Thomas Harriot’s *Artis analyticae praxis* was published in 1631, ten years after his death, edited from the manuscripts that Harriot left to his executors. No editor’s name appears on the title page or elsewhere in the book, but there is no reason to question what was regarded as common knowledge at the time, that the editor was Harriot’s longstanding friend and colleague Walter Warner.

The *Praxis* contains Harriot’s most significant contribution to the theory of equations, the discovery that polynomials can be constructed as products of linear, or sometimes quadratic, factors. This was not an early version of the fundamental theorem of algebra: Harriot never claimed that all polynomials could be generated in this way, but he constructed useful lists of many that could, and called them ‘canonical’. In multiplying out factors he also developed a number of further insights into the number and nature (positive or negative) of the roots of such equations. All this is in the first part of the *Praxis*. The second part contains examples of the numerical method of solution first taught in Europe by François Viète, whose work Harriot studied closely. The *Praxis*, as was usual at the time, was written in Latin. Seltman and Goulding have now produced for the first time a translation into English.

The translation is preceded by a brief (16-page) introduction, which analyses the book’s mathematical content. It is perhaps to be regretted that the authors did not also provide some historical analysis of the circumstances under which the book was written and published, and of its subsequent influence. I am more inclined to think now than I was the past, for instance, that Warner may have edited at least some of the book from papers no longer extant, but if so one must ask why such papers were separated from the rest of Harriot’s manuscripts, and what happened to them afterwards? When did Warner begin the work and when did he finish? Did Harriot’s mathematical colleague Nathaniel Torporley make any contribution to it? Why is no bookseller mentioned on the title page? Was this a privately printed edition, and if so how many copies were made and who would have bought or acquired them? What, if anything, do we know of the readers of the *Praxis*? To a historian, these and similar questions are as important as an analysis of the book’s contents and I would have valued the authors’ opinion on them.

The notes that follow the translation (70 pages) offer a great deal of fine textual detail. They also contain some beautiful reproductions of the original manuscripts, though hidden so far into the book that they may go unnoticed by any but the most intrepid reader.

The appendix is a little puzzling. The authors pursue a long and detailed analysis of three volumes of Harriot’s manuscripts only to infer that his system(s) of page-numbering did not necessarily imply his intentions as to future publications, a conclusion with which no-one could disagree. Harriot’s pagination can give useful information only in relation to specific topics. In his algebra, for instance, it confirms his close and systematic reading, here as elsewhere, of the treatises of Viète. I believe it also give clues to the development of his thinking on the nature of roots (whether positive, negative, or imaginary) and on his use of the term ‘canonical’ which, it seems to me, he uses differently in different circumstances. The authors of the present volume take the contrary view on this last point, and suppose that Harriot uses the word in essentially the same way each time. Since modern English editions of both the *Praxis* and the related manuscripts are now available, readers will be able to take up this and other intriguing questions about Harriot’s work for themselves.

Harriot studies are now moving steadily apace. Closer scrutiny of the British Library manuscripts and the opening up of the Macclesfield Collection at Cambridge has dispelled some of the confusions sown by Cicely Tanner in the 1970s. The Oxford Harriot lectures have produced a steady flow of scholarship of a high standard, and more and more analysis of Harriot’s work in his various fields of interest is now becoming available. Seltman and Goulding’s translation is a welcome addition to this growing body of work.

Jackie Stedall is Junior Research Fellow in mathematics at The Queen’s College, Oxford. She is the author of several books, including A Discourse Concerning Algebra: English Algebra to 1685 and The Greate Invention of Algebra: Thomas Harriot's Treatise on Equations *.*