Mathematical Thought from Ancient to Modern Times, Vol 2

Description: Mathematical Thought from Ancient to Modern Times Volume 2 by Morris Kline Oxford University Press, 1990 edition, 0195061365, Trade Paperback, VG condition, stain on one page, no creases, 421 pages. Preface to the Three-Volume Paperback Edition of Mathematical Thought The reception accorded the original edition of this book is most gratifying. I am flattered, if not a penny richer for it, by a pirated Chinese translation. Even more satisfying is a forthcoming authorized Spanish translation. This work is part of my long-time efforts to humanize the subject of mathematics. At the very beginning of my career I banded with a few colleagues to produce a freshman text that departed from the traditional dry-as-dust mathematics textbook. Later, I wrote a calculus text with the same end in view. While I was directing a research group in electromagnetic theory and doing research myself, I still made time to write Mathematics In Western Culture, which is partly history and partly an exploration of the influence of mathematics upon philosophy, religion, literature, art, music, economic theory, and political thought. More recently I have written with the general reader in mind a book on the philosophical foundations of mathematics and a book on the underlying mathematical structure of a good deal of science, most especially cosmogony and physics. I hope that students, teachers, as well as the general reader will profit from this more affordable and accessible three-volume paperback edition of Mathematical Thought. I wish to acknowledge the helpful suggestions made by Harold Edwards, Donald Gillis, and Robert Schlapp among others. My very special thanks go to Fred Pohle for his time, interest, and generosity. Having over the years taught a course based on this book, he saw a need for a multi-volume paperback version and provided the impetus for this edition. Beyond this he gave unstintingly of his time and knowledge in helping me correct errors. I am truly in his debt, as I am to my wife Helen, who undertook much of the work involved in preparing this edition. CONTENTS 18. Mathematics as of 1700, 391 1. The Transformation of Mathematics, 391 2. Mathematics and Science, 394 3. Communication Among Mathematicians, 396 4. The Prospects for the Eighteenth Century, 398 19. Calculus.in the Eighteenth Century, 400 1. Introduction, 400 2. The Function Concept, 403 3. The Technique of Integration and Complex Quantities, 406 4. Elliptic Integrals, 411 5. Further Special Functions, 422 6. The Calculus of Functions of Several Variables, 425 7. The Attempts to Supply Rigor in the Calculus, 426 20. Inhnite Series, 436 1. Introduction, 436 2. Initial Work on Infinite Series, 436 3. The Expansion of Functions, 440 4. The Manipulation of Series, 422 5. Trigonometric Series, 454 6. Continued Fractions, 459 7. The Problem of Convergence and Divergence, 460 21. Ordinary Differential Equations in the Eighteenth Century, 468 1. Motivations, 468 2. First Order Ordinary Differential Equations, 471 3. Singular Solutions, 476 4. Second Order Equations and the Riccati Equations, 478 5. Higher Order Equations, 484 6. The Method of Series, 488 7. Systems of Differential Equations, 490 8. Summary, 499 22. Partial Differential Equations in the Eighteenth Century, 502 I. Introduction, 502 2. The Wave Equation, 503 3. Extensions of the Wave Equation, 515 4. Potential Theory, 522 5. First Order Partial Differential Equations, 531 6. Monge and the Theory of Characteristics, 536 7. Monge and the Nonlinear Second Order Equations, 538 8. Systems of First Order Partial Differential Equations, 540 9. The Rise of the Mathematical Subject, 542 23. Analytic and Differential Geometry in the Eighteenth Century, 544 1. Introduction, 544 2. Basic Analytic Geometry, 544 3. Higher Plane Curves, 547 4. TheBeginningsof Differential Geometry, 544 5. Plane Gurves, 555 6. Space Curves, 557 7. The Theory of Surfaces, 562 8. The Mapping Problem, 570 24. The Calculus of Variations in the Eighteenth Century, 573 1. The Initial Problems, 573 2. The Early Work of Euler, 577 3. The Principle of Least Action, 579 4. The Methodology of Lagrange, 582 5. Lagrange and Least Action, 587 6. The Second Variation, 589 25. Algebra in the Eighteenth Century, 592 1. Status of the Number System, 592 2. The Theory of Equations, 597 3. Determinants and Elimination Theory, 606 4. The Theory of Numbers, 608 26. Mathematics as of 1800, 614 1. The Rise of Analysis, 614 2. The Motivation for the Eighteenth-Century Work, 616 3. The Problem of Proof, 617 4. The Metaphysical Basis, 619 5. The Expansion of Mathematical Activity, 621 6. A Glance Ahead, 623 27. Functions of a Complex Variable, 626 1. Introduction, 626 2. The Beginnings of Complex Function Theory, 626 3. The Geometrical Representation of Complex Numbers, 628 4. The Foundation of Complex Function Theory, 632 5. Weierstrass's Approach to Function Theory, 642 6. Elliptic Functions, 644 7. Hyperelliptic Integrals and Abel's Theorem, 651 8. Riemann and Multiple-Valued Functions, 655 9. Abelian Integrals and Functions, 663 10. Conformal Mapping, 666 11. The Representation of Functions and Excepional Values, 667 28. Partial Differential Equations in the Nineteenth Century, 671 1. Introduction, 671 2. The Heat Equation and Fourier Series, 671 3. Closed Solutions; the Fourier Integral, 679 4. The Potential Equation and Green's Theorem, 681 5. Curvilinear Coordinates, 687 6. The Wave Equation and the Reduced Wave Equation, 690 7. Systems of Partial Differential Equations, 696 8. Existence Theorems, 699 29. Ordinary Differential Equations in the Nineteenth Century, 709 1. Introduction, 709 2. Series Solutions and Special Functions, 709 3. Sturm-Liouville'Fheory, 715 4. Existence Theorems, 717 5. The Theory of Singularities, 721 6. Automorphic Functions, 726 7. Hill's Work on Periodic Solutions of Linear Equations, 730 8. Nonlinear Differential Equations: The Qualitative Theory, 732 30. The Calculus of Variations in the Nineteenth Century, 739 1. Introduction, 739 2. Mathematical Physics and the Calculus of Variations, 739 3. Mathematical Extensions of the Calculus of Variations Proper, 745 4. Related Problems in the Calculus of Variations, 749 31. Galois Theory, 752 1. Introduction, 752 2. Binomial Equations, 752 3. Abel's Work on the Solution of Equations by Radicals, 754 4. Galois's Theory of Solvability, 755 5. The Geometric Construction Problems, 763 6. The Theory of Substitution Groups, 764 32. Quaternions, Vectors, and Linear Associative Algebras, 722 1. The Foundation of Algebra on Permanence of Form, 772 2. The Search for a Three-Dimensional "Complex Number," 776 3. The Nature of Quaternions, 779 4. Grassman's Calculus of Extension, 782 5. From Quaternions to Vectors, 785 6. Linear Associate Algebras, 791 33. Determinants and Matrices, 795 1. Introduction, 795 2. Some New Uses of Determinants, 795 3. Determinants and Quadratic Forms, 799 4. Matrices, 804 List of Abbreviations Index nthdegree books

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