Mathematics of the 19th Century

The general principles by which the editors and authors of the present edition have been guided were explained in the preface to the first volume of Mathemat­ ics of the 19th Century, which contains chapters on the history of mathematical logic, algebra, number theory, and probability theory (Nauka, Moscow 1978; En­ glish translation by Birkhiiuser Verlag, Basel-Boston-Berlin 1992). Circumstances beyond the control of the editors necessitated certain changes in the sequence of historical exposition of individual disciplines. The second volume contains two chapters: history of geometry and history of analytic function theory (including elliptic and Abelian functions); the size of the two chapters naturally entailed di­ viding them into sections. The history of differential and integral calculus, as well as computational mathematics, which we had planned to include in the second volume, will form part of the third volume. We remind our readers that the appendix of each volume contains a list of the most important literature and an index of names. The names of journals are given in abbreviated form and the volume and year of publication are indicated; if the actual year of publication differs from the nominal year, the latter is given in parentheses. The book History of Mathematics from Ancient Times to the Early Nineteenth Century [in Russian], which was published in the years 1970-1972, is cited in abbreviated form as HM (with volume and page number indicated). The first volume of the present series is cited as Bk. 1 (with page numbers).

1. Geometry.- 1. Analytic and Differential Geometry.- Analytic Geometry.- The Differential Geometry of Monge’s Students.- Gauss’ Disquisitiones generales circa superficies curvas.- Minding and the Formulation of the Problems of Intrinsic Geometry.- The French School of Differential Geometry.- Differential Geometry at Midcentury.- Differential Geometry in Russia.- The Theory of Linear Congruences.- 2. Projective Geometry.- The Rise of Projective Geometry.- Poncelet’s Traité des propriétés projectives des figures.- The Analytic Projective Geometry of Möbius and Plücker.- The Synthetic Projective Geometry of Steiner and Chasles.- Staudt and the Foundation of Projective Geometry.- Cayley’s Projective Geometry.- 3. Algebraic Geometry and Geometric Algebra.- Algebraic Curves.- Algebraic Surfaces.- Geometric Computations Connected with Algebraic Geometry.- Grassmann’s Lineale Ausdehnungslehre.- Hamilton’s Vectors.- 4. Non-Euclidean Geometry.- Nikola? Ivanovich Lobachevski? and the Discovery of Non-Euclidean Geometry.- Gauss’ Research in Non-Euclidean Geometry.- János Bólyai.- Hyperbolic Geometry.- J. Bóyai’s “Absolute Geometry”.- The Consistency of Hyperbolic Geometry.- Propagation of the Ideas of Hyperbolic Geometry.- Beltrami’s Interpretation.- Cayley’s Interpretation.- Klein’s Interpretation.- Elliptic Geometry.- 5. Multi-Dimensional Geometry.- Jacobi’s Formulas for Multi-dimensional Geometry.- Cayley’s Analytic Geometry of n Dimensions.- Grassmann’s Multi-dimensional Geometry.- Plücker’s Neue. Geometrie des Raumes.- Schläfli’s Theorie der vielfachen Kontinuilät.- The Multi-dimensional Geometry of Klein and Jordan.- Riemannian Geometry.- Riemann’s Idea of Complex Parameters of Euclidean Motions.- Riemann’s Ideas on Physical Space.- The Work of Christoffel, Lipschitz. and Suvorov on Riemannian Geometry.- The Multi-dimensional Theory of Curves.- Multi-dimensional Surface Theory.- Multi-dimensional Projective Geometry.- The Terminology of Multi-dimensional Geometry.- 6. Topology.- Gauss’ Topology.- Generalizations of Euler’s Theorem on Polyhedra in the Early Nineteenth Century.- Listing’s Vorstudien zur Topologie.- Möbius’ “Theorie der elementaren Verwandschaft”.- The Topology of Surfaces in Riemann’s “Theorie der Abel’schen Funktionen”.- The Multi-dimensional Topology of Riemann and Betti.- Jordan’s Topological Theorems.- The “Klein Bottle”.- 7. Geometric Transformations.- Geometrie Transformations in the Work of Möbius.- Helmholtz’ Paper “Über die Thatsachen, die der Geometrie zu Grunde liegen”.- Klein’s “Erlanger Programm”.- Transference Principles.- Cremona Transformations.- Conclusion.- 2. Analytic Function.- Results Achieved in Analytic Function Theory in the Eighteenth Century.- Development of the Concept of a Complex Number.- Complex Integration.- The Cauchy Integral Theorem. Residues.- Elliptic Functions in the Work of Gauss.- Hypergeometric Functions.- The First Approach to Modular Functions.- Power Series. The Method of Majorants.- Elliptic Functions in the Work of Abel.- C.G.J. Jacobi. Fundamenta nova functionum ellipticarum.- The Jacobi Theta Functions.- Elliptic Functions in the Work of Eisenstein and Liouville. The First Textbooks.- Abelian Integrals. Abel’s Theorem.- Quadruply Periodic Functions.- Summary of the Development of Analytic Function Theory over the First Half of the Nineteenth Century.- V. Puiseux. Algebraic Functions.- Bernhard Riemann.- Riemann’s Doctoral Dissertation. The Dirichlet Principle.- Conformal Mappings.- Karl Weierstrass.- Analytic Function Theory in Russia. Yu.V. Sokhotski? and the Sokhotski?-Casorati-Weierstrass Theorem.- Entire and Meromorphic Functions. Picard’s Theorem.- Abelian Functions.- Abelian Functions (Continuation).- Automorphic Functions. Uniformization.- Sequences and Series of Analytic Functions.- Conclusion.- Literature.- (F. A. Medvedev).- General Works.- Collected Works and Other Original Sources.- Auxiliary Literature to Chapter 1.- Auxiliary Literature to Chapter 2.- Index of Names (A. F. Lapko).