## Chapter 6

# CONCLUSION

‘Every matter must have an end, and therefore to draw this to a conclusion.’

(Thomas Digges, 1584/5)

Throughout this thesis I have followed a small group of mathematical practitioners across a diverse geographical, social and disciplinary terrain. Dover harbour has provided an anchor for travels from Deptford dockyard to the Netherlands, from the printing houses and instrument shops of London to the trading outposts of the White Sea and the Baltic. Socially, the journeys have gone from the elevated chambers of court and Privy Council to the humble workmen and cart drivers who laboured to construct the earthwork pent at Dover harbour. Between these extremes the identity of the mathematical practitioner was fashioned as a reliable technical expert who could design and direct work, yet also authoritatively bring the phenomena of nature to rule and measure.

From astronomy to magnetism, and from ships to surveying, a culture of [page 288:] mathematical practice was created which cuts across modern boundaries between science, technology and mathematics. By following the careers of the Dover practitioners, the topography and constitution of this unfamiliar territory has been uncovered. Yet inevitably, though wide-ranging in place and activity, the journey has been highly localised in time. While tracking the careers of these practitioners and reconstructing the narrative of Dover harbour, my primary focus has been on work carried out in the 1570s and 1580s. In terms of a practitioner’s career this is a lengthy span of years; in terms of the overall development of the tradition of mathematical practice, it is only a snapshot image.

In this conclusion, I want to explore the larger significance of this local development of mathematical practice. Looking backwards several decades as well as ahead into the 17th century it becomes clear that the particular years considered in this study witnessed the establishment of a new community based around the identity of the mathematical practitioner. Moreover, expanding the temporal frame of reference reveals that English mathematical culture underwent an extraordinarily rapid development in the 16th century: literally and figuratively on the margins of Europe in the early century, the newly established tradition of mathematical practice represented a dynamic and innovative presence at its close. To slip briefly into the modern language of economic development, England caught up quickly from initial underdevelopment and, in a number of strategic areas such as navigation, magnetism and shipbuilding, was already overtaking European rivals.

Such European comparisons are not inappropriate. Though they were developing a primarily vernacular tradition, the mathematical practitioners were not insular. Most had travelled widely and had worked with ‘strangers’: their horizons [page 289:] were not restricted to only English materials. Digges, for example, had a European range of reference in astronomy and artillery, and was even able to draw on Italian authors for detailed harbour questions.1 Baker recorded important information on Venetian and Greek shipbuilding in *Fragments* and evidently learnt techniques from at least one Venetian shipwright resident in England. Borough made critical use of the maps and atlases of Ortelius and Mercator in his cartographic studies, and was able to deploy the resources of Latinate astronomy when writing on the variation of the compass. Bedwell used Tartaglia’s publications on artillery as a primary point of reference for his own ballistic investigations.

One important reason for the rapid formation of English mathematical practice was thus the possibility of assimilating and exploiting prior developments initiated elsewhere. But I also suggest that the speed of the process and its diversification were due in part to its initially minimal base line. Lacking long-established technical traditions and institutions, such as might be found among Italian engineers or teachers of the *abbacco*, the community of English mathematical practitioners began as a limited and a fragile creation. But the very fact of its small size maximised opportunities for contact and exchange.

The extent of personal interconnection and the sharing of resources is evident from even a brief review. At the intersection of all the careers that I have discussed, Dover was the site of numerous encounters, where common skills were fostered through both cooperation and the sharpening of competing proposals. Beyond the [page 290:] harbour works there were other topics of common interest, such as artillery and ballistics, on which both Thomas Digges and Thomas Bedwell worked. Shipbuilding was another point of contact. Not only was it the basis of Mathew Baker’s livelihood but it was tackled as a mathematical art by Digges and William Borough. Moreover, both Digges and Borough employed the apparatus of spherical trigonometry for astronomical, navigational and magnetic researches. Common cause was made too on an instrumental level, for example in surveying, when Borough recommended the topographical instrument devised by Leonard Digges and published by Thomas.2

In addition to this evidence of overlapping interests, there were also various borrowings between different arts. Instruments were frequently at the centre of such transfers: Baker, for instance, adapted the astronomical device of precise diagonal scales (as described and advocated by Digges) to the requirements of his paper design practice; Thomas Bedwell’s carpenter’s rule was the model for his own later artillery rule.

As well as the central figures of Digges, Baker, Borough and Bedwell, the lesser-known names in my study also provide similar evidence of commonality. Paul Ive wrote on fortification, which Digges promised to deal with, and he also translated a text on military discipline, a topic on which Digges did indeed publish. John Symonds owned and used surveying instruments, while his sophisticated deployment of flaps on architectural drawings matched the ingenious plats produced during the design of Dover harbour. Though trained as a mason, Robert Stickells expanded his claims to expertise to include questions of shipping in his portfolio of interests. Even John Hill can be mustered here, for Edward Worsop bracketed him along with Digges [page 291:] in his list of expert surveyors.

Instances of such reciprocal relations and collective interests could readily be multiplied. But the *creation* rather than the mere existence of a community of mathematical practitioners needs to be established by comparison with an earlier period. While the first vernacular texts on practical mathematics were published in the middle third of the century, I suggest that the culture of mathematical practice was principally an achievement of the later 16th century. Comparing the generation of my Dover practitioners with that represented by their fathers neatly encapsulates both the continuities and contrasts over a period of some 30 or 40 years.

There is a striking connection between the careers of various Dover practitioners and those of their fathers, especially between the mathematical authors Thomas and Leonard Digges, between the master shipwrights Mathew and James Baker, and between the navigators William Borough and John Aborough. Dover itself fits the same pattern since the harbour was redeveloped not just during Elizabeth’s reign but earlier too, by command of Henry VIII. Indeed, as part of the earlier project, John Aborough even served as co-author of an ambitious construction scheme.

Yet the continuity implied by these father-son relationships is undoubtedly more remarkable to 20th-century eyes than it would have been to those of the 16th. Occupational continuity was not just expected but indeed deliberately fostered, as an instrument of social stability and order.3 More significant in 16th-century terms are the differences between the experiences of the two generations. The chief distinction is the evidence of community: by comparison with 30 years later, it is extremely [page 292:] difficult to document contact between individuals during the 1540s and ’50s, and to uncover the productive transfer and adaptation of techniques.

Yet the creation of a community depends on more than just the piling up of detailed transactions and cross-connections. It requires a shared set of values, a common rhetoric, a joint agenda. Here again there is a useful comparison to be made between the generations of fathers and sons. While Leonard Digges (and, indeed, Robert Recorde) envisaged ‘the mathematicalls’ as an aggregate of arts and sciences with a common basis in arithmetic and geometry, the programme implied by these claims was only implemented in later decades. Whereas contemporaries of Leonard such as James Baker and John Aborough pursued the arts of shipbuilding and navigation within the distinct traditions of their trades, their sons developed and transformed these arts within the programme of mathematical practice. The ideology of the mathematical arts proclaimed the unity of subjects and practices which had previously been the province of specialists beholden only to their craft peers. Mathematical practice opened up arts to a more public sphere and re-presented them through the medium of print or, for matters of strategic, military or economic importance, within the more private chambers of the Privy Council and even sometimes of the Queen. In my final remarks, I will return to the long-term significance of mathematical practice for the construction of a public sphere of knowledge and technique. But here I want to broaden out beyond just the community of mathematical practitioners and look at their success in persuading contemporaries of the value of mathematical practice.

The expanding range of available books on the mathematical arts and sciences, and the establishment of the trade of the mathematical instrument maker, [page 293:] signal commercial success. In addition to this evidence from the operations and anonymous decisions of the market place, more readily attributable endorsements were furnished by authors outside the small circle of practitioners. For example, mathematics began to be featured approvingly in the gentlemanly courtesy literature, where it was prized not for abstruse demonstrations but for its pleasant conclusions and military utility.4 Its place as an appropriate element of education even for the youths of nobility was displayed most prominently in Sir Humphrey Gilbert’s proposal for ‘Queen Elizabethes Achademy’. Although unrealised, Gilbert’s scheme contained remarkably elaborate provision for the teaching of the mathematical arts as part of a full cycle of education. Gilbert’s emphasis was self-consciously on practice rather than mere ‘bookish circumstances’, and he embraced the programme of the mathematical arts through his emphasis on artillery, fortification, navigation, cartography and even ‘the perfect art of a shipwright, and diversity of all sort of moulds appertaining to the same’. Mathematics was here announced as a vital and virtuous element in the upbringing of those who would direct the affairs of the commonwealth.5

The mathematical practitioners also had direct success in engaging the interests of those patrons and high statesmen responsible for government. While planning for Dover harbour, deliberating with the Navy Board, working on fortifications or serving in the Netherlands, practitioners imported paper-based [page 294:] mathematical skills into design and logistical planning. In doing so, they were using the medium and the language of administration familiar to statesmen such as Lord Burghley, Sir Francis Walsingham and the Earl of Leicester. Integrating their work with the administrative procedures of state helped to secure patronage and official positions (such as those obtained by Thomas Bedwell in the Ordnance Office and William Borough in the Navy). As trusted clients, the practitioners were also assured a status distinct from that of ordinary craftsmen and labourers whose work they might oversee. Their paper skills gave them recognition and authority in such crucial settings as the chambers of council, where the mere word of a common artisan could carry little weight.

The success of mathematical practitioners as servants of the state bolstered the identification of mathematics as a worldly activity. Nor was this identification a passing idiosyncrasy of the later 16th century: mathematical practice supplied the dominant public image of mathematics in England throughout much of the 17th century and its prominence is attested by the continued presence of mathematical practitioners within the Ordnance Office and by the prestige and honours accruing to successful practitioners such as Sir Jonas Moore.6 The success of mathematical practice stands on its head the familiar modern characterisation of mathematics as abstract and otherworldly.

However, the temptation to fall back on an opposition between theory and practice, in which mathematical practice falls entirely under the latter heading, should [page 295:] be resisted. For 16th-century England, the appropriate distinction is between mathematics and philosophy rather than theory and practice. And, though scarcely articulated at the time, mathematical work on topics such as astronomy, magnetism and artillery would ultimately have important philosophical implications.

As the most developed of contemporary mathematical sciences, astronomy was frequently taken as the exemplar for other investigations. However, this was not astronomy construed as a discipline able only to ‘save the phenomena’, without regard to the physical adequacy of its hypotheses. Rather it was a mathematical astronomy interpreted in realist mode, as Thomas Digges advocated for the Copernican system in 1576. Astronomy and the mathematical sciences modelled on it were to deliver truth through geometrical models and numerically accurate prediction within a mathematically and instrumentally delimited jurisdiction. The results of such mathematical investigation were thus partial: in tackling natural phenomena, Thomas Digges and William Borough did not talk about the materiality of the heavens or the nature of magnetic attraction. Nor did Digges discourse extensively on gravity in discussing ballistics. Such questions, which fell within the domain of contemporary natural philosophy, were simply not aired by Digges or Borough.

This philosophical silence was perhaps less the outcome of a pragmatic compromise between mathematical practitioners and philosophers than the consequence of lack of exposure to the canons of contemporary natural philosophy.7 Familiarity with the standards and procedures of philosophy was most likely to be acquired at university. But of the Dover practitioners, only one can be securely identified as a [page 296:] graduate. It therefore seems significant that it was only this figure - Thomas Bedwell - who self-consciously took up philosophical issues. After giving an account of his artillery rule, Bedwell entered into explicitly philosophical discourse, taking his bearings in the discussion of ballistics and the movement of bodies not just from the mathematician Tartaglia but also from the notorious controversy between Cardano and Scaliger. Indeed, Bedwell also referred to another of his treatises then in a state of ‘some forwardness’, which he hoped would ‘contain a resolution by the causes of some hundreds of the most difficult questions in philosophy’.8

But less philosophically tutored mathematical practitioners than Bedwell rode roughshod over philosophical prohibitions on the use of artificial (instrumental) means to investigate natural phenomena. In doing so, these mathematical practitioners contributed to the long-term reconstruction of the objectives and procedures of natural philosophy characteristic of the 17th century.9

But conclusions about the 17th century cannot be plausibly inferred only from the experience of the 1570s and ’80s, especially in a tradition that was undergoing rapid development. To understand the explicit confrontation with traditional natural philosophy, it would also be necessary to study succeeding generations of mathematicians. Intriguingly, leading figures of the early 17th century, such as Edward Wright, Thomas Harriot, Henry Briggs, Edmund Gunter and William Oughtred, were all academically trained and yet, after leaving the universities, clearly [page 297:] aligned themselves with mathematical practice. In a period when scholastic philosophy was renewed in England and a return was made to the European mainstream of Latinate scholarship, these more academically learned practitioners had the intellectual background to mount explicit challenges to philosophical doctrine. But the investigation of the specific form of mathematical practice in that generation belongs to another dissertation.10

Yet there is no need to retreat entirely into agnosticism for lack of detailed studies on the period immediately after the 1580s. Excessively grandiose claims relating the Dover practitioners to traditional issues of the Scientific Revolution may be out of place here. But there are other aspects to mathematical practice which stand out for their long-term significance.

The 16th-century culture of mathematical practice was public, commercial and metropolitan. With its lectures and state service, its technological ambitions and rhetoric of improved instrumentation, the mathematical practitioners created a culture whose elements were continued into the 19th century. Moreover, though each was increasingly defined as distinct from mathematical practice, both the experimental philosophy and the technological practice of the 17th and 18th centuries were embodied in the same strongly public and commercial form.11 In the making of mathematical practice, the Dover practitioners were not just sharing skills, ideas and instruments while fashioning a new identity for themselves; they were creating a cultural form whose programme was actively pursued for more than two centuries.

## FOOTNOTES

1. On artillery, note Digges’s copy of Luys Collado’s *Practica Manual de Arteglia* (Venice, 1586), BL shelfmark C.54.k.2. For harbour construction, note his reference to Italian *palificata travata* work in the ‘Discourse’ to the Queen and in SP12/153/51, as well as the reference to Castriotto for flat-bottomed Venetian boats (SP12/152/27). (References in this chapter are only given for material not previously cited.)

2. *Discours of the Variation of the Cumpas* (London, 1581), preface.

3. Joan Simon, *Education and Society in Tudor England* (Cambridge, 1966/79), p. 294.

4. For an example, see the anonymous treatise *Cyuile and Uncyuile Life* (1579), republished in 1586 as *The English Courtier and the Cuntrey Gentleman*, and available in W.C. Hazlitt (ed.), *Inedited Tracts* (London, 1868), esp. pp. 69, 71-2, 84. For the courtesy literature, Steven Shapin, ‘“A scholar and a gentleman”: the problematic identity of the scientific practitioner in early modern England’, *History of Science* 29 (1991), 279-327, with further references.

5. Humphrey Gilbert, ‘The erection of an achademy in London for educacion of her Maiestes wardes, and others the youth of nobility and gentlemen’, in F.J. Furnivall (ed.), *Queen Elizabethes Achademy*, Early English Text Society, extra series 8 (London, 1869), 1-12, esp. pp. 4-5. Gilbert’s text is undated but was probably composed in the 1570s.

6. Frances Willmoth, ‘Mathematical sciences and military technology: the Ordnance Office in the reign of Charles II’, in J.V. Field and Frank A.J.L. James (eds), *Renaissance and Revolution. Humanists, Scholars, Craftsmen and Natural Philosophers in Early Modern Europe* (Cambridge, 1993), 117-131 and eadem, *Sir Jonas Moore. Practical Mathematics and Restoration Science* (Woodbridge, 1993).

7. For the pragmatic compromise, N. Jardine, *The Birth of History and Philosophy of Science* (Cambridge, 1984), p. 239ff.

8. The title page of Bedwell’s treatise on his artillery rule mentioned its ‘appendix of certain questions touching great artillery etc, philosophically examined’ (Bodl. MS Laud 618). The reference to Bedwell’s promised philosophical treatise is on the final page of the same text. For Cardano and Scaliger, see Ian Maclean, ‘The interpretation of natural signs: Cardano’s *De subtilitate* versus Scaliger’s *Exercitationes*’, in Brian Vickers (ed.), *Occult and Scientific Mentalities in the Renaissance* (Cambridge, 1984), 231-52.

9. For the case of magnetism, J.A. Bennett, ‘The mechanics’ philosophy and the mechanical philosophy’, *History of Science*, 24 (1986), 1-28.

10. On the English return to mainstream European scholarship, C.B. Schmitt, *John Case and Aristotelianism in Renaissance England* (Kingston, 1983), p. 27 and J.W. Binns, *Intellectual Culture in Elizabethan and Jacobean England: the Latin Writings of the Age* (Leeds, 1990).

11. Cf. Larry Stewart, *The Rise of Public Science. Rhetoric, Technology and Natural Philosophy in Newtonian Britain, 1660-1750* (Cambridge, 1992).