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