point Theorem. 5. Knots
4. The Topological Classification of Surfaces
1. The Genus of a Surface. 2. The Euler Characteristic of a Surface. 3. One-Sided Surfaces
APPENDIX
1. The Five Color Theorem. 2. The Jordan Curve Theorem for Polygons. 3. The Fundamental Theorem of Algebra
CHAPTER VI. FUNCTIONS AND LIMITS
Introduction
1. Variable and Function
1. Definitions and Examples. 2. Radian Measure of Angles. 3. The Graph of a Function. Inverse Functions. 4. Compound Func
tions. 5. Continuity. 6. Functions of Several Variables. 7. Functions and Transformations
2. Limits
1. The Limit of a Sequence an 2. Monotone Sequences. 3. Eulers Number e. 4. The Number π 5. Continued Fractions
3. Limits by Continuous Approach
1. Introduction. General Definition. 2. Remarks on the IAmit Concept. 3. The Limit of sin x/x. 4. Limits as x
4. Precise Definition of Continuity .
5. Two Fundamental Theorems on Continuous Functions
1. Bolzanos Theorem. 2. Proof of Bolzanos Theorem. 3. Weierstrass Theorem on Extreme Values. 4. A Theorem on Sequences. Compact Sets
6. Some Applications of Bolzanos Thoerem
1. Geometrical Applications. 2. Application to a Problem in Mechanics
SUPPLEMENT TO CHAPTER VI. MORE EXAMPLES ON LIMITS AND CONTINUITY
1. Examples of Limits
1. General Remarks. 2. The Limit of q. 3. The IAmit of p. 4. Discon tinuous Functions as Limits of Continuous Functions. 5. Limits by Iteration
2. Example on Continuity
CHAPTER VII. MAXIMA AND MINIMA
Introduction
1. Problems in Elementary Geometry
1. Maximum Area of a Triangle with Two Sides Given. 2. Herons Thoerem. Extremum Property of Light Rays. 3. Applications to Problems on Triangles. 4. Tangent Properties of Ellipse and Hyperboll Corresponding Extremum Properties. 5. Extreme Distances to a Given Curve
2. A General Principal Underlying Extreme Value Problems
1. The Principle. 2. Examples
3. Stationary Points and the Differential Calculus
1. Extrema and Stationary Points. 2. Maxima and Minima of Functions of Several Variables. Saddle Points. 3. Minimax Points and Topology. 4. The Distance from a Point to a Surface
4. Schwax-zs Triangle Problem
1. Schwarzs Proof. 2. Another Proof. 3. Obtuse Triangles. 4. Triangles Formed by Light Rays. 5. Remarks Concerning Problems of Reflection and Ergodic Motion
5. Steiners Problem
1. Problem and Solution. 2. Analysis of the Alternatives. 3. A Complementary Problem. 4. Remarks and Exercises. 5. Generalization to the Street Network Problem
6. Extrema and Inequalities
1. The Arithmetical and Geometrical Mean of Two Positive Quantifies. 2. Generalization to n Variables. 3. The Method of Least Squares
7. The Existence of an Extremum. Dirichlets Principle
1. General Remarks. 2. Examples. 3. Elementary Extremum Problems 4. Difficulties m Higher Cases
8. The Isoperimetric Problem
9. Extremum Problems with Boundary Conditions. Connection Between Steiners Problem and the Isoperimetric Problem
10. The Calculus of Variations
1. Introduction. 2. The Calculus of Variations. Fermats Principle in Optics. 3. Bemoullis Treatment of the Brachistochrone Problem， 4. Geodesics on a Sphere. Geodesics and Maxi-Minima
11. Experimental Solutions of Minimum Problems. Soap Film Experiments.
1. Introduction. 2. Soap Film Experiments. 3. New Experiments on Plateaus Problem. 4. Experimental Solutions of Other Mathematical Problems
CHAPTER VIII. THE CALCULUS
Introduction
1. The Integral
1. Area as a Limit. 2. The Integral. 3. General Remarks on the Integral Concept. General Definition. 4. Examples of Integration. Integration of x. 5. Rules for the "Integral Calculus"
2. The Derivative
1. The Derivative as a Slope. 2. The Derivative as a Limit. 3. Examples. 4. Derivatives of Trigonometrical Functions. 5， Differentiation and Continuity. 6. Derivative and Velocity. Second Derivative and Acceleration. 7. Geometrical Meaning of the Second Derivative. 8. Maxima and Minima
3. The Technique of Differentiation
4. Leibniz Notation and the "Infinitely Small"
5. The Fundamental Theorem of the Calculus
1. The Fundamental Theorem. 2. First Applications. Integration of x， cos x， sin x. Arc tan x. 3. Leibniz Formula for π
6. The Exponential Function and the Logarithm
1. Definition and Properties of the Logarithm. Eulers Number e. 2. The Exponential Function. 3. Formulas for Differentiation of ex， ax . 4. Explicit Expressions for e， ex， and log x as Limits. 5. Infinite Series for the Logarithm. Numerical Calculation
7. Differential Equations
1. Definition. 2. The Differential Equation of the Exponential Function. Radioactive Disintegration. Law of Growth. Compound Interest. 3. Other Examples. Simplest Vibrations. 4. Newtons Law of Dynamics
SUPPLEMENT TO CHAPTER VIII
1. Matters of Principle
1. Differentiability. 2. The Integral. 3. Other Applications of the Concept of Integral. Work. Length
2. Orders of Magnitude
1. The Exponential Function and Powers of x. 2. Order of Magnitude of log (n!)
3. Infinite Series and Infinite Products
1. Infinite Series of Functions. 2. Eulers Formula， cos x + i sin x = eix. 3. The Harmonic Series and the Zeta Function. Eulers Product for the Sine
4. The Prime Number Theorem Obtained by Statistical Methods
CHAPTER IX. RECENT DEVELOPMENTS
1. A Formula for Primes
2. The Goldbach Conjecture and Twin Primes
3. Fermats Last Theorem
4. The Continuum Hypothesis
5. Set-Theoretic Notation
6. The Four Color Theorem
7. Hausdorff Dimension and Fractals
8. Knots
9. A Problem in Mechanics
10. Steiners Problem
11. Soap Films and Minimal Surfaces
12. Nonstandard Analysis
APPENDIX： SUPPLEMENTARY REMARKS， PROBLEMS， AND EXERCISES
Arithmetic and Algebra
Analytic Geometry
Geometrical Constructions
Projective and Non-Euclidean Geometry
Topology
Functions， Limits， and Continuity
Maxima and Minima
The Calculus
Technique of Integration
SUGGESTIONS FOR FURTHER READING
SUGGESTIONS FOR ADDITIONAL READING
INDEX
^ 收 起
PREFACE TO REVISED EDITIONS
PREFACE TO FIRST EDITION
HOW TO USE THE BOOK
WHAT IS MATHEMATICS?
CHARTER I. THE NATURAL NUMBERS
Introduction
1. Calculation with Integers
1. Laws of Arithmetic. 2. The Representation of Integers. 3. Computation in Systems Other than the Decimal
2. The Infinitude of the Number System. Mathematical Induction
1. The Principle of Mathematical .Induction. 2. The Arithmetical Progression. 3. The Geometrical Progression. 4. The Sum of the First n Squares. 5. An Important Inequality. 6. The Binomial Theorem. 7. Further Remarks on Mathematical Induction
SUPPLEMENT TO CHAPTER I. THE THEORY OF NUMBERS
Introduction
1. The Prime Numbers
I. Fundamental Facts. 2. The Distribution of the Primes. a. Formulas Producing Primes. b. Primes in Arithmetical Progressions. c. The Prime Number Theorem. d. Two Unsolved Problems Concerning Prime Numbers
2. Congruences
1. General Concepts. 2. Fermats Theorem. 3. Quadratic Residues
3. Pythagorean Numbers and Fermats Last Theorem
4. The Euclidean Algorithm
1. General Theory. 2. Application to the Fundamental Theorem of Arithmetic. 3. Eulers Function. Fermats Theorem Again. 4. Continued Fractions. Diophantine Equations
CHAPTER II. THE NUMBER SYSTEM OF MATHEMATICS
Introduction
1. The Rational Numbers
1. Rational Numbers as a Device for Measuring. 2. Intrinsic Need for the Rational Numbers. Principal of Generalization. 3. Geometrical Interpretation of Rational Numbers
2. Incommensurable Segments， Irrational Numbers， and the Concept of Limit
1. Introduction. 2. Decimal Fractions. Infinite Decimals. 3. Limits. Infinite Geometrical Series. 4. Rational Numbers and Periodic Decimals. 5. General Definition of Irrational Numbers by Nested Intervals. 6. Alternative Methods of Defining Irrational Numbers. Dedekind Cuts
3. Remarks on Analytic Geometry
1. The Basic Principle. 2. Equations of Lines and Curves
4. The Mathematical Analysis of Infinity
1. Fundamental Concepts. 2. The Denumerability of the Rational Numbers and the Non-Denumerability of the Continuum. 3. Cantors "Cardinal Numbers." 4. The Indirect Method of Proof， 5. The Paradoxes of the Infinite. 6. The Foundations of Mathematics
5. Complex Numbers
1. The Origin of Complex Numbers. 2. The Geometrical Interpretation of Complex Numbers. 3. De Moivres Formula and the Roots of Unity. 4. The Fundamental Theorem of Algebra
6. Algebraic and Transcendental Numbers
1. Definition and Existence. 2. Liouvilles Theorem and the Construction of Transcendental Numbers
SUPPLEMENT TO CHAPTER II. THE ALGEBRA OF SETS
1. General Theory. 2. Application to Mathematical Logic. 3. An Application to the Theory of Probability
CHAPTER III. GEOMETRICAL CONSTRUCTIONS. THE ALGEBRA OF NUMBER FIELDS
Introduction
Part I. Impossibility Proofs and Algebra
1. Fundamental Geometrical Constructions
1. Construction of Fields and Square Root Extraction. 2. Regular Polygons. 3. Apollonius Problem
2. Constructible Numbers and Number Fields
1. General Theory. 2. All Constructible Numbers are Algebraic
3. The Unsolvability of the Three Greek Problems
1. Doubling the Cube. 2. A Theorem on Cubic Equations. 3. Trisecting the Angle. 4. The Regular Heptagon. 5. Remarks on the Problem of Squaring the Circle
Part II. Various Methods for Performing Constructions
4. Geometrical Transformations. Inversion
1. General Remarks. 2. Properties of Inversion. 3. Geometrical Construction of Inverse Points. 4. How to Bisect a Segment and Find the Center of a Circle with the Compass Alone
5. Constructions with Other Tools. Mascheroni Constructions with Compass Alone
1. A Classical Construction for Doubling the Cube. 2. Restriction to the Use of the Compass Alone. 3. Drawing with Mechanical Instruments. Mechanical Curves. Cycloids. 4. Linkages. Peaucelliers and Harts Inversors
6. More About Inversions and its Applications
1. Invariance of Angles. Families of Circles. 2. Application to the Problem of Apoilonius. 3. Repeated Reflections
CHAPTER IV. PROJECTIVE GEOMETRY. AXIOMATICS. NoN-EUCLIDEAN GEOMETRIES
I. Introduction
1. Classification of Geometrical Properties. lnvariance under Transformations. 2. Projective Transformations
2. Fundamental Concepts
1. The Group of Projective Transformations. 2. Desarguess Theorem
3.Cross-Ratio
1. Definition and Proof of Invariance. 2. Application to the Complete Quadrilateral
4. Parallelism and Infinity
1. Points at Infinity as "Ideal Points." 2. Ideal Elements and Projection. 3. Cross-Ratio with Elements at Infinity
5. Applications
1. Preliminary Remarks. 2. Proof of Desarguess Theorem in the Plane. 3. Pascals Theorem. 4. Brianchons Theorem. 5. Remark on Duality
6. Analytic Representation
1. Introductory Remarks. 2. Homogeneous Co6rdinates. The Algebraic Basis of Duality
7. Problems on Constructions with the Straightedge Alone
8. Conics and Quadric Surfaces
1. Elementary Metric Geometry of Conics. 2. Projective Properties of Conics. 3. Conics as Line Curves. 4. Pascals and Brianchons General Theorems for Conics. 5. The Hyperboloid
9. Axiomatics and Non-Euclidean Geometry
1. The Axiomatic Method. 2. Hyperbolic Non-Euchdean Geometry. 3. Geometry and Reality. 4. Poincares Model. 5. Elliptic or Riemannian Geometry
APPENDIX. GEOMETRY IN MORE THAN THREE DIMENSIONS
1. Introduction. 2. Analytic Approach. 3. Geometrical or Combinatorial Approach
CHAPTER V. TOPOLOGY
Introduction
1. Eulers Formula for Polyhedra
2. Topological Properties of Figures
1. Topological Properties. 2. Connectivity
3. Other Examples of Topological Theorems
1. The Jordan Curve Theorem. 2. The Four Color Problem. 3. The Concept of Dimension. 4. A Fixed Point Theorem. 5. Knots
4. The Topological Classification of Surfaces
1. The Genus of a Surface. 2. The Euler Characteristic of a Surface. 3. One-Sided Surfaces
APPENDIX
1. The Five Color Theorem. 2. The Jordan Curve Theorem for Polygons. 3. The Fundamental Theorem of Algebra
CHAPTER VI. FUNCTIONS AND LIMITS
Introduction
1. Variable and Function
1. Definitions and Examples. 2. Radian Measure of Angles. 3. The Graph of a Function. Inverse Functions. 4. Compound Func
tions. 5. Continuity. 6. Functions of Several Variables. 7. Functions and Transformations
2. Limits
1. The Limit of a Sequence an 2. Monotone Sequences. 3. Eulers Number e. 4. The Number π 5. Continued Fractions
3. Limits by Continuous Approach
1. Introduction. General Definition. 2. Remarks on the IAmit Concept. 3. The Limit of sin x/x. 4. Limits as x
4. Precise Definition of Continuity .
5. Two Fundamental Theorems on Continuous Functions
1. Bolzanos Theorem. 2. Proof of Bolzanos Theorem. 3. Weierstrass Theorem on Extreme Values. 4. A Theorem on Sequences. Compact Sets
6. Some Applications of Bolzanos Thoerem
1. Geometrical Applications. 2. Application to a Problem in Mechanics
SUPPLEMENT TO CHAPTER VI. MORE EXAMPLES ON LIMITS AND CONTINUITY
1. Examples of Limits
1. General Remarks. 2. The Limit of q. 3. The IAmit of p. 4. Discon tinuous Functions as Limits of Continuous Functions. 5. Limits by Iteration
2. Example on Continuity
CHAPTER VII. MAXIMA AND MINIMA
Introduction
1. Problems in Elementary Geometry
1. Maximum Area of a Triangle with Two Sides Given. 2. Herons Thoerem. Extremum Property of Light Rays. 3. Applications to Problems on Triangles. 4. Tangent Properties of Ellipse and Hyperboll Corresponding Extremum Properties. 5. Extreme Distances to a Given Curve
2. A General Principal Underlying Extreme Value Problems
1. The Principle. 2. Examples
3. Stationary Points and the Differential Calculus
1. Extrema and Stationary Points. 2. Maxima and Minima of Functions of Several Variables. Saddle Points. 3. Minimax Points and Topology. 4. The Distance from a Point to a Surface
4. Schwax-zs Triangle Problem
1. Schwarzs Proof. 2. Another Proof. 3. Obtuse Triangles. 4. Triangles Formed by Light Rays. 5. Remarks Concerning Problems of Reflection and Ergodic Motion
5. Steiners Problem
1. Problem and Solution. 2. Analysis of the Alternatives. 3. A Complementary Problem. 4. Remarks and Exercises. 5. Generalization to the Street Network Problem
6. Extrema and Inequalities
1. The Arithmetical and Geometrical Mean of Two Positive Quantifies. 2. Generalization to n Variables. 3. The Method of Least Squares
7. The Existence of an Extremum. Dirichlets Principle
1. General Remarks. 2. Examples. 3. Elementary Extremum Problems 4. Difficulties m Higher Cases
8. The Isoperimetric Problem
9. Extremum Problems with Boundary Conditions. Connection Between Steiners Problem and the Isoperimetric Problem
10. The Calculus of Variations
1. Introduction. 2. The Calculus of Variations. Fermats Principle in Optics. 3. Bemoullis Treatment of the Brachistochrone Problem， 4. Geodesics on a Sphere. Geodesics and Maxi-Minima
11. Experimental Solutions of Minimum Problems. Soap Film Experiments.
1. Introduction. 2. Soap Film Experiments. 3. New Experiments on Plateaus Problem. 4. Experimental Solutions of Other Mathematical Problems
CHAPTER VIII. THE CALCULUS
Introduction
1. The Integral
1. Area as a Limit. 2. The Integral. 3. General Remarks on the Integral Concept. General Definition. 4. Examples of Integration. Integration of x. 5. Rules for the "Integral Calculus"
2. The Derivative
1. The Derivative as a Slope. 2. The Derivative as a Limit. 3. Examples. 4. Derivatives of Trigonometrical Functions. 5， Differentiation and Continuity. 6. Derivative and Velocity. Second Derivative and Acceleration. 7. Geometrical Meaning of the Second Derivative. 8. Maxima and Minima
3. The Technique of Differentiation
4. Leibniz Notation and the "Infinitely Small"
5. The Fundamental Theorem of the Calculus
1. The Fundamental Theorem. 2. First Applications. Integration of x， cos x， sin x. Arc tan x. 3. Leibniz Formula for π
6. The Exponential Function and the Logarithm
1. Definition and Properties of the Logarithm. Eulers Number e. 2. The Exponential Function. 3. Formulas for Differentiation of ex， ax . 4. Explicit Expressions for e， ex， and log x as Limits. 5. Infinite Series for the Logarithm. Numerical Calculation
7. Differential Equations
1. Definition. 2. The Differential Equation of the Exponential Function. Radioactive Disintegration. Law of Growth. Compound Interest. 3. Other Examples. Simplest Vibrations. 4. Newtons Law of Dynamics
SUPPLEMENT TO CHAPTER VIII
1. Matters of Principle
1. Differentiability. 2. The Integral. 3. Other Applications of the Concept of Integral. Work. Length
2. Orders of Magnitude
1. The Exponential Function and Powers of x. 2. Order of Magnitude of log (n!)
3. Infinite Series and Infinite Products
1. Infinite Series of Functions. 2. Eulers Formula， cos x + i sin x = eix. 3. The Harmonic Series and the Zeta Function. Eulers Product for the Sine
4. The Prime Number Theorem Obtained by Statistical Methods
CHAPTER IX. RECENT DEVELOPMENTS
1. A Formula for Primes
2. The Goldbach Conjecture and Twin Primes
3. Fermats Last Theorem
4. The Continuum Hypothesis
5. Set-Theoretic Notation
6. The Four Color Theorem
7. Hausdorff Dimension and Fractals
8. Knots
9. A Problem in Mechanics
10. Steiners Problem
11. Soap Films and Minimal Surfaces
12. Nonstandard Analysis
APPENDIX： SUPPLEMENTARY REMARKS， PROBLEMS， AND EXERCISES
Arithmetic and Algebra
Analytic Geometry
Geometrical Constructions
Projective and Non-Euclidean Geometry
Topology
Functions， Limits， and Continuity
Maxima and Minima
The Calculus
Technique of Integration
SUGGESTIONS FOR FURTHER READING
SUGGESTIONS FOR ADDITIONAL READING
INDEX
^ 收 起
Richard Courant（1888-1972）20世纪杰出的数学家，哥廷根学派重要成员。曾担任纽约大学数学系主任和数学科学研究院院长，为了纪念他，纽约大学数学科学研究院1964年改名为柯朗数学科学研究院！成为世界上最大的应用数学研究中心。他写的书《数学物理方程》为每一个物理学家所熟知，而他的《微积分学》也被认为是该学科的代表作。
Herbert Robbins（1915－2001） 美国著名数学家和统计学家.他的研究涉及拓扑学. 测度论. 统计学等诸多领域.经验贝叶斯方法中的 Robbins引理，图论中的Robbins定理，还有Robbins 代数和Robbins问题都以他的名字命名。
Herbert Robbins（1915－2001） 美国著名数学家和统计学家.他的研究涉及拓扑学. 测度论. 统计学等诸多领域.经验贝叶斯方法中的 Robbins引理，图论中的Robbins定理，还有Robbins 代数和Robbins问题都以他的名字命名。
《什么是数学：对思想和方法的基本研究(英文版·第2版)》是世界著名的数学科普读物。它荟萃了许多数学的奇珍异宝，对数学世界做了生动而易懂的描述。内容涵盖代数、几何、微积分、拓扑等领域，其中还穿插了许多相关的历史和哲学知识。《什么是数学：对思想和方法的基本研究(英文版·第2版)》不仅是数学专业人员的必读之物，也是任何愿意做科学思考者的优秀读物。对于中学数学教师、高中生和大学生来说，这都是一本极好的参考书。
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