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.

[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)

[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.

[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.

Notes