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

Acknowledgments

1. Prelude: What is Algebra?

Why This Book?

Setting and Examining the Historical Parameters

The Task at Hand

2. Egypt and Mesopotamia

Proportions in Egypt

Geometrical Algebra in Mesopotamia

3. The Ancient Greek World

Geometrical Algebra in Euclid's Elements and Data

Geometrical Algebra in Apollonius's Conics

Archimedes and the Solution of a Cubic Equation

4. Later Alexandrian Developments

Diophantine Preliminaries

A Sampling from the Arithmetica: the First Three Greek Books

A Sampling from the Arithmetica: the Arabic Books

A Sampling from the Arithmetica: the Remaining Greek Books

The Reception and Transmission of the Arithmetica

5. Algebraic Thought in Ancient and Medieval China

Proportion and Linear Equations

Polynomial Equations

Indeterminate Analysis

The Chinese Remainder Problem

6. Algebraic Thought in Medieval India

Proportions and Linear Equations

Quadratic Equations

Indeterminte Equations

Linear Congruences and the Pulverizer

The Pell Equation

Sums of Series

7. Algebraic Thought in Medieval Islam

Quadratic Equations

Indeterminate Equations

The Algebra of Polynomials

The Solution of Cubic Equations

8. Transmission, Transplantation, and Diffusion in the Latin West

The Transplantation of Algebraic Thought in the Thirteenth Century

The Diffusion of Algebraic Thought on the Italian Peninsula and Its Environs from the Thirteenth Through the Fifteenth Centuries

The Diffusion of Algebraic Thought and the Development of Algebraic Notation outside of Italy

9. The Growth of Algebraic Thought in Sixteenth-Century Europe

Solutions of General Cubics and Quartics

Toward Algebra as a General Problem-Solving Technique

10. From Analytic Geometry to the Fundamental Theorem of Algebra

Thomas Harriot and the Structure of Equations

Pierre de Fermat and the Introduction to Plane and Solid Loci

Albert Girard and the Fundamental Theorem of Algebra

René Descartes and the Geometry

Johann Hudde and Jan de Witt, Two Commentators on the Geometry

Isaac Newton and the Arithmetica universalis

Colin Maclaurin's Treatise of Algebra

Leonhard Euler and the Fundamental Theorem of Algebra

11. Finding the Roots of Algebraic Equations

The Eighteenth-Century Quest to Solve Higher-Order Equations Algebraically

The Theory of Permutations

Determining Solvable Equations

The Work of Galois and Its Reception

The Many Roots of Group Theory

The Abstract Notion of a Group

12. Understanding Polynomial Equations in n Unknowns

Solving Systems of Linear Equations in n Unknowns

Linearly Transforming Homogeneous Polynomials in n Unknown: Three Contexts

The Evolution of a Theory of Matrices and Linear Transformations

The Evolution of a Theory of Invariants

13. Understanding the Properties of "Numbers"

New Kinds of "Complex" Numbers

New Arithmetics for New "Complex" Numbers

What is Algebra?: the British Debate

An "Algebra" of Vectors

A Theory of Algebras, Plural

14. The Emergence of Modern Algebra

Realizing New Algebraic Structures Axiomatically

The Structural Approach to Algebra

References

Index