## Chapter 1

# THE TERRAIN OF MATHEMATICS

A new form of mathematics was established in later 16th century England: the culture of mathematical practice. This new mathematical culture was articulated as a vernacular tradition and promoted through the medium of the printed book. It heavily emphasised the importance of instruments for observation, measurement and calculation, and the value of the maps, charts and plans which instruments were used to produce. The rhetoric accompanying this material culture of books, instruments and ‘plats’ stressed that mathematics was a practical and worldly activity which brought public advantage and private pleasure. In Elizabethan England, mathematical practice became a distinctive cultural form and those who practised it worked out a distinctive identity as mathematical practitioners.

This thesis studies the work and careers of a small number of these mathematical practitioners. It focuses on a period - principally the 1570s and ’80s - when the role of the mathematical practitioner was still being fashioned, when the category was fluid and open to negotiation. My prime concern is to examine just how the identity of the mathematical practitioner was established in England. [page 2:]

To do so I have selected a diverse set of individuals: a gentleman mathematician, a navigator turned naval officer, a master shipwright, a military engineer who designed instruments, a mason, an architect and an expert in fortifications. Yet despite their diversity, these practitioners form a coherent group. The common denominator which links them is their participation in the Elizabethan project to rehabilitate Dover harbour. The intersection of their careers at Dover warrants my gathering of them together as a genuinely historical collective.

By fastening on Dover harbour as a principle of selection, this study is tightly bound in its personnel and period. Dover also provides a highly specific geographical location, though not an exclusive one: following the careers of these practitioners takes us to other places too. But while it is localised in its people, time and places, this study extends broadly in other respects; it follows the wide range of subjects tackled by the mathematical practitioners and, by doing so, challenges the limitations of much existing historiography.

The subject range of 16th-century mathematics is encapsulated in the cross-section of interests and professional activities pursued by my Dover practitioners, from astronomy to shipbuilding, via geometry, navigation and surveying. The historiographic challenge posed by mathematical practice lies in the way that it confounds disciplinary divisions between the histories of science, technology and mathematics. To take even a localised slice of mathematical practice seriously, and to investigate the identity of the mathematical practitioner, is an inherently interdisciplinary task.

This introductory chapter provides the larger setting for the more specific [page 3:] chapters which follow. It outlines the character of the mathematical arts in Renaissance Europe, identifying both the historiographic obstacles to their study as well as the benefits that result from following Renaissance rather than modern conceptions of mathematical practice. Turning to the particular case of England, I then look at those forms of mathematics which existed prior to the creation of mathematical practice, with the court and the university acting as the principal venues for these earlier mathematical endeavours. Next, I sketch the specific and novel character of mathematical practice, particularly in the practitioners’ strategies for self-legitimation and in their material culture. Finally, after introducing my main characters in more detail, I consider the identity of the mathematical practitioner in general terms, and indicate the importance of this role in going beyond the simple opposition between scholars and craftsmen.

## 1. MAPPING THE MATHEMATICAL ARTS AND SCIENCES

The key feature of mathematics in Renaissance Europe was diversity: the mathematical arts and sciences were interpreted in radically divergent ways in the 15th and 16th centuries. Mathematics could be a spiritual discipline, read as a guide to meditation on the divine;1 alternatively, it could be acquired as a vocational resource by merchants, developing their bookkeeping skills.2 Mathematics was both a propaedeutic to natural philosophy in university arts courses and, through studies of [page 4:] fortification, artillery and the ordering of troops, an element of informal aristocratic military education.3 The subjects encompassed within the cycle of mathematical arts might range from arithmetic to perspective, mechanics to surveying, and navigation to astronomy, each topic with canonical texts, instruments and practices at elementary and advanced levels.

To the evident diversity of motivation and subject can be added the multiplicity of distinct roles occupied by Renaissance mathematicians: the Italian teacher of the *abbaco* who doubled as an estimator for building works;4 the humanist scholar collecting and collating classical manuscripts in order to restore the works of Greek geometers;5 the university regent master who taught a few books of Euclid before quickly passing on to higher studies such as medicine;6 and the prince fascinated by the spectacular display of astronomical clocks and ingenious devices.7 The list of such stock figures might also include the algebraist conducting intellectual duels in public mathematical contests (the context for the notorious controversy between Tartaglia and Cardano over the solution of the cubic equation), the astronomer constructing elaborate geometrical models of planetary motion, the [page 5:] instrument maker engraving scales on brass for sale from a workshop, and the astrologer who interpreted the political and military significance of celestial events, as well as the personal meaning of nativities for individual clients. All these (and many more) were recognised identities for Renaissance mathematicians and mathematical practitioners.

This extraordinary proliferation of activities, topics and individuals can be attributed in part to the absence of any overarching disciplinary formation: there was no institution or dominant community able to enforce a privileged form of mathematical practice. The resulting complexity is now compounded by the difficulty of mapping Renaissance mathematics onto modern classifications of knowledge. The mathematical arts and sciences of the Renaissance represent a terrain which has subsequently been divided up and reapportioned, its parts appearing as elements within the modern categories of science, technology and mathematics. Subjects such as astronomy, mechanics and optics have been annexed by science; machines, bridges and fortifications now belong to technology; algebra, arithmetic and geometry fall within the domain of modern mathematics. Indeed, through painterly perspective, music and architecture - all claimed as mathematical arts - the dispersion has reached much wider than just science, technology and mathematics.

Perhaps inevitably, histories of science, technology, mathematics or art have selected those aspects of Renaissance mathematics which conform to more recent conceptions of their respective fields. Probably because their primary audience is made up of mathematicians, textbooks on the history of mathematics are the most rigorous in their pruning, standardly discarding all else bar a core of arithmetic, geometry, trigonometry and, most prominently, algebra. Only rarely does the [page 6:] historian of mathematics step out beyond this textbook inner circle. One exception is Mahoney’s biography of Pierre de Fermat which, although primarily concerned with 17th century analysis, begins with a survey in which he distinguishes six broad categories of Renaissance mathematician: classical geometers, cossist algebraists, applied mathematicians, mystics, artists and artisans, and analysts.8 But, significantly, Mahoney explicitly debars *mathematici* - astronomers and astrologers - from his typology. For him, they self-evidently belong to the history of science. Despite his unusual sensitivity in acknowledging diversity, even a historian of mathematics such as Mahoney remains tied to more modern definitions of the discipline.

History of science, increasingly professionalised and with stronger ties to mainstream historiography, has been more catholic and contextual. However, interest has traditionally focused on the 17th rather than the 16th century, and recent concerns have centred more on the changing form of natural philosophy than the character of the supposedly ‘harder’ mathematical sciences. Moreover, those who have tried to provide textbook accounts of Scientific Renaissance rather than Scientific Revolution have typically juxtaposed astronomy and mechanics with other ‘scientific’ subjects such as botany and anatomy, rather than pursuing closer contemporary links with geometry, architecture, painting, or surveying. If the resulting narratives lack coherence, the absence is attributed to the period itself, diagnosed as one of creative confusion, in which previously rigid intellectual and social structures were challenged and loosened. The Renaissance is cast here as a curtain raiser to the 17th century, when convincing alternatives and new certainties take over the stage.9 [page 7:]

Not every historian of science can be targeted by such textbook strictures, but it is striking that even self-conscious efforts to chart the contours of early modern science have gone awry. Thomas Kuhn, in a pioneering attempt to trace the long-term development of physical science, persuasively identified the mathematical sciences as a coherent cluster of subjects with their own problematic, one often distinct from that of the emerging experimental sciences.10 But to the extent that Kuhn’s schema is convincing for the 17th century and afterwards, when physical science becomes at least a half-way plausible category within which to place mathematics and natural philosophy, so it is seriously flawed for earlier periods when such a category has little purchase. By importing ‘physical science’ as his focus of attention, Kuhn is forced to break his own explicit methodological prescription to respect contemporary subject definitions and boundaries.11 Moreover, the descriptive inadequacies which arise from failure to recognise the full range of Renaissance mathematical arts and sciences also give rise to more serious explanatory problems.12

Work in the history of Renaissance technology has typically been more piecemeal than studies of the history of science. In part, this is because the Industrial Revolution and thus the 18th and early 19th centuries have traditionally been its principal defining moment. But the heterogeneity of technology has also helped to [page 8:] preclude a single synoptic vision of the Renaissance. With textiles, mining, bridges, armaments, ships, building, transport, agriculture, and water- and windmills all to be incorporated, it is little wonder that no master narrative of 15th and 16th century technology has been established.

But when broader synthesis *has* been attempted it has typically sundered accounts of technology from those of science and mathematics. For example, Eugene Ferguson has stressed visual thinking as a key element for a history of early-modern machines and mechanisms, an element which distinguishes it from the presumed verbal and numerical discourse of science and mathematics.13 Alternatively, Rupert Hall has tried to keep early modern science and technology in separate boxes, in response to the perceived threat of that so-called ‘vulgar Marxism’ which sought to identify social rather than intellectual roots for the successes of early modern science.14

Modern interpretations of the Renaissance mathematical arts have thus tended to run within the fault lines of science, technology and mathematics. But faced with this background of historiographic division and disparity, how can Renaissance forms of mathematics be adequately described and explained? How can the mathematical arts and sciences be reconstituted and assessed? [page 9:]

To recapture the full character of Renaissance mathematics its different versions and varieties should be considered not just as sets of ideas or techniques but as cultural practices. ‘Cultural practice’ is meant here as a conveniently capacious grab-bag in which to assemble values, techniques, social status, procedures, the symbolic realm, material culture, ideas, and also the places in which individuals enact and instantiate culture. In this respect, though pragmatic rather than theoretically nuanced, my use of ‘culture’ is closer to that of anthropologists and ethnographers than students of ‘high culture’. Moreover, when I discuss English mathematical practice, this does not refer to an activity which is necessarily opposed to theory. When practice is treated as culture, there is no prohibition on the existence of theoretical practices.

While no single key will suffice in systematically mapping the *prima facie* anarchy of Renaissance mathematics, a study of the places of mathematics provides a good first step. On the most coarse of scales, we can readily identify a menu of ‘regional specialities’, subjects whose development is most associated with particular countries or regions. In Italy (despite the danger of generalising from the diverse experiences of different city-states and courts), there was heightened interest in mechanics,15 fortification,16 and perspective.17 In northern Europe, and particularly in Germanic lands, the university and courtly development of (technical) [page 10:] astronomy is particularly marked, from Peurbach and Regiomontanus, through Copernicus, Reinhold, and Peucer, to Tycho, Maestlin, Praetorius and on to Kepler. Spain and Portugal, on the other hand, are best known for cosmography and navigation, in connection with their voyages of exploration and conquest.

From such *grosso modo* distinctions of regional development, it is possible to pass on to more specific considerations of place. For example, when analysing the advancement of astronomy the principal institutional settings are German courts and universities, each operating within highly specific conditions.18 Likewise, the different court cultures of Italy supported particular values and styles of work, from Commandino at Urbino, to Benedetti at Milan and Galileo at the Medici court in Florence.19 Studying the places of mathematics leads from straightforward physical geography to cultural and indeed moral topography.20

Thinking about place therefore gives us one ‘way in’ to the examination of mathematical cultures. Moreover, the absence of an overarching disciplinary formation meant that the pursuit of mathematics in the Renaissance was peculiarly subject to local conditions (in complete contrast to our familiar perception of its [page 11:] universal character). This contingency and diversity of Renaissance mathematics points to the need for local studies. I thus now turn from the European stage to the particular case of England.

## 2. LOCATING ENGLISH MATHEMATICS: THE COURT & UNIVERSITIES

In the second half of the 16th century, the tradition of mathematical practice developed into the most prominent and public culture of mathematics in England. But it was not the only available form of mathematics. Before turning to the work of the mathematical practitioners, the alternative pursuit of mathematics in two specific 16th-century locations - the court and the universities - should be outlined. These were prime sites for mathematical work in contemporary Europe and, as one might expect, mathematical arts played a part in both the courtly and academic contexts of earlier 16th-century England. In this respect, there is a measure of continuity with even earlier traditions: the learned practice of astrology, for example, was strongly tied to these two locations in 15th century England.21

But the court and universities should not be relegated merely to the rank of antecedent institutions: their role was not simply to provide the cultural locale out of which mathematical practice would subsequently evolve. Indeed, mathematics in the universities would later continue in parallel with the related but separate development of mathematical practice. There was thus not a successive, linear development of English mathematical culture but rather the creation of distinctive and often co-existing [page 12:] alternatives.

The court had for long been the focal point of the political elite, but under Henry VIII there was a self-conscious effort to create a deliberately magnificent court culture intended to rival that of other European princes. The Elizabethan miniaturist Nicholas Hilliard looked back on Henry as ‘a prince of exquisite judgement and royal bounty, so that of cunning strangers even the best resorted unto him and removed from other courts to his’.22 Henry’s accumulation of people and things was extraordinary. He ordered the construction of new palaces and imported painters and sculptors from Italy to provide their decoration. Tapestries were bought up, musicians employed, books collected, and humanists of the stature of Erasmus and Vives entertained, even if only for brief stays.23

Henry’s ambitions extended beyond just the arts of peace. More martial was his foundation of an armoury at Greenwich, though the elaborate and decorative products of Henry’s Milanese, Flemish and German craftsmen were most likely to be found in tiltyard jousts rather than worn in earnest on the battlefield. Yet beyond the magnificent display of royal tournaments Henry embarked on a programme of fortress construction in which the skills of native experts were supplemented by those of strangers from Italy and Germany. While military engineers frequented his court, discussing designs for fortifications, Henry also brought in shipwrights from the Venetian Arsenal to build galleys for his newly organised and rapidly expanding navy.

Standing behind the aesthetic pleasures, learned accomplishments and [page 13:] military innovations of this manifold activity was the glorification of the prince. The universal range of Henry’s human and material acquisitions was meant to reflect back on his own universal virtues. So when Henry secured the services of scholars and practitioners with expertise in astronomy, mathematical instruments, cosmography, hydrography, and cartography, they were simply being added to his already extensive courtly collection.

Whatever the motives, Henry provided welcome employment for several figures. By 1519, Nicolaus Kratzer was his astronomer, spending time both in and around the court and also at Oxford.24 Kratzer’s instrument making skills were supplemented by those of Sebastian Le Seney who entered the king’s service in 1537 and produced at least one astrolabe for Henry.25 Another arrival from France was Jean Rotz. Appointed royal hydrographer in 1542, he presented Henry with both a variation compass (and its accompanying treatise) and a manuscript atlas, his ‘boke of ydrography’.26 Again from France came a court poet, Jean Mallard, who was obviously well advertised of his patron’s cartographic interests; Mallard’s successful petition for favour was included in his presentation copy of a French verse cosmography with its own world map.27

The systematically mathematical character of these examples of Henry’s patronage should not be overestimated: the vision of mathematics as an all-embracing [page 14:] category within which the work of a Kratzer or a Rotz might easily be accommodated was still in the future. The people and their productions were united more by their common court context than by claims as to their common mathematical character. Kratzer, the German-born astronomer, perhaps best illustrates the integration of mathematical pursuits with the other activities of the court. He repeatedly collaborated with Henry’s painter Hans Holbein, working with him, for example, in 1527 on the scheme for a cosmographical canvas ceiling to be installed in a banqueting hall. Their joint endeavours also extended to smaller-scale work: Holbein contributed illuminated capitals to the royal presentation copy of Kratzer’s manuscript *Canones Horoptri*, a text designed to accompany a new astronomical instrument. The pair were still working together on courtly gifts in the 1540s, when they cooperated on the design of a sumptuous salt-cellar encrusted with technical ornament; in addition to a clock, this elaborate goldsmith’s confection included an hour glass, two sundials and a compass.28

Mathematics and mathematical devices circulated here within the personal context of princely display. Even mundane activities such as writing and counting could be tricked out as part of a conceit for a new year’s gift: Catherine Parr once offered the king a ‘device like a cup, ... the cover having diverse small boxes with pictures and conclusions of arithmetic, the foot having three boxes for ink, dust and counters’.29 Henry’s court provided an environment in which mathematics was favoured both as ingenious and lavish display and also, in the form of scaled maps and [page 15:] plans, as a new resource for military and political strategy.30

Unfortunately for those who depended on his royal salaries and pensions, Henry VIII’s creation of a court culture in which various mathematical arts had a small but significant place did not long outlast him. For the rest of the century, the idea of an inventive and intellectually expansive court went into retreat.

The Protestant reign of Edward VI (1547-1553) and the Catholic reaction under Mary (1553-1558) were too riven by political and religious upheaval to provide a stable courtly environment. By contrast with these turbulent years, Elizabeth’s accession in 1558 was to mark the opening of a long reign in which the image of the monarch was again a powerful symbol for the nation. But whereas Henry’s tradition of princely magnificence was created through lavish expenditure and the maintenance of numerous court posts, the Elizabethan regime was financially stringent. There were to be no astronomers or hydrographers rewarded with permanent attachments to the Elizabethan court, and even the influx of Italian military engineers was allowed to dry up. Such sharp pruning destroyed the basis for an active court tradition of mathematics.31 It was not until Prince Henry became the focus of a precocious but short-lived court in the years around 1610 that the role of the mathematician was again established as a permanent and salaried fixture in an English prince’s entourage.32 [page 16:]

Of course, opportunities for patronage under Elizabeth were not entirely lacking. But it is striking that the management of patronage was chiefly in the hands of Privy Councillors such as Lord Burghley and the Earl of Leicester and that, when exercised, patronage consisted in individual acts rather than long-term appointments. Even a favoured client such as John Dee, who received personal visits from Elizabeth, petitioned in vain for a court position to formalise his occasional consultations on matters mathematical, historical and philosophical.33

A number of the most suggestive recent interpretations of the Renaissance mathematical sciences have concentrated on courts as a privileged site for significant work: Commandino and Guidobaldo at Urbino;34 Wilhelm IV at Hesse-Kassel;35 Galileo and the Medicis;36 and even the slightly anomalous case of Tycho Brahe at Uraniborg, with the island of Hven as his realm.37 But when studying the establishment of mathematical practice in later 16th-century England we are, by contrast, in an environment where the court did not represent a pinnacle of mathematical achievement and reward. Mathematics was not expected to serve as a learned or artful ornament in the image-making of the prince and, severed from the polite culture of the court, could not be magnified through royal association. Without such reflected glories, and in the absence of court opportunity, Elizabethan mathematicians had to appeal to alternative values. [page 17:]

What of that other familiar institution, the university? How did it fare as a venue for the cultivation of the mathematical arts and sciences in the 16th century? Here the story is more complex.

At the beginning of the 16th century mathematics and the mathematical sciences were of minor importance in the two English universities. In theory, mathematics had a place in the university curriculum through the quadrivial arts of arithmetic, geometry, astronomy and music, but in practice the then traditional curriculum found little space for their teaching.38 Oxford had the richer medieval heritage and there were probably a few Oxford scholars still familiar with the 14th century Merton *calculatores* and their philosophically-oriented studies of proportions.39 But such texts were more likely to be encountered in early 16th century Italy than in England.40 Likewise, the sophisticated traditions of medieval astronomy represented by the cleric Richard of Wallingford and by his peers in 14th and early 15th century Oxford had largely petered out, with only an echo surviving in occasional updates of earlier calendars.41

But from the beginning of the 16th century the universities underwent major [page 18:] change. In institutional arrangements, Oxford was particularly affected by the sweeping away of the religious houses, while in both universities there were important new college foundations: for example, Wolsey’s Cardinal College (later Christ Church) at Oxford in the 1520s, and Trinity College, Cambridge, formally founded in 1546. Indeed, this period witnessed the emergence of the college rather than the characteristically medieval hall as the focus of undergraduate life.42

In addition to these institutional changes, contemporaries noted a gradual but pronounced shift in the social composition of the student body. The stereotype of the student as a poor scholar for whom university was a path to vocational (usually ecclesiastical) advancement was challenged by the presence of increasing numbers of sons of the gentry. Many of these newcomers did not stay to complete the prescribed curriculum but were resident for only a year or two before passing on elsewhere, perhaps to acquire further social skills at an Inn of Court in London.43

The content of teaching itself changed in the first half of the century with religious and humanistic reform. Medieval philosophy and logic were largely jettisoned and more emphasis placed on the values of classical Latin and Greek. At both Oxford and Cambridge, mathematics had a place on the margins of these reforms. In Cambridge, a new university mathematical lectureship was founded around 1500, to be held by a master of arts and funded by the fees of students. The regulations stipulated that the master lecture for three years, successively on arithmetic and music, geometry and perspective and finally on astronomy. Unfortunately, [page 19:] beyond the bare list of masters who held the position throughout the century, almost nothing is known of the actual conduct of the lectures, though the Edwardian statutes of 1549 did prescribe a set range of mathematical texts for the reformed B.A. and M.A. syllabus.44

Such university posts were supplemented by the foundation of college mathematical lectureships, especially in the larger and more prestigious colleges. In Cambridge, for example, the 1516 statutes of St John’s prescribed four mathematical lecturers, while a mathematics examiner was added in 1530; Sir John Cheke introduced a mathematical tutor to King’s when he was appointed as its Provost in 1548-9; and Sir Thomas Smith endowed lectureships in arithmetic and geometry at Queens’ in 1573.45 These foundations were part of a larger trend towards the consolidation of teaching within college walls, a trend which also reinforced the rise in importance of college tutors.46

Throughout the century mathematics was therefore always available in the universities and their constituent colleges, even if the extent of its availability is not always apparent from statutes. However, the level of this academic provision and its centrality to the life of the university should not be overestimated. Mathematics’ teaching remained the province of youthful regent masters, rather than full professors. The lecturers therefore knew that they were embarking on a temporary commitment from which they would, in a couple of years at most, necessarily move on to other [page 20:] and more permanent occupations. The ephemeral nature of arts teaching (of which mathematics formed only a part) came to be seen as an obstacle to improvement. The philosopher John Case complained in 1596 that the constant rotation of junior-level arts lecturers restricted teaching to superficialities.47

Certainly, English university provision appeared weak and provincial when compared with the reformed universities of Germany, for example. Under Melanchthon’s leadership, the quadrivial arts (and especially astronomy) were given new institutional impetus by the creation of full professorships. Even if (as was also the case in Italy) the professors of mathematics were of relatively low status, the teaching of mathematics was secured within the arts faculty. Moreover, extended tenure was a prerequisite to the institutionalisation of teaching beyond elementary levels. Such conditions were not realised in England.48

However, outwith the bounds of statutory university and college provision in England, there was the possibility of informal initiative in the university pursuit of mathematics. Students seeking more advanced fare than was standardly available could probably find a sympathetic college fellow to encourage and direct their studies; in Oxford, Thomas Allen (1542-1632) seems to have acted in this capacity during the latter decades of the century.49 [page 21:]

The possibilities and the limitations of the English university environment are perhaps best illustrated by one of the most prominent academic mathematicians of the later 16th century, Henry Savile (1549-1622). As a young regent master, Savile gave a series of lectures at Oxford between 1570 and 1575 which are marked by their high technical standards and European range of reference. Savile’s principal subject was astronomy, but his surviving notes show that he placed his detailed exposition within the wider context of the mathematical sciences. Savile extended his mathematical interests during a continental tour (1579-82), when he collected and collated Greek manuscripts and established contact with European scholars who were subsequently to become his correspondents. But Savile’s impressive debut and his cultivation of mathematics were not to continue. Within the academic milieu, he progressively shifted the focus of his studies towards antiquarian, Biblical and patristic topics, and it was on these scholarly subjects that he was eventually to publish. By contrast, his work on the text of Ptolemy’s *Almagest*, and his thorough comparison of Ptolemaic and Copernican planetary models, remained in manuscript.50 The values of his academic world placed mathematics as a subsidiary activity, more appropriate to youthful regent masters than to mature scholars of riper judgement.

In a striking counterpoint to Savile’s story, those university-trained mathematicians of the late 16th and early 17th centuries who most actively and publicly developed their early mathematical interests did so outside the university. The priorities and values of academic culture, the regulations governing the length of tenure of college fellowships, even the restrictions on marriage might all count in [page 22:] personal decisions to seek preferment elsewhere.51

For those academics who stayed behind, mathematics was seen less as an object of study in its own right than as an elementary component of the full cycle of liberal education. The case of Savile notwithstanding, lectures were likely to be compiled from approved authorities, with innovation an unlikely objective for a novice lecturer. Mathematically oriented dons were more likely to collect mathematical books than to write them. Even when a fellow’s continued mathematical interests led to authorship, publication did not follow: manuscript circulation through personal contact and private correspondence remained standard. (This is the single most important reason for the ‘invisibility’ of academic mathematics to most historians.) University mathematics thus had a distinctive texture. It was largely a ‘closed shop’, its authors and audience meeting within the university’s own precincts and, through the use of Latin, remaining linguistically self-contained. By not seeking to publicly engage English contemporaries outside the university, academic mathematics was a highly localised culture.

## 3. MATHEMATICAL PRACTICE & MATHEMATICAL PRACTITIONERS

By comparison with the worlds of the Henrician court and the universities, the contrasting character of mathematical practice stands out strongly. Mathematical practice was public: it was made visible and accessible through the printing of vernacular texts on topics such as geometry, arithmetic, algebra, astronomy, surveying, and navigation. Its practitioners emphasised that their work was geared [page 23:] towards active use, rather than conspicuous display or scholarly demonstration and textual correction. These practitioners also claimed that their skills were appropriate to almost any location, whether aboard ship, in the survey of rural or urban land, or on the battlefield. The diversity of places represented on surviving maps, plans, and sea charts certainly attests to the extent of the practitioners’ geographical presence. But although mathematical practice was ubiquitous in principle, it was nevertheless closely associated with a specific place, London. The capital was the centre of the printing trade and authors who ‘went to press’ would often do so quite literally: in the absence of specialised mathematical proof readers, the author typically had to be present in the printing house during production to correct the text.52

Though tiny in comparison with the scale of the book trade, London also became the centre for the commercial manufacture of mathematical instruments.53 Of course, though their production was sited in London, both books and instruments were portable and were intended to be distributed and used more widely; London became a passage point rather than an exclusive location.54 But London’s prime position in the culture of mathematical practice was reinforced by its importance for teaching. It was in London that schoolmasters and tutors first began to publicly advertise their availability as teachers of mathematics.55 And public lectures were [page 24:] first given there in 1588 by Thomas Hood, before the opening of Gresham College a decade later.56

Integral to mathematical practice was its rich and innovative material culture. Books and instruments have already been mentioned but they were supplemented by a third major element, ‘plats’ (the contemporary generic term for maps, charts, plans and related visual artefacts). Each of these three mathematical products appeared in England in its most characteristic form during the middle decades of the century. Of course, books had been printed in England since the 15th century, but the first printed mathematical book in the vernacular was the anonymous *An introduction for to lerne to recken with the pen or with the counters* (1537); Robert Recorde’s much better-known *Grounde of Artes* followed in the early 1540s.57 Recorde planned a methodical series of textbooks, and issued several in the 1550s, when other authors such as Leonard Digges also began to publish.58

Like books, instruments have an extended prior history, but it was not until the 1550s that they began to be commercially made and sold in London. Just as mathematical authors freely borrowed and adapted from earlier continental sources, so the mathematical instrument-making trade was initially an import. The first known [page 25:] retailer was the Low Countries immigrant Thomas Gemini, who not only published Leonard Digges’ *Tectonicon* in 1556 but was advertised on the title page as ‘ready exactly to make all the instruments appertaining to this book’. A small but growing number of native English makers succeeded him later in the century, led by Humphrey Cole.

As with books and instruments, innovation in the area of maps and plans lay in a significant transition rather than creation *de novo*; again, the pattern of prior continental development is repeated. Varieties of picture map had already been known and used in medieval England, but from about 1540 ‘plats’ drawn to a consistent scale began to appear, initially for planning and recording fortifications. The transfer of the technique of measured survey to hydrography, estate mapping, and architectural design was slowly worked out in succeeding decades. Scaled plats, with the tell-tale presence of a scale bar (often surmounted by dividers), became a staple symbol of the work of English mathematical practitioners.59

Mathematical practice was thus distinguished by the new identity of its practitioners, by the places in which it was performed and taught, and by its material culture. As well as the novelty of its cultural form, the mathematical practitioners developed distinctive persuasive strategies to represent and justify their work. Such efforts were very necessary. Far from being secure or autonomous, both the status of the mathematical arts and the proper identity of the mathematician were contested in 16th-century England. In general, we only know of opposition to mathematics from the rebuttals of its advocates. But very occasionally criticism moved from informal [page 26:] and personal comment into print. For example, in his posthumously published *The Scholemaster* of 1570, Roger Ascham commented: ‘Mark all mathematical heads, which be only and wholly bent to those sciences, how solitary they be themselves, how unfit to live with others, and how unapt to serve in the world’.60

Ascham’s concern was with the moral condition of mathematics. Although his censure was directed principally at those who indulged their appetite to excess, he nevertheless clearly identified mathematics as a morally dangerous pursuit. Those who followed such studies risked losing their sense of civic duty, and thus forsaking their obligations as citizens of the commonwealth. Like other civic humanists, Ascham advocated a form of knowledge that took place out in the world and expressed itself as active service. Hence his rejection of a solitary life. Working in solitude outside society did not guarantee the purity and authenticity of one’s knowledge; on the contrary, it made one’s knowledge irrelevant and even suspect.61

Now, Ascham was not an idiosyncratic outsider but a central figure in English humanist pedagogy - he had, indeed, tutored the youthful princess Elizabeth.62 Moreover, his hostility cannot be attributed to ignorance: he had been Cambridge mathematical lecturer in the two academic years from 1539 to 1541.63 Nor did Ascham restrict his views on mathematics to merely general prescriptions; he advised against the subject’s suitability on a personal level too. In a letter of 1564 to the Earl of Leicester, Ascham reproved his patron: ‘I think you did yourself injury in [page 27:] changing Tully’s wisdom with Euclid’s pricks and lines’.64 Drawing on the moral authority of Cicero, Ascham could warn against the excessive study of mathematics.65

Aside from such ancient civic sources which could be mustered to question the propriety of mathematics, there were also other widespread contemporary concerns over the practice and powers of mathematics. In a period when astronomy and astrology were frequently treated as different branches of the same activity, mathematics could be viewed as part of a larger constellation of occult arts. From this perspective, a continuity existed between geometry, astronomy, judicial astrology, and such vertiginous arts as geomancy and spirit conjuring. When thus allied with techniques of divination, mathematics was open to attack as a black art. Moreover, its cultivation might threaten not just personal salvation but could also serve to subvert reformed religion in general, since many Protestants were quick to align occult practices with the supposed superstition of the Catholic Church.66

Mathematics was thus situated not in splendid isolation but within a cultural minefield of conflicting and controversial evaluations. Mathematical practice had to be actively promoted, and the charges of an Ascham overcome or evaded. Indeed, [page 28:] I suggest that the specific form of English mathematical practice was created and articulated in response to just such criticism. While similar formulations were developed elsewhere in Europe, their prior appearance in Italy, for example, is not a sufficient explanation of subsequent use in England. In deliberately seeking to show that their mathematics was worldly, practical and social, the mathematical practitioners were responding to the local canons of Protestant civic humanism articulated by Ascham and others. The values attributed to mathematical practice were the outcome of the practitioners’ effort to secure a moral high ground for their subject.

The best place to study the practitioners’ legitimation of mathematical practice is in the prefaces to their texts. Following the conventions of 16th century authorship, writers on mathematics frequently offered an introductory *apologia* for their subject matter. John Dee’s lengthy *Mathematicall Praeface* to the 1570 English edition of Euclid is much the best-known and the most elaborate example of this genre. Similar (though smaller-scale) defences are common in works such as Robert Recorde’s *The Path-Way to Knowledge* (1551) and Leonard Digges’s posthumous *Pantometria* (1571).

The genre of the prefatory *apologia* was well-developed in its format and content, and there was a standard repertoire of rhetorical figures which any apologist could use in promoting an art. For example, the dignity of a discipline would be bolstered by reference to its antiquity and to the eminence of its past practitioners. Hence prefaces on all subjects, mathematics included, are filled with instances of ancient, royal and biblical figures who had practised the art.

Aside from such standard virtues, there were others more particular to [page 29:] mathematical discourse. A primary asset was that of certainty: the rigorous demonstrations of Euclidean-style geometry were proclaimed as a standard of truth to which other arts could only aspire.67 However, though routinely proffered, this justification was matched in prominence (if not indeed exceeded) by claims for other virtues.

For example, in English mathematical apologetics, the range and unity of the mathematical arts were touted as a warrant for their near universal efficacy. In his *Mathematicall Praeface*, Dee enumerated and described an extraordinarily long list of arts, drawing on the riches of his library to present a digested treasury of Renaissance mathematics. Nor was Dee content with mere compilation; he sought to extend the mathematical pantheon, coining his own neologisms to identify topics and raise them to the status of independent arts. However, this intellectually imperialist expansion was not just an anarchic proliferation; in the ‘groundplat’ which diagrammatically summarised his discourse, Dee gave a powerfully graphic demonstration of the hierarchy and structure of the mathematical arts and their dependence on geometry and arithmetic as twin foundations (Figure 1.1).

Another key theme, re-echoed in innumerable prefaces and title pages, was the easiness of mathematical practice. In emphasising facility, authors avoided the humanist injunction against reprehensible obscurity and deliberate difficulty. The much-trumpeted ‘plain and easy’ didacticism of so many mathematical books was also angled towards a new audience addressed through the impersonal mechanisms of the market rather than directly encountered in patronage relations. In describing his [page 30: Figure 1.1] [page 31:] ‘Mathematical Jewel’ (a form of universal astrolabe which he published in 1585), John Blagrave nicely illustrated the language of ease and accessibility, and the important role of instruments as a medium for those values. Blagrave wrote that his Jewel was intended as a compendious ‘reduction of the arts mathematick .. from that deep difficulty wherewith hitherto they have been sequestered and closed up ... unto an easy, methodious, plain and practique discipline, lying wide open unto every ingenious practiser’.68

Along with ease went an emphasis on the pleasure and delight that could be provided by mathematical study. Leonard Digges’s *Prognostication Everlasting* opens with an *apologia* directed ‘Against the reprovers of astronomy and science mathematical’ in which he notes that

the ingenious, learned and well experienced circumspect student mathematical receiveth daily in his witty practices more pleasant joy of mind than all thy goods (how rich soever thou be) can at any time purchase.69

Painting mathematical practice with positive and attractive qualities was one strategy. Legitimation could also be sought by its inverse: excluding the negative and undesirable. To this end, barriers were erected to safeguard mathematics from irreligious contagion, and energetic efforts were made to demarcate legitimate mathematics from unlawful practices. For example, Edward Worsop, in a text ostensibly devoted to surveying, has a remarkable digression on the character of mathematics in which he pointedly bars judicial astrology (and other more fearful practices) from the realm of ‘pure mathematical knowledges’: ‘some professing [page 32:] astrology impudently usurp the names of mathematicians, as popish and superstitious priests, the names of divines’.70 Invoking purity and danger, practitioners such as Worsop marked out socially and religiously safe boundaries for mathematical practice.

However, the most highly valued virtue in the discourse of mathematical practice was undoubtedly that of utility. Practitioners wrote of mathematics as an appropriate discipline both for war and peace. Through the preparation of plans, the setting out of fortifications, the design of ships, the improvement of artillery, and the ordering of soldiers, mathematical practice could bolster military decision-making and inform martial action. Mathematics was also represented as an essential aid in more pacific contexts: for trade and merchants’ accounts; for timekeeping and the calendar; for architecture; the surveying of land; measurement of materials; and the techniques of oceanic navigation, amongst many others. At all points in the affairs of the commonwealth and its leaders, mathematics was depicted as a vital resource. The image was of mathematics as action, able to intervene in the most diverse of practices and to bring about beneficial improvement.

In his self-consciously homely dialogue on surveying, Edward Worsop vividly dramatised the practitioners’ efforts to counter opposition and enlist the support of contemporaries. Worsop sketched a scene in which gentlemanly scepticism of mathematics was replaced by belief in its value. Characteristically, the principal persuasive means to bring about this conversion was the argument from utility.

One of the characters in Worsop’s dialogue tells of a gentleman who, after witnessing a demonstration of instrumental surveying, asked what was the use of such [page 33:] ‘fine sleights’. The rebuttal listed the benefits of instrumental measurement, not just for land surveying, but for a range of military uses such as the conducting of mines under fortifications and the shooting of ordnance. In addition, the navigational determination of the heights of sun and stars further demonstrated the virtues of measurement by instrument. After this recital, one of Worsop’s gentleman interlocutors burst forth:

Call you these pretty feats and fine sleights, and such instruments knacks and jigs? Methinketh he that can do these things performeth matters of great weight in the common weal.71

The mathematician was thus to be seen not as merely a sort of intellectual juggler, someone who could perform cunning tricks and subtle shows without real consequence or lasting benefit. On the contrary, Worsop’s gentleman was persuaded that mathematics had an essential role to play in sustaining the commonwealth.

From the evidence of Roger Ascham and others, it is clear that mathematics was not seen as a necessarily virtuous pursuit in Elizabethan England. Looking at mathematical texts, and especially their prefaces, shows how practitioners sought to persuade contemporaries of the practical and moral worth of their subject. Mathematical practice was constructed as an activity compatible with the culturally dominant values of civic humanism. But this rhetorical fashioning was not just a textual process. Texts were just one aspect of the material culture of mathematical practice; instruments and plats also communicated the practitioners’ values and claims. Since subsequent chapters give case studies particularly of the symbolic uses of plats, a single instrumental example will serve as illustration here.

Humphrey Cole was the first native-born maker of mathematical instruments [page 34:] to set up shop in London, recorded in 1582 at the north door of St Paul’s (a traditional area for booksellers).72 Figure 1.2 and Figure 1.3 show a folding brass rule signed by him and dated 1574.73 Although the relative absence of engraved ornamentation gives the rule an outwardly modest and ‘practical’ appearance, the instrument nevertheless serves as an aggressive, almost extravagant, piece of propaganda for the programme of mathematical practice. The rule sets out to materially demonstrate the range and interconnection of the mathematical arts, as well as to facilitate a series of detailed operations. In it, the functional and the symbolic are inextricably intertwined.

The folding rule is enthusiastically multifunctional. Figure 1.2 shows a linear scale of inches, as well as more complex scales of board and timber measure which can be used to calculate areas and volumes, solving problems such as the number of cubic feet in a given log of wood. In addition to linear measures and computation, the instrument serves for angular measurement, for which it requires that a sight be inserted in each of the four holes in the centre of the legs. By sighting along the legs, the angular separation of objects or landmarks can be read off from the semicircular scale at the hinge.

The instrument has further surveying uses since, when straightened out to its full length, and now with just two sights (one towards each end), it can be used as a plane table alidade. A scale of equal parts (500-0-500) running along the inside edge of the legs (and across the hinge) facilitates scaled measurement directly on a plane table’s paper map. A final help to surveyors was the provision of 12 different [page 35: Figure 1.2] [page 36: Figure 1.3] [page 37:] subdivisions of the inch into equal parts, giving a variety of suitable scales for maps.

All these scales and operations were crammed on to only one side, whose observational capabilities are geared towards the horizontal measurement of angles. The reverse is more suited to measurement in the vertical plane, with the divisions of its geometrical quadrant shadow square supplemented by a scale of degrees. In this observational mode, the legs have to be pegged open at 90° (the hinge is specially equipped for this) and the thread of a plumb bob attached at a purpose-made hole.

With the legs fixed at 90°, the now non-folding rule also becomes a sundial. We move here from surveying and terrestrial measurement to astronomy and timekeeping. The gnomon, which casts the shadow, is again removable and can be fixed in a pre-positioned hole.74 Finally, while the dial requires the presence of its accompanying table (to give the day of the month on which the sun enters a given sign of the zodiac), the table on the other side is more in the way of an aide-memoire. Here Cole occupies otherwise vacant space with such handy if inessential information as the statutory definition of common measures and the dimensions of different rectangular plots of land whose area is one acre.

This rapid guided tour of Cole’s rule shows how, with a single instrument, it was possible both to measure angles, inches and elevations, as well as to draw and rescale maps, to calculate, and even to tell the time. The mathematical arts represented include at least arithmetic, geometry, mensuration, surveying, and astronomical timekeeping. The implicit claims embodied in the instrument are [page 38:] continuous with the textual rhetoric of mathematical practice: its ambition is to announce and exemplify the integration of different mathematical arts. Rather than easing or performing a single task, it suggests the general efficacy of the mathematical arts; indeed, almost their universal competence. The instrument emblematically endorses and extends the worldly presence of mathematics.75

* * *

Much of the basic evidence of mathematical practice has been collected by earlier historians. The classic text is E.G.R. Taylor’s *Mathematical Practitioners of Tudor and Stuart England* (1954), in which the historical terrain of mathematical practice was first identified and its coherence demonstrated at length. A large part of Taylor’s effort was devoted to building what was, in effect, a biobibliographic database; she brought together literally hundreds of practitioners and their texts, eventually extending to cover the period 1485-1840.76 Many of the assembled characters would previously have been considered too obscure or insignificant to merit serious study. But when gathered together, Taylor was able to reconstruct a didactic, urban and vernacular tradition, in which instrument makers and ordinary textbook writers had a place alongside better known names from the history of science.

Taylor was able to draw on earlier studies of individual arts and sciences [page 39:] such as F.R. Johnson on astronomy and, indeed, her own volumes on geography.77 Subsequently, but as part of the same generation, there was Waters’ monumental study of navigation, as well as work specifically on surveying.78 With sources such as these available, mathematical practitioners also began to enter into mainstream historiography.79 Now, several decades after the publication of these early texts, mathematical practice continues to be a fruitful area for research, through studies of its relation to natural philosophy,80 its relevance for the history of scientific instruments,81 and the context it provides for detailed biographical investigation.82

A great deal of the work of that first generation of Taylor, Johnson, Waters *et al.* still stands securely. But it should come as no surprise that there are points of detail open to correction and that new interpretations and emphases can be placed on their material. Historiography has itself moved on so that, beyond such straightforward revisions, new sources and questions can be used to interrogate mathematical practice afresh. This study returns to the early years of mathematical practice, when its character was uncertain and its future unsettled. It offers new interpretations of prominent mathematical practitioners; it extends the conventional bounds of mathematical practice by examining shipbuilding and engineering [page 40:] construction; and it emphasises the active work which the practitioners invested in ‘making mathematical practice’. I want now to introduce the Dover practitioners and their place in the remaining chapters of this thesis, before finally considering the wider significance of their identity and role.

Thomas Digges is the first to be studied in detail. The son of an author on practical mathematics, Digges was nevertheless not simply born into the role of mathematical practitioner. He was a gentleman (or, to be more precise, an esquire) whose first mathematical publications were self-consciously advanced productions, aimed at an audience appreciative of more than common accomplishments. However, in later years, Digges reoriented his interests towards active service of the commonwealth, as he worked out a role that encompassed his identity as both a gentleman and a mathematician. Digges’s interventions at Dover harbour were one consequence of his shifting commitments, while his public pronouncements prominently asserted the new role and values of the mathematical practitioner.

The next chapter shifts from the world of a gentleman to the realm of craft technology. Mathew Baker was perhaps the leading English shipbuilder of the 16th century and his reputation as a royal master shipwright extended far beyond the dockyards themselves. Making extensive use of a manuscript subsequently preserved by Samuel Pepys, this chapter investigates Baker’s design practice. Baker was one of the first shipwrights to develop paper-based design methods and the mathematical arts were his primary point of reference in this effort. While reworking the character of his technical practice, Baker was simultaneously redefining his identity as a master craftsman; not a humble figure confined to a workshop with no more than a couple of apprentices and journeymen, but a manager accustomed to leadership, whose claims [page 41:] to expertise rested on mathematics and whose characteristic place of work became a drawing office rather than the wooden world of the shipyard.

Chapter 4 is given over to the remaining practitioners, firstly to William Borough, a successful navigator, naval administrator and sometime vice admiral, who was also noted as a skilled hydrographer. Borough published an important tract on compass variation and it is through his magnetic studies that I investigate his prescription for the mathematical practitioner. After Borough comes Thomas Bedwell, an erstwhile Cambridge fellow who became a military engineer and subsequently secured the post of ordnance storekeeper at the Tower of London. The discussion focuses on the mathematical instruments that Bedwell devised and the use to which he put them in establishing both his own role and, by extension, that of the mathematical practitioner in general. The closing section of this chapter turns to three building experts: Robert Stickells, a mason-architect, John Symonds, a joiner-architect and Paul Ive, who specialised in (and published on) fortifications. I assess the varying extent to which, as expert artisans, they made use of contemporary mathematics in fashioning their own professional identities. Before concluding, one last and more shadowy figure is mentioned: John Hill, an auditor with a reputation for mathematical proficiency.

The final substantial chapter centres on Dover harbour, the common denominator linking these diverse characters. Focusing on the extended design process, I show how the practitioners participated in establishing the new form of the harbour. Characteristically, they reconstructed design as an activity taking place on paper, in which the geometry of plats and the arithmetic of money were prime tools. During this complex and often controversial construction project, the practitioners [page 42:] were creating a niche for themselves as credible and effective servants of the crown.

Dover harbour also plays a larger role in the thesis. While not at all seeking to deny the importance of London for the culture of mathematical practice, the focus on Dover gives a new perspective. It particularly avoids perpetuating both the traditional but tired opposition between London and the universities, and the debate over the character and content of the teaching carried out there. Indeed, I give relatively little attention to teaching, for none of the Dover practitioners are known to have taught in either schoolroom or lecture hall. The emphasis lies less on the maintenance and development of mathematical practice through education, than on practice as an (often vocational) activity in which problem-solving has to be accomplished at first hand.83

The presence of the practitioners at Dover also suggests the importance of large-scale civil engineering projects for the exercise and display of technical skills. Elsewhere in Europe, the use of cannon, the development of the bastion fortification and the organisation of new forms of siege warfare created exceptional opportunities for military architects. By comparison, Elizabethan England was relatively free from the ravages of war. The role of the military engineer was less recognisable, less specialised and of lower status than elsewhere. Civil sites such as Dover therefore loom proportionately larger in the creation of technical culture.84 [page 43:]

While their careers intersected at Dover, many of my practitioners also met in other circumstances or worked on the same topics. Inevitably, they crop up in each other’s chapters; Digges, for example, figures especially prominently along with Borough in the account of magnetic variation. Such cross-linking reinforces the sense that Dover provides a window onto the creation of a community rather than just onto a collection of individuals. But if Dover was a communal site, the practitioners were not a homogeneous group. Aside from their different ‘professional’ activities, they were socially diverse. The disparity in status between, for example, the gentleman Thomas Digges and the mason Robert Stickells was a major one, and highly visible within the elaborately hierarchical world of Elizabethan England. Creating the role of the practitioner was therefore a social achievement. But we should not overplay the extent to which the various practitioners were engaged in a process of social levelling. Craft practice could be as stratified and status-conscious as any other activity. Particularly for the case of the shipwright Mathew Baker, I will argue that, rather than a route through which mathematics was brought down to the ordinary craftsman, the identity of the mathematical practitioner elevated the master craftsman further above his subordinates. Whether gentlemen or expert artisans, these mathematical practitioners were ‘masters’, or sought to be regarded as such.85

The relationship between mathematical practice and craft practice was thus a complex and often ambivalent one. The practitioners’ rhetoric of utility should not be accepted uncritically as a simple statement of the ‘practicality’ of their work, or as a reliable indicator of their strong links with craftsmen. Even when the title page of [page 44:] a text such as Leonard Digges’s *Tectonicon* (1556) explicitly advertised the work as ‘most conducible for surveyors, landmeters, joiners, carpenters and masons’, we have little evidence of the actual purchasers and readers. Would the author have been able to make an accurate judgement of his text’s audience in advance of publication (or even afterwards)? Defining the practitioner’s identity in relation to that of mechanicians and artisans was a central problem throughout this period, and one whose negotiation is repeatedly traced in the case studies which follow.

Less prominent in this study is the relationship of mathematical practice to the world of learning. It has already been suggested that the primary values of mathematical practice were not those of demonstration or textual emendation. But some reference to learned authors was standard in prefaces. Arithmetic and geometry had their worth asserted by arguing that they were beneficial to the lawyer, the physician, the philosopher and even the theologian. However, such generalised defences can be read more as rhetorical tropes than as a serious attempt to occupy the territory of ‘learned professions’, as Recorde called them in his apology for geometry.86 In alleging Aristotle, Plato or Galen as witnesses to the wider relevance of mathematics, an author’s principal intention was to draw on the support of acknowledged authorities. But when it came to the substance of their books there were far fewer efforts to tackle issues that intruded into other realms. Even when a small but significant number of texts followed Thomas Digges’s public endorsement of the Copernican planetary system, there was little attempt to take more general issue with philosophical doctrine.87 [page 45:]

This disjunction between mathematical practice and academic philosophy was repeated for other topics. There is a marked contrast with contemporary Italy where, as early as the 1550s, Giambattista Benedetti pugnaciously used mathematics for an attack on philosophy, arguing ‘against Aristotle and all the philosophers’ in his account of falling bodies.88 Indeed, the Italian comparison can be extended, for there was no debate in England to parallel the proliferation of Italian texts and discourses *de certitudine mathematicarum*. This long-running controversy centred precisely on the certainty of mathematics and on the extent to which it matched the philosophical canons of demonstrative discourse. Not only intellectual issues were at stake: mathematicians and philosophers were competing for social status too.89 The absence of an English counterpart to this debate suggests that there was little interaction or conflict between mathematical practitioners and philosophers in Elizabethan England.90

However, there is one English author for whom both philosophy and mathematics figured together as vital parts of an intellectual programme. John Dee has already been mentioned for his *Mathematicall Praeface* to Euclid, now probably his most frequently cited work. But Dee had earlier published a text on astrological natural philosophy, the *Propaedeumata Aphoristica* (1558), as well as the highly arcane *Monas Hieroglyphica* (1564), most recently interpreted as an attempt at a new ‘alphabet of nature’ subsuming astronomy, the kabbalah, numerology, alchemy and [page 46:] magic.91 Through these works and his later angelic conversations, Dee is commonly identified as an archetypical Renaissance magus.92 Yet Dee was also an adviser on navigation and, in the *Mathematicall Praeface*, an advocate of avowedly ‘useful’ vernacular mathematical arts. Clearly, mathematics could be combined with philosophy and other learned disciplines.

But Dee was far from being a typical mathematical practitioner. In the *Mathematicall Praeface*, he devoted primary attention to the practical merits of geometry and arithmetic, not only in their own right but also as progenitors of a great list of mathematical arts which depended on them. In a vernacular work Dee thought it appropriate to emphasise the public, civic and practical benefits of mathematics. But he also noted the philosophical and indeed spiritual virtues of various mathematical arts and explicitly integrated mathematics within a larger metaphysical perspective, drawing on Proclus to present mathematical objects as an intermediate level of being between matter and spirit, the human and divine, sense and pure intellect. Dee also hinted at powerful occult uses of mathematics, giving just enough clues to alert the adept to his ambitions and claims.93

The philosophical and metaphysical dimensions of Dee’s mathematical work have been highlighted by recent interpretations. But in newly appreciating the coherence of Dee’s work and the interconnections between the *Mathematicall Praeface* and his other texts, we are apt to lose sight of the often less sophisticated readings that [page 47:] Dee’s contemporaries brought to bear. Certainly, the veiled projects and philosophical schemes excavated by modern scholarship were less noticeable (indeed usually invisible) to 16th century vernacular authors on mathematics.

Dee’s *Praeface* was indeed read and admired by such contemporaries as William Bourne and Edward Worsop. But their reading of his text stripped it of its philosophical and magical ambitions.94 Dee became a useful ally and a quarry for information. Taken as authoritative in his presentation of the range of mathematical arts, Dee’s *Praeface* provided a framework within which narrower and more specific work could be carried out. Bourne, in his *Treasure for Travellers* (1578), abbreviated Dee’s discussion of the mathematical sciences while acknowledging that his own acquaintance with statics was based solely on the account given in the *Praeface*.95 Likewise, Edward Worsop, whose *Discoverie of Sundrie Errours* (1582) has already been cited, relied on Dee as a point of reference for the character of mathematics, making extensive use of his discussion of astrology and astronomy. Worsop indeed called for the *Mathematicall Praeface* to be printed as a manual, assimilating Dee to the world of cheap print rather than the recondite realm of occult doctrine. Neither Bourne nor Worsop mentioned Dee’s other (Latin) publications.

Mathematical practice was thus largely distinct both from academic natural philosophy and from Dee’s philosophical and metaphysical aspirations. Generalising, one can conclude that the mathematical practitioner did not comfortably fit with contemporary expectations of scholarship. Moreover, if the practitioners were not readily identifiable as scholars neither can they be grouped together as craftsmen; the [page 48:] presence of the gentleman Thomas Digges immediately frustrates so tidy a classification. Yet scholars and craftsmen have been a staple of historiographic debate for over three decades. The identity of the mathematical practitioner thus points up the inadequacies of these categories.

Mechanicians and craft practice were given a notable role in accounts of science by historians such as Edgar Zilsel. Trading on a perception of medieval natural philosophy as static and bookish, Zilsel identified texts on the mechanical arts as the bearers of the new methods and phenomena which were to reorient natural philosophy in the 17th century.96 But such claims were vigorously rejected by, amongst others, Rupert Hall who largely denied a role for mechanicians. Though cautioning against ‘detecting polar oppositions where in reality there is a spectrum’, Hall nevertheless concluded with the influential comment that ‘it is the philosopher who has modified his attitude, not the craftsman’.97 Here, the primary role switches back from craftsmen to scholars. However, this switch is partly accomplished by assigning anyone who made lasting ‘scientific’ contributions to the camp of the scholars.98

Part of the subtext to Hall’s position was an effort to legitimate science, establishing it in an autonomous sphere of activity freed from social and political constraints. However, when an ideological rationale fades, the conclusions which it sustained often remain, to be more subtly worked into new interpretations. Hence [page 49:] Thomas Kuhn built into his account of ‘Mathematical vs experimental traditions in physical science’ an opposition between scholars and craftsmen: after outlining both a mathematical and an experimental tradition of early modern ‘science’, he then suggested that each of these two traditions was mirrored by a distinct craft tradition.99

Rather than perpetuating the stereotypes of scholar and craftsman, the identity of the mathematical practitioner can be used to explore the actual roles available in the 16th century. From gentleman geometer to mathematical shipwright, this study indicates both the range of the practitioners’ work and their active endeavours to work out a common public *persona* and agenda. Only through such a reconstruction of the practitioner’s identity can the retrospective limitations of the histories of science, technology and mathematics be overcome.

## FOOTNOTES

1. For example, Nicholas of Cusa and Charles de Bovelles; see Philip Sanders, ‘Charles de Bovelles’s treatise on the regular polyhedra (Paris, 1511)’, *Annals of Science*, 41 (1984), 513-66.

2. Warren van Egmond, *Practical Mathematics in the Italian Renaissance: a Catalog of Italian Abbacus Manuscripts and Printed Books to 1600* (Florence, 1981), Richard A. Goldthwaite, ‘Schools and teachers of commercial arithmetic in Renaissance Florence’, *Journal of European Economic History*, 1 (1972-3), 418-33 and Natalie Zemon Davis, ‘Sixteenth-century arithmetics on the business life’, *Journal of the History of Ideas*, 21 (1960), 18-48.

3. For the latter aspect, Stillman Drake, *Galileo Galilei: Operations of the Geometric and Military Compass, 1606* (Washington, D.C., 1978), introduction.

4. For an example, Nicholas Adams ‘The life and times of Pietro dell’Abaco, a Renaissance estimator from Siena (active 1457-1486)’, *Zeitschrift für Kunstgeschichte*, 48 (1985), 384-395.

5. P.L. Rose, *The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo* (Geneva, 1975). Note that, in addition to geometry, classical arithmetic and ‘algebra’ became the objects of restoration; see JoAnn Morse, ‘The reception of Diophantus’ *Arithmetic* in the Renaissance’ (unpublished Ph.D., Princeton, 1981).

6. Robert S. Westman, ‘The astronomer’s role in the sixteenth century: a preliminary study’, *History of Science*, 18 (1980), 105-47.

7. Bruce T. Moran has published a number of papers on this topic: see, for example, ‘Princes, machines and the valuation of precision in the sixteenth century’, *Sudhoff’s Archiv*, 61 (1977), 209-28, ‘Wilhelm IV of Hesse-Kassel: informal communication and the aristocratic context of discovery’, in T. Nickles, *Scientific Discovery: Case Studies* (Dordrecht, 1980) and ‘German prince-practitioners: aspects in the development of courtly science, technology and procedures in the Renaissance’, *Technology and Culture*, 22 (1981), 253-74.

8. Michael S. Mahoney, *The Mathematical Career of Pierre de Fermat (1601-1665)* (Princeton, 1973), pp. 1-14.

9. W.P.D. Wightman, *Science and the Renaissance*, 2 vols (Edinburgh, 1962), see especially I, p. 80 for the 16th century as a period when ‘relatively rigid structures became more fluid’. For the juxtaposition of mathematicians, philosophers, botanists and medics in 16th century ‘science’, see also his *Science in a Renaissance Society* (London, 1972), as well as Marie Boas [Hall], *The Scientific Renaissance* (New York, 1962), though this has a convenient survey on ‘The uses of mathematics’, ch. 7. Antonia McLean, *Humanism and the Rise of Science in Tudor England* (London, 1972) has a more limited focus but depends on the same implicit definition of science.

10. Thomas S. Kuhn, ‘Mathematical versus experimental traditions in the development of physical science’, in his *The Essential Tension* (Chicago, 1977), 31-65.

11. Cf. the preface to *The Essential Tension* (footnote 10), pp. xv-xvi.

12. See the comments on Kuhn in J.A. Bennett, ‘The mechanics’ philosophy and the mechanical philosophy’, *History of Science*, 24 (1986), 1-28, pp. 5-6.

13. E.S. Ferguson, ‘The mind’s eye: nonverbal thought in technology’, *Science*, 197 (1977), 827-836; ‘La fondation des machines modernes: des dessins’, *Culture et Technique*, 14 (1985), 183-208; *Engineering and the Mind’s Eye* (Cambridge, Mass., 1992). Note that Ferguson also has explicitly pedagogical aims: he wants to provide a historical resource for engineering educators trying to uphold the traditional values of design as against those of numerical analysis. But not all historians of technology have sought to distance their subject from mathematics. Alex Keller has given positive weight to the Renaissance mathematical arts within the history of technology; see, for example, his ‘Mathematics, mechanics and the origins of the culture of mechanical invention’, *Minerva*, 23 (1985), 348-61.

14. See, among his other essays on the topic, A.R. Hall, ‘What did the Industrial Revolution in Britain owe to science?’ in N. McKendrick (ed.), *Historical Perspectives. Studies in English Thought and Society in Honour of J.H Plumb* (London, 1974), 129-151.

15. See, for example, Stillman Drake and I.E. Drabkin (eds), *Mechanics in Sixteenth-Century Italy*, (Madison, 1969), P.L. Rose and S. Drake, ‘The pseudo-Aristotelian "Questions of Mechanics" in Renaissance culture’, *Studies in the Renaissance*, 18 (1971), 65-104, and Alex Keller, ‘Mathematicians, mechanics and experimental machines in northern Italy in the sixteenth century’, in Maurice Crosland (ed.), *The Emergence of Science in Western Europe* (London, 1975), 15-34.

16. Horst de la Croix, ‘The literature on fortification in Renaissance Italy’, *Technology and Culture*, 4 (1963), 30-50; Simon Pepper and Nicholas Adams, *Firearms and Fortifications. Military Architecture and Siege Warfare in Sixteenth-Century Siena* (Chicago, 1986).

17. J.V. Field, ‘Perspective and the mathematicians: Alberti to Desargues’, in C. Hay (ed.), *Mathematics from Manuscript to Print 1300-1600* (Oxford, 1988), and J.V. Field and J.J. Gray, *The Geometrical Work of Girard Desargues* (Berlin, 1987), ch. 2.

18. Westman (footnote 6) and Moran (footnote 7). However, court and university were not always separate. Moran notes that in the late 16th century the Marburg court controlled the local university: Bruce T. Moran, ‘Patronage and institutions: courts, universities and academies in Germany; an overview 1550-1750’, in idem (ed.), *Patronage and Institutions: Science, Technology and Medicine at the European Court 1500-1750* (Woodbridge, Suffolk, 1991), 169-84.

19. Rose (footnote 5). For more on Benedetti: *Cultura, Scienze e Tecniche nella Venezia del Cinquecento. Atti del Convegno Internazionale di Studio Giovan Battista Benedetti e il Suo Tempo* (Venice, 1987). On Galileo, see now Mario Biagioli, *Galileo, Courtier: the Practice of Science in the Culture of Absolutism* (Chicago, 1993). I do not suggest that a single location compels identical approaches from its occupants. Domenico Bertoloni Meli notes how one can distinguish differing agendas even within the so-called Urbino school of Commandino, Guidobaldo and their associates and successors; ‘Guidobaldo dal Monte and the Archimedean revival’, *Nuncius*, 7 (1992), 3-34.

20. Steven Shapin, ‘The house of experiment in seventeenth-century England’, *Isis*, 79 (1988), 373-404 and Owen Hannaway, ‘Laboratory design and the aim of science: Andreas Libavius versus Tycho Brahe’, *Isis*, 77 (1986), 585-610.

21. Hilary M. Carey, *Courting Disaster. Astrology at the English Court and University in the Later Middle Ages* (London, 1992).

22. Roy Strong, *The English Renaissance Miniature*, revised ed. (London, 1984), p. 65.

23. For this and succeeding paragraphs, see David Starkey (ed.), *Henry VIII. A European Court in England* (London, 1991).

24. J.D. North, ‘Nicolaus Kratzer - the King’s astronomer’, in E. Hilfstein et al. (eds), *Science and History. Studies in Honour of Edward Rosen*, Studia Copernicana, 16 (1978), 205-34.

25. Starkey (footnote 23), p. 152.

26. E.G.R. Taylor, ‘Jean Rotz and the variation of the compass, 1542’, *Journal of the Institute of Navigation*, 7 (1954), 9-15 and ‘Jean Rotz and the marine chart’, *Journal of the Institute of Navigation*, 7 (1954), 136-143; Helen Wallis (ed.), *The Maps and Text of the ‘Boke of Ydrography’ presented by Jean Rotz to Henry VIII* (Oxford, 1981).

27. Starkey (footnote 23), p. 151.

28. Starkey (footnote 23), pp. 65-6, 71, 135.

29. Starkey (footnote 23), p. 133. For other instruments of the Henrician court, see Mordechai Feingold, *The Mathematicians’ Apprenticeship: Science, Universities and Society in England, 1560-1640* (Cambridge, 1984), p. 196.

30. For the uses of maps in the Henrician court, see Peter Barber, ‘Pageantry, defense, and government: maps at court to 1550’, in David Buisseret (ed.), *Monarchs, Ministers and Maps: the Emergence of Cartography as a Tool of Government in Early Modern Europe* (Chicago, 1992), ch. 2.

31. There are some useful comments on Elizabethan court culture in R. Malcolm Smuts, *Court Culture and the Origins of a Royalist Tradition in Early Stuart England* (Philadelphia, 1987). The analyses of the Elizabethan court in David Loades, *The Tudor Court* (Ottawa, 1987) and David Starkey (ed.), *The English Court: from the Wars of the Roses to the Civil War* (London, 1987) are more narrowly administrative and political.

32. Roy Strong, *Henry, Prince of Wales, and England’s Lost Renaissance* (London, 1986).

33. Nicholas H. Clulee, *John Dee’s Natural Philosophy: Between Science and Religion* (London, 1988), pp. 189-99.

34. Mario Biagioli, ‘The social status of Italian mathematicians, 1450-1600’, *History of Science*, 27 (1989), 41-95, pp. 56-67, Rose (footnote 5), and Bertoloni Meli (footnote 19).

35. Moran (footnote 7).

36. Biagioli (footnote 19).

37. Victor E. Thoren, *The Lord of Uraniborg. A Biography of Tycho Brahe* (Cambridge, 1990).

38. J.M. Fletcher, ‘The faculty of arts’, in James McConica (ed.), *The Collegiate University*, vol. 3 of T.H. Aston (general ed.) *The History of the University of Oxford* (Oxford, 1986), ch. 4.1.

39. For the *calculatores*, see John E. Murdoch and Edith D. Sylla, ‘The science of motion’, in David C. Lindberg (ed.), *Science in the Middle Ages* (Chicago, 1978), ch. 7 and, for more recent literature, Edith D. Sylla, ‘Mathematical physics and imagination in the work of the Oxford Calculators: Roger Swineshead’s *On Natural Motions*’ and John E. Murdoch, ‘Thomas Bradwardine: mathematics and continuity in the fourteenth century’, both in Edward Grant and John E. Murdoch (eds), *Mathematics and its Applications to Science and Natural Philosophy in the Middle Ages. Essays in Honour of Marshall Clagett* (Cambridge, 1987).

40. Charles B. Schmitt, *John Case and Aristotelianism in Renaissance England* (Kingston, 1983), pp. 14-17.

41. J.D. North, *Richard of Wallingford*, 3 vols (Oxford, 1976) and, for some indication of the standards and form of later calendrical astronomy, Sigmund Eisner (ed.), *The Kalendarium of Nicholas of Lynn* (London, 1980).

42. James McConica, ‘The rise of the undergraduate college’, in McConica (footnote 38), ch. 1 and Joan Simon, *Education and Society in Tudor England* (Cambridge, 1966/1979), pp. 245-50.

43. Hugh Kearney, *Scholars and Gentlemen: Universities and Society in Pre-Industrial Britain* (Ithaca, 1970) and Simon (footnote 42), pp. 355-6.

44. P.L. Rose, ‘Erasmians and mathematicians at Cambridge in the early sixteenth century’, *Sixteenth Century Journal* 8, supplement (1977), 47-59. For the textbooks prescribed in 1549, see p. 51. By the end of the century, lecturers typically held the post for only one rather than the full three years; cf. Feingold (footnote 29), pp. 50-2.

45. Rose, ‘Erasmians’ (footnote 44), pp. 57-8 and Feingold (footnote 29), pp. 39-40.

46. On tutors, see Feingold (footnote 29), pp. 54-68.

47. John Case, *Apologia academiarum*, cited by J.W. Binns, ‘Queen Elizabeth I and the universities’, in John Henry and Sarah Hutton (eds), *New Perspectives in Renaissance Thought* (London, 1990), 244-52, p. 251 and, for more on Case’s text, Schmitt (footnote 40), appendix 4.

48. On Melanchthon, the German universities and the mathematical arts, Robert Westman, ‘The Melanchthon circle, Rheticus and the Wittenberg interpretation of the Copernican theory’, *Isis*, 66 (1975), 165-93. For a comparison between Germany and England, J.M. Fletcher, ‘Change and resistance to change: a consideration of the development of the English and German universities during the sixteenth century’, *History of Universities*, 1 (1981), 1-36. For mathematics in the Italian universities, C.B. Schmitt, ‘Science in the Italian universities in the sixteenth and seventeenth centuries’, in Crosland (footnote 15) and Biagioli (footnote 34), with many further references.

49. Feingold (footnote 29), p. 82 and John W. Shirley, *Thomas Harriot: a Biography* (Oxford, 1983), pp. 61-5.

50. On Savile’s mathematics, Westman (footnote 6) and Feingold (footnote 29), pp. 124-30. For recent work on Savile as a classical translator and historian, Malcolm Smuts, ‘Court-centred politics and the uses of Roman historians, c.1590-1630’, in Kevin Sharpe and Peter Lake (eds), *Culture and Politics in Early Stuart England* (London, 1994), 21-43, pp. 25-9.

51. Feingold (footnote 29), pp. 168-70.

52. For the practice of proofreading, C.L. Oastler, *John Day, the Elizabethan Printer*, Oxford Bibliographical Society, Occasional Publications, 10 (Oxford, 1975), pp. 16, 26, 29-30, James P. Hammersmith, ‘Frivolous trifles and weighty tomes: early proofreading at London, Oxford, and Cambridge’, *Studies in Bibliography*, 38 (1985), 236-51, J.W. Binns, *Intellectual Culture in Elizabethan and Jacobean England: the Latin Writings of the Age* (Leeds, 1990), appendix A, and Thoren (footnote 37), p. 185.

53. G.L’E. Turner, ‘Mathematical instrument-making in London in the sixteenth century’, in Sarah Tyacke (ed.), *English Map Making 1500-1650* (London, 1983), 93-106.

54. Instruments and vernacular books sometimes even migrated to the universities: E.S. Leedham-Green, *Books in Cambridge Inventories: Book-Lists from Vice-Chancellor’s Court Probate Inventories in the Tudor and Stuart Periods*, 2 vols (Cambridge, 1986), especially I, pp. 457-9 for Andrew Perne’s collection of maps, globes and instruments (Perne was master of Peterhouse, 1554-1589).

55. Shirley (footnote 49), pp. 73-4. For Humphrey Baker’s single sheet advert, see *STC*, no. 1209.3.

56. For Hood and the mathematical lectureship, Stephen Johnston, ‘Mathematical practitioners and instruments in Elizabethan England’, *Annals of Science*, 48 (1991), 319-44. For Gresham College, F.R. Johnson, ‘Gresham College: precursor of the Royal Society’, *Journal of the History of Ideas*, 1 (1940), 413-38.

57. The earliest surviving edition of Recorde’s *Grounde of Artes* is now thought to date from 1543 (*STC*). Such vernacular mathematical texts were preceded by an earlier Latin publication, Cuthbert Tunstall’s *De Arte Supputandi* (1522). A different version of the vernacular is represented by the Middle English of Chaucer’s treatise on the astrolabe, which appeared in editions of his collected works from 1532 onwards.

58. On Recorde, see F.R. Johnson and S.V. Larkey, ‘Robert Recorde’s mathematical teaching and the anti-Aristotelian movement’, *Huntington Library Bulletin*, 7 (1935), 59-87, Edward Kaplan, ‘Robert Recorde (c.1510-1558): studies in the life and work of a Tudor scientist’ (unpublished Ph.D., New York, 1960) and Geoffrey Howson, *A History of Mathematics Education in England* (Cambridge, 1982), ch. 1.

59. Barber (footnote 30) as well as his ‘Monarchs, ministers and maps, 1550-1625’ in the same volume, ch. 3. Also P.D.A. Harvey, *Maps in Tudor England* (London, 1993) and, for a survey of the variety of medieval maps, idem, *Medieval Maps* (London, 1991).

60. J.A. Giles (ed.), *The Whole Works of Roger Ascham*, 3 vols (London, 1864-5), III, p. 100.

61. For classical and Renaissance perceptions of the dangers of solitude, see Steven Shapin, ‘"The mind is its own place": science and solitude in seventeenth-century England’, *Science in Context*, 4 (1990), 191-218.

62. Lawrence V. Ryan, *Roger Ascham* (Stanford, 1963), pp. 102-7 on Ascham as tutor to Elizabeth.

63. Rose (footnote 44), p. 56.

64. Giles (footnote 60), II, p. 103.

65. For Cicero on the dangers of mathematics, as well as his praise of those who managed to combine moral virtue with deep learning in the mathematical arts, *De officiis*, I, vi, cited in Hannaway (footnote 20), p. 607.

66. For astrology and divination, see Keith Thomas, *Religion and the Decline of Magic* (London, 1971), especially pp. 437-8 for astrology as a popish practice. For the relation of astrology to the mathematical arts, Richard Dunn, ‘The true place of astrology among the mathematical arts of late Tudor England’, *Annals of Science*, 51 (1994), 151-63 and, for more on astrology’s relation to the occult arts, idem, ‘The status of astrology in Elizabethan England, 1558-1603’ (unpublished Ph.D., Cambridge, 1992), ch. 7. Additional material can be found in J. Peter Zetterberg, ‘The mistaking of "the mathematics" for magic in Tudor and early Stuart England’, *Sixteenth Century Journal*, 11 (1980), 83-97, but Zetterberg’s analysis is compromised by his assumption that there is an obviously ‘true’ identity for mathematics, against which ‘mistakes’ can be readily judged.

67. For example, in *Stratioticos* (1579), sig. a1r, Thomas Digges placed mathematics above philosophy, law and medicine in point of certainty.

68. John Blagrave, *The Mathematical Jewel* (London, 1585), sig. ¶2r.

69. In the edition of 1576, these remarks appear at f. 1r-v. The earliest surviving version of Digges’s book is the *Prognostication of Right Good Effect* (1555), though earlier editions were evidently published (see *STC*).

70. Edward Worsop, *A Discoverie of Sundrie Errours* (London, 1582), sigs E4r, F3r.

71. Worsop (footnote 70), sig. C2r.

72. Worsop (footnote 70), ‘An advertisement to the reader’. For the commercial character of the area round St Paul’s, and the layout of shops and stalls, see Peter Blayney, *The Bookshops in Paul’s Cross Churchyard*, Occasional Publications of the Bibliographical Society (London, 1990).

73. Science Museum, inv. 1984-742.

74. On this unusual form of dial, see Denys Vaughan, ‘A very artificial workman: the altitude sundials of Humphrey Cole’, in R.G.W. Anderson, J.A. Bennett, W.F. Ryan (eds), *Making Instruments Count* (Aldershot, 1993), 191-200.

75. For more on the Cole instrument and its context, see Stephen Johnston, ‘The carpenter’s rule: instruments, practitioners and artisans in 16th-century England’, to appear in G. Dragoni, A. McConnell and G.L’E. Turner (eds), *Proceedings of the XIth International Scientific Instrument Symposium, Bologna, September 1991* (Bologna, 1994). The 17th century fate of multi-purpose surveying instruments is discussed in J.A. Bennett, ‘Geometry and surveying in early seventeenth-century England’, *Annals of Science*, 48 (1991), 345-54.

76. E.G.R. Taylor, *Mathematical Practitioners of Tudor and Stuart England* (Cambridge, 1954) and *Mathematical Practitioners of Hanoverian England* (Cambridge, 1966).

77. F.R. Johnson, *Astronomical Thought in Renaissance England* (Baltimore, 1937), E.G.R. Taylor, *Tudor Geography 1485-1583* (London, 1930) and *Late Tudor and Early Stuart Geography 1583-1650* (London, 1934).

78. D.W. Waters, *The Art of Navigation in England in Elizabethan and Early Stuart Times* (London, 1958); A.W. Richeson, *English Land Measuring to 1800: Instruments and Practices* (Cambridge, 1966). Among earlier works on surveying, though not specifically on England, note E.R. Kiely’s *Surveying Instruments, their History and Classroom Use* (New York, 1947).

79. See the prominent use made of Taylor, Johnson and Waters in Christopher Hill, *Intellectual Origins of the English Revolution* (Oxford, 1965), especially ch. 2, ‘London Science and Medicine’.

80. Bennett (footnote 12).

81. J.A. Bennett, *The Divided Circle. A History of Instruments for Astronomy, Navigation and Surveying* (Oxford, 1987).

82. Frances Willmoth, *Sir Jonas Moore. Practical Mathematics and Restoration Science* (Woodbridge, 1993).

83. Note that I am not positing an absolute opposition between practice-as-action and the didactic realm of texts and teaching. Nor am I suggesting that one is more fundamental than the other; both are important aspects of mathematical practice. It simply happens that the figures present at Dover were more concerned with obtaining results than teaching them.

84. For the relative roles of civil and military projects in enhancing the status and specialisation of Italian engineers, see Biagioli (footnote 34), p. 47.

85. Katharine Hodgkin, ‘Thomas Whythorne and the problems of mastery’, *History Workshop*, 29 (1990), 20-41 gives a fascinating account of the Elizabethan meaning of ‘mastership’. Whythorne’s strategies for the legitimation of music and its practitioners closely parallel those being developed by contemporary mathematical practitioners.

86. *The Path-Way to Knowledge* (London, 1551), preface.

87. On English Copernicanism, see Johnson (footnote 77) and, for an account of the philosophical status of astronomy in the 16th century, N. Jardine, *The Birth of History and Philosophy of Science* (Cambridge, 1984), especially chh. 6 and 7.

88. Drake and Drabkin (footnote 15). Dee mentioned Benedetti’s conclusions in the *Mathematicall Praeface* but it was not until Thomas Harriot’s undated and unpublished notes that the topic of falling bodies become an issue for contemporary English mathematics; see Shirley (footnote 49), pp. 263-7.

89. On the Italian controversy, see Biagioli (footnote 34), with further references.

90. Note that the tradition of scholastic philosophy was at a particularly low ebb in the years from about 1525-75; Schmitt (footnote 40), pp. 23-4.

91. Clulee (footnote 33) provides the best available discussion of Dee’s philosophical and occult work.

92. Notably through the work of Frances Yates, whose suggestions were pursued in Peter J. French, *John Dee: the World of an Elizabethan Magus* (London, 1972). The most recent work on Dee places much less emphasis on Dee as magus; William Sherman, *John Dee: the Politics of Reading and Writing in the English Renaissance* (forthcoming, 1994) productively focuses on Dee’s textual practices and his geographical and historical work.

93. Clulee (footnote 33), ch. 6.

94. Cf. Clulee (footnote 33), p. 175.

95. William Bourne, *A Booke called the Treasure for Travellers* (London, 1578), sigs ***2-3 and introduction to book IV.

96. Edgar Zilsel, ‘The sociological roots of science’, *American Journal of Sociology*, 47 (1942), 544-62 and ‘The origins of William Gilbert’s scientific method’, in P.P. Wiener and A. Noland (eds), *Roots of Scientific Thought* (New York, 1957), 219-50.

97. Rupert Hall, ‘The scholar and the craftsman in the Scientific Revolution’, in Marshall Clagett (ed.), *Critical Problems in the History of Science* (Madison, 1959), pp. 5, 16.

98. Bennett (footnote 12), pp. 1-2, 6.

99. Kuhn (footnote 10).