Cover Page -- Title Page -- Copyright Page -- Preface to the Three-Volume Paperback Edition of Mathematical Thought -- Preface -- Contents -- Publisher's Note -- 34. The Theory of Numbers in the Nineteenth Century -- 1. Introduction -- 2. The Theory of Congruences -- 3. Algebraic Numbers -- 4. The Ideals of Dedekind -- 5. The Theory of Forms -- 6. Analytic Number Theory -- 35. The Revival of Projective Geometry -- 1. The Renewal of Interest in Geometry -- 2. Synthetic Euclidean Geometry -- 3. The Revival of Synthetic Projective Geometry -- 4. Algebraic Projective Geometry -- 5. Higher Plane Curves and Surfaces -- 36. Non-Euclidean Geometry -- 1. Introduction -- 2. The Status of Euclidean Geometry About 1800 -- 3. The Research on the Parallel Axiom -- 4. Foreshadowings of Non-Euclidean Geometry -- 5. The Creation of Non-Euclidean Geometry -- 6. The Technical Content of Non-Euclidian Geometry -- 7. The Claims of Lobatchevsky and Bolyai to Priority -- 8. The Implications of Non-Euclidean Geometry -- 37. The Differential Geometry of Gauss and Riemann -- 1. Introduction -- 2. Gauss's Differential Geometry -- 3. Riemann's Approach to Geometry -- 4. The Successors of Riemann -- 5. Invariants of Differential Forms -- 38. Projective and Metric Geometry -- 1. Introduction -- 2. Surfaces as Models of Non-Euclidean Geometry -- 3. Projective and Metric Geometry -- 4. Models and the Consistency Problem -- 5. Geometry from the Transformation Viewpoint -- 6. The Reality of Non-Euclidean Geometry -- 39. Algebraic Geometry -- 1. Background -- 2. The Theory of Algebraic Invariants -- 3. The Concept of Birational Transformations -- 4. The Function-Theoretic Approach to Algebraic Geometry -- 5. The Uniformization Problem -- 6. The Algebraic-Geometric Approach -- 7. The Arithmetic Approach -- 8. Thè Algebraic Geometry of Surfaces.

40. The Instillation of Rigor in Analysis -- 1. Introduction -- 2. Functions and Their Properties -- 3. The Derivative -- 4. The Integral -- 5. Infinite Series -- 6. Fourier Series -- 7. The Status of Analysis -- 41. The Foundations of the Real and Transfinite Numbers -- 1. Introduction -- 2. Algebraic and Transcendental Numbers -- 3. The Theory of Irrational Numbers -- 4. The Theory of Rational Numbers -- 5. Other Approaches to the Real Number System -- 6. The Concept of an Infinite Set -- 7. The Foundation of the Theory of Sets -- 8. Transfinite Cardinals and Ordinals -- 9. The Status of Set Theory by 1900 -- 42. The Foundations of Geometry -- 1. The Defects in Euclid -- 2. Contributions to the Foundations of Projective Geometry -- 3. The Foundations of Euclidean Geometry -- 4. Some Related Foundational Work -- 5. Some Open Questions -- 43. Mathematics as of 1900 -- 1. The Chief Features of the Nineteenth-Century Developments -- 2. The Axiomatic Movement -- 3. Mathematics as Man's Creation -- 4. The Loss of Truth -- 5. Mathematics as the Study of Arbitrary Structures -- 6. The Problem of Consistency -- 7. A Glance Ahead -- 44. The Theory of Functions of Real Variables -- 1. TheOrigins -- 2. The Stieltjes Integral -- 3. Early Work on Content and Measure -- 4. The Lebesgue Integral -- 5. Generalizations -- 45. Integral Equations -- 1. Introduction -- 2. The Beginning of a General Theory -- 3. The Work of Hilbert -- 4. The Immediate Successors of Hilbert -- 5. Extensions of the Theory -- 46. Functional Analysis -- 1. The Nature of Functional Analysis -- 2. The Theory of Functionals -- 3. Linear Functional Analysis -- 4. The Axiomatization of Hilbert Space -- 47. Divergent Series -- 1. Introduction -- 2. The Informal Uses of Divergent Series -- 3. The Formal Theory of Asymptotic Series -- 4. Summability -- 48. Tensor Analysis and Differential Geometry.

1. The Origins of Tensor Analysis -- 2. The Notion of a Tensor -- 3. Covariant Differentiation -- 4. Parallel Displacement -- 5. Generalizations of Riemannian Geometry -- 49. The Emergence of Abstract Algebra -- 1. The Nineteenth-Century Background -- 2. Abstract Group Theory -- 3. The Abstract Theory of Fields -- 4. Rings -- 5. Non-Associative Algebras -- 6. The Range of Abstract Algebra -- 50. The Beginnings of Topology -- 1. The Nature of Topology -- 2. Point Set Topology -- 3. The Beginnings of Combinational Topology -- 4. The Combinational Work of Poincaré -- 5. Combinatorial Invariants -- 6. Fixed Point Theorems -- 7. Generalizations and Extensions -- 51. The Foundations of Mathematics -- 1. Introduction -- 2. The Paradoxes of Set Theory -- 3. The Axiomatization of Set Theory -- 4. The Rise of Mathematical Logic -- 5. The Logistic School -- 6. The Intuitionist School -- 7. The Formalist School -- 8. Some Recent Developments -- List of Abbreviations -- Index.