Taming the unknown : history of algebra from antiquity to the early twentieth century

 Content: Acknowledgments xi  1 Prelude: What Is Algebra? 1  Why This Book? 3  Setting and Examining the Historical Parameters 4  The Task at Hand 10  2 Egypt and Mesopotamia 12  Proportions in Egypt 12  Geometrical Algebra in Mesopotamia 17  3 The Ancient Greek World 33  Geometrical Algebra in Euclid's Elements and Data 34  Geometrical Algebra in Apollonius's Conics 48  Archimedes and the Solution of a Cubic Equation 53  4 Later Alexandrian Developments 58  Diophantine Preliminaries 60  A Sampling from the Arithmetica: The First Three Greek Books 63  A Sampling from the Arithmetica: The Arabic Books 68  A Sampling from the Arithmetica: The Remaining Greek Books 73  The Reception and Transmission of the Arithmetica 77  5 Algebraic Thought in Ancient and Medieval China 81  Proportions and Linear Equations 82  Polynomial Equations 90  Indeterminate Analysis 98  The Chinese Remainder Problem 100  6 Algebraic Thought in Medieval India 105  Proportions and Linear Equations 107  Quadratic Equations 109  Indeterminate Equations 118  Linear Congruences and the Pulverizer 119  The Pell Equation 122  Sums of Series 126  7 Algebraic Thought in Medieval Islam 132  Quadratic Equations 137  Indeterminate Equations 153  The Algebra of Polynomials 158  The Solution of Cubic Equations 165  8 Transmission, Transplantation, and Diffusion in the Latin West 174  The Transplantation of Algebraic Thought in the Thirteenth Century 178  The Diffusion of Algebraic Thought on the Italian Peninsula and Its Environs from the Thirteenth Through the Fifteenth Centuries 190  The Diffusion of Algebraic Thought and the Development of Algebraic Notation outside of Italy 204  9 The Growth of Algebraic Thought in Sixteenth-Century Europe 214  Solutions of General Cubics and Quartics 215  Toward Algebra as a General Problem-Solving Technique 227  10 From Analytic Geometry to the Fundamental Theorem of Algebra 247  Thomas Harriot and the Structure of Equations 248  Pierre de Fermat and the Introduction to Plane and Solid Loci 253  Albert Girard and the Fundamental Theorem of Algebra 258  Rene Descartes and The Geometry 261  Johann Hudde and Jan de Witt, Two Commentators on The Geometry 271  Isaac Newton and the Arithmetica universalis 275  Colin Maclaurin's Treatise of Algebra 280  Leonhard Euler and the Fundamental Theorem of Algebra 283  11 Finding the Roots of Algebraic Equations 289  The Eighteenth-Century Quest to Solve Higher-Order Equations Algebraically 290  The Theory of Permutations 300  Determining Solvable Equations 303  The Work of Galois and Its Reception 310  The Many Roots of Group Theory 317  The Abstract Notion of a Group 328  12 Understanding Polynomial Equations in n Unknowns 335  Solving Systems of Linear Equations in n Unknowns 336  Linearly Transforming Homogeneous Polynomials in n Unknowns: Three Contexts 345  The Evolution of a Theory of Matrices and Linear Transformations 356  The Evolution of a Theory of Invariants 366  13 Understanding the Properties of ''Numbers'' 381  New Kinds of ''Complex'' Numbers 382  New Arithmetics for New ''Complex'' Numbers 388  What Is Algebra?: The British Debate 399  An ''Algebra'' of Vectors 408  A Theory of Algebras, Plural 415  14 The Emergence of Modern Algebra 427  Realizing New Algebraic Structures Axiomatically 430  The Structural Approach to Algebra 438  References 449  Index 477