## A treatise of algebra: both historical and practical... With some additional treatises, I. Of the Cono-cuneus . . . II. Of angular sections; and other things relating thereunto, and to trigonometry. III. Of the angle of contact... IV. Of combinations, alternations, and aliquot parts.

London: Printed by John Playford, for Richrd Davis, 1685.

First edition, an exceptional copy bound in contemporary red morocco gilt, of “Wallis’ last great mathematical book” (DSB). It combined a full account of the contemporary knowledge of algebra and its history, “a feat never previously attempted by any author” (DSB); it also contained many topics we would not now expect to find in a book on algebra, notably the first publication of Newton’s work on the binomial theorem (Ch. 85) and infinite series (Ch. 91), and the first attempt to give a graphical representation of complex numbers (Ch. 67), usually thought to date from the early 19^{th} century. “Of the 100 chapters, the first fourteen trace the history of the subject up to the time of Viète, with emphasis on the development of mathematical notation. The subsequent practical introduction to algebra (chapters 15 – 63) was based almost entirely on Oughtred’s *Clavis mathematicae*, Harriot’s *Artis analyticae praxis*, and [J. H. Rahn’s] *An Introduction to Algebra* (1668) … After an insertion concerning the application of algebra to geometry and geometrical interpretations of algebraic facts (chapters 64 – 72, including an attempt to give a representation of imaginary numbers), Wallis devoted the final twenty-eight chapters to … a discussion of the methods of exhaustion and of indivisibles, with reference to [Wallis’] *Arithmetica infinitorum* (1656)” (DSB), Wallis’s most important work in which he arithmetized Cavalieri’s method of indivisibles. Wallis here gives a detailed account of the *Arithmetica *and takes the opportunity (Ch. 79) to respond to Fermat’s criticisms of it. “The *Algebra* also includes an exposition of the method of infinite series and the first printed account . . . of some of Newton’s pioneering results. Wallis had long been afraid that foreigners might claim the glory of Newton’s achievements by publishing some of his ideas as their own before Newton himself had done so. He therefore repeatedly warned his younger colleague at Cambridge not to delay but to leave perfection of his methods to later editions . . . Wallis helped shape over half a century of mathematics in England. He bore the greatest share of all the efforts made during this time to raise mathematics to the eminence it enjoyed on the Continent. The center of mathematical research and of the ‘new science’ in Galileo’s time lay in Italy. It then shifted northward, especially to France and the Netherlands. Because of Wallis’ preparative work and Newton’s genius, it rested in Britain for a while, until through the influence of Leibniz, the Bernoullis, and Euler it moved back to the Continent” (DSB). “Among the most famous parts of this treatise is Wallis's discussion of the work of Thomas Harriot, especially his contention that René Descartes plagiarized Harriot’s symbolization procedure in algebra . . . After giving a list of Harriot’s discoveries in algebra, Wallis notes that there is ‘scarce anything in (pure) algebra in Descartes which was not before in Harriot.’ Most historians did not believe Wallis, because Harriot’s published work did not include a lot of what Wallis stated. But since the recent discoveries of Harriot’s algebra manuscripts, there is certainly some reason to believe that Wallis was correct. There is certainly some similarity between Harriot’s manuscripts and Descartes’ algebraic work in his *Geometry*” (MAA). It is highly unusual to find a scientific book of such importance in such a magnificent binding.

*Provenance*: The S. R. Christie-Miller – Britwell Court copy (pencil note on fly-leaf in the hand of); Charles Traylen, who purchased the book at the Britwell sale, Sotheby’s, March 30, 1971.

“Wallis had first stated his intention of writing a book on algebra in 1657 at the end of his *Mathesis universalis*, where he explained that he had hoped to include the ‘doctrine of analysis, the perfection of arithmetic’ but that he had already written more than he intended. Rather than give too short an account of ‘analysis’, or algebra, he thought it better to devote a separate volume to the subject, which he proposed to do. Given Wallis’s prolific output on other topics it is perhaps not surprising that the volume did not materialize, and there was no further mention of it until some ten years later … we may suppose that the book was not begun in earnest until 1673 and that Wallis continued to work on it until he delivered it to Collins in 1677. The dating 1673 to 1677 is confirmed by a number of mathematical letters that Wallis wrote in response to queries from Collins in 1673 and 1676, and then included in *A treatise of algebra*. Newton’s *Epistola prior* and *Epistola posterior* were written [to Leibniz] in June and October 1676, and the final third of *A treatise of algebra*, in which extracts from Newton’s letters are embedded, may have been written in its entirety in the winter of 1676–77. [It was the publication of these letters by Wallis that fanned the embers of the priority dispute between Newton and Leibniz over the discovery of the calculus.]

“Collins died in 1683 but by then the Royal Society had promised to underwrite the publication of *A treatise of algebra* and a deal had been negotiated with Richard Davis, an Oxford bookseller, who agreed to handle it if sufficient sales were guaranteed. Down payment on 100 copies seems to have been the necessary level of support, and the Royal Society undertook to buy 60 copies at 1½d per sheet and invited further subscriptions at the same rate. *A Proposal to publish A treatise of algebra* was circulated in 1683; it invited subscribers to send a deposit of five shillings before December 1683 and promised to print at a rate of two sheets a week from 1 August 1683. (The book eventually required four quires, or 96 sheets, of paper, and so cost twelve shillings to subscribers but sixteen shillings or more to later buyers.) The book was printed by John Playford in London, who possessed the necessary range of type, and it was eventually completed not in 1684, as had been hoped, but in 1685, some twelve years after Wallis first began to write it.

“The delay between writing and printing gave Wallis plenty of opportunity to add to his text and he continued to do so up to the last possible moment, the final ‘Additions and Emendations’ being inserted in 1684 when much of the book was already printed. The opportunity to publish was too valuable to waste and prompted Wallis to include as much as he could of the work deposited with Collins, whether directly related to algebra or not. [Owing to the reluctance of English publishers to print mathematical works, Wallis, Barrow and others had adopted the habit of depositing their papers with the Royal Society until an opportunity for publishing arose; Collins had held some of Wallis’s papers for several years.] Some of it was relegated to appendices, but some was integrated at what Wallis considered appropriate points in the main text. This makes *A treatise of algebra* read at times like an anthology of results and ideas with little obvious relation to one another. Running through the text, nevertheless, is a clear historical thread. At times it seems in danger of vanishing but Wallis always managed to bring it back into focus: the history that was incidental to his *Mathesis universalis* thirty years earlier had now become his main theme” (Stedall 2002, pp. 8-12).

“When John Wallis (1616–1703) wrote his great historical study, *A treatise of algebra both historical and practical *(1685), he devoted three-quarters of its 100 chapters to exploring the work of five 17th-century English mathematicians: Oughtred, Harriot, Pell, Newton, and himself. His pride in the achievements of his own countrymen in his own century was from the beginning a prime reason for writing his book not just as mathematics but as history, a history designed to make the English contribution plain. The first 14 chapters of *A treatise of algebra *are a prelude to Wallis’s larger plan: in them he explored the origins and early development of algebra and the evolution of the modern number system without which, he argued, algebra could never have progressed. But above all, Wallis began to set his story in an English context, by pointing to the existence of mathematical learning in England since early postclassical times, and especially during the later medieval period 1100–1450. Commentators on *A treatise of algebra *have tended to pass over these early chapters, yet they reveal Wallis at his finest as a historian, for here he displayed greater objectivity and a truer sense of the complexities of historical development than in almost anything else he wrote, and in his investigations into the origins of the number system, he was the first to apply modern historiographical standards and methods to both primary and secondary mathematical sources.

“A unique combination of circumstances in 17th-century Oxford made Wallis’s research possible. Histories of mathematics are perhaps written only when mathematicians perceive marked changes in the nature and scope of their subject, and by the second half of the 17th century it was plain that mathematics was steadily liberating itself from the constraints of its classical past and taking on a life and momentum of its own. Wallis had seen this revolution at first hand during his long tenure of the Savilian professorship, and indeed had done much to bring it about. He was also well placed in a second and more material way, through his access to the unprecedented accumulation of books and manuscripts in Oxford’s Bodleian Library. From the opening of the library in 1602 there had been energetic and wide-ranging efforts to collect and preserve texts from England and abroad and the concentration of this wealth of material in a single place both reflected and encouraged new attitudes to historical study. Wallis’s … longstanding interests in grammar, etymology, cryptanalysis, music, astronomy, calendar reform, and general history all informed his account of the medieval period. He knew his classical sources thoroughly but also recognized, thanks to the new proliferation of oriental studies in Oxford, the debt of European mathematics to Indian and Islamic sources, and the main theme of Chaps. 2–4 of *A treatise of algebra *was the transmission of learning from Islamic Spain to northern Europe” (Stedall 2001, pp. 73-74).

“When it came to his own day, it was quite clear to Wallis that mathematics, far from being a relatively fixed body of knowledge, was taking off in new directions, and that history could no longer be regarded as the handing on of tradition, but as the story of growth and change. Wallis’s title itself indicates as much: ‘shewing the original, progress, and advancement thereof, from time to time; and by what steps it hath attained to the heighth at which now it is’. There was perhaps nobody better able than Wallis to understand and record the ‘Changes and Alterations’ that were taking place in seventeenth-century mathematics. Everything from Chapter 15 onwards is about the work of seventeenth century English mathematicians, with most of whom Wallis was personally acquainted” (Stedall 2002, p. 14).

One of the most important, and controversial, aspects of Wallis’s *Algebra* is his treatment of the work of Thomas Harriot (c. 1560-1621). The only part of Harriot’s work to be published was his *Artis analyticae praxis* (1631), but there was much manuscript material on algebra, and other subjects, to some of which Wallis had access. “Because Harriot never published any of his scientific or mathematical findings in his lifetime it has been difficult to establish his true place in the intellectual history of the period or to judge the extent of his mathematical influence on those who came after him. Wallis thought that Harriot should have been more acclaimed than he was, and devoted no less than a quarter of *A treatise of algebra* to extolling him. Furthermore, he repeatedly accused Descartes of having made use of Harriot’s algebra without acknowledgement, and thereby inflamed a controversy that has not been satisfactorily settled since” (*ibid*., p. 88).

“Wallis saw as clearly as Pell the true magnitude of what Harriot had done, and was concerned, with Pell's encouragement, to put the record straight. His description of Harriot’s algebra was intended not just as straight rendering of content but as an assessment of its place in the history of the subject, and of its potential for development. In this he can only be said to have succeeded: his exposition of what could be built on Harriot’s foundations was essentially correct, and his list of twenty-five ‘Improvements of Algebra to be found in Harriot’ was a fair summary of what could be found in Harriot’s work or easily deduced from it. Wallis’s account was for over 300 years the most thorough and detailed analysis of Harriot’s algebra available and, had he not been so aggressive towards Descartes, or so secretive about his sources, might have stood as the fine testament to Harriot that he intended it to be.

“What should we now consider to be the ‘Improvements of Algebra to be found in Harriot’? The first and most obvious must be his notation, the use of lower case letters, with repetition to indicate multiplication. The only significant difference between modern notation and Harriot’s is in the use of superscripts for exponents … Harriot’s notation made possible his second great achievement, the handling of equations at a purely symbolic level. His interest in the structure and manipulation of equations was firmly in the tradition of Cardano, Bombelli and Viète, but Harriot was the first to have a notation that actually helped rather than hindered his investigations. His third outstanding achievement was his crucial insight into the way polynomials could be built up as products of linear or quadratic factors and to see that such composition was in turn a powerful analytic tool. In Harriot’s clear layout, results about the number and kind of roots, and the relations between the roots and the coefficients became immediately obvious. Wallis pointed this out repeatedly … Harriot was not in the habit of describing his mathematics verbally, but if he failed to state such results explicitly it did not mean he was not aware of them, for his entire work on equations was based on the way coefficients were composed from roots. Such relationships between roots and coefficients were to become the foundation of all subsequent work on polynomial equations, and were to help to lead eventually to the development of modern abstract algebra …

The tragedy was that Harriot published nothing, and those who afterwards tried to do so failed to do him justice. Instead, from 1637 onwards, Descartes’ *La Géométrie* became the foundation and inspiration for continental mathematicians, and eventually became the dominant influence in England too. No wonder that Wallis, already deeply suspicious of the French, became bitter as he came to know Harriot’s algebra better at what he, like Pell, saw as Descartes’ usurpation of Harriot’s rightful place. Wallis … did his best to ensure that Harriot was given the recognition he deserved, but later readers of Wallis’s account, lacking the supporting evidence of the manuscripts, saw only his apparent exaggeration and polemic, and dismissed it” (*ibid*., pp. 123-125).

“Seventeenth-century mathematicians knew that what they were doing was both new and exciting: Harriot, liberated by his notation, took flight in previously unimaginable ways; Wallis was visibly moved as he closed in on his fraction for 4/π; Newton’s delight in his infinite series for logarithms spurred him to page after page of calculation; Brouncker threw into the mathematical arena some extraordinary gems of mathematical invention. They would not have considered Oughtred’s description of the new mathematics as ‘the discovery of wonders’ any exaggeration, and it was exactly this sense of inspiration and progress in English mathematics that Wallis wanted to convey. That he himself had created part of the history he was describing, and that he wrote into it his personal preferences and prejudices, does not invalidate his account but draws us all the more fully into his world” (*ibid*., p. 18).

Wallis also includes a discussion of work by one of the most enigmatic of seventeenth-century English mathematicians, John Pell (1611-85), who published almost nothing (at least under his own name). Pell is best known today for his commentary in the English translation of Rahn’s *Teutsche Algebra*, which is described by Wallis in Chs. 57-63. But Stedall (2002) points out several other contributions by Pell in *A treatise of algebra* which are not explicitly acknowledged. “There is one further section of Wallis’s *A treatise of algebra* that contains work that can possibly be linked to Pell: four chapters on the geometrical representation of complex numbers (Chs. 66-69). Wallis suggested that just as a real number, *a*, may be thought of as a geometric mean between two positive numbers *b* and *c* (that is, *a* = √*bc*), so an imaginary number may be considered as a geometric mean between a positive number *b* and a negative number –*c* (that is, *a *= √−*bc*), and thus the geometric construction of the former can be adapted to give a corresponding construction of the latter. This idea enabled Wallis to interpret roots of positive and negative squares as sines and tangents, respectively. In further constructions he showed that if real roots were represented by points on a circle, imaginary roots could be represented on a related hyperbola. There is nothing in Wallis’s text that overtly links any of this with Pell, but in his final summary Wallis explained that the value of these constructions was to indicate just how far complex or ‘impossible’ solutions deviated from the real or ‘possible’: ‘[The constructions], while declaring the case in Rigor to be impossible, shew the measure of the impossibility; which if removed, the case will become possible.’Compare these words with a paragraph from a letter written to Collins in 1674: ‘These impossible roots, saith Dr. Pell, ought as well to be given in number as the negative and affirmative roots, their use being to shew how much the data must be mended to make the roots possible.’ It seems, therefore, that both Wallis and Pell in the early 1670s were interested in what Wallis calls ‘the measure of the impossibility’ of an equation, and whether one could somehow adjust an ‘impossible’ equation to make it ‘possible’. Here again there seems to be evidence of greater collaboration between Pell and Wallis than meets the eye in the pages of *A treatise of algebra*. It is perhaps no coincidence that Wallis inserted his work on complex numbers immediately after his other references to Pell” (*ibid*., pp. 150-151).

“There are four appendices to *A treatise of algebra*, adding half as much again to the length of the book. All were written between 1662 and 1672, and were probably papers Collins had been holding for Wallis awaiting publication …

*Cono-cuneus or the shipwright’s circular Wedge*. A treatise written by Wallis in 1662 to consider shapes circular at the base, wedge shaped at the top, of potential use in ship building.

*Treatise of angular Sections*. Begun by Wallis in 1648 after reading Oughtred’s *Clavis,* and completed in 1665.

*A Defense of the Treatise of the Angle of Contact*. A discussion of the dispute between Peletier and Clavius arising from Euclid III.16, on whether the angle between a circle and its tangent at the point of contact could be said to have any magnitude. Wallis sided with Peletier whose view was that it did not. Wallis had written his original treatise on this subject in 1656, and he took up cudgels again after Leotaud in his *Cyclomathia *of 1662 came out in favour of Clavius. A long letter from Wallis to Leotaud written in 1667 forms the greater part of Wallis's new *Defense*. The penultimate chapter also contains an interesting discussion of magnitudes which ‘are nothing; yet are in the next possibility of being somewhat … And may very well be called *Inchoatius* or *Inceptives* of that somewhat to which they are in such possibility,’ an idea which in some ways prefigured later discussions on the nature of fluxions.

*A Discourse of Combinations, Alternations and aliquot Parts*. This treatise was first mentioned in Wallis’s discussion of Harriot, in Chapter 37 entitled ‘The composition of coefficients’, where having noted how the coefficients of polynomial equations were composed from the roots Wallis set out to detail the permitted combinations … The fourth and final chapter contains Wallis’s work on one of the number problems put to him by Fermat in 1657, in modern notation, to solve *x*^{3} + *x*^{2} + *x* + 1 = *y*^{2} and *x*^{2 }+ *x* + 1 = *y*^{3} in integers. Both he and Frenicle had long ago found several solutions, but Wallis’s anger was aroused by the posthumous publication of Fermat’s work in 1679 which reprinted the challenge, specifically addressed to Wallis, but without either Wallis’s solutions or Fermat’s acknowledgment of them. Wallis reworked both problems in greater detail than in 1658 [in his *Commercium epistolicum*] and this time made sure by adding them to his own book that his achievement was publicly recognised” (Stedall 2000, pp. 321-322).

Also appended to Wallis’s *Algebra *is his student John Caswell’s *A Brief **(but full**) Account of the Doctrine of Trigonometry, both Plain and Spherical**. Caswell (1654/5 – 1712) *held the Savilian Chair of Astronomy at Oxford from 1709 to 1712, the chair that Wallis himself held from 1649 until his death in 1703.

Stedall, A large discourse concerning algebra: John Wallis’s 1683 *Treatise of Algebra*. PhD Thesis, Open University, 2000. *Ibid*., ‘Of Our Own Nation: John Wallis’s Account of Mathematical Learning in Medieval England,’ *Historia Mathematica* 28 (2001), pp. 73-122. *Ibid*., *A Discourse Concerning Algebra*, 2002.

Folio (325 x 198 mm), pp. [xvi], 374, [iv], 17, [i], 76 [i.e., 176], [i], 17, with engraved portrait frontispiece by Loggan and 10 folding engraved plates, diagrams in text. Contemporary red morocco, gilt panelled covers, spine richly gilt, gilt edges. A superb copy.

Item #5004

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Price:
$20,000.00
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