Taming the unknown : history of algebra from antiquity to the early twentieth century / Victor J. Katz and Karen Hunger Parshall.

Format:

Book

Author/Creator:

Katz, Victor J., author.

Parshall, Karen Hunger, 1955- author.

Language:

English

Subjects (All):

Algebra--History.

Algebra.

History.

Physical Description:

xiii, 485 pages ; 25 cm

Place of Publication:

Princeton, New Jersey : Princeton University Press, [2014]

Contents:

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

Proportion 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

Indeterminte 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

René 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 Unknown: 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.

Notes:

Includes bibliographical references and index.

ISBN:

9780691149059

0691149054

OCLC:

852745720