# Like father, like son? John Dee, Thomas Digges and the identity of the mathematician

## Stephen Johnston

Museum of the History of Science

University of Oxford

In early 1573 two English mathematical books were being prepared for the press. Though produced by different printers they were issued as a pair and today are usually found bound together. John Dee’s *Parallaticae commentationis praxeosque nucleus quidam* and Thomas Digges’s *Alae seu scalae mathematicae* were both prompted by the new star of 1572. The material fact of their joint publication neatly echoes the sentiments of familiarity expressed by the two authors. Digges supplied a preface to Dee’s work, explaining the extent to which the two texts had been composed independently, while also praising Dee’s learning and the benefits of their collaboration and discussion. Both Dee and Digges further specified their relationship in the prefaces to their own works. The bond between them was avowedly close, indeed paternal: for Dee, Digges was ‘my most worthy mathematical heir’, while Digges repeatedly referred to Dee as a ‘revered second mathematical father’ and acknowledged the pleasure of their intellectual intimacy.1

These comments have often been noted and, in light of Thomas Digges’s Copernicanism, occasionally been incorporated in attempts to establish Dee’s views on heliocentric cosmology.2 Nevertheless, Digges has not figured prominently in studies of Dee - no doubt because the scarcity of additional evidence has seemed to preclude any extended analysis of their association. Yet the relationship was more than simply a passing alliance inflamed by the excitement of the new star. In his all too brief comments in the preface and proemium to *Alae*, Digges indicated that the connection with Dee stretched back much further. Thomas recorded that his mathematical education had been begun by his father Leonard, himself a mathematician in his own right, and the author of two popular vernacular texts published in the 1550s. But according to Thomas, Leonard had only been able to plant certain seeds of elementary mathematical learning in his son, and after his death it was left to Dee to cultivate and supplement these with further instruction.3

The mutually acknowledged paternal relationship between John Dee and Thomas Digges therefore gives us a remarkable window onto Dee’s work and significance. Digges certainly provides an opportunity to assess Dee’s role in forming the next generation of mathematicians. I argue that Dee offered the youthful Digges not only specific mathematical instruction but that he also supported his pupil’s vision of the character and value of mathematics. In particular, I suggest that Dee’s prior investigations underlie the studies embodied in Digges’s first mathematical publication, in 1571. Moreover, Dee’s early career in the 1550s as a mathematical client in noble households provided an exemplar for emulation as Digges first fashioned his own role in the early 1570s.

As well as illuminating the issue of Dee’s contemporary influence, Digges can also be used to examine Dee’s own mathematical values and commitments. Over the course of the 1570s and 1580s, Digges reworked the terms of his mathematical identity, shifting away from a commitment to advanced and novel topics and prioritising instead the active service of prince and commonwealth. However, rather than representing a break with the pattern of Dee’s career, I suggest that Digges’s civic turn helps us to understand the changing character of Dee’s own role as he advocated and practised a vernacular ethic of mathematical service in the 1570s.

While seeking to emphasise the importance of this series of connections between the careers of Dee and Digges, I do not conclude that they pursued identical ambitions. Digges never followed Dee’s broadest conceptions of the terrain of mathematics and its relationship to other areas of learning. I examine the significance of this difference through the lens of Copernicanism and argue that the different responses of master and pupil reflect a fundamental divergence in their respective conceptions of the identity of the mathematician.

Mathematics has long been crucial to evaluations of Dee’s work. Since E. G. R. Taylor’s *Tudor Geography* of 1930, Dee’s conception and practice of the mathematical arts have been a touchstone for those seeking to rescue his reputation, whether in order to proclaim his significance within Elizabethan culture or to instate him in the pantheon of Scientific Revolution. Conversely, Dee has also been enrolled in the arguments of those sceptical of the positive impact of occult philosophies in the late Renaissance. Querying Dee’s attainments as a mathematician has provided one means of undermining claims for Neoplatonism, Hermeticism and related magical traditions. At the extremes, these approaches become undiscriminating apology on the one hand and dismissive anachronism on the other, in either case blurring the possibility of a critical evaluation of Dee himself. By using Digges as a constant point of comparison, we can establish a perspective on Dee which is distanced and yet close to contemporary categories, in which we seek to recover the integrity of Dee’s own enterprise while retaining a sensitivity to its differences and distinctions.

The obvious place to begin is Dee’s *Mathematicall Praeface* to Euclid of 1570. Now probably his most frequently cited work, this text integrates a philosophical account of mathematics with a richly ramified classification of the various mathematical arts and sciences, elaborated and combined to create a mathematical manifesto whose claims are both disciplinary and yet personal to Dee. But despite its seeming timeliness - Digges’s first publication is dated from the following year - I do not want to take this apparently obvious route.

Although Digges recorded Dee’s mathematical instruction in print he gave no details or dates. Julian Roberts and Andrew Watson have recorded a small but vital fragment of independent evidence which suggests that Dee became Thomas’s mathematical master soon after Leonard’s death, which most likely occurred in 1559 when Thomas was only about 13 years old. Dee had signed his copy of the 1544 Basel edition of Archimedes’ *Opera* on 1 January 1550 but, in addition to Dee’s signature, the title page also carries the revealing note ‘Thomas Diggius 1559’.4 The striking coincidence of dates hints strongly that Dee assumed the mantle of tutor immediately after Leonard’s death. Certainly, the language of fatherhood which both Dee and Thomas Digges used in 1573 would be most readily explicable if there was little interruption in paternal role between Leonard Digges and Dee. Given this probability that Thomas had first studied with Dee at the end of the 1550s, we therefore cannot be sure that Dee’s *Praeface* to the 1570 Euclid is the most appropriate or relevant text for understanding Digges’s early work. Rather than Dee, Digges himself can serve more securely as our starting point.

Thomas Digges (c.1546 - 1595) is most familiar within the history of science as the first advocate of Copernicanism in England.5 However he appears in many more narratives than that of Renaissance astronomy and cosmology. He is to be found in accounts of navigation, ballistics, surveying and harbour engineering, as well as military strategy and administration, and parliamentary politics.6 In many of these accounts he is portrayed as a key figure in the development of the tradition of mathematical practice in England.

Digges’s first publication was *Pantometria* (1571). At first sight this text appears to link him more closely with the outlook of his natural father rather than his ‘second mathematical father’. *Pantometria* was actually written by Leonard Digges as a ‘geometrical practise’ divided into three books, dealing respectively with the reckoning of heights and distances, areas, and volumes. This treatment of instrumental and computational techniques for surveying and mensuration was justified in terms of civic and military utility as well as personal pleasure. In securing the posthumous publication of his father’s most important extant work, Thomas was undoubtedly exercising the duties of filial responsibility. But he did more than rescue his father’s practical geometry from oblivion. At the end of *Pantometria*, Thomas appended a vernacular text of his own, a *Mathematicall Discourse of Geometricall Solids*.

Digges’s *Mathematicall Discourse* provides us with the best witness to his earliest mathematical commitments and values. Although his editorial work saved Leonard’s *Pantometria* from obscurity, Thomas’s own contribution to the volume has been entirely forgotten. Yet it is a remarkable text, with a range and ambition quite unlike any other English mathematical work published in the 16th century. The *Mathematicall Discourse* proclaims the value of advanced mathematical study not just in the realm of lofty rhetoric but through the disciplined endeavour of elaborating hundreds of new theorems. There is no evidence that Leonard Digges had grappled with the kind of mathematical material that here engaged his son; we should look instead to John Dee for the origins and motivation of Thomas Digges’s efforts.

The *Mathematicall Discourse* is primarily concerned with the properties, dimensions, and interrelations of the five regular (Platonic) solids. Its text gives several hundred theorems dealing with such topics as the mutual inscription and circumscription of these solids. The final section of the text investigates similar questions but does so by studying five ‘transformed’ bodies - semi-regular Archimedean solids generated by the metamorphosis of each of the five Platonic solids. The *Mathematicall Discourse* covers its subject in just over one hundred pages, but its brevity is deceptive. The amount of labour invested in its preparation is disguised by Digges’s decision to omit proofs of his mass of theorems for the sake of brevity.7

John Dee was one of the few people - if not indeed the only person - in Elizabethan England who could have helped to set the agenda of this work and inform its detailed choices. Although Dee is nowhere named or referred to (Digges cites only Euclid), there are a number of significant points of overlap with his mathematical interests and style. Firstly, there is Digges’s principal concern with solid geometry; this was the area of Euclid’s *Elements* on which Dee focused his published annotations.8 Digges does however begin with some preparatory material on the plane geometry of polygons and circles, and their mutual inscription and superscription. Dee had already taken up this topic, as well as having given specific attention to the solid geometry of inscribing the Platonic solids in a sphere.9

Although both Dee and Digges tackled and sought to extend the Euclidean corpus, neither was bound by the demonstrative form of the *Elements*. Dee was often interested in finding ‘useful’ mechanical techniques rather than rigorous proofs.10 Even in his more formal mathematical work he rarely sought to prove geometrical properties; rather, as Marshall Clagett has characterised Dee’s work on conics, he typically propounded his propositions as problems in which a given magnitude is the starting point from which another magnitude is sought.11 Such numerically formulated ‘data problems’ also make up the staple content of Digges’s *Mathematical Discourse*. Moreover, this stylistic affinity is complemented by a parallel concern with identifying and classifying irrational magnitudes in the terms of Euclid’s *Elements* book X.12

These detailed connections strongly suggest the extent to which Digges’s early mathematical endeavours were rooted in Dee’s prior concerns, and probably derive from problems which had occupied Dee at the end of the 1550s. Only through the support of a figure such as Dee could Digges have believed that there was in England an audience for his studies of polyhedra, which were expressly intended for ‘the satisfaction also of such as delighting in matters only new, rare and difficult, seek to reach above the common sort’.13

If the content and character of Digges’s *Mathematicall Discourse* indicates the significance of his formative period with Dee at the end of the 1550s, so does a consideration of his early role as a mathematician. Digges differed from Dee in both birth and upbringing: he appears not to have attended university and, rather than a scholar, was a gentleman who inherited substantial holdings of land and property.14 Though he had no need to seek preferment to a living or court office, his relationships with patrons in the early 1570s nevertheless echo the noble and courtly service of Dee’s early career.

Dee acted as tutor and consultant to a succession of patrons in the late 1540s and 1550s, with mathematics as his key vehicle for credit. While at Louvain, Dee tutored Sir William Pickering in arithmetic and mathematical instruments. Back in England, he successfully dedicated texts of 1550 and 1551 on the celestial globe and on the distances and magnitudes of planets and stars to Edward VI. When Dee entered household service with William Herbert, the Earl of Pembroke, in February 1552 it was presumably as a *mathematicus*. Certainly, the basis of Dee’s subsequent service with the Northumberland family was mathematical. Dee gave advice on the voyage to Cathay of 1553, an enterprise in which the Duke of Northumberland was heavily involved. Dee also wrote vernacular tracts on mathematical topics for the Duchess, one on ‘The Philosophicall and Poeticall Original occasions, of the Configurations, and names of the heavenly Asterismes’ and the other on ‘The true cause, and account (not vulgar) of Fluds and Ebbs’, a title whose emphasis presages Digges’s early concern to reach beyond the common sort.15 These poetic and philosophical considerations were balanced by the tuition in military mathematics offered to Northumberland’s son John, Duke of Warwick and which Dee would subsequently mention in the *Mathematicall Praeface*.16

Digges’s principal patron of the early 1570s was William Cecil, elevated to the title of Lord Burghley in 1571 and created Lord Treasurer in 1572. Burghley received the dedication of *Alae seu scalae mathematicae* and he had also privately solicited advice from Digges about the new star. But while Digges’s published work dealt with the mathematical determination of place, distance, and magnitude, Burghley’s concern was with the astrological meaning of the exceptional event.17 In 1574 Digges designed a polyhedral garden sundial for Burghley and also presented him with a manuscript text to accompany his ‘Frame Astronomical’, a celestial ceiling with mechanical sun installed in Burghley’s newly-built house of Theobalds.18 Digges’s lost treatise included tables to determine the positions of stars in relation to the horizon, meridian, sun, and moon, ‘whereupon sundry conclusions both pleasant for variety of knowledge and necessary for common use are grounded. Whereof I have in 50 conclusions digested the greater part, with their Histories Poetical and Judgements Astronomical.’ This work was clearly similar in genre and style to Dee’s 1553 text on the constellations for the Duchess of Northumberland.

A generation apart, both Dee and Digges studied and published on deliberately elevated and novel mathematical matters, far exceeding the reach of ‘the common sort’. But Digges recapitulated broader elements of Dee’s early mathematical role. Just as Dee served the Northumberland family and other patrons in the 1550s, so Digges provided mathematical services for Lord Burghley in the early 1570s. Noble service revolved around advice on the patron’s interests, the provision of texts and ingenious devices, and probably some household tutoring. Dee thus seems to stand behind not only the topic and detailed form of Digges’s earliest mathematical work but also Digges’s earliest fashioning of his identity as a mathematician.

Dee may also have prompted Digges to investigate matters beyond the purely mathematical. For example, an alchemical manuscript passed through Digges’s hands in which he wrote out Walter Haddon’s poem in praise of Thomas Norton and digested or copied alchemical schemata and classifications.19 Digges displayed no other sign of interest in alchemy and this otherwise puzzling evidence may be a trace of his close relationship with Dee.

When presenting the *Mathematicall Discourse* in 1571 Digges clearly expressed the values which sustained his investigation of polyhedra. While the edition of his father’s *Pantometria* was portrayed as a work embodying the practical virtues of utility, the defence of the *Mathematicall Discourse* was rhetorically constructed around intellectual elevation. Digges feigned to ignore those who might castigate his advanced study of polyhedra as ‘a fond toy, a mere curious trifle, serving to no use or commodity’. Unless a detractor genuinely valued the study of ‘hard and difficult’ matters, persuasion would be useless. Digges rounded on potential critics as ‘two-footed moles and toads whom destiny and nature hath ordained to crawl within the earth, and suck upon the muck’; such men ‘may not possibly by any vehement exhortation be reduced or moved to taste or savour any whit of virtue, science, or any such celestial influence’.20 Digges’s robust language enforced a stark division of men into either virtuous followers of Euclid, Archimedes, and Apollonius or ignorant acolytes of Epicurus and Midas, content with the realm of lucre and mere worldly pleasure. The *Mathematicall Discourse* embodied mathematics not as a useful or vocational pursuit adapted to military or civic ends but as the work of a gentleman who primarily prized intellectual nobility.

Yet the terms of this seemingly forceful apology for advanced mathematics were not to be sustained by Digges. In his later career he substantially redefined his identity and public persona as a mathematician. This self-conscious shift is most vividly displayed in *Stratioticos* (1579), a text on military mathematics. Digges there offered some autobiographical reflections on his work of the early 1570s, and confessed that ‘the strange variety of inventions in the more subtle part of these mathematical demonstrations did breed in me for a time a singular delectation’. However, with maturer judgement, he had turned from subtlety and delight to practicality and utility. Digges stated that he had latterly ‘spent many of my years in reducing the Sciences Mathematical from Demonstrative Contemplations to Experimental Actions, for the Service of my Prince and Country’.21

This advertisement of reformed commitments was no mere rhetorical flourish; Digges’s subsequent activities show him immersed in the *vita activa* - as a so-called ‘man of business’ in the House of Commons, as an engineering advisor, and as an administrator in the Earl of Leicester’s 1585 expeditionary force to the Netherlands. Yet these civic and military commitments did not represent an abandonment of mathematics by Digges: in the early 1590s he was still advocating the study of ballistics and artillery as a high mathematical art with immediate relevance to the nation’s security.22

Was this a turn away from the example of Dee? Certainly, Dee did not actively pursue the particular forms of military and technical service in which Digges distinguished himself. But there are nevertheless significant resonances between the new values of mathematical practice espoused by Digges and those of his erstwhile mentor. Digges’s efforts to articulate an identity more as mathematical practitioner than *mathematicus* direct us towards the civic turn in Dee’s own career. Rather than a shift away from Dee, Digges’s later work can be interpreted as a move in the same direction, for the 1570s were the crucial decade for Dee’s promotion of the civic values of mathematics.

During this decade, Dee’s published claims for the value of mathematics were expanded to embrace a new rhetoric beyond that of its philosophical significance and demonstrative certainty. In both the *Mathematicall Praeface* (1570) and the *General and Rare Memorials* (1577) Dee presented himself in vernacular form as a benefactor of the commonwealth through the medium of useful mathematics.

Dee’s two main publications before 1570 were the *Propaedeumata aphoristica* (1558 and 1568) and the *Monas hieroglyphica* (1564); both were Latin texts dealing with relatively recondite topics and expressed in often obscure aphoristic form. Dee had of course been involved in a range of practical activities prior to the 1570s, his navigational consultations of the 1550s being perhaps the most prominent instance. But these were private transactions and it was only in the 1570s that Dee articulated such activity in a self-conscious effort to shape his public image. Much of this self-fashioning was conducted through his personal apologetics, where Dee sought to defend himself firstly from the charge of conjuring and latterly from the accusation that he was unwilling to share the results of his labours with his countrymen.23

Though his name was removed from subsequent editions, the branding of Dee as a conjuror in Foxe’s *Book of Martyrs* permanently marked him as a suspect figure.24 Moreover, his protestations of legitimacy and innocence are not merely symptoms of oversensitivity or paranoia; it is possible to identify a surprising number of individuals whom Dee considered had slandered or falsely accused him.25 Dee had a well-rehearsed defence against such adversaries for, prior to his own ‘Digression Apologeticall’ in the *Mathematicall Praeface*, he had composed an *apologia* for Roger Bacon in the 1550s, in which he defended Bacon from the vulgar sort who believed that he had acted with the help of demons.26 Dee himself sought to shake off the charge of conjuror by asserting that he proceeded only by natural and lawful means. The lengthy recitation of the mathematical arts which occupies so much of the *Mathematicall Praeface* bolsters this claim by displaying the wide-ranging powers and effectiveness of mathematics. Yet though the manifold benefits of practical mathematics served the commonwealth, Dee’s role was to furnish the ‘groundplat’ for the work of others rather than to personally devote himself to that end. Rather than duty to the commonwealth, Dee assigned himself prime responsibility to God and the attainment of divine wisdom. His self defence in 1570 therefore declared his Christian piety and ardent desire for truth as a means of rebutting the slanders which had impugned his good name.27

By the time Dee published the *General and Rare Memorials* in 1577 a second charge also required his urgent attention, namely that he had deliberately withheld material from his countrymen. Dee had genuine work to do here to clear his reputation. Although he had listed the names of various of his works in the *Propaedeumata aphoristica*, 20 years later none of these had yet been published.28 Moreover, Dee had openly doubted the wisdom of spreading his ideas too widely. The second edition of the *Propaedeumata aphoristica* cautioned the reader against letting the work fall into the wrong hands: ‘you must not reveal [it] openly to unworthy and profane persons … lest, to your shame and mine, it should be turned to great harm’. Dee expressed similarly wary sentiments in his letter to the printer Sylvius at the beginning of the *Monas hieroglyphica*.29

Dee’s supposed reticence was not just the object of anonymous carping but of specific accusations and incitements. In the *General and Rare Memorials* Dee cited the case of an unnamed scholar who had apparently agitated for Dee’s banishment on the grounds that ‘to no Man of this Realm, he did at any tyme, or yet doth, or will, communicate any part, of his learned Talent, by word or writing: But is wholy addicted, to his priuate commodity only avancing, by his own Studies and practises very secret’.30 Although he had subsequently secured a recantation from his accuser, Dee used the opportunity of publication to proclaim his status as a dutiful citizen.

Dee now made it clear that he served more than only a divine calling to knowledge, when he referred to his ‘faythfull enterprises: vndertaken chiefly, for the Advancement of the wonderfull Veritie Philosophicall: And also, for the State Publik of this Brytish Monarchie, to become florishing, in Honor, Wealth, and Strength’.31 Indeed, if it were not for the parenthetical restriction of his comments to worldly matters, Dee would seem to be claiming that all endeavours should be subordinate to the needs of the commonwealth: ‘All true Subjects, their Chief Intent, and principall purpose, (in all worldly their affayres, Artes, Sciences, and Studyes, &c.) ought to be, the procuring, furdering, mainteyning and encreasing of the weal and Commodity Publik, so much as in them lyeth, and as, they decently and dutifully may’.32 This theme was elaborated and reiterated, as Dee sought to reinterpret all of his prior labours under the heading of public service. Dee considered that he should ‘receyve great publick thanks, comfort, and ayde of the whole Brytish state, to the honour, welfare, and preservation wherof (next unto his duty doing unto God) he hath directed all the course of his manifold studies, great travailes, and incredible costs’.33 Whatever the plausibility of this highly charged retrospective verdict, Dee’s memorandum on the navy and his proposed volumes on navigation were meant as further proof of his earnest service.34

Dee’s civic turn may have been motivated by highly personal circumstances, but the resulting texts expressed not merely a personal *apologia* but a general programme for the development of the mathematical arts. Moreover, Dee’s commitment to active service was not restricted to the realm of print. It was during the later 1570s and early 1580s that Dee achieved his greatest influence and standing in and around the court.35 He was frequently consulted for advice on political, geographical and historical matters and, whether supplying navigational instruction or mustering historical and cartographic sources to support Elizabeth’s legal claims to foreign territories, Dee presented himself as a faithful servant of the commonwealth.

There is therefore a strong parallel between the mathematical careers and shifting identities of Dee and Digges through the 1570s and into the early 1580s. Yet their paths were to diverge: while Digges fulfilled military and civil duties within the Elizabethan state and journeyed to the Netherlands as part of a military intervention in support of Protestantism, Dee secured access to the spiritual realm through the agency of Edward Kelley, leaving England in 1583 with Albrecht Laski. Dee’s departure has often been interpreted as the beginning of a sad decline, a retreat into mysticism and delusion. Yet there can be little doubt that Dee himself believed that through these spiritual conferences he was able to achieve the most potent and universal forms of action and intelligence. The angelic conversations, with their political and irenic dimensions, were surely for Dee not only a culmination and transformation of his philosophical concerns, but a prolongation and deepening of the civic and worldly activism which he had developed in the 1570s. However, Dee’s horizons expanded beyond those of service only to the British commonwealth. For Dee, the existence of spirits was not only literally interpreted and routinely granted but access to spiritual intelligences was philosophically justified within a metaphysics linking human nature and the supercelestial realm. His conversations with angels were a means to achieve closer contact with the God who provided the basis and authority for all mortals, including temporal rulers.36

Yet however we might interpret the angelic conversations - and it is striking that Dee refers to them not just as dialogues but as actions - Digges provides a contemporary vantage point from which Dee’s distinctive choices are thrown into relief. While mathematics allows us to identify the substantial areas of common ground between Dee and Digges, I suggest that it can also provide a window onto the differences that are most vividly highlighted by their divergent paths in the mid-1580s. The case of astronomy illustrates the points where their conceptions of the proper terrain and status of mathematics conflicted rather than converged.

Mathematical astronomers of the 16th century adopted a variety of strategies to minimise and defuse potential conflict with natural philosophers. While the hybrid Ptolemaic-Aristotelian scheme of the medieval tradition of *Theorica planetarum* remained current, mathematical astronomers increasingly resorted to what Nicholas Jardine has termed the ‘pragmatic compromise’. This compromise gave the mathematician the liberty to use whatever mathematical means were deemed necessary to ‘save the phenomena’, in the classic phrase, while questions of the physical reality and character of the heavens fell under the gaze of the philosopher.37 Dee presents an interesting case for the operation of these interrelations because he had both mathematical *and* philosophical ambitions. Yet he did distinguish between mathematical astronomy and a more philosophically based account of the cosmos, and he placed them on distinct epistemic levels. In the *Mathematicall Praeface*, Dee first charged astronomy with determining the sizes and distances of the earth, sun, moon, and fixed stars. As is clear from the *Propaedeumata aphoristica* this information was of primary use in Dee’s vision of a reformed astrology based on perspective.38 However, Dee used biblical authority to determine the principal remit of the astronomical art: God ‘made the Sonne, Mone, and Sterres, to be to us, for Signes, and knowledge of Seasons, and for Distinctions of Dayes, and yeares’.39 Although Dee enigmatically asks his readers to weigh the significance of the word ‘signs’, it is clear that a principal duty of the astronomer is to the calendar; he continues by saying that only through diligent observation and calculation of the celestial motions can there be

the distinct Course of Times, dayes, yeares, and Ages: aswell for Consideration of Sacred Prophesies, accomplished in due time, foretold: as for high Mysticall Solemnities holding: And for all other humaine affaires, Conditions, and covenantes, upon certaine time, betweene man and man: with many other great uses.40

Thus, although the title ‘John Dee on Astronomy’ has been given to the modern edition and translation of the *Propaedeumata aphoristica*, Dee’s major work on astronomy is actually his treatise on calendar reform of 1583 rather than his optically-based theory of astrological influence.

That the mathematical sciences could contribute to natural philosophy is evident from the *Propaedeumata aphoristica*. But, for Dee, the mathematical astronomer did not have the sole or even primary authority to pronounce philosophically on the heavens. Dee’s cosmology was based on much more than just mathematical astronomy. Astrology and, indeed, alchemy were equally at stake in questions concerning the cosmos. Alchemy as ‘inferior astronomy’ presupposed not just an identification of the metals with the planets, but also the conventional order of the planets. Moreover, as Clulee has made particularly clear, in the *Monas hieroglyphica* Dee went well beyond such standard doctrines and correspondences, not only integrating astronomy with disciplines such as alchemy but subsuming it within a wider programme which sought deeper and more primordial truths about the world and its God-given structure.41 When juxtaposed with his programme of hieroglyphic writing and ‘real kabbalah’, the specifically *mathematical* art of astronomy was for Dee a limited and partial enterprise in its determination of celestial dimensions, positions and motions.

The significance of Dee’s conception of the limited status of mathematical astronomy can be highlighted by contrast with Digges’s vision, and is particularly evident in their respective responses to Copernicus. Both Dee and Digges were fulsome in their praise for his work, but Dee lauded Copernicus as a restorer of mathematical astronomy without discussing the physical reality of the heliocentric hypothesis, whereas Digges adopted the Copernican theory of the planetary order on mathematical grounds.

The topic of Dee and Copernicanism has been repeatedly examined, since it has provided a test case for those who wish to argue either that Dee was one of those working towards modern science or, conversely, that his mystical interests precluded any significant affiliation with progressive developments. The majority of recent studies have denied Dee’s belief in the heliocentric theory and used this conclusion to undermine the Yates thesis that Hermeticism contributed to the adoption of Copernicus’s world-system.42 Here I want to move beyond the terms of this debate –sufficiently discussed by now – and to look instead at what Dee’s astronomical work tells us about the identity of the mathematician and the relation of mathematics to philosophy and other areas of learning.

Dee saw Copernicus as the most recent in a long line of major mathematical astronomers, restoring a science whose predictions and parameters were manifestly inaccurate. Copernicus’s work was valuable because it provided the basis for new and improved tables which better predicted celestial motions. Dee was unstinting in his praise of Copernicus in this role. In his preface to John Feild’s ephemeris of 1556 Dee referred to Copernicus’s dazzling brilliance, his divine studies and his Herculean labours in restoring the celestial discipline.43 Likewise, Copernicus figures prominently in Dee’s treatise on calendar reform, where Dee places him as the most recent and notable astronomer in his historical review aimed at establishing values for the most slowly changing of astronomical parameters.44 Both these texts fall within the domain of mathematical astronomy and Dee makes it clear that this terrain is not the appropriate one on which to consider the reality of Copernicus’s proposals. In the preface to Feild’s tables, Dee comments simply that ‘this is not now the place to discuss [Copernicus’s] hypotheses’. The calendar proposal refers to the ‘newe paradoxall Hypotheses’ of Copernicus and Dee says that his account depends ‘chiefly upon the said Copernicus his Calculation, and Phaenomenies: excepting his Hypotheses Theoricall; not here to be brought in question’.45

It is not clear to what place Dee thought a consideration of Copernicus’s physical and philosophical claims could be deferred. In the *Monas hieroglyphica*, Dee did discuss celestial order beyond the delimited domain of mathematical astronomy, but there he sought to transcend the detailed concerns of astronomy. In that text, rather than observation as the prime means of determining the underlying arrangement of the heavens, Dee was tempted by the prospect of deriving superior truth from his reconstitution of primordial symbols.

Will not the astronomer be very sorry for the cold he suffered under the open sky, for [all his] vigils and labours, when here, with no discomfort to be suffered from the air, he may most exactly observe with his eyes the orbits of the heavenly bodies under [his own] roof, with windows and doors shut on all sides, at any given time, and without any mechanical instruments made of wood or brass?46

Yet Dee did not despise astronomical observation, or believe that it was entirely redundant. He had carried out programmes of observation with large-scale instruments in the mid-1550s and also at the time of the new star.47 But precise observation delivered accurate parameters rather than establishing principles; cosmological principles were for Dee rooted in a wider disciplinary constellation than mathematics alone.

Digges on the other hand adopted and advocated the Copernican world system as the best representation of the actual order of the planets. In his *Alae* of 1573 he already expressed a preference for Copernicus, repeatedly echoing Copernicus’s condemnation of the ‘monstrous’, botched arrangement of the Ptolemaic spheres.48 However, he withheld an absolute acceptance that Copernicus had restored the perfect anatomy of the heavens, wondering whether some adjustments were still required and whether the new star might indeed provide evidence that would prove Copernicus’s case. Digges returned to the issue of Copernicanism in his *Perfit Description of the Caelestiall Orbes*, appended to the 1576 edition of his father’s popular almanac, the *Prognostication Everlasting*. The *Perfit Description* provided an augmented translation of the major chapters in the first book of Copernicus’s *De revolutionibus* and was preceded by Digges’s much-reproduced image of the cosmos in which the stars extend infinitely outwards from the solar-centred planetary system. Although his earlier hopes for the new star as a cosmological arbiter had remained unfulfilled, he now nevertheless expressed his views more emphatically in favour of the heliocentric doctrine.

Digges’s advocacy of Copernicanism rested on an extraordinary elevation of the power of mathematics. Not only were mathematical techniques of investigation and argument to be accounted the best available for the domain of astronomy, but mathematics was to be considered as a means to arrive at the truth rather than just a tool to save the phenomena. Mathematics could demonstratively restore the perfect order of the celestial spheres without deferring to philosophy, traditionally placed above it in disciplinary hierarchies. Digges considered that, in contrast to the demonstrative certainty of mathematics, philosophy could offer only plausible or probable arguments.49 This relative evaluation of mathematics and philosophy helps to explain the structure and sequence of the *Perfit Description*. Digges inverted the order of presentation which Copernicus had adopted for *De revolutionibus* and began his adapted translation with Book I, chapter 10: ‘On the order of the celestial spheres’. Here Copernicus had presented his new arrangement of the planets and given a mathematical justification for his scheme. Only after reading this principal (and, for Digges, self-sufficient) argument was the reader then taken back to *De revolutionibus* I, 7-9 where Copernicus had stated and then disputed the standard philosophical reasons against the motion of the earth. Digges made his expository strategy quite clear:

because the world hath so long a time been carried with an opinion of the earth’s stability, as the contrary cannot now be very impersuasible, I have thought good out of Copernicus also to give a taste of the reasons philosophical alleged for the earth’s stability, and their solutions, [so] that such as are not able with geometrical eyes to behold the secret perfection of Copernicus’s Theoric may yet by these familiar, natural reasons be induced to search further, and not rashly to condemn for fantastical so ancient doctrine revived and by Copernicus so demonstratively approved.50

The realm of philosophy is only for those who lack ‘geometrical eyes’ and remain stuck with probable argument and the deceptive evidence of the senses.51 Digges’s conception of a geometrical vision sufficient in itself to overturn traditional geocentric doctrines endowed mathematical astronomy with independent authority and freed it from subjection or subordination to philosophy or, indeed, any other discipline. Dee had opted for a differently instructed vision to uncover celestial truth:

Raising toward heaven our cabbalistic eyes (that have been illuminated by speculation on these mysteries) we shall behold an anatomy precisely corresponding to that of our monad, which, in the light of Nature and life, will at all time reveal to us as is here shown, and will, by its pleasures, quite openly discover the most secret mysteries of this analysis of the physical world.52

The contrast between Dee’s cabbalistic and Digges’s geometric eyes neatly encapsulates their differing visions of the scope and competence of mathematics. However, the contrast should not be overdrawn. Digges’s geometric enterprise could scarcely be self-founding: inevitably it drew on moral resources and intellectual assumptions from beyond the strict realm of geometrical demonstration. Digges took as axiomatic the simplicity and order of the heavens, and accepted a fundamental opposition between the celestial and sublunary realms.53 In both *Alae seu scalae mathematicae* and the *Perfit Description* he contrasted the perfection and regularity of the heavens with the terrestrial world of generation and corruption. Above, there is the immutable empire of uniform, eternal, and pure substance, equated with the sacred temple for the Calvinist elect, while below, mortal sinners live out their days on the dark star of the earth. The celestial spheres attract the mind upwards away from the dregs of the body and their resplendent beauty entices noble reason with the prospect of joy and felicity, removed from all worldly troubles. All that is blessed finds its true home above, while mutability and decay are the destiny of the profane beings who dwell here below. Celestial order, simplicity, and harmony are opposed to sublunary irregularity and the uncertain, base, and brutish cares of humanity.54 Digges’s geometrical vision was thus grounded in a rich stratum of metaphor, moral evaluation and religious commitment. But he did not seek to provide a firm disciplinary foundation for his mathematical astronomy through a systematic philosophical or theological justification of these assumptions.

Dee on the contrary did attempt to elaborate a comprehensive philosophical position, within which mathematics could be situated. In the *Mathematicall Praeface*, he presented mathematics as just one division of a tripartite ontology and epistemology. Dee drew particularly on Proclus to present mathematical objects as an intermediate level of being between matter and spirit, the human and divine, sense and pure intellect.55 For Dee, mathematics was thus always placed in explicit relation with other learned disciplines, as part of a larger hierarchy of knowledge. While distinguishing different realms of being and understanding, Dee did not see each as being sealed off from the others. He sought to connect and integrate disciplines in order to straddle the boundaries between natural, mathematical and esoteric knowledge. His objectives in mathematics were not inwardly self-referential, but were continuous with his other intellectual ambitions. By emphasising the practical, philosophical and spiritual virtues of various mathematical arts, Dee could move from the mathematics of the calendar or of navigation through to considerations of supercelestial intelligences without a break, as part of the same enterprise.

Hence while Dee placed each limited mathematical art within the broad horizons of Renaissance scholarship, Digges restricted the mathematical arts to a narrower intellectual terrain. Dee moved freely within a self-assigned sphere of learning which incorporated mathematics, magic and natural philosophy. Digges did not accept the values underlying this continuum but elevated mathematics above other disciplines. Digges expressed this sense of segregation from the beginning of his career. He wished that nothing be allowed to sully the purity of his *Mathematicall Discourse* of 1571. Not just vulgar concerns with utility and profit were excluded by this prescription. Digges would not even allow any philosophical ‘pollution’ of the certain and separate domain of mathematics. In the preface to his treatise Digges noted that he would not ‘discourse of [the regular solids’] secret or mystical appliances to the elemental regions and frame of the celestial spheres, as things remote and far distant from the method, nature and certainty of geometrical demonstration’.56 The most obvious object of this implied reproach was the Platonic theory of the elements, as expounded in *Timaeus*. But Digges may have had a much more contemporary target in mind: the astrological *De divina astrorum facultate* (1570) of Dee’s former associate Offusius, in which the regular solids were used to determine cosmic and elemental proportions.57 Evidently Digges had no wish to meddle with uncertain areas of natural or astrological philosophy. Yet whoever Digges meant to censure, his comments serve chiefly to highlight his contrasting image of geometry and his distinct identity as a mathematician.

Digges’s deliberately limited conception of the legitimate terrain of the mathematician did not of course sever his ambitions entirely from the realm of philosophy. The case of Copernicanism shows that mathematics could, when secure on its own territory, challenge established philosophical conclusions. Dee however had a more expansive and positive conception of philosophy, and frequently identified himself as a philosopher rather than a mathematician. Mathematics was central to his programme but achieved much of its significance when brought to bear on philosophical issues: mathematically-informed natural philosophy provided insight into God’s creation while metaphysics carried the philosopher into supercelestial realms, offering ascent towards the divine.

Much of the contrast between the identities of Dee and Digges as mathematicians therefore turns on their respective conceptions of philosophy. Whatever they shared in mathematics, Dee and Digges evidently had quite a different exposure to philosophy. From his time at Trinity College, it is clear that Dee was steeped in philosophical literature, and particularly in the familiar and conventional terrain of Aristotelian natural philosophy. Nicholas Clulee has indeed interpreted the *Propaedeumata aphoristica* as the outcome of an early naturalistic and Aristotelian phase of Dee’s intellectual career.58 The product of a mainstream academic education, Dee could almost take familiarity with Aristotelian natural philosophy for granted.

Digges’s intellectual upbringing was quite different. There is no evidence that he attended university and the only references to his education are to the mathematical instruction of his father and Dee. It may be that Digges saw so little through philosophical eyes because he had received no systematic grounding in academic philosophy. Digges’s distinctively different evaluation of philosophy certainly suggests that Dee’s instruction of the youthful Digges was largely limited to mathematical matters. Trained up in mathematics, it was therefore on mathematics that Digges rested his criteria of intelligibility and evidence.

I began this account of Dee and Digges with their texts prompted by the new star of 1572. Their conclusions on the star exemplify both the common elements in their mathematical work, and also the extent to which their visions of mathematics diverged. The published works provided demonstrative treatments of parallax as a means of establishing the distance and thus location of the new phenomenon. The texts were issued as soon as possible in order to encourage other European astronomers to make the necessary measurements. Digges’s volume also dealt extensively with observational practice, discussing the correction of instrumental errors and the judicious choice of computational procedures.59

The two texts were published before either Dee or Digges had arrived at definitive verdicts on the nature of the new star, though Digges was clearest in identifying it as a genuinely celestial phenomenon and recording its position in the constellation of Cassiopeia. The similarities between their investigative strategies and chosen expository forms also appears to have been matched by the conclusions they reached after the star had finally disappeared from view. Although neither published their final judgements, Dee’s conclusion that the star had been let down and then withdrawn is evident from the title of his lost work on the star: *De stella admiranda in Cassiopeiae Asterismo, coelitus demissa ad orbem usque Veneris, iterumque in coeli penetralia perpendiculariter retracta lib. 3*.60 That this conclusion was shared by Digges is suggested by the evidence of the antiquary William Camden, who recorded in his *Annales* that both Dee and Digges believed the star to have gradually faded as it disappeared further from the earth.61

Yet despite the extent of their agreement, Dee and Digges made quite different uses of the new star. As we have seen, for Digges it was a first occasion for Copernican reflections as well as potential evidence for the superiority of the heliocentric world-system. Dee likewise considered the star to be of great significance but, with his quite different conception of the purpose and scope of mathematical astronomy, he interpreted it not as a cosmological but as a calendrical revelation. For Dee, the new star was literally an epoch making event: he used its appearance to mark a new year zero of the same status as the birth of Christ and the creation of the world. He dated the ‘Necessary Advertisement’ to the *General and Rare Memorials* as ‘Anno, Stellae (Coelo Demissae, rectaque Reversae) Quinto: Julij vero, Die 4 et Anno Mundi 5540’.62 Dee also interpreted the new star as a sign or portent, just as the *Mathematicall Praeface* had hinted. In his copy of Manilius’s *Astronomica* he noted:

I did coniecture the blasing star in Cassiopeia appering ao 1572, to signify the fynding of some great Thresor or the philosphers stone … This I told to Mr. Ed. Dier. at the same tyme. How truly it fell out in ao 1582. Martij 10 it may appere in tyme to come ad stuporem Mundi.63

For Dee, the new star was enrolled in his conception of mathematical astronomy as a calendrical and chronological art which also revealed portents; for Digges, the new star announced a celestial reformation in which mathematics triumphed as the key to heavenly truth.

My title posed the formulaic question ‘like father, like son?’ for Dee and Digges. Despite Dee’s evident importance for the content and direction of Digges’s mathematical career, it is clear that he did not foster a mathematical son altogether in his own image. Digges emerged from Dee’s tutelage with a vision of mathematics and its legitimate ambitions markedly different from that of his mentor. In this mathematical relationship, father and son ultimately followed different paths. Dee either did not try to instruct Digges in the more philosophical and recondite dimensions of his own work or, if he did, his teaching was ineffective. Dee had greater long-term impact, both on Digges and on other more humble contemporaries, through his prescription for the role of the mathematical practitioner set out in the *Mathematicall Praeface*. Yet Dee himself sought a more profound and prestigious identity beyond that of the mathematical practitioner and, even in the vernacular and outwardly practical *Mathematicall Praeface*, he hinted at the more potent and esoteric arts which only the adept might practise. Ironically, just as Dee was forsaking earthly avenues to wisdom in favour of spiritual access to more ‘radicall truthes’ in the 1580s, the prosaic part of his public programme was being successfully adapted by figures such as Digges who refashioned their identities as mathematical practitioners.64

Digges thus helps to clarify the thorny problem of Dee’s contemporary significance and influence. The scale of that problem is perhaps most vividly illustrated by Peter French’s deeply ambivalent response to the issue. In his intellectual biography of Dee, French wrote that Dee ‘was decidedly out of place in sixteenth-century England’. Yet only three pages before he had decided that ‘Dee and the diverse contemporary attitudes towards him epitomize the English Renaissance’.65 Rather remarkably, Dee was meant to be both awkwardly isolated and yet also representative. The uniquely privileged perspective on Dee provided by a detailed examination of his relations with Digges indicates how Dee could be both close to and distant from his contemporaries. Yet, however suggestive, an account which uses Digges as an exclusive vehicle for approaching Dee is obviously too narrowly limited to carry complete conviction. I therefore want to conclude by showing how Digges’s ultimately double-edged relationship to Dee is matched by the ironies of the latter’s reputation among more humble mathematical practitioners.

The most sensitive of recent interpretations of Dee have sought to recover his own self-understanding and intellectual practice. As a result we can better appreciate the coherence of Dee’s work and the rich interconnections between texts and plans which were often avowedly composed in haste and for highly specific occasions. Yet, given this historiographical finesse, 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 *Mathematicall Praeface* was indeed read and admired by such mathematical contemporaries as William Bourne and Edward Worsop. But their reading of his text stripped it of its philosophical and magical ambitions. Dee instead 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*.66 Likewise, Edward Worsop, in writing on surveying, 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 abstruse realm of occult doctrine. Neither Bourne nor Worsop mentioned Dee’s other (Latin) publications.67

If mathematical practitioners such as Bourne and Worsop failed to fathom the depth of Dee’s studies, they nevertheless demonstrate Dee’s importance for the development of practical mathematics. Whereas Dee sought for himself an elevated role as a philosopher - in one instance, even as a Christian Aristotle - he was a key figure in establishing the identity of the mathematical practitioner and in promoting mathematics as worldly, instrumental, practical, vernacular and public.68

Dee cannot of course be assigned exclusive paternity for the creation and growth of the tradition of mathematical practice in England. Yet he was clearly not averse to casting himself in a paternal role; whether he would have extended it beyond the specific case of Digges is however uncertain. But what of Dee’s own parentage? If every father was once a child, how did Dee portray himself as offspring rather than as parent, and what did he make of his own intellectual genealogy?

As Sherman has stressed, Dee was passionately concerned to establish and display his genealogy.69 Among the remarkable features of these endeavours were his attempts to link himself with Arthur and the early Welsh kings and to connect his lineage with that of Elizabeth. Yet perhaps the most extraordinary element in his genealogy was Dee’s effort to forge a connection with Roger Bacon. Dee may have recognised many intellectual authorities but, as Clulee has persuasively argued, Bacon was the figure to whom Dee evidently felt closest.70 Bacon provided Dee with a role model who advocated the centrality of mathematics within a broadly-based philosophical programme. Dee’s composition of an *apologia* for Bacon in 1557 gives a hint of his sense of personal identification. But Dee set out not only to style himself on the intellectual example of Bacon but to actively construct a descent through which he could count Bacon as a blood relation. A substantial section of Dee’s calendar proposal is devoted to proclaiming the merits of Bacon’s own work on the topic and Dee wrote that no-one had made a better diagnosis and case for reform than this other British subject, ‘named (as some thincke) David Dee of Radik: But otherwise, and most commonly, (upon his name altered, at the alteration of his state, into the Fryerly profession) called Roger Bachon’.71 Not content to magnify the achievement of his 13th-century predecessor, Dee was suggesting that he and Bacon actually belonged to the same family, for David Dee features in Dee’s own genealogical constructions.72 Whatever the ultimate rationale for this identification it powerfully demonstrates Dee’s sense of pedigree and suggests how much was at stake in his acceptance of Digges as his mathematical heir.

## Notes

1. John Dee, *Parallaticae commentationis praxeosque nucleus quidam* (London, 1573), sig. A2v and Thomas Digges, *Alae seu scalae mathematicae* (London, 1573), sig. A2r. Digges repeats the sentiment in his preface to Dee’s *Nucleus* at sig. A2r.

2. For example, Peter J. French, *John Dee: the World of an Elizabethan Magus* (London, 1972), pp. 98-9.

3. Digges comments on his mathematical education and associations in *Alae* (note 1), sigs A2r, B3r and in *An Arithmeticall Militare Treatise, named Stratioticos* (London, 1579), p. 190. On Leonard Digges, see Joy B. Easton, ‘Leonard Digges’, in C. C. Gillispie (ed.), *Dictionary of Scientific Biography*, 16 vols (New York, 1970-80), IV, p. 97.

4. Julian Roberts and Andrew G. Watson, *John Dee’s Library Catalogue* (London, 1990), pp. 43 and 82 n. 68.

5. F. R. Johnson made the first serious study of Digges’s Copernican text and the bulk of his careful account still stands; see Francis R. Johnson and Sanford V. Larkey, ‘Thomas Digges, the Copernican system, and the idea of the infinity of the universe in 1576’, *Huntington Library Bulletin*, 5 (1934), 69-117 and Francis R. Johnson, *Astronomical Thought in Renaissance England: a Study of the English Scientific Writings from 1500 to 1645* (Baltimore, 1937), esp. chh. 5 and 6. For another classic but skewed interpretation, Alexandre Koyré, *From the Closed World to the Infinite Universe* (Baltimore, 1957), pp. 34-9. For a more recent example of Digges’s place within the story of astronomy, René Taton and Curtis Wilson (eds), *Planetary Astronomy from the Renaissance to the Rise of Astrophysics: Tycho Brahe to Newton (General History of Astronomy*, vol. 2A) (Cambridge, 1989), pp. 22-3. For an example of Digges’s place in an account of the Scientific Revolution, A. R. Hall, *The Scientific Revolution* (London, 1954), p. 104.

6. D. W. Waters, *The Art of Navigation in Elizabethan and Early Stuart Times* (London, 1958), A. R. Hall, *Ballistics in the Seventeenth Century* (Cambridge, 1952), A. W. Richeson, *English Land Measuring to 1800: Instruments and Practices* (Cambridge, Mass., 1966), John Summerson in H.M. Colvin (ed.), *History of the King’s Works*, 6 vols (London, 1963-82), IV, 755-764, H. J. Webb, *Elizabethan Military Science: the Books and the Practice* (Madison, 1965). For the political career, see his entry in vol. 2 of P.W. Hasler (ed.), *The House of Commons 1558-1603*, 3 vols (London, 1981) and, for more recent work, Michael Graves, ‘Managing Elizabethan Parliaments’, in D. M. Dean and N. L. Jones (eds), *The Parliaments of Elizabethan England* (Oxford, 1990), 37-63 and Patrick Collinson, ‘Puritans, Men of Business and Elizabethan Parliaments’, *Parliamentary History*, 7 (1988), 187-211.

7. Leonard and Thomas Digges, *A Geometrical Practise, named Pantometria* (London, 1571), sig. Aa1r.

8. Dee’s additions to the Euclidean text appear in books X-XIII. For the character and content of these additions, see John Heilbron’s introduction to Wayne Shumaker’s edition and translation of the *Propaedeumata aphoristica:* *John Dee on Astronomy* (Berkeley, 1978), pp. 22-27.

9. British Library, MS Cottonian Vitellius C.VII, ff. 270r-273r, 278v-279r.

10. Note especially the mechanical methods presented in Dee’s *Mathematicall Praeface* to Euclid’s *Elements* (London, 1570) in the section on ‘Statike’ (sigs cjr-ciijv), as well as the general rationale for his additions to Euclid at f. 371r-v of the 1570 edition. Corpus Christi College, Oxford, MS 254, f. 188r preserves another mechanico-mathematical ‘Inventum Johannis Dee’.

11. Marshall Clagett, *Archimedes in the Middle Ages*, 5 vols (Madison/Philadelphia, 1964-84), V, part 4, appendix 2, p. 493.

12. Dee’s lost *Tyrocinium Mathematicum* was largely concerned with the theory of irrational magnitudes: Euclid, *Elements* (London, 1570), f. 268r-v. There are some surviving fragmentary notes on the topic in British Library, MS Cottonian Vitellius C.VII, f. 274ff. The terminology of irrational majors, binomials and apotomes of various orders recurs throughout Digges’s *Mathematicall Discourse*.

13. Preface to *Mathematicall Discourse*: *Pantometria* (note 7), sig. S4v.

14. For Digges’s inherited land, see the list of his father’s lands in *Calendar of the Patent Rolls, Philip and Mary*, vol. 2, *1554-1555* (London, 1936), p. 270 and also note the wealth indicated by his will, PRO PROB11/86, ff.164r-166v.

15. Nicholas H. Clulee, *John Dee’s Natural Philosophy: Between Science and Religion* (London, 1988) provides a convenient resume of the material of this paragraph at pp. 27, 29-32. For Dee and Pembroke, see the fuller discussion in Roberts and Watson (note 4), pp. 3-4. The titles of the texts for the Duchess of Northumberland appear in Dee’s *Letter Containing a most briefe Discourse Apologeticall*, in James Crossley (ed.), *Autobiographical Tracts of Dr. John Dee*, Chetham Society, 24 (1851), p. 75. Note that Dee may not have provided exclusively mathematical service at this time; Roberts and Watson (p. 5) note that Dee seems to have served as chaplain in Bishop Bonner’s household.

16. *Mathematicall Praeface* (note 10), sigs *iiijv-ajr.

17. Public Record Office, SP12/90/12, letter of Thomas Digges to Lord Burghley dated 11 December 1572.

18. British Library, Lansdowne MS 19/30, printed in James Orchard Halliwell (ed.), *A Collection of Letters Illustrative of the Progress of Science in England* (London, 1841), pp. 6-7. The precise form of Burghley’s ‘frame’ is not known. It is mentioned by Jacob Rathgeb, who recorded the 1592 visit to England by Frederick, Duke of Wurttemburg: *Kurtze und Warhaffte Beschreibung der Badenfahrt* (Tubingen, 1602), ff. 32v-33r, translated in W.B. Rye, *England as seen by Foreigners in the Days of Elizabeth and James the First* (London, 1865), p. 44. For Theobalds, John Summerson, ‘The building of Theobalds, 1564-1585’, *Archaeologia*, 97 (1959), 107-26.

19. Bodleian Library, Ashmole MS 1478, ff. 1-60. Digges did not retain the manuscript, which was acquired by Simon Forman in 1594. In addition to inserting alchemical material, Digges repeatedly signed his name and added a ‘questio geographica’, some algebraic workings, and a list of books which is almost exclusively mathematical.

20. *Pantometria* (note 7), sig. A4r.

21. *Stratioticos* (note 3), sigs A3r, A2r.

22. For a fuller account of Digges’s changing priorities and his later activities, see Stephen Johnston, *Making Mathematical Practice: Gentlemen, Practitioners and Artisans in Elizabethan England *(unpublished Ph.D., Cambridge, 1994), chapter 2.

23. William H. Sherman, *John Dee: the Politics of Reading and Writing in the English Renaissance* (Amherst, 1995), pp. 10-12, 22-3 has recently emphasised the importance of Dee’s personal apologetics for an interpretation of his role.

24. Clulee (note 15), p. 35.

25. See David Gwyn, ‘John Dee’s *Arte of Navigation*’, *The Book Collector*, 34 (1985), 309-322, pp. 312, 314-5 and Clulee (note 15), pp. 193-4.

26. For the full title of the ‘Speculum unitatis: sive Apologia pro Fratre Rogero Bachone Anglo’, see Dee’s dedicatory letter to Mercator in *Propaedeumata aphoristica* (note 8), pp. 116-7.

27. *Mathematicall Praeface* (note 10), especially sigs Ajv and Aijr-v.

28. *Propaedeumata aphoristica* (note 8), pp. 116-7. Both editions listed 11 unpublished works, though Dee revised the list to include some different texts in the second edition. None of the works was ever published.

29. *Propaedeumata aphoristica* (note 8), p. 120-1; C. H. Josten, ‘A translation of John Dee’s ‘Monas hieroglyphica’ (Antwerp, 1564), with an introduction and annotations’, *Ambix*, 12 (1964), 84-221, pp. 150-3.

30. *General and Rare Memorials pertayning to the Perfect Arte of Navigation* (London, 1577), sig. eijr.

31. *General and Rare Memorials* (note 30), sig. ?iiijv.

32. *General and Rare Memorials* (note 30), p. 11.

33. *General and Rare Memorials* (note 30), sig. e*iijv; see also sig. e*jr and p. 65.

34. Roberts and Watson (note 4), p. 10 seem too charitable in accepting that, from early in his career, Dee believed mathematics and navigation should ‘be made known as widely as possible for the good of the state’. As an explicit theme this seems to belong only to the 1570s.

35. Sherman (note 23), pp. 4, 150, 152, 173-5.

36. C. L. Whitby, ‘John Dee and Renaissance scrying’, *Bulletin of the Society for Renaissance Studies*, 3, no. 2 (1985), 25-36 and Clulee (note 15), ch. 8, esp. p. 220ff.

37. N. Jardine, *The Birth of the History and Philosophy of Science* (Cambridge, 1984), esp. ch. 7.

38. For Dee’s astrology, see Clulee (note 15), chapter 3. See also Richard Dunn’s paper in this volume.

39. *Mathematicall Praeface* (note 10), sig. bijv. The reference is to Genesis I, 14.

40. *Mathematicall Praeface* (note 10), sig. bijv.

41. Clulee (note 15), chapter 4.

42. In addition to Clulee (note 15) and Heilbron (note 8), pp. 34-49, see also Robert S. Westman, ‘Magical reform and astronomical reform: the Yates thesis reconsidered’, in Robert S. Westman and J. E. McGuire, *Hermeticism and the Scientific Revolution* (Los Angeles, 1977) and J. Peter Zetterberg, ‘Hermetic geocentricity: John Dee’s celestial egg’, *Isis*, 70 (1979), 385-93.

43. John Feild, *Ephemeris anni 1557 currentis iuxta Copernici et Rheinhaldi canones* (London, 1556), sig. Air.

44. Bodleian Library, Ashmole MS 1789; see, for example, f. 6v for ‘Nicolaus Copernicus, the sixth, and most notable lyne of our Astronomicall Dyall’.

45. Bodleian Library, Ashmole MS 1789, ff. 6v, 31r, and note also f.. 8r: ‘Copernicus imagineth the Theoricall cause hereof ... whereof, here, is no place to reason’.

46. *Monas hieroglyphica* (note 29), pp. 130-1.

47. For the work of the 1550s, see the *Compendious Rehearsall* in *Autobiographical Tracts* (note 15), p. 28 and Bodleian Library, Ashmole MS 1789, f. 10r; Digges reports on Dee’s observations of the new star in *Alae* (note 1), sig. B3r.

48. *Alae* (note 1), sigs A4v, 2A3r, 2A4v, L2v.

49. Cf. *Stratioticos* (note 3), sig. a1r. This high evaluation of the status of mathematics, particularly in relation to astronomy, has ancient precedent in the opening section of Ptolemy’s *Almagest*.

50. Thomas Digges, *Perfit Description* in Leonard Digges, *Prognostication Everlasting* (London, 1576), sig. N4r; also sig. N2v for the persuasiveness of Copernicus to ‘any reasonable man that hath his understandinge ripened with Mathematicall demonstration’.

51. While Digges frequently vaunted (mathematical) reason at the expense of the senses (for example, in the *Perfit Description* (note 50), sig. M1r-v), in *Alae* (note 1), sig. H.4r he nevertheless acknowledged that the proper practice of astronomy depended on two complementary components: disciplined sensible experience as well as mathematical demonstration.

52. *Monas hieroglyphica* (note 29), pp. 174-7.

53. In *Alae* (note 1), sigs A3v, B1v, Digges referred to extra-geometrical starting points as physical foundations. In stressing Digges’s conviction of the radical divide between the terrestrial and celestial, I differ from Johnson and Larkey’s interpretation (note 5), pp. 101-2. Wishing to present Digges as an exemplary anti-Aristotelian, they attempted to explain away his use of the distinction, seeing it as no more than a sop to his readers. This is to misconceive an essential element of Digges’s intellectual order.

54. I have run together the characterisation in this paragraph from various passages in *Alae* (note 1) and the *Perfit Description* (note 50). *Alae*: sig. A1v (the beautiful order of the heavenly bodies), Ar (the unchanging pure aether), A3v (no substantial change in the heavens), and L2v (our troubled life on this dark and obscure terrestrial star). *Perfit Description*: the diagram and its captions (the earth as the globe of mortality compared to the perfect joy of the habitacle for the elect), sig. M2r (quotations from Palingenius’s *Zodiacus Vitae*), N4r (‘our Elementare corruptible world’ matched against ‘the glorious court of the great god’). Note that, in the preface to *Pantometria*, Digges had earlier contrasted Atlas’s imprisonment in a mortal carcass here in this most inferior and vile part of the universal world with the pleasant and beautiful frame of celestial orbs (sig. A3r).

55. For Dee’s reading of Proclus, see Clulee (note 15), chapter 6.

56. *Pantometria* (note 7), sig. S4v.

57. Jofrancus Offusius, *De divina astrorum facultate in larvatam astrologiam* (Paris, 1570), ff. 3r-5r. For Offusius’s connection with Dee, see Owen Gingerich and Jerzy Dobrzycki, ‘The master of the 1550 radices: Jofrancus Offusius’, *Journal of the History of Astronomy*, 24 (1993), 235-254.

58. Clulee (note 15), chapters 2 and 3.

59. John Roche discusses Digges’s reformation of the astronomical cross-staff in ‘The radius astronomicus in England’, *Annals of Science*, 38 (1981), 1-32.

60. *Compendious Rehearsall* (note 47), p. 25.

61. William Camden, *Annales rerum Anglicarum, et Hibernicarum, regnante Elizabetha, ad annum salutis MDLXXXIX *(London, 1615), p. 232. In the text to his frontispiece in *Alae*, Digges says that he has recorded the position of the new star in case it retreats back again by divine command before the end of the world.

62. *General and Rare Memorials* (note 30), sig. e*iiijr.

63. Roberts and Watson (note 4), p. 85 (catalogue no. 251). March 10, 1582 was the date of Dee’s first session with Kelley. For Dee on the significance of the anniversaries of the new star, see *ibid*. p. 157 (catalogue no. D20).

64. The term ‘radicall truthes’ comes from Dee’s *Libri mysteriorum*, cited by Clulee (note 15), p. 209.

65. French (note 2), pp. 22, 19.

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

67. Edward Worsop, *A Discoverie of Sundrie Errours and Faults Daily Committed by Landemeaters* (London, 1582), sig. G3v. The examples of Worsop and Bourne confirm Clulee’s suggestions on the significance of the *Mathematicall Praeface*: ‘It may have been Dee’s ironic fate to have contributed to the progress of science among those who were ignorant of the magical direction which Dee thought was the highest level of science’ Clulee (note 15), p. 175. See also French (note 2), pp. 173-4 for further contemporary references to the *Mathematicall Praeface *in the same vein.

68. For Dee’s self-identification as a Christian Aristotle, see *General and Rare Memorials* (note 30), sig. e*jv. On the formation of the culture of mathematical practice, see Stephen Johnston, ‘The identity of the mathematical practitioner in 16th-century England’, in Irmgard Hantsche (ed.), *Der ‘mathematicus’: zur Entwicklung und Bedeutung einer neuen Berufsgruppe in der Zeit Gerhard Mercators* (Bochum, 1996), 93-120.

69. Sherman (note 23), especially pp. 10, 87.

70. The theme recurs throughout Clulee (note 15), but see the summary statement at pp. 232-4.

71. Bodleian Library, Ashmole MS 1789, f. 26v.

72. See, for example, the abbreviated lineage added in the margin of Dee’s copy of *The Laws of Hywel Dda*, reproduced in Sherman (note 23), p. 108.