| The Theory of Numbers in the Nineteenth Century | p. 813 |
| Introduction | p. 813 |
| The Theory of Congruences | p. 813 |
| Algebraic Numbers | p. 818 |
| The Ideals of Dedekind | p. 822 |
| The Theory of Forms | p. 826 |
| Analytic Number Theory | p. 829 |
| The Revival of Projective Geometry | p. 834 |
| The Renewal of Interest in Geometry | p. 834 |
| Synthetic Euclidean Geometry | p. 837 |
| The Revival of Synthetic Projective Geometry | p. 840 |
| Algebraic Projective Geometry | p. 852 |
| Higher Plane Curves and Surfaces | p. 855 |
| Non-Euclidean Geometry | p. 861 |
| Introduction | p. 861 |
| The Status of Euclidean Geometry About 1800 | p. 861 |
| The Research on the Parallel Axiom | p. 863 |
| Foreshadowings of Non-Euclidean Geometry | p. 867 |
| The Creation of Non-Euclidean Geometry | p. 869 |
| The Technical Content of Non-Euclidian Geometry | p. 874 |
| The Claims of Lobatchevsky and Bolyai to Priority | p. 877 |
| The Implications of Non-Euclidean Geometry | p. 879 |
| The Differential Geometry of Gauss and Riemann | p. 882 |
| Introduction | p. 882 |
| Gauss's Differential Geometry | p. 882 |
| Riemann's Approach to Geometry | p. 889 |
| The Successors of Riemann | p. 896 |
| Invariants of Differential Forms | p. 899 |
| Projective and Metric Geometry | p. 904 |
| Introduction | p. 904 |
| Surfaces as Models of Non-Euclidean Geometry | p. 904 |
| Projective and Metric Geometry | p. 906 |
| Models and the Consistency Problem | p. 913 |
| Geometry from the Transformation Viewpoint | p. 917 |
| The Reality of Non-Euclidean Geometry | p. 921 |
| Algebraic Geometry | p. 924 |
| Background | p. 924 |
| The Theory of Algebraic Invariants | p. 925 |
| The Concept of Birational Transformations | p. 932 |
| The Function-Theoretic Approach to Algebraic Geometry | p. 934 |
| The Uniformization Problem | p. 937 |
| The Algebraic-Geometric Approach | p. 939 |
| The Arithmetic Approach | p. 942 |
| The Algebraic Geometry of Surfaces | p. 943 |
| The Instillation of Rigor in Analysis | p. 947 |
| Introduction | p. 947 |
| Functions and Their Properties | p. 949 |
| The Derivative | p. 954 |
| The Integral | p. 956 |
| Infinite Series | p. 961 |
| Fourier Series | p. 966 |
| The Status of Analysis | p. 972 |
| The Foundations of the Real and Transfinite Numbers | p. 979 |
| Introduction | p. 979 |
| Algebraic and Transcendental Numbers | p. 980 |
| The Theory of Irrational Numbers | p. 982 |
| The Theory of Rational Numbers | p. 987 |
| Other Approaches to the Real Number System | p. 990 |
| The Concept of an Infinite Set | p. 992 |
| The Foundation of the Theory of Sets | p. 994 |
| Transfinite Cardinals and Ordinals | p. 998 |
| The Status of Set Theory by 1900 | p. 1002 |
| The Foundations of Geometry | p. 1005 |
| The Defects in Euclid | p. 1005 |
| Contributions to the Foundations of Projective Geometry | p. 1007 |
| The Foundations of Euclidean Geometry | p. 1010 |
| Some Related Foundational Work | p. 1015 |
| Some Open Questions | p. 1017 |
| Mathematics as of 1900 | p. 1023 |
| The Chief Features of the Nineteenth-Century Developments | p. 1023 |
| The Axiomatic Movement | p. 1026 |
| Mathematics as Man's Creation | p. 1028 |
| The Loss of Truth | p. 1032 |
| Mathematics as the Study of Arbitrary Structures | p. 1036 |
| The Problem of Consistency | p. 1038 |
| A Glance Ahead | p. 1039 |
| The Theory of Functions of Real Variables | p. 1040 |
| The Origins | p. 1040 |
| The Stieltjes Integral | p. 1041 |
| Early Work on Content and Measure | p. 1041 |
| The Lebesgue Integral | p. 1044 |
| Generalizations | p. 1050 |
| Integral Equations | p. 1052 |
| Introduction | p. 1052 |
| The Beginning of a General Theory | p. 1056 |
| The Work of Hilbert | p. 1060 |
| The Immediate Successors of Hilbert | p. 1070 |
| Extensions of the Theory | p. 1073 |
| Functional Analysis | p. 1076 |
| The Nature of Functional Analysis | p. 1076 |
| The Theory of Functionals | p. 1077 |
| Linear Functional Analysis | p. 1081 |
| The Axiomatization of Hilbert Space | p. 1091 |
| Divergent Series | p. 1096 |
| Introduction | p. 1096 |
| The Informal Uses of Divergent Series | p. 1098 |
| The Formal Theory of Asymptotic Series | p. 1103 |
| Summability | p. 1109 |
| Tensor Analysis and Differential Geometry | p. 1122 |
| The Origins of Tensor Analysis | p. 1122 |
| The Notion of a Tensor | p. 1123 |
| Covariant Differentiation | p. 1127 |
| Parallel Displacement | p. 1130 |
| Generalizations of Riemannian Geometry | p. 1133 |
| The Emergence of Abstract Algebra | p. 1136 |
| The Nineteenth-Century Background | p. 1136 |
| Abstract Group Theory | p. 1137 |
| The Abstract Theory of Fields | p. 1146 |
| Rings | p. 1150 |
| Non-Associative Algebras | p. 1153 |
| The Range of Abstract Algebra | p. 1156 |
| The Beginnings of Topology | p. 1158 |
| The Nature of Topology | p. 1158 |
| Point Set Topology | p. 1159 |
| The Beginnings of Combinational Topology | p. 1163 |
| The Combinational Work of Poincare | p. 1170 |
| Combinatorial Invariants | p. 1176 |
| Fixed Point Theorems | p. 1177 |
| Generalizations and Extensions | p. 1179 |
| The Foundations of Mathematics | p. 1182 |
| Introduction | p. 1182 |
| The Paradoxes of Set Theory | p. 1183 |
| The Axiomatization of Set Theory | p. 1185 |
| The Rise of Mathematical Logic | p. 1187 |
| The Logistic School | p. 1192 |
| The Intuitionist School | p. 1197 |
| The Formalist School | p. 1203 |
| Some Recent Developments | p. 1208 |
| List of Abbreviations | |
| Index | |
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