As with the technical terms of any field, the word ‘algebra’ has undergone several changes of meaning throughout its history, as the subjects to which it has been attached have developed. The book under review traces this process from ancient times up to the modern day. The word ‘algebra’, we are told, first emerged in western European languages as a corruption of part of the title of a ninth-century text by the Islamic scholar al-Khwārizmī (fl. 800–847). Since this work concerned the solution of polynomial equations (in modern notation, any equation of the form

$$a_{n}x^{n} + a_{n - 1}x^{n - 1} + \ldots + a_{2}x^{2} + a_{1}x + a_{0} = 0,$$

which is to be solved for x, and where n is an integer and a _n, …, a ₀ are known numbers), the term ‘algebra’ subsequently became the new name for the (much older) subject in which solutions of such equations are sought.

Although al-Khwārizmī limited himself to quadratic equations (those of degree 2: that is, n = 2 in the above equation), the centuries following his work saw the extension of his methods to equations of higher degree: solutions for degrees 3 and 4 were discovered in sixteenth-century Italy, for example. However, efforts to extend known methods still further failed, and, by the eighteenth century, mathematicians were beginning to suspect that such higher-degree equations were not in fact soluble in general – at least not in the desired form; that is, in terms of elementary operations performed on the coefficients a _n, …, a ₀: addition, subtraction, multiplication, division and root extraction. Then, in 1824, the Norwegian mathematician Niels Henrik Abel (1802–1829) published a proof that there is indeed no such general solution for an equation of fifth degree. His findings gave impetus to a new direction in the study of polynomial equations: the determination of which equations may be solved in this way. A comprehensive answer to this question was provided by the French mathematician Evariste Galois (1811–1832) in 1831. Galois's approach was to study certain permutations of the solutions of a polynomial; in doing so, he noted that the collection of these permutations forms a structure that he termed a ‘group’. This notion subsequently became the cornerstone of the theory of polynomial equations. Moreover, as the nineteenth century progressed, mathematicians began to notice that this same structure appears elsewhere in mathematics, and might therefore be studied in an entirely abstract setting, divorced from any specific interpretation: theorems proved about an abstractly defined group would then be applicable in any specific instance. Owing to its origins in the study of polynomial equations, the investigation of such abstract groups, and of the other similarly abstractly defined structures that emerged in the late nineteenth and early twentieth centuries, was given the name ‘modern algebra’, or ‘abstract algebra’, or, latterly, simply ‘algebra’. Modern mathematicians thus employ the word ‘algebra’ in a variety of senses, ranging from the solution of equations to something rather more abstract.

The book under review, written by two leading authorities in the history of mathematics, is a history of algebra (in all its senses) from its origins in the solution of practical (and, indeed, not-so-practical) word problems in the ancient world, through the works of al-Khwārizmī, Abel, Galois and many others, to the more recent development of abstract algebra. The reader requires a basic mathematical competence, but need not be an expert. Indeed, the book should be entirely accessible to a (mathematical) undergraduate readership. It will certainly be a useful resource for teaching courses in the history of mathematics, and will also supply the interested student reader with the historical context that straight mathematics courses often lack.

An impressive feature of this book is its comprehensiveness, not only in time span but also in subject matter and in geography. With regard to subject matter, for example, it is good to see linear algebra (originally the solution of systems of simultaneous linear equations) being treated here. The history of this subject can be rather difficult to trace: in certain respects, it is so basic that it has arisen independently within a broad range of apparently unrelated mathematical contexts; moreover, the topics that linear algebra treats are so varied that they have not historically been connected, and were only brought together under the heading ‘linear algebra’ in the twentieth century. Thus the subject's somewhat tortuous history means that it does not always feature (at least, not in any great detail) in general histories, so the authors of the book under review are to be congratulated for providing an accessible integrated overview. In relation to geography, we are not simply presented with a Eurocentric view of the subject (although the later developments certainly took place in Europe): we learn also of ancient and medieval Indian, Chinese and Arabic developments. It is nice, for example, to see the so-called Chinese remainder theorem, a result well known to mathematicians, in its Chinese context. Another highlight of the book is Chapter 8, in which the authors give an overview of the transmission of ancient mathematical knowledge via the Islamic world to Renaissance Europe.

All of the mathematical content of the book is, quite self-consciously, converted into modern terms: for example, symbolic notation is used in the description of problems that pre-date the introduction of such symbolism by several centuries. Generally speaking, such conversions of notation can have the effect of disguising the thought processes of the original authors, and of inadvertently imposing modern ideas onto historical mathematics. In the case of the present book, however, I believe that this is entirely justified: this is not a book to go to in order to find original formulations of historical mathematics. Instead, it is an accessible introduction, which provides enough references for the interested reader to pursue matters further. The balance of references seems about right for a book written at this level, with a mixture of primary and secondary sources cited. In summary, this is a very readable introduction to an important topic within the history of mathematics.