# The identity of the mathematical practitioner in 16th-century England

## Stephen Johnston

Museum of the History of Science, Oxford

Perhaps the key feature of mathematics in Renaissance Europe was diversity: the mathematical arts and sciences were interpreted in radically different ways in the 15th and 16th centuries. Mathematics could be a spiritual discipline, read as a guide to meditation on the divine; alternatively, it could be acquired as a vocational resource by merchants, developing their bookkeeping skills. Mathematics was both a preliminary to natural philosophy in university arts courses and, through studies of fortification, artillery and the ordering of troops, an element of informal aristocratic military education.1 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; the humanist scholar collecting and collating classical manuscripts in order to restore the works of Greek geometers; the university regent master who taught a few books of Euclid before quickly passing on to higher studies such as medicine; and the prince fascinated by the spectacular display of astronomical clocks and ingenious devices.2 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 [page 94:] cubic equation), the astronomer constructing elaborate geometrical models of planetary motion, the 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.

One of these roles was that of the mathematical practitioner. Since being brought to prominence by the work of E.G.R. Taylor, the mathematical practitioner has been a useful (if often ill-defined) character in the writings of historians.3 Taylor identified a didactic, vernacular and usually urban tradition of English practical mathematics, in which instrument makers and ordinary textbook writers had a place alongside better-known names from the history of science. This ‘mathematical practice’, as it was known, was promoted through the medium of the printed book and heavily emphasised the importance of instruments for observation, measurement and calculation, as well as 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.

Many of the characteristic features which Taylor attributed to English mathematical practice were also common to practical mathematics elsewhere in Europe. Indeed, early English work was often derived from prior Italian, German and Iberian sources. Moreover, active links can be found between English mathematicians and their continental contemporaries, with Gerard Mercator being one of the more notable of such contacts. But despite the European scale of the phenomenon of the mathematical practitioner, there remains a rationale for regional and national studies of this role. Although faced in principle with an extraordinary proliferation of mathematical activities, topics and identities, the actual possibilities open to individual mathematicians [page 95:] were strongly shaped by the opportunities, expectations and roles available in their local settings. Local incentives and constraints were vital in encouraging certain varieties of mathematics at the expense of others.

In England, the tradition of mathematical practice developed into the most prominent and public culture of mathematics in the second half of the 16th century. In this paper I want to explore the role of the mathematical practitioner as it was formed and developed in this period. However, mathematical practice was not the only available option for those with mathematical inclinations. Before turning to the work of the mathematical practitioners, I will therefore begin with a comparative survey of the alternative pursuit of mathematics in two specific locations – the court and the universities. These were prime sites for mathematical work in contemporary Europe and, as one might expect, the mathematical arts played a part in both the courtly and academic contexts of 16th-century England. In this respect, there is a measure of continuity with earlier traditions: the learned practice of astrology, for example, was strongly tied to these two locations in 15th-century England.4

What emerges from this review of court and university is that, although each provided distinctively different conditions for the practice of mathematics, they can both be readily allied with the role of the *mathematicus*. I use the term here in distinction to that of the mathematical practitioner. Although there are significant areas of overlap it is at least heuristically useful to differentiate between the vernacular world of the mathematical practitioner and that of the *mathematicus.* Certainly* *for the English context, the image of the *mathematicus *carries connotations either of courtly accomplishments or of Latin learning. By juxtaposing the pursuit of mathematics as a courtly or academic activity with the tradition of mathematical practice, the distinctive and novel features of the mathematical practitioner’s identity can be more sharply delineated. [page 96:]

## Mathematics at court

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’.5 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.6

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 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. [page 97:]

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.7 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. 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’.8 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.

The systematically mathematical character of these examples of Henry’s patronage should not be overestimated: in England the vision of mathematics as an all-embracing 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. [page 98:]

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’.9 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 plans, as a new resource for military and political strategy.10

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. 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.11

Of course, opportunities for patronage under Elizabeth were not entirely lacking. But it is striking that the management of patronage [page 99:] 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.12 Thus when studying the establishment of mathematical practice in later 16th-century England we are in an environment where the court did not represent a pinnacle of mathematical achievement and reward. Under Elizabeth, 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.

## The academic environment

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. 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. But such texts were more likely to be encountered in early 16th-century Italy than in England. Likewise, the sophisticated traditions of medieval astronomy [page 100:] 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.13

But from the beginning of the 16th century the universities underwent major 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.14

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.15

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, beyond the bare list of masters who held the position throughout the century, almost nothing is known of the actual conduct of the lectures, [page 101:] though the Edwardian statutes of 1549 did prescribe a set range of mathematical texts for the reformed B.A. and M.A. syllabus.16

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. 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.17

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 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.18

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 [page 102:] 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.19

However, outwith the bounds of statutory university and college provision, 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.20

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.21 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 [page 103:] 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 personal decisions to seek preferment elsewhere.22

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, with its authors and audience typically meeting within the university’s own precincts and, through the use of Latin, remaining linguistically self-contained.

## Mathematical practitioners and mathematical practice

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 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, [page 104:] 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.23

Though tiny in comparison with the scale of the book trade, London also became the centre for the commercial manufacture of mathematical instruments. 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. 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. And public lectures were first given there in 1588 by Thomas Hood, before the opening of Gresham College a decade later.24

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. Recorde planned a methodical series of textbooks, and [page 105:] issued several in the 1550s, when other authors such as Leonard Digges also began to publish.25

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 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.26

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 and [page 106:] 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’.27

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.28

Now, Ascham was not an idiosyncratic outsider but a central figure in English humanist pedagogy – he had, indeed, tutored the youthful princess Elizabeth. Moreover, his hostility cannot be attributed to ignorance: he had been Cambridge mathematical lecturer in the two academic years from 1539 to 1541. 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 changing Tully’s wisdom with Euclid’s pricks and lines’.29 Drawing on the moral authority of Cicero, Ascham could warn against the excessive study of mathematics.

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, [page 107:] 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.30

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, the specific form of English mathematical practice can be seen as a response to just such criticism. In deliberately seeking to show that their mathematics was worldly, practical and social, the mathematical practitioners were responding to the canons of Protestant civic humanism articulated by Ascham and others. The values attributed to mathematical practice were thus 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 [page 108:] 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 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.31 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.

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 ‘Mathematical Jewel’ (a form of universal astrolabe which he published in 1585), [page 109:] 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’.32

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.’33

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 astrology impudently usurp the names of mathematicians, as popish and superstitious priests, the names of divines’.34 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 [page 110:] 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 ‘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.’35 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. [page 111:]

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. Together they underpinned the identity of the mathematical practitioner.

However, the practitioners were not a homogeneous group. Aside from their different ‘professional’ activities – as surveyors, navigators, authors and teachers – they were socially diverse. The disparity in status between a gentleman mathematician such as Thomas Digges and an artisan such as Robert Norman, the compass-maker and author on magnetism, 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 as well as an intellectual achievement. However, 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. Rather than a route through which mathematics was brought down to the ordinary craftsman, the identity of the mathematical practitioner often elevated the master craftsman further above his subordinates. Whether gentlemen or expert artisans, the mathematical practitioners were ‘masters’, or sought to be regarded as such.36

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 112:] page of 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 mathematical practitioner’s identity in relation to that of mechanicians and artisans was a central and recurrent problem not just during the Elizabethan period but also throughout the 17th century.

What of 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.37 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.

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. Indeed, the Italian comparison can be [page 113:] 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. 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.38

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 dedicated to Mercator, 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 magic. Through these works and his later angelic conversations, Dee is commonly identified as an archetypical Renaissance magus.39 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 his 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 [page 114:] 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.40

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 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. 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’.41 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. [page 115:]

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 presence of gentlemen such as Thomas Digges immediately frustrates so tidy a classification. Distinct from these categories of scholar and craftsman, and also distant from the role of the courtly *mathematicus*,* *the mathematical practitioner could be positioned as a published author, a deviser of instruments or a technical advisor – a figure active in the market place and an intermediary between the mechanical arts and the realm of patrons and statesmen. The dynamism of mathematical practice – its enthusiastic promotion and rapid expansion – was rooted precisely in the novelty and the ambiguities of this identity.

## Notes

1.For examples of these contrasting approaches to mathematics, see *Philip Sanders*, Charles de Bovelles’s treatise on the regular polyhedra (Paris, 1511), in: Annals of Science 41 (1984), p. 513–66, *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, in: Journal of European Economic History 1 (1972–3), p. 418–33, *Natalie Zemon Davis*, Sixteenth–century arithmetics on the business life, in: Journal of the History of Ideas 21 (1960), p. 18–48 and *Stillman Drake*, Galileo Galilei: Operations of the Geometric and Military Compass, 1606, Washington, D.C., 1978, introduction.

2. For examples of these roles, see *Nicholas Adams,* The life and times of Pietro dell’Abaco, a Renaissance estimator from Siena (active 1457–1486), in: Zeitschrift für Kunstgeschichte 48 (1985), p. 384–395, *P.L. Rose*, The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from [page 116:] Petrarch to Galileo, Geneva, 1975, *Robert S. Westman*, The astronomer’s role in the sixteenth century: a preliminary study, in: History of Science 18 (1980), p. 105–47 and several papers by *Bruce T. Moran*: Princes, machines and the valuation of precision in the sixteenth century, in: Sudhoff’s Archiv 61 (1977), p. 209–28, Wilhelm IV of Hesse–Kassel: informal communication and the aristocratic context of discovery, in: T. Nickles (ed.), Scientific Discovery: Case Studies, Dordrecht, 1980 and German prince-practitioners: aspects in the development of courtly science, technology and procedures in the Renaissance, in: Technology and Culture 22 (1981), p. 253–74.

3. *E.G.R. Taylor*, The Mathematical Practitioners of Tudor and Stuart England, Cambridge, 1954.

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

5. *Roy Strong*, The English Renaissance Miniature, revised ed., London, 1984, p. 65.

6. For this paragraph and the succeeding details on the court of Henry VIII, see *David Starkey* (ed.), Henry VIII. A European Court in England, London, 1991.

7. *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, p. 205–34.

8. *Helen Wallis* (ed.), The Maps and Text of the ‘Boke of Ydrography’ presented by Jean Rotz to Henry VIII, Oxford, 1981.

9. *Starkey,* 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.

10. 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, Ill., 1992, ch. 2.

11. On Elizabethan court culture, see *R. Malcolm Smuts*, Court Culture and the Origins of a Royalist Tradition in Early Stuart England, Philadelphia, Penn., 1987. For Prince Henry: *Roy Strong*, Henry, Prince of Wales, and England’s Lost Renaissance, London, 1986.

12. *Nicholas H. Clulee*, John Dee’s Natural Philosophy: Between Science and Religion, London, 1988, p. 189ff.

13. *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, *John E. Murdoch and Edith D. Sylla*, The science of motion, in: David C. Lindberg (ed.), Science in the Middle Ages, Chicago, Ill., 1978, ch. 7, *Charles B. Schmitt*, John Case and Aristotelianism in Renaissance England, Kingston, 1983, p. 14ff., *J.D. North*, [page 117:] Richard of Wallingford, 3 vols, Oxford, 1976 and, on later calendrical astronomy, *Sigmund Eisner* (ed.), The Kalendarium of Nicholas of Lynn, London, 1980.

14. *James McConica*, The rise of the undergraduate college, in: McConica, ch. 1 and *Joan Simon*, Education and Society in Tudor England, Cambridge, 1979, p. 245ff.

15. *Hugh Kearney*, Scholars and Gentlemen: Universities and Society in Pre-Industrial Britain Ithaca, NY, 1970 and *Simon,* p. 355f.

16. *P.L. Rose*, Erasmians and mathematicians at Cambridge in the early sixteenth century, in: Sixteenth Century Journal 8, supplement (1977), p. 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*, p. 50ff.

17. *Rose*, Erasmians, p. 57f. and *Feingold*, p. 39f and 54ff..

18. *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, p. 251 and, for more on Case’s text, *Schmitt*,* *appendix 4.

19. On Melanchthon, the German universities and the mathematical arts, *Robert Westman*, The Melanchthon circle, Rheticus and the Wittenberg interpretation of the Copernican theory, in: Isis 66 (1975), p. 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, in: History of Universities 1 (1981), 1–36.

20. *Feingold*, p. 82 and *John W. Shirley*, Thomas Harriot: a Biography, Oxford, 1983, p. 61ff.

21. On Savile’s mathematics, *Feingold*, p. 124ff. 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, p. 25ff.

22. *Feingold*, p. 168ff.

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

24. *G.L’E. Turner*, Mathematical instrument-making in London in the sixteenth century, in: Sarah Tyacke (ed.), English Map Making 1500–1650, London, 1983, p. 93–106. Instruments and vernacular books sometimes even migrated to the universities: [page 118:] *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, p. 457ff. for Andrew Perne’s collection of maps, globes and instruments (Perne was master of Peterhouse, 1554–1589). On London teachers, *Shirley*, p. 73f.; on Hood and the mathematical lectureship, *Stephen Johnston*, Mathematical practitioners and instruments in Elizabethan England, in: Annals of Science 48 (1991), p. 319–44; on Gresham College, *F.R. Johnson*, Gresham College: precursor of the Royal Society, in: Journal of the History of Ideas 1 (1940), p. 413–38.

25. The earliest surviving edition of Recorde’s *Grounde of Artes* is now thought to date from 1543. 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. On Recorde, see *F.R. Johnson* and* S.V. Larkey*, Robert Recorde’s mathematical teaching and the anti-Aristotelian movement, in: Huntington Library Bulletin 7 (1935), p. 59–87 and *Geoffrey Howson*, A History of Mathematics Education in England, Cambridge, 1982, ch. 1.

26. See *Barber*, and also 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.

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

28. 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, in: Science in Context 4 (1990), p. 191–218.

29. *Giles*, II, p. 103. On Ascham as tutor, *Lawrence V. Ryan*, Roger Ascham, Stanford, Calif., 1963, p. 102ff and as mathematical lecturer, *Rose*, Erasmians, p. 56.

30. For astrology and divination, see *Keith Thomas*, Religion and the Decline of Magic, London, 1971, especially p. 437f. for astrology as a popish practice. See also *Richard Dunn*, The true place of astrology among the mathematical arts of late Tudor England, in: Annals of Science 51 (1994), p. 151–63.

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

32. *John Blagrave*, The Mathematical Jewel, London, 1585, sig. ¶2r.

33. In the edition of 1576, these remarks appear at f. 1r-v. The earliest surviving version of Digges’s book is the ‘Prognostication [page 119:] of Right Good Effect’ (1555), though earlier editions were evidently published.

34. *Edward Worsop*, A Discoverie of Sundrie Errours, London, 1582, sigs E4r, F3r.

35. *Worsop*,* *sig. C2r.

36. On the Elizabethan meaning of ‘mastership’, *Katharine Hodgkin*, Thomas Whythorne and the problems of mastery, in: History Workshop 29 (1990), p. 20–41. Whythorne’s strategies for the legitimation of music and its practitioners closely parallel those being developed by contemporary mathematical practitioners.

37. The Path-Way to Knowledge, London, 1551, preface.

38. For Benedetti, see *Stillman Drake and I.E. Drabkin*, Mechanics in Sixteenth-Century Italy, Madison, Wisc., 1969. John 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*, p. 263ff. For the Italian controversy on the certainty of mathematics, see *Mario Biagioli*, The social status of Italian mathematicians, 1450-1600, in: History of Science 27 (1989), p. 41-95. Note that the tradition of scholastic philosophy was at a particularly low ebb in England in the years from about 1525–75; *Schmitt*,* *p. 23f.

39. *Clulee* provides the best available discussion of Dee’s philosophical and occult work. For Dee as magus, *Peter J. French*, John Dee: the World of an Elizabethan Magus, London, 1972, following the earlier work of Frances Yates.

40. *Clulee*, ch. 6.

41. *William Bourne*, A Booke called the Treasure for Travellers, London, 1578, sigs ***2–3 and introduction to book IV.

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