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Mathematics in the 19th Century: Mathematical Logic, Algebra, Number Theory, Probability Theory

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Table of Contents

One Mathematical Logic.- The Prehistory of Mathematical Logic.- Leibniz's Symbolic Logic.- The Quantification of a Predicate.- The "Formal Logic" of A. De Morgan.- Boole's Algebra of Logic.- Jevons' Algebra of Logic.- Venn's Symbolic Logic.- Schroeder's and Poretski-'s Logical Algebra.- Conclusion.- Two Algebra and Algebraic Number Theory.- 1 Survey of the Evolution of Algebra and of the Theory of Algebraic Numbers During the Period of 1800-1870.- 2 The Evolution of Algebra.- Algebraic Proofs of the Fundamental Theorem of Algebra in the 18th Century.- C.F. Gauss' First Proof.- C.F. Gauss' Second Proof.- The Kronecker Construction.- The Theory of Equations.- Carl Friedrich Gauss.- Solution of the Cyclotomic Equation.- Niels Henrik Abel.- Evariste Galois.- The Algebraic Work of Evariste Galois.- The First Steps in the Evolution of Group Theory.- The Evolution of Linear Algebra.- Hypercomplex Numbers.- William Rowan Hamilton.- Matrix Algebra.- The Algebras of Grassmann and Clifford.- Associative Algebras.- The Theory of Invariants.- 3 The Theory of Algebraic Numbers and the Beginnings of Commutative Algebra.- Disquisitiones Arithmeticaeof C.F. Gauss.- Investigation of the Number of Classes of Quadratic Forms.- Gaussian Integers and Their Arithmetic.- Fermat's Last Theorem. The Discovery of E. Kummer.- Kummer's Theory.- Difficulties. The Notion of an Integer.- The Zolotarev Theory. Integral and p-Integral Numbers.- Dedekind's Ideal Theory.- On Dedekind's Method. Ideals and Cuts.- Construction of Ideal Theory in Algebraic Function Fields.- L. Kronecker's Divisor Theory.- Conclusion.- Three Problems of Number Theory.- 1 The Arithmetic Theory of Quadratic Forms.- The General Theory of Forms; Ch. Hermite.- Korkin's and Zolotarev's Works on the Theory of Quadratic Forms.- The Investigations of A.A. Markov.- 2 Geometry of Numbers.- Origin of the Theory.- The Work of H.J.S. Smith.- Geometry of Numbers: Hermann Minkowski.- The Works of G.F. Vorono-.- 3 Analytic Methods in Number Theory.- Lejeune-Dirichlet and the Theorem on Arithmetic Progressions.- Asymptotic Laws of Number Theory.- Chebyshev and the Theory of Distribution of Primes.- The Ideas of Bernhard Riemann.- Proof of the Asymptotic Law of Distribution of Prime Numbers.- Some Applications of Analytic Number Theory.- Arithmetic Functions and Identities. The Works of N.V. Bugaev.- 4 Transcendental Numbers.- The Works of Joseph Liouville.- Charles Hermite and the Proof of the Transcendence of the Number e; The Theorem of Ferdinand Lindemann.- Conclusion.- Four The Theory of Probability.- Laplace's Theory of Probability.- Laplace's Theory of Errors.- Gauss' Contribution to the Theory of Probability.- The contributions of Poisson and Cauchy.- Social and Anthropometric Statistics.- The Russian School of the Theory of Probability. P.L. Chebyshev.- New Fields of Application of the Theory of Probability. The Rise of Mathematical Statistics.- Works of the Second Half of the 19th Century in Western Europe.- Conclusion.- Addendum.- 1. French and German Quotations.- 2. Notes.- 3. Additional Bibliography.- Bibliography (by F.A. Medvedev).- Abbreviations.- Index of Names.


"...The book, indispensable for historians of mathematics, can be warmly recommended to every working mathematician." --EMS Newsletter (on the first edition)

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