UNKNOWN QUANTITY: A REAL AND IMAGINARY HISTORY OF ALGEBRA by John Derbyshire Joseph Henry Press, 2006, 374 pp. ISBN 0-309-09657-X
Unknown Quantity is the story of algebra from its beginnings in mathematical cuneiform texts to modern category theory. Along the way, the reader learns about the mathematics, the cultural and historical settings, and the personalities that made modern abstract algebra. Interspersed with the historical narrative are several "Math Primers" that outline the relevant mathematics.
Derbyshire divides his story into three parts, marking what he considers to be the major developments in algebra. The first part takes us from "the earliest times to the adoption of a systematic literal symbolism letters representing numbers - around the year 1600." (p. 3) Here, we read about Otto Neugebauer (the great translator of mathematical cuneiform texts), Ahmes (the Egyptian scribe who compiled the Rhind Papyrus), Hypatia (the first woman mathematician whose name we know), al-Khwarizmi (whose work and name gave us the terms "algebra" and "algorithm"), Omar Khayyam, Leonardo of Pisa (Fibonacci), Cardano (whose Ars Magna revealed the solutions to the cubic and quartic equations), Tartaglia, Viète, and Descartes. The "Math Primers" here tell us about numbers and polynomials, and the general solution to cubic and quartic polynomials.
Part 2 tells the story of "the first mathematical victories of (algebraic) symbolism and the slow detachment of symbols from the concepts [of] traditional arithmetic and geometry, leading to the discovery of new mathematical objects." (p. 3) The "Math Primers" get a little more advanced as they explain roots of unity, vector spaces, and the idea of an abstract algebra. The major characters in this part of Unknown Quantity are Newton, Vandermonde, LaGrange, Ruffini, Abel, William Rowan Hamilton, Grassman, Leibniz, Takakazu Seki (a Japanese contemporary of Leibniz), Cauchy, Boole, Cay ley and JJ. Sylvester. We also encounter the technique of solving systems of equations given in the Chinese mathematical text The Nine Chapters of the Mathematical Art. We call the method Gaussian elimination after Gauss who discovered it 2,000 years after the authors of the Nine Chapters. …