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Editorial — Numbers in Early Modern Writing

Editorial — Numbers in Early Modern Writing

Katherine Hunt and Rebecca Tomlin

[1] ‘Here’s another of your ciphers to fill up the number’. John Ford’s choice of insult, a derogatory description of a gallant, spoken by the nurse Puttana in ’Tis Pity She’s a Whore (c.1629; I.ii.110-111) is revealing in a number of ways. The ‘cipher’ here is a nothing, a man of little worth, making up the numbers of a young lady’s gaggle of suitors, but he seems nevertheless to have some substance, because he ‘fill[s] up the number’. In this throwaway line is contained the derogatory dismissal of this useless gallant; the partial reversal of the gendered implications of the cipher (the empty zero more usually being associated with woman, and the phallic 1 with man); and also a hint at a relatively new understanding, in popular mathematics, of just what a cipher might be able to do. The cipher, or zero, in Hindu-Arabic notation, is worthless alone but, placed after another number in written notation, it fills up the number by multiplying it by a factor of ten. This function of the cipher is not a new idea; nor is Ford’s a novel application of it to literature. But the number joke is telling nonetheless, indicating the increasing familiarity, in this period, with Hindu-Arabic notation and the new mathematics, in which numbers were calculated in writing rather than using counting aids; indeed, the word ‘ciphering’ began in this period to be applied not just to writing zeroes but to numbers more generally. Not only does Ford expect the audience to understand an arithmetical joke about the place value of the cipher, but he gives the joke to a low-status, female, domestic servant. In fact, in Ford’s play the well-bred lady, Annabella, does not respond to Puttana’s comment or engage in the numerical chatter and Puttana, as her name indicates, turns out to be a thoroughly unwise advisor to her young mistress. But Puttana’s jibe nevertheless allows us to trace the outlines of ambivalent early modern attitudes to number: it is at once a fluid and vivid source of metaphors and jokes, especially ones which rely on a sophisticated level of knowledge about the written language of number in order to be understood; but it is also the material of whores and servants, tradesmen and the lower classes.

[2] This special issue of the Journal of the Northern Renaissance investigates how contemporary developments in written, and particularly printed, works on and of numbers both reflected and shaped the creative works of the period and how such novel concepts as the cipher, unity and equivalence, fractions, and the spatial aspects of written number were given significance beyond arithmetic in poetry, on stage, and in prose. The numbers that fill early modern books oscillate between practicality and mystery, materiality and abstraction, application and theory. What were the practical issues arising from printing numerical texts, and how were numbers represented on the page? How were early modern literary writers influenced by developments in theoretical and applied arithmetic and, conversely, is it possible to trace the influence of what seemed to be purely literary forms on the development of numerical writing? Such questions were the basis of ‘Working it out: numbers in early modern writing’, a multidisciplinary conference we organised at Birkbeck in 2013, and from which this collection of essays has developed. Some of the papers included here were first given at the conference; all show the rich and diverse readings of early modern numerical culture developing in recent work from both sides of the Atlantic.

[3] A thread running through many papers at the conference, and the chronological and intellectual starting point for this issue, is the work of Robert Recorde. His The Ground of Artes, probably first published in 1543, was the first arithmetical textbook to be written originally in English (translated books had appeared earlier in the century); it introduced Hindu-Arabic notation and the new mathematics to the English vernacular. The book proved popular and influential and was issued in forty-five editions, with various additions but relatively few subtractions, until 1699. Over the century and a half that divides the first and the last editions of the Ground, the English writing of numbers changed. With the help of Recorde’s self-consciously didactic textbook, English people, among others, took up en masse what he called ‘Arithmetyke with the penne’ (Recorde 1543: sig.116v). Hindu-Arabic notation took over from Roman as the principal figures used for calculation and Roman letters-as-numbers slowly lost their practical value, leading to new relationships between letters and numbers. Writing about numerical calculation reflexively concerned itself with calculations done in writing, not manually with counters or an abacus; although Recorde did include a section on calculating with counters it was tucked away at the end of the first volume of his book, always somewhat marginal to his main project, intended either for ‘them that can not write and rede’ or for those who ‘have not a some tymes theyr penne or tables redye with them’ (Recorde 1543: sig.116v). This section persisted in later editions of the book, until the final edition in 1699 when the instructions for reckoning by counters were at last left out. ‘Arithmetyke with the penne’, or ciphering, had definitively won; although people surely continued to use counters or fingers to reckon with, numbers were now things primarily to be written and read, and manipulated on the page. Calculation was no longer an ephemeral activity, lost as soon as the counters that embodied it were moved, but was now a process of writing and, by writing, recording. The publication history of The Ground of Artes provides, then, both a start and an end point for this special issue: it bookends the period in which, in England, numbers moved from the counter table to the page, from the material object to the written symbol.

[4] Any understanding of numbers in the period must grapple with the spread of their use: who used numbers and to what purpose. Given that this was a period in which numbers turned into things that were principally written, it is tempting to draw up a standard by which to measure numeracy analogous to that used to determine literacy (although the latter term is also problematic). In fact, as Keith Thomas argues in an important article, it is much harder to assess popular numeracy than literacy (Thomas 1987). Numeracy is, in one sense, innate: one doesn’t need to be able to read and write to tell that two apples are more than one, for instance. But such examples are in fact very limited: the number sense is culturally and historically determined, and has to be learned; the innate knowledge that two is more than one only goes so far. And in any case degrees of early modern numeracy were very varied, and many people would know only the number work that was immediately relevant to their professional needs; as Thomas suggests, while many early modern people would have had a basic ability to add and subtract, knowledge of multiplication and division was, to modern sensibilities, surprisingly restricted. Arithmetic was not part of the usual grammar school curriculum (Thomas 1987: 109-110) and, though it was part of the quadrivium, much of what we would now class as basic mathematics went unlearned by students at the universities. This is not to suggest that the tradesman was somehow more progressive than the scholar: Mordechai Feingold (1984) has conclusively shown the complex links between the two, and the sophistication of the work being done in the universities. A recurring theme of the articles in this special edition is the tension over number work and the extent to which it was the preserve of an intellectual elite, or utilised by the trades and the applied arts. Much of the number work discussed here is related less to the mathematical discoveries of the universities and more to tradesmen’s shop arithmetic: but also to the relationship, in this period, of the latter to the former. Our authors show how the middle ground between the two was appropriated by the literary imagination and the material book culture of the early modern period.

[5] Practical, applied mathematics put numbers to work in the world. As knowledge of the new mathematics was disseminated — as printed numbers became much more commonplace, and reckoning by pen overtook mercantile computation with abacus or counting board — the material qualities of these numbers-in-the-world took on new or augmented forms. Understanding of the features of written numbers and the ways in which they might be manipulated permeated the wider culture, arithmetic books with their pages of printed numbers could be found in many more homes, and mathematical concepts became available to an ever-widening range of people. Recent work by John Denniss (2009), Kathryn James (2011), Benjamin Wardhaugh (2012, 2014), and Travis Williams (2012, 2013), has explored the pedagogical, the literary, and the material characteristics of books in which mathematics was taught and learned, from Recorde onwards. As Kathryn James suggests, over the course of the sixteenth century ‘an informal vernacular popular literature of mathematics emerged’ in England (James 2011: 4); articles collected in the present issue investigate different aspects of the implications of this increasing familiarity with written numbers for literary and material culture, as written numbers became more common in both manuscript and print.

[6] Given that the popular understanding of written numbers changed over this period, how did this affect numbers in writing, both literary and less so? Looking as we are from the twenty-first century, from a world described by binary code and big data and with an emphasis on metrics, accounting, quantifying and digitising, is it possible to comprehend the strangeness and novelty of the new mathematics introduced to England in the sixteenth century? In recent years, as part of the turn in literary studies towards the history of science and perhaps inspired by our contemporary drive to quantify and to measure, scholars have become interested in the intersection of mathematics and early modern literature. Whereas twentieth-century numerological readings of Renaissance texts discussed the mystical potential of numbers, this new work investigates how practical, applied mathematics interacted with literary production. Recent studies have examined the intersection of literature with calculation and quantification (Glimp and Warren 2004); measurement and mis-measurement (Blank 2006); the zero or cipher (Ostashevsky 2004, Parker 2009); geometry and spatial reasoning (Mazzio 2004, Turner 2006); developments in algebra and arithmetic, counting and accounting, including double-entry bookkeeping (Poovey 1998, Raman 2008 and 2010, Smyth 2010, Wilson-Lee 2013, Woodbridge 2010); and financial trends more generally, in studies which move on from New Economic Criticism (see for example Korda 2002, 2009, and 2011; Sullivan 2002).

[7] What these works share, with each other and with the present collection, is a conviction that modes of numerical thought — and the ways of thinking and doing that characterized practical mathematics — influenced the literature of the early modern period. These studies insist that number work was productive for literature; the essays collected here argue too that literature, and other cultural forms, was productive for number work. From the poetry of Shakespeare and Donne, to accounting and fencing and cryptography, the essays we present here traverse the boundaries between literary and numerical writing. The papers share an epistemological concern with what is known about number and how that is reflected in the various cultural products that are their subject matter. Together, they show us that forms of literary production— conceptual and material — affected the writing of number, as well as the other way round.

[8] The shift to ciphering brought to the workings of numbers both clarity and new kinds of complication. Ephemeral numbering by gesturing hands and moving counters was pinned to the page by the new mathematics, but this was not a straightforward shift from messy body to tidy book. Puttana suggests as much in her description of the empty-headed gallant as a ‘cipher’, his value as a man limited to his ability to ‘fill up the number’, connecting the place value of numbers to their spatial and embodied dimension in the early modern imagination. The numbers on the page, a material manifestation of the abstract, were a way to comprehend the material world, but also, paradoxically, to articulate its bewildering intangibles. The material world is not always tractable and these essays, countering the anticipated orderliness of number work, explore the tensions that sometimes arise between the things that are countable and abstract numeration.

[9] Our opening article, by Lisa Wilde, examines just such complexity in the text that forms the starting-point of this issue, Robert Recorde’s The Ground of Artes (1543). Wilde employs a close reading of this book to investigate and to question some ideas about popular numeracy in the middle of the sixteenth century, and evaluates the importance of this book in the effort to shape algorism into a body of knowledge accessible to the early modern reader. The pedagogical tactics of the Ground, Wilde argues, situate it within the late-medieval scholastic tradition which places emphasis on reason as the basis of understanding. However, the text also sometimes puts reason to one side in favour of ‘practical logic’ — here is arithmetic as practical craft, related to techne rather than episteme; something corporeal, and at times openly illogical. In uncovering some of the ‘messy realities of […] popular numeracy’, Wilde allows Recorde’s text to emerge as one which is often problematic, showing up the tensions and potential antagonisms involved in the early modern adoption and practical application of Hindu-Arabic notation.

[10] Ken Mondschein’s article, on Camillo Agrippa’s Treatise on Arms (1553), takes us beyond English writing to explore the relationship between arithmetic in the abstract and its application to the physical world. This innovative fencing manual adopts, Mondschein argues, the principles of Euclidean geometry (number in space), and Aristotelian concepts of time (number in time), and applies them to the human body in action. He shows how Agrippa reflected and contributed to the vernacularisation of the mathematical conception of the world and the idea of number as the underpinning of reality. Agrippa’s application of a set of core geometrical and numerical principles to a complicated physical practice is based on experience, not theory, and acts as something of a bridge between an allegorical deployment of numbers and a scientific one. The article offers a brief overview of numerical conceptions in Italian fencing books as a context for Agrippa’s contributions and legacy and argues that these, together with the works of Agrippa’s successor Girard Thibault, were enumerations of the universe and the human operatives within it which reflected and disseminated the ‘Scientific Revolution’.

[11] In her article, Rebecca Tomlin considers the self-fashioning of early modern authors of works on numbers. She offers a material-text-based reading of James Peele’s two books on double entry book-keeping, The maner and fourme (1553) and The Pathe waye to perfectness (1569), which were among the earliest works on the subject to be published in English. Tomlin proposes that The maner and fourme is an attempt to integrate applied number work into the reformed humanist publication strategy of the King’s Printer, Richard Grafton. In The Pathe waye to perfectness Tomlin finds signs of Peele’s assertion of his authorial identity as humanist scholar. This article focuses on the title pages of Peele’s works, and considers them alongside other works printed by Grafton, to examine the ways in which these two texts position accountancy as a practice and seek to fashion the cultural capital of this particular form of number work.

[12] James Beaver’s article returns us to Robert Recorde and offers an assessment of how developments in mathematical symbolic language, and especially Recorde’s contribution to novel notational figures, including ‘=’ ‘+’ and ‘-‘, influenced John Donne’s poetics. Donne’s poetry suggests a sustained investment in quantitative language as a mode of articulation; Beaver shows that Donne’s use of the developing semiotic system of symbolic mathematics highlights the sympathies and disjunctions between this system and verbal language. He argues that quantitative language is a means of orientation for Donne, a way to organize language and perceptions, and yet, as the article explores, one that gives rise to ambiguities and anomalies. As the poet seeks to make a framework from numbers by which the abstract can be measured, number breaks down or becomes inadequate.

[13]  Stephen Deng discusses another aspect of the relationship between the new mathematics and the literary imagination, reading Shakespeare’s sonnets as an engagement with the mathematical techniques introduced to England in the sixteenth century. Deng argues that the new modes of numerical representation, and the mathematical properties that accompanied them, enabled Shakespeare to think in a varied and sustained manner through the complexity of identity, gender and sexuality. The zero, or cipher, can multiply any number by 10 simply by being added to its end, and the sonnets that seek to persuade the ‘fair youth’ to reproduce are read through this abstracted procreative power of zero. Deng shows how the cipher’s converse function, to subsume other numbers into itself through the simple operation of multiplying any number by zero, is reflected in the destructive sexuality of the ‘dark lady’ sonnets. The sonnets’ pre-occupation with the notion of identity as singular and unbroken is expressed through the newly introduced ‘broken numbers’, or fractions, and their challenge to classical notions of unity. Deng discusses these various ways in which Shakespeare used the abstracted, imaginative capacity of numbers to engage with material concerns of gender and sexuality.

[14] ‘Oh…Millions of deaths’ complains the Duke, extravagantly, as he expires in Middleton’s The Revenger’s Tragedy. Observing that quantities recur with surprising frequency in the genre of revenge tragedy, Derek Dunne draws on Middleton’s play as well as Antonio’s Revenge and Hamlet in order to quantify the reciprocal, and escalating, nature of revenge in these plays. On the one hand this can be linked to the competitive intertextuality of the genre itself, where each author tries to outdo his predecessor — the logical conclusion, perhaps, of Renaissance emulatio. But, Dunne argues, this might also point to a deeper psychology of revenge which struggles to equate life with life, and refuses to accept numerical parity.

[15] The article by Katherine Hunt considers the material qualities of numerical texts in the later seventeenth century. She investigates the numerical tables commonly found in early modern texts that were intended to be used by tradesmen, and which provided the fruits of complicated calculations at one’s fingertips. These pages full of numbers provide, Hunt argues, a material familiarity with number that was unprecedented; their structure can be compared to other varieties of printed table in the period which were used to organise and to corral knowledge. Hunt takes as her focus William Godbid and his successors, printers of mathematical and other numerical texts in late seventeenth-century London. Godbid was renowned as the best printer of high-end mathematical books, and he seems to have paid similar attention to the books he printed for merchants and tradesmen. The breadth of his output, spanning mathematical and didactic texts with profound attention to the particularities of working with number, suggests continuities — material, if not always intellectual — between texts aimed at different ends of the market, between abstract and applied number work.      

[16] The final essay completes the arc travelled by number from the mid-sixteenth to the late-seventeenth century, from exotic intellectual challenge to fashionable reading material. Katherine Ellison explores the ways of reading that cryptography manuals of the middle and late seventeenth century invited and set forth. These texts refused to confine textuality to the alphabet and promoted, Ellison argues, ciphering and deciphering as mathematic modes of reading that should be adopted in one’s everyday, and particularly domestic, life. Ellison finds these manuals to be surprisingly, and deliberately, disorderly. They are, on the one hand, concerned to portray ciphering as very learnable; on the other, they are not always concerned with conveying secret codes and secret messages, but instead often contain unsolved (or unsolvable) problems, and stress the interpretative liberty available to readers of cryptography. John Wilkins, Samuel Morland, and other authors of these books revelled in numbers: both in the quantities of ciphers that were possible, and in the use of digits to create their codes. They encouraged a familiarity with number that displays a profound understanding of Hindu-Arabic notation, of a sort which would have been barely imaginable a century earlier.

[17] Collectively, these articles point to a developing understanding of the integral place of numbers in early modern literary culture. They show how ciphers ‘filled up the number’ but also demonstrate the many ways in which number filled up the early modern page. The development of the new mathematics over the course of our period brought changes to the writing of number — from Robert Recorde’s lessons discussed in Wilde’s article, to the alphanumeric codes described in Ellison’s — which brought about profound changes to the relationships between letters and numbers in the popular imagination. We close with an afterword by Carla Mazzio which asks whether we have, in our own moment, undergone a comparable shift in our collective understanding of the relationship between numbers and writing. As literary study begins to explore the implications of our developing ability to grapple with big data, will writing be read using numbers rather than letters? There is considerable disquiet among literary scholars at such a proposition and, as Mazzio suggests, the essays collected here argue in favour of the continuing value of ‘zooming in’, of microscopic attention to written texts as well as the wide-angled possibilities of the digital humanities. Mazzio’s afterword — and her own forthcoming book, Trouble with Numbers: The Drama of Mathematics in the Age of Shakespeare (under contract with University of Chicago Press) which examines the aesthetically, dramatically, and emotionally productive dimensions of troubled forms of computation, calculation, and processes of thinking about (as well as through) mathematics in the early modern period— emphasises the complexity of writing with and of numbers, and suggests some exciting directions for the kind of work we propose in this issue. The meeting of numbers and writing continues to be a fruitful point of exchange; as the essays here show, it is at moments in which the contact between the two is marked by upheaval and flux that new modes of thinking emerge. Perhaps, then, we are entering a new stage of the relationship between numbers and writing, an analogue to Recorde’s new world of ‘arithmetycke by the penne’. The writing and the reading of number — for Recorde and his successors, but also for us — is unruly, but generative; written numbers prove excitingly disruptive when the merest cipher can ‘fill up’ and produce something from nothing.

[18] Many people have helped this special issue to come about. For their advice and support over the course of this project we would like to say thank you to Steven Connor, Vanessa Harding, Adam Smyth, and Sue Wiseman. We are very grateful to everyone who presented at the conference at Birkbeck in 2013, particularly the keynote speakers: Stephen Clucas, Natasha Glaisyer, and Emma Smith. For their help in the practicalities of planning and running the conference, our thanks go to Catherine Catrix, Sue Jones, Gillian Knight, Simon Smith, and Jackie Watson; we are extremely grateful to Lina Hakim for designing the conference poster and other materials. We would also like to recognise the generous sponsors of the conference: the Society for Renaissance Studies, ICAEW’s Charitable Trusts, the Royal Historical Society, and Birkbeck, University of London. We have found editing this special edition to be an exemplary process of scholarly dialogue, patience, and generosity and would like to thank the authors of all the articles collected here, and the anonymous peer reviewers for their incisive and constructive comments. We would especially like to thank Carla Mazzio for her characteristically perceptive afterword and for her verve, her encouragement and her insight over the course of this project. Finally, we would like to thank the editors of the Journal of the Northern Renaissance for their support and good humour throughout.

Katherine Hunt is a Career Development Fellow in English Literature at The Queen’s College, University of Oxford. Rebecca Tomlin is currently completing her doctoral thesis at Birkbeck, University of London.


Blank, Paula. 2006. Shakespeare and the Mismeasure of Renaissance Man (Ithaca: Cornell University Press)

Denniss, John. 2009. ‘Learning arithmetic: textbooks and their uses in England 1500-1900, The Oxford Handbook of the History of Mathematics ed. Eleanor Robson and Jacqueline Stedall (Oxford: Oxford University Press)

Feingold, Mordechai. The Mathematician’s Apprenticeship: Science, Universities, and Society in England, 1560–1640 (Cambridge: Cambridge University Press, 1984)

Ford, John, 2011. ’Tis Pity She’s A Whore, ed. by Sonia Massai. London: Methuen (Arden Early Modern Drama)

Glimp, David and Michelle R. Warren. eds. 2004. Arts of Calculation: Numerical Thought in Early Modern Europe (Basingstoke and New York: Palgrave Macmillan)

James, Kathryn. ‘Reading Numbers in Early Modern England’, BSHM Bulletin: Journal of the British Society for the History of Mathematics 26:1 (2011), 1-16.

Korda, Natasha. 2002. Shakespeare’s Domestic Economies: Gender and Property in Early Modern England (Philadelphia: University of Pennsylvania Press)

_____.  2011. Labors Lost: Women’s Work and the Early Modern English Stage (Philadelphia: University of Pennsylvania Press)

Mazzio, Carla. 2004. in David Glimp and Michelle R. Warren, eds, Arts of Calculation: Numerical Thought in Early Modern Europe (Basingstoke and New York: Palgrave Macmillan), pp. 39-65

Ostashevsky, Eugene. 2004. ‘Crooked Figures: Zero and Hindu-Arabic Notation in Shakespeare’s Henry V’, in David Glimp and Michelle R. Warren, eds, Arts of Calculation: Numerical Thought in Early Modern Europe (Basingstoke and New York: Palgrave Macmillan), pp. 205-228

Parker, Patricia. 2009. ‘Cassio, Cash, and the “Infidel 0”: Arithmetic, Double-entry Bookkeeping, and Othello’s Unfaithful Accounts’, in A Companion to the Global Renaissance, ed. by Jyotsna G. Singh (Oxford: Blackwell).

Poovey, Mary. 1998. A History of the Modern Fact. (Chicago: Chicago University Press)

Raman, Shankar. 2008. ‘Death by Numbers: Counting and Accounting in The Winter’s Tale’, in Alternative Shakespeares 3, ed. by Diana E. Henderson (London and New York: Routledge), pp. 158-180.

_____. 2010. ‘Specifying Unknown Things: The Algebra of The Merchant of Venice’, in Making Publics in Early Modern Europe, ed. by Bronwen Wilson and Paul Yachnin (London and New York: Routledge), pp. 212-231.

Recorde, Robert. 1543. The grou[n]d of artes (London: R. Wolfe)

_____.  1699. Arithmetick; or, The Ground of Arts, ed. by Edward Hatton. (London: Printed by J.H. for Charles Harper and William Freeman, 1699).

Smyth, Adam. 2010. Autobiography in Early Modern England. (Cambridge: Cambridge University Press)

Sullivan, Ceri. 2002. The Rhetoric of Credit: Merchants in Early Modern Writing (Madison/London: Associated University Presses)

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Wardhaugh, Benjamin. 2012. Poor Robin’s Prophecies: A curious Almanac, and the everyday mathematics of Georgian Britain (Oxford: Oxford University Press)

_____. 2014. ‘Consuming Mathematics: John Ward’s Young Mathematician’s Guide (1707) and its owners’, Journal for Eighteenth-Century Studies. 

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_____. 2013a. ‘Procrustean Marxism and Subjective Rigor: Early Modern Arithmetic and Its Readers’, in ‘Raw Data’ Is an Oxymoron ed. by Lisa Gitelman (Cambridge, Mass.: MIT Press), pp. 41-59.

_____. 2013b. ‘The Dialogue of Early Modern Mathematical Subjectivity’, Configurations 21:1, 53-84.

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‘Whiche elles shuld farre excelle mans mynde’: Numerical Reason in Robert Recorde’s Ground of Artes (1543)

‘Whiche elles shuld farre excelle mans mynde’: Numerical Reason in Robert Recorde’s Ground of Artes (1543)

Lisa Wilde

[1] In a much-discussed article on the nature of human thought, philosopher and cognitive scientist Tim Van Gelder sets out to unsettle the ‘contemporary orthodoxy […] that [cognition] is computation: the mind is a special kind of computer, and cognitive processes are the rule-governed manipulation of internal symbolic representations’, arguing instead for an organic and ‘dynamical’ interplay between ‘the brain, the body and the environment’ as common sites of cognition (Van Gelder 1995: 345, 379). I will have occasion to return, in passing, to this question of dynamical cognition (and in particular to the theories of ‘distributed cognition’ that can be derived from such claims); but for now, I should primarily like in this essay to examine the unspoken assumption behind Van Gelder’s and similar investigations: the notion that computation is itself unproblematic as a form of cognition, and specifically that it can be wholly defined by ‘the rule-governed manipulation of internal symbolic representations’. In particular, I will consider to what extent this can be said to be true for the ‘average’ English user of numbers in the Renaissance, especially during the slow rise in popular numeracy that followed upon the increasing penetrance of Hindu-Arabic computational practices, as set forth for the first time in readily available vernacular works of instruction. England’s society-wide transition away from clumsy Roman forms of number and toward the novel practices of Hindu-Arabic numeration occupied much of the hundred-year span from the middle decades of the sixteenth century to the closing years of the seventeenth; but while the conversion proceeded at different rates depending on locality and on social echelon (with mercantile classes, for instance, being considerably quicker to attain basic competency in the new number forms), its result everywhere was to create a new presence for number in the popular mindset, one where widely differing levels of actual individual competence in arithmetic (a 29-year-old Pepys, for instance, describes himself in 1662 seeking out tutoring in basic multiplication) nonetheless coexisted with a growing awareness of practical computation as a form of reasoning and a fact of life (Thomas 1987: 112; Denniss 2009: 453). The focus of this paper, then, will be on the messy realities of this strictly popular numeracy, in possible contradistinction to the more coherent practice of numerical reasoning among specialists and experts. What did the early modern English citizen intend when carrying out processes of elementary calculation? And what did (s)he understand when accepting computational ‘facts’ derived by others?

[2] As Louis Karpinski notes, the sixteenth-century growth of popular numeracy in England was propelled in large part by a flood of newly-available printed works offering vernacular instruction in Hindu-Arabic numeration and computation (a set of practices referred to collectively as ‘algorism’, via a corruption of the name of ninth-century Persian scholar Muhammad ibn Mūsā al-Khwārizmī, whose recovered mathematics texts were central to the early modern revival of practical arithmetic) (Karpinski 1925; Denniss 2009: 448). While other forms of more traditional learning and teaching certainly had a role to play in the transmission of arithmetical knowledge, then, the sheer volume of vernacular publishing on arithmetic — forty-five distinct works issued in the last half of the sixteenth century alone — suggests that autodidacticism via the consumption of cheap ‘algorisms’ was sufficiently widespread to constitute an important force in the ongoing shaping of the popular quantitative imagination, particularly among more casual users of number (Karpinski 1925: 70; Denniss 2009: 453; Denniss and Smith, 2012: 26). I should like, in consequence, to explore what we can learn about English numerical sensibility in the process of acquiring computational skills both through and with these early works of arithmetic instruction; and in doing so, I will focus here on the foundational work of this genre, Robert Recorde’s The Ground of Artes, Teachyng the Worke and Practise of Arithmetike (1543).[1]

[3] The Ground attracts our attention partly through the sheer magnitude of its impact: while not quite the first published English-language arithmetic, it was easily the most popular and influential, running through fifteen editions before 1600, and a total of forty-seven by the time of its last printing in 1699 (Howson 1982: 13; Easton 1967: 515). More critical for an investigation of numerical mentality, however, is the remarkable subtlety of psychological insight that shapes the manual’s mathematical pedagogy. The work of a sometime arithmetic tutor (as well as physician and later comptroller to the Royal Mint), Recorde’s Ground is framed as a set of lively dialogues between a learned ‘Master’ and his eager ‘Scholer’ (Williams 2011: 5). But while for the most part both the raw material and the broader structure of the Ground are derived from earlier sources — notably Sacrobosco’s De Arte Numerandi, by then a standard text at English universities, and Gemma Frisius’s Arithmeticae practicae methodus facilis (1539) — Recorde’s English version is remarkable for the care with which it elaborates the bare framework of existing Latin-language arithmetical theory into a thoughtful and richly detailed instruction in computational practice (Williams 2011: 87–90). His Master is an attentive and skillful teacher who carefully tailors his lessons to his student’s capacity (‘by cause it is somewhat harde for you, yet I wyll let it passé for a whyle’) and motivation (‘best it is to omitted no tyme, lest some other passyone coole this great heate’) (6, 102). Likewise, conscious of the potential for ‘many forms of working’ to ‘trouble [the] mynd’ of a beginner, the Master sifts out only the most generally useful of the currently-available calculation techniques, and carefully arranges his material so that the student, following topics ‘in that order as I did reherse’, will naturally ‘learne them spedely and well’ (7). Indeed, Recorde’s prefatory discussion of educational practice not only recommends general attention to a pupil’s intellectual needs, but actually presents ‘right teaching’ as a process actively shaped by the agency of the student: he has ‘wryten in ye fourme of a dyaloge’, he explains, ‘bycause I judge that to be the easiest waye of enstruction, when the scholer may aske every doubte orderly, and ye mayster may answere to his question plainly’ ([vi]). In this commitment to posing, as well as answering, the doubtful student’s questions, then, the text of the Ground frames itself, in some sense, as an ‘orderly’ image of the learning mind itself. While the lively intimacy of the pupil-interlocutor relationship was characteristically used — for instance, by Erasmus — to ‘allure’ and engage student readers of instructional dialogues, here the ventriloquism of the student seems to have an epistemological, as well as a merely emotive or persuasive, value: the vocal presence of the student-persona gives concrete shape to the evolving structures of mathematical knowledge that the text aims to create (Erasmus 1900: 276).

[4] Within the dialogue itself, this commitment to a pedagogy of response is enshrined in the Master’s promise to demonstrate the basis of algorism in ‘reason’: before he even begins his lesson, he carefully assures the Scoler that he ‘desyre[s] no […] credence […] except I shewe reason’ for the principles of calculation he imparts, since it is ‘to moch, and mete for no man, to be beleued in al thynges without shewynge of reason’ (6). While the grounding of mathematical principle in demonstrative reason has been a familiar part of the disciplinary ethos from classical times onward, Recorde’s notions of arithmetical reason in practice prove considerably more pliable than our post-enlightenment notions of mathematical logic might lead us to suspect. The Ground, indeed, opens its prefatory discussion with two vivid images of arithmetical cognition, neither one of which bears much relation to formal logic: Recorde describes human numeracy first as a category of descriptive language (‘yf nombre be lackynge, it maketh men dumme, so that to most questions, they must answere mum’), and subsequently as the intuitive ability to perceive meaningful quantitative distinctions in real life (the capability lacking, for instance, in a mother hen who does not perceive that her brood has diminished from 4 to 3 chicks) (2,[v]). As the text moves on from the introduction, through the basics of positional notation in a Hindu-Arabic system (here, an astonishingly laborious process of figure-by-figure transcoding involving multiple reference charts) to the fundamental operations of addition, subtraction, multiplication and division, and onward to techniques of practical arithmetic like reduction, proportion and alligation, we catch glimpses of the ‘shewynge of reason’ in a rich and varied tapestry of explanatory metaphor: figures vs. letters as Frenchmen vs. Englishmen; computational direction oriented to hands and fingers; computational units as parcels, bundles, and rooms; and above all computation as a process of spatialized motion, upward, downward, and sideways through almost every corner of the page (9-10, 15, 39b-40b).

[5] In his generous understanding of responsive pedagogical ‘reason’, Recorde operates squarely in continuity with the wider traditions of late-medieval scholastic instruction. Indeed, the standard early fifteenth-century school-text on dialectic, Petrus Hispanus’s Summulae Logicales, had defined ratio as that by which argument ‘creates faith in doubtful matters’ (rei dubiae faciens fidem), an essentially rhetorical view that the Master’s speech about ‘shewing reasons’ seems merely to recast, with an instructional twist that extends ‘fidem’ from mere conviction to comprehension (Hispanus 1947: 44). (Elsewhere in the Ground, the Master’s definition of (instructional) ‘reason’ verges still closer to Hispania’s, as when he promises the Scoler to ‘declare thynges unto you so playnely that you shall not need to doubte’ (Recorde 1543:38)). Even in the Summulae, this sensitivity to the phenomenal nature both of dubiety and of understanding draws the strictly technical meaning of ‘ratio’ into an implicitly dialogic relation with a long list of additional senses for the term: Hispanus notes that ‘reason’ can refer to a mental ability, to a spoken explanation for something, or even to the shape that forms a material object, as iron is cast into the form of a knife (Hispanus 1990: 49, 1947: 44). ‘Reason’ in its colloquial totality, then, seems less akin to the recognisable classical edifice of pure, syllogistic logic than to what we might call a cognitive ‘model’: a structure of knowledge, sensory or otherwise, that makes something make sense, and against which new information can be judged either true or not-true, consistent or anomalous. As Walter Ong notes, the recognition that this space of meaning could be filled by a number of different constructs — not only demonstrative or syllogistic logic, but also narrative structure, geometric and other visual forms, aesthetic values, or intuitions about practical causality — is characteristic of early fifteenth-century works on logic, many of which even attempted to concretise the laws of symbolic logic in the form of diagrams or storybook-pictures, forms of meaning which were more easily assimilated by students (Ong 1958: 74–91). Given this rhetorical focus on the experience of conviction or understanding, a promise to ‘shewe reason’ before requiring ‘credence’ effectively shades into a statement about meaningfulness: the instructor guarantees that the knowledge he conveys will ‘make sense’ within one (or more) of the cognitive structures by which the student makes meaning of his universe, and is thus freed, should necessity arise, to court students’ comprehension by simultaneously invoking as many separate models of reason as may seem apposite.

[6] It is also the case, though, that in the Ground the more artful varieties of ‘reasoned’ elucidation frequently seem to operate in opposition to classic forms of syllogistic demonstration. Mere pages after proposing ‘reason’ as the essential prerequisite to arithmetical understanding, for instance, the Master seemingly reverses that characterization:

SCHOLER: But I do not se the reason of this.

MASTER: No, no more do you of many thynges els, but hereafter wyll I shewe you the reasons of all Arithmeticcall operations, for this I juge to be ye best trade of teaching, fyrst by summe brefe preceptes to enstructe a learner sumwhat in the use of the arte, before he learne ye reasons of the arte, and then maye you afterwarde more soner make hym to perceaue the reasons: for harde it is for to occupye a yonge learned wytte wt both the arte and the reasons of it all at ones. (32)

While Recorde may appear to renege here on his earlier promise to show reason before expecting a student’s ‘credence’, the apparent contradiction between the two passages turns on a conflation of two separate sorts of ‘reason’, which the actual instructional context helpfully teases apart. The Scoler, at the opening of the passage, is objecting because he ‘does not see the reason of ‘a complex double-checking procedure that verifies the results of currency addition via a bizarrely modified form of ‘casting out nines’. As it turns out, the Master never does fulfill his promise to ‘shewe you the reasons’ of the procedure — and understandably so; since the checking method in question relies on points of modular arithmetic that would not be explicitly formulated for two hundred years, it is questionable whether even Recorde himself would have had a strong grasp of its theoretical basis. Seemingly, though, not all forms of rational modelling are alien to the immediate pedagogical purpose here, since the Master’s indefinite postponement of mathematical ‘reasons’ is actually followed directly by an extended passage of practical logic:

SCHOLER: Yet at the least, I praye you show me why did you write your no[m]bre that remayneth (after you had withdrawen al the nynes) at the end of a lyne, for I sawe no reason why yt dyd serve?

MASTER: Dyd you ever marke a carpenter when he wrought?

SCHOLER: Yea, many tymes.

MASTER: And haue you not sene hym when he hath taken measure of a borde, that he hath pricked it, and hath with a twyche of his hande drawen a lyne from the pricke that he made?

SCHOLER: Yes I haue marked that, and haue sene some make .iii. or .iiii. lynes by the pricke, and some also haue I seen make a crosse by it, but that I perceaued was for the easy finding of theyr pricke.

MASTER: and euen so is this lyne, for the easy fyndynge of your re[m]ayner… (34)

The key distinction, seemingly, is between systematic or theoretical ‘reasons’, which are to be only ‘afterwards […] perceav[ed]’, and purely local or practical ones fit for immediate dissemination: the algorismic model being inculcated in the Ground foregrounds the internal syntactic logic of the computational process (‘a line here makes it easy to find the remainder; make sure the term on one side balances the other’), while a broader semantic understanding of the system as a whole (why cast out nines and not eights? how do the remainders come out the same?) is actively subordinated to the immediate demands of present practice. This view is consistent with the text’s wider interest in arithmetic as a practical craft, more techne than episteme; but it also, I would argue, speaks to the conceptual distance that divides Hindu-Arabic calculation from existing structural models of number and quantity in the early-modern mathematical mentality. Not only is the wider theory of algorism seemingly inessential to the working understanding the text aims to create, but the Master’s words even suggest a certain conceptual antagonism between the two so that it is felt to be ‘harde’ ‘to occupye [the] witte’ with arithmetical theory and practice simultaneously. As a result, the same explanatory gestures that strengthen a learner’s syntax-level comprehension (by linking computational technique with the familiar experiential context of practical mensuration) turn out to entail a sort of renunciation of meaning in a wider structural sense, as the Scoler turns away from questions of mechanism to buckle down to the immediate business of learning to reckon.

[7] To be sure, the Ground’s official line is that this distance is a temporary matter only, pending a subsequent lesson at ‘another more conveniente tyme’ (never realised) in which all reasons will be shown (36, 73). As in the Master’s hand-waving over the ‘reasons’ for casting out nines, though, it is clear that this postponement slides almost inevitably into a sort of uneasy agnosticism regarding the underlying structural basis for the helpful results algorism yields. Instead, chary of reasoning about algorism in a more global fashion, the Ground turns eagerly to rationalisation at the more local level of practical routine; indeed, one of the work’s more notable educational achievements is its success in weaving the capricious twists and turns of computation into a larger fabric of seeming reasonableness and purpose. Even the instructional meta-structure of the lesson is arranged, carefully, to fit with the learner’s natural thought-process: in the passage quoted above, for instance, the Scoler’s stream-of-consciousness musings on carpenter’s marks— sometimes mere pricks, but sometimes also ‘.iii. or .iiii. lynes’, or ‘crosses’ — are made into a transition-point to lead into a discussion of alternative double-checking protocols that likewise rely on crosses and stacked lines.

[8] As we have seen in the instance of the carpenter’s ‘pricke’, one important source of meaning in the arithmetical world of the Ground is the active association of mathematical computation with the idea of a craft or technical operation — meaning that otherwise-mysterious features of the system can be explained in strictly functional terms, as logically tending to make calculations ‘more reddye’, ‘more certayne’, or less ‘deceavable […] yf a mans memory be other dulle, other troubled’ (54, 32). The influence of this model appears, too, in the consistent tendency of both Master and Scoler to refer to successful calculations as ‘well done’ (vs. our present-day right or correct), as though the computation itself were literally something ‘worked’ by the arithmetician, chiefly accountable to standards of good craftsmanship rather than adherence to any objective external necessity.

[9] If numbers gain meaning partly from the agency and artistry of human calculators, though, they also seem, in many parts of the Ground, to act in allegiance to internal rules of their own, repetitive patterns of expression and motion that are to be learned and followed rather than controlled. Chief among these principles is a sort of spatialised sense of hierarchy and correspondence, investing the ordering of written marks on the page with a logical actuality in their own right — such that like numbers are felt (almost a priori ) to belong with like, greater above lesser, many-placed to the left of fewer placed, and so forth. This is arithmetical reason (to return to Petrus Hispanus’s taxonomy in the Summulae Logicales) construed as material form — and although it has some links to real mathematical exigency, as cultivated in the Ground it ultimately verges closer to a purely aesthetic sense of balance or propriety. We meet it first, for instance, at the opening of the section on addition, when the Scoler proposes to add ‘.ii. droves of cattell’, the first containing ‘848 shepe’ and the second ‘186 other beastes’, and begins the calculation by writing:

[+] 186

The Master first reproves him for ‘goyng about to adde together .ii. summes of sondry thynges’ — here a somewhat finical, if legitimate objection, since both sheep and beasts have after all previously been grouped under the common heading of ‘cattell’, but nonetheless instructionally useful in paving the way to extend the same principle to the arithmetic problem itself. In fact, as the Master points out, the Scoler has been ‘twyse deceaued’ in his calculation: not only does the addition of unlike objects produce a ‘confused su[m]me’ at best, but the ‘wrytinge 14 […] under 6 & 8’ is likewise ‘unpossyble: for howe can two figures of two places, be writen under one fygure, and one place?’ (18-19).

[10] We see here at work the rhetorical slippage between causal or explanatory and merely formal ‘reason’: there may indeed be some logical justification for objecting to the addition of sheep to beasts, but the act of writing a ‘14’ in the ones column (which the Scholar has demonstrably just performed) is not intrinsically ‘unpossyble’ in the same thoroughgoing sense. Instead, the Master’s explanation draws in the language of natural causality to describe purely artificial or conventional norms.  While a deeper semantic explanation does exist for the stricture in question (i.e., that since Arabic-number addition proceeds piecewise, extra units of ten from the addends still remain to be incorporated even after the units of one have been summed), the Master opts not to cite it here. Rather, he skirts the tricky issue of positional reasoning altogether, via a local appeal to a set of phantom ‘possibilities’, a numerical decorum that apparently dictates that only like entities stack together in the same written column. When next we meet this principle, in a discussion of currency-addition, the language places a still stronger emphasis on the simple, abstract pairing of like with like:

MASTER: Yf your denominations be poundes, shyllyns, & pennes, wryte poundes under po[ndes], shyllynges under shyllynges, and pennys under pe[n]nys: And not shyllynges under pennys, nor pe[n]nys under poundes. (26)

This is still not unreasonable from a common-sense point of view, but the Scoler’s response— agreeing that indeed ‘it were agaynste reason so to confou[n]de su[m]mes’, but also worrying that ‘yet yf you had not spoken of it, peraduenture I shuld haue ben deceaued in it’ — makes it clear that the proposition is read as stating a formal rule, rather than a self-evident proposition about the absurdity of totalling different coins as the same unit. Evidence from the text, indeed, suggests that it is also internalised as such, for example when the Scoler proposes to continue the half-complete multiplication

[x] 29

by ‘writ[ing] the 4 [of 9*6=54] under the 3, and the 5 under the next place (as reason wylleth me) thus’:

[x] 29

This is a computational move that adheres admirably to ‘reason’ in the sense of the orderly spatial sorting of corresponding figures, but that actually produces a sort of nonsense (the random ‘536’ that has not resulted from any multiplicative operation; the lone floating 4) as regards the arithmetical content of the problem itself. To be fair, the Master seems equally willing to allow numerical decorum to dictate practice even in the absence of underlying arithmetical reality, as when he judges it ‘the beste waye’ that ‘ever […] ye greatest nombre be written hyghest’, even though it is not strictly ‘necessary’ for the calculation (22). Through repeated appeals along these lines, computational practice in the Ground verges closer to something resembling a numerical art of dispositio — per contemporary rhetoric-manuals, the art of ‘bestowyng matter, and placing it in good order,’ according to ‘what is meete for every parte’—and written number becomes itself the concrete ‘matter’ of calculation, rather than a mere shorthand tracking some separate arithmetical reality (Wilson 1553:84).  The conventional ‘bestowyng’ of numerical matter thus becomes a self-evident part of the reason of well-wrought calculation, quite independently of any links to the wider quantitative reality being assayed. Indeed, errata to subsequent editions of the Ground meticulously track and regulate the spatial arrangement of its figures, specifying by line and page (vs. computational function) where a 9-1/4 should be flipped to make 6-1/4, or where ‘a pricke’ should be put in a figure ‘to sever him 1/3 asunder’ (Recorde 1582: sig. [Yy.8]).[2]

[11] Within this framework of computational disposition, columns, rows and lines, the visual markers of numerical decorum, quickly come to act as a sort of dynamic framework of meaning, through which numbers move, divide and recombine with all the ‘orderly’ arbitrariness of a quadrille. Here these figures’ rhetorical function in supplying instructional reason (quod rei dubiae faciens fidem) is aided immeasurably by the Ground’s textual layout, which consistently supplements in-text narratives of calculations with marginal figures showing the corresponding working at each stage. While earlier arithmetics — notably Fibonacci’s Liber Abaci (1202) — had also included some worked problems, the Ground is exceptional in the thoroughness and extent of its illustrations; within the dialogue, their very ubiquity gives them an oddly ambiguous relation to the mathematical content itself (Fibonacci 2002). Rather than merely illustrating calculation procedures imagined as taking place elsewhere, such figures come, at different times, to stand in for the fictional space of the staged ‘lesson’; for the fictional writing-surface being marked in the course of that lesson; for the thought-space of the Scoler’s calculations; for the concrete objects (including sheep and beasts) being quantified; and, in the end, for a hybrid amalgam of all of these that becomes the natural habitat of arithmetical operation itself.

Fig. 1. 'Examples of addition’, with ‘profes’, The Ground  of Artes p. 37.15

Figure 1: ‘Examples of addition’, with ‘profes’, The Ground of Artes p. 37.15

[12] Within this pattern, the succession of stereotyped computational forms on the page achieves, over time, a sort of determinative force, helping to build up through repetition a prescriptive visual sense of how a particular sort of problem should look. Arithmetical adherence to these norms is almost universally described as the ‘dewe workyng’ of sums ‘dewly set’, just as Arabic number itself was defined at the Ground’s outset as ‘conceau[ing number] by fygures and places dewe’. In fact, the notion of computational figures as observing a sort of duty — the term evoking ideas of rule or decorum, but also claiming (through the connection with devoir) a normative connection to some wider quantitative reality — offers a surprisingly apt description of the signifying status of such forms (38). The positioning of numbers on the page and within written figures is, as we have seen, partly a representation of their real arithmetical properties, but in the Ground that positioning seems also to be felt as possessing some power to direct the operation itself so that formal principles of balance, antithesis, correspondence, symmetry and so forth achieve status as valid forms of mathematical causation in themselves. The manual’s back-checking procedure for verifying additive results, for instance, is explained wholly as a matter of visual symmetry: the end-results of various remainder-calculating formulae are to be written at opposite poles of the various bars in a cross-like figure (see fig. 1), at which point the Scoler must ‘consyder euery nomber, comparynge it to the nomber that is agaynst it: and bycause I fynde them to be euer one lyke his matche, I knowe that I haue well done’ (36). At the end of the section, several more such balanced figures are offered as ‘profes’ of the accompanying ‘Examples of addition’, the visual symmetry alone apparently invested with a compelling logical force. Such figural logic frequently seems capable of guiding the procedures of algorism, as well as assesssing its results — as in the section on multiplication, for instance, where the wider ‘reason’ the Master cites has become a simple question of abstract spatial positioning:

MASTER: Marke fyrst the ordre of the places in this fygure, and so shall you perceaue the reason of getherynge them into a su[m]me. The slope barres do parte the places, so that the fyrst place is the loweste corner (in all suche fygures) of the nethermost square, and all the halfe squares betweene ye barre, and the next standeth for the seconde place, and so the roume betwene that and the nexte barre is the thyrd place, and so forth. (58; see fig. 2)

Fig. 2 Final working of a "checkerboard-style" multiplication procedure, The Ground of Artes p. 58.

Figure 2: Final working of a “checkerboard-style” multiplication procedure, The Ground of Artes p. 58.

It is perhaps significant that Recorde’s most enduring contribution to conventions of mathematical notation is the equals sign (=), a symbol that similarly uses visual forms to suggest simultaneously a fixed mathematical reality and an imperative towards a particular computational action. The perennial tendency among schoolchildren to confuse the operational and the relational significances of the sign, what it does versus what it means — a subject of much concern, and a good deal of discussion, among present-day educational theorists — seems, in this light, less an unaccountable error and more like an instance of unconscious cognitive archaeology, in which mathematical naiveté ends up resurrecting an originary mode of understanding that has been lost in the course of subsequent centuries’ ‘corrections’(Capraro et al., 2011)

[13] This tendency to conflate representation with production (or, in Hispanus’s terms, reason as form with reason as causal explanation) becomes increasingly critical in cognitive terms as we turn to the third major source of syntactic ‘reason’ in the Ground. With intermediate stages of the computational process given concrete and highly spatialised embodiment via arithmetical figures, the transitions between those states are naturally envisioned in terms of motion; and Recorde, essentially in a position to pioneer a formal language of arithmetical operation in English, turns not unnaturally to the vocabulary of concrete physical manipulation to describe the computation process. Thus, figures are ‘gethered’ and ‘putte together’ with one another, ‘taken’, ‘bated’ and ‘withdrawn’ from larger sums, and even ‘held’, ‘kepte’ or ‘reseru[ed] in [the] mynde’ of the calculator; in the Ground, too, is formalised the still-popular figure by which in subtraction an extra 10 is said to be ‘borowyd’ from an adjacent digit of the minuend (though it does not originate from this book) (24, 38, 22, [35b]). Particular figures are also consistently identified by their location within the written frame of the calculation (as ‘the nether fygures’; the number ‘at the foote of the crosse’; the ‘ouernumbre’; or ‘the laste figure nexte ye lefte hande’), instead of by arithmetical function or quantitative identity (62). The Master is generally careful to flag and define technical terms (mostly, like digit, article and multiplicator, legacies of Boethian arithmetic) that are non-spatial, even going so far as to explain, painstakingly, that in English ‘the nomber, by whyche multyplycation is made […] is […] allways put before this worde, tymes’ (50, my italics). By contrast, spatial and manual terms like those we have cited here generally pass without comment, as though the literal signification of a phrase like ‘the abiected penes’ were indistinguishable — or perhaps simply not worth distinguishing — from its technical meaning within the computational process (33). Recorde is, in effect, embodying the calculation process, less a matter of indulging in metaphor than of actively creating a cognitive model in which the learner’s confidence in computation is strengthened via association with existing spatial and material knowledge. With such a model in place, it becomes intuitively plausible that the focus of calculation should proceed ‘orderly’ along in a row, that a sum should accumulate down a column and be ‘set’ at the bottom, or that part of a number should be split off and written down, while the remaining part bounces back up to lodge elsewhere in the figure.

[14] In working to shape algorism into a body of knowledge accessible to the early modern reader, the Ground thus draws on three major categories of ‘reason’: ideas of practical craftsmanship, of spatial or formal balance, and of the concrete physics of bodies and motion. It is important to recognise that none of these forms of meaning would, presumably, seem novel to the work’s audience: like all mathematical instruction, the manual’s explanations work not by creating or introducing arithmetical understanding ex nihilo, but by extending and modifying forms of ‘reason’ already long ago internalised by the learner. Since this essay began with an argument for embodied cognition, it is worth noting that the effect of the whole is generally to provide a historical case-study in what scientists of situated cognition George Lakhoff and Mark Johnson call the ‘theory of primary metaphor’: the notion that more complex forms of conceptual modeling are built up molecularly from ‘minimal… metaphorical parts’, themselves putatively formed by conflations made during childhood between ‘experiences and judgments […] and sensorimotor experiences’ (Lakoff and Johnson 1999: 46). In this case, the fluid motion of numerical action from printed page to mental space (‘reseru[ed]… in mynde’) and back again would also seem to offer intriguing evidence toward one more radical extension of situated-cognition theories like Lakhoff’s and Van Gelder’s: the hypothesis of the ‘extended mind’, which posits that cognitive processing like that at work in the Ground may extend itself functionally beyond the corporeal bounds of the thinking subject, making computational thought-work something that really does take place both across and through the spaces of brain, body and written page (Van Gelder 1995: 380–81).

[15] For our purposes, however, the emerging reality of a concretely embodied, or even concretely extracorporeal, practice of computational thinking is equally interesting for its effects on the wider popular mentality of arithmetic more generally: that is, of the newly-numerate individual’s experience of employing and accepting the conclusions of arithmetic. The dance-like logic of ordered computational motion may here be intended chiefly as a teaching device, but the Scholer’s errors — again, an important part of the Ground’s exposition — mark these forms of alternative numerical ‘reason’ as different in important ways from mere temporary pedagogical crutches. The Scoler’s enthusiastic participation in a rhythmic additive-carrying algorithm, for instance — digit on paper, article ‘kepte in your mynde’ to sum with the next place — meets with an unexpected check:

SCHOLER: Then muste I adde 6 to 8 whiche maketh 14 […] therfore must I take the dyget, whiche is 4, and wryte it under 6 and 8, kepynge the article 1 in my mynde, thus. Then do I come to the second figures […] that maketh 13, of whiche nomber I write the dyget 3 under 8 and 4, & kepe ye article in my mynde, thus. Then come I to the thyrde figures […] [which] maketh 10 […] Then of 10 I wryte the cyphar under 1 and 8, and kepe ye article in my minde.

848 848 848
[+] 186 [+] 186 [+] 186
4 34


MASTER: What nede that, seynge there foloweth no more figures?

SCHOLER: Syr I hadde forgotten, but I wyll remember better hereafter. (10)

The Scoler here deserves more sympathy than blame, since the exposition preceding this passage does indeed present additive carrying as a bouncy, repetitive back-and-forth business: digit on paper, article ‘kepte in your mynde’ to sum with the next place, repeat, repeat, repeat. Indeed, both his calculation and the Master’s correction (entirely technical, we note) register arithmetical truth as something that is less perceived than it is performed, via a series of physical moves whose operational momentum sweeps the Scholer past the bounds even of the calculation itself. We see how easily numbers and, more importantly, the quantitative information they encode, can settle back to become mere undifferentiated material for a strictly performative reckoning, a cognitive process grounded not (as Van Gelder would have it) in transparently rational ‘manipulation of internal symbolic representations’ but in something akin to a rhetoric or even a poetics of figure, where considerations of form, balance, ornament and decorum jostle alongside strict accuracy to determine the ‘dewe workyng’ of any given sum.

[16] As in any investigation of collective mindset or intentionality, of course, it is important to be realistic about the limits of generalization from particular data. The peculiar computational orientation that we have described is likely to have persisted only in relatively casual early-modern users of arithmetic; certainly we would not expect to find such attitudes in groups for whom more sustained encounter with number had had the chance to build a hard-won intimacy of numerical understanding. Nonetheless, for a general population only slowly moving toward numeracy, members of the former class — poised, like Pepys, somewhere past complete innumeracy but well shy of anything so abstruse as multiplication, and possessing a ‘sense’ of right reckoning but possibly only a hazy understanding of the real basis for computational problem-solving — may have been sufficiently numerous to substantially affect the way arithmetical thinking was produced and received in public settings. I would like to close with a set of computational examples from a later text, issued a full four decades into the Tudor revolution in Hindu-Arabic numerical practice that had been touched off by the appearance of books like Recorde’s. In A discoverie of sundrie errours and faults daily committed by landmeaters (1582), London surveyor Edward Worsop offers a sustained and self-conscious critique of practical arithmetic’s potential for corruption and misuse. Written, like the Ground itself, in dialogue form, the pamphlet begins in Recorde’s own vein with an extended scene of friendly quiz-style interlocution, in which an authorial stand-in, one ‘Worsop a Surveyor’, tests two less-expert companions (Messrs ‘Peter. Jhonson a Clothier’ and ‘Watkins. Steven a Servingman’) on their grasp of rent-reckoning, the calculation of areas, and such similar points of practical land-mensuration as might be expected to engage business-minded laymen (Worsop 1582: B1r). Indeed, the extent of Johnson’s and Watkins’s engagement turns out to be surprisingly impressive, given the characters’ deliberate assignment to conspicuously non-technical professions. Clothier and servingman enter readily and with considerable confidence into questions of (for instance) the effect of a shift in linear units on the resulting numbers for area, or the relationship between measured perimeter and the quantity of ground contained within; in response to Worsop’s queries, their eager calculations rehearse and re-rehearse a sequence of ritualised computational process and quasi-oracular numerical solution that by this time should seem very familiar:

WORSOP: Tell mee I pray you, howe many acres a close of foure sides conteineth, if everie side be iust xl. Perches in length?

JOHNSON: Fourtie times, fourtie pence, is xx. Nobles, and xx. Nobles, is ten acres. (D2v)

[17] Johnson’s glib reckoning here relies on a standard procedural approach to the problem, supplemented by the sorts of strictly numerical shortcuts that make sense for a man of his occupation. Prior to this exchange, he has declared himself as ‘lik[ing] best the old manner of measuring [area], by laying head to head, and side to side, taking their halfes, and that ways to cast up the contents’ — that is, by adding up the two long sides of a plot, adding up the two short sides, dividing by two to average, and multiplying out the resulting two numbers to yield a total area. And indeed, his answer performs precisely this process: first, finding an average length and average width (an easy point, since each side measures ‘40’), then multiplying the two together to yield an area. That every step but the last is nominally performed in terms of currency (‘pence’ and ‘nobles’) instead of surveying-measure appears to be a peculiarity of Johnson’s mercantile outlook (and an interesting point of characterization on the author’s part); understandably more accustomed to reckoning with money than with spatial measures, the clothier takes advantage of the fact that the relation of pence to nobles (1d=1/80 noble) resembles the relation of square perches to acres (1 perch2= 1/160 acre) to work most of the calculation as though it were a money-problem, simply dividing by two at the end to convert his fictive ‘nobles’ to more appropriate units of ground.

Fig. 3 The crooked close. Edward Worsop, A discoverie of sundrie errours and faults daily committed by landmeaters (London, 1582), E1r.

Figure 3: The crooked close. Edward Worsop, A discoverie of sundrie errours and faults daily committed by landmeaters (London, 1582), E1r.

[18] Johnson’s computation, as computation, is perfectly correct: his miraculous transmutation of land into gold and back occurs within the framework of a perfectly decorous working of the formula for calculating areas. Critically, though, the performative ‘dewe working’ of sums ‘dewly set’ here fails to provide a guaranteed path to final truth. An accompanying diagram (conveniently supplied by  ‘Worsop’ from a book he happens to be carrying) neatly clarifies the mistake that has occurred: for irregularly curved figures, the length of the sides is no certain clue to the area enclosed, so that ‘those crooked hedges being cast up as though they lay straight, will yeelde a farre greater quantitie of grounde, then the square close a.b.c.d. doth: and yet the square is greater, for the close with crooked hedges, is included within it’ (Worsop 1582: C1r). The processual magic of ‘cast[ing] up’ has indeed ‘yeeld[ed]… a quantitie of grounde, ‘as promised; but now, startingly, it is a quantity with a patently fictitious existence, bearing no discernible relation to the physical reality of the plot before us.

[19] This gap between numerical seeming and quantitative reality is at the heart of Worsop’s critique; indeed, in his complaints of arithmetic’s capacity for mere facile persuasion in the absence of real truth, its moral flexibility and tendency toward special pleading, the author verges surprising close to contemporary critiques of rhetoric itself (Mack 2002: 9–10). Such is ‘the ignorance of the time’, Worsop warns, that for the naïve layman, the mere forms of computation have become deeply persuasive; so much so that merely ‘to talk by roate of measure… procureth great credit’ to various fraudulent measurers. Since the external performance of computation offers no sure way to distinguish ‘ignorant’ from ‘learned practitioners’, the public remains vulnerable to the flashy ‘shows and brags’ of unskilled landmeaters who ‘winne unto themselves greate good opinions, and… cary away the doings from the learned’ (Worsop 1582: E3v). As the Discoverie continues, these lines of complaint build to a climax in Steven’s recounted story of four competing surveyors, each hired respectively by one of four sisters wishing to arrive at a just division of a parcel of inherited land. It is decided that each surveyor should measure the land and compute the area separately, and ‘that one should not tell another howe much hee made it, because it was thought good to see howe they would agree’ (Worsop 1582: C2v). To everyone’s dismay, however, ‘when their reckonings were compared together, they disagreed very much […] [w]hereupon rose great contention and wagering’ as to which might be correct, and ‘such lustie barganing on all sides, that crownes, and angelles were but trifling layes’ (Worsop 1582: C2r–v). The interesting point for our purposes is that this dispute over right reckoning is ultimately not resolved by arithmetical means. While every surveyor staunchly defends his own figures, in the end popular consensus simply places its faith in one Master Morgan, the most well-respected of the reckoners, so that ‘at last all gave place’ to him — so thoroughly, indeed, that when another surveyor discovers that he had slightly ‘misreckoned himself’ in his initial calculation, he actually declares himself to be nonetheless ‘glad thereof, because by that occasion his content came nearest to Master Morganes’ (Worsop 1582: C2v). Even Steven rebukes himself for not having ‘durst to have aduentured at the first’, since ‘[he] could haue gayned twentie nobles by laying on master Morgane his doings’ (Worsop 1582: C2v). For the semi-numerate citizen, then, distinguishing right versus wrong numerical reasoning becomes partly a matter of sheer ethos — discerning the true expert’s quiet confidence from the flashy ‘sleights’ of the arithmetical charlatan —and partly just the luck of the draw. Small wonder that Worsop’s own solution is simply to avoid placing final faith in tricksy arithmetic: no man’s casting-up should be trusted, he argues, unless he can also ‘proove’ his figures in more reliable geometrical terms, via a diagrammatic ‘plat’ that makes quantitative relationships immediately ‘apparent to sense’ (Worsop 1582: E3r, F3r).

[20] In this essay, I have argued that early English algorisms can provide a wealth of fruitful insight into the evolving dynamics of numerical cognition at the individual level, during a period of rapid change in the state of popular numeracy across the nation more generally. Our investigation of these dynamics at work in Recorde’s Ground highlights the complicated realities of the novice’s arithmetical thinking, where theoretical principles of positionality and operation are in reality vested in existing forms of corporeal, material or spatial understanding, and the real business of numerical thought often seems to be located (in ways largely predicted by modern theories of embodied cognition) as much in the body or on the page as in the pristine space of the Cartesian mind. Critically, such an analysis also points up ways in which the imperfect realities of individual competency could impact popular perceptions of and engagements with new forms of mathematical practice. Historians have long acknowledged the importance of investigating early science and mathematics using decentred approaches, ones that take an expanded view of intellectual community and preserve an ‘appreciation of the full range of… activities that constituted past sciences’ (Jardine 2004: 262; Cifoletti 2006: 370). Evidence from early arithmetics suggests that such categories — particularly in the case of mathematics — might profitably be expanded still farther to include more nuanced accounts of inexpert understanding along a continuum of differing competencies and use-levels, as well as of the inevitable nodes of mutual influence where differing levels of practice intersect or overlap; given the frequency with which basic manuals turn up in the libraries even of advanced practitioners, their shortcuts and assumptions may also speak interestingly to a hidden cognitive infrastructure undergirding even what has traditionally been deemed ‘expert’ mathematical thought. Recorde himself contends that numerical thinking is an universal part of the basic business of being human, such that non-engagement with number ‘declareth [one]…unworthy to be counted in the felowshyp of men’ ([iv]). If we accept his point, then the evolution of numerical cognition at all levels, not just at the forefront of elite practice, becomes emphatically an account worth reckoning.

Princeton University


[1] Unless otherwise specified, all references will be to the first (1543) edition of the Ground of Artes, published in London by R. Wolfe.The initial edition of the Ground is somewhat idiosyncratically paginated. None of the introductory material (14 total pages, including including a table of contents and dedication to Sir Richard Whalley) is numbered, and the body of the text is numbered by leaves rather than pages (so that facing recto and verso pages are designated by the same number, at the top right-hand side of the recto). In addition, the numbering after page 40 inexplicably dips back to 35 and begins counting up again, so that the volume actually contains two separate sets of pages designated 35-40. For the purposes of this essay, I have numbered the 14 sides of introductory material using roman numerals i-vii, counting facing sides as a single leaf, in accordance with the practice of the rest of the book. The second set of pages from 35-40 will be cited as ‘35b’, ‘36b’ and so forth. [back to text]

[2] My thanks, here, to the anonymous reviewer who brought these errata to my attention.[back to text]


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_____, and Fenny Smith. ‘Robert Recorde and his remarkable Arithmetic’, in Robert Recorde: The Life and Times of a Tudor Mathematician, ed. by Gareth Roberts and Fenny Smith (Cardiff: University of Wales Press, 2012), pp. 25-38.

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The Number of Motion: Camillo Agrippa’s Geometrical Fencing and the Enumeration of the Body

The Number of Motion: Camillo Agrippa’s Geometrical Fencing and the Enumeration of the Body

Ken Mondschein

[1] Camillo Agrippa (c. 1520–1600) was a sixteenth-century architect, engineer, and natural philosopher. Born in in Milan, he spent the majority of his career in the hothouse of patronage and politics that was Renaissance Rome, where he was associated with Farnese and Medici circles and was also a member of the Confraternity of St. Joseph of the Holy Land, itself a centre for artisans and artists (Anglo 2000; Mondschein 2014: xxi–xxiii; Lincoln forthcoming 2014). It is not Agrippa’s hydraulic engineering, his plan to move the obelisk to St. Peter’s, his navigation, or his tomes of natural philosophy that proved his most enduring legacy, but rather his first-published work, the 1553 Treatise on the Science of Arms. Though written, like all his books, in Italian, the Treatise on Arms, which was dedicated to Cosimo I de’ Medici, was not only popular and read throughout Europe — several copies appear, for instance, in the library of the dukes of Saxony (von Bloh 2012: 207)— but also a major influence on the fashion and style of the northern Renaissance. In this work, Agrippa explains a new system for using the ‘wearing sword’ then carried as a sidearm and indispensible article of dress by all men of rank. Within a generation, works appeared in France, Holland, Germany, and England explaining fencing after Agrippa’s principles, and the fashionable sidearm had lengthened and narrowed into what we know today as the rapier. Though usually not as explicitly mathematical as Agrippa’s original treatise, the education of a young man of means would still include instruction on how to act as Mercutio describes Tybalt in Romeo and Juliet: as someone who ‘fights as you sing prick-song: keeps time, distance, and proportion. He rests his minim rests: one, two, and the third in your bosom’ (II.3.18-20).

[2] The Bard’s jibing was rooted in reality: Agrippa explained his system of fencing by reducing not just all possible actions, but the human body itself, to mathematical symbols, giving possible actions and responses in the language of Euclidian geometry. As he tells us, ‘this pursuit is ultimately governed by points, lines, times, measures, and so forth, and comes from thinking in a mathematical — which is to say, a geometrical — fashion’ (Agrippa 1553: I.2; trans. Mondschein 2014: 10: …in fine questa Professione si governa solamente con punti, linee, tempi, misure, et simili, et nascono in certo modo da consideration’ mathematica, o sia pur sola Geometria). Furthermore, since as Shakespeare noted, fencing takes place not only in space (‘distance’), but also in time, Agrippa explains when to perform one’s operations by using the Aristotelian conception of time as the ‘number of motion with respect to the before and after’ (Physics, IV.11).

[3] Moreover, in the astronomical dialogue appended to the treatise, he makes the implicit argument that because of his mastery of number in space and time (that is, the classical quadrivium), he has the authority to speak on any subject whatsoever. According to Agrippa’s way of thinking, both astronomy and the movements of the human body are the union of number in space (that is, geometry) and number in time. Number, in other words, unites the macrocosm and the microcosm — what Steven Shapin in his synthesis The Scientific Revolution calls the ‘animistic’ tendency of Aristotelian knowledge (Shapin 1996: 29). By applying this use of number to a subject of concern to both the ruling classes and those who aspired to such status, Camillo Agrippa both reflected and contributed to the vernacularization of a mathematical conception of the world and the idea of number as the underpinning of reality. This accords well with the ‘Zilsel thesis,’ which posits that the scientific revolution was enabled by formally educated members of the elite coming into contact with the upper strata of craftsmen. Pamela Long, in her recent and acclaimed revitalization of this idea, identifies ‘artisan-practitioners’ as key to the production and spread of scientific knowledge — a category that certainly encompasses fencing masters (Long 2011).

[4] Agrippa participated in the pedagogical changes of the sixteenth century in other ways, as well. Whereas earlier authors had their students follow patterns, much as medieval artists copied models or writers copied letters, Agrippa emphasizes a deductive approach to fencing pedagogy. He also rejects dogmatic authority, replacing it with his own experience and reason and so we can see in fencing a parallel to changing ideas of education. Likewise, with the increasing emphasis on fencing as a ‘science’, masters after Agrippa presented their works as ‘discourses’ or ‘reasonings’ (ragiomento or ragione) — a whole new way of thinking that presents an argument, discourse, or dialogue on a subject, rather than merely acting as a memory-book, as had earlier fencing works.

[5] Of course, Agrippa did not come ex nihilo. Rather, he was situated not only in scholastic and humanist traditions, but also in a court culture that had for several generations merged the martial with the numerical. Writers on arms in Italy from the fifteenth century onwards had deployed conceptions of a mathematical construction of the universe in their explanations of martial arts. (Writers in Germany did so as well, and at an earlier date, though the German style of fencing was neither as fashionable as the Italian, nor came to use such a wide array of mathematical conceptions.) I do not propose in this brief paper to trace the entire history of fencing, or to provide the reader with detailed arguments hinging on obscure technical details, but rather to give a brief overview in plain language of numerical conceptions in Italian fencing-books, and to then discuss Agrippa’s contributions and legacy. Finally, I will discuss his successors, especially Girard Thibault and his geometrical system of fencing — and how these enumerations of both the universe and of human operators within the universe, while situated in traditional knowledge, also reflected and helped to spread the on-going Scientific Revolution. Just as Johan Huizinga spoke of chivalry as ‘an aesthetic ideal assuming the appearance of an ethical ideal’ (Huizinga 1924: 58), Agrippa and his followers invested in an aesthetic-moral apparatus that saw the human world as linked to the divine, operating within a paradigm that expressed itself through instruments as diverse as astronomy, music, and fencing.

Writers before Agrippa

[6] The first Italian fencing writer who explicitly invokes ideas of number is Filippo Vadi, who was born to a noble Pisan family, served as a counsellor to Borso d’Este, Duke of Ferrara, and produced at least one illustrated manuscript dedicated to Guidobaldo da Montefeltro, Duke of Urbino in the 1480s (National Central Library of Rome Cod. 1324: f. 15r; Mele and Porzio 2002: 4­–5). Vadi’s work is clearly derived from the earlier manuscripts of Fiore dei Liberi (fl. c. 1350–1420) in the Estense library (Mondschein 2011). However, whereas dei Liberi only makes passing reference to scientific theory, such as that ‘heavy things are great impedances to light ones’ (Aristotle, Physics VIII.4; Mondschein 2011: 47–d), Vadi gives a detailed argument that fencing, like music, is a science, arguing that the sword is subject to Euclidian geometry:

Geometry divides and separates
with infinite numbers and measures
that fill pages with knowledge.
The sword is under its purview
since it is useful to measure blows and steps
in order to make the science more secure.
Fencing is born from geometry


Music adorns this subject
song and sound together in art
to make it more perfect by science.
Geometry and music together
combine their scientific virtue in the sword
to adorn the great light of Mars.

La geometria che divide eparte
Per infiniti numeri emisure
Che impie di scientia le sue carte.
La spade e sotto posta a le sue cure
Convien che si mesuri i colpi e i passi
Acio che la scientia tasecure
Da geometria lo scrimir se nasce


La musica ladorna esa sugetto
Chel canto elsono senframette in larte
Per farlo di scientia piu perfecto
La geometria e musica comparte
Le loro virtu scientifiche in la spada
Per adornare el gran lume de Marte

(NCLR Cod. 1324: f. 4r; Mele and Porzio 2002: 42–43)

[7] One might rightfully ask how, exactly, we are supposed to ‘measure’ the chaotic movements of a sword-fight: The measure that saved you one day might kill you the next. The answer, not explicitly given in any treatise but understandable in context if one is an experienced fencer, is that we measure space not absolutely, but relatively —that is, space is measured not with some fixed metric (which is an Enlightenment idea in any case), but rather by comparing one measurement with another. Thus, no matter what angle at which a blow arrives, it should always be crossed obliquely. This is measurement in the sense of the reckoning of magnitude, not in the sense of referencing an absolute metric — in other words, Shakespeare’s ‘proportion’. It is geometrical measurement of the sort used in architecture and, as Fiore had before him, Vadi deploys dividers, the icon of rationalized measurement, to represent the mental skills required to become a proficient fencer.  These are seen over the head of the figure in his segno, an allegorical diagram showing the attributes a swordsman must possess.

Fig. 1 Vadi’s segno, showing the qualities needed by a swordsman. Courtesy Wikimedia Commons and the Wiktenaeur.

Figure 1: Vadi’s segno, showing the qualities needed by a swordsman. Courtesy Wikimedia Commons and the Wiktenaeur.

[8] Dividers thus symbolize not only the measurement of space, but also of time, since one must also measure one’s actions and execute them at the proper moment. Feint to the left; in the length of time created by the adversary covering the imagined attack, strike him on the right. In other words, like space, we are to measure one time relatively against another, after the Aristotelian dictum that time is ‘the number of motion with respect to the before and after’ (Physics IV.11). Agrippa’s Florentine contemporary, Francesco Altoni, who worked in the Medici court, even explicitly says that ‘time is nothing more than the space of motion’ (Altoni 2007: 76: il tempo non è altro che spatio di moto). To be successful, an action must be made in a shorter ‘space’ than the opponent’s counteraction, and Altoni, Vadi, and other writers all make use of terms describing the Aristotelian proportional measurement of time such as ‘half-time’ and ‘double time’in other words, measuring the space of time relatively, one against the other.

[9] Aristotelian ideas of the measurement of time are seen in fencing literature as early as Germanisches Nationalmuseum Nuremberg Codex 3227a, a commonplace book dated to 1389 and containing not only magic spells and recipes for food, alchemy, and the hardening of iron, but also several fencing texts. Notably, it is the first record of the teachings of the enigmatic (and possibly apocryphal) fencing master Johannes Liechtenaeuer, whose merkverse (teaching poem), repeated in a German manuscript and print tradition that lasted well into the sixteenth century, makes use of explicit Aristotelian terminology: ‘Before, after, weak, strong, ‘at the same time,’ you must remember that word’ (GNM 3227a: 17r: Vor noch swach sterke | yndes das wort mete czu merke). On the reverse side of the folio, the anonymous scribe makes the Aristotelian connection even more clear:

Motus das worte
schoneist des fechtenseyn
hort und krone

Motion [motus], that beautiful word
is to fencing
a heart and crown

(GNM 3227a 17v)

The manuscript context of the fencing book makes its intent clear: By mastering the principles on which the universe operates and learning useful skills, one empowers oneself—and key to learning the art of fencing is undertaking an Aristotelian analysis of time and movement as the ‘number of motion with respect to the before and after’.

[10] The similarity of the proportional measurement of time and space is fully congruent with late Scholastic natural philosophy. For instance, William of Ockham (c. 1287–1347) likens measurement of time to measurement of space by saying that we can know a duration of time against a conventionally determined period, just as a yard is a measure of length of cloth. Ockham tells us that as a rough estimate, we can estimate the time in reference to a pre-known quantity — though this second way, however, presupposes familiarity with the first, more precise method (Ockham 1634: IV, 3). Likewise, Jean Buridan (fl. c. 1320–1358) says much the same thing as Ockham: ‘by time and by motion, which is time, we indeed measure other motions’ (Buridan 1964: IV.13: per tempus et per motum qui est tempus mensurant bene alii motus).

[11] These Scholastic glossae of the Physics are ultimately derived from Arisotle’s observation that, like lines, we must have two times—that is, two ‘numbers of motion’ — to compare one duration against another (Physics IV.12), as well as St. Augustine’s observation that we can only know time as the ratio of the duration of observed things as perceived by the intelligent soul: ‘In you, O my soul, I measure time’ (Confessions XI.26: In te, anime meus, tempora metior). This sort of comparative time reckoning was common in a world without mechanical timepieces. For instance, medieval people measured how long to cook something by the amount of time it took to say certain prayers, such as occurs many times in Le ménagier de Paris; for instance, “boil it in sweet water for the space [of time] it takes to say a misere” (boulir une onde en eaue doulce par l’espace de dire une miserelle) (Anonymous 1846: 2.244). This is why Agrippa and other fencing writers do not discuss time in distance in terms of integers or formulae — ‘number’ as we would recognize it: Aristotle and his Scholastic followers saw the geometrical proportional measurement of space and time, which include infinite divisions and irrational measurements, as different from the arithmetical use of number (Evans 1955). Rather, we may consider it as more similar to the comparisons of magnitudes.

[12] Though by Agrippa’s era the idea of ‘tempo’ had become a common term of art in Italian fencing — just as, north of the Alps, fencing writers continued to describe actions as happening in terms of the Aristotelian ‘before’ and ‘after’ (vor and nach) —this is not to say that all Italian fencing-book writers incorporated ideas of measurement into their works. Even though the best-selling writer of the early sixteenth century, Achille Marozzo, who published his Opera nova in Modena in 1536, came both from the university town of Bologna and from a line of fencing mathematicians, he was not particularly ‘scientific’ in a sense that a modern writer would recognize. His teacher’s teacher, Filippo di Bartolomeo Dardi, was a professor of arithmetic and geometry at the University of Bologna before his death in 1464 (Pantanelli 1930: 45–49). Though Marozzo does use the common idea of ‘tempo’ (which he assumes the reader understands), he is not a theorist. To learn to fence from Marozzo was to be initiated into a craft-guild, or mestiero, and having to swear oaths to God, the Virgin, and St. George. Like a medieval memory-palace brought to life, Marozzo has his students run through a series of postures with colourful mnemonic names, such as the ‘guard of the long and extended tail,’ ‘head guard,’ ‘face guard,’ and ‘iron door guard of the boar,’ and then put them together into a series of lessons (trans. Mondschein 2014: xvii). In this, it is similar to works such as a Florentine fragment, MS 01020 in the Fisher Rare Book Library at the University of Toronto, from the 1420s, or Royal Armouries MS I.33 from 1320s southern Germany (Forgeng 2010). Thus, though we have fencing books sometimes invoking ideas of number, we do not have enumeration. The quantitative turn that placed control of the universe in the hands of the mathematician-practitioner was absent in this genre until Agrippa’s treatise of 1553.

Agrippa, Fencing, and Number

[13] Agrippa’s endeavour to reduce fencing to ‘mathematical’ or ‘geometrical’ way of thinking went far beyond his use of Aristotelian ideas of time and his application of Euclidian geometry. Not only did he reduce the earlier multiplicity of guard positions to four numbered positions that can cover all contingencies — four being the Pythagorean tetracys — but he reduced all the possible positions of the body to a finite number labelled by the letters of the alphabet (trans. Mondschein 2014: 8).

[14] As far as fencing goes, what does Agrippa actually say to do, and how was this different from other writers? To begin with, Agrippa tells us that, on the theory that the closest distance between two points (i.e., one’s sword’s point and one’s enemy’s body) is a straight line, the best guard position is with the point threatening the target and the arm held in front of the body (trans. Mondschein 2014: 16). All four of Agrippa’s primary guard positions follow this principle. This is in contrast with other author-practitioners of his generation such as Altoni, who, even if they favoured keeping the point forwards, cock the arm back behind their shoulder to make a stronger, full-arm thrust. Agrippa says that, despite the seemingly exposed position his guard leaves us in, any attempt by the adversary to remove the threatening point will give his student a tempo in which to strike. It also allows one to act in a smaller tempo.

[15] To facilitate this action and minimize one’s tempo, Agrippa advocated keeping the right foot forwards and using a large step to carry the thrust home — in other words, a fencing lunge. This contrasts with the left-foot forward stance often used by other writers with the aim of attacking with a forceful ‘passing’ step in which one steps forward with the rear (that is the right) foot. In the following diagram, Agrippa gives us a geometric proof of the efficacy of this manoeuvre: The further one extends the arm and bends the knee, the further one reaches (trans. Mondschein 2014: 10–14).

Fig. 2 Agrippa’s geometrical proof of the lunge. Courtesy Malcolm Fare

Figure 2: Agrippa’s geometrical proof of the lunge. Courtesy Malcolm Fare.

[16] Agrippa also tells us that if the adversary does make contact with one’s sword, then there are a variety of ways to regain leverage and riposte. There are also ways to respond to an adversary’s attack in a single tempo, which Agrippa says is best, as the two tempi represented by a parry and a riposte would give the adversary a chance to perform some other action before he himself is struck (trans. Mondschein 2014: 46). Finally, in the second part of the treatise, Agrippa gives a number of tactical scenarios in which his theories are applied.

[17] So, what we have here is reason applied to ordering and training the human body to perform optimally in a real situation. As Evelyn Lincoln (forthcoming 2014) points out, Agrippa can be seen in the context of a Milanese tradition of artist-practitioners who applied their theories to practical arts. In this mathematical imagining, Agrippa was perhaps influenced by Niccolo Tartaglia’s Nova Scientia (1537) and the nascent science of ballistics; his brother, Giorgio, was an artilleryman (Lincoln forthcoming 2014). Tartaglia’s aim was to prove at what angle a cannonball would achieve its furthest range; the comparison between Agrippa’s diagram and Tartaglia’s ballistic parabolas is obvious.

Fig. 3 Ballistic parabola from the 1558 print run of the 1550 edition of Tartaglia. Courtesy Max Planck Institute for the History of Science, reproduced by Creative Commons Share-Alike license.

Figure 3: Ballistic parabola from the 1558 print run of the 1550 edition of Tartaglia. Courtesy Max Planck Institute for the History of Science, reproduced by Creative Commons Share-Alike license.

[18] Another likely influence was Cesare Cesariano’s edition of Vitruvius. Cesariano, a Milanese military engineer, published a profoundly illustrated and well-received edition of the Roman architectural manual in Milan in 1521, and it is difficult to imagine that Agrippa, who lived in the midst of the construction of St. Peter’s Cathedral, was not familiar with it. Cesariano’s woodcuts, dealing with constructing the ideal, circular temple from the proportions of the ideal human body, impose a figure upon a grid. In doing so, Cesariano did more than just take the measure of man—he made man into a metric.

Fig. 4 Cesariano’s Vitruvian man. Courtesy Wikimedia Commons.

Figure 4: Cesariano’s Vitruvian man. Courtesy Wikimedia Commons.

[19] Agrippa’s numerical fencing also mirrored contemporary ideas of art. His contemporary and fellow Milanese, the art theorist Giovan Paolo Lomazzo, claimed that Carlo Urbino, to whom Erwin Panofsky attributed the Codex Huygens, was the engraver of copperplates for Agrippa’s fencing treatise (Marinelli 1981: 218; Panofsky 1940). The Codex, a copy of Leonardo’s notebooks, contains not only perspective studies, but mechanical studies of human motion. The human body, measured according to the Vitruvian schema, is considered mechanically, its movements considered according to geometrical analysis. Agrippa takes this artistic study and applies both number and morality to it by means of a stick and a geometrical diagram:

….let me explain that it is there to encourage by word and example those people who, because of their makeup or some other inherent indisposition, think themselves unfit for this exercise. A piece of wood, taken unfinished from a tree or shrub and not having had any work done to it, provided that it is straight and strong enough to be used with a light hand, is quite sufficient to make all sorts of geometrical figures such as circles, squares, triangles, octagons (from which you can similarly make a proportional sphere), which you can see alongside the figures of the four guards, and so on. Similarly, anyone who has their eyes open will see that I am right when I say that a man, governing himself with reason and art, ought to perform this activity well.

….Mi pare il dovere che si notifici il Perche: et cosi facendo, dico, haverla messa quì per questo fine, ciò è per inanimire in questo principio con tal essempio molte persone à la profession’ de l’Arme, le quali per la complessione, o per altra indisposition’ naturale, paiono à se stessi inhabili per tal essercitio: perche si come un’legno simile senza industria alcuna, o ragione di qual arte si volglia, tolto cosi rozzo, & incomposto da l’arbore, o sterpe, o qual altra cosa che sia, pur che tanto stia retto, & saldo in se quanto possi sustentare una mano leggerissima per effettuar l’intento suo, basta, & è bono, anzi in proposito, per fare una moltitudine di figure di Geometria, come sono Circolo, Essagono, Triangolo, Ottangolo (dal qual si fa con esso medisimamente una Sfera proportionatissima) & diverse alter, le quali si postranno veder’ in compagnia e le figure de le Quattro Guardie, cosi intromesse à posta, accio’ che (venendo capriccio à à qualch’uno di farne la prova) potesse vedere che di quallo ch’io dico non sia altro, che parte di verità, debitamente un’homo governandosi con ragione, & con arte, potra fare in questa professione cio che si conviene.

(Agrippa 1553: I.4)

Fig. 5 Agrippa’s First Guard. Courtesy Malcolm Fare.

Figure 5: Agrippa’s First Guard. Courtesy Malcolm Fare.

[20] In other words, just as the stick is sufficient to draw all the geometrical figures, which in turn give us the principles for finding out the structure of the universe, the human body, that is the proportional mirror of the universe, is good enough to execute all the necessary actions of fencing. It is, in other words, a sort of divider, the very essence of proportional enumeration. Agrippa is here taking the theoretical Vitruvian ideas of geometry and the unity of the macrocosmic and microcosmic and turning them into a sort of technology.

[21] Agrippa was also certainly familiar with contemporary optical theory, such as that of Alberti, as he also makes use of the mathematical ‘science’ of perspective. In this diagram, he shows that, just as the eye-rays can only look in one direction, so, too, can one move one’s body out of the way of an oncoming attack.

Fig. 6 Agrippa’s counterattack. Courtesy Malcolm Fare.

Figure 6: Agrippa’s counterattack. Courtesy Malcolm Fare.

[22] In describing how to move out of the way of an attack, Agrippa likens the human body to a ball (trans. Mondschein 2014: 52). In this, he uses a idea of the spherical human form, an idea found not only in Vitruvius, but also in Alberti’s Intercenales: ‘Nothing is more capacious [than a circle], nothing more whole in itself, nothing stronger, nothing more able to shrug off shattering blows because of its angles, nothing freer in its motion. Therefore, we must remain within the circle of reason, that is of humanity, which is connected and complicit with virtue, and God to virtue, which comes from God’ (Alberti 1890, 232: circulo nihil capacious, nihil integrius, nihil robustius: nam est quidem ex se totus angulus ad omnes impetus fragendos accomodatissimus, suoque in motu omnium liberrimus; quasi igitur in tutissim liberrimoque circulo rationis ipsos non habendos nobis, hoc est humanitati, cui connexa et complicita virtus est; virtuti vero Deus, nam ex Deo est). The idea of the circular nature of the human body is also found in the Hippocratic text De Locis in Homine, which had been published in Latin translation in Rome in 1525 by Fabius Calvus: ‘The beginning of a circle cannot be found. If one wishes therefore to find a first and absolute beginning in the human body, then all is the beginning, and all the end…’ (De Locis in Homine I.1: Circulo enim descripto principium non reperitur. Vult siquidem in humano corpore nullum reperiri principium primum & absolutem, sed Omnia principium esse, & omnia finem…) The circular conception of the human body is also very similar to the use of geometrical analysis in the Codex Huygens to consider the human body from different perspectives.[1]

[23] Agrippa also deploys Aristotelian ideas of tempo. Though he is not explicit on this account, this is made clear in context as he explains his ideas on how to use a sword. The idea of tempo, as explained above, had long since been part of the technical language of fencing, and remains so today. One can easily see why Agrippa was so interested in astronomy, which is more than the measure of objects in space and time— it also reveals the structure of the macrocosmic universe, and thus, the microcosm of men (and their duels). His astronomical treatise is, unsurprisingly, non-Copernican, and gives no new revelations to historians of astronomy: rather, he explains the equant, the Ptolemaic idea of the centre of the planetary epicycles. It serves, however, the rhetorical purpose of showing that Agrippa has the mastery of natural philosophy to speak ‘reasonably’ on any subject whatsoever. Rather than relying on tradition, Agrippa proudly tells us, he is an independent operator, able to figure out things for himself by observation and reason without formal university education. Such an attitude, as examples as diverse as Galileo and Menocchio (Ginzberg 1980) show us, would have been dangerous after the Council of Trent, but Agrippa’s disinclination to say anything heterodox ensured his book’s being kept from the Index.

[24] Agrippa’s attitude is encapsulated by two allegorical scenes, one used as a frontispiece, and one before the dialogue. In the former, Agrippa disputes with a group of confounded-looking university professors, who have nothing to back them up save for their tomes of ancient knowledge. Agrippa, however, has an armillary sphere and a pair of dividers — a model of the world and the tool for measuring at it. A sword is at his side, his foot is perched on a globe, and a geometrical diagram and sword are on his side of the floor. Above him are measuring instruments — dividers, as well as a protractor or quadrant of the sort that a geometer or artillerymen might use. Governing all is the hourglass of time. As Anglo says, ‘The author… is using both pure and applied mathematics to place personal combat upon a scientific basis’ (Anglo 2000: 25). Even if he was in actual fact indebted to traditional knowledge as much as any medieval philosopher had been, Agrippa’s rhetorical stance was that he was a new sort of man rejecting the dogmatism of the past in favour of a new sort of experiential learning—a learning that was, as in surveying, ballistics, and other emerging technologies, based on an enumeration of the world.

[25] In the scene before the dialogue, Agrippa is beset by academics dressed in shabby robes who seek to clobber him with a quadrant as his fashionable and sword-armed supporters come to his rescue. In the background is an obelisk inscribed with emblematic hieroglyphics. It is in reference to this that he mentions, in the dialogue, his two sources—‘if some students of Euclid or of Aristotle want to drag my name through the mud, I will defend myself as best I can, both on my own and with the help of my patrons’ (Agrippa Dialogo; trans. Mondschein 2014: 103–4: se non che forse alcuni allevi di Euclide, o di Aristotile, vorrano imputar mi, di quel ch’io dico, & io col mio aiuto, & d’altri miei Patroni mi diffenderò).

Figure 7: Allegorical Scene. Courtesy Malcom Fare.

Figure 8: Allegorical Scene. Courtesy Malcom Fare.

[26] Who were these patrons? Who was Agrippa writing for? Some clues are provided by the men he name-drops in his book, all of whom were artists and intellectuals in the circles of Cardinal Alessandro Farnese: Alessandro Corvino, Francesco Siciliano, Gerolamo Garimberto, Alessandro Ruffino, Alessandro Cesati, Francesco Salviati, Fillipo Archinto, and Annibale Caro, who is Agrippa’s interlocutor in the astronomical dialogue. These are men whose habitus included reverence for antiquity, a taste for art, a knowledge of hieroglyphic emblems (albeit inaccurately derived from Horapollo), and the Vitruvian geometry that was then being used to plan St. Peter’s.

[27] They were not, however, either on the cutting edge of natural philosophy or men who wanted to topple the structure of the world. We must therefore take Agrippa’s self-proclaimed revolutionary nature with a grain of salt. Despite his use of number, Agrippa is not Copernican. Nor is he even particularly mathematical. He is vernacular, writing on a subject of interest to the aristocracy and deploying the fashionable paradigm of the day to explain his method. Despite his ‘proofs,’ no mathematics are needed to follow him (though an understanding of geometry helps if one is to follow the first part of the dialogue). Rather, we are dealing here with a symbolic use of number in an almost animistic sense — a sort of pneuma of the world-spirit, connecting the human and heavenly realms, how that which is above is like that which is below. Furthermore, unlike astronomers, Agrippa does not give us a quantitative analysis in his analysis of fencing, but rather a relativistic one — we are still dealing with ‘number’ in the sense of proportional measurement, rather than as an absolute quantity. (Even in astronomy, concepts of absolute space and time would not be widely accepted until Newton, but measurements of degree and time against the celestial sphere do give us a sort of absolute yardstick.)

[28] Though Agrippa’s is an early metaphorical deployment of number as the bridge between macrocosm and microcosm in a work on a physical art, he is hardly unique in his conceptions. Leonardo’s notebooks and the works derived from them, such as the Codex Huygens, are filled with such conceptions. Similarly, in the unattributed portrait of Luca Pacioli (1445–1517) below, the mathematician, flanked by a noble patron or student, sketches a triangle (evoking the Trinity) in a circle (evoking the unity of God and man) — one hand on a book, one on his chalk, his eyes fixed on a heavenly geometrical figure half-filled with water so as to refract its surroundings, and instruments for measurement, including dividers and an angle, before him on the table. Pacioli, besides being an acquaintance of Leonardo’s, also notably worked in the court of Urbino (the ducal palace is even reflected in the pendant rhombicuboctahedron), and this is, the same milieu from which Vadi came.

Fig. 7 Luca Pacioli, c. 1495. Courtesy Wikimedia Commons.

Figure 9: Luca Pacioli, c. 1495. Courtesy Wikimedia Commons.

[29] On the other hand, we must avoid seeing Agrippa’s fencing as solely the manifestation of fashionable ideas of number lying behind the structure of reality — all thought, and no practical action. The duel of honour was a very real phenomenon in Agrippa’s lifetime. At the risk of committing a syllogism, the reason why his work was so popular is because it presents a very practical method of using a sword in personal combat. This is not to say that he was not a mirror of his times, but also that we must see his invention as something meant to be used in the real world. His fencing system is both fashionable and practical.

[30] While Agrippa may be more groundbreaking in fencing than he is in natural philosophy, what he does do successfully is provide a very cogent analysis of fencing actions, reducing a very complicated practice to a set of core principles that are seen in a numerical and geometrical light. He, in other words, gives an effective analysis of a natural phenomenon in order to reproduce effects according to his will — the very essence of theory as applied to technology. He applied this approach to a matter of interest to the European aristocracy, and so helped to spread an idea that mathematical analysis is a powerful tool for understanding the world. Agrippa’s science is also very much an applied one — experiment in the sense of actual sense experience, as opposed to thought-experiment. He is thus a bridge of sorts between an allegorical deployment of number and a scientific one.

Writers after Agrippa

[31] As the first real fencing theorist, Agrippa’s impact on the field was profound. Not only did he articulate the basis for what would eventually become codified in the modern sport of fencing, but no fencing book after him was complete without some discussion of the nature of art and science — though most showed the application of this theory rather than its causes. Ridolfo Capo Ferro, in his treatise of 1610, even argued that fencing was an art, not a science, because fencing does not examine ‘eternal and divine things going beyond the will of humans,’ but an elevated ‘art of doing’ (as opposed to a craft or trade) whose products are ephemeral and whose rules are both universally true and well-ordered (Capo Ferro 1610: 5; trans. Leoni 2011: 8). His teaching consists mostly of examples of tactical actions. Likewise, Salvator Fabris, fencing master of Christian IV of Denmark, whose Lo Schermo was printed in Italian in Copenhagen in 1606, protested his poor learning and that he would not use fancy geometrical terms and proofs — though the art was founded in geometry — and that he would, instead, explain it in plain language (Fabris 1606: A4; trans. Leoni 2005: 2).

[32] Most late sixteenth and early seventeenth-century Italian fencing books only had the barest traces of numerical and geometrical imaginings, such as Capo Ferro’s lunge, above, and his instructions that the sword should be the length of the lunge, which is turn based off the proportions of the body. Even Giacomo di Grassi, who is very un-Agrippan in his fighting system, gives us a geometrical diagram to show that in certain circumstances a cut is more direct than a thrust and another to show that, like a gun-sight, holding a buckler far away gives more cover than holding it close to one’s body.

Fig. 8 Capo Ferro’s lunge. Courtesy the Wiktenaeur.

Figure 10: Capo Ferro’s lunge. Courtesy the Wiktenaeur.

Fig. 9 Di Grassi’s ‘gunsight’. Courtesy the Wiktenaeur.

Figure 11: Di Grassi’s ‘gunsight’. Courtesy the Wiktenaeur.

[33] What Fabris, Capo Ferro, and other writers do share in common with Agrippa is a sense of ordering their pedagogy from first principles to complex actions. Most earlier writers did not explain principles or define terms of art, but rather simply explained chains of action. (Vadi, who has some preliminary material, is the one exception). Later rapier masters almost universally give a sense of the basic building blocks —distance, timing, etc. — and only then proceed to how these are applied in combat. They, in other words, fulfil the Aristotelian ‘knowing a thing by its cause’ — the logical analysis of physical phenomena (Physics I.1; Carranza 1582: 12r). This is a particularly Western way of looking at a problem: first the principles (which, according to Capo Ferro, are universally true), then application. This mentality can be contrasted to the pedagogy of Chinese martial arts, which emphasize rote repetition of technique and forms in memorized traditional choreographies.

[34] One Italian who made great use of geometrical proofs was Frederico Ghisliero, a military man who also wrote (now-lost) works on the mathematical arts of siege craft, fortification, and artillery, but who is perhaps best-remembered today for hosting Galileo at a dinner party during the latter’s period of Copernican crusading. In 1587, Ghisliero published a book with a geometrical consideration of fencing derived from Agrippa — though Sydney Anglo considers him more a student of Jerónimo de Carranza than of Agrippa (Anglo 2000: 68–71). Showing a great deal of Vitruvian influence, Ghisliero uses radii of circles to describe distance, and gives us images of his fencer in ‘scientific’ perspective. He even begins (as Copernicus did, Newton would later, and my writers did in between) with two chapters on geometrical principles.

[35] The Carranza mentioned above is the inventor of the Spanish ‘destreza’ school of fencing — an amazing late-Scholastic, intertexual, Aristotelian edifice. Influenced by Agrippa’s work, Carranza came up with his own geometrical system of fencing in the 1560s. This school was thereafter continued by his disciple-cum-critic Luis Pacheco de Narvaez (Anglo 2000: 67–69; Fallows 2012: 218–235; trans. Mondschein 2014: ixxx–xxx). Carranza was a captain in the Spanish army, a client of the Duke of Medina-Sedonia, and associated with the School of Seville. His connections sufficed to earn him the governorships of his hometown of Sanlúcar de Barremeda and of Honduras. Narvaez, for his part, later became the chief fencing master of Spain, in charge of examining other masters. Again, we are dealing with writers who, far from wishing to challenge the orthodox structure of the world, rather wished to appeal to those in power by translating one element of elite habitus — the mathematical underpinnings of the world — into another sphere — martial performance.

[36] What did Carranza and Pacheco teach? Unlike the knees-bent postures taught by the Italians, they felt that the swordsman should stand erect, this being the most dignified position. Combat takes place within an imaginary circle described by the diameter of the swords, with the fencers’ movement described as radii, chords, and arcs and an elaborate taxonomy of all possible motions rationalized by the degrees of leverage on the sword. As befitting the conservative Spanish milieu, their explanation of movement is entirely orthodox Aristotelian—an upwards movement is ‘violent,’ whereas a downwards one was ‘natural’. In other words, the Spanish school describes fencing entirely in geometrical and Aristotelian terms. Needless to say, dividers appear both Carranza and Pacheco’s author portraits.[2]

[37] Carranza and Luis Pacheco were widely known in Europe, and mentioned —usually derisively — by several authors. For instance, Ben Jonson alludes to their geometrical conception of fencing in his The New Inn:

TIPTO: But doth he teach the Spanish way of Don Lewis?

FLY: No, the Greeke Master he.

TIPTO: What cal you him?

FLY: Euclide.

TIPTO: Fart upon Euclide, he is stale, and antique, | Give me the modernes.

FLY: Sir he minds no modernes, Go by, Hieronymo! [an Italian fencing teacher who worked in London in the Elizabethan era]

TIPTO: What was he?

FLY: The Italian, That plaid with Abbot Antony, in the Friars, | And Blinkin-sops the bold.

TIPTO: Aye mary, those, Had fencing names, what is become of them?

HOST: They had their times, and we can say, they were | So had Caranza his: so had Don Lewis.

TIPTO: Don Lewis of Madrid, is the sole Master | Now, of the world.

HOST: But this, of the other world | Euclide demonstrates! he! He is for all! | The only fencer of name, now in Elysium.

FLY: He does it all, by lines, and angles, Colonel. | By parallels, and sections, has his Diagrammes!

(The New Inn: II.5)

The characters then go on to give odds on imaginary fencing contests of philosophers in Elysium. The Spanish school of fencing is also referred to by Quevedo and Cervantes, and the former actually fought a duel with Pacheco wherein he knocked off the master’s hat.

[38] Both the Spanish method of fencing and the Agrippan geometrical turn was taken to their ultimate and most explicit extent by a Dutchman, Girard Thibault, whose book The Academy of the Sword (L’Académie de l’espée) ranks as one of most sumptuous printed works ever created. As Kate Van Orden points out, we should see it as a counterpart to Antonie de Pluvinel’s (also geometrical) L’Instruction du Roy en l’exercice de monter à cheval, since both were worked on by the same artists, both were colossal ‘atlas’ editions, and both were associated with the circle around Louis XIII (Van Orden 2004: 57). Thibault had apparently learned the Spanish school of fencing in Sanlúcar while working as a wool merchant. Besides his skill in fine arts, architecture, and medicine, he studied mathematics at Leiden and, beginning in about 1610, taught his own version of Carranza’s school (de la Verwey 1978). This was acclaimed by Dutch fencing masters in 1611, and earned Thibault introductions to aristocratic circles. The Academy of the Sword was published posthumously in Paris in 1628 with royal imprimatur.

[39] Thibault makes no bones about the numerical relationship of microcosm to macrocosm when he says:

Man is the most perfect and excellent of all the creatures of the world, in whom is found the other marks of divine wisdom, a most excellent representation of the whole universe, in his whole being and his principle parts, so that he is rightfully called the Macrocosm by the ancient philosophers—that is, the Small World. For besides the dignity of the soul, which has great advantages over all that is perishable, his body contains an abridgement not only of that which can be seen here down on earth, but also yet that which is in Heaven itself, representing all with a harmony so sweet, beautiful, and whole, and with a just accord of Numbers, Measures, and Weight which correspond so marvellously to the virtues of the Four Elements, and to the influence of the Planets, that one can not find anything similar.

The most perfect number of Ten is continually shown before the eyes, in its entirety by his own fingers, and broken equally into two parts by the two hands, each one with five fingers, which are broken into two unequal parts by the thumb and the rest into One and Four, of which on is composed of Two things, and the Four of Three. In this way, this structure always shows him the premier and most excellent numbers 1, 2, 4, 5, and 10, which the illustrious philosophers such as Pythagoras and Plato, and all of their students, held so highly, that they chose to hide in them, and deduce from them, the greatest mysteries of their doctrine.

L’Homme est la plus parfite & la plus excellent de toutes les Creatures du Monde, auquel se trove parmy les autres marques de la sagesse divine, une si exquisite representation de tout l’Univers, en son entier & en ses principales parties, qu il en a esté appellé à  bon droit par les anciens Philosophes Microcosme, c’est à dire, le Petit Monde. Car outre la dignité de l’ame, qui a tant d’avantages par dessus tout ce qui est perissable, son corps contient an abbregé, non seulement de tout ce qu’on voit icy bas en terre, mais encores de ce qui est au Ciel mesme ; representant le tout avec une harmonie, si douce, belle, & entiere, & avec une si juste convenance de Nombres, Mesures, & Poids qui se rapportent si merveilleusement aux vertus des Quatre Elements, & aux influences des Planetes, qu’il ne s’en trouve nulle autre semblable.

Le tres-parfait nombre de Dix luy est continuellement representé devant les yeux, en son entier sur ses propres doigts ; & derechef in deux moitiez egales sur ses deux mains, á chascune par le nombre de Cinq doigts; qui sont derechef partis inegalement par le poulce, & par le reste en Un & Quatre, dont l’Un est composé de Deux articles, & les Quatre de Trois : de façon que ceste structure luy met tousiours en veue les premiers & plus excellents Nombres dont tant d’Illustres Philosophes, comme Pythagoras, & Platon, & tout ceux de leurs Escholes, ont fait tant d’estime, qu’ils y ont voulu cacher, & en deduire les plus grands mysteres de leur doctrine.

(L’Académie de l’espée, I.1)

[40] Thibault then proceeds to cite the Vitruvian rule of constructing a temple according to the measure of the human body, even linking this to the dimensions of the Temple of Solomon and of Noah’s Ark. After a short oration on the dignity and utility of human proportion, which recalls the study of anatomy then going on at Leiden, he then extols the use of reason in self defence, by which man, seemingly the most helpless of creatures, renders himself master of all.

Therefore, all the abovesaid Artists, Architects, Perspectivists, and others have sought to prove the foundations of their rules by the proportions of the human body, and I have similarly taken the same course, but with better results, and have found with the help of this same compass the true and proportional measure of all the Movements, Times, and Distances necessary to follow my Practice, as will be declared to you in a moment in the explanation of my Circle, where the measures and proportions of man are applied to man himself and to the movements he makes with his own limbs, where the aforesaid proportion is found, and without which it is impossible to perform the least action in the world.

Tout ansi donc que les susdits Artistes, Architectes, Perspectivistes, & autres ont tasché de prover les fondements de leurs regles par les proportions du corps de l’homme, ansi avons nous pareillement tenu la mesme course, mais avec meilleure adresse, & avons trouvé à l’aide de ceste mesme buxole la vraye & proprtionnelle mesure de touts les Mouvements, de touts les Temps, & Distances, necessaires á observer nostre Practique: comme il vous sera semonstré tout á l’instant en la declaration de nostre Circle; où les mesures & proportions de l’homme sont appliquées à l’homme mesme, & aux mouvements qu’il fait avec ses propres membres, où ladite proportion se trouve, & sans laquelle il luy est impossible de faire le moindre action du Monde.

(L’Académie de l’espée : I.3)

[41] Thibault, like Agrippa, then tells us the human body is a circle, and goes on to advise us on the construction of his ‘mysterious circle,’ by which we learn to perform the proportionate movements of fencing. The circle is based on the proportion of the sword, which is equal to the radius and the cross of which, if the point is placed between the wielder’s feet, should reach his navel. The sword itself — the symbol of the enfranchised and potent male, created by his own genius, just as God fashioned his natural limbs, proportional to his body to aid him in self-defence — is thus a sort of measuring-tool. The author’s sigil, repeated several times in the art in front matter, unsurprisingly contains a pair of dividers.

[42] Relationships of leverage between the two adversary’s swords were conceived of as numerical relationships, with the sword, continuing the proportions as the body, divided into twelve parts. Higher numbers, closer to the hand, have mechanical advantage over lower numbers, closer to the point. Van Orden summarizes, ‘Like Kepler and Newton, Thibault conceived of physics according to the precepts of musica speculativa’ (Van Orden 2004: 62) — in other words, he sees all the possible motions in fencing as a harmonic relationship between two numbers. While I concede that Van Orden is correct in a metaphorical sense, I do not see any explicit deployment of musical theory here — if anything it is more the case that Thibault’s description of leverage is more what Vadi made explicit: music and fencing share a common root in number (Anglo, 2007).

[43] Though the martial art expressed in Thibault’s atlas-sized edition may seem overly complicated to us, the masters of Amsterdam seem to have found it efficacious as well as aesthetic. The whole school was based on a mathematical understanding of both the world and of fencing. By obeying the numerical principles of time and of proportion — in other words, fencing scientifically — the fencer cannot but conquer his foe. It is a way of explaining how to operate in space and time in accordance with a ‘system of the world’ — in other words, a technology.

Figure 12: Fencing school at Leiden. Image courtesy Wikimedia Commons.

Figure 13: Anatomy theatre at Leiden. Image courtesy Wikimedia Commons.

Figure 14: Thibault’s circle.  The influence of a geometrical floor plan for fencing in the Spanish manner at the University of Leiden (Figure 12), c. 1610—pre-dating Thibault’s residence there—and the studies of anatomy taking place at the university (Figure 13), can all be seen in Thibault’s circle. Image courtesy Wikimedia Commons.

[44] The contrast between Thibault, the style of Louis XIII, and the style of Louis XIV is extreme. The sceptical turn of mind of the later seventeenth century would find the hermetic constructs of Thibault ponderous and ridiculous. If we had Descartes’ lost fencing treatise, it would perhaps be an excellent illustration of this tendency; however, all we know of this book is a mention by Descartes’ biographer Adrian Baillet (Baillet 1691: I.35). We do have a work on fencing published in Rennes in 1653 by Charles Besnard, who was possibly acquainted with the philosopher as a young man, since the 18-year-old Descartes had spent the winter of 1612–13 in Rennes practicing the military arts (Baillet 1691: I.35; Brioist, Drévillon, and Serna 2002: 168). However, the tradition that Besnard was Descartes’ master is rather specious, and Besnard’s treatise is quite different from what Descartes’ could have been. Judging from Descartes’ having spent much of his life in the Netherlands, his geometrical inclinations, and Baillet’s dismissal, the philosopher’s fencing might well have been more similar to Thibault’s than Besnard’s — in fact, Baillet tells us the former ‘completely wasted his time’ studying riding and fencing in Rennes (Baillet 1691: I.35: On peut juger par son petit traité d’Escrime s’il y perdit entiérement son temps). Let us, rather, take Besnard’s treatise as an example of what physical education would have been like for a young man in the mid-seventeenth century.

[45] Though Besnard’s The Liberal Master-at-Arms (Maître des Armes Liberal) is ultimately based on Agrippa’s work (as would be most later fencing), it is greatly simplified. The art is still rationalized, but didactic rather than argumentative, giving principles and best means of operating much as Capo Ferro and Fabris did before him. His postures and actions are greatly simplified; one trusts that there is theory there, as Besnard insists, but the exact details are left for the master. The student’s job is to have his body trained and disciplined. In all, Besnard strives for the uniformity and universalism of definition that characterized the Enlightenment. The limits of enumeration have been realized, the idea of sacred harmony has fallen out of favour, and ‘Augustinian’ fencing, as Peter Gay would have put it — the whole Scholastic-hermetic basis of Agrippa and Thibault—was replaced by a rationalized system of training the body (Gay 1966). The idea that a fencing master could ensure patronage and fame by showing how his system mirrored the cosmos was no more. Fencing, as Diderot and D’Alembert would later characterize it, was not a science, but an art (L’Encyclopédie 21:6:1).


[46] Though the sixteenth and early seventeenth centuries were enamoured of enumeration, there was some misplaced enthusiasm about what could be explained by number. The fencing masters of this time invested in an aesthetic-moral apparatus that saw the human world as linked to the divine, operating within a paradigm that expressed itself through instruments as diverse as astronomy, music, and fencing. Its substance was humanist, but its principles were still that of the Aristotelian Middle Ages, and its aims were not an objective knowledge, but to show how the operator could use the likeness of that which was above and that which was below to control their world. On the broader scale, this tends to complicate our thinking on the Scientific Revolution, which was much more than just the Copernican Revolution — it was also a revolution of enumeration, of ways of thinking about the world, and of the growing acceptance of physical experiment. It was also not always so revolutionary: As Agrippa and his followers show, one could be a defender of Ptolemy and employ a relativistic, geometrical method ultimately rooted in the Scholastic treatises of the thirteenth and fourteenth centuries, yet position oneself as a rebel against traditional authority. Fencing texts thus give us insight into how ‘progress’ is historically situated, contingent, and non-linear.

[47] They also give us insights how knowledge is spread. For these ideas of enumeration to be transmitted to society at large required conduits between centres of intellectual production, such as Renaissance Rome, and the rest of Europe. Fencing masters, who ultimately wrote to please their audience and who certainly fit into Long’s idea of ‘artist-practitioners’, performed this task admirably. By producing treatises that successfully applied the numeric turn to the subjects important or fashionable to those of status, or who aspired to status, writers such as Camillo Agrippa helped to popularize new ideas of human knowledge—at least until the scepticism of the later seventeenth century demanded a new, more didactic method. Still, the proportional methods these writers employed, the enumeration that linked the microcosm and the macrocosm — the enumeration of the dividers, not the meter-stick — was well-suited to fencing and other physical arts, and, as a conceptual tool, was highly successful.

All images in this article reproduced from Wikimedia Commons and the Wiktenaeur (http://wiktenauer.com) are used by Fair Use and/or Creative Commons Share-Alike License.


[1] The Morgan Library’s images of the Codex Huygens is copyright-protected, and so we cannot reproduce any figures; however, the entire manuscript is viewable at http://www.themorgan.org/collections/works/codex/default.asp.[back to text]

[2] A true appreciation of la verdadera destreza is beyond the scope of this article, and will have to wait for the publication of Mary Dill Curtis’ 2012 Ph.D. thesis.[back to text]


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Sixteenth-Century Humanism, Printing and Authorial Self-Fashioning: The Case of James Peele

Sixteenth-Century Humanism, Printing and Authorial Self-Fashioning: The Case of James Peele

Rebecca Tomlin

[1] ‘I have a book on accountancy that is often mistaken for a Bible’, Hilary Mantel’s Thomas Cromwell tells Harry Percy in Bring Up the Bodies (2012: 358). ‘Especially by you’, Percy responds. In placing an accounting book in her fictional Cromwell’s library, and in linking it to the Bible, Mantel gestures towards a relationship between accounting books, the printing of the Bible in English, the writing of Tudor history, and English humanism, a relationship which this article will suggest is closer and more complex than she probably realised. Some of those connections are explored here through a study of the works of James Peele, The maner and fourme how to kepe a perfecte reconying (1553 – henceforth ‘The maner and fourme’), which is the earliest surviving work on double-entry book-keeping by an English writer, and The Pathe waye to perfectnes, in th’accomptes of debitour and creditour (1569 — henceforth ‘The Pathe waye to perfectnes’). This article proposes that Peele’s first work, The maner and fourme, is an attempt to integrate number work into the reformed humanist publication strategy of the printer Richard Grafton. As twenty-first century readers, we have a distinct sense of authorial identity; the opening sentence of this article refers to ‘Hilary Mantel’s Thomas Cromwell’ in the expectation that the reader will know immediately that Mantel is referred to as an author, and as such, is the key figure identified as producing the book in question. In the mid sixteenth-century, acknowledgement of authors was rare, except in the case of legal and medical texts (Hellinga and Trapp 1999: 86). Read for the signs of Peele’s assertion of his authorial identity as a humanist scholar, The Pathe waye to perfectnes illustrates the developing importance of authorship in the sixteenth-century. Following the ‘material turn’ in book history, this article focuses on the title pages of Peele’s works and other volumes published by Grafton to examine the ways in which these two texts position accountancy as a practice and seek to fashion its cultural capital.

[2] When writing about early books on accounting, academics have generally taken one of two paths. Histories that deal with the technical development of accounting practice and with commercial matters are usually not concerned with the impact of a wider cultural context on the production of books on double-entry book-keeping. Conversely, among sociologists, literary scholars and cultural historians, a profound (and, in some cases, possibly over-stated) relationship has been posited between double-entry book-keeping and its contribution to religious and social change in early modern Europe.

[3] The association between nascent capitalism, ‘the protestant work ethic’ and early accounting texts was first made by Max Weber, who argued that ‘capital accounting, according to the methods of modern bookkeeping and the striking of a balance’ (double-entry book-keeping), was one of the ‘general requisites for the existence of capitalism’ (1905, 1923: 276). The connection made by Weber between double-entry book-keeping and religious, social and economic developments of the early modern period, although contentious, remains the starting point for much recent thinking about the impact of early modern book-keeping texts by writers including Mary Poovey (1998) and Natasha Korda (2011). The link theorised by Weber between rational capitalism, Protestantism and double-entry book-keeping has been countered by James A. Aho (2005), who points out that the system originated in pre-capitalist medieval Italy and links it instead to the Catholic practise of confession and a neurotic ‘scrupulism’ in accounting for one’s deeds to God. Studies that embrace a less wide-ranging target than capitalism and ‘the protestant ethic’ suggest, for example, that the careful process of writing which double-entry book-keeping requires was a creator of mercantile credit (Sullivan 2002) and a model for early modern life-writing (Smyth 2010).

[4] Historians of accounting have expressed reservations regarding the feasibility of tracing a significant early modern cultural impact of what was, until the eighteenth century, actually a little-known and rarely-used practice. Their objections arise partly because they perceive in some scholarship an imprecise lack of distinction between the double-entry system of keeping books and other ‘single-entry’ accounting systems which also use the terminology of ‘debit’ and ‘credit’. The accounting historian Basil Yamey challenges the developing interest in the cultural impact of the accounting system by expressing a sceptical inability to discern ‘that inherent in double-entry book-keeping there is some crucial rationalizing effect, some scientific or mathematical principle or notion, or some abstract ideas which could not fail to influence ideas, attitudes, behaviour or practices outside the narrow confines of record-keeping and simple administration.’ (1989: vii; see also his highly sceptical 2005 article.)

[5] This article directs its attention to an aspect of the works of James Peele that has not previously received critical attention either from those interested in the technical developments of accounting practice, or from the cultural historians seeking evidence of the impact of double-entry book-keeping on the wider culture. Noting that ‘the non-textual features of a document carry their own semantic weight’ (Lesser 2004: 16), it uses the paratextual material of Peele’s books, and especially their frontispieces, as, in Gérard Genette’s term, ‘thresholds of interpretation’ through which to consider the material and cultural conditions of their production (1997). In doing so, Peele’s works are placed more precisely in their historical and cultural moment, while also being made to point to the wider significance of the number work that they do. As with all books, the works of James Peele were shaped by what Daybell and Hinds term ‘the material concerns’ of their production; that is, both the physicality of the texts and ‘the materiality of the socio-cultural contexts in which they were produced, transmitted and consumed’ (2010: 2). Although, as Genette remarks, paratext considered in the absence of its text is ‘a mahout without an elephant, a silly show’ (1997: 410), this article will nevertheless attend to paratextual aspects of Peele’s works to argue that they were not intended to be read solely as texts of applied number work for utilisation by artisans. These works make a claim for the intellectual place of numbers and number work among the liberal arts and for the status of the specialised form of writing that is double-entry book-keeping alongside other works of Elizabethan literary scholarship.

[6] Elizabeth Eisenstein argues that the claims made by early printers of accounting manuals that double-entry book-keeping was good for the common-weal and one’s own soul were mere posturing in order to increase sales; ‘one more variation on a theme’, she states, ‘which had been used to promote accountancy books from the first Renaissance blurb writers on’ (1979: 384). In response this article will take seriously the claims made by these books about their place in the intellectual moment from which they emerged. While accepting that printers, of course, needed to make a profit from the works that they produced, it follows studies by, among others, Zachary Lesser (2004), Peter McCullough (2008) and Kirk Melnikoff (2009), in arguing that a printer’s own religious and/or political ideology also influenced the publication choices he made. The role of authorship in the sixteenth century remains opaque; the contrasting presentation of Peele’s two works provides a striking case study of the shifting importance accorded to the author. Where The maner and fourme appears to form part of a larger humanistic programme directed by Richard Grafton, this article will argue that in his second work, The Patheway to perfectness, printed some sixteen years later, Peele was able to assert himself as author.

The maner and fourme (1553)
[7] The starting point for this reading of Peele’s books is the impressive title page of  The maner and fourme (see figure 1, below). The rebus of its printer, Richard Grafton, appears prominently at the bottom of the page, depicting a ‘tun’ or barrel, from which issues a grafted tree, probably the tree of knowledge, the whole being a pun on his surname.

Figure 1: James Peele, The maner and fourme (1553), Title Page.

Figure 1: James Peele, The maner and fourme (1553), Title Page.

[8] As is not unusual in a mid-sixteenth century book, the printer’s rebus is large and striking, while Peele’s name does not appear anywhere on the title page; this book is intended to be seen as the work of its printer, rather than that of its author. In 1553, when Peele’s text was being prepared for print, Richard Grafton was the King’s Printer, a major figure who had been pivotal in the promotion of the English Reformation through print, and in shaping the evolving relationship between the Court and the City of London (Gadd and Ferguson 2004). Grafton had a long association with Protestant reformers including Thomas Cranmer and Thomas Cromwell. His ‘sometimes incautious combination of reformist commitment and commercial activity made him one of the most eye-catching evangelicals of the period’ according to Alec Ryrie (2003:19). Imprisoned three times during the 1540s for printing religious books, at a time when Henry VIII was struggling to exert control over the process of reformation, Grafton was closely involved, with his partner Edward Whitchurch, in the production in Antwerp of Coverdale and Tyndale’s English Bible and in the printing of the first official bible in English, the Great Bible of 1539 (Devereux 1990; Ryrie 2004; Ferguson 2004). The famous title page of the Great Bible demonstrates that Grafton understood the importance of paratextual images in controlling the way in which the contents of a book were understood. There, Henry VIII is depicted as the embodiment of temporal and spiritual authority, received directly from God, and also as the distributor of the word of God in English. It conveys in a single image everything that the King wanted to say about the Church of England and Henry’s place in a reformation that was to be achieved, at least in part, by the Bible to which the frontispiece was attached.

[9] Following his return to England in 1539, Grafton, with Whitchurch, had been given space at the dissolved Greyfriars’ property to set up a print business, as well as monopolies on the production of English scripture and service books (Sisson 1930, Ferguson 2004). During Edward VI’s reign he was the King’s sole printer of year books, statutes and other official documents, and of the authorised works closely associated with the dissemination of religious writings in English, including the earliest editions of The Book of Common Prayer (Cummings 2011). Peele’s first work was therefore prepared for production by a famous and prestigious printer whose entire career was shaped around producing books that supported Protestant reform in England, for which he had, more than once, gone to prison.

[10] It has not been previously noted (as far as the author is aware) that The maner and fourme may have been the book that Grafton had on the press when he was imprisoned for printing the proclamation of Lady Jane Grey as Queen on 10 July 1553. McKerrow and Ferguson note, without comment, that although the title page is dated ‘1553’, the printing of The maner and fourme appears to have been completed in 1554 by John Kingston (Grafton’s former apprentice) and Henry Sutton (1932: 28). The STC and ESTC agree that Grafton printed only the title and verso of the book and note that the assumption that the work was completed in 1554 is essentially conjecture. Assessments of Grafton’s motivations for printing the proclamation of Jane Grey vary; David Womersley describes it as an example of Grafton’s ‘recklessness of political consequence when religion was the cause’ (2010:46), while James Raven merely shrugs it off as a misjudgement that ‘hardly proved his wisest undertaking’ (2007:34). E.J. Devereux dryly notes that nine days later Grafton ‘learned that he had effectively resigned his office’ as Queen’s Printer (1990: 41); he was imprisoned by Queen Mary and his printing business disbanded.

[11] Only one complete copy of The maner and fourme, held by The Institute of Chartered Accountants in England and Wales, is known to survive. The other copy listed in the ESTC, held at the British Library, is incomplete: it contains the explicatory text but has lost all of its model ledger pages. There are several possible explanations for the low rate of survival of Peele’s work. Texts that were used, particularly as learning aids, may have become worn and annotated to the extent that they were eventually discarded. For example, only eight copies of what must have been very many Elizabethan printings of the basic educational text, the ABC, survive, presumably because they were used to destruction (Farmer and Lesser 2013). Perhaps few copies of Peele’s first book have survived because they were used to learn from and eventually became worn out. In 1588 John Mellis produced A Briefe Instruction and maner how to keepe bookes of accompts after the order of debitor and creditor in which he claimed to reproduce the first book in English about double-entry book-keeping, Hugh Oldcastle’s 1543 A Profitable Treatyce called the Instrument or Boke to learn to know the good order of the keepying of the famouse reconynge called in Latyn, Dare and Habdare, and in English, Debitor and Creditor. No original copy of Oldcastle’s book has survived, and Mellis, who was a teacher of accounting and mathematics, claims that he has reproduced it because his own ‘ancient old copie’ was thirty years old and he wanted to make the work more widely available (1588: A2v). Evidence that Peele’s works were used as working text books is offered by the copy of The Pathe waye to perfectnes held at the Huntington Library, which has annotations (in the hand of the second Earl of Bridgewater) that appear to show a reader engaged with learning from the text (Travitsky 1999). The maner and fourme includes a large number of almost blank pages, marked up as ledgers. Perhaps, paper being expensive, they were appropriated and used as a working ledger, a conjectural explanation that would account for the pages missing from the British Library copy.

[12] It is possible that few copies of The maner and fourme survive because few were printed to start with. Some comparison might be made with Pacioli’s Summa de arithmetica, geometria, proportioni et proportionalità (1494) of which Alan Sangster has estimated that between one and two thousand copies of the first edition were printed (2007). The comparison is not very helpful, however, because the Summa, which was produced at a much earlier stage of the development of print technology, was a complete anthology of mathematics of which the accountancy element formed only a part, and it was made for the Italian market; each of these is sufficient reason to frustrate direct comparisons between the two books. Franklin B. Williams, Jr. estimated that some four thousand entries in the STC are known only by a single copy; he also noted that it is rare for valuable books cared for in libraries to entirely disappear (1978). Information about the price of mid-sixteenth century books is scarce and there is no indication of how much either of Peele’s books would have sold for, but both are folios and are printed on high quality paper, and in comparison to, say, A Briefe Instruction, which is a quarto volume, and of lower quality paper, they are expensive books. Although the rate of attrition for early books is high, the scarcity of surviving copies of The maner and fourme, a large and relatively expensive book, suggests that its initial print run was small, perhaps because Grafton’s successors were not as interested in it as he was. The maner and fourme is dedicated to the ‘the right worshipfull Sir William Densell, knight’ and the company of Merchant Adventurers of which he was Governor. Usually referred to as ‘Damsell’, Sir William was knighted at Queen Mary’s coronation, despite signing the devise altering the succession in favour of Lady Jane Grey, and this dedication to him as ‘knight’ is further evidence that printing of the book was completed after Grafton lost his business (Bisson 1993: 6). Pacioli’s Summa was probably produced for the instruction of wealthy merchants and their sons educated through the Italian abbaco system (Sangster, Stoner & McCarthy 2008). Although, as noted above, comparisons between the two books are far from conclusive, it seems likely that Peele’s The maner and fourme was similarly intended for a market of wealthy merchants in London, including the Merchant Adventurers.

[13] If Peele’s book was intended as an instruction book for merchants, then it disrupts the assumption, commonly made about books of this type, that they were low-status trade manuals with little ideological or intellectual ambition. Although James Raven argues that a surprisingly large output of ephemera and hack work was an important source of funding for many printing houses, in Grafton’s business the profits were supported by his monopolies as King’s Printer (2007). This lucrative position enabled Grafton to specialise in high-quality works; taken together with his ideological commitments, it would be reasonable to conclude that The maner and fourme was not an anomaly in his output, a lone example of a low status trade manual amongst his other, more ambitious work. Grafton’s biography strongly indicates that the texts he selected for printing were meaningful to him, and that his output forms a collective body of work sharing certain qualities that confer that meaning. It is therefore appropriate to consider The maner and fourme alongside Grafton’s other publications, and also to consider whether it reflects an ideological commitment, beyond simple commercial gain.

[14] Scholars of early modern books with no interest in double-entry book-keeping may nonetheless recognise the title-page compartment used by Grafton for The maner and fourme. It depicts the youthful Edward VI with his courtiers of both Lancaster and York allegiances who, as Hall’s Chronicles (1548) described, had been united in the person of Edward’s father, ‘the high and prudent prince kyng Henry the eight, the undubitate flower and very heire of both the sayd linages’ (McKerrow 1913). Woodblocks of the type used for this page, unlike metal engraved plates, allowed the title inside the decorative frame, or compartment, to be changed, so that the block could be used for different books and titles (Corbett & Lightbown 1979: 6). This folio-sized block was used frequently by Grafton, including for the title pages of the first edition of Edward Hall’s The Union of the Two Noble and Illustrate Famelies of Lancastre and Yorke (1548 – henceforth ‘Chronicle’); John Marbeck’s Concordance to the English Bible (1550); the second, most radically Protestant, version of The Book of Common Prayer (1552); and seven editions of yearbooks and statutes between 1548 and 1553. Indeed, this woodblock was used so often by Grafton that the bibliographers McKerrow and Ferguson rather dejectedly comment that owing ‘to the difficulty of distinguishing and identifying the various editions, it has been thought better to make no attempt at a complete list here’ (1932: 70). The image of the King in council suggests that the block was prepared primarily for the official legal works which were the products of the council meetings that it depicts. The other books for which Grafton used it were also important to a regime that consciously promoted the use of print to maintain itself (Alford 2002: 116). Hall’s Chronicle, edited and issued by Grafton after Hall’s death, is a history of England that confirms the validity of the Tudor reign over England and Henry VIII’s role in the reformation of its church; and the Concordance and the Book of Common Prayer were instrumental in the Edwardian project of Protestant reformation (Devereux 1990). Richard Totell, Grafton’s son-in-law, acquired some of Grafton’s equipment and materials in 1553 and continued to use the woodblock for statutes and year-books until at least 1575. The last example of its use available on Early English Books On-Line appears to be Totell’s edition of the statutes of Henry IV, printed in 1575 (STC 9609).

[15] There is some evidence that Grafton did have access to other folio-sized title page compartments around 1553. A 1552 edition of the Book of Common Prayer held at the Bodleian (STC 16286.5, via EEBO) has two additional title pages preceding the one featuring the ‘King in council’. These are much more elaborate in style, and include four putti, caryatids, shields and the royal coat of arms. This block apparently belonged to Grafton’s long-term business associate, Edward Whitchurch, whose initials ‘E’ and ‘W’ appear in the lower corners. Whitchurch had used the block for the title pages for his two-volume edition of The paraphrases of Erasmus, printed in 1548 and 1549. The 1552 Book of Common Prayer includes a sub-title that seems to have been adapted for Peele’s work the following year: ‘the fourme and maner of makynge and consecratynge Bisshoppes, Priestes and Deacons’ (A2r). Other than this apparent borrowing from Whitchurch, Grafton’s folio-sized works from 1548 onwards most often utilise the ‘King in council’ block.

[16] The use and re-use of an image of Tudor authority for many different texts illustrates how practical, commercial and ideological matters all influence the material qualities of an early book. Large and decorative woodblocks were a valuable asset in a printing business and needed to be used as intensively as possible. As a businessman, Grafton would have been concerned to ensure that his prestigious and valuable position as King’s Printer was maintained and enhanced. Although The maner and fourme is the only work that does not have an overt religious or political content for which the title page is used, at first glance an unbound copy would have looked just like one of Grafton’s editions of the law of the land or one of the key texts of Tudor authority and Edwardian religious reformation.

[17] A further example confirms the association between Grafton, books on double-entry book-keeping, and Hall’s Chronicle. Prior to printing The maner and fourme, and at around the time he had become King’s Printer to Edward VI, Grafton had printed the English version of Jan Ympyn Christoffels’ book on double-entry book-keeping, A notable and very excellente woorke (1547). This is the earliest surviving work on double-entry book-keeping in English; if Thomas Cromwell did indeed own a book on accountancy in 1536, it was most likely to have been written in Italian or possibly Dutch, and the translation history of Christoffels’ work, from Italian to Dutch, and then into French and subsequently English, broadly traces the progress of the double-entry book-keeping system itself across Europe (Sullivan 2002). The title page of Grafton’s English translation of Christoffels’ book uses the well-known ‘Tudor Rose’ title-page compartment which symbolically represents Henry VIII as the culmination of the union of the red and white roses of Lancashire and York (see figure 2, below).[1] The design of the pedigree as a vine may be that referred to by John Foxe when he writes that ‘certayne there were which resorted to hym, [Grafton] of whom some were drawers for his petigree and vineat, some were gravers, the names of whom were John Bets, and Tyrrall’ (Foxe 1576: 581). Agreeing with the conclusion of the nineteenth-century bibliographer William Lowndes, Graham Pollard attributes the design to Betts, and the woodcarving to Tyrrall (Pollard 1933:17). This compartment, with its emphasis on the genealogy of Henry VIII, was surely prepared for Hall’s Chronicle, but was not used for the first (1548) edition, appearing instead in the 1550 edition.

Figure 2: Jan Ympyn Christoffels, A Notable and very excellente woorke (1547), Title Page.

Figure 2: Jan Ympyn Christoffels, A Notable and very excellente woorke (1547), Title Page.

[18] The long and complex publishing history of Hall’s Chronicle is beyond the scope of this article, but editions completed and edited by Grafton after Hall’s death appeared in 1548 and 1550 (Pollard 1933, Devereux 1990). Assuming that the date given on the Christoffels’ title page is correct, then the title-page block had been made by 1547, and it is not clear why Grafton would have used this title page for a book on accounting but not for the first edition of the Chronicle for which it was so carefully prepared, which appeared in 1548. For this, he selected the ‘King in Council’ block that he also used for The maner and fourme. The title-page compartments that Christoffels’ and Peele’s works share with the Chronicle, a work of major propaganda significance to the Tudor monarchy, suggest that Grafton regarded books on double-entry book-keeping as intellectually significant and perhaps even as instrumental in the Tudor reformist project.

[19] In addition to his official religious and legal output, Grafton also published books that would appeal to buyers with an interest in humanism. In 1542 he had printed Nicolas Udall’s English translation of Erasmus’s Apophthegmes, and he seems to have embarked on a programme of publishing humanistic texts, perhaps following the model of the successful Aldine Press of Venice. Further translations of Erasmus and of Aristotle’s Ethics, and An abridgement of the notable woorke of Polidore Vergile appeared between 1545 and 1547, along with A prognostication for this yere M.D.xlvi written by the German humanist, Achilles Permis Gasser (1547). In the period from 1551 until the effective end of his business in 1553, Grafton also produced a set of secular books that included John Caius’ Counseill against the Sweate (1552); Thomas Wilson’s The Rule of Reason (1551) and The Art of Rhetorique (1553); and Peele’s The maner and fourme (1553). These books share the common characteristic of being concerned with translating into English the latest ideas emerging from Renaissance Europe, either in the form of rediscovered classical texts (Wilson) or contributions to the ‘common weal’ (Caius and Peele).

[20] Wilson’s The arte of rhetorique is a synthesis and adaptation of the major ancient rhetorical works, including Cicero and Quintilian, together with various works of Erasmus. It is cited by Charles Nauert as an English example of ‘the victory of humanism as an essential part of the education of all who aspired to hold influential positions in the royal courts of northern Europe’ (2006: 192). The dedication of Wilson’s The rule of reason describes a close relationship between the writer and Grafton:

Notwithstandyng I must nedes confesse, that the Printer hereof your Majesties servaunt provoked me firste hereunto, unto whome I have ever founde my selfe greately beholdyng, not onely at my beyng in Cambrige, but also at all times els, when I moste neded helpe. (1551:A3V)

The references to being ‘greately beholdyng’ and in need of help suggest a specific obligation beyond the usual rhetorical trope of submission to a patron. In addition, Wilson’s dedication is to his printer as patron, whereas the standard appeal is to a member of the nobility or other eminent personage, and often seems to seek to mitigate the stigma of print. Wilson, a master of rhetoric, would have been well aware of the implications of this expression of gratitude to Grafton. A pupil of Nicholas Udall’s at Eton College, Wilson remained his friend after leaving for Cambridge, apparently supported by Grafton, where he was part of the Protestant humanist circle that surrounded John Cheke (Doran & Woolfson 2008). Until their deaths from the sweating sickness in 1551, Wilson was tutor to Henry and Charles Brandon, sons of the Duke of Suffolk, who were members of the young humanist-educated elite surrounding Edward VI. He edited an obituary for the Brandon brothers written in Latin by another member of the Cambridge circle, Walter Haddon, which was printed by Grafton in 1551. Caius’ work on the latest treatments for sweating sickness, printed by Grafton in 1552, may also have been a response to their deaths. Also in 1552, Grafton published Haddon’s Sive Exhortatio ad literas, a Latin oration which publicly celebrated the Protestant erudition of the five daughters of Anthony Cooke, the eldest of whom, Mildred, was married to William Cecil (Goodrich 2008). Both Haddon and Grafton were eventually buried, in 1571 and 1573 respectively, at Christ’s Church, Greyfriars’, close to where Grafton, and later James Peele, lived and worked (Sisson 1930). The works printed by Grafton throughout this period indicate his association with a network of authors who shared social links and intellectual concerns, and through Grafton, Peele is also connected to this group of English humanist intellectuals.

[21] Given that Grafton was imprisoned on at least three occasions for the ideological content of the works he printed, Eisenstein’s evaluation of the positioning of both printers and texts of ‘business arithmetic’ seems unfairly dismissive, at least as far as Grafton and The maner and fourme are concerned:

Early printers in their prefaces did all they could to reinforce the impression that theirs was an unusually elevated calling. At the same time they catered to the needs of other merchants by issuing handbooks and manuals that were also dignified by the addition of poetical prefaces and an abundance of classical allusions. By artful references to Boethius, Pythagoras, and the muses, business arithmetic could be elevated to the rank of a liberal art and linked to the wisdom of the ancient philosophers. (1979: 393)

Eisenstein implies that the humanistic paratextual apparatus of early modern books on accounting is inappropriately used to dignify handbooks, manuals and business arithmetic for merchants, commercial works that she feels should not aspire to be ‘elevated to the rank of a liberal art.’ The likelihood of making a profit on a book was an undeniably important part of any publishing decision, and Grafton may have commissioned a new and improved book on accounting from Peele, based on a successful venture in publishing the Christoffels’ text. The profit motive and other reasons for publishing a work are not, however, mutually exclusive.

[22] Tracey Hill has observed that ‘the production of culture in early modern London invariably went on in ways which have been stigmatised as those relating to “hack” writing’, that is, in collaboration and for financial reward (2010: 6). The snobbery about commercially inflected civic culture, which Hill identifies as anachronistically diminishing the artistic significance of city pageants, has possibly also served to reduce the intellectual status of Peele’s works. It would seem difficult to make the argument that Peele’s book is a lone exception to Grafton’s collection of religious, official and humanist works, tainted by its commercial applications and produced for an intellectually inferior audience. Wilson’s dedication points to Grafton’s role in ‘provoking’ the author into preparing a text for publication, and it is possible that Grafton encouraged Peele to write The maner and fourme because a book on double-entry book-keeping was a suitable addition to his series of texts that aimed at disseminating the humanities in a socially useful way. Just as Wilson’s The arte of rhetorique and The rule of reason aimed to strengthen the English vernacular by making the useful arts of rhetoric and dialectic reasoning available to those not able to study them in Latin (Mack 2002: 79), so The maner and fourme made available the practical application of mathematics to any reader of English.

Figure 3: John Caius, Counseill against the sweate (sig. E8r)

Figure 3: John Caius, Counseill against the sweate (sig. E8r)

[23] An image of ‘Reason’ and the other liberal arts that appears in both the Counseill against the sweate (Caius 1552: sig.E8r) and the 1553 edition of The Rule of Reason (Wilson 1553: sig.Aaiv) illustrates the place that the number work of The maner and fourme takes among the humanistic texts published by Grafton. The use of the image in Caius’ and Wilson’s texts shows that they were intended to be read in this larger intellectual context and it also reminds us that arithmetic holds equal status to the other liberal arts. In the emblem (see figure 3) the central personification of Reason, or Logic, holding an open book, is supported by Grafton’s mark and surrounded by figures representing the other arts. Arithmetic takes a prominent place, supporting both Grafton’s mark and Logic, and opposite Grammar in the geometrical layout.

[24] Wilson’s works on rhetoric and logic are easily located in this scheme, and Caius’ Galenist medicine is also reflected in the arts of logic and astrology. A verse from Wilson’s The Rule of Reason suggests that Peele’s work on business arithmetic belongs, with Logic, among the liberal arts, and, more specifically, that ‘reckoning’, as the particular application of arithmetic that can make things ‘even’, enjoys an equal status with rhetoric and logic.

A brief declaration in meter, of the vii liberal artes, wherin Logique is comprehended as one of them:
Grammer dothe teache to utter wordes.
To speake bothe apt and playne,
Logiquely art settes furth the truth,
And doth tel what is vayne.
Rethorique at large paintes wel the cause,
And makes that seme right gaie,
Whiche Logique spake but at a worde,
And taught as by the waie.
Musike with tunes, delites the eare,
And makes us thinke it heaven,
Arithmetique by number can make
Reconinges to be eaven.

Geometry thinges thicke and brode,
Measures by Line and Square,
Astronomy by sterres doth tel,
Of foule and eke of fayre.
(Wilson 1551: sig. B4, author’s italics)

[25] Wilson’s description of arithmetic, which ‘by number can make reconinges to be eaven’ is an evocation of the ‘reckoning by pen’, the use of written numbers, which was in the process of superseding the old-fashioned method of ‘reckoning by counters’, and which is described in Peele’s instructions on ‘how to kepe a perfect reconying’. The use of number arithmetic, and the move away from counters, allowed for the use of zero and the development of a more abstracted form of arithmetic through the sixteenth century (Thomas 1987; Rotman 1987; Jaffe 1999).

[26] Peele’s first book, then, was a product of both of its author’s applied knowledge of accounting and Grafton’s ideological project of producing a collection of high quality humanist texts, directed at a readership that was not university educated, and therefore wanted to read these works in the vernacular. In addition to being a work that effected the practical application of numbers, attention to the paratextual and contextual information supplied by the material form of The maner and fourme directs us towards reading it as part of the humanist intellectual current of the mid-sixteenth century.

The Pathe waye to perfectnes (1569)
[27] The maner and fourme illustrates the dominant role of the printer in producing mid-sixteenth century books. A close study of the paratextual and material qualities of Peele’s second book, the optimistically titled The Pathe waye to perfectnes, in th’accomptes of debitour and creditour, shows how Peele’s self-fashioning as a humanist author exemplifies the transition towards the assertion of authorial identity during the late sixteenth century. In elevating himself to the level of scholar, Peele makes the claim that his text is also scholarly. This article proposes that the features that Eisenstein dismisses as ‘artful references’ are integral to Peele’s text and that this reading enables us to reassess Peele as an illustrative example of the possibility of humanist education to enable self-fashioning. Further, in contrast to the interweaving of royal images of power and accounting texts in Grafton’s work, the exclusive focus of The Pathe waye to perfectnes on the City and its social structures and institutions shows that self-fashioning in the sixteenth century did not always happen in reference to the Court and royalty.

[28] By 1569, when The Pathe waye to perfectnes was published, Grafton’s printing business had collapsed, although he continued to live on the old Greyfriars’ site, most of which had been converted into Christ’s Hospital after 1553 (Sisson 1930). Grafton was instrumentally involved in the setting up and running of the Hospital, where he lived until he died in 1573. Having been appointed Clerk of the Hospital in 1562, Peele was his neighbour, and lived there until his own death in 1585 (Prouty 1952). Grafton may have helped Peele get the job as Clerk, and they would both have worshipped at Christ’s Church, where they were buried and Peele’s children were christened and married. As well as being Clerk, Peele was paid to teach arithmetic and writing at the Christ’s Hospital school and seems also to have continued to keep a private school on the premises. (Prouty 1952).

[29] The Pathe waye to perfectnes was printed and sold by Thomas Purfoote, a less prestigious printer with more diverse commercial interests than Grafton, but even so an established businessman who had been one of the freemen listed in the Charter of the Stationers’ Company when it was incorporated in 1557. Purfoote had some business dealings with Grafton’s son-in-law, Richard Tottel, but no further relationship with Peele has been identified (Pantzer 1991). Purfoote judged that it was worth investing in the bespoke design of the title page of The Pathe waye to perfectnes (see figure 4, below), which signals that its printer is up to date with the latest fashions in book styling and also serves to advertise the prestige of the book within. The frontispiece displays several coats of arms connected to Peele and his work, ranked by their relative importance, with those of Elizabeth I shown at the top, and below them and slightly smaller, the arms of London. The remaining coats of arms are those of the Merchant Adventurers, the Merchants of the Staple, the recently founded Russia Company and those of Peele’s own Salters’ Company. A coat of arms related to the Peele family is shown in the lattice window behind the figure of Peele himself (Prouty 1953). The frontispiece celebrates the various merchant companies to which the work is dedicated and whose members were among its intended purchasers, and Peele’s place among them. Six complete or partial copies of this book are listed in the ESTC. The copy held at the Huntington Library appears to have been annotated by John Egerton (1623-1686), the second Earl of Bridgewater (Travitsky 1999), indicating that it retained its interest some considerable time after being printed in 1569, and also that its circulation had spread beyond the merchants of the City.

Figure 4: James Peele, The Pathe Waye to Perfectnes (1569), Title Page.

Figure 4: James Peele, The Pathe Waye to Perfectnes (1569), Title Page.

[30] As with Grafton’s title-page blocks, this one also enjoyed a long life after the publication of the book for which it was first made. For an edition of The Life and Death of Hector (STC 5581.5) printed by Thomas Purfoot (junior) in 1614, Elizabeth’s arms were replaced by those of James I, Tudor roses and thistles were added to the top of the page, and the company arms were replaced with animal symbols representing the four continents, Europe, Asia, Africa and America. For one of the seven editions of The Whole Book of Psalms printed in 1615 for the Stationers Company (STC 2550), the animals at the corners have been replaced with the four evangelists. In both cases, Peele remains, and with his identifying coat of arms erased, he has come to represent the generic figure of the author.

[31] Placed on the frontispiece as if guarding the threshold to The Pathe waye are classical personifications of Wisdom and Science who, the motto below states, ‘prevent indigence.’ A portrait of Peele is placed in the midst of the web of the various London institutions that are represented in the design, suggesting that he was a known figure who, in himself, is a guarantee of the quality of his work. In contrast to his total absence from the title page of The maner and fourme, both Peele’s name and some biographical details are given: ‘James Peele, Citizen and Salter of London, Clercke of Christes Hospitall, Practizer and teacher of the same’. The portrait of the author is an unusual feature in books of this period, and its presence represents an effort to fashion Peele as a public intellectual, and to elevate his work with numbers to a status above that of a practical manual. Peele’s image appropriates the social status made available to those without wealth or aristocratic lineage through teaching and learning by English humanism. Peele is shown writing, surrounded by books and implements, his attention focused on his work. The iconography of the portrait can be read in part through the absence of certain signifiers. Peele is shown alone, in a closet too small and confined to admit others, and certainly in the absence of any pupil, adult or child, other than the reader. This is not Peele portrayed as a teacher of book-keeping. Although the books in front of Peele may be read as ledgers, the other elements of the emerging iconography of sixteenth century merchants’ portraits in London identified by Tarnya Cooper are absent (2012). These include books, and especially prayer books and ledgers; loose papers such as manuscripts, letters and bills; memento mori, including skulls and clocks, symbols of the vanity of worldly goods; money boxes and coins; and, in particular, a recognisable expression of watchfulness, caution, and perhaps mistrust. Represented as neither teacher nor merchant, Peele’s image aligns more closely with the tradition of portraits of scholarly saints like Jerome, and most significantly, it adopts the iconography of portraits of Erasmus, the most influential humanist in Northern Europe, whose image was widely circulated throughout the sixteenth century in the form of engravings. As is usual in depictions of Erasmus, Peele is depicted in a confined space, a closet or a study, his eyes cast down as he writes, his attention focused on his work, rather than the viewer. There is none of the merchant’s watchfulness, and certainly no money or the reminders of the vanity of wealth and of impending death that are seen in the portraits of pious London merchants. Substituted for memento mori are the works that will confer immortality on the writer after his death. This is Peele portrayed as an aspiring humanist scholar, and, by extension, the book on which he works is presented as a humanist text.

[32] In addition to the links between Peele and other humanist scholars that can be traced through Richard Grafton, some connections between Peele and sixteenth-century literary culture more generally can be made. An anonymous commendatory poem at the front of Path waye to Perfectness appears to have been written by Arthur Golding, author of the highly influential translation of Ovid’s Metamorphoses which had been published two years previously in 1567 (Tomlin 2012; Pincombe 2013). In 1566 and 1569, Peele was paid by the Ironmongers’ Company to prepare the ‘posies, speeches, and songs that were spoken and sung by the children’ in the Lord Mayor’s show (Prouty 1953; Bergeron 1971: 128; Hill 2010). Peele’s contribution does not survive, but the use of speech in the mayoral shows was a recent innovation in 1566; the earliest pageant text seems to date only from 1561, and so Peele’s would have been among the first few of what Lawrence Manley describes as ‘new attempts to disseminate, more widely and in verbal form, the ideology of the city elite’ (1995: 266). Peele’s son George, who was to become the successful playwright, was born in 1556 and educated at Christ’s Hospital and later at Oxford (Prouty 1953). Probably as a consequence of his father’s earlier involvement in the shows, and his connections in the City, George Peele wrote the texts for up to six Lord Mayor’s shows between 1585, the year that his father died, and 1595 (Hill 2010). Based on the marriage of James Peele’s daughter Isabel to a Matthew Shakespeare in 1569 (the year in which The Pathe waye to Perfectness was published) it has been suggested that the Peele family may have been distantly related to William Shakespeare (Salkeld 2012).

[33] Although Peele nowhere makes reference to his humanist predecessor, Luca Pacioli, in his second book he aligns the method he describes with the other rediscovered ancient texts recovered by the humanists and translated into the vernacular:

This order is both auncient and famous: and doubtles grounded altogether uppon reason, for tyme out of mynde, it hath bene and is frequented, by divers nacions, and chiefelye by suche as have bene and be the most auncient and famous Merchauntes. (1569: sig. *3v)

On the links between humanism and Pacioli, Richard Macve has argued that the treatise on double-entry book-keeping was included in the Summa because it is a closed and balanced system, that reflected Pacioli’s ‘Neo-Platonic and mystical interest in “perfect forms” that contain the quintessence of the cosmos’ (1996:8, see also Yamey 1994). Close links between double-entry book-keeping as a method, and rhetoric, the central art of humanist practice, have been explored by other scholars. Aho (2005) traces the origins of the debit and credit balancing system of double-entry book-keeping back to Ciceronian rhetorical techniques. Mary Poovey (1998) and Ceri Sullivan (2002) both argue, to different ends, that the process of book-keeping is itself a rhetorical exercise, in which the merchant engages in order to create credit, an indispensable asset in a culture of trade (Muldrew 1998). Elizabethan humanism was strongly influenced by Cicero’s conceptions of learning and its effect on the human character, and English theorists of the sixteenth century were particularly exercised by the idea that liberal education must bear ‘fruit’ that is of practical value to the state (Crane 1993, Pincombe 2001).

[34] The aphoristic banners which frame Peele’s portrait on the frontispiece of The Pathe waye to perfectnes read ‘Wisdome and Science, Prevent Indigence’ and ‘Practise procureth perfection.’ They both control, and confer authority on, Peele’s image, demonstrating his capacity to utilise the axiomatic form central to the humanist education process. (Crane 1993; Smyth 2004, 2010). These are not the mere ‘artful references to Boethius, Pythagoras, and the muses’ that Eisenstein dismisses; for those who had the education that enabled them to produce the appropriate aphorism in the right time and place, ‘aphoristic citations of classical and sacred texts demonstrated intellectual and doctrinal credentials’ and created a form of intellectual capital (Crane 1993: 95). ‘Wisdome and Science, Prevent Indigence’ is an axiom that captures the humanist understanding that the combination of ‘wisdom and science’ in the form of knowledge was a kind of capital that could compensate for the lack of material wealth or aristocratic birth. The same sentiment is expressed in the first stanza of Peele’s prefatory poem to The maner and fourme, ‘An exhortation to learne sciences….’:

As lacke of Science causeth povertie
And dooeth abate mans estimation,
So learnyng dooeth brynge to prosperitie
Suche as of goodes have small possession (1553, B4v)

Peele presents double-entry book-keeping as a technique that might be learned, and as such, a process by which cultural capital, good credit, and financial capital or ‘prosperitie’ can be acquired, even by those who ‘of goodes have small possession’. It promises social mobility, or at least financial security, as the reward for diligence and learning.

[35] Besides its obvious application to the smooth-running of the merchants’ businesses, Peele suggests that his own contribution to the ‘common weal’ is the avoidance of disputes among friends and neighbours that a proper system of accounting will enable:

For emongest althynges nedefull in any nacion, touchying worldly affaires, betwene man and man, it is to be thought that true and perfect reconying, is one of the chief, the lacke wherof, often tymes causeth, not onely great discencion but also is an occaision of greate losse of time, and empoverishment of many, who by lawes, seke triall of suche thynges, as neither partie is well hable to expresse, and that for lacke of perfecte instruccion in their accompt, whiche thyng might, if that a perfecte ordre in reconyng were frequented of all men, right well be avoided [….] Wherefore my desire is that this my travaill herein taken, might be so beneficiall to all menne, that at all tymes eche man with other, frendly maie conferre their reconynges, and therby to staie suche variances, as els maie ensue. (1553, sig.A3)

Peele promotes orderly book-keeping as a contribution to civic order because it will prevent dissension, suspicion and litigation between business men, who he, crucially, terms ‘frende or neighbour’. Craig Muldrew’s influential study described an early modern community held together by a personal and ubiquitous network of credit relations (1998). Even in London, with its rapidly growing population, business relationships were based on family and kinship networks, and neighbourhoods and communities policed relationships of trust and obligation (Raven 2007). Grafton and Peele appear to have been neighbours at Christ’s Hospital for many years, as well as business contacts, and their relationship exemplifies the communal nature of business relationships in sixteenth century London. Close bonds and social connections are not, of course, necessarily harmonious and business in pre-capitalist, early modern London was not usually based on impersonal, alienated transactions but instead was deeply personal and potentially divisive; Peele recognises that far from being a force for cohesion, the social connections created by poor book-keeping may be those of resentment and dispute. The wish to avoid disputes among neighbours was more than an abstract desire to contribute to the ‘common-weal’; curates were instructed not to allow parishioners to take communion if they were in dispute with their neighbours: ‘those betwixt whom he perceiveth malice, and hatred to reigne, not suffering them to be partakers of the Lordes table, until he knowe them to bee reconciled’ (Cummings 2011: 124). Good neighbourly relations were not only financially and socially desirable, they were a spiritual requirement.

[36] Peele points out that where records are badly kept, it is not even possible for a man to know for certain if he has been cheated. The festering resentment of someone who thinks he has been done wrong but cannot prove it, or indeed, of one who cannot prove his honesty, destroys the desired fellowship of man:

For often times the lawes is attempted of some one man against his frende or neighbour, but even of suspicion. For that his reconynges, through want of a perfecte ordre, have been negligently kepte, fearyng that he hath been deceived, when that he is not thoroughly hable to saie (with clear consience) whether he have been deceived in any thyng at all, or not. (1553: sig.A3)

The uncertainty caused by imperfect reckonings could cause suspicion and unease, not just between neighbours, but within a man’s own mind when he suspected but could not prove that he had been cheated. Ordered accounts enable ‘friendly’ reconciliation of differences and avoid ‘descencion’; they also permit a man to pursue his debts with a clear conscience, certain that his reckonings are in perfect order. The fictional Merchant of The pathe waye to perfectnes, whose accounts are in disorder, cannot be at peace with himself; he says that he is ‘at discorde with my selfe’ and that he needs the Schoolmaster ‘to helpe me to renewe the frendship betwene me and my selfe.’ (1569: A1)

[37] Peele writes of his system of book-keeping as an instrument to eliminate discord and, like any utopian scheme aimed at ‘perfectnes’, it is almost certainly doomed to fail. Smyth comments on the gulf between the theory articulated by writers including Peele and the actual financial records that he has studied: ‘after such encomiums of accounting methods, extant manuscript accounts often appear bathetic, disordered, half-hearted, thin’ (Smyth 2010: 100). Despite Peele’s efforts, and the appearance of at least six manuals of double entry book-keeping in English by 1600, the system was not widely adopted until several centuries later (Yamey 1982, 1999).

[38] The double-entry system, by which an authoritative set of financial records is constructed from ‘parcels’ of transactional information, mirrors the educational method of ‘gathering’ and ‘framing’ aphoristic extracts of canonical works in order to create the authoritative humanist subject (Crane 1993). The manner in which Peele instructed the books of account to be prepared would seem familiar to those educated in the grammar schools, where the schoolmaster provided texts of Latin classics from which sentences were translated to supply the controlled and authoritative fragments that filled pupils’ commonplace books. In the accounting system described by Peele, the original record is the Memorial, which is a common book which ‘serveth for every servant of the house, to write therin all suche thynges as are by them received or delivered in the absence of the master, or the accompt keeper’ (1553: A4). The compilation of the Memorial as a comprehensive book of record is required, ‘wherinto all things for trafique in occupyinge are to be entered at large uppon the present doing: eyther by the master him selfe, thaccompte keper, or anie other of the servantes, as often and when so ever anie of them have occasion to deale therin […] to thintent, that nothinge should slippe oute of memorie, or be left unwritten.’ (1569: A5v) From this source book, which contains the records of all members of the household who are able to write and to receive or deliver goods, the account keeper condenses his entries ‘oute of the saide memoriall boke into percelles with the phrase and order of debitour and creditour’ (1553: A5v). The Journal or daily book kept by the account keeper is an ordered list of fragments, or ‘parcelles’, taken from the larger and more digressive Memorial, and is therefore, like the book of common-places compiled by the humanist scholar, a distillation of the larger text into its nuggets of essential information. Double-entry book-keeping is an attempt to realise a stable and authoritative financial record, by processes of ‘gathering’ parcels and ‘framing’ them using a prescribed methodology, just as the humanists tried to establish a stable and authorative use of language by gathering and framing axiomatic extracts from classical works in their commonplace books (Crane 1993). This process of writing and re-forming of data in the double-entry system has been described as ‘an exemplary model of production that informed and shaped the movement of other kinds of texts’ (Smyth 2010:10). Integrated into, rather than separate from, a literary production of ‘self’ through life-writing, accounting manuals like Peele’s ‘established a strong link between particular methods of arranging financial records, and ideas of reliability and truthfulness’ which served as a template that life-writers could adopt when selecting relevant facts, shaping and forming them into a narrative (Smyth 2010: 4).

[39] The dialogue form that Peele adopts in The Pathe waye to perfectnes was a fairly common expository device in sixteenth century instructional texts, used, for example, by Robert Recorde in The Ground of Arts (1543). Peele’s, however, has a more than usual amount of characterisation; in his text, the exposition is enlivened by a tetchy schoolmaster, a rueful merchant and a bumptious student, whose dialogue defines their entries and exits, almost as if in a play. Peele uses his fictional characters to convey the message that acquiring social capital through education is laborious; the serious demeanour of the humanist is seen to confer and justify power, in contrast to the inborn ease with which aristocratic courtiers were expected to dignify their status. The Merchant who has lost control of his accounts is rebuked when he inappropriately adopts courtly sprezzatura in claiming to effortlessly understand the method that has just been explained to him. He is sharply rebuked by the Schoolmaster who ‘loves no jest’:

Marchaunte. This is profitable in dede, and as playne & pleasaunt as maye be, and utterlye voyde of the tedious circumstances that I have harde hath belonged therunto, but if ther be no more curiositie then herin apeareth, I understande this verie well.
Scholemaster. I am glad that you have so quicklie conceyued it, but yet I praye you let me heare how you understand of the same?
Marchaunt. Nay soft sir you shoute verie swyft: I perceave you love no jest.
(1569: A4v)

The humourless Schoolmaster is only returned to equanimity when the Merchant defers to his role as teacher and the precedence of laborious learning over his inappropriate courtly ease is restored. Peele emphasises his own labour when he writes of ‘my peines and diligence herin’ and instructs his reader to ‘read therefore and use this my labour for thy commoditie, I doubte not but it shalbe as profitable to thee as to me painfull.’ (1553: A3); ‘Practise’ is required to ‘procure perfection’ declares the title page, while the Preface to The pathe waye to perfectnes refers to ‘my great and paynfull travell’ (1569: *4) and in the verse ‘A Brefe admonition’, ‘Godes bountie’ is earned ‘with diligent care and studious paine.’ (1569: *2V).

[40] In the ‘Epistle to the Reader’ of The Pathe waye to perfectnes, Peele responds to the Ciceronian duty to share his knowledge for the common good:

In consideration (gentle Reader) that no man is borne to benefite him selfe onelye: But that all men in there callinge are bounde to further others, as theye wolde others sholde further them: Therfore to the intente that I woulde not bee idle and negligente amonge the reste, which travelith to profet the common weale. I have (in the facultie, which I professe) travayled and set forthe this worke. (1569: *3)

Peele stresses the laborious nature of the work and his lack of self-interest; he had already made similar claims in the ‘Epistle to the Company of Merchant Adventurers’ in The maner and fourme: ‘I do not so muche seke my private commoditie, as I have respect to the commune profite of my countrey, and your thankefull love’ (1553: A2V). Of course, it would not be unreasonable to regard this rather generic self-deprecation with a certain amount of scepticism; no doubt Peele’s private commodity and the communal profit were not mutually exclusive and the Merchants’ ‘thankefull love’ may well have been expressed financially.

[41] Aspects of the contents and form of The Pathe waye to perfectnes support the notion that Peele was attempting to fashion himself as a humanist scholar. His stated intention to contribute to the common weal, the use of humanist educational techniques such as the dialogic form and the parcel method, and the serious diligence of his Schoolmaster avatar all support the hypothesis that the image of Peele on the frontispiece is intended to be read as a portrait of a writer engaged in a serious intellectual project.

[42] In conclusion, while we are unlikely to confuse them with the Bible, the number work that Peele’s books promote, as a specialised form of writing, is elevated to the status of the other sixteenth-century humanist texts and reformist works. The material features of The maner and fourme, which are shared with Richard Grafton’s other publications, direct the careful reader to see double-entry book-keeping as part of a wider intellectual programme. Further, they demonstrate the possibilities offered by English humanism for self-fashioning, both to an author like Peele, and to the students to whom he hoped to teach his method. In practical application their influence was limited, but as representatives of sixteenth-century intellectual discourses Peele’s works have been unfairly neglected. Reading Peele’s works with attention to their material manifestation as books establishes their previously over-looked position in the cannon of English humanism.

Birkbeck, University of London



My thanks are due to Adam Smyth, Richard Macve and the two anonymous reviewers for their help in preparing this article; all faults remaining are, of course, my own.

[1] There are two copies of Christoffels’ book (ESTC S95937) listed in the English Short Title Catalogue. The copy listed as being held at the library of Columbia University is in fact a reverse negative photostat* of the apparently unique copy is held in the State Public Historical Library, Moscow. This usage of the compartment does not appear in McKerrow & Ferguson’s catalogue, probably because they were not able to access the Moscow copy in 1932. A high quality photocopy is held by the ICAEW in London. At the present time, the book does not appear on Early English Books Online. *My thanks to Jane Siegel of the University of Columbia Rare Books Library for her confirmation of this point. [back to text]


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Donne, by Number: Quantification and Love in ‘Songs and Sonnets’

Donne, by Number: Quantification and Love in ‘Songs and Sonnets’

James Beaver

[1] What does counting have to do with love? Elizabeth Barrett Browning’s ‘Sonnet LXIII’, which opens, ‘How do I love thee? Let me count the ways’ (2007: l.1), offers perhaps the most well-known correlation between the two. Yet the epistemological surety Browning finds in quantifying love is no given for a love poet. In ‘Love’s All (Lovers’ Infiniteness)’, John Donne, for example, refuses Browning’s inclination to count, worrying over what, exactly, a sum total would mean for his love: ‘Yet I would not have all yet, / He that hath all can have no more’ (Donne 2008: ll. 23-24. This edition used throughout). Compared to Browning, Donne appears a quantitative sceptic: a basic principle of number and multitude tells him that once a quantity is exhausted, ‘no more’ can be added. John Carey remarks of Donne’s anxiety over measurement: ‘These are conflicts we stumble into once we try to quantify emotions—yet wanting them quantified is the most natural thing in the world’ (1981: 126). Although we might empathise with Carey’s assertion on a personal level, the suggestion that quantifying love is natural seems to overlook the mathematical and poetic discourses in which these poets participate when they presume, refuse, or worry over what it means to ‘count the ways’. In other words, to what extent is Carey’s ‘inclination to quantify love’ a product of cultural experience?

[2] While criticism since the mid-twentieth century has explored Donne’s relation to early modern scientific developments (Williams 1935; Coffin 1957; Empson 1957), more specific attention to the relationship of Donne’s work to mathematical developments in his time needs to be developed. This essay offers an assessment of how developments in mathematical symbolic language — specifically, Robert Recorde’s contribution to novel notational figures — in mid-sixteenth century England may have influenced Donne’s poetics. While Carey concludes that ‘[Donne] seems to have regarded mathematics as, at best, a curiosity’, and that, ‘like most Elizabethans, Donne had virtually no interest in using numbers for the purpose of serious computation’ (1981: 128-29), Donne’s poetry suggests, alternatively, a sustained investment in quantitative language as a mode of articulation. Poems like ‘The Computation’ and ‘The Primrose’ serve as obvious examples of ways in which Donne could explicitly rely on counting to articulate love and loss, with the former tallying time passed since a lover’s absence, and the latter, operating within a neo-Pythagorean numerology to calculate an equation by which men and women add together. In that poem’s closing lines, ‘Numbers are odd, or even, and they fall / First into this, five, women may take us all’ (ll. 29-30), A.J. Smith notes that Donne is calculating the sum of the gendered numbers ‘2’ (for female) and ‘3’ (for male) to equal the five petals of the primrose (1996: 396). Such abstract number games, in the end, are used to represent lovers’ experience, whether the physical absence of a partner for a lengthy period of time or the erotic suggestiveness of a world in which, hypothetically speaking, ‘women may take us all’ (l. 30).

[3] Even more compelling than these specific examples, a brief survey of Donne’s love poetry reveals the striking frequency with which his poetics make use of quantitative language. Thirty-four of the fifty-four ‘Songs and Sonnets’ contain some reference to number, such as ‘one’, ‘two’, ‘a hundred’, or ‘thrice’. Donne is even more preoccupied with relationships between ‘all’ and ‘nothing’: he refers to ‘all’, ‘nothing’ or ‘none’ in thirty-six of the fifty-four poems. Overall, fifty-two poems contain some type of quantitative language, whether a number, ‘all’ or ‘nothing’, or less specific quantitative references like ‘increase’ and ‘proportion’. The sheer volume of such references hints that Donne has a vested interest in quantitative thought as a means of representation. He may not be a ‘serious’ student of mathematics, as Carey observes, but his language is especially computative. As defined by Aristotle, ‘quantum’ is concerned with that which is ‘numerable’ or ‘measureable’ (1020a10), and, as Paula Blank observes (2006: 41), it can be viewed in contrast to ‘quality,’ which, for Aristotle, refers both to ‘the differentia of the essence’ (1020a33) and ‘the sense in which numbers have a certain quality’ (1020b3-4). Quantitative language, in my definition, consists of verbal expressions of numerical degrees and spatial or temporal magnitudes, in contrast to verbal expressions of kinds or essences.

[4] What does it mean for a poet to have numbers on his mind in turn-of-the-century early modern England? In one sense, as Margaret Ferguson observes, it simply means being a poet: ‘As a verbal practice marked both by a concern for “measure” (as in syllable or line counting) and by a tendency toward “license” (as in rule-breaking and a love for excess), poetry enacts and reflects on many meanings of “numbers”’ (2013: 78). In another sense, though, this question requires not only our awareness of the broader critical narratives which chart a dramatic shift from medieval structures of knowledge to scientific modernity occurring in the fifteenth, sixteenth and seventeenth centuries (Daston 1991; Dear 1995; Crosby 1997), but also an assessment of both the numerological traditions and particular mathematical developments in England which contributed toward defining early moderns’ relationships to numbers. My focus here lies in what purchase Donne gains in his poetics by relying so frequently on quantitative language as a mode of articulation, insofar as such language alludes to a developing symbolic mathematical language that, in the seventeenth century, would assume epistemological claims that the English vernacular could not. As Paula Blank observes, in addition to ‘mimetic theories of poetry’ and ‘ones that assimilate verbal art to the visual arts or to visual representation more generally’, ‘sixteenth-century poets […] additionally thought of their writing as an instrument of measure, one that proceeded quantitatively rather than qualitatively, deciding relations rather than depicting “nature”’ (2006: 41). In this view of poetry, questions of numerological correlations between poetic structures and the order of the cosmos become secondary to questions of taxonomy. Symbolic mathematics and the language of words constituted two developing, related semiotic systems at the time, with shifting epistemological claims to the world and varying capacities for representing the realms of experience. Donne’s predilection for quantification highlights both the sympathies and disjunctions between these developing systems.

[5] As J.L. Lemke says of the relation between the semiotic systems of mathematics and verbal language:

No mathematical treatise entirely avoids the connective tissue of verbal language to link mathematical symbolic expressions, to comment on the process of development of arguments, and so on. All our applications of mathematics, in the context of which most of our present commonly used mathematics evolved historically — in the natural sciences, engineering and design, commerce and computing — require verbal language to link mathematical tools to specific real-world things and events. (2002)

Lemke’s understanding of the relation between mathematical and verbal languages is informed by a theoretical framework drawn from C.S. Peirce and pragmatism (Buchler 1955; Houser and Kloesel 1992; Halliday 1978; Hodge and Kress 1988; Lemke 1995).[1] Despite his observation of the ‘link’ which verbal language provides for mathematical expressions, Lemke’s primary focus is on how mathematics functions as an independent semiotic system, capable of producing meaning which verbal language cannot. My own approach diverges from the theoretical investments of pragmatism but uses Lemke’s observation as a frame for considering economies between mathematical and verbal languages in a period when distinct boundaries between the two systems were not yet well defined. Although a range of valuable work has studied the effects of the increased cultural reliance upon quantification in the early modern period (Crosby 1997; Reiss 1997; Poovey 1998), the ‘link’ between ‘the connective tissue of verbal language’ and ‘mathematical symbolic expressions’ opens a particularly valuable critical space to historicize relations between the two languages. Specifically, it allows us to study how notational developments in mathematics — on the order of syntax and symbols — are managed, or framed, through what Lemke calls ‘verbal language’, or, in the present essay, I deem the English vernacular.

[6] Building from the seminal work on the history of mathematical notation by Florian Cajori, Lemke observes, ‘In most mathematical writing before modern times, symbolic expressions were rare; they were integrated into the running verbal text, and they were clearly meant to be read out in words as part of complete sentences that also included ordinary words’ (2002). Despite this long-standing historical relation between words and mathematical symbols, Jacob Klein, in a remarkable study of the algebraic innovations from the mathematicians Simon Stevin and François Viète, argues that a new conception of number emerges in the early modern period, in which ‘the fundamental ontological science of the ancients is replaced by a symbolic discipline whose ontological presuppositions are left unclarified’ (1968: 184). In other words, mathematics began to develop into a coherent, autonomous semiotic system, as disciplinary practice shifted from classical questions of the ontological status of numbers (arithmos) toward semiotic considerations on notation (Klein 1968; Rotman 1987; Hodgkin 2005). My intention here is to interrogate one example of the formulation of mathematics as an autonomous symbolic discipline in sixteenth-century England, in order to then consider how the poetic text relies repeatedly on the ‘connective tissue of verbal language’ to allude to symbolic mathematical expressions. When Donne, in ‘The Primrose,’ concludes, ‘if half ten / Belong unto each woman, then / Each woman may take half us men’ (ll. 25-26), he is using the vernacular to articulate an equation linked to mathematical symbolic expressions. What does it mean when a poetic text, like ‘The Primrose’, relies on verbal language to facilitate such a relation to mathematical semiotic fields?

[7] One can pose this question, I think, while keeping in mind, as Keith Thomas does in his seminal essay on numeracy in England, ‘the importance of numbers in religious symbolism and allegory or of numerology in poetry and philosophical speculation’ (1987: 124). In ‘The Primrose’, for example, Donne is quite clearly appealing to symbolic numerical values as he calculates relations between men and women, such that they equal ten. Robin Robbins observes that ten is a ‘triangular number, 1 + 2 + 3 + 4, symbolic of holiness and perfection’ (Donne 2008: 237). As Thomas notes of number in the period, ‘At all levels of society, number remained as much a language of quality as of quantity’ (1987: 124). As an early modern test case, Donne, no doubt, makes use of both dimensions of number in his poetics. And yet, while acknowledging the qualitative—the symbolic or allegorical—dimensions in Donne’s numbers, as studied by mid-to-late twentieth-century numerological criticism (Fowler 1964, 1970, 1971; Patrides 1958; Heninger, Jr. 1974; Røstvig 1994), we may still consider to what extent his poetics engage features of a quantitative numeration derived from the developing symbolic mathematics.

[8] With respect to developments in mathematics during the Renaissance, England was well behind the Continent. As Cajori observes, “Up to the seventeenth century, mathematics was cultivated but little in Great Britain. During the sixteenth century, she brought forth no mathematician comparable with Viète, Stifel, or Tartaglia’ (1991: 146). Thomas confirms England’s rather slow start, noting, ‘Arabic figures had come relatively late to England; it was only between the mid-sixteenth and mid-seventeenth centuries that they established themselves in most forms of account-keeping’ (1987:120). Furthermore, because of this delay, ‘Arabic figures had to be explained to those who found them strange and unfamiliar’ (Thomas 1987: 120). Along with John Dee, who provided the first English translation of Euclid, Robert Recorde emerged as a key figure in sixteenth-century English mathematics, publishing the first arithmetic, algebraic and geometric books in English.

[9] Gareth Roberts and Fenny Smith have recently emphasised the significance of Recorde’s place in the early history of mathematical development in England. Notably, they do not associate his value with theoretical contributions: ‘Recorde himself made no claim that he was pushing forward the frontiers of mathematics. He was, rather, a communicator of mathematical ideas who sought to explore ways in which to make mathematical knowledge and skills available to a wide population’ (2012: 1). Recorde’s achievements lay in the semiotic realm, specifically, in his dual emphases on the vernacular and symbolic notation. Recorde exemplified what Richard Mankiewitz identifies as ‘the tendency to write in the local vernacular in preference to Latin,’ a practice which ‘made mathematical textbooks accessible to a wider public’ (2000: 70). During his lifetime, Recorde published five mathematical texts, all in English.

[10] In The Pathway to Knowledg [sic] Containing the First Principles of Geometry (1551), he explains in his dedicatory epistle to Edward VI his civic aspirations for a treatise written in English:

And I truste (as I desire) that a great numbre of gentlemen, especially about the courte, whiche vnderstand not the latin tong, or els for the hardnesse of the mater could not away with other mens writyng, will fall in trade with this easie forme of teachyng in their vulgar tong, and so employe some of their tyme in honest studie, whiche were wont to bestowe most part of their time in triflyng pastime. (1551: sig. B3v)

Here, Recorde’s appeal to his sovereign invokes the humanist argument that knowledge of the classics would yield more effective servants of the state. If Recorde’s justification for the vernacular followed this long-standing humanist argument, his emphasis on symbolic mathematical notations, with respect to Arabic numerals, was a bit more novel. His best-known work, The Whetstone of Witte (1557), contains the first recorded use of  (=) for the equals sign in mathematical notation, in both England as well as the Continent (Mankiewitz 2000: 71). In the same text, Recorde uses the now-familiar modern notation of the addition (+) and subtraction (-) symbols for the first time in an English text. Recorde’s pedagogical contribution to symbolic mathematics reflects his aim to communicate its practices and principles to a broader English public.

[11] Of a particularly complex, detailed section on algebra in Whetstone, Ulrich Reich says the following on Recorde’s approach:

In retrospect we witness in this section part of the detail of the laborious process as mathematicians struggled over several centuries to get to grips with equations and their solutions, their efforts often frustrated and encumbered by unwieldy symbolism. Recorde visibly strives to improve the processes, by introducing enabling symbolism, by dismissing the ‘idle bablying’ of overelaboration, and by diligently searching for simplifications. (2012: 113)

If Latin was a linguistic barrier to certain English gentlemen’s ability to learn mathematics, so was this ‘unwieldy symbolism.’ The (=) symbol arises out of Recorde’s drive for notational simplicity: ‘And to auoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke vse, a pair of paralleles, or Gemowe [twin] lines of one lengthe, thus: ===== bicause noe. 2. thynges can be moare equalle’ (1557: sig. Ff2r). Like mathematicians before him, Recorde recognizes that symbolic mathematical notation becomes necessary as calculations become more frequent and complex. He could be called a notational pragmatist, in this sense, and, as such, he is well aware of the needs of his English readership, who may be unfamiliar not only with Latin, but with Arabic numerology and the basic syntax of mathematical operations. His ‘[twin] lines of one lengthe, thus: =====’ invokes a direct visual appeal to the reader, ‘bicause noe. 2. thynges can be moare equalle’.

[12] Recorde’s work as a communicator of mathematics means he spends considerable effort educating his audience on how to read the new semiotic system before it. This requires a basic hermeneutic skill set that acknowledges the fundamental disparity between the semiotic fields of symbolic mathematics and the vernacular. In a dialogue at the opening of the 1582 edition of The Grounde of Artes Teaching the Perfect VVorke and Practise of Arithmetike (originally published in 1543), the Master explains to his Scholar the relationship between (numerical) value and (signifying) figure (i.e., ‘6,’ ‘2,’ ‘8’), emphasizing both the link between numbers when they are connected together as well as the disjunction between numbers and words:

MAYSTER [sic]: Nowe then take héede, these certaine valewes euerye figure representeth, when it is alone written withoute other Fygures ioyned to him. And also when it is in the firste place, though manye other doe followe: as for example: This figure 9 is ix. standing now alone.

SCHOLER [sic]: Howe? is he alone and standeth in the middle of so many letters?

MAISTER [sic]: The letters are none of hys fellowes. For if you were in Fraunce in the middle of a M. Frenche men, if there were no English man with you, you woulde recken your selfe to be alone.

SCHOLER [sic]: So it is. Then 9 without more figures of Arithmetike, betokeneth ix, whatsoeuer other letters be aboute it. (1582: sig. D1r-D1v)

In this exchange, the Master delineates the fundamentals of numeration: value, figure, and place. Recorde is aware that the novice in mathematics needs to understand the value of a figure which ‘is alone written withoute other Fygures ioyned’ (i.e., ‘the figure 9 is [in value] ix.’) with respect to the vernacular. The Scholar is initially baffled by the disjunction between numerical figure and alphabetical letters: ‘Howe?’ he asks of the value of 9, ‘Is he alone and standeth in the middle of so many letters?’ The Master answers in a way we might expect from the author who justifies his use of the vernacular for the good of the English state: ‘The letters are none of hys fellowes,’ he says, just as an Englishman, surrounded by a thousand Frenchmen in France, ‘recken[s]’ [sic] himself alone. Recorde’s attention to this basic differentiation indicates that before he can expound on the syntax of symbolic mathematics, he must first clarify that there is a different syntax — or language — available to the student in mathematical signification, and, furthermore, that numerical figures and vernacular figures form separate sets of communities, as distinct, or so Recorde would have it, as national identities.

[13] Let us briefly pause to examine two features of the relationship Recorde establishes between mathematical language and the English vernacular. On the one hand, the substitution of the (=) symbol arises both from a pragmatic aim to ‘auoide the tediouse repetition of these woordes: is equalle to’ and because Recorde often uses the symbol in his own work. On the other hand, when it comes to the basic principles of numeration and the vernacular, a distinct boundary is posited. ‘Howe is he [the 9] alone and standeth in the middle of so many letters?’ the Scholar asks, only to hear the Master present a rule which emphasises starkly divisions in national identities: ‘The letters are none of hys fellowes’ (1583: sig. D1v). The division is, of course, ideologically charged, as Recorde plays upon the reader’s sense of his own Englishness as a discrete cultural category precisely at the moment he urges the reader to ‘recken’, or count: if you can understand how an Englishman ‘recken[s]’ [sic] himself alone in the middle of a thousand Frenchman, you can recognize the isolation of this symbol from words surrounding it. If, as Lemke notes, ‘No mathematical treatise entirely avoids the connective tissue of verbal language to link mathematical symbolic expressions’ (2002), we can observe here how economies between this connective tissue alter depending on what is at stake.

[14] Like mathematicians before him, Recorde can easily collapse words into a symbol to achieve linguistic elegance. In A History of Mathematics, Cajori demonstrates, by reviewing studies of medieval manuscripts, that the emergence of the (+) symbol for addition followed a similar operation: ‘the sign + comes from the Latin et, as it was cursively written in manuscripts just before the time of the invention of printing’ (1991: 140). Thus, a calculation of ‘5 et 7’, over a longe period of time, eventually became ‘5 + 7’. Here, then, the Latin literally collapses into symbol. Recorde explains his use of (+) and (-) in Whetstone as follows: ‘There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made – and betokeneth lesse’ (sig. S2v). Though the substitution of symbols for words may appeal to common sense, the tradition of notation in which Recorde engages in this moment is, in fact, fraught with tensions: specifically, tensions between words and symbols. As Cajori notes, the symbolic notation for addition and subtraction was also configured in letters which more directly corresponded to verbal counterparts: ‘In [Luca Pacioli’s] Summa de Arithmetica, Geometria, Proportioni et Proportionalità, the words ‘plus’ and ‘minus,’ in Italian piú and meno, are indicated by and ’ (1928: 107). These abbreviations spread across the Continent after their introduction, and ultimately rivalled, in representational capacity, the (+) and (-). ‘The + and –, and the and , were introduced in the latter part of the fifteenth century, about the same time,’ Cajori observes. ‘They competed with each other for more than a century, and and   finally lost out in the early part of the seventeenth century’ (1928: 236). The transition from the words ‘plus’ (più) and ‘minus’ (meno), to, respectively, the abbreviations and , and, then, in their eventual erasure and displacement from modern mathematical language, the (+) and (-) symbols, exhibits over the course of a long period of time a remarkable semiotic fluidity between the vernacular and mathematical symbolic language. It is the possibility afforded by this fluidity that allows Recorde to introduce, quite unremarkably, the substitution of (=) for equivalence.

[15] In this relatively early stage in the development of modern symbolic mathematical language in England, Recorde’s emphasis on notation is important. While not at the forefront of theoretical developments in mathematics, he does occupy a place at the forefront of the expansion of a symbolic mathematical language—in emphasising a consistent vocabulary and syntax apart from the vernacular, while simultaneously promoting the accessibility of mathematical texts in the English vernacular. Reich describes his legacy as follows:

This legacy includes an extensive mathematical vocabulary, largely based on borrowings from Latin and continental European languages, notably Italian, German and French. […] The legacy also extends to Recorde’s use of symbolism as part of a process of simplifying mathematical expressions: his use of the sign of equality together with the signs for addition and subtraction for the first time in English, coupled with use of the cossic signs and signs indicating square and other roots borrowed from Scheubel, established an unprecedented style of mathematical writing.   (2012: 116-17)

Recorde’s stress on ‘simplifying mathematical expressions’ to achieve a symbolic language that could simultaneously appeal to a broader English public and perform more complex operational functions is echoed by the more prominent English mathematicians who followed him, especially those in the first half of the seventeenth century.

[16] England’s most famous mathematician of the early seventeenth century William Oughtred published Clavis Mathematicae (1631), a landmark mathematical treatise at the time, and introduced the (X) symbol for multiplication, replacing the symbol of a cross (Smith 1958: 404-405). According to Cajori, ‘[Oughtred] laid extraordinary emphasis upon the use of mathematical symbols; altogether he used over 150 of them’ (1991: 157). Like Oughtred, John Wallis was an advocate of expanding the scope of mathematical symbolization, as Gordon Hull describes:

Thinkers such as Wallis, innovators in mathematical developments, expressed fewer or even no such worries about the scope of symbolization. Thinkers who straight-forwardly adhered to a premodern understanding of science tended to downplay the role of construction in knowledge acquisition. (2004: 121)

Such enthusiasm might be said to reflect an attitude held by many of those at the forefront of mathematics in the English Renaissance. An increased complexity of computations was accompanied by a more nuanced language, as evident in the expansion of symbolic representation to more complex ideas, such as infinity. In 1655, Wallis published De Sectionibus Conicis, the text widely credited with the first use of the lemniscate, the modern symbol for infinity (∞) (Cajori 1928: 214; Maor 1991: 11). Wallis’s work, especially his Arithmetica Infinitorum (1656) in which the lemniscate symbol appears again for infinity, is often considered a precursor for the development of calculus in the latter half of the century. Infinity, both as symbol and concept, reflected the increased level of sophistication in symbolic mathematical language. It also, of course, had deep symbolic implications in the realms of philosophy and theology, as Brian Rotman has demonstrated in wonderful detail (1987). The lemniscate, itself, channels symbolic traditions, both ancient and Christian. John Barrow observes, ‘The ribbon like figure-eight on its side is an ancient symbol, a shadow of the ancient ourobos symbol of the snake eating its tail’ (2008: 339). It also ‘provided the mysterious cross of St. Boniface in early Christian tradition’ (Barrow 2008: 339). Whether Wallis had either in mind is uncertain, but the difficulty in ascertaining the answer reflects Wallis’s unconcern with ‘the scope of symbolization’, as Hull phrases it (2004: 121). Rather than considering deeply the symbolic roots of the lemniscate, Wallis, like Recorde before him, was probably mostly concerned with introducing a functional, useful sign that could accommodate more difficult operations.

[17] As the breadth of symbols expanded, mathematical relationships were also becoming more crucial to cultural articulation in the early modern landscape. As William Bouwsma succinctly puts it, ‘quantity now tended to be substituted for quality as the essential principle of orientation’ (1980: 234). In other words, quantification afforded a means of articulation that extended beyond the mathematical. In a poem to the preface of Thomas Hylles’s The Arte of Vulgar Arithmeticke (1600), an anonymous poet finds in numbers nothing less than the foundation of all language: ‘No state, no age, no man, nor child, but here may wisdome win / For numbers teach the parts of speech, where children first begin’ (1600: sig. C3v). In this celebration of number’s expressive capacities, we may well wonder: what of words?

[18] The tension arising in what Christopher Johnson identifies as the ‘epistemological confusion’ in the period (2004: 67) plays out, on at least one level, between the developing mathematical language of symbols and the language of words. Hylles’s poet’s tribute to number reveals in its very lines precisely this tension: the poet’s ‘number’ shares its etymology with the ‘number’ of ‘metrical periods or feet; lines, verses’ (OED, ‘number’, n. 17a). As Blank and Ferguson have studied, this sense of ‘number,’ now rare, was significant to conceptions of poetry in the sixteenth and seventeenth centuries (2006; 2013). The OED lists two entries from the end of the sixteenth century: E.K. glosses line 110 in the October Eclogue of The Shepheardes Calendar (1579) as, ‘The numbers rise so ful, and the verse groweth so big, that it seemeth he hath forgot the meanenesse of shepheards state and stile’; Shakespeare, too, in Love’s Labour’s Lost (1598), writes, ‘I fear these stubborn lines lack power to move / O sweet Maria, empress of my love, / These numbers will I tear, and write in prose’ (IV. 3. 53-55). Such examples reflect the dual role ‘number’ played in the early modern period. Its root, from the Latin numerus, meant ‘sum, total, numeral,’ but also ‘rhythm in words or music, grammatical number, metrical foot, (plural) metrical lines, musical strains’ (OED, ‘number’, n.). In the Old French and Middle French nombre, the OED dates the quantitative definition earlier than the metrical: ‘sum, total (early 12th century as numbre), grammatical number (13th cent)’ (‘number’, n.). Traced from the Middle French, ‘number’ exhibits a tension still very much at play in the sixteenth and seventeenth centuries.  Examining George Puttenham’s mathematically-inflected treatment of meter in The Art of English Poesie (1589), Blank contends that ‘his work is representative of an early modern habit of confusing arithmos (for Puttenham, a variant of “arithmeticall”), rithmos, arithmetic, and the Latin ars metrica’ (2006: 42). Moreover, this confusion or tension in ‘number’ thrives precisely in the period in which symbolic mathematical language takes shape. How does the ‘number’ of verse relate — or compete — with mathematical symbols?

[19] Writing at the turn of the century, John Donne was keenly aware of the tensions at the foundations of early modern thought. In his wide knowledge base, he was somewhat of an anomaly: a former aspiring courtier (‘Jack Donne’) who became an Anglican priest (‘Dr. Donne’), and, throughout, remained well-read in the most recent scientific and mathematical developments of his time. Herbert Grierson wrote that ‘no other poet in the seventeenth century known to me shows the same sensitiveness to the consequences of the new discoveries of traveler, astronomer, physiologist and physician as Donne’ (1912: 189). Donne’s understanding of the implications of scientific and mathematical developments of the period underlies the quantitative references in his poetry, as he illuminates both the anxieties and paradoxes that the early modern period experienced with what Carla Mazzio identifies as the ‘newly assertive regime of standard measurement’ (2004: 60). Donne folds quantitative language effortlessly into his lovers’ experiences, suggesting its expansive representational capabilities; and yet, he repeatedly interrogates what it means to ‘count the ways’ of love: how quantification represents and orients experience, as well as how it fails to do just that. When the quantifiable enters Donne’s verse, it is always teetering under the instabilities of hyperbole and competing forms of measurement. In this view, he figures as the counterpart to Blank’s depiction of Shakespeare in the Sonnets, who, she contends, ‘imagines unsettled, unstable, and uncertain relationships among the parts of his created works’ (2006: 43). Most of all, although he demonstrates an interest in what symbolic numerology offers his accounts of love, Donne persistently refuses the expanded scope the quantifiable extends towards the regulation and systematization of experiences in space and time: Donne’s lovers do not live on grids.

[20] In both ‘The First Anniversary’ and ‘The Triple Fool,’ to take examples of his probing of the disparities between quantification and experience, he sets quantitative language in relationship with emotion, specifically ‘grief’. In ‘The First Anniversary’ grief is that which is ‘without proportion’: ‘Since euen griefe it self, which now alone / Is left vs, is without proportion’ (ll. 307-8). In ‘The Triple Fool’ grief is unruly, that which the narrator proposes to tame in ‘numbers’: ‘Grief brought to numbers cannot be so fierce, / For, he tames it, that fetters it in verse’ (ll. 10-11). Just as proportion is lost in grief in ‘The First Anniversary’, so is there a hope to enclose grief in ‘numbers’ in ‘The Triple Fool’ — perhaps to regain proportion?

[21] These examples have two implications. First, in Donne’s poetics, quantitative language is made to engage the language around it. In ‘The Triple Fool’, Donne’s speaker sets ‘numbers’ beside ‘grief’ in a misguided attempt to impose order. This is critical: for Donne, ‘numbers’ do not exist in a vacuum, surrounded by symbols and signs, as in the mathematical equation, ‘9+13=22’, for instance. Donne’s ‘numbers’ are not separate from the terms that surround them; they must, as Recorde advises against, meet the other’s ‘fellowes’ (1583: sig. D1v). Like Recorde’s Frenchmen and Englishman, numbers and words exist uneasily side by side, but in Donne’s example, those ‘numbers’ are forced into conversation with the words that surround them—in this case, ‘grief.’ The result is a pairing of disparate modes of experience, orientations to the world that are incompatible with one another. ‘Numbers’ are removed from the isolation of mathematical language and made to engage with the resonances of an emotion, and they do so in the vernacular.

[22] The second implication of these examples is that quantitative language does something with our world and our ways of thinking: ‘numbers’ do not sit benignly beside ‘grief’ — they work upon it. Quantitative language becomes in ‘The Triple Fool’ a way of organizing our world that lays representational claims upon the non-mathematical. In this sense, Donne is quite aware that the discoveries in mathematics and sciences at the time have very real implications on our everyday language and perceptions.

[23] For Donne quantitative language is ultimately a means of orientation. Even the non-mathematical can be defined based on its relationship to the mathematical: grief is simply that which is ‘without proportion’. Yet, if the quantitative is such an active, pervasive means of orientation, from where does it derive its authority? As a language, or semiotic system, mathematics is invention, built upon notational innovations like Recorde’s (=), (+), and (-) symbols. In this sense, orientation derived from mathematical language is artificial. Yet, mathematics is also a means of interpreting the natural world. In this latter sense, it lays claims upon reality — as Galileo’s ‘language’ of God suggests.

[24] Donne recognizes how easily the distinction between mathematics as invention and mathematics as the ‘language’ of God can be blurred. In ‘The First Anniversary’, a tribute to his patron Sir Robert Drury’s young daughter Elizabeth, who died in 1610, Donne takes the occasion of Elizabeth Drury’s death to critique a way of seeing and interpreting nature that is a product of developments in science. Ostensibly, this is a criticism of empiricists, but when mathematics is taken for the language of empirical reality, it is implicated as well:

We thinke the heauens enioy their Sphericall,
Their round proportion embracing all.
But yet their various and perplexed course,
Obseru’d in diuers ages doth enforce
Men to finde out so many Eccentrique parts,
Such diuers downe-right lines, such ouerthwarts,
As disproportion that pure forme. (ll. 251-57)

In this passage, proportion is linked with human thought, which is then interrupted by observation: ‘We thinke’ the heavens to be ‘Sphericall’, and yet a more ‘perplexed course’ is ‘obseru’d’. Donne is not clear whether proportion is natural phenomenon or human concept, but one might argue that the ambiguity is precisely the point: the cosmological order and human thought are intricately connected in this epistemological framework. Human perception, though, disrupts the connection. Observations lead ‘men to finde out’ those ‘Eccentrique parts’ and ‘downe-right lines’ that ‘disproportion that pure forme’ — ‘disproportion’ deriving from a relationship between a way of seeing and nature itself. Importantly, both ‘pure forme’ and ‘disproportion’ are produced as ways of thinking and seeing that blur human cognition and the natural world, invention and discovery. With the latter, though, we encounter a confused tangle of mathematical words, both numerical and geometric — ‘many Eccentrique parts’ and ‘divers downe-right lines’ that ‘men finde out’ — which function as middlemen between the boundary of things natural and human constructions. The ambiguity here is more problematic than the ambiguity between human thought and the cosmological order in the first example, because we have introduced the terms, or vocabulary, of a different epistemological engagement with the natural world, while a question over that very framework remains: have we produced ‘Eccentrique parts’, or have they indeed been discovered, in nature?

[25] For Donne, the answer is hardly as clear as we might hope. In the wake of developments in mathematics and science, things in nature exist within the human cognitive structures imposed upon them:

They haue empayld within a Zodiake
The free-borne Sunne, and keepe twelue signes awake
To watch his steps; the Goat and Crabbe controule,
And fright him backe, who els to eyther Pole,
(Did not these Tropiques fetter him) might runne. (ll. 263-7)

Donne’s imagery of the ‘free-borne Sunne’’s losing battle with the ‘Zodiake’’s ‘twelue signes’, ‘the Goat and Crabbe’, and ‘these Tropiques’ sets the dramatic tension between an epistemological order gone-wrong and a natural world unable to escape the human eye. The cognitive structure has ‘empayld’, its astronomical ‘signes’ ‘watch’, ‘controule’, and ‘fetter’ the natural world. Number is integral and alive in this scenario: the ‘twelue signes awake / To watch his steps’ (emphasis mine). ‘Controule’ is realized through the union of symbols and human perception, as observers rely upon ‘signs’ to ‘watch’. In the paradox at play for Donne, the world is disproportioned precisely when we attempt to confine it entirely within the parameters of human cognition, of signs and perception.

[26] ‘The First Anniversary’ is as much about the threat to language as it is about the natural world. Donne knows that the way of seeing affects our understanding of reality, and, in turn, the relationship of words to that reality. His use of quantitative language highlights the ways in which the epistemological crisis — or, in Johnson’s terms, “confusion” (2004: 67) — resides in a tension of value at the level of competing languages. Straddling the boundary between things natural and human constructions, mathematical ‘signs’ achieve a tremendous capacity for explaining the world, and, in doing so, deprive verbal language of its own capacity for representing the order of that world. The lament in ‘The First Anniversary’ concerns, in this sense, a loss of meaning in words.

[27] Proportion is merely one linguistic casualty. His emphasis upon it in ‘The First Anniversary’ reflects not only a loss in the natural world, but a loss in language as well. In the wake of recent scientific discoveries, the paths of all heavenly bodies are thrown off: ‘All their proportion’s lame, it sinkes, it swels’ (l. 277). Proportion constitutes a relationship between man and world. It is not only the heavenly bodies whose ‘proportion’s lame’, but our very relationship to them. In this sense, ‘proportion’, the word itself, has gone lame in language. It is ‘lame’ precisely because its discursive value is constrained within the parameters of a scientific or mathematical taxonomy within which the natural world is also contained: ‘Man hath weau’d out a net, and this net throwne / Vpon the Heauens, and now they are his owne’ (ll. 279-280). Is it possible that the proportion Donne considers here is that of symbolic mathematics, represented in our modern nomenclature with (:)? Though the symbol had yet to be introduced, Donne anticipates the implications of the mathematician’s language on his own. The mathematician lays claim to ‘the Heauens’ in his language, and yet distances this claim from the language of words, entrenching it instead within symbols.

[28] It is no wonder, then, that for Donne, ‘beauties best, proportion, is dead’ (l. 307). With the death of proportion, comes the end to its meaning:

And that, not onely faults in inward parts,
Corruptions in our braines, or in our harts.
Poysoning the fountaines, whence our actions spring,
Endanger vs: but that if euery thing
Be not done fitly’nd in proportion,
To satisfie wise, and good lookers on,
(Since most men be such as most thinke they bee)
They’re lothsome too, by this Deformitee. (ll. 329-336)

‘Proportion’, here, means something quite different than the ‘round proportion embracing all’ earlier in the poem (l. 252). Deflated in value, it becomes ‘Deformitee’. Curiously, this ‘Deformitee’ arises from corruption within and corruption without. There is something wrong with our cognitive models — ‘corruptions in our braines’ —yet also with our way of seeing. This is a proportion borne of observation: ‘To satisfie wise, and good lookers on’. We do things in a proportion not necessarily in accordance with nature, but in accordance with the terms of convention. This is ‘proportion’ that is a product of convention, a term framed by and for ‘good lookers on.’

[29] The temptation to satisfy ‘good lookers on’ is, perhaps, unending. The compulsion to measure is difficult to resist. Once ‘Man hath weau’d out a net, and this net throwne / Vpon the Heavens, and now they are his owne’ (ll. 279-280), what more was left but to describe how this net is weaved, to build the language? Wallis introduces the symbol for infinity (∞) in 1651, in what we might perhaps deem a literal signification of ‘the Heavens […] are his own’. Though it is beyond the scope of this essay, one could argue, along a constructivist reading of mathematics, that the use of (∞) for the development of calculus that followed did not require empirical verification, but, rather, agreement; that is, mathematical developments occurred not necessarily through discovery in the empirical or exploratory senses, but rather by notational innovations, as symbols and syntaxes were gradually accepted or revised by the broader community of mathematicians.

[30] This emphasis on the primacy of mathematics made its way outside equations. It bled into ways of thinking, and even intimate moments. Donne removes ‘numbers’ from the symbolic field and places it in conversation with everyday language, in order to emphasize its effect upon thought as well as its orientation of experience. Quantitative language, he says, is doing something — with how we are thinking, how we are seeing, in short, how we are orienting ourselves in the world. And this comes at a cost, to perception and to language. For all of his reliance upon quantitative language in his love lyrics, Donne’s most memorable moments are distinguished by his resistance to measurements of desires and emotions. In ‘Negative Love’, for example, he rejects measurement in favor of that ‘Which can by no way be express’d’ (l. 11); the speaker of ‘The Relic’, too, refuses to measure, stating, ‘All measure and all language I should pass, / Should I tell what a miracle she was’ (ll. 32-3). This reticence toward measurement is reflected especially in Donne’s use of quantitative language, which, while frequent, exhibits a deep scepticism over what counting will achieve for his lovers. In the Neoplatonic musings of ‘Air and Angels’, for example, Donne allows ‘That [Love] assume thy body’ (l. 13), but, then, immediately acknowledges the impossibilities of enumerating his lover’s parts in the Petrarchan fashion: ‘Ev’ry thy hair for Love to work upon / Is much too much: some fitter must be sought’ (ll. 19-20). The indescribable is figured as the impossibility of enumerating a love which inheres neither ‘in nothing nor in things / Extreme and scatt’ring bright’ (ll. 21-22), an entity, that is, neither consisting of the null set nor the multitude. If Donne’s lover observes and counts, he says, he restrains love to a representational order with which it is inherently incompatible.

[31] The problem of enumerating love using the Petrarchan blazon is echoed in Donne’s concerns over tracking love through time. Again, although he does count love in time, he more often suggests that love obeys a time outside of the systemized quantification of temporality afforded by the clock. He states this thesis most directly in ‘The Sun Rising’: ‘Love, all alike, no season knows nor clime, / Nor hours, days, months, which are the rags of time’ (ll. 9-10). Because of the self-identity it possesses in the nature of its unity, love refuses a representational knowledge produced in ‘the rags of time’. At times, a love which does, and does not, belong in time is depicted within a single poem. ‘The Legacy’ observes, rather precisely, ‘an hour’ since the lovers parted (l. 3), while then claiming, more abstractly, that ‘lovers’ hours be fully eternity’ (l. 4). Similarly, ‘A Lecture upon the Shadow’ remarks upon the ‘three hours which we’ve spent / In walking here’ (ll. 3-4), even as Donne’s speaker goes on to suggest love resides in a different timescale altogether: ‘Love is a growing or full constant light, / And his first minute after noon is night’ (ll. 25-26). In the final lines of ‘A Fever’, Donne’s speaker trades in eternity for one hour: ‘For I had rather owner be / Of thee one hour than all else ever’ (ll. 27-8). The sexual, physical suggestiveness in this closing seems at odds with the timelessness Donne allows his lovers elsewhere. Yet, rather than contradicting, ‘Love, all alike, no season knows nor clime, / Nor hours, days, months, which are the rags of time’ (ll. 9-10), it is quite possible Donne is simply using this ‘one hour’ as another play on how ‘lovers’ hours be fully eternity’ (‘The Legacy’, l. 4). We expect timelessness; he offers an hour. It is in the hyperbole of this enumeration that lovers’ time inheres.

[32] Christopher Johnson contends that ‘hyperbole ([…] literally means a ‘throwing-beyond’) is Donne’s signature trope’ (2004: 75). In the etymology to which Johnson points, the question of measurement is underscored. Blank’s reading of Puttenham is again helpful for understanding the taxonomic register to tropes: ‘tropes that are based on identifications, such as metaphor, may thus be understood “mathematically” as well as visually: rather than “likenesses” they create “equalities”’ (2006: 42). When Donne engages in hyperbole as a play on measurement, what does he ‘throw’ ‘beyond’? ‘Beyond’ what, exactly?  ‘Throwing-beyond’, I want to argue, posits both a limit point and its transgression, and, in so doing, taxes the very notion of measurement. Puttenham names hyperbole ‘the Overreacher’ or ‘Loud Liar’, and admonishes its application for challenging measure: ‘For although a praise or other report may be allowed beyond credit, it may not be beyond all measure’ (2007: 276). Puttenham labels the first example he presents of the trope ‘ultra fidem and also ultra modum’ (2007: 276-7). Donne’s use of hyperbole, then, is at odds with that ‘net throwne / Upon the Heavens’ of ‘The First Anniversary’. In other words, his ‘measurement’ is not ‘throwne / Upon’, but rather stretches the net, calling our attention to it, emphasizing both its conventionality and fallibility. And, ultimately, it serves as recognition of those aspects of the world and experience that cannot be brought under measurement in any conventional sense, or, rather, which obey alternative orientations, representational schemes and taxonomies altogether.

[33] When he does use calculation or measurement, it is to emphasize the varying, and at times incompatible, orientations that quantification provides for lovers’ experiences. Just as he removes ‘numbers’ from the symbolic mathematical field in ‘The Triple Fool’, so he makes calculations outside of mathematical isolation. As Johnson says, ‘numbers and calculation are also used […] to give flesh to abstract ideas’ (2004: 75). In ‘The Computation’, calculation stretches days and years, a physical life ‘thrown-beyond’ its limits:

For my first twenty years, since yesterday,
I scarce believed thou couldst be gone away;
For forty more I fed on favours past,
And forty on hopes that thou wouldst they might last;
Tears drown’d one hundred, and sighs blew out two;
A thousand, I did neither think nor do […]  (ll. 1-6)

The hyperbole at work builds until the reader loses count — in fact, is overwhelmed by enumeration. Robbins comments on the sum: ‘By now the “years since yesterday” total 2,400, implying that every hour has seemed like a hundred years’ (Donne 2008: 158). The speaker reflects on this protracted life of the lover in the final lines, ‘Yet call not this long life, but think that I / Am by being dead immortal — can ghosts die?’ (ll. 9-10). The suggestion of immortality, as Robbins notes, is especially appropriate for the poem’s tenth line: ‘Ten is a circular number (all its powers, 102, 103, etc., beginning with itself), thus symbolising eternity’ (Donne 2008: 158). Donne’s hyperbolic addition here yields, then, an entirely different sense of temporal experience than we expect in the conventional lifespan. Number either becomes broken, as it overwhelms and surpasses any sense of a singular human experience, or it folds into symbolic, allegorical numerology, allusive of the very ontologies — modes of being — which Klein argues are no longer part of theoretical inquiry with the increased focus of mathematics as a semiotic discipline (1968: 184).

[34] ‘The Computation’, an explicit reference to mathematics, is a reflection on how one measures experience. The relationship between the abstract and material, so critical to Donne’s poetics, was, in his time, beginning to find expression in symbolic mathematical language. Mathematical concepts were made material in symbols, and these symbols, in turn, provided the framework for a language to discern reality. Of course, ‘The Computation’ is not only about measuring time, but also about measuring love. Love, like time, is both abstract and physical, drawing it dangerously close to the field of symbolic mathematical language. In acknowledgement of this, lovers must resist measurement, as Donne suggests in ‘A Valediction: Forbidding Mourning.’ Do not mourn, he says, less our love be exposed. The empiricists are at the doorstep, prepared to make measurements, even in this intimate space. They are the ‘good lookers on’, like those who stand over virtuous men as they die: ‘some of their sad friends do say, / “Now his breath goes,” and some say, “No”’ (ll. 3-4). But where to escape? As symbolic representation expands, how can the lovers avoid being absorbed by it?  Donne attempts to collapse the relationship of two into one, a favourite numerical trick of his early lyrics:

Our two souls therefore, which are one
Though I must go, endure not yet
A breach, but an expansion,
Like gold to aery thinness beat. (ll. 21-24)

This seems an acceptable solution: if all relationships can be absorbed by measurement, then remove relationship from representation. The problem, of course, is that these lovers are not abstract ideas, not mathematical concepts, but physical things, incapable of such reduction. Donne is wary of reducing the physical to the abstract — of removing these lovers from a perhaps messy reality to ‘aery thinness’. He reconsiders: ‘If they be two, they are two so / As stiff twin compasses be two’ (ll. 25-26). This metaphor proves more satisfying, preserving the physical. Here, there is no abstruse reduction: two is two. But now, in the physical form, each becomes an accompanying symbol of measurement: ‘twin compasses’. In transforming each, Donne is ostensibly taking back measurement from the ‘good lookers on’, and restoring it to the lovers, who are alone capable of estimating their relationship. The compass is an interesting choice here for Donne, because, like a mathematical symbol, it is a tool for measurement — abstract mathematics made physical. It is this restoration of measurement that Donne celebrates at the close of the poem. The restoration of measurement, notably, parallels the restoration of proportion. ‘Thy firmness’, Donne says, ‘makes my circle just.’

Brown University


[1]Lemke references these sources directly in his essay, and the central element of each work is developing Peirce’s conception of social semiotics. For further reading on the semiotic perspective that Lemke draws upon, see especially C.S. Peirce, 1960. Collected Papers of Charles Sanders Peirce, vols. 1 and 2 (Cambridge, MA: Belknap Press of Harvard University). See also the selections ‘Logic Semiotic: The Theory of Signs’ and ‘The Nature of Mathematics’ (Buchler 1995: 98-119, 135-149). [back to text]


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Sexual and Poetic Figuration and the New Mathematics in Shakespeare’s Sonnets

Sexual and Poetic Figuration and the New Mathematics in Shakespeare’s Sonnets

Stephen Deng

[1] Shakespeare famously apologizes in the prologue to Henry V for the inability of his play to ‘cram / Within this wooden O the very casques / That did affright the air at Agincourt’ (1996: Henry V, Prologue, 12-14), calling attention to a key problem in accurately representing the drama of history, especially in terms of the sheer magnitude of an event, the specific figures involved. In this admission about the inadequacy of staging, the playwright invokes an unusual relation between literary and quantitative representation hinging on a ‘crooked figure’ common to both commercial theater and the new mathematics based on Hindu-Arabic numbers and positional notation:

O, pardon! since a crooked figure may
Attest in little place a million,
And let us, ciphers to this great accompt,
On your imaginary forces work.
Piece out our imperfections with your thoughts;
Into a thousand parts divide one man,
And make imaginary puissance.
(1996: Henry V, Prologue. 15-25)

The shape of the ‘wooden O’ and the ultimate emptiness of theatrical representation call to mind for Shakespeare the peculiar ability of another nothing, the Hindu-Arabic cipher or ‘0’, a placeholder that could magically multiply any number by ten just by its appropriate placement, part of the new mathematics that was gradually becoming accepted in early modern England. Only a handful of these ciphers, requiring but ‘little place’ in notation, can quickly turn one figure into a million. The actors in this formulation become ciphers that combine with the audience’s imagination — implicitly compared to non-zero numbers — in order to recreate the accurate scale at Agincourt. However, in the Prologue’s subsequent reformulation, it is the actors themselves who are the non-zeros combining with the ciphers of the audience’s imagination. He asks the audience to imaginatively divide any given figure on stage ‘into a thousand parts’ so that a few bodies can authentically represent the great battle at Agincourt by tacking the ciphers of the audience’s imaginations on to each figure on stage. [1] But regardless of whether the cipher in this operation is the actor or the audience’s imagination, Shakespeare conceives of the two assuming properties similar to the magic of zero, in being able to represent large magnitudes within a small space and thus produce a more convincing depiction of historical reality.

[2] This particular attempt at conjuring history through dramatic and quantitative representation intimates a relationship between literary and mathematical representation more broadly. First, since enumeration is a practice of counting objects, numbers tend to materialize implicitly what is otherwise immaterial, an ability that calls to mind the world-making capacity of literary representation, like the poet king Richard II employing his ‘still-breeding thoughts’ to ‘people this little world’ of his cell at Pomfret Castle (1996. Richard II, 5.5.8-9). And yet, rules within the system introduce the possibility of dematerialization, such as in the operation of subtraction or multiplication by zero: what is something can quickly become nothing. Such transformations are indicative of an imaginative capacity that enumeration makes possible: it allows one to count abstractly even in the absence of specific corresponding material objects.[2]  Moreover, because it constitutes a self-contained system, there is always the possibility for number to be abstracted from that which it is purported to count. This latter property raises the question of whether enumeration of material objects is akin to the substitutive effects of literary tropes. Note, for example, that the prologue asks the audience to ‘cram / Within this wooden O the very casques’, the soldiers’ helmets, as synecdochal, or perhaps metonymic, figures for the soldiers themselves, much like contemporary military references to ‘boots on the ground’. Is the abstraction of number in the representation of history any more reductive and dehumanizing than the abstraction of associating individual soldiers with their common headwear or footwear? With both enumeration and trope, and perhaps for any mathematical or literary figuration, the lack of personal and individual detail leaves such depictions of historical reality insufficient at best. The problem is most pronounced in cases such as this: the figuration of people and especially representations of complex identities.

[3] And yet numeric representation and the mathematical properties that accompany it allow Shakespeare, especially in his sonnets, to think in a sustained manner through the complexity of identity and the potential for imaginative enumeration, the possibility of creation through number.  In this article I will argue that, in early sonnets, Shakespeare alludes to the multiplicative power of the cipher as a placeholder in his expressions of desire for the propagation of beauty embodied in the ‘fair youth’.[3] Sonnet 6 in particular exploits this mathematical property to conceive generations of beautiful offspring from this solitary source of beauty, a calculus that assumes the very possibility of multiplying identities and therefore a rigid correspondence between father and child, leaving the mother a mere ‘nothing’ within the sexual operation. Moreover, like the prologue to Henry V, sonnet 20 extends this procreative power of zero to the imaginative realm, positing the potential for various ‘no things’ to produce multiple versions of the fair youth through means other than human reproduction, and once again calling attention to the possibility of ‘identity’ as multiple. But Shakespeare also introduces a more destructive power of zero within the context of the ‘dark lady’ sonnets, a dark side of the cipher that carried cultural resonance, not only because of distrust about the numerical magic associated with Arabic practices (Parker 2013: 223-26; Kaplan 1999: 102),  but also from the recurrent sexual associations with the cipher as a woman’s procreative ‘nothing’, combining with and at times threatening the unity or identity of her male partner. While the cipher could multiply other figures surrounding it, in other forms of combination, like multiplication and division with the cipher, it has the capacity to subsume other figures into itself. Later sonnets also consider potential paradoxes that emerge when different conceptions of number clash, especially between the new mathematics of the medieval and early modern periods and classical mathematics of Greece and Rome (Raman 2008: 168; Wilson-Lee 2013: 460-64).  In sonnets such as 135 and 136, Shakespeare introduces a classical conception of unity, which treated one as a non-number (and therefore comparable in kind to the relatively new figure of the cipher) and denied the possibility of fractions or ‘broken numbers’, only to counteract these principles with more modern conceptions of ‘one’ as both a countable unit critical to maintaining accurate accounts and as a divisible unit belying its status as the principle of all number. The convergence of these two systems gives rise to multiple, contradictory outcomes for ‘will’ in the sonnets, who either maintains himself as a unified entity, becomes fragmented as merely part of a larger whole, or becomes entirely subsumed or rewritten by association with the feminized cipher. In all of these instances, Shakespeare assumes an intimate connection between the figurations of number and person in order to assess the very notion of ‘identity’, and the article will conclude with an exploration of this term, that was emerging at the time within discourses on mathematics as well as with respect to the ‘identity’ of an individual. However, Shakespeare also recognizes the possibility of conflict that arises between these two discourses over the question of whether ‘identity’ is singular or multiple.

The new mathematics

[4] In his sonnets Shakespeare revisits the seeming paradox, exemplified in his prologue to Henry V, of the cipher being at once empty in itself but multiplicative in combination with other figures. The cipher was a key component of the Hindu-Arabic number system, which was imported, or perhaps re-imported, into late medieval Europe from the east (Kaplan 1999: 11-12, 17). Starting in the eleventh century, translations of the great Arab mathematician Al-Khwarizmi’s Arithmetic — by the Englishman Robert of Chester, who studied mathematics in Spain, and the Spanish Jew John of Seville — began to circulate in the Latin world, introducing the new method of reckoning to a scholarly audience (Menninger 1969: 411). This became the basis of what I am calling the new mathematics in medieval and early modern Europe in contrast to classical mathematics of Greece and Rome, which did not employ a notion of zero or any place-based system of notation, a concept I discuss further below. It was especially the advocacy of Leonardo of Pisa (1180-1250) — better known in the mathematics world by his designation as the son of Bonaccio, ‘Fibonacci’ — that made its practice more acceptable within the business community. Leonardo had learned arithmetic under the tutelage of an Arab master in the Pisan colony of Bugia, in present day Algeria. Believing this system to be superior to that of Roman numerals, he promoted its use in his 1202 manuscript, which he called Liber abaci, ‘the book of the abacus’, despite the text’s introduction of a system eliminating the need for an abacus (Swetz 1987: 11-12).

[5] Although Leonardo had important influence on the business community, it took several generations for the foreign system to become conventional in Europe, partly from xenophobic distrust due to its association with Arabic culture, partly from concern about fraud since it was easier to falsify the new numbers (which of course overlaps with the first reason), and partly from general confusion about the method in a time before printing. Fraud concerns appear to be the main purpose for a 1299 edict in Florence forbidding bankers from using the numbers, and for a requirement in 1348 at the University of Padua that booksellers have their prices marked ‘“non per cifras, sed per literas clara” (not by figures [or more accurately “ciphers”, a term used for all numbers], but by clear letters, i.e. in Roman numerals)’ (Pullan 1968: 34). Such suspicion would forestall widespread adoption of the system for several hundred years. Although references to the numerals appear in various medieval manuscripts, Roman numerals and tally marks continued in general use in Italy until the late fifteenth or early sixteenth century (Jaffe 1999: 33). Similarly, in England, there are various early allusions to zero — for example in 1399 William Langland refers to a ‘sifre…in awgrym, That noteth a place, and no thing availith’ — but most people persisted in using counter-boards and Roman numerals (Pullan 1968: 34). Even in Shakespeare’s day, account books such as those of Philip Henslowe recorded transactions using Roman numerals despite the fact that at the time ‘algorism’ had become more accepted and more useful to the business community. Information about the new computational methods had also become more widespread by this period. During the sixteenth century nearly a thousand arithmetic primers were published, including Robert Recorde’s influential The Ground of Artes, published in 1543, to explain the new mathematics to a broad audience of practitioners (Jaffe 1999: 34).

[6] The eventual adoption of the system occurred primarily because of its advantage for business purposes. In medieval Italian the Hindu-Arabic numerals are even referred to as ‘figura mercantesca’ (mercantile figures), pointing to their particular connection to business (Edler 1934: 121). Many proponents of the new mathematics claimed to be able to compute figures more quickly than those using counting-boards or abacuses and Roman numerals, which had been the most common method of counting in medieval Europe. In ‘The table of Verbes’ within his dictionary, John Palsgrave (1530: sig. [2]B6v) included as an example of ‘I reken’ the boast, ‘I shall reken it syxe tymes by aulgorisme or you can caste it ones by counters’. By the end of the seventeenth century, John Arbuthnot (1700: 27) could claim without apparent irony, ‘I believe it would go near to ruine the Trade of the Nation, were the easy practice of Arithmetick abolished: for example, were the Merchants and Tradesmen oblig’d to make use of no other than the Roman way of notation by Letters, instead of our present’. Pragmatism would eventually overcome xenophobia, convention and the initial confusion over the new methods, and even lead to European dependence on this foreign system.

[7] The greatest advantage to the system was in its positional notation for which the cipher played a critical role. Indeed, without the ability to represent an empty column, the system could not function because a practitioner would be unable to distinguish between numbers such as ‘12’ and ‘102’. In The Ground of Artes, Robert Recorde explains the importance of ‘ii. Valewes’ in the system, the values of individual figures and the values derived from their position in notation:

For the valewe is one thyng, and the figures are an other thynge, and that cometh partely by the dyuersite of fygures, but chefely of the places, whereby thei be sette… But here muste you marke that every figure hath .ii. valewes. One alwayes certayne, that it sygnifieth properly, which it hath of his forme. And the other vncertayne, whiche he taketh of his place. (1543: sig. A6v)

The ‘1’ in both ‘12’ and ‘102’ is of course the same and thus has the same value in itself — this is the first value described by Recorde. However, because of the zero placed between 1 and 2 in 102, the 1 in this case takes on a higher value from its position in the hundreds column rather than in the tens column. Zero is peculiar in that its individual value signifies nothing, and therefore in order to have any value at all it relies on the presence of other figures around it, which explains the Fool’s reference to Lear as ‘an O without a figure’ (1996: King Lear, I.4.191-92).[4] But despite its dependence on other figures, the zero’s positioning has the power to transform the positional value of all other numbers. In one of the first references to the ‘cipher’ in the English language (according to the OED), Thomas Usk notes that ‘Although a sipher in augrim have no might in signification of it selve, yet he yeveth power in signification to other’ (‘Cipher’ 2014). As I point out below, this codependency of zero and non-zero figures becomes an important quality for representing the procreative process in Shakespeare.

[8] Most importantly, the cipher tends to increase the value of other figures around it, as the anonymous Treviso Arithmetic (1478), the first printed arithmetic book, explains: ‘The tenth figure, O, is called cipher or “nulla”, i.e. the figure of nothing, since by itself it has no value, although when joined with others it increases their value’ (Swetz 1987: 41-42). As with the ‘12’ and ‘102’ example, placement of zeros increases the value of all figures to their left and therefore the value of the number overall. Robert Recorde explains how this continual process of placing zeros can quickly expand the figure:

[zeros] are of no valewe them selfe, but they serue to make vp nomber of places, and so maketh the figure folowynge them to be in a forther place, and therfore to signifie the more valewe, as in this example, 90 the cyphar is of no valewe, but yet he occupieth the fyrst place, and causeth 9 to be in the seconde place, and so to signifie .X . tymes 9 that is .XC. so [that] ii. cyphars thrusteth the fygure followyning them, into the .iii. place, & so forth. (1543: sig. B3v-B4r)

Every zero placed to the right of any given number will multiply the entire number by ten, quickly increasing its magnitude, like Shakespeare’s ‘ciphers’ in the prologue to Henry V, in only a ‘little place’ (1996: Henry V, Prologue, 16-17). With rapid increases in the scale of commerce and trade within early modern Europe, the advantage of such an efficient system of quantitative representation becomes clear.

The Procreative Power of Zero

[9] This multiplicative capacity lends itself to representations of human procreation. The most explicit reference to this procreative power of zero within Shakespeare’s sonnets can be found in sonnet 6, which builds on the exhortatory message of the early sonnets for the propagation of beauty. This sonnet extends the financial language of its preceding sonnets by contrasting the ‘vse’ of one’s substance with ‘forbidden vsery’:

That vse is not forbidden vsery,
Which happies those that pay the willing lone;
That’s for thy selfe to breed an other thee,
Or ten times happier be it ten for one,
Ten times thy selfe were happier then thou art,
If ten of thine ten times refigur’d thee,
Then what could death doe if thou should’st depart,
Leauing thee liuing in posterity? (6. 5-12)

According to Aristotle, tokos, the Greek term for interest meaning ‘offspring’, signifies its status as unnatural breeding, which partially explains the connection Shakespeare makes between numbers and procreation. Usury continued to be denounced because of biblical prohibition despite a general acceptance in England of an interest rate of ten percent following the Usury Act of 1571 and the specific mention of ‘ten’ calls to mind for critics the accepted rate of interest in England at the time. For example, Natasha Korda relates the ‘legally tolerated’ rate of ten percent to the cipher, which ‘was associated with the calculation of interest and indebtedness; its power to increase or decrease by a factor of ten suggested the gains and losses of creditors and debtors, respectively’ (2009: 145; Hawkes 2001: 105; Herman 2000: 269-70).  As Stephen Booth points out, however, the ten-fold increase associated with the cipher would amount to a 1000 percent, not a 10 percent rate of interest (1997: 142). Therefore, while the sonnet evokes both usury, and the multiplication by ten associated with the cipher, the two are only indirectly related in terms of the general connection between finance and sexual reproduction.

[10] Most importantly, criticism relating the sonnets to usury has tended to subsume the new mathematics, which I believe should be a separate point of interest in the poems. Although there is no explicit reference to the cipher in sonnet 6, as in the prologue to Henry V, the specificity of ten children, who could themselves each produce another ten, clearly points to the cipher’s multiplicative power: placing a ‘0’ next to a ‘1’ produces ‘10’ children who would also be ‘ten times happier’ than the initial ‘1’ when they produce another ‘100’ children in total. Moreover, the notational placement of ones and zeros plays on the correspondence between numbers and gender that critics have noted in sonnet 20: ‘one thing’ or 1 = male; ‘nothing’ or 0 = female (Callaghan 2007: 75; Booth 1997: 164). The persistence and proliferation of beauty in the male youth requires his productive combination with a female. The youth would therefore be ‘refigur’d’ in being transformed from his initial ‘1’ into ‘100’ once his own ten children each produced ten more of their own, and despite his own subtraction from the equation upon death, he would be left ‘liuing in posterity’.

[11] Notice, however, that such procreative mathematics assumes continuity in identity between father and child: all children become mere ‘refigur’d’ versions of the father. Like the ‘casques’ at Agincourt in the prologue to Henry V, any sense of differentiation between people counted becomes erased by number and figurative representation. It was common to consider the offspring as close copies of the parent, with appearance an important indicator to prove one’s legitimacy. Nevertheless, we generally perceive clear demarcation in Shakespearean depictions of fathers and sons, differences that disappear in the procreative arithmetic he employs in sonnet 6. Identity moves from something individual and unique to something multiple and reproducible, but numbers alone in procreation clearly obfuscate some defining characteristics for the particular identities represented. Moreover, there is an assumption here that the woman is ‘nothing’ in the procreative process other than the ability to procreate. Like the cipher, she is valueless without any combination with the male’s ‘one’, and once she performs her part in procreation, her own identity becomes annihilated beyond the total number of ‘refigur’d’ versions of the father that she helps to create. Therefore the identity of both mother and child become subsumed into the identity of the father after such procreative arithmetic is employed.

[12] And yet this problem of subsumed identities does not prevent Shakespeare from exploring a more imaginative form of procreation in sonnet 20 based on a similar model. The final lines of the sonnet mention the ‘one thing’ and ‘nothing’ that have become a source of critical controversy, especially about Shakespeare’s (or the speaker’s) sexual proclivity (Callaghan 2007: 76):

And for a woman wert thou first created,
Till nature as she wrought thee fell a dotinge,
And by addition me of thee defeated,
By adding one thing to my purpose nothing.
But since she prickt thee out for womens pleasure,
Mine be thy loue and thy loues vse their treasure. (20. 9-14)

The main contention is over the interpretation of ‘nothing’. If we read the line to mean that the ‘one thing’ added to the youth by nature is of no purpose to the speaker (‘to my purpose nothing’), critics may interpret a rejection of homoerotic possibilities. On the other hand, if the ‘one thing’ added is a ‘nothing’, which could connote any orifice and thus not be gender-specific, one could read the final lines as an expression of homoerotic desire. Booth (1997: 165) suggests this double meaning in his citation of Martial’s epigram: ‘divisit natura marem: pars una puellis, / una viris genita est (Nature has divided the male: one part is made for girls, one for men)’. The first ‘thy love’ in the final line of sonnet 20 could then refer to the speaker’s ‘one thing’ (‘Mine’), and the second to the youth’s, the ‘vse’ of which would serve as ‘treasure’ for women.

[13] Therefore, while the final ‘vse’ of the youth for ‘womens pleasure’/‘treasure’ would indicate the combination of ‘1’ and ‘0’ as in sonnet 6, the allusion to the youth’s own ‘nothing’ suggests some other form of procreation represented in the poem, if not human reproduction. Despite the impossibility of human offspring from the relationship between two men, perhaps Shakespeare conceives of figurative offspring, maybe even the sonnets themselves, from this numerical magic of combining ones and zeros. The act of writing — the one thing that is the pen combined with the nothingness of the blank page — is of course central to the self-conscious creativity of the sonnets. Similarly, the centrality of writing to the new mathematics — and its key uses in accounting systems in order to keep track of material reality— suggests the possibility of materializing what is immaterial in other forms of representation. Multiplication in the new mathematics is, after all, mere figuration on a page. Indeed, it is the emptying out of the material in accounting — moving from elaborate verbal descriptions that identify particular things to abstract number — which allows for the multiplicity of representation. John Dee (1570: sig. []4v) notes this unusual status of ‘Things Mathematical’ as in between ‘material’ and ‘immaterial’ within his preface to Euclid: ‘For, these, beyng (in a maner) middle, betwene thinges supernaturall and naturall: are not so absolute and excellent, as thinges supernatural: Nor yet so base and grosse, as things naturall: But are thinges immateriall: and neuerthelesse, by materiall things hable somewhat to be signified’.[5] Thus despite the speaker and youth’s incapacity for human reproduction, methods of quantitative representation would lead Shakespeare to conceptions of other forms of abstracted procreation, which would be able to replicate the beauty of the young man even if not in human form.

[14] However, the procreative female who would serve to immortalize the young man (at the same time that her own identity would be subsumed) contrasts with the destructive sexuality of the ‘dark lady’ in later sonnets, a destructive capacity that is in line with other properties of the cipher. Even the productivity of the female zero could arouse negative connotations (Traub 2000: 443; Daileader 1998: 135). For example, in Cymbeline, Posthumus, upon being asked by Iachimo whether he wants to hear more about Innogen’s purported infidelity, tells him to ‘Spare your arithmetic, never count the turns. / Once, and a million!’ (1996: II.4.142-3). The ignominy of one illicit sexual act assumes an equivalency to a million with the multiplicative power of the female cipher, a crucial factor in the ‘arithmetic’ that Posthumus attempts to avoid. And in Middleton and Rowley’s The Changeling, De Flores contemplates in an aside the burgeoning relationship between Alsemero and Beatrice, imagining that one adulterous affair would quickly multiply to others (and hoping to be included among those numbers):

                                                                          if a woman
Fly from one point, from him she makes a husband,
She spreads and mounts then, like arithmetic,
One, ten, a hundred, a thousand, ten thousand –
Proves in time sutler to an army royal.                                                                                              (1990: II. 2. 60-64)

Such instances show that the multiplicative power of the cipher in the context of gender could be employed for other numbers (lovers, sexual acts, diseases, etc.) besides progeny. Once a number becomes abstracted from what it originally counts or accounts for, it can be attached to any other countable entities instead. So the potential for replication extends even across multiple categories of objects.

The Dangerous Cipher

[15] But in addition to being able to multiply various quantities, the cipher tends to subsume other numbers into itself, an operation that counters the procreative power of zero discussed earlier, which tends to leave the zero as ‘nothing’ other than its ability to multiply ones. According to the multiplicative property of zero, any number times zero produces zero, and according to the property of division, zero divided by any number maintains itself. Therefore, the cipher cannot be transformed by other figures in such operations. While some mathematicians following Aristotle insisted on the indivisibility of one since it is the principle of unity, the truly indivisible unit proved instead to be zero. This appropriative tendency emerges in sonnet 134 by the reference to the lady as a ‘vsurer’ who has claimed the youth and ‘mortgag’d’ the speaker to her ‘will’ (1-2). While once again the language of usury becomes prominent here (Korda 2009: 129),  the transformation in meanings of ‘will’ in this and the following two sonnets suggests that the source of her appropriative power is from the same ‘nothing’ that was a source of creative power in earlier sonnets.

[16] It is this promiscuity and sexual desirability of the lady that gives rise in sonnet 136 to concerns about the destruction of male identity. The ‘one’ will mentioned early in the sonnet eventually becomes a ‘nothing’ by the end, or merely the name ‘Will’:

Will, will fulfill the treasure of thy loue,
I fill it full with wils, and my will one,
In things of great receit with ease we prooue,
Among a number one is reckon’d none.
Then in the number let me passe vntold,
Though in thy stores account I one must be,
For nothing hold me so it please thee hold,
That nothing me, a some-thing sweet to thee.
Make but my name thy loue, and loue that still,
And then thou louest me for my name is Will.                                                                                (136. 5-14)

Valerie Traub (2000: 444) finds in the ‘phallic “one”’ of the first two lines of this passage a ‘fantastic desire to return to a state of undifferentiation: not the merger of infant with maternal body, but the cramming of the mistress’s womb with male bodies, and in so doing, eliminating all space for independent female desires’. It is significant that whereas in sonnet 135 ‘will’ assumes the connotation of both male and female sexual organs, in sonnet 136 the male ‘will’ predominates. However, Eve Kosofsky Sedgwick (1985: 38) observes that in the sonnet ‘the men, or their “wills”, seem to be reduced to the scale of homunculi, almost plankton, in a warm but unobservant sea’. Both readings, the male body as both dominant and insignificant, are in fact consistent with Shakespeare’s own mathematical inconsistency. The sonnet, especially the line ‘Among a number one is reckon’d none’, alludes to the classical principle that one is not a number but the principle of number, so that as Leonard Digges writes, ‘Number is the multitude of Unites sette together’ (1579: sig. B1r). The speaker therefore utilizes the status of one as a non-number in order to compare himself to another non-number, the cipher: ‘For nothing hold me so it please thee hold’. Natasha Korda (2009: 144) notes that while the reference to one as a non-number sets up the ‘gendered opposition between the male one and the female “nothing”’ as in sonnet 20, this very opposition ‘is destabilized […] by the female creditor’s “will” — construed as both the enormity of her wants and the spaciousness of the receptacle in which her sexual and monetary “treasure” (l. 5) is stored’. I would agree that this opposition becomes destabilized, but it does so primarily because of the arithmetic properties of the figures involved, especially the cipher in the equation which tends to subsume all other figures. While the speaker hopes that he will remain ‘some-thing sweet’ to the lady and even just ‘one’ in her ‘stores account’, the realization is that he has become a mere nothing within the context of another all-encompassing nothing. The statement would therefore suggest a collapsing of identity between the male one and female cipher as both are either ‘all one’ or ‘as good as none’ (Booth 1997: 469; Blank 2006: 143-44).

[17] However, the reference to the ‘reckon’d’ one as none introduces a complication since it could mean either ‘consider’ or ‘count’. He acknowledges the fact that in her ‘stores account I one must be’. That is, despite its status as non-number, one must be reckoned/counted within the accounts — it is a component, indeed the basis, of the composite number that represents the sum (of lovers in the metaphor). This recognition is illuminated by the fact that ‘Among a number one is reckon’d none’ does not stand alone as a statement, but is preceded by ‘In things of great receit with ease we prooue […]’. The classical comprehension of the one’s role, the sonnet points out, is suggested (though not definitively proven) by the fact that one among many is essentially none. His desire, then, is not to be considered as ‘nothing’ but as close to nothing or as if he were nothing because the ‘one’ of his account is insignificant compared to the ‘number’ within which he would like to ‘passe vntold’. This is also the reason he can remain ‘some-thing sweet’ (perhaps with a pun on the ‘sum’ of all the lovers) despite his appeal to be ‘reckon’d’ or considered as nothing. Sonnet 136, therefore, puts pressure on the classical conception of one as a non-number by introducing the fact that in modern accounts, every ‘one’ matters in establishing a full record of transactions and calculations. All must eventually add up. It was for a similar reason that in 1585 the Flemish mathematician Simon Stevin insisted that one should be considered a number, rejecting the classical view of its special status (Ostashevsky 2004: 208).

Broken Numbers

[18] Sonnet 135 poses a similar challenge to its classical principles by introducing the concept of fractions, another key mathematical component which, by being considered as numbers, negated the idea that one is the basis of all numbers. In the Metaphysics, Aristotle had written that ‘to be one is to be indivisible…a unity is the principle of number’ (1966: 1052b16-22). Some mathematicians such as John Dee held on to this view in rejecting the new idea of fractions as numbers: ‘vulgar Practisers’ (from which he of course excludes himself) ‘extend [the] name [of Numbers] farder, then to Numbers, whose least part is an Vnit. For the common Logist, Reckenmaster, or Arithmeticien, in hys vsing of Numbers: of an Vnit, imagineth lesse partes: and calleth them Fractions’ (1570: sig. *2r). However, Shakespeare, in his prologue to Henry V, alludes to fractions when he asks the audience, ‘Into a thousand parts divide one man, / And make imaginary puissance’ (1997: 24-25). By linking this division of one actor to the multiplication of imagined soldiers on his insufficient stage, the playwright encourages his audience to ignore the classical indivisibility of one and to consider this act of division a conjuring of numbers.

[19] Whether the classical or more modern understanding of ‘one’ predominates will determine how we interpret sonnet 135:

Who euer hath her wish, thou hast thy Will,
And Will too boote, and Will in ouer-plus,
More then enough am I that vexe thee still,
To thy sweet will making addition thus.
Wilt thou whose will is large and spatious,
Not once vouchsafe to hide my will in thine,
Shall will in others seeme right gracious,
And in my will no faire acceptance shine:
The sea all water, yet receiues raine still,
And in aboundance addeth to his store,
So thou beeing rich in Will adde to thy Will,
One will of mine to make thy large Will more.
Let no vnkinde, no faire beseechers kill,
Thinke all but one, and me in that one Will.       (135. 1-14)

In addition to the play on his name and the general term for sexual desire (as in Twelfth Night’s subtitle ‘What You Will’), ‘will’ assumes particular references to both male and female sexual organs. In fact, the wordplay evokes the combination of the two, especially with the references to ‘adding’ (‘To thy sweet will making addition thus’) but at the same time leaves uncertain what quantity results from the sexual operation. In the middle of the sonnet, the speaker reiterates the claim that his own will would be an insignificant part of the ‘large and spatious’ will of the lady’s, a veritable drop in the ocean. And yet he first describes his own ‘Will in ouer-plus’, alluding not only to his excessive sexual desire but also to his own abundance of semen. His multitudinous seed thereby challenges his own described unity, the ‘One will of mine’ that he wishes to be added to her ‘spatious’ will. But even if he can restore unity through his one ‘will’ that will be added to ‘make thy large Will more’, he recognizes that his own unity represents only part of the larger collective. Once again he attempts unification by asking that she ‘Thinke all but one, and me in that one Will’. Still, the ability for him to remain unified in the end assumes the classical conception of one as indivisible. In that case, when he is ‘in that one Will’, he has completely appropriated not only her will but the will of all her other lovers (consider it me when you think of that one will). If, on the other hand, we permit the possibility of fractions and therefore a divisible one, then his being ‘in that one Will’ means that he is merely a part, and a small fraction of the whole at that.

[20] While ‘will’ in these sonnets primarily bears connotations of sexual desire or the sexual organs, the potential for its unification, fragmentation or dissipation suggests that it extends more broadly to the notion of identity, especially because of the potential for puns on the poet’s own name: what effect does ‘will’, in its various sexual associations, have on ‘Will’? According to the OED, the mathematical meanings of ‘identity’ — for example, in Billingsley’s translation of Euclid, ‘Proportionalitie is a likenesse or an idemptitie of proportions’ — emerged in the late sixteenth century, around the same time as the meaning of ‘identity’ as ‘The sameness of a person or thing at all times or in all circumstances; the condition of being a single individual; the fact that a person or thing is itself and not something else; individuality, personality’ (‘Identity’ 2014). Note, however, that Billingsley’s ‘idemptitie’ in the context of ‘proportionalitie’ implies a comparison between two objects, so that even the notion of ‘identity’ in this context works against the potential of ‘identity’ as individuality or uniqueness. In the sonnets, Shakespeare employs such emergent and convergent conceptions — albeit by employing the term ‘will’ instead of the more modern ‘identity’ — to consider how the mathematics of sexuality might produce tension and contradiction in the establishment of male identity. Of course the identity under consideration is only the male identity, the phallic oneness that becomes either maintained (through propagation of future generations) or threatened by its interaction with the female cipher. The mathematics of gender identity in the sonnets clearly privilege the male at the expense of the female, as if the very possibility of ‘female identity’ is excluded.[6] Moreover, the female principle understood in mathematical terms can be understood only in relation to the male principle since in itself it remains, to quote Lear’s fool, merely ‘an O without a figure’: the female ‘O’ always serves as the dependent ‘other’ for the male ‘figure’ even as it has the capacity to affect the male.

[21] How exactly this ‘O’ affects the ‘figure’ becomes an important concern within the sonnets. The dangerous cipher of the later sonnets becomes an infinitely divisible and cooptive entity threatening the unified identity (i.e., the ‘oneness’) of the speaker (and the youth) even as he relies on it in the earlier sonnets to maintain, through its reproductive power, the beautiful identity of the ‘fair youth’, an identity in itself that proves potentially multiple. The later sonnets’ emphasis on classical mathematics, and especially classical conceptions of unity, would tend to encourage interpretations that find no potential fragmentation of identity, though it does leave open the possibility of comparison and relation between the cipher and the one as special ‘non-numerical’ figures. On the other hand, their inclusion of more modern conceptions of the function of one would encourage interpretations of fragmentation as well as resistance to the collapsing of binaries. Of course this very process of abstracting identity to number already entails a significant loss, if not absolute annihilation of identity. Once numbers are abstracted from what they count, those same numbers can be used to count various other entities (implying the potential for multiplicity even across various categories), and the original referent for the numbers may be forgotten. Part of the enumerative process is this ability to separate number from materiality. But the problem of identity in the sonnets also reminds us that numbers are not mere abstractions, that they have a cultural history with significance beyond the technological facilitation of more efficient counting and accounting, a history that often engages with material concerns such as those of commercial and political culture, and gender and sexuality.

Michigan State University


[1] Ostashevsky (2004: 212) notes that this passage not only invokes the paradox of multiplying by dividing but also the ontological indivisibility of one as each part resulting from the division of each actor would still be considered one thing and not a fraction of a thing.[back to text]

[2] On the cipher as the limit of such ability to associate number with object see Rotman 1996: 13.[back to text]

[3] In all references to particular sonnets, I follow the numbering of the 1609 printing. All citations of sonnets are from Booth’s (1997) edition of Shakespeare’s Sonnets.[back to text]

[4] Jaffe (1999: 41) points out that ‘O’s status as somehow different from other numbers is emphasized in the new math primers by the fact that it is always described separately, set apart’. For more on Lear and the cipher, see Rotman 1987: 78-86.[back to text]

[5] The signature for the citation from John Dee is an unusual character showing a finger of a hand pointing to the right (called a “fist” or “manicule”). The only font I have found that can represent it is Wingdings 2. The Unicode for it I believe is U+1F449 (white right pointing backhand index), which wordpress does not support. More can be read on Wikipedia.[back to text]

[6] An interesting counter-example to this negation of female identity might be seen in Wilson-Lee’s (2013: 459-60) reading of Cressida’s fragmentation or fractioning in Troilus and Cressida. The very possibility of fractioning implies an original unity of identity.[back to text]


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Booth, Stephen (ed.) 1997. Shakespeare’s Sonnets (New Haven: Yale UP)

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‘Cipher’. The Oxford English Dictionary. 2nd ed. Oxford: Oxford UP, 2014. 6 October 2014 <http://dictionary.oed.com>.

Daileader, Celia R. 1998. Eroticism on the Renaissance Stage (Cambridge: Cambridge UP)

Dee, John. 1570. “Mathematicall Praeface” in The Elements of Geometrie of the Most Auncient Philosopher Euclide of Megara. Trans. H. Billingsley (London)

Digges, Leonard. 1579. An Arithmeticall Militare Treatise, Named Stratioticos (London)

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‘Superfluous Death’ and The Mathematics of Revenge

‘Superfluous Death’ and The Mathematics of Revenge

Derek Dunne


[a] ‘Oh…Millions of deaths’ complains the Duke as he expires in Middleton’s The Revenger’s Tragedy (3.5.186), thus fulfilling Vindice’s fantasy of ‘constant vengeance’ (3.5.109). John Kerrigan and others have long recognised the excessively reciprocal nature of revenge on the early modern stage. This article seeks to quantify that process, or rather to survey the quantities that recur with surprising frequency in the genre of revenge tragedy.

[b] In Antonio’s Revenge, Pandulpho wishes to prolong his enemy’s death ‘till he hath died and died/ Ten thousand deaths in agony of heart’ (5.5.78), while Hamlet’s Laertes curses whoever has made his sister mad: ‘O, treble woe/ Fall ten times double on that cursed head’ (5.1.235). On the one hand this can be linked to the competitive intertextuality of the genre itself, where each author tries to outdo his predecessor – the logical conclusion of Renaissance emulatio. But might it also point to a deeper psychology of revenge, that struggles to equate life with life, and refuses to accept parity?

[c] In King Lear Shakespeare demonstrates how love cannot be quantified, through the motif of Lear’s diminishing train of knights: ‘Thy fifty yet doth double five and twenty,/ And thou art twice her love’ (2.2.448). Revenge tragedy is similarly interested in arithmetic, where the variable is not love but vengeance.


[1] In its simplest form, the equation of revenge is A injures B, and so B injures A. If the injury happens to be fatal, then B can no longer retaliate, and so must (often from beyond the grave) recruit C to take his or her place. Kerrigan says of this need for a third party that ‘[t]he displacement of revenge from one character to another creates a structure of obligation which modifies the economy of vengeance’ (1996: 3). The suggestive phrase ‘economy of vengeance’ implies that human lives are in some sense quantifiable, as codified in the so-called lex talionis — an eye for an eye. Similarly, Linda Woodbridge has argued persuasively that ‘revenge speaks a language of debt and obligation’ (2010: 84). As England experimented with Italian double-entry ledgers of debit and credit, revenge tragedies too can be seen to ‘balance the books’, meticulously matching injury for injury, death for death. While Woodbridge is right to identify the genre’s ‘numeric idiom’ (2010: 83), I wish to focus on a particular instance of numeracy in revenge tragedies that cannot be reduced to the logic of book-keeping. Revenge by its very nature exceeds the double entry system, upsetting any desire for ‘balance’ through its refusal to allow one debt/death to cancel another. In what follows, I seek to quantify what is at stake when characters on the early modern stage engage in the mathematics of revenge.

[2] From Titus Andronicus’s complaint that he is ‘as woeful as Virginius was,/ And have a thousand times more cause than he’ (5.3.49) to Laertes’s curse, ‘O, treble woe/ Fall ten times double on that cursed head’ (5.1.235), stage revengers are drawn to expressing themselves mathematically. This may be attributable to the fact that a body count is such an integral part of how these plays are plotted; reciprocity requires a tally:

As dear to me was my Horatio,
As yours, or yours, or yours, my lord, to you.
My guiltless son was by Lorenzo slain,
And by Lorenzo and that Balthazar
Am I at last revenged thoroughly.
(The Spanish Tragedy, 4.4.167)

Where revenge differs from mathematics is that while most totals can be reached by both addition and subtraction, the sum total of revenge can only ever go one way: upwards. A death toll, like any other toll, is accumulative. In the early modern theatre it both necessitates and creates a constant supply of stage revengers. In Hamlet, Laertes may demand of Claudius ‘Give me my father’ (4.5.116), but the redundancy of his request highlights the impossibility of any real reciprocity in revenge. In exchange for Polonius, Laertes will only receive the death of Hamlet, a perverted form of gift-exchange. While subtraction may not be an option for dramatists of revenge — characters can only come back to life momentarily in their ghostly, minus, form — division and multiplication do have an important part to play. That is, by dividing the revenge up in some manner, it becomes possible to multiply the ‘vengeances’ exacted. In The Tragedy of Hoffman, when the victims turn revengers, it is natural for them to envisage the potency of their vengeance in numerical terms: ‘And ’twould more vex him…/…/…than a hundred deaths’ (5.1.283). It is in the revengers’ attempts to make a single life suffer multiple deaths, rather than in the sheer number of dead bodies at the play’s climax, that the genre shows its abiding concern with the quantification of revenge. Furthermore, in undermining the parity of one character’s life for another’s, the genre reflects on its own laws of composition: lex talionis, an eye for an eye.

[3] This essay will focus on John Marston’s Antonio’s Revenge and William Shakespeare’s Hamlet, both revenge plays from the turn of the seventeenth century that are so intertwined that the order of their composition still cannot be definitively established (Neill and Jackson 1998). The dialogue between these two is well documented (Mercer 1987), but what has not been noted previously is the equations that unite the revengers in both. Paying attention to the mathematics of revenge sheds new light on the relentless intertextuality of the genre, and also makes sense of certain mathematical problems embedded in the plays, such as why Hamlet kills Claudius twice. I touch on Thomas Middleton’s The Revenger’s Tragedy as a coda of sorts; its protagonist Vindice returns to the problem of quantifying revenge in such a way as to undermine the very process of quantification. But first, it is necessary to establish the connection between revenge, mathematics, and hyperbole, which are each intimately bound up with the self-referentiality that these plays are known for (Kerrigan 1996). It is not my intention to engage in numerology, or to construct a grand equation of revenge, but rather to put forward a theory (as opposed to a theorem) that accounts for the abiding connections between mathematics and revenge on the early modern stage.

Intertextuality by Numbers

[4] Rosalie Colie describes the early modern author as being ‘in conscious competition with the very best that tradition could offer him’:

[T]here was an insistence on outdoing and overgoing earlier achievements, each man newly creating out of and against his tradition. (Colie 1974: 5)

Revenge tragedy offers a perfect example of just such a tradition, making the excesses of each play essentially emulatory in nature. From the mythological Atreus who feels ‘For yet even this too little seems to me’ (Thyestes, 5.3.85), stage revengers have always sought to make their revenges stand out against their historical and literary background: ‘For worse than Philomel you used my daughter,/ And worse than Progne I will be revenged’ (Titus Andronicus, 5.2.194). The intermeshing of Classical precedent and violent comparison is key to Shakespeare’s method in Titus, as the protagonist ‘self-consciously strives to surpass his classical models’ (Miola 1992: 29).

[5] Hallett and Hallett rightly point out that ‘[r]evenge is itself an act of excess’ (1980: 11), but this does not account for the constant cross-referencing of a genre in competition with itself. As Piero puts it in Antonio’s Revenge: ‘Is’t to be equalled thinkst thou?’ (Antonio’s Revenge, 1.1.79). The quotation encapsulates the way in which the concept of revenge challenges the very notion of equality, while also betraying a certain insecurity on the part of the character. At the same time, these plays are filled with numbers and equations. Take for example Aaron’s boastful admission of guilt in Titus Andronicus:

Tut, I have done a thousand dreadful things
As willingly as one would kill a fly
And nothing grieves me heartily indeed
But that I cannot do ten thousand more.


In the course of four lines, Aaron’s crimes multiply from a thousand to ten thousand effortlessly, exemplifying the exponential use of numbers and figures permeating these plays. By the time of Middleton’s revenge tragedy to end all revenge tragedies, knowingly titled The Revenger’s Tragedy, revengers are not satisfied with thousands and tens of thousands. Rather, the Duke is subjected to ‘[m]illions of deaths’ (3.5.188) by the arch-revenger Vindice. Such a claim is consciously exaggerated, pointing as it does to the intersection of revenge, hyperbole, and numerical superiority.

[6] While hyperbole was only one of any number of rhetorical figures taught and practiced in the grammar schools of early modern England (Mack 2002), the figure is yoked to — and transformed by — a distinctive language of revenge, as with Aaron’s boast quoted above. The rhetorical figure of hyperbole was famously called the over-reacher by Puttenham in his Art of English Poesie (1589: 159, sig. Y2r), which aptly describes stage revengers such as Titus, Hoffman, and Vindice.[1]

Chettle’s Hoffman shows himself so eager to be revenged that he hopes to ‘pass those of Thyestes, Tereus,/ Jocasta, or duke Jason’s jealous wife’ (1.3.21): personal revenge is measured against classical precedent. Titus Andronicus similarly models himself on Priam (1.1.83), Coriolanus (4.4.66), and Virginius (5.3.49), explicitly inviting comparison with those who have gone before. The eponymous Antonio of Marston’s play sounds a different note when he complains ‘Let none outwoe me: mine’s Herculean woe’ (2.3.142).

[7] When such comparisons are scattered so liberally through the revenge genre, it is clear that the impulse to compare goes deeper than any particular play. The motif exceeds the explanation offered by straightforward intertextuality, and in fact offers a critique of intertextuality per se. Having been trained to copiously imitate Classical authors in the schoolroom, revenge dramatists show the violent fruits of such a labour. Shakespeare’s Titus is the clearest example: it is Chiron and Demetrius’ familiarity with Ovid that teach them to lop off Lavinia’s hands as well as her tongue. Thus the literary competitiveness inherent in Renaissance emulatio leads to bloodshed, showing its kinship with revenge. As protagonists strive to be ‘peerless in revenge’ (Antonio’s Revenge, 3.5.29), dramatists draw attention to this through Classical reference and vengeful precedents, with Senecan tragedy acting as a touchstone of sorts: ‘scelera non ulcisceris, nisi vincis’, translated by Jasper Heywood as ‘Thou never dost enough revenge the wrong/ Except thou pass’ (2.1.20). The intertextuality of such an excessive genre is indicative of early modern dramatists’ efforts to compare and quantify vengeance itself. However, such violent hyperbole belies the revenger/author’s own fear that revenges cannot be ranked so easily. Therefore, rhetorical superlatives are combined with numerical superiority to ensure that one’s revenge is excessively notable and notably excessive. This is best illustrated by John Marston’s Antonio’s Revenge, first performed by the Children of the Chapel at the turn of the seventeenth century.

3:1 in Antonio’s Revenge, or the Problem of Piero

[8] Typical in its plot, Marston’s play provides its central character, Antonio, with a host of reasons for seeking the death of his duke, Piero: the latter has murdered Antonio’s father Andrugio, falsely accused Antonio’s fiancé Mellida of inchastity, and framed Antonio’s recently-murdered friend Feliche as Mellida’s lover. In the course of the play Mellida too succumbs to Piero’s plots, giving Antonio a plethora of reasons to be revenged. In a sense, this is where the revenger’s problem lies: Antonio must take separate revenges for father, fiancé and friend in order for balance to be restored. If he were only to cite Andrugio in the final scene, does this leave Mellida and Feliche unrevenged? The deaths of Andrugio, Feliche and Mellida are greater than Piero’s single life, so that Piero’s multiple homicides create a crisis of revenge that can be thought of symbolically as: 3 > 1. Furthermore, Marston gives his protagonist Antonio numerous accomplices, including Pandulpho, the father of Feliche, and Alberto, mutual friend of Antonio and Feliche. But if each of these would-be revengers is to achieve a ‘full’ revenge on Piero, then surely they would have to kill their enemy single-handedly. Indeed the title seems to suggest just such a proposition. How then can the sole enemy Piero be equally divided among a range of revengers, while still conforming to the quid pro quo of revenge tragedy?

[9] A solution begins to emerge after Antonio has been visited by the ghost of his father Andrugio, who reveals all to his son. Andrugio accuses Piero of murder and exhorts, seven times in the course of a twenty-line speech, his son to be revenged (3.1.32-51). Thenceforth Antonio’s mind is fully focused on revenge: ‘May I be cursed by my father’s ghost/…/If my heart beat on aught but vengeance’ (3.2.35). He is soon presented with the opportunity to take an indirect vengeance, when Piero’s young son Julio enters. Antonio’s slaying of the innocent Julio has been called ‘one of the worst excesses in revenge literature’ (Hallett and Hallett 1980: 170), but surely this is precisely the point. Antonio is then ‘forced into a gruesome rationalisation of his savagery’ (Mercer 1987: 78), as he attempts to make a spurious distinction between Julio’s ‘soul’ and Piero’s blood that runs in the boy’s veins:

He is all Piero, father all. This blood,
This breast, this heart, Piero all,
Whom thus I mangle. Spirit of Julio,
Forget this was thy trunk. I live thy friend.

It is difficult to accept such specious reasoning from the revenger. Nor, I would argue, are we expected to. Antonio’s fantasy of slaying Piero in the person of his son Julio highlights the problem of Piero having only one life to lose. But due to the nature of Piero’s crimes, a single death is not sufficient. Thus Antonio seeks to prolong Piero’s life only to prolong his death, so to speak: ‘I’ll force him feed on life/ Till he shall loathe it’ (3.2.89).

[10] In the aftermath of the murder of Julio, Antonio appears almost indistinguishable from his enemy Piero. Just as Piero had earlier boasted of his ‘unpeered mischief’ (1.1.10), Antonio is urged by his father’s ghost to ‘be peerless in revenge’ (3.5.29), which is precisely what he has done. Marston creates visual correspondences between the two: Antonio comes onstage with ‘his arms bloody, [with] a torch and a poniard’ (SD 3.5.13) just as Piero’s first entrance was ‘smeared in blood, a poniard in one hand bloody, and a torch in the other’ (SD 1.1.1). But the play’s two final acts significantly modify this image of a protagonist dripping with blood and crying out for ‘vindicta’, as Marston retires the Senecan model of a single revenger bent on the destruction of his enemies, replacing it with a more inclusive form of retribution. In bringing together all those wronged by Piero in the course of the play, Marston must necessarily lessen the protagonist’s role in the final revenge. But as this is revenge tragedy, a dramatic genre as opposed to a real crime, the literal can give way to the literary; multiple homicides require multiple deaths.

[11] It is notable that while the group revenge against Piero begins as meticulously balanced, this gives way to an excessiveness close to sadism. The band of revengers exact a protracted and cruel revenge on Piero that includes taunting him with the dead body of his son Julio (5.5.52). Their justification comes from Piero’s own excesses, which they rehearse for the benefit of both Piero and the audience:

Antonio: My father found no pity in thy blood.
Pandulpho: Remorse was banished, when thou slew’st my son.
Maria: When thou empoisoned’st my loving lord,
Exiled was piety.
Antonio: Now, therefore, pity, piety, remorse
Be aliens to our thoughts: grim fiery-eyed rage
Possess us wholly.

This is followed by a further twenty lines of torture, insults and curses for Piero, including the curious stage direction ‘They offer to run all at Piero, and on a sudden stop’. This action, seemingly merciful in its import, is in fact further evidence of their sadism. It is designed to prolong Piero’s pain, as Pandulpho makes clear:

let him die, die, and still be dying.
And yet not die, till he hath died and died
Ten thousand deaths in agony of heart.


The multiplication of deaths for Piero suggested by Pandulpho’s grim lines goes beyond the ‘eye for an eye’ logic of lex talionis. It plays with the possibility of inflicting innumerable deaths, by prolonging the moment of revenge indefinitely.

[12] On the point of death, Marston structures the final blow to be highly stylised and communal:

Antonio: This for my father’s blood.
[He stabs Piero]
Pandulpho: This for my son.
Alberto: This for them all.

The balanced lines here replicate the restoration of order which Piero’s death brings about. At the same time, the lineation cannot do justice to the simultaneity of revenge, as the printed text can only approximate the united revenge action being staged. Indeed more than one stage direction is needed here, to give equal weight to each revenger in the scene. To return to the initial problem of Piero’s multiple homicides being represented as 3 > 1, Marston solves this by making one Piero die in triplicate.

[13] In the aftermath of Piero’s death, the revengers do not try to hide their crime, but rather jostle with each other to take full credit for the murder. When asked by a Venetian senator, ‘Whose hand presents this gory spectacle?’ (5.6.1), they reply:

Antonio: Mine
Pandulpho: No: Mine
Alberto: No: Mine
Antonio: I will not lose the glory of the deed.

This competitiveness is allayed by Alberto’s ‘Tush, to say troth, ’twas all’ (5.6.11); again individuality gives way to inclusiveness: ‘Mine’ becomes ‘all’. This word is picked up by the senator who responds in kind with, ‘Blessed be you all, and may your honours live/ Religiously held sacred, even for ever and ever’ (5.6.12). Instead of having his revengers die for their part in Piero’s death (thus unbalancing the symmetry achieved), Marston exonerates Antonio and his accomplices for having successfully solved the problem of Piero. In a sense, the title of Antonio’s Revenge is misleading, in that it suggests a single protagonist. Marston multiplies not only the motives for revenge, but also the revengers whose duty it is to carry out vengeance. Faced with only one enemy, they must join forces in such a way that manifold revenges can be achieved.

40,000:0 in Hamlet, or ‘Forty thousand to love’

[14] As a play, Hamlet is most famous for its protagonist’s deep philosophical cogitations, yet there are also a surprising number of equations running through the text that merit closer analysis. These are tied up with Hamlet’s concern to quantify and measure the world around him, from the celestial (‘What a piece of work is man’, 2.2.269) to the sub-human (‘a slave that is not twentieth part the kith/ Of your precedent lord’, 3.4.95). This may account for Shakespeare’s above-average usage of the word ‘quantity’ in the play, an inherently ambivalent term that can mean both ‘a specific or definite amount’ and ‘an indefinite (usually large or considerable) amount’ (OED 1.a; 1.b). The word ‘quantity’ appears in only ten plays, and in just two of these is it used more than once. Timon of Athens, that most economic of plays, uses the term twice (5.1; 5.4). Yet in Hamlet the count rises to four, twice as many as in Timon. From Hamlet himself, we hear of the clown who will laugh only to ‘set on some quantity of barren spectators to laugh too’ (3.2.39). In the same scene, the Player Queen equates love and fear, seeing them as proportional to each other:

For women fear too much, even as they love,
And women’s fear and love hold quantity –
Either none, in neither aught, or in extremity.


This obscure passage struggles to equate fear and love, even suggesting that the two are somehow interchangeable. Shortly afterwards in the closet scene, Hamlet resorts to the word once again in questioning his mother’s powers of judgement, for going from Old Hamlet to Claudius, ‘this to this’ (3.4.69):

sense to ecstasy was ne’er so thrilled
But it reserved some quantity of choice
To serve in such a difference.


Finally, we have Hamlet’s impassioned declaration of love:

I loved Ophelia – forty thousand brothers
Could not with all their quantity of love
Make up my sum.


When it comes to comparing — or struggling to compare — like with like, then whatever is being compared must first be made quantifiable; hence the quantity of ‘quantities’ in Hamlet.

[15] The term ‘quantity’ is also strangely apposite to the paradox of revenge, where deaths are measured one against another; as Laertes bluntly puts it, ‘Give me my father’ (4.5.116). Throughout Hamlet, Shakespeare meticulously balances parallel revenge plots, inviting the audience to make comparisons. Claudius has killed Old Hamlet, and so the younger Hamlet must be revenged. Hamlet mistakenly kills Polonius, setting in motion Laertes’ own revenge plot. Characters within the text even appear to be aware of this equivalence, as when Hamlet tells Horatio that he is sorry for having offended Laertes: ‘For by the image of my cause I see/ The portraiture of his’ (5.2.77 Folio). However, it is this very similarity that gives rise to the play’s interrogation of equivalence per se, and the problem of quantification. If Claudius is as inferior to Old Hamlet as we are led to believe (‘not twentieth part the kith/ Of your precedent lord’ (3.4.95)), how can his death possibly ‘equal’ that of Old Hamlet’s? More to the point, are the two characters who share the name Hamlet in some sense equals? Both problems come together in Hamlet’s first soliloquy. ‘My father’s brother (but no more like my father/ Than I to Hercules)’ (1.2.152): in seeking for equivalence, Hamlet only finds difference. And there’s the rub. In the hall of mirrors that is Hamlet, Shakespeare makes his audience painfully aware of the difficulty in the very notion of comparison. In the first instance, I want to problematise critics’ use of Laertes as a ‘foil’ for Hamlet, by analysing Shakespeare’s use of comparisons in the graveyard scene. I then move on to the final revenge(s), to see how characters mete out vengeance in the wake of the play’s ambivalent equivalences.

[16] Ahead of the fencing match that will precipitate the end of Hamlet/Hamlet’s revenge, we witness verbal sparring of a different sort between Hamlet and Laertes. Ophelia is quite literally the grounds of the argument, in the graveyard scene that opens the final act. Insults and taunts quickly accumulate on both sides, and this is underlined by an incessant use of arithmetic. Consider Laertes’ curse on the person responsible for his sister’s madness: ‘O, treble woe/ Fall ten times double on that cursed head’ (5.1.235). 3 x (10 x 2) = 60. What are we to make of this bizarre equation of woe? It is accompanied by the hyperbolic gesture of leaping into the grave, ‘Till I have caught her once more in mine arms’ (5.1.239). Laertes’ larger than life demonstration of grief continues as he wishes to be buried until the grave mound reaches higher than ‘old Pelion or the skyish head/ Of blue Olympus’ (5.1.242). It is this last, with its combination of histrionic excess and classical emulatio, that prompt Hamlet to declare himself:

                                               What is he whose grief
Bears such an emphasis, whose phrase of sorrow
Conjures the wandering stars and makes them stand
Like wonder-wounded hearers? This is I,
Hamlet the Dane.


That this battle is a rhetorical one is evident from Hamlet’s terms of reference: ‘emphasis’, ‘phrase of sorrow’, and shortly, ‘I will fight with him upon this theme’ (5.1.255). Ophelia is no longer the ground of the argument, but rather argumentation is:  ‘Nay, an thou’lt mouth,/ I’ll rant as well as thou’ (5.1.272). At this point Hamlet picks up on Laertes’ paltry quantifications, as if to degrade his mathematics through numerical superiority:

I loved Ophelia – forty thousand brothers
Could not with all their quantity of love
Make up my sum.


The presence of numbers in this highly charged scene, while unexpected at first, ties in with larger networks of comparative and competitive language. Just as making Ossa like a wart can be achieved only through moving ‘Millions of acres’ (5.1.270), Hamlet’s choice of forty thousand is as arbitrary as it is meaningless.[2] Clearly this is as much a linguistic combat as it is a genuine outpouring of emotion. And as in any combat, someone must keep score. While in the fencing match of 5.2 that task falls to Osric, here it is the audience who must keep score between the two. Laertes’ language is held up for ridicule, in a game of rhetorical point-scoring — ‘A hit, a very palpable hit’ (5.2.262).

[17] Hamlet leaves behind the rhetoric of numbers in the final scene, only to be faced with a more difficult problem than Antonio’s in Marston’s play. How is he to be revenged on Claudius, who is both doubly guilty — ‘a father killed, a mother stained’ (4.4.56) — and infinitely unworthy of Old Hamlet, ‘a slave that is not twentieth part the kith/ Of your precedent lord’ (3.4.95)? A more fundamental question is whether or not Hamlet’s slaying of Claudius can be deemed revenge at all, an argument that has found some favour over the years (Ure 1974: 42; Mercer 1987: 247; Gurnham 2009: 13). It is my contention that the mathematics of revenge traced thus far can help us to interrogate this proposition, by framing the question in terms of why Hamlet kills Claudius twice. Our familiarity with the text of Hamlet should not blind us to the strangeness of Hamlet’s choice to poison Claudius by two different means, within the space of a few short lines. Does this suggest that Claudius is doubly guilty and, if so, of what crimes?

[18] Let us then turn to the final fencing match to establish exactly what happens, in order to see the logic of Hamlet’s choice in those last crucial moments. When the instruments for the fencing match are brought forward, it must be remembered that Hamlet has no knowledge of Claudius’s and Laertes’s foul play. The fact that he does not think twice about entering a contest organised between his enemy and the son of the man he has killed is of a piece with his trust in providence, while also confirming Claudius’s claim that he is ‘[m]ost generous and free from all contriving’ (4.7.133). Throughout the scene Hamlet plays his part unwittingly, and it is not until Laertes informs him of what has actually been happening (5.2.298-305) that he can act with full knowledge. On learning that ‘the King’s to blame’ (5.2.305), Hamlet immediately points his weapon in the direction of his uncle: ‘The point envenomed too? Then venom to thy work!’ (5.2.306). But this is not sufficient for Hamlet, and he proceeds:

Here, thou incestuous, damned Dane!
Drink of this potion. Is the union here?
Follow my mother.


Even in the earliest printed version of Shakespeare’s play, the quarto of 1603 which is half the length of the ‘enlarged’ 1604 edition, Claudius is subjected to multiple deaths; first Hamlet stabs the king saying ‘The poisoned instrument within my hand?/ Then venom to thy venom — die damned villain!’ (17.95). Hamlet’s speech continues uninterrupted as he forces the king to drink from the poisoned chalice: ‘Come drink – here lies thy union, here!’ (17.97). Thompson and Taylor point out that the ‘union’ pun is somewhat defunct since the pearl is only present in the longer version of the play. However, the ‘double-killing’ of Claudius remains intact, a constant feature of Hamlet’s revenge, if revenge is the right word for it.

[19] Claudius is fatally poisoned twice by Hamlet, but whose deaths precisely are being requited? Hamlet’s final word to Claudius, along with the punning on union, seems to suggest that the poisoned chalice is commended to his own lips for having poisoned his wife, albeit unintentionally. Another mirror image is created, as the poisoner becomes the poisoned. Does this mean that the stabbing of Claudius with a poisoned rapier equates with Hamlet’s revenge for his father? If we look at Laertes’s speech which prompted Hamlet’s actions, we see that having first absolved Hamlet of murder (‘I am justly killed with mine own treachery’ (5.2.292)), Laertes then tells Hamlet that his own death is imminent:

                                      Hamlet, thou art slain.
No medicine in the world can do thee good:
In thee there is not half an hour’s life;
The treacherous instrument is in thy hand.


Hamlet is dead, but yet still living (Sale 2007); which means that his revenge on his uncle is not for Old Hamlet, but for his father’s namesake, Hamlet himself. Hamlet has been stabbed on Claudius’ instructions, and for that he stabs Claudius: an eye for an eye.

[20] In the aftermath of the King’s death, it is important to note the reasons Hamlet gives for killing Claudius. Where we might conventionally be reminded of the various crimes and motivations that have led up to this moment, Hamlet’s silence on the matter is deafening. Hamlet delivers twenty-eight lines between poisoning Claudius and his own death, yet he neglects to mention the two regicides that form the core of the play’s action. De Grazia observes that Hamlet’s death speech ‘manages to cram in a great deal’ (2007: 203), which makes the omission all the more significant. One of the ways in which Hamlet’s killing of Claudius is remarkable is how little reference is made to the actions and motivations that started the revenge tragedy. In the Folio text of Hamlet, the word ‘father’ appears over seventy times in total. Except for mentioning his ‘father’s signet in my purse’ when recounting his sea-voyage (5.2.49), Hamlet does not once use the word ‘father’ before, during, or after the duel. This in a play ‘whose common theme/ Is death of fathers’ (1.2.103).

[21] What is lacking in all three early printed versions of the text is any indication that Hamlet’s killing of Claudius can be directly related to the ghost’s command to be revenged. As a literary critic, it is dangerous to play the numbers game with Shakespeare. Yet Hamlet is a play that is overtly conscious of numbers and their signification, or lack thereof. Earlier in the play, before Hamlet is faced with split-second decisions about the death of Claudius, he observes the army of Fortinbras crossing through Danish territory. This prompts his ‘How all occasions do inform against me’ soliloquy (4.4.31), as he meditates on the value of the soldiers’ lives in relation to the ‘little patch of ground/ That hath no profit but the name’ (4.4.17). He had been told by the Norwegian captain that he personally has no interest in the land:

To pay five ducats — five — I would not farm it
Nor will it yield to Norway or the Pole
A ranker rate should it be sold in fee.


The strange repetition of ‘five’, along with the unmistakeable economic language of ‘yield’, ‘rate’ and ‘fee’, would appear to conform to Woodbridge’s theory that economic instability underwrites much of the drama of the period (2010). But it should also be noted that in Hamlet’s private response to this state of affairs, he wonders at a situation ‘[w]hereon the numbers cannot try the cause’ (4.4.62). Obscure in its phrasing, Hamlet’s line suggests that there is a limitation to the uses to which numbers can be put. When it comes to valuing a human life, it is hard indeed to try the cause using only numbers.

‘We cannot justly be revenged too much’ (The Revenger’s Tragedy, 5.2.9)

[22] A final example of the intersection of revenge and numeracy comes from Thomas Middleton’s The Revenger’s Tragedy. This is revenge tragedy at its most refined, as attested to by the protagonist’s full and enthusiastic identification with the role laid out for him: ‘’Tis I, ’tis Vindice, ’tis I’ (3.5.167). At once generic self-identification and intertextual echo of Hamlet, the line is indicative of a play that is well aware of the genre in which it is operating. This awareness extends to the multiplication of revenge plots before our very eyes: Vindice has a double motivation provided by father and fiancé (and is seconded by his brother Hippolito), the Duke’s four sons plot each others’ demise, while a noble Antonio apparently recruits ‘five hundred gentlemen in the action’ (5.2.29) to be revenged for the rape of his virtuous wife. In the play’s opening scene, the protagonist even suggests that the seven cardinal sins are too few for this drama:

the uprightest man – if such there be,
That sin but seven times a day – broke custom,
And made up eight with looking after her.


Taken together, such details suggest that Middleton’s play is far more generically aware than anything we have seen hitherto on the early modern stage; and Middleton shows his interest in using numbers to prove it.

[23] The play culminates in an elaborate double masque in the final act where two sets of revengers dance their way to death. Rather than focusing on the intricate symmetry of the final scene, where four revengers replace four revengers who have killed four victims and are killed in turn — (4+4)-(4+4) = 0 — I want to examine a single death, which contains ‘[m]illions of deaths’ (3.5.188) at the play’s centre. Vindice’s revenge on the Duke is both coup de théâtre and commentary, as Middleton/Vindice seek to present a revenge that cannot be equalled. For Vindice to outdo not only the many competing revengers within his own play, but also his generic predecessors, his revenge on the Duke must be both dramatically unique and excessively violent, while still remaining quantifiable. For if revenge is beyond measure, then how are we to know whose is the most excessive?

[24] Vindice’s innovation is to use the skull of his beloved fiancé, poisoned by the Duke, as the murder weapon. Smearing the lips with poison, he invites the Duke to meet him in an ‘unsunnéd lodge/ Wherein ’tis night at noon’ (3.5.18), where he promises to introduce the lecher to a country lady with ‘a grave look’ (3.5.139). Vindice has a sharp eye for reciprocity, and he is keen that his love ‘shall be revenged/ In the like strain, and kiss his lips to death’ (3.5.104). Hippolito’s response, ‘Brother, I do applaud thy constant vengeance,/ The quaintness of thy malice above thought’ (3.5.108), moves us towards the ‘witty violence’ and aestheticisation of revenge that is such a staple of the play’s critical reception (Brucher 1981: 270; Dollimore 1989: 149; Hirschfeld 2010: 208). In case the quaintness of his malice is lost on the Duke, Vindice spells it out for him in his dying moments:

Vindice: ’tis the skull
Of Gloriana, whom thou poisonedst last.
Duke: O, ’t’as poisoned me.

Vindice’s crude moralising continues: ‘Then those that did eat are eaten’ (3.5.162). Direct reciprocity is most dominant here, a ‘witty literalisation of eye-for-an-eye justice’ (Hirschfeld 2010: 205). But balance shortly gives way to excess, as Vindice compounds the Duke’s physical pain with mental agony:

Vindice: Puh, ‘tis but early yet. Now I’ll begin.
To stick thy soul with ulcers; I will make
Thy spirit grievous sore: it shall not rest,
But like some pestilent man toss in thy breast.
Mark me duke:
Thou’rt a renowned, high, and mighty cuckold.
Duke: Oh!
Vindice: Thy bastard, thy bastard rides a-hunting in thy brow.
Duke: Millions of deaths.

Where other revengers voiced the hope that their revenge would be numerically adequate, here the victim himself freely admits it from his own lips. Leaving no room for speculation, the success of Vindice’s revenge is presented as a statement of fact. The lack of specificity (how many millions?) only underlies how total this revenge has been.

[25] Middleton’s use of 1,000,000 is both ridiculous and wry — how far we have travelled from the overreacher Marlowe, whose Helen could launch only a thousand ships. Considering the multiplication of revenge plots invented by Middleton in this play, revenge as a term here has become practically meaningless. Can this make sense of the play’s playful self-reflexive title, ‘The Revengers Tragedie’, lacking (as was common) a possessive apostrophe? Just as the movement of a decimal point changes the value of a given number, the early modern printed text’s lack of apostrophe is capable of multiplying ownership and responsibility: one revenger’s tragedy becomes many revengers’ tragedies. The character of Vindice may be deadly serious in inflicting millions of deaths on his opponent, but Middleton appears to be less certain how total one revenge play can really be, positing an inverse relationship between mathematical rhetoric and generic triumph.

‘His epitaph, thus: Ne Plus Ultra’ (Antonio’s Revenge, 2.3.132)

[26] Before the final act of Hamlet, on hearing the news of Laertes’ return, Claudius turns to his queen and says:

O my dear Gertrude, this,
Like to a murdering piece in many places
Gives me superfluous death.
                                                                    Hamlet, 4.5.94

When Claudius talks of superfluous death, what precisely is he trying to say? In Thompson and Taylor’s Arden 3 edition of Hamlet, they gloss the curious phrase as ‘i.e. kills me many times over’. While this is a perfectly accurate paraphrase, it does not address the paradoxical nature of what Claudius is suggesting; to die more than once is a contradiction in terms. Yet the paradox of suffering multiple deaths is not unique to Hamlet’s antagonist, and indeed recurs with surprising frequency in the genre of revenge tragedy. Variations on the theme of ‘superfluous death’ emerge across a broad spectrum of authors in relation to the staging of revenge, most hyperbolically in the Duke’s million deaths in The Revenger’s Tragedy. In the end, Claudius too succumbs to a ‘double’ death at the hands of the titular hero. The king’s proleptic fear alerts us to the precisely calculated fatalities that characterise the genre of revenge tragedy, while simultaneously suggesting a superfluity to so many of the revengers’ equations of death.

[27] This necessarily brief survey of revenge by numbers has demonstrated that revenge as a concept may aim towards excess, which in its early modern literary manifestation was rigourously quantified. Alongside the multiplication of deaths necessitated by revenge is the desire to intensify those deaths, whether by a factor of a hundred, a thousand or a million. The reasons for this are themselves double: on the one hand, authors’ conscious intertextuality lead them to numbers as a way of keeping score between those writing in the genre, while on the other, the insufficiency of many of the numbers in the examples given undercuts the very notion of quantifiability that characters are so intent to prove. Early modern revenge tragedies problematise any easy equation of life for life, questioning the very notion of parity between one death and another. This leads to a situation in which the quantification of revenge requires not an economic model of supply and demand, but rather an algebraic one. If X is the death of a loved one, what then is the value of Y, the death of one’s enemy, in the equation X x Y = Revenge?

University of Fribourg


[1] I would like to thank Laurie Maguire for directing me towards Puttenham for this definition.[back to text]

[2]In Q1, the number chosen to represent Hamlet’s love is twenty (16.153). Q2’s ‘forty thousand’ is two thousand times greater, as even the multiple playtexts seem to compete.[back to text]


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Convenient Characters: Numerical Tables in William Godbid’s Printed Books

Convenient Characters: Numerical Tables in William Godbid’s Printed Books

Katherine Hunt

[1] In a prefatory epistle to The Countrey Gaugers Vade Mecum, or Pocket Companion (London, 1677) Richard Collins, Supervisor of Excise in Bristol, explained to the excise men in England and Wales what his book was for. It contained tables which ‘will be of great use’ to them because, he wrote, ‘there are [sic] no Cask or Brewing Vessel that you will meet withall, but they may be Gauged by the following Tables by any person, though he be a stranger to the Art of Arithmetick’ (Collins 1677: sig.B1r; italics reversed).[1] The book was intended literally as a vade mecum to be carried around by the men in charge of calculating excise duties and was designed to enable them, with the help of the tables provided, to calculate the volume of the containers they inspected and, therefore, the money they were to collect. The slim octavo volume is composed of prefatory material, including dedications and a table of contents (16 pages); explanation and instructions on how to use the tables contained within the book (23 pages); and a coda at the back (7 pages) in which Collins described and explained the new instrument he invented for the gauging of small vessels for brewing. The remainder and the majority of the pages, however — 66 in all — are taken up with Collins’s numerical tables: pages which are filled with numbers, arranged in ordered grids. Take, for example, the opening of pages 56-57, which show how to calculate the volume, given in ale gallons, of cylinders of various sizes [see fig. 1, below]: each page contains six columns of numbers, which increase the lower down in the page they appear; the top row and left-hand column are both scales by which the reader can find the particular desired figure (diameter of the cylinder along the top; depth down the left-hand side). The pages present a rather forbidding mass of numbers: the reader needs instructions, usually included in separate pages at the end of each table, in order to discover precisely how that table is to be used.

Figure 1: Richard Collins, The Countrey Gaugers Vade Mecum: Or Pocket Companion (London: Printed by William Godbid, 1677), pp. 56-7 (STC/Wing C5383). Image from Early English Books Online, from a copy in Bristol Reference Library.

Figure 1: Richard Collins, The Countrey Gaugers Vade Mecum: Or Pocket Companion (London: Printed by William Godbid, 1677), pp. 56-7 (STC/Wing C5383). Image from Early English Books Online, from a copy in Bristol Reference Library.

[2] Collins’s printed numerical tables, and others like them, are the subject of this essay. Early modern mathematical and technical books often contain tables full of numbers: tables of logarithms, of ephemerides, of interest, of sines and tangents, of straightforward multiplication of currency. They have antecedents in (and themselves interwove with) the almanacs which proliferated after the advent of printing and, indeed, in earlier handwritten tables too. The tables on which I will principally focus here are often, as in Collins’s book, appended to a manual: they are related to the action that a how-to book describes and directs. In such books these tables proliferated and took over the books from which they came, filling pages and spreading out onto folded-out sheets. If Hindu-Arabic notation had to be taught in the sixteenth century (see, for example, the essay by Lisa Wilde in this issue), just a hundred years later such numbers were prolific and profuse. The tables of numbers in early modern printed books brought a material familiarity with number that, simply in the sheer number of numbers they contained, was unprecedented.

[3] While such tables provided numbers in bulk, they were never meant to be read from start to finish; rather, picked on and picked through to pinpoint the figure that was important in a particular moment. They were numbers that were carried around in pockets but that remained largely unread. In this paper I explore the ways in which these masses of numbers were composed, printed, and described, and the directions that were given for their use. I take as my focus Richard Collins’s publisher, William Godbid, who as well as printing many of these companions for tradesmen was also an important publisher of more high-end mathematical books. The familiarity with number that Godbid showed in his printing practices, along with his range and accuracy, makes him a particularly good figure through which to examine these masses of numbers; his popularity among authors of different kinds of works provides a connection between high and low numerical books of the late seventeenth century.

[4] The printed numerical tables discussed in this paper shared visual and functional characteristics with other contemporary kinds of table, though they are distinct from them, too. The table was an important organising tool for early moderns, and one which has been frequently linked to the epistemological enterprises of the period; I will discuss some of these other tables, before focussing in more detail on Godbid’s numbers. Most importantly, the table was a useful and a common way to represent information: Michel Foucault even argued, in The Order of Things, that ‘[t]he centre of knowledge, in the seventeenth and eighteenth centuries, is the table’ (Foucault 2002 [1966]: 82; italics in original). For Foucault the table was the principal space in which knowledge could be organised, and copiousness corralled into order. More recent, and more specific, investigations into the organisation of early modern knowledge on the page have similarly taken the table as a focus. Steffen Siegel, in his study of the many and various tables produced by Christophe de Savigny in his Tableaux accomplis de tous les arts libéraux (first published 1587), has shown the systematic relationship between the ordering of knowledge and the ordering of the visible elements of the table (Siegel 2009). Tables were widespread and diverse; indeed, as Ann Blair argues in her study of information overload, ‘[t]he notion that tabulae of various kinds (tables and diagrams) were self-explanatory because they brought the material in view in summary form was widespread among early modern pedagogues and neither challenged nor defended in specific detail’ (Blair 2010: 146). Thomas Urquhart’s definition of a table in his mathematical treatise of 1645 The trissotetras: or, a most exquisite table for resolving all manner of triangles seems to illustrate Blair’s point: a table is ‘an Index sometimes, and sometimes it is taken for a Briefe and summary way of expressing many things’ (Urquhart 1645: sig.P1r). The table could summarise and bring together, centralising knowledge in the way that Foucault suggested; also, as Blair argues and as Urquhart implies, its definition in this period was rather vague, the table ‘sometimes’ having one function, and at other times another.

[5] The table is marked by its two-dimensionality, by its flatness. In classical Latin a tabula, originally a flat plank or board, became intimately linked to writing both letters and numbers. It was a flat surface to write on, or a space in which to set out accounts. The sense of a table as a gaming board is present in the Latin too: the tabula was also an expanse on which to play games, the flatness providing a literal and a figurative levelling out of difference between the gamers, making their play fair. The table as furniture, with legs, is a later use of the word, given by the OED to ca. 1050 in English, taking over from the Saxon word ‘board’ (OED, ‘Etymology’ section of ‘table, n.’). The multifarious meanings of the word persist but flatness is the main characteristic of the table, or tabula, in the early modern use of the word: its two-dimensionality is more important than the literal three dimensions of the four-legged table, or indeed of the board alone. Flatness is significant because it permits both writing and, crucial for our concerns here, the schematic laying out of information: flatness provides the link between the material and the abstract, the three- and the two-dimensional, forms of the table.

[6] Tables frequently were, and continue to be, employed to perform counting and accounting. A recent volume on the history of mathematical tables traces the form from, as the subtitle of the book proclaims, ‘Sumer to Spreadsheets’ (Campbell-Kelly et al. 2003). Ancient Sumerian accounts provided information in tables around 2000 BCE; in medieval and early Renaissance iconography, the female figure representing Arithmetica was sometimes depicted sitting at a three-dimensional table, using a gridded two-dimensional table and counters to reckon with. Tables were useful for non-numerical calculation and reference, too: the canon tables attributed to the Roman theologian Eusebius, showing relationships between passages in the gospels, were popular from their invention in the fourth century onwards; closer to William Godbid’s period, the printer and astrologer Regiomontanus produced complex and precise tables from his printing house in fifteenth-century Nuremberg. Indeed gridded tables seemed suited to the technologies of printing: the manner in which movable type is composed lends itself to the grid, to neat corners and orderly rows, to straight lines that (at least in theory) match up. However, many tables— such as the table of contents and the ready reckoner— were, as Ann Blair has shown (Blair 2007: 21), not new or unique to print, but were rather continued from the manuscript tradition. David Murray discusses a ready reckoner ‘of extraordinary usefulness’ from ca. 457 CE: a recognisable antecedent of those printed tables for calculating prices of items that, he argues, ‘came into use in England about the middle of the seventeenth century’, and are very similar to those on which I shall focus (Murray 1930: 296, 298).

[7] Some examples that are roughly contemporaneous with Godbid’s work show the ways in which the table was used in this period to gather and to organise. One such was by being synoptic, bringing knowledge together to show the whole of it at once. This is what John Graunt, London haberdasher and proto-statistician, emphasised in the preface to his Natural and Political Observations […] made upon the Bills of Mortality (1662). In this publication Graunt compiled the bills, produced weekly by the London parish clerks for each area of the city, into large numerical tables to show their combined results. From these tables Graunt proposed an (actually incorrect) theory about life expectancy, which in revised form would become essential to the calculation of annuities and, by extension, constituted a significant moment in the history of the concept and calculation of probability. More relevant here, however, is the method and the tools that Graunt used to collate and to organise his information. He had ‘look[ed] out all the Bills I could’, he wrote, and ‘furnish[ed] my self with as much matter of that kind’; he took this information and ‘reduced [it] into Tables […] so as to have a view of the whole together’ (Graunt 1662: 2). He collated individual, slowly built-up results, ‘reducing’ them into tables in order to make them comparable. The overview this provided not only confirmed Graunt’s ‘Conceits, Opinions, and Conjectures, which upon view of a few scattered Bills I had taken up; but did also admit new ones, as I found reason, and occasion from my Tables’: the form of the table itself brought about new knowledge (Graunt 1662: 2). In the table, Graunt was able to condense time into a visible whole which he could ‘examine’ (Graunt 1662: 2) in order to make his Observations; they were a space in which the patterns of the numerical information became visually and intellectually apparent, forming a totality from which Graunt could draw new conclusions.

[8] Graunt’s tables corralled copiousness so that the combined results would, visually, emerge. But tables could break information down as well as mould it together. In the dichotomous tables, or branching diagrams, that were so important to the method made popular by Peter Ramus, written propositions are divided up, getting smaller and smaller from principal heads down to the finest detail. These tables emphasise the hierarchy, the cascade from the top term to the bottom, rather than relationships in general. Seeing from top to bottom, or left to right, we see cause and, subsequently, effect; if we reverse the direction in which we read the table we are presented with a kind of explanation, an ancestry, of the term with which we start. Instead of right-angled grid lines, as in Graunt’s table, dichotomous tables use curled brackets to enclose the ‘children’ of a particular heading. As Noel Malcolm writes, the Ramist method provided a new and ‘satisfying master-plan of the sciences’ which ‘gave new impetus to […] mapping and categorizing tendencies’ that invigorated chronology, genealogy, and much else (Malcolm 2004: 215). These branching diagrams show the whole at once, but in a different way to Graunt’s ‘whole together’. Here, we see the whole because the information can continue indefinitely onwards. The tables are limited, however, because they are unable to show interrelationships between terms; an item at the bottom, or the right-hand side, of the table can only have one ‘parent’, and the bringing together of new connections that so excited Graunt has no place in the system.

[9] The information in a two-dimensional branching diagram is limited to a single direction of movement. The schematised form of the grid, on the other hand, best exploits the relational possibilities of the two-dimensional table. In the table as grid, lines divide categories; vertical and horizontal dimensions represent separate, but combinable, information; movements — straight lines and right-angled turns — make connections between terms. The gridded table is visually similar to that other early modern repository of information and site of ordered knowledge, the cabinet of curiosities: ‘a world in miniature’, as Claire Preston argues, which aimed ‘to recreate by spatial analogies the supposed likeness between things’ (Preston 2000: 172). Three-dimensional objects were arranged in a two-dimensional grid; the horizontal and the vertical, and the meeting-points between the two dimensions, were epistemologically charged.

[10] If the table arranged written objects as the cabinet did physical ones, an invention by the English clergyman Thomas Harrison combined the two. In his note closet, which he invented in the 1630s, Harrison devised a system for taking notes and organising pieces of written information using the kind of spatial, moveable form that we can see in the cabinet of curiosities. The note closet was a gridded frame on which to hang slips of paper bearing written commonplaces: a kind of tabulated card index. The invention was, Noel Malcolm writes, a product of ‘the Renaissance enthusiasm for creating, on a human, microcosmic scale, a physical arrangement of materials that might illustrate or represent the world’ (Malcolm 2004: 217). This microcosmic arrangement was itself tabular: Harrison described his card-index system as his ‘Tables’ (Malcolm 2004: 204): the table being the best way to describe this gridded organisation of knowledge. So in Harrison’s note closet the table was a place in, on, or through which pieces of information were collected, before they could be selected and finally digested. The table was an in-between stage, rather like the writing tables to which Hamlet is referring when he calls out for ‘My tables! Meet it is I set it down’ (Hamlet 1.5.107), which were used for commonplacing (Stallybrass et al. 2004). The table was an intermediary step towards properly-digested knowledge.

[11] In these examples — the synoptic table, the dichotomous branching diagram, the flat grid, and the writing tables — the table condenses information: collapsing time or space to make it visible in a single, flat expanse. They are not straightforwardly the centre of knowledge that Foucault describes, and they certainly demonstrate the heterogeneity that Urquhart suggested, but all work by centralising, and the table is an intermediate step to an understanding of a given totality. The tables on which I’m going to focus are rather different. They do not attempt the relational display of ordered knowledge we found in the cabinets; they don’t show the whole of knowledge, broken down as in the Ramist dichotomous tables or as synopsis in Graunt’s; nor do they quite help with an in-between digestion of knowledge. The grid-like form of the table was a space which could summarise and organise information, as in the examples I’ve just discussed. But it could also operate as a structure to expand and multiply it.

[12] The tables on which I will focus were for reference. They were not the centre of knowledge, but rather an appendix to it — an appendix that was crucial to the dissemination of practical knowledge and experience that was promised in the main body of the book. I will discuss here different kinds of numerical tables: among them logarithms, and other trigonometric tables; astronomical tables; tables of interest; and tables for calculating volume — small books, useful and portable, made to be used by surveyors, navigators, traders, excise-men, and seamen in their day-to-day life and work. These books often contained instructions for use, and examples of how to use the tables, but a large proportion of the volume, as in Richard Collins’s book, was taken up with pages of full of tabulated, massed number. The numerical tables were shortcuts, marketed in many cases as trustworthy calculations that tradesmen could count on and count with, and by which they could avoid having to do the workings-out themselves. These were pages that nobody would read through, but which must nevertheless be absolutely complete. In terms of page count, they threaten to overwhelm the books from which they come. These common kinds of early modern table were encountered at the back of books, or in larger fold-outs pasted into them: marginal and yet materially dominant, they offer shortcuts to and distillations of others’ learning.

[13] That these tables were shortcuts aligns them with other kinds of mathematical table of the period, such as those contained within the books of logarithms invented and produced by the mathematician John Napier. These were first published in his Mirifici logarithmorum canonis descriptio (1614), and added to in subsequent editions by Henry Briggs; the first English translation was published in 1616, with many further editions thereafter. The logarithm tables, like Collins’s tables for gauging, were designed as aids: making complicated calculations very much easier. ‘This new course of Logarithmes’, as Napier wrote in a dedicatory epistle to the English translation of his book (1616),

doth cleane take away all the difficultie that heretofore hath beene in mathematicall calculations […] and is so fitted to helpe the weaknesse of memory, that by meanes thereof it is easie to resolve mo[r]e Mathematical calculations in one houres space, then otherwise by that wonted and commonly received manner of Sines, Tangents, and Secants, can bee done even in a whole day. (Napier 1616: sig.A4r; italics reversed).

The English translation of Napier’s book was explicitly designed to disseminate such calculations to those ‘Countreymen in this Island’, and for ‘the more publique good’, the tables that Napier had originally ‘set forth in Latine for the publique use of Mathematicians’ (Napier 1616: sig.A5v).

[14] The tables in Napier’s and Collins’s books share visual characteristics and a common purpose as shortcuts, and yet I draw a distinction between the two. Napier and Briggs were concerned with new ways of making calculations which would undoubtedly help practitioners, in part a product of what Gerald Turner has called the ‘massive expansion in the practice of the mathematical arts’ in the sixteenth century (Turner 2000: 4); the tables also showed, however, the fruits of novel operations and ways of knowing. The relationship between mathematical texts and writing for the trades is not always easy to draw. There was something of a ‘trickle-down’ effect, as many academic mathematicians simplified their rules and calculations so that they could be put to use in navigation, surveying, cartography, and the like (see for example Bennett 1991; Turner 2000). But one must be careful, as Mordechai Feingold warns, not to assume that such books, just because they were written in English, ‘bridged the gap between practice and theory and made such information available to the run-of-the-mill carpenter, mariner or gunner’. Instead, he argues, ‘the most important works went unappreciated by the vast majority of practitioners’ (Feingold 1984: 178). Napier offered the English translation of his logarithms to his non-Latin-speaking countrymen, but knowledge did not necessarily flow straightforwardly down the social scale.

[15] Collins’s book did not pretend to be condensing esoteric knowledge from the universities; it can, instead, be described as what Natasha Glaisyer has called ‘popular didactic literature’. This genre, which dramatically expanded in the middle of the seventeenth century (Glaisyer 2011: 510), variously aimed to teach skills — angling, cooking, measuring — to amateurs, and also to aid craftsmen and tradesmen in their practical work. The books on which I focus fit into the category identified by John Denniss as a new trend in the seventeenth century: ‘the publication of ready reckoners, which enabled those with little or no knowledge of arithmetic to find answers to practical problems, and those who did have some knowledge to obtain the answer more rapidly and with less labour’ (Denniss 2009: 456). If Napier’s logarithms were at one end of the hierarchy, Richard Collins’s tables were at the other. The similarities, but also the differences, between books like Collins’s and books like Napier’s are themes that run through this paper.

[16] Collins’s gauging manual was aimed explicitly at country gaugers, providing ready reckoners to help them perform their work. The focus of its paratexts cascades in a way which suggests the organisational structure of the collection of excise in England and Wales: the book’s main dedicatory epistle is aimed at Collins’s ‘Honoured Masters’, the farmers of excise, those who were in charge of collecting the tax (Collins 1677: sig.A2r); a subsequent letter is addressed to Collins’s ‘Brethren’, his fellow supervisors for the duty of excise in England and Wales (Collins was responsible for Bristol); and, thirdly, the author addresses the book’s target audience, the country gaugers themselves, who went about the everyday task of measuring barrels of ale and calculating how much their makers should pay. A portrait of Collins himself appears in a printed frontispiece to the book: he is rather portly, resplendent in a wig and lace collar, modelling the instrument for gauging small vessels that he has invented, and which is described at the back of the book [see fig. 2, below]. When a place of manufacture was given in an engraved portrait like this one, in the vast majority of cases it was made in London. The portrait is unusual in this regard, then, for its signature: drawn and engraved by one Joseph Browne, in Tetbury in Gloucestershire. This ties Collins, and Browne, very securely to the west of England, the provincial area in which the former lived and held influence.

Figure 2: Joseph Browne, ‘Vera Effigies Richardi Collins’, print, 1676. © The Trustees of the British Museum (museum no. 1892,1201.170)

Figure 2: Joseph Browne, ‘Vera Effigies Richardi Collins’, print, 1676. © The Trustees of the British Museum (museum no. 1892,1201.170)

[17] The collection of excise on beer and malt, which Collins monitored, had been introduced in 1643 as a way for the Long Parliament to raise war revenues but was so profitable a tax that it continued into the Restoration and beyond. Collins’s was one of several books on gauging published in the later seventeenth century which, D’Maris Coffmann argues in her recent work on the administration of excise in the Interregnum and the later seventeenth century, suggests ‘a measure of the extent to which gauging was established as a profession by the Restoration’ (Coffmann 2013: 1448 n.135). This kind of gauging was, then, a relatively new occupation, one for which a handy manual, a vade macum, might well be helpful. Collins explained that ‘Many other Books there are, and accurately written, upon the Subject of GAUGING, for instructing the Understanding’; his book, however, ‘is only for the guidance of the hand, and is equally useful to the skilful and unskilful in the Art it self; to the first for dispatch, to the other for safety, that in his want of knowledge in the Art it self of Gauging he may not want certainty in the Charge he is to make’ (Collins 1677: sig.A2v-A3r). Collins’s book is printed in octavo, easy to carry and to use: it is literally a manual — ‘for the guidance of the hand’ — and the tables it contains are simply and straightforwardly to help the gauger locate the correct amount of ‘Beer, Ale, or Worts, in small Tubs, Keelers, Tuns, and Coppers’, and much else (Collins 1677: 62). The many pages of tables are interspersed with instructions on their use, often with examples in which the process of calculation is worked through, so the reader can check that their process is correct.

[18] In addition to Collins’s, I will discuss some of the other manuals published by William Godbid and his successors, including Samuel Morland’s The Doctrine of Interest (1679), which introduced simple and compound interest, and decimal fractions, ‘for all merchants and others’ (Morland 1679: titlepage); The Sea-Man’s Kalender (1674), based on the navigational treatise addressed to ‘the ingenious sea-man’ and written by John Tapp earlier in the century, but here reissued and revised by Henry Philippes; and John Smith’s Stereometrie: Or, The Art of Practical Gauging (1673), another book, like Collins’s, that aimed to teach gauging and was dedicated to the farmers of excise, but which also gave some more general, and abstract, geometrical problems. All of these books are aimed at tradesmen of some kind, and all employ pages of numerical tables in order to help these men go about their business. Godbid was by no means unique in his printing of these tables, or indeed these how-to books, which were very common. He was more unusual, however, because he combined a skill in printing mathematical books with a readiness to print the kind of manuals that were likely to include tables and other calculating devices as part of a vade mecum.

[19] William Godbid was registered as an apprentice in 1646 and was active between 1656 and 1677, working from premises in Little Britain which he had taken over from Thomas Harper (details from the British Book Trade Index, www.bbti.bham.ac.uk). A survey of Godbid’s shop in 1668 recorded three presses, five workmen, and two apprentices (Plomer 1907: 83). On Godbid’s death his business and stock were taken over jointly by his widow Anne and his former apprentice John Playford, nephew of the music publisher of the same name — who had himself worked closely with Godbid. In 1683 John Playford the printer took over the business from Anne and ran it until his death in 1685; his equipment was sold in 1686 (Kidson 1918: 533). This sale marks an end point to the combined Godbid-Playford output; I treat as a unit the works printed by William Godbid, Anne and John Playford together, and then Playford alone, covering in total the years 1656-1685.

[20] The English Short Title Catalogue gives the Godbids and Playford somewhere over 400 works in the period up to 1683 (though this figure includes later editions of the same titles); further books were printed by John Playford alone after that. Their catalogue included John Evelyn’s pamphlet against smoke, Fumifugium (1661), and other works by members of the Royal Society such as Boyle’s Tracts: containing suspicions about some hidden qualities of the air (1674); poetry, including works by Thomas Bancroft, and also a 1674 edition of George Herbert’s The Temple; the first books on the new practice of change-ringing; almanacs and astrological calendars; theological works; books teaching English grammar; the odd play and sermon; a book about the virtues of coffee; some translations from classical authors; and a number of books about fishing, almost certainly influenced by the mathematician John Collins, an important client and associate of the printing house, who from 1677 was accountant to the Royal Fishery Company.

[21] Godbid’s output marks him out as a versatile printer and one who was skilled in, and in demand for, very different types of work. One of his most beautifully-printed books was the translation of Aesop’s Fables (1666) printed for, and in close collaboration with, the painter and draughtsman Francis Barlow, who etched the illustrations (see Hodnett 1978; Flis 2011). In this book the etchings are printed alongside text in both engraving and letterpress. Barlow’s collaboration with Godbid was successful and in Aesop’s Fables their work, and the mixture of relief and intaglio printing, is particularly well integrated. This work with Barlow proved Godbid to be extremely adept at managing text and image, or words and non-words, in a way that is echoed in his other printing too.

[22] Over and above all these other works, Godbid and his heirs were considered specialist printers in two fields in particular: music and mathematics. In fact Thomas Harper, from whom Godbid had bought his shop, had published in both disciplines, too (books of psalms, for instance; and treatises by Jonas Moore and William Oughtred), but Godbid built up this part of the business much further. His reputation as a printer of music rested largely on the work he did with John Playford, the music publisher, whose titles included books of psalms, musical primers, and the first books describing country dancing; the latter ‘dominated’ the publishing of music in the second half of the seventeenth century (Carter 2013: 88). The majority of Playford’s output was handled by the printing shop in Little Britain, and it is their association with Playford that has garnered the most attention for Godbid and his successors (see, for example, Kidson 1918, Carter 2013, Herrisone 2013).

[23] In an advertisement at the back of his 1679 edition of his Introduction to the Skill of Music, the publisher commended his printers (Anne Godbid and John Playford Jr) as ‘the ancient and only Printing-House in England, for Variety of Musick and Workmen that understand it’ and also as ‘the usual House for printing Mathematical Books, witness the difficult Works of Dr. Pell, Dr. Wallis, Dr. Barrow, Mr. Kersie, &c. there printed’ (Playford 1679: sig.M2r).[2] Godbid’s printing house had not only the movable type necessary to print music, but the special characters for mathematical printing, too. In May 1686, following the death of John Playford the printer, his sister Ellen (or Eleanor) placed an advertisement in the London Gazette to sell the equipment from the shop, which was ‘well known and ready fitted and accommodated with good presses and all manner of letter [sic] for choice works of Musick, Mathematicks, Navigation and all Greek and Latin books’ (quoted in Kidson 1918: 533).

[24] Mathematicians, authors of the ‘difficult Works’ that John Playford invoked, seem to have agreed that Godbid’s establishment was the best place to have their work printed. In a letter of 10 February 1676/7 John Collins wrote to his fellow mathematician Thomas Baker that ‘in truth we have but one printer, namely, Mr. Wm Godbid in Little Britain, that is accustomed [to] and fitted for such [mathematical] and music work, who besides is a very worthy honest person’ (Rigaud 1841, II: 15). Collins wrote elsewhere, too, of his admiration for Godbid’s printing. In a letter to another mathematician, John Gregory, in 1670, he advised about getting the latter’s work printed. ‘There is not any Printer now in London accustomed to Mathematicall worke’, Collins claimed, ‘or indeed fitted with all convenient Characters, and those handsome fractions but Mr Godbid where your Exercitations were printed, and at present he is full of this kind of worke’ (quoted in McKitterick 1992: 374). John Wallis was similarly keen to have his Treatise of Algebra (1685) printed by John Playford. ‘It is not every Printing-house,’ Wallis explained in his preface,

that is provided with such variety of Characters as would be necessary to suit such an occasion as this. And, to have all such cast a-new for this purpose; would be a matter of great charge. For preventing of which, I judged it most expedient […] to make use of that of Mr. John Playford (in London;) which, by Mr. William Godbid (while he liv’d) and since by himself, is plentifully supplyed with such Furniture, on purpose to be ready for such occasions. (Wallis 1685: sig.B1v).

The shop in Little Britain was the only printing house which had the stuff, the ‘Furniture’, the ‘variety of Characters’ and the ‘handsome fractions’ that were needed to print sophisticated mathematical texts.

[25] Wallis and John Collins were referring to the large, expensive mathematical books that Godbid’s printing house produced. Richard Collins’s gauging manual was considerably less grand, its tables representative of a more modest kind of numerical printing. Books such as Collins’s needed not ‘handsome fractions’ but rather a large stock of regular numbers: quantity rather than quantity; enough characters, enough numbers, to print the pages of tables. Even without the problem of unfamiliar or unusual characters, printing these tables was laborious work: the (unknown) printer of the Philosophical Transactions in 1683 complained that a relatively straightforward tide table took five days to compose (Johns 1998: 90). John Pell described how laborious an undertaking was the printing of his 32-page book of mathematical tables, Tabula numerorum quadratorum decies millium […] A Table of Ten Thousand Square Numbers (printed by Thomas Ratcliffe and Nathaniel Thompson, 1672). From delivering the manuscript to the printer to having the finished volume took, Pell wrote in his diary, ‘From September 13 to March 21 following’: that is, he continued, ‘191 dayes, or 27 weeks, 2 dayes. […] A long time for printing of 8 sheets’ (Malcolm and Stedall 2005: 291).

[26] Such delays were perhaps due to the fact that as well as simply having the requisite number of sorts with which to print, the printer had to be trusted to print them in the correct order. The reputation of Godbid’s printing house seems often to have been invoked when the authors emphasised one of the key characteristics of their tables, and mathematical books more generally: their accuracy (see McKitterick 2003: 124). In his Treatise of Algebra (1685) John Wallis suggested that numbers were more easily mistaken for each other than letters. The content of his book was, he wrote in the preface, ‘so different from the Printers common Road’, that errors were more likely to creep in. Because Wallis was in Oxford and therefore unable to oversee the printing of his book, he enlisted Edward Paget, Master of Mathematics at Christ’s Hospital, to ‘see to the Correcting of the Press; especially as to what is peculiarly Mathematical, wherein the ordinary Correctors were less acquainted’. Nevertheless there were some mistakes in the book: small errors ‘such as in another Book would not have been worth the noting’, which ‘the Eye would (either not see, or) easily Correct’: the ‘mistake or misplacing of a letter’ in Wallis’s book, for example, would be equivalent in another book to ‘the omission or mistake of a Word’. Because they are mathematical, mistakes are here both easier to make, and more significant (Wallis 1685: sig.B1v). Those errors that remained Wallis collected into an errata list and, as was common (see Blair 2007), urged readers to make the corrections themselves; he himself corrected by hand the mistakes in the copies in the Bodleian and Savilian Libraries in Oxford, and the copy in the Royal Society.

[27] Wallis produced this errata list not, he explained, ‘to the Printers disparagement, whom I have no great cause here to blame’, but purely ‘for the Readers ease’ (Wallis 1685: sig.B1v). Indeed, whereas some authors took to the errata list to berate their printer’s carelessness, and despite the difficulties of printing mathematical works, authors in their prefatory material frequently emphasised the accuracy of the printing that Godbid (and his heirs) guaranteed. In his Elements of that Mathematical Art (1673), John Kersey also praised Godbid, writing that ‘the Faults of importance escaped in this Impression of the First and Second Books are only these fourteen’. The errors are a mixture of numerical and alphabetical, mathematical and linguistic. Some seem to be the kind of typos that take on new significance when applied to number (‘3ddde’ to be corrected to ‘3ddee’, for instance, or ‘19’ instead of ‘9’); others are verbal corrections with semantic implications, such as the instruction to exchange ‘not exceed’ for ‘be less than’. These few and corrected faults showed, Kersey wrote, ‘the exact care of the Printer’ (Kersey 1673: sig.b4r; italics reversed).

[28] The mathematicians’ demands for accuracy were echoed by the authors of books intended for tradesmen. Indeed, in a study of Restoration texts that taught how to calculate interest (including books by Morland and Playford), Natasha Glaisyer has shown that for many authors claims to, and proofs of, the accuracy and trustworthiness of their tables were essential: the calculation of credit required the parallel cultivation of credibility among one’s readers (Glaisyer 2007). Richard Collins promised his audience that ‘The Tables are Calculated and Printed with so much care, that you may safely confide in them’ (Collins 1677: sig.A5r). In his gauging manual, Stereometrie (1673), John Smith reassured his readers, too, that:

Concerning all which Tables, I may further confidently say, such and so great hath been my own and the Printers vigilance and care, that I find no one numeral Mistake in them, and am not in the least conscious to my self of any remissness in their Calculations; yet of some Mistakes elsewhere in the Book you have Advertisement, which I desire (before reading the Book) to be corrected. (Smith 1673: sig.A5r)

The verbal content, and the examples given to show how to do calculations, might need altering — but Collins and Smith proclaimed their masses of numbers to be immaculate.

[29] Similar claims to precision are frequently given in paratexts to how-to books, but are equally often found to be unfounded, and the books filled with mistakes. Indeed, many authors chose in their prefaces to point out the errors in tables by their contemporaries and rivals. In his Doctrine of Interest of 1679, a copy of which was presented to Charles II, Samuel Morland asserted that his

Tables are Calculated with greater care, and are much more correct than those that have been Published of late years. For instance, all those Tables in Mr. Newton’s Book, Printed 1667 [John Newton’s Scale of Interest] are full of Errors and mistakes; and which is very remarkable, the Tables which Mr. Dary has Published as his own [Michael Dary’s Interest Epitomized, 1677], are only transcribed out of Mr Newton’s Book, and that with all the Errors, which are so many, that they must needs mislead and discourage either young or old Practitioners from trusting to, or making use of them. (Morland 1679: sig.A2v)

Morland’s book was, he claimed, free from the iterative errors and lazy copying that plagued those of his careless contemporaries. In Morland’s own book, on the other hand, ‘it is presumed that there will hardly be found one false Figure’ (Morland 1679: sig.A3r). As Natasha Glaisyer argues, in this passage Morland extends mere self-promotion to attacks on his peers (Glaisyer 2007: 694).

[30] For all the assertions by Morland and the others, how precise could their tables really be? The printer might print the wrong numbers, or the author could supply incorrect numbers to begin with; numerical errors are, as Morland and Wallis both suggest, easy to create and transmit but difficult to catch. Morland attributed the accuracy of his book to his own checking and calculation and to, he wrote, ‘the more than ordinary care and diligence of Mr. John Playford, Printer, (whom I have found the most ingenious and dexterous of any of his Profession, in Printing of Tables, and all sorts of Mathematical Operations)’ (Morland 1679: sig.A3r). Printing accurate tables required ingenuity and dexterity, knowledge and skill, and a collaboration and a degree of trust between author and printer; Morland’s idea of trust hinged on the user of the book believing in its accuracy, implying that both author and printer could be relied upon. (In fact, Dary’s book, which Morland criticises, had been published by Godbid — the same printing shop that was now run by Playford).

[31] As well as errors with individual digits, bigger mistakes could be disastrous — as an example from another printing house, that of the great Joseph Moxon, displays. A major mistake in William Oughtred’s Trigonometrie, published in a Latin and then an English edition, both in 1657, and with tables printed by Moxon, undermined the many pages of tables it contains. In the English edition, a dedicatory epistle by Richard Stokes pointed out that ‘the number of Figures in the Tables’ ‘[fell] short of that required and used in the rules’. The calculations in the explanatory rules were given to 7 decimal places, but the tables only gave these figures to 6 decimal places [see fig. 3, below]. This, he continued,

sprung from the intention of Printing it in octavo, for which volume the number of Figures was resolved on, and upon the changing the Volume [to quarto] forgot to be altered, The revered Author has both discovered and amended the errour, in the appendix, as farre as could be, as you may there perceive. (Oughtred 1657: sig.A3v)

Because the pages were set up to be printed in the narrower octavo format, they did not correspond to the rules meant to explain them: an appendix attempted to redress the balance but these tables lack that confidence that Godbid’s authors proclaimed.

Figure 3: William Oughtred, Trigonometria (London: printed by R. and W. Leybourn for Richard Stokes and Arthur Haughton, 1657; tables printed by Joseph Moxon), pp.90-91 (STC/Wing O589). Image from Early English Books Online, from a copy in the Huntington Library, San Marino, California.

Figure 3: William Oughtred, Trigonometria (London: printed by R. and W. Leybourn for Richard Stokes and Arthur Haughton, 1657; tables printed by Joseph Moxon), pp.90-91 (STC/Wing O589). Image from Early English Books Online, from a copy in the Huntington Library, San Marino, California.

[32] Authors, publishers and printers frequently and (in some cases baselessly) exaggerated the accuracy of their books, but in the publications that the printer John Playford wrote and compiled himself, he explained at some length precisely how his tables were made to be so correct. In the miscellany of useful tables that he compiled under the title The Vade Mecum, or the Necessary Companion (first published 1679 and frequently reissued — this is from the second edition, of 1680), he noted ‘two things many times the cause why Books of this nature appear abroad not so correct as they should be’. The first of these common errors is, he wrote,

Because they are too much hastened from the Press, and not time enough allowed for the strict and deliberate examination of them; which in all Books ought to be done, especially in these, for as much as one false Figure in a Mathematical Book, may prove a greater fault than a whole word mistaken in Books of another kind. (Playford 1680: sig. A2r)

Accuracy required careful attention to the figures, and not rushing through the press. The second error was, he continued, when ‘Persons take Tables upon trust without trying them, and with them transcribe their Errors, if not increase them’. This is what Morland accused Dary of doing: of simply copying someone else’s table, without checking it first. Playford professed to have avoided both of these faults. He was careful to double-check all the calculations he included:

for not trusting to my first Calculation of them, I new Calculated every Table when it was in Print by the first Printed Sheet, and when I had so done, I strictly compar’d it with my first Calculation, from which care I hope there is not one false Figure among them. (Playford 1680: sig.A2r-A2v)

Playford describes a process which moves between the calculations in his head or with a pen to the printed page and back again, checking for errors and for precision. Recalculating, comparing to the final printed version, and seeking out ‘false Figures’ — and note that same phrase, ‘one false figure’, that Morland used: Playford emphasised the iterative processes involved in trying to make sure these tables were correct.

[33] Edmond Halley noticed that, for books with tables to aid navigators at sea, ‘the first Editions have generally been the best; frequent Copying most commonly vitiating the Originals’ (quoted in Johns 1998: 31). While Morland pointed out that errors were transmitted when they were copied, Halley suggested that error could creep in in simple reprinting, too. But reprinting could also represent a chance to improve accuracy and correct mistakes: both in second editions and as when Playford re-calculated and re-checked the existing tables he borrowed from elsewhere to include in his compendium. The books often include those kinds of entreaties to the reader often found in works of this kind such as those we have already seen from John Wallis (see also Blair 2007). In The Sea-Man’s Kalender (1674), a compendium of useful tables for navigators, John Tap asked ‘the courteous Readers to do me that favour, as to correct what they shall find amiss, either in the Printer’s over-sight or mine own errour: I shall not only endeavour the mending of them in my next Impression, but be very thankful for them’ (Tapp and Philippes 1674: sig.A2r). This entreaty was particularly relevant because future editions of this book had been guaranteed by William Godbid, its printer. Henry Philippes, in his address to the seamen who were the intended audience for the book, explained that

these [astronomical] Tables [for navigation, that were at the heart of the book] are subject to grow old, and wear out of date; yet such hath been the good fortune of the Book, and the care of the Stationer, that the quick sale of the Book hath encouraged him still to renew the Tables; for this means, the Book hath not only been preserved in its first excellency and exactness, but hath from time to time received the Friendly Additions of Mr. Henry Bond, an Antient Professor of these Arts. (Tapp and Philippes 1674: sig.A2v)

Here, then, frequent renewal aided rather than impeded the accuracy that the paratexts of these books proclaimed.

[34] If accuracy was of prime importance, the next thing that the authors of these books emphasised was the ease of using them. Although they should be as self-explanatory as possible, these were nevertheless tables that readers had to be trained to see and to read. The language with which the authors of these books describe how the straight lines and right angles of the tables were to be used suggests a kind of directed treasure hunt, in which the reader is led by the author’s instructions: they are invited to come with me, a vade mecum on the page. In a table of sines, Henry Philippes directed the reader to ‘look for the Min[ute] at the left side of the Table, and carrying your eye downwards from the Deg[ree] till you come right against them in the number which you find in the common Angle to them both, is the right Sine of your given Arch desired’ (Tapp and Philippes 1674: 121). For Richard Collins the horizontal and the vertical speak to each other: in one of his gauging tables, ‘The Number in the other Columns is the Content in Ale Gallons, and Hundred Parts, answering to the Diameter on the top of the Table, and the Depth in the first Column in the side of the Table.’ (Collins 1677: 62) Jonas Moore, in his A new systeme of the mathematicks (1681), explained how these numbers, intended to aid navigation at sea, themselves required a kind of navigation. He explained the way of reading the tables in terms of movement and selection: finding the correct row, or column; ‘turning’; ‘looking downwards thereunder till I come right against’ the number he is looking for; locating the ‘common meeting’ between the two co-ordinates selected (Moore 1681, II: 354). Samuel Morland proclaimed his tables to be easy to use in part because they didn’t strain the eye: ‘being performed by Addition only’, and therefore ‘less subject to error; and not only so, but whereas all other Operations of Multiplication do extreamly distort the Eyes by looking stedfastly upon Figures placed Diagonally, by this Tariffa the Eye looks on them always in a straight Line, and no otherwise’ (Morland 1679: sig.A4v-B1r). The grid helped the eye and the hand to use the tables in order to perform calculations, thereby made simpler for the user of the book.

[35] Ease of use was important because many of these books were, as Richard Collins said, made to be ‘plain and easie to the meanest Capacities’ (Collins 1677: 62). Collins wrote that in the tables he provides, calculations are given ‘with as much ease as the Interest of any Sum of Money in your common Almanacks, the use of which almost every Countrey Man knows’ (Collins 1677: sig.B1v; italics reversed): these tables built on older tables, which were familiar from almanacs. It was not uncommon in popular didactic texts to appeal explicitly to the less educated consumer (Glaisyer 2011: 516), and such appeals provided an explicit separation between books for practical mathematics and more academic texts. Often such books were explicitly designed to simplify other, more complicated books: Mordechai Feingold believes this to be true of Henry Philippes’s navigational text, which condensed other books that were ‘either too complicated for, or not sufficiently applicable to, the needs of the vast majority of their London practitioners’ (Feingold 1984: 179). Samuel Morland, indeed, advertised his book as ‘more plain and easie than that of other Men; and those things which they have left intricate and difficult to be understood, are here made evident by clear Demonstrations, obvious to the meanest capacity’ (Morland 1679: sig.A2r). Collins, like Morland, made it clear that he ‘hath endeavoured to reduce the Doctrine [of how to calculate volume] to Tables, to avoid both those Rods and Arithmetical Calculations’ (Collins 1677: sig.A7r), and that his book ‘is only intended for Practitioners, who may by help of Tables shun or avoid intricate or laborious Arithmetical Operations’ (Collins 1677: sig.A6r). The tables were intended to excuse their users from doing the actual calculations themselves: the labour involved in calculating the tables themselves was omitted.

[36] John Smith, in his Stereometrie, recommended that his readers have ‘a proper Genius, not only ready to conceive Mathematical Notions, but apt likewise to take a kind of pleasure in them’ (Smith 1673: sig.A6r), but nevertheless stated that his work ‘hath none of those Embellishments, which a Polish’d or Learned Pen might have adorned it with’ (Smith 1673: sig.A4v), precisely because the book was supposed to be handy:

if it be objected, That of divers Multiplicators, the Rise and Fabrick is not given [that is, the calculations are not shown in full]: To this I answer, I did indeed at first intend the inserting of the way for constituting every particular Multiplicator […] but finding many of them very much complicated, the denodation [denotation; writing out] and unravelling them would cost many words and much paper, and so not only render the Book more chargeable, but voluminous, and beyond the bulk of a pocketable Vade-Mecum, (Contraction having been all along designed.) (Smith 1673: sig.A5r-v)

Expediency has forced Smith to exclude the detailed workings out, has prevented him from being able to ‘unravel’ the ‘fabrick’ of his calculations. These are tables that show the end results of knowledge and of calculation, but none of the complicated path by which they got there. Some writers alluded to the missing parts of their books, in which their intellectual operations are described and explained. Richard Collins, for example, wrote that he wanted ‘to add a Second Part, in which shall be explained the Reason and Manner of Calculating these Tables […].’ (Collins 1677: sig.A6r); needless to say, this second part never appeared. The vade mecum books were curtailed, doing away with ‘many words and much paper’ to show only what was necessary and practical. The reader’s interaction with these tables did not generate new knowledge as did, for example, the use of Graunt’s tables from the bills of mortality. Rather, these tables simply allowed their readers to avoid doing troublesome calculations themselves.

[37] What might this way of using the tables say about these numbers and the labour of calculating them? In the preface to the English edition of Napier’s logarithms, Henry Briggs described the generation and the ancestry of numerical tables: how many men through time

have laboured much, and some of them bestowed very great cost, both of their owne estate, & also from the liberall contribution of sundry great Princes upon the maintenance of divers men, who for many years together have wholly employed themselves to calculate these Tables. (Napier 1616: sig.A6v)

Napier’s tables, produced by him alone, rank alongside these others, the calculation of which was the result of careful intellectual labour. The labour of these learned scholars was then passed down in basic form for the use of those of the ‘meanest capacity’ (Morland 1679: sig.A2r). In the late seventeenth century, however, scholars had begun to wonder about who should undertake the calculations of such tables, and how they should do it. Leibniz invented his calculating machine in the 1670s as a response to his feeling that that ‘it is unworthy of excellent men to lose hours like servants in the labor of calculation which could safely be relegated to anyone else if machines were used’ (quoted in Blair 2010: 111). Indeed, in the following century, the burden of laborious calculation was inverted — at least somewhat. In her study of the immense logarithm and trigonometric tables overseen by Gaspard Riche de Prony in post-Revolutionary France, Lorraine Daston observes that the production of these tables, done (de Prony himself explained) according to Adam Smith’s principles of the division of labour, marked an important moment at which calculation became something mechanical rather than intellectual — or, at least, which could be mechanical rather than intellectual. The tables contained, Daston explains, the ‘calculation of ten thousand sine values to twenty-five decimal places and some two-hundred thousand logarithms to at least fourteen decimal places’ (Daston 1994: 186) and were made, under de Prony’s direction, by non-mathematicians. (According to Ivor Grattan-Guinness some of the calculators were unemployed hairdressers, short on work now that elaborate hairdos had been replaced by straitened revolutionary styles [Grattan-Guinness 2003: 109]). De Prony’s tables differ from those that Godbid printed in their form and function as well as in their creation. Never published in full, they remained for the most part in manuscript form: monumental, to be sure, but — as Daston argues — ‘a symbolic if not a practical landmark in the history of calculation’ (Daston 1994: 189). Daston is keen to point out that the calculation of these logarithms was not mechanical: it was more like an expert production process than a proto-industrial factory. Nevertheless, these tables were produced by hired assistants and then checked over by more knowledgeable mathematicians, in a reversal of the flow of knowledge we see in Morland’s, Collins’s, and Napier’s books, in which the hard labour is done by the more learned man, and then handed down to be sent out into the world.

[38] Ian Hacking has described the ‘avalanche of printed numbers’, and of ‘printed and public tables’ that were produced in the years following the Napoleonic wars in Europe, and which gave rise to modern statistics as we know it (Hacking 1990: viii; 73). The earlier tables, including Godbid’s, did not quite constitute an ‘avalanche’, but nevertheless their handiness was constantly challenged by their own efforts to achieve completeness. Despite their claims to ‘contraction’, however, such as John Smith’s promise not to use ‘many words and much paper’, there remained a bulk to these books and these tables (Smith 1673: sig.A5v). This spreadingness happened not just over pages and pages, but exceeded the limits of the small book, too. John Graunt, like Collins, talked of ‘reducing’ his numbers to tables, but the tables always threatened to increase wildly; in the Graunt’s tables, showing the whole together, the neatly-printed fold-out table dwarfed the book itself. In Godbid’s vade mecum tables, the calculations could be given to more and more decimal places, or to smaller and smaller degrees of accuracy. Rather than summarising, discriminating, and organising, as other kinds of tables did, these tables were potentially endless: they spread out rather than shrunk down.

[39] The profusion of numbers in these tables provided visual and material similarities between different kinds of mathematical, or numerical, texts. There was undoubtedly a distinction between learned and tradesmen’s books, and the books that we might think aimed to disseminate mathematical learning did not, simply by virtue of being in English, necessarily reach the ‘common practitioner’ (Feingold 1984: 180). And yet the printing house of Godbid and his successors provides a link between different ends of the market. In their handling of numerical tables these printers showed a familiarity with numbers, and a careful and accurate way to handle them, that pleased John Wallis in Oxford just as much, and for the same reasons, as it pleased John Collins in Bristol.

[40] Whereas other tables encouraged a processing of information, in the reference tables I’ve been considering here knowledge is laid out but not explained; given to be used rather than, necessarily, understood. These tables don’t represent a relational gathering of knowledge in as the tables Foucault invoked; they are not synoptic patterned numbers as are Graunt’s; and they have none of the digestion of Hamlet’s writing tables or the splitting into parts of the dichotomous. The direction of interpretation here is outwards: they send knowledge out into the world, for convenience and use. Their material form, as pages of unreadable number, is unwieldy and rather forbidding, but they are framed by paratexts that suggest how to read them by using directed methods of navigation. These ubiquitous, everyday tables formed an important and very common form of numerical printing, as well as offering another variety of early modern tabular thinking. And for all their material links to more academic mathematical books, the texts that Godbid printed for Collins and the others were literally manuals, intended (as Collins promised) ‘for the guidance of the hand’. These were books, and numbers, with which to go about interacting with the world.


[1] I have modernised i/j and u/v spellings throughout. [back to text]

[2] This, the eighth edition of An Introduction to the Skill of Music (1679; Wing P2481) is made up of three parts. First, Playford’s ‘The Grounds and Rules of Music’; second, ‘The Art of Descant: or Composing of Musick in Parts’, by Thomas Campion; and lastly ‘The Order of Performing the Divine Service in Cathedrals and Collegiate Chappels’. Each of the three sections begins with new pagination. The third and final part begins on sig.L7r at p.1; the advertisement appears on what is pp.7-8 of this section, sig.M2r-v (quote taken from p.7, sig.M2r). Please also note that the EEBO copy of this book (from the copy in Bristol public libraries) is wrongly listed as dating from 1697, rather than 1679.[back to text]


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‘1144000727777607680000 wayes’: Early Modern Cryptography as Fashionable Reading

‘1144000727777607680000 wayes’: Early Modern Cryptography as Fashionable Reading

Katherine Ellison

[1] Cryptography, considered one of the occult sciences until the early seventeenth century, found new status in Britain during and after the Wars of the Three Kingdoms as a respectable, useful scholarly discipline publicly embraced by early modern mathematicians as well as political and military strategists. In manuals published for both general readership and for specialist, court audiences, cryptographers John Wilkins, Noah Bridges, Samuel Morland, and John Falconer launched a rhetorical campaign to legitimize ciphering and deciphering as practical as well as theoretical benefits to English national and financial security. Yet, England’s political and economic confidence was not the sole or even primary objective of the manuals published in unprecedented volume from the 1640s through the 1680s. Though their uses of numbers vary, implicit in Wilkins’s Mercury; or the Secret and Swift Messenger (1641), Bridges’s Stenographie and Cryptographie (1659) and its anonymously published sequel Rarities: or the Incomparable Curiosities in Secret Writing (1665), Morland’s New Method of Cryptography (1666), and Falconer’s Cryptomenysis Patefacta: Or the Art of Secret Information Disclosed without a Key (1685), is the promotion of ciphering and deciphering as a mathematic mode of reading that should be adopted in one’s everyday, particularly domestic, life.[1]

[2] In their recent studies of early modern reading habits, David Scott Kastan and Heidi Brayman Hackel warn scholars not to make assumptions about genre and literacy and convincingly argue that theories of literacy must be backed by archival evidence (Kastan 1999; Brayman Hackel 2005: 9; see also Jardine and Grafton 1990, Sherman 2002, and Sherman 2008). Certainly, I could not jump to conclusions about genre when I first read pedagogical cryptography texts while searching for the key to some ciphers in Sir Thomas Scot’s letters. Mercury, for example, resembles a writing manual yet is also like an arithmetic textbook; it is a study of natural philosophy and also a book of secrets; it is vaguely similar to stenography manuals and yet also reads like a historical romance. It has poetry and teaches ciphers in music and geometry. Bradin Cormack and Carla Mazzio categorize how-to books of the period generally as those that instruct readers ‘how to express yourself’, ‘how to do things’, ‘how to be somebody’, ‘how to look after yourself’, and ‘how to find your way’, and cryptography manuals teach all of these skills. They are guides to rhetorical gesture, but like surgery manuals they also ‘are concerned with emergent professional fields and identities’ (Cormack and Mazzio 2005: 85). The manuals promote new practitioner disciplines, help define the parameters and protocol of specialization, and also dictate modes of civility and self-fashioning. Promoting the recreational popularity of cryptography, the manuals also emphasize self-control of one’s intelligence as a marker of personhood. And certainly, learning to cipher or decipher also means learning to look after oneself.

[3] Though writings on cryptography during the seventeenth century are different enough from one another that they must be attended to separately, manuals between 1641 and the 1680s all widen the definitions of writing and reading to include the analyses of numbers and other computational symbolic systems used in the sciences. Each of the manuals, explicitly or implicitly, addresses contemporary anxieties about communal reading and an author’s lack of control over who reads, or is read, extending even to private correspondence. All are frustrated with confining textuality to the alphabet. As material objects mobile in the community in both manuscript and print form, letters and other documents exchanged hands known and unknown. At the same time that cryptographers propose methods for keeping messages secret within this world of publicity, then, they also offer — or claim to offer — methods not reliant upon a reader’s individual ability to write in alphabetic characters or in languages like English, Latin, or Greek. These manuals do tell readers what to do, but their instruction is not as directive as one might assume: they also tell readers that writing and reading are not activities defined by or restricted to words, sentences, or generic conventions. A citizen who cannot spell can still cipher his or her communication by experimenting with numbers or various symbolic systems. The archives yield much that can help current scholars understand that literacy, during the seventeenth century, was not a simple binary, as rates based on signatures or spelling imply (Cressy 1980; Bennett 1970).

[4] Wilkins’s foundational manual establishes cryptography as an experimental science and natural magic. ‘Experimental science’ and ‘natural magic’ are terms that William Eamon defines as synonymous for some seventeenth-century thinkers, like Wilkins, following the rhetorical footsteps of Roger Bacon: ‘The manipulation of nature by the application of art, the medieval conception of what today we would call scientific technology’ (Eamon 1994: 51; see also Macrakis 2010). By experimenting with and on nature, as through repeatable procedures or processes, one could not only learn the secrets of the universe but harness its powers in such a way that those unaware of the procedures would think they see unexplained magical phenomena. Ciphers were and are, in this sense, scientific technologies to assist human communication. Eamon notes how Wilkins was a transitional figure in the world of seventeenth-century science, like Sir Thomas Browne, but unlike Browne did not publicly decry books of secrets. Rather, Wilkins was well known for his efforts to popularize science. Eamon cites Wilkins’s Mathematical Magick (1648) as an example of one influential effort to increase public interest in mechanical engineering. Wilkins openly combined his knowledge of experimental science, such as through the Oxford Experimental Philosophy Club, with activities that demonstrated how dazzling and entertaining science could be (Eamon 1994: 309). Where cryptography manuals differ from other books of secrets on, for example, potions, alchemy, medicine, or the supernatural is in their focus on reading practices and the potential uses of mathematic knowledge for more effective communication. Mercury, I find, is Wilkins’s attempt before Mathematical Magick to situate mathematics as the foundation of a flexible language of secrecy. Mercury begins to build a new reputation for cryptography as an experimental discipline and a demonstration of natural magic, in contrast to its previous image as a practice of black magic and incantation, such as in Johannes Trithemius’s influential manuscript ‘Steganographia’ (circulated widely in 1499) and his longer work, Polygraphia libri sex, posthumously printed in 1518.[2] Though Polygraphia was largely practical, pushing cryptographers to use multiple media as well as new mathematical approaches, the final step in encryption required chanting for angels to deliver the messages.

[5] Wilkins, Bridges, the Rarities author who only goes by G.B., Morland, and Falconer revised the public image of cryptography, and through it communication by numbers and various numbering systems, as useful in nonmilitary and nonpolitical life as well as in times of war and scandal by using three main rhetorical strategies. First, they present ciphers as learnable, no matter one’s exposure to alphabetic reading and writing systems. In fact, cipher is offered as a language uniquely accessible to readers of all classes and even ethnicities and cultural backgrounds. Second, they present cryptography as most effective when it is experimental and creative, denying that ciphering and deciphering are strictly finite thought processes that follow set steps without imagination. Manuals teaching ciphering also emphasize that the mathematic and geometric foundations of early modern cryptography allow for interpretive liberty. By creative and interpretive, I mean that readers are invited to learn the foundations of ciphering and deciphering but then required to build upon their own personal experiences and observations to reach solutions from various paths. In other words, there is more than one way to cipher, and media as diverse as music, drawing, geometry, and gesture can be as effective as alphabets. Third, music, drawing, geometry, gesture, and other approaches are innately computational and arithmetic in nature. Early modern cryptography was a multimodal discipline in which numbers could be symbolically expressive.

[6] As Karen Britland and Sarah Poynting have recently shown, political use of cipher was relatively uninspired during the wars. Queen Henrietta Maria, Charles I, their treasurer Sir Richard Forster, Charles II, and other important players relied mostly on substitution ciphers, which by contemporary cryptographers’ standards were notoriously flimsy methods. Nadine Akkerman finds this to be the case in the correspondence of Elizabeth Stuart as well. Elizabeth used seven substitution systems across her letters to various correspondents, and though the systems would be altered when there was suspicion of a breach, ‘keys were used for decades’ and, Akkerman suggests, Elizabeth ‘placed rather too much trust in them’ (Akkerman 2011: 1055). The instruction occurring in contemporary manuals is more sophisticated. The manuals promote ciphering logic as a kind of reading that requires more than mere substitution. And finally, they make this argument while highlighting the status of cryptography as a former occult practice. In other words, they do not deny the occult history of the discipline but rather capitalize upon the popularity of occult publications and books of secrets.

[7] Though non-specialists use the terms ‘code’ and ‘cipher’ interchangeably, as did some seventeenth-century cryptographers, my use distinguishes codes as meaningful substitutions (for example, the number 28 is code for King and so has meaning) and ciphers as the process or steps involved in arriving at meaning. Codes require a codebook; ciphers require following algorithmic instructions determined in a key. In other words, ciphers are mathematic, in the general sense that mathematics is the study of patterns and structures, while codes can be whatever an encoder says they are. Seventeenth-century cryptographers were more interested in ciphering than coding because the existence of a codebook meant that all of a party’s messages could be decoded easily if the book was discovered or captured during battle. Keeping keys near ciphers (or even writing the keys down), of course, was also irresponsible and dangerous: this was a lesson learned the hard way four years after the publication of Mercury, when Charles I’s private papers, including his cipher keys, were taken during the Battle of Naseby on June 14, 1645 and published as The Kings Cabinet Opened. The most secure ciphers, the manuals explain, require ingenuity, cannot be recorded simply as a sequence of clear steps, and yet can be swiftly solved. Geometric shapes one might recognize at a glance but that are not codes, hand gestures in particular sequences, calculable musical notes, and even patterned flashes of light can communicate secretly and swiftly, as Wilkins’s title boasts, and demand a kind of reading that recognizes arithmetic functions as literacy.

[8] National or political security and maneuvering were not the only objectives of early modern cryptography manuals. David Underdown and, more recently, Geoffrey Smith, have discussed the necessity and popularity of ciphering during the Wars of the Three Kingdoms, but the manuals have not been discussed as influential in military or court practice as, for example, the Scottish Army made plans to invade England or Charles I maneuvered locations during hiding (Smith 2010: 101-03). Only Lois Potter grants the publication of Mercury a prominent position in contemporary politics, noting that its appearance in 1641 not coincidentally paralleled the dissolution of the Star Chamber and George Thomason’s decision to start collecting every publication he could starting that same year (1989:1-2; see also Woolf 1990). She sees Mercury more as a theoretical statement than a political one, however, as a sign of the public’s desire for openness. She does suggest that since Wilkins had sympathies with Parliament, Mercury could be an exposure of Charles I’s opacity. Yet Wilkins’s political alliance is not clear. Like most cryptographers, including Morland and John Wallis, Wilkins served under multiple regimes. He was a Royalist who served Cromwell and Charles II (Shapiro: 1969: 3). Morland served Cromwell but worked as double agent to organize Charles II’s return (Dickinson 1970). John Wallis, arguably the most skilled cryptographer of the era but unpublished in that discipline until the eighteenth century, was employed by Cromwell, Charles II, and William and Mary with apparently seamless transition. Much less is known about Bridges, a clerk for Parliament, beyond clues in Stenographie and Cryptographie that he may have been promised a post in Charles II’s office. Little has been written about Falconer. David Kahn notes that Falconer was entrusted with James II’s private cipher before he ascended to the throne, according to a genealogical book marked ‘Falconer’s Writings’ on the spine in the New York Public Library (1996: 1016). Regarding politics perhaps all we can conclude, as Wilkins’s biographer Barbara Shapiro does, is that Wilkins (and I would add all the major cryptographers of the period) ‘were [always] in the right place at the right time’ (1969: 3).

[9] Wilkins, Bridges, G.B., Morland, and Falconer were most interested in the instructional process of creating and reading secret messages with mathematic precision and everyday use. The strengths of each manual differ, however. Mercury demonstrates the use of what Wilkins calls ‘philosophical numbers’, a precursor to his universal language in An Essay Towards a Real Character, and a Philosophical Language (1668) twenty-seven years later. A philosophical number is universal; it represents ‘any such measure, whereby we judge the differences betwixt severall substances’ (Wilkins 1641: 107; see also Heeffer and Van Dyck 2010). All cultures use philosophical numbers, he notes, and with little effort any two correspondents of different cultures, even if they speak wildly different native tongues, can understand the meaning of their philosophical numbering systems. Similarly, cultures with linguistic difference may use musical notation, chemical symbols, or astronomical signs similar enough to be understood by non-natives. Philosophical numbers are the inspiration for Wilkins’s growing obsession with a universal language scheme built upon ‘things’ that would finally unite the ‘seventy-two Languages of the first confusion’ (1641: 110).[3] The ciphers Wilkins is most known for, then, are geometric and spatially dynamic. Shapes that appear random in nature may be ciphers to be solved using a series of steps familiar only to the insider correspondents. The shape of a hand gesture, the pattern of movement in a person’s gait, or even the shape traced by the eye can all be ciphers and thus require algorithmic solution.

[10] Though with a different rhetorical agenda than Mercury and through a string of bizarre examples, Bridges’s Stenographie and Cryptographie also offers an approach to the discipline that foregrounds numbers as expressive media for secret communication. Bridges’s Stenographie and Cryptographie opens as a primer in shorthand and ends as a critique of the false claims of accessibility in cryptography manuals. The first section, which includes chapters of linguistic sophistication on how to write single and double consonants, dipthongs, prepositions, clauses, and other syntactic elements stenographically, is quite unlike Mercury in that it requires a high level of alphabetic literacy. The dedication to the reader stresses not only that readers must be able to write well but that they must excel at ‘faire writing’, or elegant penmanship, and the lessons in stenography exclude any reader who is not familiar with the intricacies of English grammar in writing (Bridges 1659: ‘In Commendation of Faire Writing’). The second section on cryptography begins with praise of Wilkins and the lessons of Mercury and then quickly unfolds as a critique of Wilkins’s rhetoric of accessibility. Bridges acknowledges Wilkins’s contributions to the discipline and the sophistication of his ciphers, but his main impetus is to distinguish between ciphering and deciphering as distinct skills and counter Wilkins’s open invitation to the everyday person, even without mathematic or linguistic education, to aspire to decipher. Bridges excerpts examples from Mercury and walks readers through the mathematic variations possible; the number of combinations one might have to sort through to arrive at one of Wilkins’s solutions, Bridges shows, proves that only those with mathematic ability and training in cryptography can truly decipher. Ciphering can indeed be done without advanced literacy in numbers or alphabets, but most alleged decipherers, Bridges argues, have been frauds ‘in quest of a picklock’ (1659: 31). ‘Yet with submission to the Authors [Wilkins’s] great judgment and learning’, Bridges writes, ‘I take leave to say there are those who pretend an ability of unfolding any Character’ (1659: 30). Pretenders attempt to impress political authorities, and as the authorities are also not experts, often succeed in securing courtly confidence and undeserved positions. Though even a non-expert can create a cipher, only the truly mathematically skilled can solve it. Bridges’s argument is based on the astounding number of transmutations possible when ciphers have upward of nine to twenty four letters. He uses the example of the following verse:

            Quod sat sit (sors) da, fed ne post tu                                                                                                                           rape de me.

Bridges notes that this verse ‘is variable 39916800 wayes’ (1659: 33). He does not offer any definition of transmutation, but his examples demonstrate that the concept is similar to Wilkins’s philosophical numbers in that both involve the use of one symbolic system to articulate the inexpressible. Though Blaise Pascal and Gottfried Leibniz are credited with the first theory of transmutation as a foundational principle of trigonometry, Bridges and other mathematicians were also experimenting with the concept during the same decades. The applicability of transmutation in encryption and decryption is particularly evident in manuals like Stenographie and the work later published by cryptographer and mathematician John Wallis. Transmutation is, generally, the expression of intangible or intractable quantities in rational numbers; in other words, the translation of very large, voluminous, and even potentially infinite possibilities in terms that can be more easily grasped using analogies familiar to a culture. ‘If the said verse were written in all its varieties’, for example, ‘it would make at the least 3323 such volums [sic] as is Ovids Metamorphosis’ (Bridges 1659: 33). Further, given that Metamorphoses contains 12,012 verses, ‘if they were all written one under another, in one roll of paper in the form of a long square, allowing but an inch of the roll in length to every 6 verses, that roll would extend in length 105 English miles’ (Bridges 1659: 33). Only after his mathematic exercise in transmutation does he note that his point is simple: ‘methinks if these great pretenders [to deciphering] did but consider what a prodigious number the transmutation of the 24 letters will produce, and how skill in number doth furnish us with mysterious and hidden things, which present a world of confusion, they should be strangely abasht’ (Bridges 1659: 32-3).

[11] Though the 1665 Rarities, authored by G.B., is considered a sequel to Stenographie and includes some of the same examples, it frames itself as a commentary on the inaccessibility of Stenographie’s lessons and, like Mercury, revises examples for the non-expert reader. Similarly to Mercury, which focuses on multimodal and even bodily geometric cipher — the creation of signs using hand shapes, for example — Rarities opens the potential graphic field of cryptography to include everyday objects and movements familiar to all of his readers. Rarities also incorporates demonstrations of transmutation, but those demonstrations are so hyperbolic as to become satirical. G.B. begins by explaining ‘easie Examples’ of transposition using the phrase ‘God with us’ in order to build reader confidence (1665: 7). The first two are indeed easy: they are basic substitution ciphers in which, for instance, the key ‘2345’ is lined up with the phrase, each number indicating the alphabetic character shift (G.B. 1665: 4):

G o d w i t h u s.
2 3 4 5 2 3 4 5 2

If the agreed upon alphabet omits j and v (and therefore is a b c d e f g h i k l m n o p q r s t u w x y z), the cipher will appear as:

i r h b l x m a u

The correspondent would know the key and therefore to shift the letters backward to reach the original message. G. B. understands that such ciphers are not secure, so having helped the reader see that ciphering is not necessarily difficult he shares more complicated cases that ‘I must confess my self much delighted in’ (1665: 5). He goes on to describe an example that appears in an unnamed previous manual, which is most likely Stenographie, but without, he finds, adequate instruction: ‘Under this Cypher I found the Example set down, but no Directions thereon, nor could I possibly find out the contrivement till I met with the Authors Directions in the Second Edition of the following Tracts, as I formerly hinted’ (G.B. 1665: 12). I will consider the implication of G.B.’s work across cryptography publications in the next sections, as well as the fact that the manual he consults is missing instructions, but here it suffices to emphasize that G.B.’s mathematical purpose is to appear to clarify those other editions and demonstrate his ability, even as a self-proclaimed novice, to illuminate what was most complex in past examples. Also valuable to G.B. is using computation to prove that complex ciphers require independent thinking beyond rote instruction: the untrained mind can learn how to fill in the blanks in the exercises and how to use numbers even without formal education. Ciphering and deciphering, in other words, do not follow set instructions that one need merely memorize. The interpretive process can vary by individual and even by cipher, and multiple paths may lead to the same solutions.

[12] Morland’s New Method (1666) represents the most computationally challenging of the mid-century manuals, though it shares with Rarities and Mercury an emphasis on creative problem-solving. New Method least resembles contemporary conduct manuals, and Morland’s purposes are clearly political and military, not domestic literacy, reform. I include this manual because it is in conversation with the others, however; Morland was aware of and explicitly resistant to the popularization of the discipline for everyday use. He criticizes previous manuals without naming them, remarking that ‘the ordinary Methods of Cryptography, (vulgarly termed Writing in Cypher) where they are weakly contrived, or carelessly written’ may ‘with much pains and industry’ be solved by unintended recipients (Morland 1666: 7). Published expressly for Charles II and his court, this short pamphlet works immediately through what Morland sees to be the most secure ciphers, all of which require complex steps using various mathematic principles. Morland’s ciphers are based entirely upon the concept of a ‘numerical alphabet’ that requires lengthy instructions and knowledge of geometry (1666: 8). Instructions may determine whether a message should be deciphered using a parallelogram, triangle, trapezoid, rhombus, or a polygon simple or central in transcription in ascending columns, diagonally, or even in a serpentine pattern. For one example, Morland notes that there are ‘295288899039603018847 618609643519999999 false ways’ to cipher the message but only one that is true (1666: 8). Morland muses that no kind of genius could decipher a message with this complexity of instructional steps: ‘Now how any Mortal should hope in all his life time, (were he sure to live Quadrillions of Trillions, of Billions of Methusalem’s years!) to hit upon so many Heterogeneous Quasita mentioned in the premises, for which he has in a manner no Data, it cannot enter into my Understanding’.  He sarcastically notes that only the ‘Black Art’ could possibly illuminate such a full-proof method (Morland 1666: 8). New Method is also an advertisement for Morland’s Machina Cyclologica Cryptographica, a series of connected cipher disks for computing that was not innovative in basic design but built from impressive metals and marketable to a fashionable public. Though he scolds other cryptographers for carelessly written manuals that damage the credibility of the discipline, he also engages in profitable outreach to non-specialists. His cipher disk even features a red rose at its center. Morland invented several pocket-sized machines in gold, silver, and bronze that helped mathematics appeal to eager consumers, his Cyclologica Cryptographica among the more useful (Ellison 2013).

[13] Cryptography historians are perhaps most interested in Falconer’s Cryptomenysis, credited with the first use of columnar transposition (Kahn 1996: 155). Columnar transposition entails writing a message in columns of pre-determined length according to a keyword. In fact, this was not Falconer’s innovation but already noted in Rarities and Morland’s manual, with Morland creating newer approaches to that method than Falconer demonstrates. However, Falconer does include other ciphers that are not found in earlier manuals, among them a witty letter that reads one message when whole and a second, contradictory message, when folded in half. Falconer is, perhaps not surprisingly, most influenced by Wilkins and even includes a number of Wilkins’s examples, but Falconer is more invested in contemporary shortcomings in deciphering than innovations in ciphering. He does not share Wilkins’s fascination with the potential of a universal language built upon the concept of philosophical numbers, but his project to popularize mathematics as a mode of everyday reading, and through it to legitimize cryptoanalysis as a respectable discipline necessary for the success of a nation, is evident from the opening pages of Cryptomenysis. Falconer acknowledges that ‘secret intelligence’ has certainly worked in the background of most human conflict and that even during England’s recent military conflicts, ‘People of both Kingdoms, studyed a part of this BLACK A*t’ (Falconer 1685: ‘To the Reader’).[4] Yet, he quickly notes, there is no denying that secure methods of private communication are absolutely necessary during times of war and peace. Nodding toward the popularity of the cryptography manual as a genre, he observes that ciphering has received much attention, and many manuals have boasted the innovation of ciphers that cannot be solved without keys. Deciphering has received less attention, but he is confident that with experience, those who solve ciphers can be as innovative as those who write them. He is careful to point out that he disagrees with those who make fabulous claims about their abilities to solve any cipher no matter its difficulty, as I will show Morland taunting in his New Method. Falconer is not unrealistic: he believes not that any reader can solve any cipher with experience but, rather, that once one understands the rules of deciphering, s/he can apply those rules across languages and, with creativity and much practice, solve riddles without access to keys. ‘It is true, the unridling of such Mysteries, is more immediately the Province of those who sit at the Helm of Affairs, Military and Civil’, he writes, ‘Yet if a private Sentinel, by deciphering an intercepted Epistle, should save an Army, &c. ‘tis no Crime, I hope, that he be more clear sighted than his Superiours’. Emphasizing the need for such skills to be taught to a general public, he notes that ‘Knowledge of Uncyphering should not be confined to a Corner’ (Falconer 1685: B4).

[14] Looking across this sample of manuals published during and after the Wars of the Three Kingdoms, one can see that even if their political goals are vague or secondary, the immediate historical contexts may in part explain each manual’s interest in promoting cryptography as a mathematic discipline well-suited for domestic literacy. Wilkins, writing as conflict is clearly underway, carefully addresses two audiences. He is sensitive to specialists desiring better methods for the immediate communication needs of the wars, and he is also mindful of nonmilitary readers frightened by the turmoil and seeking more control over their private correspondence. Stenographie and Cryptographie is published in 1659, and in it Bridges hints that undeserving frauds have impressed political decision-makers and secured eminent positions while those with true expertise, rigorous training, and value to the new regime, perhaps like himself, are overlooked. Rarities and New Method are of a different political decade and published when one could look back in hindsight at past successful and failed alliances and communication methods. While both G.B. and Morland continue promotion of cryptography, and while both seek to profit from teaching the discipline, G.B. mediates between specialist and non-specialist audiences and Morland writes strictly for the court. Falconer’s manual balances complex ciphers for expert users with methods that even the least expert could use for recreational correspondence, such as hiding a message within the text of an unfolded letter.

Ciphering as Accessible Pedagogy

[15] By the early seventeenth century, literate audiences were made up of diverse populations of readers. Readers were of different skill levels, demonstrated mastery of more or fewer languages, and sought out texts based on different tastes and expectations. Jennifer Andersen and Elizabeth Sauer, like Roger Chartier, focus on the ways in which ‘modes of reading become more various and distinct’ during the early modern period (Andersen and Sauer 2002: 4; Chartier 1989: 174; See also Sharpe 2000: 55 and Ingram 2002: 169). Examination of epigrams and other paratextual elements of printed books that address and even attempt to typify readers, in particular, characterizes one of the approaches to the history and theory of reading that Jennifer Richards and Fred Schurink survey in the 2010 special issue of Huntington Library Quarterly. While the trend in book history has been toward materiality and emphasis on ‘use’ as articulated by, for example, William Sherman in Used Books (2007), genres that invite interactivity are also attracting scholars of reading history. Cryptographers found a subject that appealed to remote, anonymous, numerous readers from diverse religious backgrounds, a spectrum of social situations, and from both sides of the political divide, and their manuals provide a ‘model for utilitarian reading’, to use Richards and Schurink’s phrase (2010: 351).

[16] On the title page of Rarities, G.B. reaches across this broad audience to promote his book: his readers include ‘Ministers of State’, ‘Ladies’, and ‘every ordinary person’. He notes there that his manual is ‘publish’d to promote the Publick, to delight the Ingenious, and encourage the Industrious’. Cecile Jagodzinski agrees that the ‘target audience for these letter-books and cipher keys was wide, ranging from the upwardly mobile soldier to the pining lover’. ‘Ciphers and letters’, she writes, ‘became the conveyors of everyday life and familiar relationships for commoners as well as kings’ (1999: 86). I find that male and female readers of all social ranks and professions were invited by the authors to find their own uses for the manuals, and I would add that more than being merely students, the readers of these manuals were required to navigate the genre, to make connections, and to even enter debate about what had been and would continue to be the uses of the craft in the past and future.

[17] What does it mean, though, for ciphers to be ‘the conveyors of everyday life’ (Jagodzinski 1999: 86)? Answering this question reveals the differences between cryptography manuals and the related but not synonymous genre of the arithmetic textbook. While both arithmetic and cryptography manuals share the rhetoric of universal reader usefulness, arithmetic textbooks often, but not always, welcome only the experienced student of arithmetic, and demand (but not necessarily demonstrate) absolute linear reading. It is important that problems are solved in a standardized way; accounting methods for merchants, for example, should be consistent across the profession. Cryptography manuals are not linear reading experiences and emphasize multiplicity in problem solving. Even their instructions omit steps or move quickly between lessons, whereas early modern math textbooks proceed step-by-step, scaffolding knowledge. Travis D. Williams outlines the basic structure of the arithmetic textbook beginning with even the earliest primers of the sixteenth century. The genre is largely ‘scholastic’ in content and organization, he finds, moving logically through basic operations to fractions and other more complex functions (2012: 171). Cryptography manuals emphasize that readers need no prior knowledge of the discipline to succeed, whereas many arithmetic textbooks caution that readers need a solid foundation in, at the very least, addition and subtraction. Karen Britland convincingly summarizes that ‘cipher invites a kind of non-linear and comparative reading that encourages one to look beneath the veil of cipher in the hope of uncovering a truth’ (2014: 23; see also Edwards 2003). My only addition to Britland’s point here would be that the manuals provide alternative reading models and, by publicly embracing the formerly ‘occult Art’ of cryptography, as Falconer refers to it, the manuals popularize a multimodal kind of computational thinking that allows for interpretive flexibility (Falconer 1685: ‘Introduction’ B). Cryptography manuals are also not training readers to be professional cryptographers. Citizens can adopt various methods, tweak them, and use them for serious purposes or personal entertainment.

[18] Addressed to a wide audience of novice and specialist puzzlers, cryptography manuals proclaim the accessibility of their craft. Even as Morland boasts that only dark magic could help eavesdroppers solve messages written in his ciphers, he still assures readers that his instructions are accessible and the ‘learned Reader will easily perceive that it may yet be read and transcribed in a very great number of other different ways and Methods’ (1666: 4). Certainly, claims of easy learning and multiple pathways to the same solution are common in instructional manuals on various subjects during the seventeenth century. Yet, there are distinct differences between these claims when one compares arithmetic textbooks and cryptography manuals. In a 1629 copy of Arithmaticke: Or, an Itroduction [sic] to learne to reckon with the Pen, or with Counters, in whole Numbers or broken, republished numerous times throughout the sixteenth and seventeenth centuries, the (unknown) author stresses the necessity of mathematics for ‘all manner of persons’ (1629: ‘To the Reader’ A2). The author acknowledges that most arithmetic manuals are not accessible to a general readership’s understanding, so the goal of Arithmaticke is to be clearer in instruction. The reader address does not, however, claim the education will be easy and cautions that readers must proceed in the precise order of the lessons: ‘For if you leape to the second part before you have perfected the first, or to the thirde, before you have seene the second: you shall never prosper nor profit in this Arte’ (1629: ‘To the Reader’ A2). Likewise, William Barton’s Arithmeticke abreviated Teaching the art of tennes or decimals to worke all questions in fractions as whole numbers (1634) addresses the reader to establish the discipline’s usefulness and accessibility. Though Barton notes he will try to be clear in his instruction, however, the textbook requires sophisticated past knowledge and is designed for serious mathematic scholasticism. Henry Phillippes’s edited version of Baker’s Arithmetick: Teaching The Perfect Work and Practice of Arithemetick both in Whole Numbers and Fractions (1670), also published often during the period, assumes the need for all citizens to understand mathematics but warns readers in the epistle dedicatory that it is not accessible to everyone: ‘Let none enter here that is ignorant in Arithmetick’. Unlike cryptography manuals that require no previous mathematical or linguistic knowledge, Baker’s textbook is only for those with prior ‘school-day’ or merchant experience (1670: ‘The Epistle Dedicatory’).

[19] Dedicatory poems that precede Mercury resemble the praise readers would have been accustomed to in conduct manuals. In couplets that open one dedicatory poem in Mercury’s front matter, Tobias Worlrich is so confident that the material is easily learnable that he remarks, ‘I’m loth to tell thee what rare things they be,/ Read thou the book, and then thou’lt tell them me’ (Wilkins 1641: ‘To the Reader’). Cryptography is the ‘Mint of Knowledge’ and any ‘man that deals in Traffick’ and ‘who profess the Knowledge of Nature or Reason’ can learn to write and read code. Though poetry ‘is not in every man’s power’ and ‘requires such a natural Faculty as cannot be taught’, ciphering has universal potential and can be learned quickly (Wilkins 1641: 11, 20). With the system he begins to envision during Mercury and that he will continue in his Essay Towards a Real Character, Wilkins promises that citizens from around the world will be able to learn a common, universal language in a very short time. Cryptography manuals emphasize the accessibility of the craft to non-specialist readers, and as the essential skill of the modern citizen. To discount cryptographers’ claims as mere convention would be to underestimate the literacy reform project that they explicitly champion.

[20] Throughout Mercury, Wilkins emphasizes how easy it is to learn ciphering and how, as the teacher, he has streamlined his instructions for more efficient learning. ‘It were an easie matter for a man that had leasure and patience for such enquiries’, he writes, ‘to find out sundry Arguments of this kind for any purpose’ (1641: 65). As he works through one complex cipher, he assures the reader that ‘for the easier apprehending of this, I shall explain it in an example’ (Wilkins 1641: 57). Occasionally Wilkins even stops his instructions to assure readers that they now know so much they do not need him: ‘There may be divers other ways to this purpose, but by these you may sufficiently discern the nature of the rest’ (1641: 60). Wilkins and Morland emphasize that there is more than one way to follow a particular cipher, and anyone ‘from the Arctick to the Antartick Star’ will be able to learn the art of cryptography and ‘freely traffick through the Universe’ (Wilkins 1641: ‘To Mercury the Elder’).

[21] To further prove how learnable cryptography is, Wilkins frames his manual as the product of his own learning experience, stating at the very beginning that Mercury was inspired by Francis Godwin’s Nuntius Inanimatus (1629), which described a hypothetical telegraph system.[5] Wilkins notes that he at first questioned the validity of Godwin’s outrageous claims to teach readers how to ‘discourse with a Friend, though he were in a close Dungeon, in a besieged City, or a hundred miles off’ (1641: ‘To The Reader’). Yet the credibility of Godwin persuades him to read further and to study the history of cryptography and secret writing; with studious diligence, he ‘attained mine own ends’ and masters the language (Wilkins 1641: ‘To the Reader’). Exhilarated by his own ability to learn an ancient practice that he at first thinks is a type of magic, he writes Mercury to distill the information from his extensive studies and efficiently pass his learning onto the everyday citizen without the time or leisure to read Aristotle, Polybius, Julius Africanus, Frontinus, Isaac Casaubon, Johannes Walchius, Gerardus Vossius (the Latin name for Dutch scholar Gerrit Janszoon), and the many others who make up his bibliography. Wilkins presents himself as a student who has easily mastered the craft and is now passing it onto his readers.

[22] From 1641 until at least the Revolutionary War and the beginnings of American intelligence, authors of cryptography manuals presented themselves as imperfect readers, not as masters passing on their craft to students. In 1685, Falconer’s epistle to his reader explains that ‘a few years ago, having had some discourse with a Gentleman, concerning the Possibility of Resolving any Writing in Secret Character, and the means to perform it; I was taken with the Novelty of the Thing’. Cryptomenysis is thus presented as a kind of journal, as ‘the account of my Discoveries in this progress’ (Falconer 1685: ‘To the Reader’). G.B. also creates ethos through his identification with the reader in Rarities. He tries to serve as a mediator between the curious audience and cryptographers of the past, who were more skilled than they revealed. He begins his epistle to the reader recounting that he is about to publish ciphers that were ‘plunder’d from the Author of the following Tracts’ and ‘came shortly after to my hands, and have lain by me to little purpose’. He explains that he kept the papers so long without publishing them because ‘my great pains to make some progress therein came to nothing untill he in his second Edition gave the Learner Directions how to proceed’ (G.B. 1665: 1).[6] He then describes his efforts to understand the ciphers and stresses that ‘I have no ends beyond the kindness I owe to the common capacities, and the publick’ in sharing ‘that rare Art’ (G.B. 1665: 2). His loyalty to his reader is so strong, in fact, that he suggests that he has risked his life in letting them see the ciphers: ‘I mean well to all, and so consequently have the less to account for: but however I am concluded, I shall shortly (if I live) be Thy more useful Friend G.B.’ (1665: 2). Repeatedly throughout the manual he reiterates his innocence and stresses that he is merely trying to help the reader with those ‘easie Examples (of which I lately thought my self as ignorant as any that are to learn)’ (G.B. 1665: 3). This intimate relationship with the reader draws audiences to the manuals and helps make the genre so popular that some historians and literary scholars have observed that ‘it became quite the fashion to learn to write in cipher’ (Jagodzinski 1999: 85).

[23] Admittedly, the pose that Wilkins, Falconer, and G.B. adopt as students of cryptography sharing information with their similarly novice readers is immediately challenged by the knowledge they display. The poems that follow Wilkins’s epistle dedications, for example, praise his learning and reputation. The first poem, by Sir Francis Kinaston, represents him as the offspring, ‘Mercury the yonger’, of Trithemius and Gustavus Selenus, the pseudonym of August, Duke of Braunschweig-Lüneberg. Both were accomplished cryptographers of the early sixteenth and early seventeenth centuries, respectively. Wilkins is a genealogic inevitability, a cryptographer so talented that even ‘the winde,/ Should it contend, would be left farre behind’ (Wilkins 1641: ‘To Mercury the Elder’). In a second poem, Anthony Aucher addresses Wilkins as an ‘unknown God’ and in the third, Richard West emphasizes Wilkins as author and artist with timeless authority, more genius than Plato or any other philosopher or scientist in history: ‘This Dutchman writes a comment, that Translates,/ A Third Transcribes; Your pen alone Creates/ New necessary Sciences’ (Wilkins 1641: ‘To the Unknown Author; Wilkins 1641: ‘To his honour’d Friend I.W. on his learned Tract’). Ironically, these poetic dedications, and even West’s declaration that Wilkins is neither student nor compiler but creator, help establish the persona of cryptographer as curious reader. Because others assert Wilkins’s talent and trustworthiness, Wilkins need not do it himself. Further, the poems emphasize the experimental, magical, and natural. Cryptography is a marvel to admire and enjoy.

[24] Wilkins and the flattering poets that populate the beginning of Mercury assure readers that cryptography is a common sense language; gestures, sounds, movements, glances, and even nods can communicate according to predetermined systems, yet they are mathematically accessible because they are bodily. In fact, repeated emphasis on cryptography as corporeal help the authors build the reputation of cryptography as natural. It is not the subject of this essay to explore how cryptography assumes the living body as mathematic form — with multiple parts for counting, repeatable and calculable movements, and geometric shapes and lines to be computed — but all of the manuals operate according to this implicit theory. Richard West stresses that:

You tell us how we may by Gestures talke:
How Feet are made to speak, as well as walke:
How Eyes discourse, how mystique Nods contrive;
Making our Knowledge, too, Intuitive. (Wilkins 1641: ‘To his honour’d Friend’)

The ability to cipher begins not with scholastic mastery of math but with a kind of common sense understanding of how patterns function in nature. Richard West claims that Wilkins can provide a coping mechanism for individual readers operating within a wide audience, giving them strategies for recognizing and creating patterns to maintain their privacy while remaining part of a diverse crowd. Patricia Meyer Spacks helpfully notes that privacy ‘has little to do with the much-debated split between “public” and “private”; indeed, using codes or ciphers to communicate is not simply the opposite of making that communication public (2003: 3). Language as articulated through arithmetic provides attractive textual solutions to eavesdropping because it can circulate freely in the public yet hide thoughts that at least two people want to keep between themselves. Patterns hide and yet express. Cipher is not simply private language; it moves in wide circles, read by many but only fully understood by a few. It is the language of ‘psychological privacy’ which Spacks distinguishes from physical privacy (2003: 7). West’s italicized emphasis on ‘intuition’ shows that Mercury promotes cryptography as a skill anyone can learn to protect their thoughts because it is a natural ability that can be mastered with tuned senses and heightened awareness of one’s surroundings, a part of being human and interacting with the world. It is not just the special talent of mathematical prodigies working in isolation from others. By publishing their secrets (or pretending to), they also affirm that privacy should not only be a privilege of the powerful but the commoner as well. 

Cryptography as Experimental Science

[25] Ciphering would seem to be the one type of writing, and cryptanalysis the one type of reading, that eliminates interpretive plurality by purposely splitting audiences into those with the key and those without. Yet in early modern cryptography manuals, interpretive unpredictability begins as early as the pedagogical introduction to the craft. Cryptographers open readers up to an infinite textual territory, giving them new tools and then setting them free in a landscape of numbers and letters with potentially meaningful patterns they could never see before. G.B., for example, challenges readers to work independently, to interpret freely a type of language that one would assume is not open to diverse interpretations. When G.B. decides to ‘only hint that method’ and consequently let go of the reader’s hand, he creates a cognitive moment of parataxis, a situation in which readers must take information merely presented or juxtaposed without explanation and make their own meanings through experimentation (1665: 6). On the fifth page of his lessons, G.B. quits two more codes, the first because it is ‘not sufficiently secure against an artificial scrutiny’ and the next because it is a rather delightful code but too ‘obvious to the meanest capacities’ (1665: 5-6). Readers willing to work to understand those abandoned codes must, in the first case, repeat the directions many more times and, in the second case, look back to the second edition G.B. refers to often. The only assumption these authors make about their wide audiences, in fact, is that they are familiar with the genre of the cryptography manual. G.B.’s references to the unnamed second edition assumes prior reader knowledge. Wilkins does not make such ambiguous references, probably because Mercury is the first of many manuals to follow, yet he, too, builds from other works like Godwin’s. As well as linking unidentified texts, Rarities also has sketchy connections between chapters — G.B. moves abruptly from codes to invisible ink, for example. The logical gaps in the sources, chapter topics, and ciphering, however, are less problematic the more one understands the genre’s conventions. It is as though cryptographers draw and depend upon a collective text, and readers who want to understand a cipher more fully know they can look at previous manuals to find similar (or identical, since the manuals commonly replicate the same material) example. Cryptography manuals are intertextual, demonstrating awareness of other works in the genre and referencing them clearly. Little unattributed lifting of material occurs; copied examples are usually cited. Parataxis extends beyond the pages of the manual immediately in front of readers because they are encouraged to look across the genre.

[26] Both G.B. and Morland emphasize the interpretive liberty computational ciphers grant them even as they instruct toward ‘true’ or right answers. Recalling Bridges’s use of transmutation to prove that untrained decipherers could never discover the solution to ciphers written in more than nine characters, G.B. takes on the Latin verse example that Bridges provides, though G.B. uses two lines of the verse instead of just one. Bridges had computed that the one line of verse (‘Quod sat sit (sors) da, fed ne post tu rape de me’) has 39,916,800 transmutations. G.B. recalculates that Latin line and a second to conclude that it is more accurately ‘variable 1144000727777607680000 wayes, the words rape & neque standing alwayes as they are’ (1665: 15). G.B. then goes further:

But if they change places (as they may once) then the former number of varieties will be doubled thus 2288001455555215360000. Those written in all varieties according to 6 Verses or 3 Disticks in the depth of an inch, each Verse 3 inches long, would fill a roll of Paper of 3 inches breadth 62444484876533760000 feet long, whose weight would make 99857918246056 of our ordinary Cart loads. (1665: 15)

G.B. continues this series of computations, all prompted by musing upon how many variations a reader might find in the process of solving one cipher, for another page and a half. ‘If written on a piece of paper’, he notes, the variations ‘would cover the face of the earth 3,055 times’ and the ‘ink that should write the varieties would load 27 billion, 159 million, 778 thousand, 763 Carts and would fill a pool of 1 74/100 foot depth as broad as the whole Mediteranean [sic] Sea’ (1665: 16). G.B.’s intent in providing these familiar analogies to express intractable quantities is not to reveal fraud, as is the case in Bridges’s manual. With a wider audience and a more playful, even comic, tone, G.B.’s transmutations stress the number of possibilities as interpretively liberating.

[27] Morland also humors the reader with semantic multiplicity when he stresses that his example code has ‘Millions of Millions, of Millions, of Millions, of Millions, of Millions, of distinct Orders’ (1666: 7). Similarly, Morland explains the creative decisions a cipherer and decipherer can make:

It is evident, that every such Figure containing but 9 Columnes, admits of no less than Three Hundred Sixty two Thousand, eight Hundred and eighty different Transpositions. And others of them, as Fig.VI. and VII. Containing 18 Columnes, admit of Six Thousand four Hundred and two Millions of Millions, of different Transpositions; More by Three Hundred seventy three Thousand, seven Hundred and five Millions; More by Seven Hundred twenty and eight Thousand; And there being in the Writing 81 Letters, they make an Oblong containing 27 Columnes and three Lines, And consequently such a Figure will admit of 10888869450418352160768000000 distinct Transpositions. (1666: 4)

Morland’s pains to illustrate the creative possibilities open to cryptographers serve as much for self-flattery as they do advocacy for the popular uses of mathematics as multimodal reading. Yet, such illustrations of the interpretive possibilities of cryptographic thinking become conventions in the manuals. Falconer makes a similar, less boastful argument twenty years later that further legitimizes cryptography, but he reveals even more explicitly that alphabetic thinking is and always has been, in a sense, mathematic in nature. He uses the same kind of logic, showing readers how the number of transpositions possible using a 24-letter alphabet would reach ‘to the Man in the Moon’ (Falconer 1685: 5). ‘To be serious’, he continues, ‘you find twenty four Letters have 620448401733239439360000 several Positions’ (1685: 5). To a cryptographer, letters and numbers are interchangeable communicational characters, but numbers, in a sense, are even more expressive than letters because they can illustrate the enormous scale on which the human mind is capable of thinking. Falconer uses this convincing example, of course proven through computation: ‘If one Writer in one day write forty Pages, every one containing forty Combinations, 40 multiplied by 40, gives 1600, the Number he compleats in one day, which multiplied by 366, the Number (and more) of Days in a Year; a Writer in one Year shall compass 585600 distinct Rows’ (1685: 5). ‘Therefore in a thousand million of years’, Falconer concludes, ‘he could write 585600000000000, which being again multiplied by 1000000000, the number of Writers supposed, the Product will be 585600000000000000000000, which wants of the number of Combinations no less than 34848401733239439360000’ (1685: 5-6). This computation, of course, does not take into account that cipherers can work not only with letters and numbers but also with invented characters, ‘arithmetical figures’, and other symbols to express themselves.

Cryptography and Occult Mystery

[28] Many of the lessons in the manuals omit necessary instructions, end without resolution, or pair problems with the wrong solutions. G.B. explicitly admits he will ‘only hint that method’ (1665: 6). Kahn, a historian of cryptography, notes in a short section on this period that seventeenth-century manuals ‘stained cryptology so deeply with the dark hues of esoterism that some of them still persist, noticeably coloring the public image of cryptology’ and regrettably, because of seventeenth-century cryptography manuals, ‘People still think cryptanalysis mysterious’ (Kahn 1996: 93). Certainly, at times, the manuals seem esoteric even though they claim their appeal to a wide, novice readership. G.B. writes in the final lines of his first chapter, ‘The rest of the Discourse is in shorthand and Algebraical Characters, which I cannot make out’ (1665: 17). Such a dismissal prevents readers not educated in algebra to pursue the cipher any further, yet it also supports G.B.’s claim to be an unskilled cryptographer finding his way alongside the readers. From the abandoned ciphers to missing alphabets and typographical oversights, the manuals foreground their topics as mysterious even as they stress their accessibility. I have argued elsewhere that this self-reflexive awareness of and playfulness with reader expectations, and the relationship between the narrating novice voice and the reader as student, represents one nonliterary example of growing skepticism about idealized history and romance that would root the early novel (Ellison 2008). Here, I am more interested in how purposeful or accidental error promotes arithmetic as a domestic mode of reading by utilizing the language of mystery that is part of cryptography’s occult past. Eamon’s work helps to clarify the relationship between experimental science and the occult during the seventeenth century, when Royal Society members including Wilkins worked to establish credibility for experimentation and reposition experimental science as academic. John Henry explains how certain aspects of the occult became absorbed by the soon-to-be mainstream sciences of the seventeenth century. The result, he notes, is that the occult became fragmented. What would have once passed as magic in a previous age was renamed and repositioned: alchemy became chemistry; astrological principles grounded astronomy; incantation was legitimized as cryptography (Henry 2008: 1; see also Moran 2005).  Michael McKeon makes a similar but less specific claim when he recognizes that the increased publication of books promising the disclosure of secrets, some complete with recipes, could ‘be seen as a subtle reconception of the parallel between experimental technique and the natural phenomena it sought to know’ (2006: 65). The experiential techniques of occult sciences like alchemy were useful for Royal Society scientists, as was occult performativity. Anthony Grafton posits a similar argument but in reverse, arguing that the occult enveloped the practices and rhetoric of contemporary mathematics and engineering to bolster its credibility (2005: 17). Texts such as Wilkins’s own Mathematical Magick also demonstrate, for Grafton, a ‘technological brand of magic’ in which the new sciences did not replace the occult sciences but rather could justify their disciplines as legitimately useful as well as divinely inspired (2005: 18).

[29] I believe that cryptography manuals sacrifice accuracy, in part, because ciphers that work are not necessary for the genre to participate in the contemporary agenda to situate numbers and patterns, in all their functions, as useful as well as magical. Cryptography manuals invite a diverse readership to entertain itself with novelty, topicality, variety, secrecy, and the promise of an interactive experience, all conventions Barbara Benedict cites as important to the early anthology (1996: 5). If readers did not learn how to write or read ciphers or codes, they could content themselves with believing that they were part of a secret. That readers could not always make the ciphers work only supported the status of cryptography as a ‘marriage of magic and engineering’, in Grafton’s words (2005: 35).

[30] G.B. critiques the ways in which previous manuals are riddled with error and misleadingly complicated instructions. If readers missed those errors while enjoying earlier manuals like Wilkins’s, G.B.’s assessment no doubt opens their eyes to the inaccuracies of the genre. In the ‘Epistle to the Reader’, G.B. claims that when he was reading an earlier manual ‘my great pains to make some progress therein came to nothing’ (1665: 1). G.B. is correct: previous manuals exhibit so many problems with the codes and ciphers that the marginalia in some copies show reader work that stops abruptly in mid-solution. I argue elsewhere that these mistakes, misleading explanations, missing elements in the codes, and even misnumbered pages in Mercury, Cryptomenysis, and New Method cause readers to lose their way in the complicated pages and, in some cases, give up on learning the craft (Ellison 2011). I also find that the manuals challenge reader comfort with textual conventions; as the position of page numbers, the order of sections, the appearance of particular genres like dedications and paratext, and even margin size became more standardized with print, users of texts developed blind spots and assumptions that disallowed them from processing error as significant. For example, blots of ink were so common that ciphers could easily be hidden in their patterns, and instructions paired with the wrong examples could be written off as mere printing flaws. In his dedication to Charles, Earl of Middleton, Falconer admits that his manual has ‘defects’ but begs that readers not mistake those as a reflection upon his patron. Falconer then as much as admits that he has made significant errors, apologizing for using an entirely wrong alphabet for one cipher, though he does not specify which. G.B. promises that he has therefore corrected mistakes and distilled the confusing directions and will ‘lead thee by the hand from familiar Examples, and the beginnings of that Art (as I find them in his Papers) to the sublime curiosities thereof’ with ‘easie Examples (of which I lately thought my self as ignorant as any that are to learn)’ (1665: 2-3). But then, right away, G.B. breaks his promise. On the very next page, when he notes that the manual he is citing ‘has purposely disordered the Letters in their ranks to perplex the inquisitor’, he says he will ‘only hint that method and proceed’ to the next code that ‘is more ready and pleasant’ (G.B. 1665: 6). He does not disorder the letters – he just stops in the middle of his directions.

[31] Although missing information and error would seem to damage the credibility of the authors, it may instead strengthen the relationship the authors work to build with their readers. On the one hand, readers searching the manual for scandalous secrets and reading for ‘leasure studies’, as Wilkins expects, will not necessarily pause to try each example; the façade of technicality lends the text enough authenticity to make the intrigue interesting (1641: ‘To the Right honorable George, Lord Berkley’). On the other hand, readers who do discover the inaccuracies are further convinced of the difficulties surrounding secret writings. Readers of cryptography manuals expected instability in their texts; that was, indeed, one of the attractions of the subject. Cryptography also makes visible the many ways in which language can be easily manipulated and adapted to different audiences; that, too, is a defining characteristic of the craft. Integral to the effectiveness of error in keeping the attention of readers is the way in which it creates, rather than undermines, the authority of the author and, in the end, gives readers what they expect while still surprising them along the way. As we know from Sharpe’s convincing study of early modern reading, by the end of the seventeenth century authors no longer needed kings or other important political patrons to establish their authority (2000: 31). And as Jagodzinski finds, authority, when it is attainable through authorship alone, becomes a fragile construction, an authorial characteristic that must be earned through an established relationship with the reader rather than asserted as fact (Jagodzinski 1999: 10).


[32] The consensus thus far in historical studies of early modern cryptography and cryptology, specifically from the viewpoint of literary scholars, is that the use of ciphering to create and solidify social ingroups and outgroups, to position oneself in relation to political peers, was as much if not more important than the content of the messages and whether or not they could be solved (See Akkerman 2011 and Britland 2014). To focus too much on the political self-fashioning at stake in showing one could cipher, however, is to overlook the practical distinctions cryptographers like Wilkins, Bridges, G.B., Morland, and Falconer make when they favor one type of symbolic system over another. While Wilkins advocates for philosophical numbers, Bridges demonstrates the potential of transmutation. Morland presents a rigorous numerical alphabet and Falconer presents advancements such as columnar transposition at the same time that he offers more novice-friendly methods that entail object manipulation such as folding. These preferences are part of a larger campaign — not necessarily coordinated amongst them but building rhetorically over time — to rethink the notion of modern reading in multimodal communication systems.

[33] Where early modern cryptography manuals differ from arithmetic textbooks of the same decades is in their invitation to readers to participate in the intimate secrets of political intrigue while, at the same time, they allow or even orchestrate obstacles that prevent readers from fully learning those secrets. The impenetrability of seventeenth-century cryptography that Kahn cites is not an exaggeration, but it is arguably also not an accident. The manuals do not teach exactly what they proclaim to teach, but they have another lesson: arithmetic methodology is and should be experimental, creative, and interpretative, a continuous negotiation with a text whose author works in contract with a wide, diverse audience with varied expectations, skill levels, and commitment to the material. Ciphering, which might seem the most formulaic of all linguistic practices, becomes the stage upon which cryptographers can most effectively prove the complexity of literacy.

[34] Seventeenth-century cryptographers were both mathematicians and rhetoricians, both skilled masters of narrative and technological pioneers. Wilkins even claims matter-of-factly that his subject ‘doth also belong unto one of the liberall Arts’ (1641: 11). The stories they embed in their manuals have been so riveting, in fact, that nearly all twentieth- and twenty-first-century histories of cryptography, and even a large number of technical textbooks and course syllabi, repeat their stories verbatim, often without any idea who the original authors were. Recognizing contemporary repercussions of more diverse literacy and the move toward liberal reader interpretation, Wilkins, Bridges, Falconer, G.B, and their colleagues helped popularize mathematics and mathematic thinking, presenting communication by ‘philosophical numbers’ and ‘numeric alphabets’ as interpretive. They embraced the historical occult mystery and power of numbers yet stressed that even the most daring mathematical adventures could be learned from the safety of anyone’s desk.

Illinois State University


[1] References to all of the cryptography manuals except Rarities are from first editions viewed in person in the following collections. The relationship between Bridges’s Stenographie and Cryptographie and Rarities is worth noting. In 1659, Noah Bridges published Stenographie and Cryptographie, with a second edition in 1663. In 1665, Rarities appeared anonymously. Both are printed by J.G. The scan of the 1659 Stenographie scholars access on Early English Books Online is from the Bodleian Library and is bound with Rarities, so Rarities has been attributed to Bridges; however, Stenographie and Rarities are separately paginated, have their own signatures, and are on different paper stocks. A note on the back of Stenographie’s title page indicates the two works have been grafted together, presumably by a later binder or owner. Thank you to Katherine Hunt and Rebecca Tomlin for verifying the Bodleian copy. Copies of Stenographie in the Philip Mills Arnold Semeiology Collection as well as at the Rare Book and Manuscript Library at the University of Illinois at Urbana-Champaign also provide evidence that the two texts are separate publications.[back to text]

[2] Trithemius was a pseudonym for Johann Heidenberg.[back to text]

[3] Cryptography predates universal language schemes but is a personal inspiration for Wilkins’s later project. See Wollock 2011, Markley 1993, Cohen 1977, Cram and Maat 2001, Knowlson 1975, Lewis 2007, and Glidden 1987: 191.[back to text]

[4] Falconer does not indicate which ‘kingdoms’ he is referring to here when he uses ‘both’, but his examples imply he is referencing England and Scotland, not Ireland.[back to text]

[5] Some sources, like Mercury, spell Godwin’s text Nuntius Inanimatus, while others, such as William Poole in his recent 2006 article in The Seventeenth Century, spell it Nuncius Inanimatus. I use Wilkins’s spelling for this essay. See Poole 2006.[back to text]

[6]This statement would seem to provide evidence that G.B. is referring to Noah Bridges’s two editions of Stenographie and Cryptographie, yet it is still unclear why Bridges’s papers ended up in G.B.’s hands, if they are different people and G.B. is not an abbreviation for Bridges, or why, if they are the same person, Bridges publishes the first two editions under his own name but disguises his identity in the third.[back to text]


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Epilogue — Writing in the Aftermath: Digital Humanities, c. 1600?

Epilogue — Writing in the Aftermath: Digital Humanities, c. 1600?[1]

Carla Mazzio

The little matter of distinguishing one, two and three—in a word, number and calculation:  do not all the arts and sciences necessarily partake of them?

— Plato, The Republic

The bureaucratization of knowledge is above all an infinite excrescence of numbering.

— Alain Badiou, Number and Numbers

[1] ‘Without words, there is no possibility of reckoning of Numbers,’ wrote Thomas Hobbes in his Leviathan, ‘much less of Magnitudes, of Swiftnesse, of Force, and other things, the reckonings whereof are necessary to the being, or well-being of man-kind’ (Hobbes 1651:14). Such a seemingly simple insight into the relationship between verbal expression and numerical computation is worth remembering at this point in history, particularly given the accumulated rise of symbolic notation, computer coding, and algorithms for the navigation of big data in the 21st century — when numbers and mathematical symbolism seem to have taken on lives of their own. Before the divide between the ‘two cultures’ of science and the humanities, facilitated by the development of disciplinary and professional specialization as well as shifting cultural conceptions of knowledge, truth and meaning, numbers and words were understood to have a great deal more in common than they do today.  Indeed, whereas Hobbes insisted that ‘the use of words in registring our thoughts, is nothing so evident as in Numbring’ (14), the equation seems in some ways to have reversed. The use of various forms of numbering in registering our thoughts — even or perhaps especially as humanities scholars involved in various arts and practices of words — may well evince, consciously or unconsciously, the power of numbering at work in our words and world today. We need only allude to the power of quantitative assessment over practices of scholarly evaluation and production in the UK (and increasingly America) to suggest the extent to which numbers have come to matter in the lives of departments, institutions, and scholars focused, in various ways, on the arts and practices of language and mediation across the humanities.  The extent to which numbers have come to matter, that is, cannot be underestimated in the critical turn to understand numbers in material as well as historical terms.

[2] If this volume is any indication, not to mention the collective bibliography that it represents, ‘numbers’ have become newly charged as signs, symptoms, and forces to be reckoned with. Indeed, given the ‘numerical turn’ in early modern studies in recent years, we might wonder exactly where, when, how, and why numbers became so charged and, by the same token, exactly what numbers have been charged with.[2] The epigraph above from the contemporary philosopher Alain Badiou, claiming that ‘the bureaucratization of knowledge is above all an infinite excrescence of numbering,’ may well resonate with those now confronted with numerical models of assessment across the humanities and social sciences (Badiou 2008:2). But this is just part of a much larger and more philosophical charge that Badiou has made against the ‘despotism of number’ over all aspects of knowledge acquisition and production across the arts and sciences, not to mention the mental habits of everyday life (4). Number as ‘today’s fetish,’ he writes, conceals more serious problems at stake in accessing substantial modes of thought and being (Badiou 2007: 26). Thinking about numbers in relationship to forms of counting is for Badiou a form of not thinking at all. He thus aims to turn to a pre-modern ‘ontology of number’ to counteract epistemologies of number integral to methodologies as well as institutional and cultural forces across the humanities, social sciences and sciences; integral to the numerical organization of lives, minds, and souls in forms of quotidian life, and thus the reduction of the human to numbers that count or do not count. [3]

[3] For Badiou, as for other thinkers for whom numbers used for purposes of counting have become inseparable from large-scale social transformations or re-constellations, approaches to number can often lead to very large generalizations. There is something about numbers today that almost seems to call out for a return to large-scale thinking, be it a return to longue durée historiography (if with attention to issues of socio-economic inequality) recently advocated by historians Jo Guldi and David Armitage or the shift to “distance reading” through graphs, maps and trees, posited by literary scholar Franco Moretti (Guldi and Armitage 2014; Moretti 2007, 2013). Many readers in fact drawn to this Special Issue, Numbers in Early Modern Writing, may well come equipped with various large scale narratives about the power of numbers as newly important vehicles of social and epistemological transformation in the early modern period. For longstanding debates about the rise of capitalism, rationalism, empiricism, objectivity, probability, ‘modernity,’ or governmental, bio-political regimes all clearly involve questions that centre around the status and power of number, calculation, and the work of writing, or discourse more broadly, in the world.

[4] The specific call by Hunt and Tomlin to explore numbers in writing over the course of this volume, however, opens up space for a different set of questions, not least by unhinging ‘numbers’ from those large conceptual categories so often associated with them, and thus from the often now reflexive ethical imperative to wrest particular kinds of authority from them. Numbers — in this volume and in scholarship to be produced in its wake — can thus roam as freely in the historical imagination as they did in early modern writing. For then, as now, numbers emerged in every genre of writing imaginable; from almanac to atlas, bible to budget, chronicle to cookbook, or diary, epic and fable to grimoire. They emerged in paratexts as well as texts in fascinating ways, with the capacity to disconcert and confuse as well as to orient and clarify the order of things.[4] The fact that numbers emerged in fundamental ways in texts concerned with the arts of language (logic, rhetoric, and grammar) as well as in those concerned with arts of calculation (arithmetic, geometry, music, and astronomy) has encouraged recent scholars to think more seriously and more locally about rhetorical dimensions of number and quantitative aspects of language in the early modern period.[5]  This movement alone, and the essays in this volume that develop upon it, seem to me to offer creative and intellectually far-reaching responses to the oft assumed antinomies between the power of numbers and vulnerability of humanistic inquiry in the present day.

[5] To take seriously the generic and taxonomic lability of early modern ‘numbers in writing’ opens up the possibility of rethinking contemporary models of generalization about — or based upon — numbers in scholarship, culture, and society. The recent impulse to turn or return to the distinctly material dimensions of numbers (as entities in and of themselves or as vehicles for the shaping of material worlds), moreover, can help to ground a more capacious understanding of the relationship between numbers, writing and the production of knowledge in both the present and the past. For if numbers were promiscuous in affiliation with genres of writing, modes of expression, and categories of knowledge, they were no less mobile in relationship to representational forms and material surfaces. They were of course inscribed or printed as Roman numerals, Hindu-Arabic numerals, words, marks, or dots among other forms. They could be found on clock-faces, tally-sticks, coins, dice, calendars, armillary spheres, gravestones and numerous other objects peopling the landscape of early modernity. Accordingly, like other kinds of text, numbers and numerical thinking alike often relied upon material technologies ranging from abacus, pen, and navigable book to scientific instruments within and beyond the book and page. Although numbers as symbols of quantity can often tend toward further abstraction, attending to the local conditions and the material surfaces upon which they existed or were imagined to exist (from parchment and erasable paper to canvas, stone, bone, wood, linoleum, metal, wax and sand) can help to draw attention to the inter-animation of media, materiality and modes of cognition across a range of disciplines.

[6] In the context of early modern incarnations of number, it is important to simply note that numerical modes of thinking often crossed the material/immaterial boundary in ways that energize many essays in this volume. Indeed, it is striking to note the extent to which numerical manipulation or mental calculation in this volume is understood to be embodied, be it through the logic of textbook arithmetic, the work of numbers in the art of fencing, or the sexual economies of zeros and ones in Shakespeare’s poetry. The attention to physical as well as semiotic dimensions of numbers across these essays attests, in part, to the power of material studies to bring even those seemingly abstract symbols of quantity down to earth, to the surfaces of pages, the tips of fingers, and into the play of minds and worlds compatible with recent approaches to embodied thought and extended cognition. To consider the history of numbers and writing in the present day is thus also to continue to complicate an ongoing history of oppositions between material and immaterial, concrete and abstract, mental and physical worlds through which numbers have come to mean.[6]

[7] Since the general subject of ‘numbers’ in the present day has the capacity to raise questions if not eyebrows about the making of credible or objective knowledge, the turn to exploring numbers in and as writing opens up timely questions about particular rhetorical ‘figures,’ as it were, that were and may continue to be integral to the making of fiction and the making of fact.[7] How, for example, might numbers have operated as vehicles of social as well as aesthetic power by conjuring illusions of material reality from a distance? Claims about the size of an army, the scale of one’s property, about a particular number of sheep or ducats can all serve as a means of ‘conjuring’ persons and things. The ways in which numbers can seem to make persons and things or actions materialize — a point emphasized by Stephen Deng in his exploration of numbers as part of the imaginative writer’s tool-kit as well as by James Beaver in his analysis of John Donne — offers an important counterbalance to distinctions often drawn between abstract number and material things, not to mention between the arts of number and the arts of language.

[8] While numbers could be mobilized, as Christopher Johnson has put it, ‘to give flesh to abstract ideas,’ the opposite was also often the case (2004: 74). The use of physical bodies to illustrate the logic of both numbers and letters in early mathematical textbooks is one case in point. Of course the practical orientation of many early vernacular mathematics led to necessary visual or rhetorical insertions of the practitioner or practitioners into the scene of calculation. But at the same time, the strategic verbal and visual representation of bodies, situations, and events in early mathematical texts of all kinds could conjure whole worlds of numerical engagement oriented toward individual and collective action.

Figure 1. From Simon Stevin’s De Weehgdaet, Wisconstige Gedachtenissen (Leiden, 1608), Reproduced by Permission of the Huntington Library.

Figure 1. From Simon Stevin’s De Weehgdaet, Wisconstige Gedachtenissen (Leiden, 1608), Reproduced by Permission of the Huntington Library.

[9] The illustration from the Flemish Mathematician Simon Stevin’s De Weehgdaet (a text on weighing first published in 1586 and reprinted in his 1608 ‘Mathematical Memoirs’) is but a single example of the visually appealing and pedagogically useful depictions of bodies in action in sixteenth and early seventeenth-centuries vernacular texts [Figure 1]. Such forms of action and practice were not always, of course, textual—nor distinct from innovations in higher mathematics, as the case of Galileo, who was trained in engineering and military science, might suggest. But it remains notable that ‘the Everyman of early English practical mathematical literature’, as Kathryn James has recently put it, ‘is resolutely present on the battlefields, on ships, tramping the bounds of estates in the company of not always trusting or trustworthy tenants’ (2011: 6, italics added).[8] Whereas the issue of rank and status implicit in James’s description of bodies in action in English mathematical texts opens onto a well-known field of debate about relationships between artisans and aristocrats, for Ken Mondschein in this volume, it is the already idealized, elite body-in-motion of the fencer in sixteenth-century Italy that provided a ground for the placement (and thus transmission) of all manner of mathematical abstractions.

[10] Part of the larger question at stake here is how numbers may have been ‘conjured’ or made to matter as objects of social and epistemological value through various dimensions of writing and print. What kinds of authority, in other words, could numbers convey or disrupt? If numbers now have the power to convey authority in the establishment of fact or scale, numbers in early modernity in many ways depended upon a range of authorizing functions in order to matter. The close attention to the history of print and publication in considering the status of numbers on pages in essays by Tomlin and Hunt seems to me to further expand upon issues of authorship and authority. Tomlin’s demonstration of James Peele’s attempt to construct himself as an ‘author’ and to construct accountancy as part of a distinctly humanist publishing agenda by the King’s printer indexes the lability of early modern conditions of authority and authorship, not simply for readers but for writers and transmitters of technical knowledge. Hunt’s essay complicates authority of another kind: the authority of Michel Foucault in approaching — or generalizing about — tables in early modern books. Hunt not only exposes the immense amount of labour and typographic specialization that went into the printing of those ‘masses of numbers’ in numerical tables, but shows us how numerical units of information become detached from the mental processes that created them, in fact functioning as a kind of ‘cheat-sheet’ for those who wanted to avoid the often difficult labour of calculation. For Hunt, we need not all follow in the wake of Foucault and others who turn to the formal properties of textual or graphic culture (in this case the table) as indices of larger cultural models of thought. Some tables, like contemporary calculators, exist to eliminate rather than shape the process of thought.

[11] Numbers in Early Modern Writing thus seems to me to offer a kind of collective counter-response to some contemporary assumptions and large generalizations about numbers at work in relationship to both history and writing. In addition to the essays alluded to above, in those focused largely on numbers at play in poetic and dramatic literature, we see how productively imaginative fictions could draw upon the operational as well as symbolic and linguistic dimensions of arithmetic. We see this in the fascinating interplay of zeros and ones in relationship to issues of reproductive futurity and vulnerable masculinities in Shakespeare’s sonnets (Deng); in the quantitative dimensions of John Donne’s verse (Beaver); and in the quantitatively cast hyperbole in the revenge tradition, itself offering a model of exponential increase rather than an eye-for-eye equivalence of promised violence in relationship to an original crime (Dunne). So too, we have seen how rhetorical and logical modes can be understood to inform the composition of arithmetical knowledge, from the idea that the rhetorical category of dispositio (or arrangement) may be understood as an ordering principle of numbers in arithmetic to the attention that Robert Record pays to justifying numbers as integral to the art of speech itself (Wilde). And finally, we have seen how numbers could operate to embed messages in the form of codes that linked number-work with the very rhetoric of the ‘secret’ once linked with the occult (Ellison).[9] Throughout the essays, then, we see how deeply entwined the arts of rhetoric, grammar and logic were with worlds of numerical cognition in ways that nicely complicate, if often implicitly, large scale generalizations about numbers as vehicles of changing epistemes, mentalities, or epochs across time and space. At the same time, the fascination with the hyperbolic dimensions of number in several essays in this volume might be considered in light of the peculiar affinity between ‘number’ and overstatement, or between numbers and abstractions that can seem to unite otherwise incommensurable things, ideas, practices, and discourses. As John Dee (citing Giovanni Pico della Mirandola) put it in 1570, ‘By Numbers, a way is had, to the searchyng out, and vnderstandyng of euery thyng, hable to be knowen’ (sig j r-v).

[12] The slippery symbolic economy of ‘number’ and ‘numbers’ in the making of generalizations across distinct fields of thought, practice and locution has a powerful socio-economic correlative today. In Ted Porter’s formulation, ‘in intellectual exchange, as in properly economic transactions, numbers are the medium through which dissimilar desires, needs, and expectations are somehow made commensurable’ (Porter 1995: 86). The potential of ‘number’ to eliminate variability and complexity in language as well as in other conditions of material and mental life is one way to consider relationships between numbers and writing, not simply in the historical terms of the drive to create a universal language in the wake of the development and power of vernacular languages in the later seventeenth century but in terms of our own practices of thought, association, word choice and generalization as scholars interested in numbers and writing in both past and present. For what do we make of the powerful attention granted to the function, history and power of numbers represented by the largely early career scholars in this volume, particularly given the palpable presence of numbers at work across the humanities, involving the economics of scarcity, scale, and shifting cultural values? The answers to such a loaded question of course depend upon one’s approach to those numbers in writing. For me, the rise of the rhetoric of (as well as the various transformations constellated under) the ‘digital humanities’ may well be seen as a kind of unspoken dimension informing the work in this volume. Indeed, a close reader might note a trend across the essays that in fact departs from what Franco Moretti has termed ‘distance reading’ enabled by the massive digitization of texts that can now be sorted and assembled in ways that exceed the human capacity for textual comprehension. In contrast to the scale of textual meta-analysis enabled by the digital humanities, the essays in this volume arrive at generalizations through the careful examination of the numerical world of a single author or printer, within a single text, or across a delimited number of plays within a specific genre — all of which depend upon acute powers of close reading. This volume, that is, welcomes readers into a capacious understanding of ‘numbers’ but also into various forms of ‘large scale’ inquiry enabled by microscopic analysis in which details of history, texts and cultures come to matter as the texture of context becomes enlarged, more granulated. It is worth noting, in other words, the extent to which the essays within in effect counter a new economy of scale offered by digital manipulations that enable the kinds of ‘distance reading’ through graphs, maps and trees, aligning information in new ways to arrive at conclusions otherwise difficult to detect.

[13] Before making a further observation about the contribution constituted by this volume, here it is worth pointing out the problematic language of value so often aligned with vocabularies of measurement integral to scholarship: ‘big data’ can render other bases of analysis ‘small’, subjective or random. The ‘micro-history’ that once countered ‘macro-historical’, large-scale narratives can easily be rhetorically undermined as offering a form of knowledge that is limited in scale and thus in impact, as we have recently seen in the controversial The History Manifesto (Guldi and Armitage, 2014). Even the ‘close-reading’ that remains a staple of literary pedagogy and a continuing and essential ingredient of powerful writing and scholarship across the humanities can easily seem to imply, particularly in contradistinction to ‘distance-reading’, a lack of perspective or objectivity enabled by critical distance. Without buying into a logic in which size matters in the scope of analysis, argument, subject matter explored, it is worth remembering that ‘micro’ analysis and ‘close reading’ both involve processes of enlargement, magnification, a careful process of finding otherwise unobserved or underexplored networks of meaning within seemingly ‘small’ units of representation, be it in the form of image, text, sound or any other media form. Whether we invoke metaphors of microscopic, macroscopic or telescopic analysis, of close or of distance reading, we need to be fully aware of assumptions about the value of forms of thought and inquiry as delimited by models of measure, scope, scale, impact, or other forms of quantitative assessment.

[14] The word ‘digital’ has of course carried a great deal of rhetorical and symbolic capital in recent years, particularly when paired with ‘humanities’, for the ‘addition of the high-powered adjective to the long-suffering noun’ can suggest to many, as Adam Kirsh has put it, ‘nothing less than an epoch in human history’ (2014). Generalizations about ‘digital’ technology no less than numbers themselves as constituent elements of humanistic practices, institutions and systems of organization are often hyperbolic, an overstatement of a case that we are just beginning to understand. What is so refreshing about this volume is that it represents a new generation of scholars working within historical and literary contexts to come to grips, in various ways, not with a ‘digital condition’ of early modernity or of the present moment, but with varied and often surprising conditions of digits, by which I index both senses of the term ‘digits’, alluding to numbers and fingers, symbols of quantity that might also lead us back to our own writing, reading, and working hands. Putting digits in the hands of new generations of students and scholars as objects and subjects of humanistic inquiry seems to me central to the necessary process of arriving at new generalizations and forms of understanding that confront — rather than enact — fears, fantasies, or displaced anxieties about agency encoded in a turn to numbers, to digits, as instruments of thought, perception, and social organization. If this volume represents a new understanding of the ‘digital humanities’, c.1600, this is but the beginning of a movement through which we may reckon with the place and power of numbers as they come to matter in the aftermath, as it were, of a powerful cultural awareness of numbers as central to our understanding of just what counts and what does not.


[1] I would like to thank the editors of this volume, and also Heidi Brayman Hackel, Dympna Callaghan and Heather James for helpful comments on this afterword.[back to text]

[2] See especially Glimp and Warren(2004), xv-xxix.[back to text]

[3] The vivid image of numerical excrescence that conjures a deforming social and epistemological body with no end in sight indexes, if in small, how Badiou at once reveals and relies upon the power of numbers to infiltrate whole systems of thought. Although Badiou postulates a pre-modern ‘ontology of number’ as one way to escape or counter the tyranny of numbers as vehicles of counting, he seems to at times to be caught within the very prison-house of number that he at once describes and constitutes.[back to text]

[4] For an early seventeenth-century dialogue concerning the common confusion among students ‘almost ready to go to university’ about basic numbers within books (in both letters and numbers, Roman Numerals and Hindu-Arabic ‘figures’) see Brinsley (1613) 25-26.[back to text]

[5]  On the relationship between the arts of the trivium and arts of the quadrivium in early modern France and Italy, see especially Reiss (1997); on rhetorical categories of invention and disposition in relationship to algebra, see Cifoletti (2004), 132-153; on Humanist and aristocratic readers of early vernacular mathematical texts in England, see James (2011), 1-16, and for a seminal approach to the power of zero across a range of early modern forms of representation, see Rotman (1987). For a volume that deals with the ‘imbricated histories of numbers and letters’ in medieval and early modern culture, see Glimp and Warren (2004), xvi. Glimp and Warren, in turn, call attention to earlier scholarship on this ‘mutual imbrication’ that will be of particular interest to readers of this volume, by Clanchy (1979), in which we find ‘technologies of writing’ deployed by institutional administrators and poets alike and by Jed (1979), who examines ‘graphic habits’ shared by Florentine humanist and mercantile cultures.[back to text]

[6] For a range of approaches to inscription, materiality and mathematic and scientific texts, see Lenoir (1998) and Chemla (2004). As Lenoir puts it at the outset of ‘Inscription Practices and the Materiality of Communication’ in Inscribing Science, Metaphors of inscription and writing figure prominently in all levels of discourse in and about science. The description of nature as a book written in the language of mathematics has been a common trope since at least the time of Galileo, a metaphor supplemented in our own day by the characterization of DNA sequences as a code for the book of life, decipherable in terms of protein semantic units. An important recent direction in the fields of science and literature studies is to consider such descriptions as more than metaphoric, as revelatory of the process of the process of signification in science more generally’ (1).[back to text]

[7] On numbers and the emergence of the concept of the ‘fact’ from early modern English accounting practices and discourses, see Poovey (1998) and also Porter (1995).[back to text]

[8]On the promotion of vernacular mathematics through print technology and appeals to various models of practice, see Johnson (1991).[back to text]

[9] In contrast to mathematical textbooks of the seventeenth century that so often disavowed affiliations with the occult through the stress on ‘easy’, plain and practical instruction (Neal, 1999), books on cyphering, as Ellison observes, capitalized on the history of the secret within the tradition of the occult.[back to text]


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_____. 2008. Number and Numbers, trans. Robin Mackay (Cambridge: Polity Press)

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Chemla, Karine ed. 2004. History of Science, History of Text (Dordrecht: Springer)

Cifoletti, Giovanna C. 2004. ‘The Algebraic Art of Discourse: Algebraic Dispositio, Invention and Imitation in Sixteenth-Century France’, in History of Science, History of Text, ed. Karine Chemla (Dordrecht: Springer): 132-153

Clanchy, T. M. 1979. From Memory to Written Record: England 1066-1307 (Cambridge: Cambridge University Press)

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Hobbes, Thomas. 1651. Leviathan, or, The matter, forme, and power of a common wealth, ecclesiasticall and civil (London: Printed for Andrew Crooke)

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Johnson, Christopher. 2004. ‘Clavius’s Number and its Afterlife in Donne, Wilkins, and Kircher’, in Arts of Calculation: Quantifying Thought in Early Modern Europe, eds. David Glimp and Michelle R. Warren (New York: St. Martins/Palgrave MacMillan), pp. 67-91

Johnson, Stephen. 1991. ‘Mathematical Practitioners and Instruments in Elizabethan England’, Annals of Science, 48, 331-344

Kirsch, Adam. 2014. ‘The False Promise of the Digital Humanities’, The New Republic (May 2, 2014): http://www.newrepublic.com/article/117428/limits-digital-humanities-adam-kirsch (accessed January 15, 2015)

Lenoir, Timothy, ed. 1998. Inscribing Science: Scientific Texts and the Materiality of Communication, (Stanford: Stanford University Press)

Mazzio, Carla. 2004. ‘The Three-Dimensional Self: Geometry, Melancholy, Drama, in Arts of Calculation: Quantifying Thought in Early Modern Europe, eds. David Glimp and Michelle R. Warren (New York: St. Martins/Palgrave MacMillan), pp. 39-66

Moretti, Franco. 2007. Graphs, Maps and Trees: Abstract Models for Literary History (London: Verso)

Moretti, Franco. 2013. Distance Reading (London: Verso)

Neal, K. 1999. ‘The Rhetoric of Utility: Avoiding Occult Associations for Mathematics through Profitability and Pleasure’, History of Science, 37, 151–178

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Reiss, Timothy. 1997. Knowledge, Discovery and Imagination in Early Modern Europe: The Rise of Aesthetic Rationalism (Cambridge: Cambridge University Press)

Rotman, Brian. 1987. Signifying Nothing: The Semiotics of Zero (Stanford: Stanford University Press)