Mathematical Thought from Ancient to Modern Times: Volume 2

Cover......Page A001.djvu
Title Page......Page __sk_0001.djvu
Preface to the Three-Volume Paperback Edition of Mathematical Thought......Page __sk_0005.djvu
Preface......Page __sk_0007.djvu
Contents......Page __sk_0011.djvu
Publisher's Note to this Three-Volume Paperback Edition......Page __sk_0015.djvu
1. The Transformation of Mathematics......Page __sk_0019.djvu
2. Mathematics and Science......Page __sk_0022.djvu
3. Communication Among Mathematicians......Page __sk_0024.djvu
4. The Prospects for the Eighteenth Century......Page __sk_0026.djvu
Bibliography......Page __sk_0027.djvu
1. Introduction......Page __sk_0028.djvu
2. The Function Concept......Page __sk_0031.djvu
3. The Technique of Integration and Complex Quantities......Page __sk_0034.djvu
4. Elliptic Integrals......Page __sk_0039.djvu
5. Further Special Functions......Page __sk_0050.djvu
6. The Calculus of Functions of Several Variables......Page __sk_0053.djvu
7. The Attempts to Supply Rigor in the Calculus......Page __sk_0054.djvu
Bibliography......Page __sk_0062.djvu
2. Initial Work on Infinite Series......Page __sk_0064.djvu
3. The Expansion of Functions......Page __sk_0068.djvu
4. The Manipulation of Series......Page __sk_0070.djvu
5. Trigonometric Series......Page __sk_0082.djvu
6. Continued Fractions......Page __sk_0087.djvu
7. The Problem of Convergence and Divergence......Page __sk_0088.djvu
Bibliography......Page __sk_0094.djvu
1. Motivations......Page __sk_0096.djvu
2. First Order Ordinary Differential Equations......Page __sk_0099.djvu
3. Singular Solutions......Page __sk_0104.djvu
4. Second Order Equations and the Riccati Equations......Page __sk_0106.djvu
5. Higher Order Equations......Page __sk_0112.djvu
6. The Method of Series......Page __sk_0116.djvu
7. Systems of Differential Equations......Page __sk_0118.djvu
8. Summary......Page __sk_0127.djvu
Bibliography......Page __sk_0129.djvu
1. Introduction......Page __sk_0130.djvu
2. The Wave Equation......Page __sk_0131.djvu
3. Extensions of the Wave Equation......Page __sk_0143.djvu
4. Potential Theory......Page __sk_0150.djvu
5. First Order Partial Differential Equations......Page __sk_0159.djvu
6. Monge and the Theory of Characteristics......Page __sk_0164.djvu
7. Monge and Nonlinear Second Order Equations......Page __sk_0166.djvu
8. Systems of First Order Partial Differential Equations......Page __sk_0168.djvu
9. The Rise of the Mathematical Subject......Page __sk_0170.djvu
Bibliography......Page __sk_0171.djvu
2. Basic Analytic Geometry......Page __sk_0172.djvu
3. Higher Plane Curves......Page __sk_0175.djvu
4. The Beginnings of Differential Geometry......Page __sk_0182.djvu
5. Plane Curves......Page __sk_0183.djvu
6. Space Curves......Page __sk_0185.djvu
7. The Theory of Surfaces......Page __sk_0190.djvu
8. The Mapping Problem......Page __sk_0198.djvu
Bibliography......Page __sk_0199.djvu
1. The Initial Problems......Page __sk_0201.djvu
2. The Early Work of Euler......Page __sk_0205.djvu
3. The Principle of Least Action......Page __sk_0207.djvu
4. The Methodology of Lagrange......Page __sk_0210.djvu
5. Lagrange and Least Action......Page __sk_0215.djvu
6. The Second Variation......Page __sk_0217.djvu
Bibliography......Page __sk_0218.djvu
1. Status of the Number System......Page __sk_0220.djvu
2. The Theory of Equations......Page __sk_0225.djvu
3. Determinants and Elimination Theory......Page __sk_0234.djvu
4. The Theory of Numbers......Page __sk_0236.djvu
Bibliography......Page __sk_0240.djvu
1. The Rise of Analysis......Page __sk_0242.djvu
2. The Motivation for the Eighteenth-Century Work......Page __sk_0244.djvu
3. The Problem of Proof......Page __sk_0245.djvu
4. The Metaphysical Basis......Page __sk_0247.djvu
5. The Expansion of Mathematical Activity......Page __sk_0249.djvu
6. A Glance Ahead......Page __sk_0251.djvu
Bibliography......Page __sk_0253.djvu
2. The Beginnings of Complex Function Theory......Page __sk_0254.djvu
3. The Geometrical Representation of Complex Numbers......Page __sk_0256.djvu
4. The Foundation of Complex Function Theory......Page __sk_0260.djvu
5. Weierstrass's Approach to Function Theory......Page __sk_0270.djvu
6. Elliptic Functions......Page __sk_0272.djvu
7. Hyperelliptic Integrals and Abel's Theorem......Page __sk_0279.djvu
8. Riemann and Multiple-Valued Functions......Page __sk_0283.djvu
9. Abelian Integrals and Functions......Page __sk_0291.djvu
10. Conformal Mapping......Page __sk_0294.djvu
11. The Representation of Functions and Exceptional Values......Page __sk_0295.djvu
Bibliography......Page __sk_0297.djvu
2. The Heat Equation and Fourier Series......Page __sk_0299.djvu
3. Closed Solutions; the Fourier Integral......Page __sk_0307.djvu
4. The Potential Equation and Green's Theorem......Page __sk_0309.djvu
5. Curvilinear Coordinates......Page __sk_0315.djvu
6. The Wave Equation and the Reduced Wave Equation......Page __sk_0318.djvu
7. Systems of Partial Differential Equations......Page __sk_0324.djvu
8. Existence Theorems......Page __sk_0327.djvu
Bibliography......Page __sk_0335.djvu
2. Series Solutions and Special Functions......Page __sk_0337.djvu
3. Sturm-Liouville Theory......Page __sk_0343.djvu
4. Existence Theorems......Page __sk_0345.djvu
5. The Theory of Singularities......Page __sk_0349.djvu
6. Automorphic Functions......Page __sk_0354.djvu
7. Hill's Work on Periodic Solutions of Linear Equations......Page __sk_0358.djvu
8. Nonlinear Differential Equations: The Qualitative Theory......Page __sk_0360.djvu
Bibliography......Page __sk_0366.djvu
2. Mathematical Physics and the Calculus of Variations......Page __sk_0367.djvu
3. Mathematical Extensions of the Calculus of Variations Proper......Page __sk_0373.djvu
4. Related Problems in the Calculus of Variations......Page __sk_0377.djvu
Bibliography......Page __sk_0378.djvu
2. Binomial Equations......Page __sk_0380.djvu
3. Abel's Work on the Solution of Equations by Radicals......Page __sk_0382.djvu
4. Galois's Theory of Solvability......Page __sk_0383.djvu
5. The Geometric Construction Problems......Page __sk_0391.djvu
6. The Theory of Substitution Groups......Page __sk_0392.djvu
Bibliography......Page __sk_0398.djvu
1. The Foundation of Algebra on Permanence of Form......Page __sk_0400.djvu
2. The Search for a Three-Dimensional “Complex Number”......Page __sk_0404.djvu
3. The Nature of Quaternions......Page __sk_0407.djvu
4. Grassmann's Calculus of Extension......Page __sk_0410.djvu
5. From Quaternions to Vectors......Page __sk_0413.djvu
6. Linear Associative Algebras......Page __sk_0419.djvu
Bibliography......Page __sk_0422.djvu
2. Some New Uses of Determinants......Page __sk_0423.djvu
3. Determinants and Quadratic Forms......Page __sk_0427.djvu
4. Matrices......Page __sk_0432.djvu
Bibliography......Page __sk_0440.djvu
Abbreviations......Page __sk_0441.djvu
Name Index......Page __sk_0443.djvu
Subject Index......Page __sk_0453.djvu