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