Unknown Quantity: A Real and Imaginary History of Algebra

Derbyshire's previous book, Prime Obsession, was a tour de force whose goal was to explain the Riemann Hypothesis to non-mathematicians. The book was widely acclaimed as a success, an opinion that I share: I read the book with pleasure and learned some mathematics along the way. I thought Derbyshire succeeded in telling the interesting history behind the mathematics in pleasant and engaging prose, and in explaining (at least some of) the mathematics to non-specialists, a feat not to be sneered at given that the mathematics of the Riemann Hypothesis is certainly non-trivial.

So, having read Prime Obsession, I was curious and motivated to read Derbyshire's newest book, Unknown Quantity, which he describes as "a history of algebra, written for the curious nonmathematician". In this sense, Unknown Quantity has the same goal as Prime Obsession. However, the scope of Unknown Quantity is much larger, because algebra is a vast subject with a long history, dating back to the Babylonians, whose coherence is hard to see, especially for nonmathematicians.

Like Prime Obsession, Unknown Quantity contains large sections that describe the mathematics. In Prime Obsession this was done in the odd-numbered chapters, with the even-numbered chapters focusing on the history. In Unknown Quantity, Derbyshire chose to sprinkle "Math Primers" along the way (there are six primers among 15 chapters). There is a fair amount of mathematics in the main text as well. I have a feeling that a nonmathematician will need a lot of motivation to go through all this material. But such a reader will be rewarded with a reasonable sense of how algebra evolved from concrete (and theoretical!) problems handled by the Babylonians to the solution of polynomial equations, and then will get at least an overview of how algebra become abstract and pervaded all areas of mathematics.

Some high points in the book are: an engaging account the romantic story of the solution of cubic and quartic equations by Tartaglia, Cardano, and Ferrari; the use of complex numbers by Bombelli; the early attempts at a theory of equations via invariants and symmetric functions; the role of Lagrange (or should we say Vandermonde?) resolvents for solving the quintic. The book even contains a convincing "proof" of the Fundamental Theorem of Algebra.

On other topics, Derbyshire has not been as successful. Galois theory, despite the romantic aura around the short life of Galois (which Derbyshire argues is not totally warranted), is not a light topic, despite (or perhaps because of) its great beauty. It is hard to get the point across to nonmathematicians. The same can be said, even more strongly, of Kummer's ideals and Noether's ring theory. In particular, I think Derbyshire has failed to give a good account of Emmy Noether's work on invariants.

The book goes as near as possible to contemporary mathematics, discussing Klein's Erlangen Program, algebraic topology, algebraic geometry, algebraic number theory, category theory, and even Grothendieck's work on the modern foundations of algebraic geometry (Derbyshire concentrates mostly on Grothendieck's "colorful" life rather than on his mathematics, which is just as well, given how hard it is, even for mathematicians).

The prose itself is quite pleasant, as we have come to expect from Derbyshire. The book contains over 170 informative and sometimes entertaining endnotes, a good index, and 32 pictures of the main characters in the history of algebra.

In summary, I think Derbyshire has done at good job at portraying algebra and its journey toward abstraction from its roots in early civilizations. All interested readers will learn something about mathematics and its history. Readers with the right background will then be able to enjoy more mathematical accounts such as The Beginnings and Evolution of Algebra by Bashmakova and Smirnova and van der Waerden's classic A History of Algebra.

Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.