Books on the history of algebra, all the way from the Babylonians, are probably not high on most readers' lists of books that must be read. Still, this one could be the exception. Bashmakova and Smirnova catch one's attention right off by explaining in very simple terms the Babylonian sexagesimal system — base 60 with only two digits! Of course, along with the economy of symbols, there come a few weaknesses in the system: there is no zero and, though the system is positional, the representations are not unique. Lest one discount the achievements of the Babylonians for this unsettling non-uniqueness, one should keep in mind that they could solve, for example, quadratic equations, and find Pythagorean triples and Babylonian triples (solutions to x^{2} + y^{2} = 2z^{2}). Further, they knew that the sum of two squares times the sum of two squares is again the sum of two squares, the essential lemma for the proof of the two squares theorem in number theory. And they knew all of this around 4000 years ago!

For all their impressive achievements, the Babylonians only dealt with tables and with specific problems. It remained for the Greeks to develop the idea of proof and in the material in this second phase of algebra — the authors break up the history into five phases — the work is highly geometrical. Here most readers will be on more familiar ground — a discussion of the controversies surrounding incommensurability and the questions of the possibility of carrying out certain geometrical constructions, with interesting material on what happens when one changes the rules to include something other than straightedge and compass.

The third phase of the history of algebra, a long period, ran from the time of Diophantus to the time of Viète and Descartes. [This chapter was reprinted in the *American Mathematical Monthly*, volume 106 (1999), pages 57-66.] This was the beginning of what the authors call "literal algebra" (i.e., algebra with letters). Diophantus posed, in his *Arithmetica*, a series of problems with solutions, and used a form of symbolic notation — but we can be thankful we're not still using his symbols, given an example here of a cubic equation written in Diophantus' notation. If students are intimidated by algebra now… We learn here that Diophantus did far more than inspire Fermat to consider sums of two *n*th powers of integers as an *n*th power and to claim to have proved Fermat's Last Theorem. Diophantus discovered ways of finding two right triangles with the same hypotenuse: 39, 52, 65 (there's a misprint in the book at this point) and 25, 60, 65, for example. This led to Fermat's work determining the number of ways an integer can be expressed as the sum of two squares.

The period from Diophantus to Fibonacci, during which we see the Arabic contributions, though covering nearly 1000 years, was not very eventful. But during the Renaissance, discoveries in algebra came quickly. Cardano is quoted as having said, "I was born in this century [1501] in which the whole world became known; whereas the ancients were familiar with but little more than a third part of it…". In 1591 Viète introduced the language of formulas into mathematics. During this fourth phase in the history of algebra, the period between Viète and the 1830s, one of the principal motivating problems was the solvability of polynomial equations by radicals.

This book gives one of the most satisfying accounts of this history that I have seen. Lagrange's remarkable observations on his resolvents are here. The authors identify the gap in Ruffini's attempted proof of 1799 of the insolvability of the quintic (something usually not addressed in other texts) and describe Abel's contributions. Without going into the technical details of Abel's proof the authors lead us through the fundamental ideas involved. Also during this period Gauss in his doctoral dissertation proved the Fundamental Theorem of Algebra.

With Galois came a new way of thinking in algebra, as Kolmogorov claimed, "the algorithm of formulas was replaced… by the algorithm of concepts." The authors close with chapters on problems of algebraic number theory and the development of commutative algebra, then on linear and noncommutative algebras, featuring Kummer's work on ideals in the former, the work of Hamilton and Grassmann in the latter. In the problems of algebraic number theory one had to forfeit unique factorization. In linear algebra and its offshoots, one had to abandon commutativity and associativity: multiplication of Hamilton's quaternions lacks commutativity and with Cayley numbers (octonions) multiplication is neither commutative nor associative. This breaking with traditional assumptions, all rather startling in the 19th century, opened the way for the extraordinary growth of "modern" or "abstract" algebra in the 20th.

In addition to having one of the most beautiful book covers of this or any year, this book presents the history of algebra with great clarity and elegance — with no small amount of credit due to Shenitzer, who provided a smooth and idiomatic translation from the Russian. There are informative notes at the end of each chapter. If one were inclined to quibble, one might wish that some references to Russian sources had been replaced or supplemented by references to works more readily available to an English speaking audience. But why quibble? The book shows us a splendid panorama of the development of a discipline with a fascinating history stretching over four millennia. It's a winner!

G. L. Alexanderson (

galexand@math.scu.edu) is Valeriote Professor of Science and Chair of the Mathematics and Computer Science Department at Santa Clara University. He is a former editor of

*Mathematics Magazine* and has served both as Secretary and as President of the MAA. This review originally appeared in the "

Read This!" column.