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What is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. Taming the Unknown considers how these two seemingly different types of algebra evolved and how they relate. Victor Katz and Karen Parshall explore the history of algebra, from its roots in the ancient civilizations of Egypt, Mesopotamia, Greece, China, and India, through its development in the medieval Islamic world and medieval and early modern Europe, to its modern form in the early twentieth century.

Defining algebra originally as a collection of techniques for determining unknowns, the authors trace the development of these techniques from geometric beginnings in ancient Egypt and Mesopotamia and classical Greece. They show how similar problems were tackled in Alexandrian Greece, in China, and in India, then look at how medieval Islamic scholars shifted to an algorithmic stage, which was further developed by medieval and early modern European mathematicians. With the introduction of a flexible and operative symbolism in the sixteenth and seventeenth centuries, algebra entered into a dynamic period characterized by the analytic geometry that could evaluate curves represented by equations in two variables, thereby solving problems in the physics of motion. This new symbolism freed mathematicians to study equations of degrees higher than two and three, ultimately leading to the present abstract era.

Taming the Unknown follows algebra's remarkable growth through different epochs around the globe.

Victor J. Katz is professor of mathematics emeritus at the University of the District of Columbia. Karen Hunger Parshall is professor of history and mathematics at the University of Virginia.



TABLE OF CONTENTS:

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
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 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