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).

Klappentext

This book is the second volume of a study of the history of mathematics in the nineteenth century. The first part of the book describes the development of geometry. The many varieties of geometry are considered and three main themes are traced: the development of a theory of invariants and forms that determine certain geometric structures such as curves or surfaces; the enlargement of conceptions of space which led to non-Euclidean geometry; and the penetration of algebraic methods into geometry in connection with algebraic geometry and the geometry of transformation groups. The second part, on analytic function theory, shows how the work of mathematicians like Cauchy, Riemann and Weierstrass led to new ways of understanding functions. Drawing much of their inspiration from the study of algebraic functions and their integral, these mathematicians and others created a unified, yet comprehensive theory in which the original algebraic problems were subsumed in special areas devoted to elliptic, algebraic, Abelian and automorphic functions. The use of power series expansions made it possible to include completely general transcendental functions in the same theory and opened up the study of the very fertile subject of entire functions. This book will be a valuable source of information for the general reader, as well as historians of science. It provides the reader with a good understanding of the overall picture of these two areas in the nineteenth century and their significance today.

Inhalt

1. Geometry.- 1. Analytic and Differential Geometry.- 2. Projective Geometry.- 3. Algebraic Geometry and Geometric Algebra.- 4. Non-Euclidean Geometry.- 5. Multi-Dimensional Geometry.- 6. Topology.- 7. Geometric 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).