A Discourse Concerning Algebra: English Algebra to 1685

History has not been kind to John Wallis' early attempt at writing a history of algebra up to his time. Starting with Montucla, most historians of mathematics have been rather dismissive about the value of the various editions of his Treatise of Algebra. Truth be told, parts of the work are just plain bad writing and at times he is so focussed on the cause of the English that he loses all objectivity. Yet much of the criticism regarding the Treatise dates from long after Wallis' death and, until recently, few have sought to appreciate the book in the context of Wallis' work and the state of mathematics during his lifetime. It is exactly this that Stedall sets out to do in the book under review here and the results of her findings make for a riveting read.

As Stedall makes abundantly clear, it is hard to see how to appreciate or even understand Wallis' historical work without an intimate knowledge of the state of algebra and the networks among those dedicated to its furtherance in England during the first three quarts of the seventeenth century. To understand Wallis' work, one has to consider the pitiful state of the study of algebra in early seventeenth-century England and the dearth of homegrown textbooks. One has to discuss how William Oughtred's poorly written Clavis (1630) could become the requisite introduction to algebra for a whole generation of English mathematicians. Stedall takes up all of these strands and weaves them together with a firm hand to form the backdrop to Wallis' Treatise. Against this background it becomes clear that Wallis' aim was not only to write a history of algebra, but also to bring a fair amount of algebra to the notice of the general public which otherwise would have gone largely unnoticed. In three separate chapters, Stedall gives three examples of this — one regarding the work of Thomas Harriot, one regarding the work of John Pell in algebra and one regarding the work of William Brouncker.

In the case of Harriot, Wallis has often been accused of touting the achievements of the secretive polymath over those of Descartes. If one compares what was publicly known of Harriot's accomplishments with Descartes' publications, one cannot but agree that Wallis' claims that much of Descartes' results had already been found by Harriot are overstated and hard to defend. In a chapter that reads like a mini detective story, however, Stedall convincingly demonstrates that in fact Wallis had access to many of Harriot's unpublished manuscripts. As Stedall has shown elsewhere, these manuscripts do contain many results that seem to anticipate Descartes' work. Far from being a blatant overstatement, Wallis' description of Harriot's achievements was actually fairly accurate.

Like Harriot, Pell never published much during his lifetime. More accurately, perhaps, Pell was loath to see his name in print. Basing her argument on a number of fascinating discoveries, Stedall makes a strong case that in fact a large part of Wallis' Treatise was directly copied from one of Pell's manuscripts, including work on the approximation of π and on complex numbers. Wallis in no way acknowledges Pell's substantial input, but Stedall also convincingly argues that Pell might not have wanted to receive any public acknowledgement.

As for William Brouncker, Wallis did acknowledge his contributions and also gave a fair assessment of the latter's work. As Stedall notes, however, in many ways the mathematical styles of the two men were too different for Wallis to do full justice to the depth of Brouncker's work. Wallis undeniably did help to make Brouncker's work better known. Ironically, it also was Wallis' work that helped rob Brouncker of any kudos for solving a particular kind of Diophantine equation. After learning about these equations and Brouncker's solution from the Latin translation of Wallis' Treatise, Euler erroneously attributed this work to Pell and the equations has been known as Pell's ever since.

In the concluding chapter of her book, Stedall makes a casual reference to work by Fowler and Whiteside on Wallis' Treatise. Fowler does not receive any special mention in the acknowledgements, but I suspect his approach to history may have been a source of inspiration to Stedall. Her style of doing history certainly shows some similarity to Fowler's, although most of the time she seems to be on much firmer ground — mostly because there is just much more material than the relatively small number of sources that Fowler can work from. Inspiring and exciting as this kind of style might be, there can be no doubt that Stedall's work is for the specialist (as in the case of Fowler). Not being a specialist in 17th century algebra myself, I learned a lot from the book, but I also was not always sure how to tell received knowledge from speculation. Stedall presents the history of 17th-century algebra as a much richer topic than has traditionally been the case. I most certainly welcome this kind of approach and I would like to see more of it in history of mathematics. At the same time, however, I wish I would feel more comfortable as to how much of her version is solid history and how much gifted `Hineininterpretierung'. I suspect there is much more of the former than the latter, but if I ever have time, I might just hole up with Walls' massive Treatise and read it all for myself. At the very least, Stedall made me wish that I had the leisure to do so — in spite of her assertion that Wallis can pretty boring at times.

Eisso Atzema (atzema@math.umaine.edu) is Lecturer in Mathematics at the University of Maine at Orono.