‘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)
 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?
 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).
 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).
 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).
 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.
 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.
 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.
 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.
 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:
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).
 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
by ‘writ[ing] the 4 [of 9*6=54] under the 3, and the 5 under the next place (as reason wylleth me) thus’:
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]).
 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.
 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)
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)
 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.
 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).
 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.
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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.
 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)
 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.
 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.
 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).
 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.
 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]
 My thanks, here, to the anonymous reviewer who brought these errata to my attention.[back to text]
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