xii PREFACE The sources have been indicated in the bibliographies of the various chapters.The interested reader can obtain much more information from these sources than I have extracted.These bibliographies also contain many references which should not and did not serve as sources.However,they have been included either because they offer additional information,because the level of presentation may be helpful to some readers,or because they may be more accessible than the original sources. I wish to express thanks to my colleagues Martin Burrow,Bruce Chand- ler,Martin Davis,Donald Ludwig,Wilhelm Magnus,Carlos Moreno, Harold N.Shapiro,and Marvin Tretkoff,who answered numerous ques tions,read many chapters,and gave valuable criticisms.I am especially indebted tomy wife Helen for her critical editing of the manuscript,xtensive checking of names,dates,and sources,and most careful reading of the galleys and page proofs.Mrs.Eleanore M.Gross,who did the bulk of the typing, was enormously helpful.To the staff of Oxford University Press,I wish to express my gratitude for their scrupulous production of this work. New York M.K. May 1972
Contents 18.Mathematics as of 1700,391 IThe Transtormation of Mathematics.391 2 Mathematics and Scienc.394 19.Calculus in the Eighteenth Century,400 1 Inuoduction.400 2 The Function Concept,403 3 The'Techniqueof Integratior and Complex Quantitics.406 4 Elliptic Integrals,411 Further Spccial Functions 422 6 The (akulus of Functions of Several Variables,425 Supply Rigor in the Calculus,426 20.Infinite Series,436 1 Introduction,436 2 Initial Work on Infinite Series.436 3.The Expansion of Functions,440 4 'I he Manipulation of Series,422 5 Trigonometric Serics,454 6 Continued hractions,459 7 The Problem of Convergence and Divergence,460 21.Ordinary Differential Equations in the Eighteenth Century,468 cond Order Ea tations,484 6 The Method of Series,488 7 Systems of 22.Partial Differential Equations in the Eighteenth Century,502 6 heory.52 Monge and the haxtemo Systems of Fi Mathematical Subject,542 23.Analytic and Differential Geometry in the Eighteenth Century,544 5.555 6
xiv CONTENTS 24.The Calculus of Variations in the Eighteenth Century,573 Least Action,579 5 Lagrange and Least Action,587 6The Second Variation,5 25.Algebra in the Eighteenth Century,592 I 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.Ihe 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 Gcometrical Representation of Complex Numbers,628 4 The Foundation of Complex FunctionTheory,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 Fouricr Integral,679 4.The Potential Equation and Grcen's 'Theorem, 681 5 Curvilinear Coordinatcs,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 ccial Functions,709 3.Sturm- laritie unctions. Equations,730 onlinear Differential Equations The Qualitative Theory,732 30.The Calculus of Variations in the Nineteenth Century,739 of Va Variations Proper.745 31.Galois Theory,752 1.Introduction.752 2.Binomial E ork on the Solution o y Radic als,7544 quations.752.Abel's W onstruction Problems,763 6 The bstitution (;roups,764
CONTENTS XV 32.Quaternions,Vectors,and Linear Associative Algebras,722 I I'he 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 I Introduction,795 2 Some New Uses of Determinants,795 3.Determinants and Ouadratic Forms.799 4 Matrices.804 List of Abbreviations Index
Publisher's Note to this Three-Volume Paperback Edition Mathematical Thought from Ancient to Modern Times was first published by Oxford University Press as a one-volume cloth edition.In publishing this three-volume paperback edition we have retained the same pagination as the cloth in order to maintain consistency within the Index, Subject Index,and Notes.These volumes are paginated consecutively and,for the reader's convenience,both Indexes appear at the end of each volume