Until around 1800 ‘algebra’ meant what we now often call ‘common algebra’, a theory in which letters are used to represent constants and variables associated with real and complex numbers and/or geometrical magnitudes; it played a leading role especially in the theory of equations, functions, series and the calculus. Then algebra became algebras, with new ones created that dealt with objects of other kinds; usually they were more abstract than those handled in common algebra, such as operations of various kinds rather than the objects upon which they were operating, or structural features of collections of mathematical objects of no specified kind.

This book, the printed outcome of a workshop held at Berkeley in 2003, tackles several aspects of these extensions. The authors have written carefully scholarshipped pieces, several of them drawing on archival as well as printed sources, and their work is complemented by a good index.

The bulk of the book consists of ten articles on developments over the period, with some emphasis laid upon the decades from 1900 to 1930. We see algebras being introduced or used, and many interactions taking place between abstract algebras, algebraic number theory, algebraic geometry, and aspects of topology; an impressive mass of sophisticated pure mathematics is revealed. The authors write clearly, but in places they assume quite a lot of technical knowledge of the reader. They also make clear the prominence in many cases of German mathematicians and, partly because they were under their influence, also the Americans, who came to register significantly in mathematics only in the 1890s but then made up for lost time rather speedily.

Left rather isolated are the first two pieces, which treat aspects of the early non-common algebras, first in the hands of French mathematicians and philosophers from around 1770, and then with British mathematicians (especially the pioneer work by Babbage and John Herschel in the 1810s) up to mid-century. However, the later story is omitted, and, moreover, in silence. But why not treat, for example, Boole in the 1840s helping to launch formal logic with an algebra that was created heavily under the influence of the calculus of differential operators – one of the first non-common algebras to be developed? Again, where do we site the central place of algebra in analytical (meaning algebraic) mechanics already with Lagrange by the 1770s and elaborated in the nineteenth century by Hamilton and various successors? The editors confine their own introduction to a nice summary of each of the following articles but pass over other territory without comment. Thus, while their volume is an important contribution to the history of the development of modern algebra, the location of this subject within mathematics as a whole needs further exegesis.