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 to my wife Helen for her critical editing of the manuscript,extensive checking of names,dates,andsources,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 34.The Theory of Numbers in the Nineteenth Century,813 1.Introduction,813 Number Theory,829 35.The Revival of Projective Geometry,834 1.The Renewal of Interest in Geometry,834 2.Synthetic Euclidean Geometry,837 3.The Revival of Synthetic Projective Geometry,840 4.Algebraic Projective Geometry,852 5.Higher Plane Curves and Surfaces,855 36.Non-Euclidean Geometry,861 1.Introduction,861 2.The Status of Euclidean Geometry About 1800,861 3.The Research on the Parallel Axiom,863 4.Foreshadowings of Non-Euclidean Geometry,867 5.The Creation of Non-Euclidean Geometry,869 6.The Technical Content of Non-Euclidian Geometry,874 7.The Claims of Lobatchevsky and Bolyai to Priority,877 8.The Implications of Non-Euclidean Geometry,879 37.The Differential Geometry of Gauss and Riemann,882 1.Introduction.882 2.Gauss's Differential Geometry.882 3.Riemann's Approach to Geometry,889 4.The Successors of Riemann,896 5.Invariants of Differential Forms.899 38.Projective and Metric Geometry,904 of Non-Euclidean Ge o D ive and M ometry,904 etry,906 4. odels a roblem 913 No-LudldcCcomuy. 39.Algebraic Geometry,924 1.Background,924 2.The Theory of Algebraic Invanants,925 3.The Conceptof Birational Transformations,932 4.Ihe Fu ction-Theoretic Approach to Algebrai Geometry,934 5.The Uniformization Problem,937 6.The Algebraic-Geometric Approach,939 7.The Arithmeuc Approach,942 8.The Algebraic Geometry of Surfaces,943
xii CONTENTS 40.The Instillation of Rigor in Analysis,947 4 The Inte of Analysis 972 41.The Foundations of the Real and Transfinite Numbers,979 al the RealN ber Sys nce n hnite S 99g on he Th 998 9.The Transfinite Cardinals and Ordinals. Status of Set Theory by 1900.1002 42.The Foundations of Geometry,1005 1.The defects in Euclid.1005 2.Contributions to the foundations of proiective Geometry.1007 3.The Fo ndations of Euclidean Ge onal Work,1015 5.Some Open Que 43.Mathematics as of 1900,1023 5 Mathe m3ti sas the Study of Arbitrary Structures,1036 6.The Problem of Consistency,1038 7.A Glance Ahead,1039 44.The Theory of Functions of Real Variables,1040 Mea ns.1050 45.Integral Equations,1052 ction.1052 2.The Beginning ofa Gene eral Th 1056 3.The Work of Theory,1073 46.Functional Analysis,1076 1.The Nature of Functional Analysis,1076 2 The Theory of Functionals,1077 3.Linear Functional Analysis,1081 4.The Axiomatization of Hilbert Space,1091 47.Divergent Series,1096 1.Introduction,1096 2.The Informal Uses of Divergent Series,1098 3.The Formal Theory of Asymptotic Series,1103 4.Summability.1109 48.Tensor Analysis and Differential Geometry,1122 I The Origins of Tensor Analysis,1122 2.The Notion of a Tensor,1123 3 Covariant Differentiation,1127 4.Parallel Displacement,1130 5.Generalizations of Riemannian Geometry,1133
CONTENTS xiii 49.The Emergence of Abstract Algebra,1136 1.The Nineteenth-Century Background,1136 2.Abstract Group Theory,1137 3. The Abstract Theory of Fields,1146 4.Rings,1150 5.Non-Associative Algebras, 1153 6.The Range of Abstract Algebra,1156 50.The Beginnings of Topology,1158 1.The Nature of Topology,1158 2.Point Set Topology,1159 3.The Beginnings of Combinational Topology,1163 4.The Combinational Work of Poincare,1170 5.Combinatorial Invariants,1176 6 Fixed Point Theorems,1177 7.Generalizations and Extensions,1179 51.The Foundations of Mathematics,1182 setThe Re tMche Th ch istic School, 1192 6.The Intuitionist School,1197 malist School,1203 8.Some Recent Developments,1208 List of Abbrevations 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