Unfortunately but inevitably John Derbyshire's "Unknown Quantity" is nowhere near as good as his previous pop math book "Prime Obsession", but in saying this I am praising the latter and not disparaging the former. "Unknown Quantity" still rates the full five stars.
"Prime Obsession" is laser focused on one problem (Riemann's hypothesis on an important characteristic of the zeta function) and on one man (Bernhard Riemann). Derbyshire then expands on the mathematical topic twice, first by explaining the context in which the hypothesis mattered (its relationship to prime numbers) and finally by taking a general look at the branch of mathematics called analysis. He does the same thing with the human side of his story: he presents Riemann's friends and colleagues and then he presents earlier and later analysts. This makes for a near perfect book, both topically and dramatically.
"Unknown Quantity" on the other hand gives us a more general history of its subject, which is algebra. Derbyshire takes us from its beginnings in ancient Babylon and describes in detail how to decipher a sample problem found on cuneiform tablet. He then goes to Alexandria where Diophantus used the first notation using something like "x" to represent unknown quantities. It didn't catch on, and Arab mathematicians developed the field during the middle ages using word problems again.
Early in the Renaissance Italians then took over the search for roots and found methods to solve roots of third and fourth degree polynomials. Descartes then invented analytic geometry and standardized the usage of x-y-z as our notation for unknown quantities.
Newton contributed to the field, but his invention of calculus (yes, yes along with Leibniz) diverted attention away from algebra for two centuries. It was in the nineteenth century that algebra as we know it today took off. Mathematicians were getting comfortable with the square root of minus one and the Complex field of numbers extended from the Real field. An Irishman named Hamilton worked hard at extending fields again from Complex numbers to the theory of Quaternions. Quaternions never caught on but as a by-product, Hamilton laid down ideas that directly lead to the development of vector spaces and linear algebra.
Along with this, later mathematicians developed group theory, ring theory, and field theory. A group is simply a set and an operation on that set that meets certain requirements. Rings and fields are groups with more requirements. These theories form algebra as it is studied today. In a nutshell, Derbyshire presents an excellent history of the origin and development of much of algebra and of most of the important figures that have contributed to the field. But of course, to be general he's had to sacrifice focus.
Perhaps instead of a single chapter, Derbyshire could have written his whole book around the story of Evariste Galois. Galois, a politically radical young Frenchman, died fatally wounded in a duel at twenty years of age. Despite his youth, he solved a problem that ultimately changed algebra from simply searching for roots of polynomials to the more abstract pursuit of studying groups and permutations. There's plenty of drama in >>that<< story! But it has also been done to death. Derbyshire made the right choice by giving us a less exciting but more informative book.
Vincent Poirier, Tokyo
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Fascinating History of Algebra
Fascinating History of Algebra"Unknown Quantity: A Real and Imaginary History of Algebra" by John DerbyshireReaders who enjoyed "Prime Obsession" will find "Unknown Quantity" irresistible. In this very readable text John Derbyshire covers the broad history of modern algebra. The history starts four thousand years ago in Egypt and Mesopotamia.The author tells the lives of the men and women who created modern algebra. Their stories are fascinating.The people who make up the history of algebra include (from the photographic plates after page 184):01 - Otto Neugebauer - found algebra in old Babylonian tablets02 - Hypatia03 - Omar Khayyam - wrote poetry and tackled the cubic equation04 - Girolamo Cardano - found a general solution for the cubic05 - Francois Viete - separated things sought from things given06 - Rene Descartes - algebrized geometry07 - Sir Isaac Newton - saw symmetry in solutions08 - Gottfried von Leibniz - found relief for his imagination09 - Joseph-Louis Lagrange - carried symmetry forward10 - Paulo Ruffini - believed the quintic was unsolvable11 - Augustin-Louis Cauchy - made an "arithmetic" of permutations12 - Niels Abel - proved Ruffini right13 - Evariste Galois - found permutation groups in equations14 - Arthur Cayley - abstracted the group idea15 - Ludwig Sylow - delved into the structure of finite groups16 - Camille Jordan - wrote the first book on groups17 - Sir William R. Hamilton - found a new algebra18 - Herman Grassman - explored vector spaces19 - Bernard Riemann - launched two geometric revolutions20 - Edwin A. Abbot - took us to Flatland21 - Julius Plucker - based his geometry on lines not points22 - Sophus Lie - mastered continuous groups23 - Felix Klein - mastered the group-ification of geometry24 - Henri Poincare - algebraized topology25 - Eduard Kummer - used algebra on Fermat's Last Theorem26 - Richard Dedikind - discovered ideals27 - David Hilbert - a geometry of tables, chairs and beer mugs28 - Emmy Noether - pulled it all together29 - Solomon Lefschetz - harpooned a whale30 - Oscar Zariski - refounded algebraic geometry31 - Saunders Mac Lane - attained a higher level of abstraction32 - Alexander Grothendieck: - as if summoned from the voidJust as before, the author takes a field of mathematics interesting for expert and layman alike. This is a very fresh perspective on the history of algebra.See Also:Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in MathematicsI thoroughly enjoyed and highly recommend "Unknown Quantity."
Top reviews from other countries
Charles stewart
Good balance of history and detail for the amateur
Reviewed in the United Kingdom on March 30, 2015Dennis Mobberley
Very Worth Your While
Reviewed in the United Kingdom on February 10, 2018
Growltiger
An unusual distribution
Reviewed in the United Kingdom on March 12, 2010
