‘For the Use and Benefit of such as are wholly Ignorant of the very first Rudiments (Or have not the least Notion) of Mathematicks.’

John Ward, *The Young Mathematician’s Guide*, Sig A3r.

John Ward, *The Young Mathematicians Guide.* *Being* *a plain and easie Introduction to the Mathematicks* (London, 1719), title page.

The title page of John Ward’s mathematical textbook gives us a good indication not only of its contents but also of the reasons the book became a mathematical bestseller in Georgian England and Ireland. The clarity of the title page, with its careful delineation of the chief contents of the book and its emphasis on simplicity and innovation, was attractive not only to school boys trying to learn mathematics for the first time, but also proved appealing to adult readers eager to have an easy, portable mathematical compendium in the vernacular. That it was a popular mathematical textbook is plain: as Benjamin Wardhaugh (2015) points out, the text was printed no fewer than twelve times at London alone and he estimates that by 1725 there were c. 25,000 copies of it in existence. It was reprinted at Dublin in 1731, 1755 and 1769, and a French translation was published in 1756; it continued to be used as a textbook until the nineteenth century. Since this mathematical exhibition is primarily the work of a group of young twenty-first century mathematicians we have deliberately chosen Ward’s five-fold mathematical division of ‘Arithmetic’, ‘Algebra’, ‘Geometry’, ‘Conic Sections’ and ‘Infinites’ as the basis of the structure of the initial part of this online exhibition.

John Ward, *The Young Mathematicians Guide.* *Being* *a plain and easie Introduction to the Mathematicks* (London, 1719), portrait.

Little is known of the author himself for he gave scant information about his origins. We have some approximation of his age from the frontispiece portrait to this work where it states that the portrait was produced when he was aged 58, in the year 1706 – he was, therefore, born sometime circa 1648. We know also from indirect references that he died just a few years before Edward Worth, c 1730. He is not mentioned in either the matriculation lists of the University of Oxford or Cambridge. Indeed most of the little we know of him comes from his own publications: the 1695* The Compendium of Algebra* (London, 1695), *Synthesis et analysis Vulgo algebra*, a broadsheet of the same year, and the 1707 *The Young Mathematician’s Guide*, bought by Worth in the corrected third edition of 1719. Ward’s *Clavis Usuræ, or a key to interest*, followed in 1710 and in 1714 his last work, *A practical method, To Discover the Longitude at Sea*, appeared in print in London. From the title pages of these works we find nuggets of information. He mentions a number of times that he had previously been ‘a general gauger in the revenue of excise’ – clearly this employment took place before his first publication of 1695. This claim is repeated and augmented in Worth’s 1719 copy of the* The Young Mathematician’s Guide*, where Ward describes himself as ‘Professor of the Mathematicks in the City of Chester’ and says that he had previously been ‘Chief Surveyor and Gauger-General in the Excise’. However, since the title page of the third edition did not change from that of the first edition in 1707 it tells us little of what he had been doing in the intervening years.

His 1695 broadsheet, our first encounter with him in print, advertised his course in mathematics. It promised much: ‘*A Learned*, in a Month or Six Weeks, may know more in *Arithmetick* and *Geometry*, by help of this *Analytics*, than ‘tis possible he should* Ever* comprehend by any other Method.’ The broadsheet sheds light on Ward’s intended audience: youths about twelve or thirteen years of age who would be housed and taught at Ward’s own house in Fanchurch Street in London (he offered to teach the sons of the nobility and gentry at their own houses). Ward was clearly offering an intensive, onsite course and was eager to give demonstrations of the course at a number of venues, including ‘Mr Warner’s, a Mathematical Instrument-maker, in little Lincolns-Inn Fields’ and at the ‘Sign of the Kings-Arms and Glober, near Lincolns-Inn Gate’, as well as at his own house. Unfortunately, we do not know how successful this particular venture proved to be but the fact that his *The Compendium of Algebra*, probably a textbook for the 1695 course, was reprinted in 1698, suggests that he was at least moderately successful. The title page of the 1698 edition tells us that Ward was, at that time, ‘teacher of the mathematicks, at the Globe i[n] Fleet street near Fetter-lane end’; his 1707 *The Young Mathematician’s Guide* informs us that by October 1706 he had moved to King’s gate in Red-Lyon Fields’.

The text of *The Young Mathematician’s Guide* throws light on some parts of his mathematical friendship circle: Ward’s inclusion of a laudation from two famous Newtonian mathematicians, Joseph Raphson (fl. 1689–1712), and Humphrey Ditton (1675–1714), strongly suggests links with the rising Newtonian movement within the Royal Society, though Ward himself was not elected a Fellow. Raphson’s version of Ozanam’s *Dictionnaire mathématique* appeared as *A mathematical dictionary* in 1702, while his translation of Newton’s *Arithmetica *universalis was published posthumously as *Universal arithmetic* in 1720 (O’Connor and Robertson, 1996). Ward had dedicated his *The Compendium of Algebra* (London, 1695), to Raphson not only because he was familiar with Raphson’s *Analysis æquationum universalis* (London, 1690), a work on algebra which he highly praised, but also because he hoped to acquire Raphson’s patronage. As the approbation in Worth’s 1719 edition makes clear, in this Ward had been successful. Equally, Ward’s dedication of *The Young Mathematician’s Guide* to Sir John Wentworth of North Elm’s Hall, demonstrates that Ward had close links with the baronet who had not only encouraged the publication of the work but has also (unlike many other dedicatees), actually read the manuscript before publication.

Unfortunately Ward gives little more information than this and his preface to the reader of *The Young Mathematician’s Guide* is equally brief: not for Ward a long discursive passage on the importance of mathematics, a subject he felt to be so self-evident as not to require elucidation. Instead he explains his chief selling point – that, though many other mathematical authors had published on the subject, they all too often over-estimated the mathematical learning of their audience. By contrast, Ward’s text was ‘purely for the Use and Benefit of such as are wholly Ignorant of the very first Rudiments (Or have not the least Notion) of Mathematicks.’ Ward’s aim was to ensure a good grounding of the basics which would be a handy guide for aspiring mathematicians.

A handy guide in more ways than one, for the work was published in octavo format. This was for two reasons: first, it was cheaper to print octavos than the more costly quartos and folios; and, secondly, because Ward intended his textbook to be as portable as possible. He makes the point himself that few previous mathematical authors had published ‘in any small Volume’, an innovation of which he was justly proud (indeed he mentioned the same point in his preface to his *The Compendium of Algebra*, 1695). For Ward, *The Young Mathematicians Guide* was a book which should be constantly referred to as an essential aid. The format of the book and its content were well matched. Ward started with ‘the very First Foundation or First Principles’ (i.e. Arithmetic and Geometry), and gradually moved the reader forward to deal with more complex formulations. The aim was to provide the young reader with a basic mathematical education which would enable him to ‘either Qualify him for Business; or fit and prepare him for higher Speculations without the Trouble and Charge of other Books.’

Ward was able to incorporate some useful notation and equations from the *Teutsche Algebra* (Zurich, 1659) of Johann Heinrick Rahn (1622-1676), which was translated by the English mathematician Thomas Brancker (1633-1676) and published at London in 1668 (with additions by John Pell (1611-1685)), under the title *An Introduction to Algebra*. There are some idiosyncrasies in Ward’s notation: many eighteenth-century mathematical commentators used two forms to signify “therefore”, both including 3 dots but with the dots in a different order. Ward eschewed the use of three dots and instead used two dots, in line with Oughtred’s *Opuscula mathematica*. His used “>” to indicate angles and a symbol of a circle with a dot in the middle of it as a sign for a circle. Ward was one of the few writers who used the colon to mark ratio, a symbol which became more common by the end of the century. Beyond these examples, the vast majority of his notation for addition, subtraction, multiplication, division, involution (or raising to a power) and evolution (or extracting roots) is similar to what is in use today.

John Ward, *The Young Mathematicians Guide.* *Being* *a plain and easie Introduction to the Mathematicks* (London, 1719), p. 143.

Ward’s style was designedly ‘Plain and Homely’, fitted for a younger reader (or, for that matter, a reader unskilled in mathematics). His approach to the teaching of mathematics clearly developed between 1695 and 1707. His broadsheet of 1695 (not collected by Worth), demonstrates that at that time he was busy advertising his services as a teacher of mathematics and was focusing on algebra. In 1695 he assured his readers that ‘*The Sciences Mathematical* have their Rise and Foundation in Algebra; This then ought first to be Learned, in order to a True and Easie obtaining the Rest.’ It is likely that his *A compendium of algebra. Consisting of plain, easie and concise rules for the speedy attaining to that art. Exemplified by various problems, with the solution of their æquations in numbers. By a new and general method of resolving all kind of æquations, both pure and adfected, with great ease and expedition; very different from all others yet extant. Applied to squaring the circle, making of sines, tangents, secants, and logarithms with great facility. Also an appendix concerning compound-interest and annuities* (London, 1695) was written as a textbook for this 1695 course. By comparing these works with his later *The Young Mathematician’s Guide*, we can see that Ward subtly changed his approach to the pedagogy of mathematics between 1695 and 1707, the year the first edition of *The Young Mathematician’s Guide* was published. By 1707, as the title page of his *The Young Mathematician’s Guide* makes clear, he had decided to start with arithmetic before moving on to ‘Algebra or Arithmetick in Species’, then the ‘Elements of Geometry’ followed by Conic Section and finally, he aimed at explaining ‘the Arithmetick of Infinites’.

In his *The Young Mathematician’s Guide* Ward was interested in two things: mathematical innovation and the provision of a clear synthesis of what was already known. It is evident that he was interested in expanding mathematical knowledge as much as he could: Cohen and Shannon (1981) draw attention to his attempt in his *The Young Mathematician’s Guide * ‘to use geometric methods to calculate pi’ – the last attempt of its kind, where Ward indicated his desire to find 2π to 15 places (16 places of figures). His treatment of the subject demonstrates that he was aware of earlier work on the topic by mathematicians such as Ludolph Van Ceulen and John Wallis whose works were likewise collected by Worth. Ward’s last work, *A practical method, To Discover the Longitude at Sea, By a New Contrived Automaton. Freed from all the Various Effects of Air in different Climates, &c. And not Liable to Disorder by the Irregular Motion of a Ship. The whole Method Rendered Plain and Easie to be Understood by every Mariner. With an account of the author’s new instrument for taking the* *latitude more accurately at sea, than hath hitherto been practised. Humbly Offer’d to the Consideration, and Use of the Publick* (London 1714), demonstrates his innovations in another area, the invention of instruments for navigation – and reminds us of the close connections between teachers of mathematics in Georgian England and instrument makers. Ward’s status of a bestselling author of mathematical works was not only demonstrated by the many reprintings of the above texts, but also by the decision in 1730 to bring out a posthumous collection of his works, entitled *The posthumous works of Mr. John Ward, Author of the Young Mathematician’s Guide. In two parts. Part I. Containing, His New Method of Navigation by Parallel Parts, by which all Questions in Sailing may be answered with great Expedition and Truth, in a different Manner from Plain Mercator, and Great Circle Sailing, by the Solution of a plain Triangle only. Also, Compendiums of Practical and Speculative Geometry, and of Plain Trigonometry, with their Application to Plain Mercator, and Middle Latitude Sailing, with several curious Questions in Surveying. Part II. Containing, The Doctrine of the Sphere, and the Demonstrations and Calculations of Spherical Trigonometry, in which the Construction of the Figures are New, and drawn so as to represent Solids, by which the Demonstrations are made easy to the meanest Capacity. Published by a particular friend of the author’s, from the original manuscript, and revised by Mr. George Gordon, Mathematician in London* (London, 1730). The lengthy title gives us some indication of his writings following his *The Young Mathematician’s Guide*, works primarily of a geometrical nature.

**Sources**

Cohen, G. L. and G. Shannon, ‘John Ward’s Method for the Calculation of Pi’, *Historia Mathematica,* 8 (1981), 133-144.

O’Connor, J. J. and E. F. Robertson, ‘Joseph Raphson’,* MacTutor History of Mathematics* (University of St. Andrews, 1996).

Ward, John, *Synthesis et analysis. Vulgo algebra* (London, 1695).

Ward, John, *A compendium of algebra. Consisting of plain, easie and concise rules for the speedy attaining to that art. Exemplified by various problems, with the solution of their æquations in numbers. By a new and general method of resolving all kind of æquations, both pure and adfected, with great ease and expedition; very different from all others yet extant. Applied to squaring the circle, making of sines, tangents, secants, and logarithms with great facility. Also an appendix concerning compound-interest and annuities* (London, 1695)*.*

Ward, John,* The Young Mathematician’s Guide. Being a Plain and Easie Introduction to the Mathematicks. In Five Parts*. (London, 1719).

Wardhaugh, Benjamin, ‘Consuming Mathematics: John Ward’s* The Young Mathematician’s Guide, *(1707) and Its Owners’,* Journal for Eighteenth-Century Studies* 38. no 1, (2015), 65-82.

**Text:** Elizabethanne Boran and Alan Noone.