'on the Origins of Dee’s Mathematical Programme' the John Dee–Pedro Nunes Connection

Studies in History and Philosophy of Science 43 (2012) 460–469

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On the origins of Dee’s mathematical programme: The John Dee–Pedro Nunes connection Bruno Almeida CIUHCT – Centro Interuniversitário de História da Ciência e Tecnologia, Pólo da Universidade de Lisboa, Faculdade de Ciências, Edifício C4, Piso 1, Gabinete 28, Campo Grande, 1749-016 Lisboa, Portugal

a r t i c l e

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Article history: Available online 9 January 2012 Keywords: John Dee Pedro Nunes Nautical science Mathematical programme

a b s t r a c t In a letter addressed to Mercator in 1558, John Dee made an odd announcement, describing the Portuguese mathematician and cosmographer Pedro Nunes as the ‘most learned and grave man who is the sole relic and ornament and prop of the mathematical arts among us’, and appointing him his intellectual executor. This episode shows that Dee considered Nunes one of his most distinguished contemporaries, and also that some connection existed between the two men. Unfortunately not much is known about this connection, and even such basic questions such as ‘What could John Dee know about this Portuguese cosmographer’s scientific work?’ or ‘When, why and where did this interest come about?’ still lack proper answers. In this paper I address this connection and examine Nunes’ influence on Dee’s mathematical work. I argue that Dee was interested in Nunes’ work as early as 1552 (but probably even earlier). I also claim that Dee was aware of Nunes’ programme for the use of mathematics in studying physical phenomena and that this may have influenced his own views on the subject. Ó 2011 Elsevier Ltd. All rights reserved.

When citing this paper, please use the full journal title Studies in History and Philosophy of Science

1. Introduction In 1558, John Dee (1527–1609) published his first work, Propaedeumata aphoristica.1 At the beginning of the book, Dee included a dedicatory letter (dated 20 July 1558) to his friend, the Flemish cartographer Gerard Mercator, in which he recalled the good times they spent ‘philosophizing’ together in Louvain. He also justified the delay in the completion of his awaited scientific work, declaring that had fallen severely ill in the previous year. Then, he makes an unexpected statement: You should know that, besides the extremely dangerous illness from which I have suffered during the whole year just past, I have also borne many other inconveniences (from those who, etc.) which have very much hindered my studies, and that my

strength has not yet been able to sustain the weight of such exertion and labor as the almost Herculean task will require for its completion. And if my work cannot be finished or published while I remain alive, I have bequeathed it to that most learned and grave man who is the sole relic and ornament and prop of the mathematical arts among us, D. D. Pedro Nuñes, of Salácia, and not long since prayed him strenuously that, if this work of mine should be brought to him after my death, he would kindly and humanely take it under his protection and use it in every way as if it were his own: that he would deign to complete it, finally, correct it, and polish it for the public use of philosophers as if it were entirely his. And I do not doubt that he will himself be a party to my wish if his life and health remain unimpaired, since he loves me faithfully and it is inborn in him by nature, and reinforced by will,

E-mail address: [email protected] Dee (1558). This book had one more known edition in the sixteenth century (Dee, 1568), and was recently edited by Wayne Shumaker, with an introductory essay by J. L. Heilbron: Dee (1978). The text can be considered his first attempt to bring forward a manifold system of inspection of ‘the true virtues of nature’: in the words of Shumaker, this was ‘in the main, a fully intelligible series of recipes for applying arithmetic and geometry to a standard scholastic physics and astronomy.’ Dee (1978), ‘Praeface’, p. ix. 1

0039-3681/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.shpsa.2011.12.004

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industry, and habit, to cultivate diligently the arts most necessary to a Christian state.2

which he describes as in his possession were graduated on the principle of the Nonnius (Taylor, 1938, p. 8).

At first sight, one cannot help but be struck by Dee’s intention to appoint the Portuguese cosmographer and mathematician Pedro Nunes (1502–1578) as his literary executor in case of premature debility. Moreover, Nunes would not only have been the executor, but could have taken Dee’s works as his own. It is significant that of all of the potential candidates ‘existing’ at the time (including Mercator himself), Dee expresses complete confidence in the ability of his Iberian friend to make his works available to the public and even, if needed, carry on his unfinished studies. This preliminary reading is significant enough to catch investigators’ attention. Yet as far as I can tell, Dee’s biographers have mentioned this passage, if at all, merely as a curiosity: for example, Nunes is not discussed by Fell-Smith (1909), while Woolley (2001) refers to him only briefly.3 In general, other English historians formulate similar notes or do not go much further. For instance, in Heilbron’s ‘General notes’ (Dee, 1978, p. 205): ‘Dee submits his unpublished writings to a literary executor, the mathematician Pedro Nuñes, for editing if his own life should be cut short.’ Baldwin (2006), in an article devoted to Dee’s interest in nautical science and its applications, includes a brief reference to the cosmographer’s name and his influence on the Englishman’s vision:

Taylor thus highlighted an existing friendship (of which, as far as I know, there is no confirmation from Nunes’ side), and went further by stating that Dee applied some of Nunes’ ideas on nautical science, and suggesting important clues to follow.4 Interestingly, however, the connection between Dee and Nunes has not been missed in popular culture: Umberto Eco, in his novel Foucault’s pendulum, develops a fiction in which Dee plays a significant role in a conspiracy theory referred to as ‘The Plan’, and sets Nunes working as his cosmographer (Eco, 1989). Among Portuguese scholars, the connection has been highlighted by Costa (1933), and the letter was translated into Portuguese by Rua (2004). Although Rua’s study is more oriented towards Dee and Nunes’ (possibly) shared astrological interests, the author makes a careful approach to the issue, listing other evidence of the links between both men, some of which will be discussed below. Returning to Dee’s letter, it seems clear that further explanation is needed as to why Dee addresses Nunes as a friend and literary executor. From his words, it is possible to infer common intellectual (and even moral) interests—otherwise why would he trust his works to the Portuguese, and name him ‘the sole relic and ornament and prop of the mathematical arts’? Dee’s reference to the promotion of ‘the arts most necessary to a Christian state’ may reveal a broader set of shared interests that deserves a deeper study. Nevertheless, if one compares the works of both men up to 1558, it is not easy to establish a connection between them. In his letter, Dee lists works on pure mathematics, astronomy, perspective, cosmography, religion and other topics that may be classified as ‘occult’, but only one work on navigation. Hence, it appears that what needs to be clarified is more than an influence on nautical science. The letter raises a number of questions that demand more thoughtful answers. Was the reference in the letter to Mercator an isolated episode? How far did John Dee’s knowledge of Nunes’ scientific work did go? When, why and where did this interest come about? Indeed, does Dee’s own mathematical programme reflect Nunes’ ideas in any way? In sum, to what extent was the relationship between both men important in the shaping of Dee’s thought? In this paper I address these questions, review what is known and provide new evidence for the influence of Pedro Nunes’ work on John Dee’s scientific production.5

Dee’s actions have to be viewed as determined by his own version of an interdisciplinary, technological and mathematical vision of the Habsburg Empires. . . . It was a structured system of quasi-colonial thought developed initially in the Armazéns da Guiné [in Portugal]. . . . The whole notion was brought to its academic apogee by his great friend and fellow mathematician, Pedro Nuñez. (Ibid., p. 108) Since Baldwin’s analysis focuses on Dee’s impact on British navigation rather than Nunes’ (possible) influence, this passage does not provide further details of their relationship. Nevertheless, historians like Taylor (1963, 1968, 1971) and Waters (1958), highlighted the link between the two men in their important work on the history of navigation. Taylor, for example, wrote: John Dee had formed (under circumstances that are quite unknown) a close friendship with his great Portuguese contemporary Pedro Nunes, and throughout his career as mathematical adviser to a long succession of English explorers he is found to be applying the principles laid down in Nunes’ important works upon nautical science. Three of Nunes’ books, De Erratis Orontii, De Crepusculis, and De Navigatione were in Dee’s library, and it is possible that the five-foot Quadrant and ten-foot cross-staff

2. The work of the Portuguese cosmographer As noted above, the work of Dee and Nunes is not obviously connected, and, judging from his printed works, many of the topics

2 ‘. . . me Scias, praeter periculosissimum, quo toto iam proxime elapso anno laboravi, morbum, alia etiam multa (ab illis, qui. &c.) esse perpessum incommoda, quae mea studia plurimum retardavere: viresque etiam meas, nondum posse tantum sustinere studii laborisque onus, quantum illud, Herculeum pene (ut perficiatur) requiret opus. Unde si mea haud queat opera, vel absolvi, vel emitti, dum ipse sim superstes, Viro illud legavi eruditissimo, gravissimoque, qui Artium Mathematicarum unicum nobis est relictum et decus et columen: nimirum D. D. Petro Nonio Salaciensi: Illumque obnixe nuper oravi, ut, si quando posthumum, ad illum deferetur hoc meum opus, benigne humaniterque sibi adoptet, modisque omnibus, tanquam suo, utatur: absolvere denique, limare. ac ad publicam Philosophantium utilitatem perpolire, ita dignetur, ac si suum esset maxime. Et non dubito. quin ipse (si per vitam valetudinemque illi erit integrum) voti me faciet compotem: cum et me tam amet fideliter, et in artes, Christianae Reip[ublicae] summe necessarias, gnaviter incumbere, sit illi a natura insitum: voluntate, industria, ususque confirmatum.’ Dee (1978, pp. 114–115). The letter is also cited by Van Durme (1959, pp. 36–39). Excerpts from this letter, in Portuguese, can be accessed in Costa (1933, p. 233), and Tarrio (2002, pp. 96–108). For a full Portuguese translation, see Rua (2004). 3 Woolley (2001) includes two brief references to Nunes: ‘Dee also met Pedro Nuñez, then the leading navigator in Lisbon, from where Columbus had set off in 1492 in search for a western passage to the Indies. Nuñez evidently became a close and important intellectual friend. When Dee was struck down with a serious illness in the late 1550s, he appointed Nuñez his literary executor’ (p. 20); ‘He [John Dee] arranged for a draft of Propaedeumata to be published, and handed over the rest of his literary affairs to his friend Pedro Nuñez’, (p. 51). There are some mistakes in this appreciation: there is no evidence that the two men ever met in person; Nunes was not ‘the leading navigator in Lisbon’, since he was not a navigator but the king’s cosmographer; and Columbus did not set off from Lisbon on his first voyage, but from Palos de la Frontera (Huelva, Spain). 4 A reference to this connection also appears in Johnson & Nurminen (2007); while Krücken (2002) is a website that presents a deep analysis of Dee’s work on rhumb lines and also mentions how this relates to Nunes’ and Mercator’s works. Leitão & Almeida (2009) provide an English-language website dedicated to Pedro Nunes, including information on the Dee–Nunes connection at http://www.pedronunes.fc.ul.pt/episodes/john_dee/john_dee.html. 5 Some of these questions had already been set in Leitão (2007).

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Fig. 1. Difference between a rhumb line or loxodrome (ab) and a great circle (ed), Nunes (2002, p. 113).

that attracted Dee did not catch Nunes’ attention. By the time the young Dee was in Louvain in the late 1540s, the Portuguese cosmographer had already published three books, establishing a reputation as a fine mathematician throughout Europe.6 Nunes’ first book, generally known as the Tratado da sphera, was published in Lisbon in the end of 1537.7 This consisted of translations into Portuguese of Sacrobosco’s Tractatus de sphaera, Book I of Ptolemy’s Geography, and the chapters on the sun and moon in Peuerbach’s Teoricae novae planetarum. This edition also included two original treatises on theoretical navigation—Tratado que ho doutor Pêro nunez fez sobre certas duuidas de nauegação (‘Treatise made by doctor Pedro Nunes about certain doubts on navigation’), and Tratado que ho doutor pêro nunez Cosmographo del Rey nosso senhor fez em defensam da carta de marear (‘Treatise made by doctor Pedro Nunes, king’s cosmographer, defending the nautical chart’). While working as a cosmographer, Nunes realised that a lack of mathematical knowledge had resulted in many errors and problems in navigation, and therefore claimed that seamen should be trained in mathematics. In these two vernacular treatises, he pioneered the use of geometrical and trigonometrical tools to solve navigation problems. This approach was uncommon in the ‘art of navigation’ literature of his time, which presented mostly straightforward sets of rules, tables, mnemonics and simple introductions to the Sphere.8 Nunes’ first texts already expressed a clear distinction between what he later termed ars navigandi and ratio navigandi: the ‘ars’ denoting common seamanship based on known sets of rules, procedures and instruments, and the ‘ratio’ referring to nautical activity based in the understanding and use of mathematical principles, eventually leading to something similar to what could be called today a ‘scientific seamanship’ (Leitão, 2006, pp. 188–189). In the process, Nunes also presented a thorough study of the nautical chart, identifying its problems, and was the first to develop the concept of a rhumb line, known today as the loxodromic curve (Fig. 1).9

However, Nunes did not confine his work to nautical science. In 1542 he published De crepusculis, in which he answered a question posed by a noble pupil about the problem concerning the length of twilights for different regions, and showed how an atmospheric phenomenon could be explained using a deductive Euclidean structure.10 As in his previous book, he based his mathematical explanations on real questions made by real people, expressing his concern to reconcile mathematics with physical reality.11 Throughout the book, he also presented a great deal of relevant information for astronomers, including an interesting suggestion for a graphical solution destined to improve instrumental measures, known today as the nonius scale. Nunes’ valuable suggestions later led to this book being highly regarded by, among others, Christopher Clavius (1538– 1612) and Tycho Brahe (1546–1601). In the dedication to the King, Nunes also announced his intention to complete a translation of Vitruvius’ De architectura, while in the final pages he announced several works to be published in the future. These included treatises on the astrolabe, proportions or globes and nautical charts, establishing him as an interesting author, worthy of notice by the international scholarly community. In 1546, Nunes published the book that would establish his scientific reputation as one of the leading European mathematicians. De erratis Orontii Finaei12 revealed errors in the mathematical demonstrations of Oronce Fine (1494–1555), the famous and highly regarded professor of mathematics at the Collège de France, who had tried to solve three classical problems (to double a cube, to trisect an arbitrary angle, and to square a circle) as well as some gnomonic problems. After this, Nunes published twice more. In 1566 he published his Opera, which included his most advanced ideas on different aspects of navigation, astronomy, mechanics, and other topics, with a major printer in Basel.13 In 1567, the Libro de Algebra appeared in Antwerp.14 These two titles concluded the mathematical work that Nunes had started to develop in the early 1530s as an expression of his scientific thought and of his ‘program for the mathematization of the real world.’15 As claimed above, this programme is most apparent in the case of nautical science, since its main purpose was to achieve a full practice of seamanship ‘by art and by reason’, in such a way that it would be possible to merge the study and practice of mathematics with the natural skills and craftsmanship of seamen. However, the programme also extended to astronomy and mechanics. The topic of the minimum twilight was mentioned earlier, but Nunes also confronted peripatetic mechanics in a mathematical explanation of the movement of a rowing boat in a text included in his Opera (Annotationes in Aristotelis Problema Mechanicum de Motu nauigij ex remis).16 3. New evidence John Dee is now viewed as one of the most interesting personalities of the Elizabethan intellectual world. Within the social context of his time, he developed a national and international network

6 Gemma, Mercator and Dee were among the first to study and apply some of Nunes’ ideas, but are far from being the only ones. In my Ph.D. research I address the influence and transmission of Nunes’ works and ideas (considering texts in vernacular) on navigation in the sixteenth and seventeenth centuries, mainly in Portugal, Spain and England. For more on the European diffusion of Nunes’ work see Leitão (2002) and (2007). 7 First edition, Nunes (1537). The latest edition is Nunes (2002). 8 Three known examples are Faleiro (1535), Medina (1545) and Cortés (1551). 9 The word ‘loxodrome’ was introduced by Willebroord Snell; see Snell (1624). 10 First edition, Nunes (1542). The latest edition is Nunes (2003). 11 As Nunes states in his dedication to the King: ‘I was persuaded to clearly explain this matter [i.e. twilights] with help of the most certain and most evident principles of mathematics. So, meditating and investigating, I have discovered things that I have read nowhere else and would not be worth of credit, had they not been demonstrated . . . ’ The translation from Portuguese is mine; see Nunes (2003, p. 142). 12 First edition: Nunes (1546). See the latest edition, Nunes (2005). 13 First edition: Nunes (1566). See the latest edition, Nunes (2008). 14 First edition: Nunes (1567). See the latest edition, Nunes (2010). 15 The ‘Nunes program’ is presented and discussed in Leitão (2006). 16 This text was very influential at the time. For example, Henri de Monantheuil (1599) and Giuseppe Biancani (1615) included important comments on it.

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Fig. 2. Left: Frontispiece of John Dee’s copy of de Sá’s De navigatione. It is possible to read ‘Joannes Deeus: 1552:’ in the centre of the page. Right: highlight of the last page with an early depiction of Dee’s monas hieroglyphica inserted into a human representation. By permission of Cambridge University Library.

of contacts, extending from the Court to the university and to other institutions that promoted the sharing of knowledge about the sciences. He collaborated with the Muscovy Company, and his trainees and collaborators included John Davis, Richard Hakluyt, Walter Raleigh, Francis Drake, Thomas Digges, Thomas Harriot and others. At the international level, he established personal and scientific contacts with many learned men of his time, including Girolamo Cardano, Oronce Finé, Federico Commandino, Abraham Ortelius, Gemma Frisius and Gerard Mercator. Considering his own written indications in the aforementioned letter, the Portuguese Pedro Nunes may well be included in this list. It is not clear when Dee’s interest for Nunes’ work first came about. While a young man, he studied at Cambridge University and, in his pursuit of knowledge, later headed to mainland Europe in 1547. The period spent in Louvain was significant in shaping his early natural philosophical ideas. In fact, it was during his stay in the Low Countries, first in 1547 and then later between 1548 and 1550, that the young Dee must have first heard of Nunes’ work. It is reasonable to speculate that Nunes’ writings would have reached Louvain—the city where Gemma Frisius (1508–1555) had gathered a group to whom he taught private lessons on geometry and astronomy17—without difficulty. The region had strong commercial connections with Portugal, and there was also a large community of Portuguese Jews. One of these, Diogo Pires (or Didacus Pyrrhus Lusitanus), had links to Portuguese intellectuals at Louvain, such as Amato Lusitano and Damião de Góis. He also wrote a dedicatory poem included in Gemma’s 1540 edition of Apian’s Cosmography. In the absence of direct evidence, it is not possible to establish a stronger correlation between Nunes, Gemma and Pires, but it is interesting to note that Pires also studied at Louvain, and that he was a friend of the publisher Rutger Ressen (Rutgerus Rescius), himself a friend of Gemma.18 There is little doubt that Gemma

and Gerard Mercator (1512–1594) would have paid attention to Nunes’ writings.19 Mercator made a globe picturing rhumb lines in 1541, and in 1545 Gemma included a description of a rhumb line in his edition of Apian’s Cosmography (Apian, 1545, Ch. XV, fols. 23v–25v). In 1550/1, John Dee was in Paris, where he met Oronce Finé. It is very likely that Dee became aware of Nunes’ book on the Frenchman’s mathematical errors during this period. Even without direct evidence for this (other than noting that he later had a copy of this book at Mortlake), we might speculate that a reading of this text influenced his interest on the unresolved problems of squaring of the circle and doubling the cube, and even further work on Euclid’s Elements. Dee owned copies of all of Nunes’ books, with one exception: the Tratado da sphera.20 The reason for his failure to acquire this book, aside from its linguistic relevance, may relate to a clue recently found at Cambridge University Library. Clulee (1977, p. 640, n. 27) mentions the existence of a book on navigation in Dee’s library, together with the date 1552. Possible candidates are few in number, and it is highly likely that either the work referred to is either Portuguese or Spanish. It is here worth noting that the catalogue of Dee’s library in Roberts and Watson (1990) includes Diogo de Sá’s De nauigatione (Paris, 1549), which is actually one of the most aggressive attacks on Nunes’ ideas.21 Dee acquired his copy of this work—now at Cambridge University Library—in 1552, after his return to England, and devoted a good deal of study to it (Fig. 2).22 Despite its allusive title, this work covers much more than navigation. It is composed of three books: in the first, the author opens a discussion concerning the certainty of mathematics and its efficacy in producing true knowledge. The subject had an ontological nature, for it was claimed that mathematics dealt only with accidents of substances, rather than with substances themselves, thus

17 ‘[I]l [Gemma Frisius] donnait depuis 1543 et au moins jusqu’en 1547, à son domicile, des leçons privées qui portaient sur la géométrie et l’astronomie. Elles étaient fréquentées notamment par Gérard Mercator, l’Espagnol Juan de Rojas, l’Anglais John Dee et le Frison Sixtus ab Hemminga.’ Hallyn (2008), p. 16. 18 Diogo Pires (1517–1599) studied medicine in the University of Salamanca, the same place where Nunes had studied years before. He did not finish his medical studies but became a well known poet. For more on Pires, see António Manuel Lopes Andrade (2005). 19 Mercator knew most of Nunes’ work, possessing both printed and manuscript copies of his books in his personal library. See Cherton & Watelet (1994). 20 In Roberts & Watson (1990): De erratis Orontii Finaei (entry 100), Petri Nonii Salaciensis Opera (entry 189), De crepusculis (entry 674), Libro de algebra (entry 769). 21 De Sá (1549). Dee’s copy of this book is today at Cambridge University Library, shelfmark R⁄.5.27 (F). See note [B 154] in Roberts & Watson (1990). 22 Dee’s signs his name in Latin and writes ‘1552’, but it is possible that he knew the book earlier than this, while in Paris, since it was published there.

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Fig. 3. De nauigatione, fols. 17v and 18r. By permission of Cambridge University Library.

producing a rather ‘incomplete’ knowledge of nature.23 Accordingly, de Sá also discusses the classification and hierarchy of sciences. In the second book, he explicitly attacks Nunes’ treatises by means of a dialectic dispute between the philosopher and the mathematician, in which the first repeatedly exposes the limitations and superficiality of mathematics from a philosophical point of view. De Sá was a scholar of humanist and scholastic training, who claimed that true and certain knowledge, as sustained by Aristotle in his Posterior analytics, was obtainable only by means of the philosophical study of nature, rather than through mathematical study. In the third book, de Sá continues his critiques while suggesting some solutions to simple nautical issues, such as the sun’s regiment.24 The reading of this work indicates that de Sá did not object to the use of mathematics in some cases, although for him mathematics dealt only with formal causes and therefore produced a ‘limited’ knowledge of reality. This was not a novel intellectual position, given that this discussion was very much alive in sixteenth century Europe (and is today known as the quaestio de certitudine mathematicarum).25 Mota suggests that de Sá was well aware of the discussion taking place in Paris, and played an important role in exporting it to Portugal (2008, p. 176). In my opinion, De nauigatione is more interesting in what it attacks than in what it proposes. The author analyses Nunes’ first treaty step by step according to his own agenda, and launches an all round assault on what he perceived to be the cosmographer’s programme: the use of mathematics as a basis for all certain knowledge about nature, and the mathematization of scientific subjects. Yet Nunes never directly alluded to the quaestio in his works, although he argued that the strength of mathematics lay in its demonstrations and on the logical progression that permitted it to explain natural phenomena. However, it may be argued that de Sá’s attack also backfired, since the second and third books included fine translations of

23 24 25

much of Nunes’ early work, thus providing Latin versions that could be read by any interested European scholars who were unable to read Portuguese. John Dee acquired the book in 1552, and therefore was an early reader of de Sá’s ideas. His copy and his annotations (mainly underlining, short marginal comments, marginal pointers and manicules) reveal more about his interests in the text. In Book One he was above all interested in the discussion concerning the certainty and application of mathematics, and on the hierarchy of sciences (Fig. 3). Throughout, de Sá demonstrates that he knows the basic sources for the discussion on the quaestio very well and, from his underlining, it seems that Dee also benefited from this. In Book Two, Dee continues to reveal his interest in the hierarchy of the sciences and in the mathematical principles of nautical science. On fol. 23r, he appears to suddenly lose interest as de Sá brings theological arguments into the discussion, only to assume interest again at fol. 28v, when de Sá turns to cabala, Pico della Mirandola and astrology (Fig. 4). On fol. 30r (when de Sá reiterates the theological arguments), Dee once more stops underlining and only starts again when the author begins his translation of Nunes’ work. Dee then concentrates on technical aspects of the theory of rhumb lines. In Book Three, he again shows interest in more technical aspects (of cartography, for example), paying close attention to the discourse of the ‘mathematician’ (that is, Nunes’ words and ideas translated and adapted by de Sá) and showing virtually no interest in the practical applications proposed by Diogo de Sá. For instance, Fig. 5 provides a good example of Dee’s study of de Sá’s partial translation of Nunes’ first treatise (Nunes, 2002, pp. 114–115). De Sá’s diagram is also adapted from the same first treatise (ibid., p. 115). At the top right corner, Dee’s note, ‘Nonnius’, calls attention to an important selection of the cosmographer’s theory of rhumb lines.

‘Mathematica non sunt substantia rerum, sed accidentia superuenientia substantiis’, de Sá (1549), fol. 13r. For more on Nunes and de Sá dispute see, for example, Albuquerque (2002). On the quaestio in Portugal and the role of Diogo de Sá’s book in this discussion, see Mota (2011).

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Fig. 4. De nauigatione, fols. 28v–29r (the circle around Pico’s name is mine). By permission of Cambridge University Library.

Fig. 5. De nauigatione, fols. 83v–84r. By permission of Cambridge University Library.

In conclusion, it seems that Dee was interested in the power of mathematics to construct valid knowledge, and in Nunes’ articulation of this subject within his texts on nautical science. Even if embedded within a philosophical discussion, the correct translation and adaptation of Nunes’ treatises seems to have enabled 26

Namely Faleiro (1535), Medina (1545) and Cortés (1551).

Dee to use these works as a textbook on the topic. In fact, in 1552 only three other significant treatises on navigation were printed, none of which were available in the English language.26 The mathematical content of those treatises was also very elementary. Nonetheless, in addition to the more advanced material on

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Fig. 6. Nunes’ first representation of a rhumb line (left: the centre is a pole, the outer circle is the equinoctial, the red line is a rhumb line) compared with a modern representation of a ‘Paradoxal Compass’ (right: image from Krücken, 2002, with permission). Examples of rhumb lines are given in red (note: the two diagrams are not for the same rhumb). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Nunes’ ideas, de Sá’s text also included some basic notions on navigation that could be useful. The acquisition of De nauigatione may also echo Dee’s interest in the certitudine mathematicarum while he was still in Paris, and his interest in the classification of sciences, as later seen in his Mathematicall preface to Euclid (Dee, 1570). 4. Enter rhumb lines All this considered, one can speculate that the reading of De nauigatione was a decisive factor in Dee’s decision to contact Nunes. Unfortunately, it is not possible to confirm the existence of an epistolary correspondence since the Portuguese cosmographer never alluded to it. This is not an unusual situation, since Nunes never refers to any of his mathematical contacts, but, if such a correspondence existed, we might expect that a much discussed mathematical topic of the time—the construction of rhumb tables—was not left out of these discussions. Nunes was the first to write about rhumb lines, but his first treatises did not provide a method for calculating rhumb tables. In 1541, and without indicating a mathematical process for doing it, Mercator devised a globe representing rhumb lines. By this time, Nunes was already developing his mathematical theory of rhumb lines and was aware of the need to calculate tables to use in navigation, although it is not known if he already had devised a good mathematical method for doing so. What is known is his involvement in another polemic against a scholar who had proposed an

alternative (and incorrect) description of the rhumb line, and who had tried to calculate rhumb tables. This episode shows that the discussion was alive in Portugal in the 1540s.27 Only in 1566 would Nunes present a mathematical method for calculating rhumb tables. Between 1556 and 1558, John Dee was interested in resolving the cartographic problems introduced by the navigation in high latitudes, and calculated a set of tables known as Canon gubernauticus or an arithmeticall resolution of the paradoxall compas.28 In E. G. R. Taylor’s opinion, this Canon, was in fact a practical development on the teaching of Pedro Nuñez on this subject and its invention belongs to a period when Dee is known to have been in personal touch with the great Portuguese.29 From the tables was possible to draw rhumb lines, or ‘paradoxal’ lines, on a ‘Paradoxal compass’ which Taylor defined as a zenithal equidistant projection chart. In his Canon gubernauticus, Dee calculates the latitude and longitude for the seven classical rhumbs spiralling across the globe from a point at the equator to a point at 80° latitude, resembling Nunes’ own images (Fig. 6). Early stages of the development of the calculus of rhumb lines implied the resolution of series of spherical triangles, used by both Nunes and Dee. Dee’s method added technical nuances to that first published by Nunes in 1566, although, interestingly, it was similar to the one Edward Wright published in 1599.30 This coincidence raises some questions about the ‘debate’ over rhumb lines between

27 Until now was not possible to identify either the scholar’s identity or his text attacking Nunes’ ideas on rhumb lines. However, the cosmographer responded to this attack with a manuscript defence. This document is presently kept at the Biblioteca Nazionale di Firenze (Codice palatino no. 825). For a modern transcription and study on this manuscript, see de Carvalho (1953). See also Almeida (2006) for a more technical study of the development of rhumb lines in Nunes’ work. 28 This manuscript can be found in Oxford, Bodleian Library MS Ashmole 242, No. 43. It has no date and includes many rectified values. These tables were in fact aids to what Dee called ‘paradoxal navigation’, a subject later also addressed by John Davis and others. See, for example, Davis (1595), fol. K2v. The definition provided suggests a superficial study, and Davis certainly owed much of it to the documents taken from Dee’s library. 29 Taylor (1968, p. 95). For more on this subject see Krücken (2002). 30 Wright (1599). According to Waters (1958, pp. 372–373), Dee probably discussed these tables and the nautical triangle solution with Thomas Harriot, who developed an independent solution to Mercator’s projection. Nevertheless, there is a similarity between Dee’s and Wright’s tables suggesting the usage of the same calculus method. It also suggests that Wright’s Certaine errors in navigation may have benefited from these contacts.

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Dee, Mercator and Nunes. First, it questions the outline of this a shared knowledge, a challenge supported by the lack of documentary evidence; second, if there was a debate, it must be assumed that the Englishman and the Portuguese diverged in their methods for calculating rhumb lines. In effect, Nunes’ method did not have much influence in England, where Wright’s and Harriot’s would eventually succeed. Nevertheless, I would suggest that only knowledge of Nunes’ ideas permitted the advances made by those men. In fact, none of them omitted to give the cosmographer some credit for these developments.31 With or without an epistolary connection, John Dee would in time express his own vision of the nautical science in his Mathematicall preface to Euclid: The Arte of Nauigation, demonstrateth how, by the shortest good way, by the aptest Directio[n], and in the shortest time, a sufficient Ship, betwene any two places (in passage Nauigable,) assigned: may be co[n]ducted: and in all storms, & natural disturbances chauncyng, how, to vse the best possible meanes, whereby to recouer the place first assigned. What nede, the Master Pilote, hath of other Artes, here before recited, it is easie to know: as, of Hydrographie, Astronomie, Astrologie, and Horometrie. Presupposing continually, the common Base, and foundacion of all: namely Arithmetike and Geometrie. So that, he be hable to vnderstand, and Iudge his own necessary Instrumentes, and furniture Necessary: Whether they be perfectly made or no: and also can, (if nede be) make them, hym selfe. . . . And also, be hable to Calculate the Planetes places for all tymes . . . . Sufficiently, for my present purpose, it doth appeare, by the premisses, how Mathematicall, the Arte of Nauigation, is: and how it nedeth and also vseth other Mathematicall Artes.32 In Dee’s opinion, the ‘modern’ sea pilot should base his everyday art on knowledge supported by mathematical methods and tools. This intellectual position coincides with Nunes’ own vision of the ratio nauigandi, reinforced in his 1566 work: Everything that we write on these [nautical] subjects must be received without any hesitation, since nothing exists more exact, nothing more certain and nothing more evident then mathematical demonstration, which certainly nobody will ever be able to oppose.33 In fact, the ideas that both men shared would echo throughout Europe and were included in many sixteenth- and seventeenthcentury navigation textbooks.34 Dee’s role in the reception, appropriation, diffusion and transmission of these ideas has to be underlined: in England (and, indeed, in Europe) he was one of the first to consider navigation as a mathematical discipline. This intellectual position reinforces his important role in what both Waters and Taylor called ‘the English awakening’ to maritime affairs.

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5. A wider influence? While it is clear that Nunes influenced Dee’s vision of nautical science, this connection still does not explain why Dee considered the Portuguese to be ‘the sole relic and ornament and prop of the mathematical arts among us.’ In fact, much less has been said by historians about Nunes’ influence on Dee’s use of mathematics to study nature and on his mathematical views in general. Besides the already noted convergence of the role and use of mathematics, a few other clues are worth mentioning. The first is Dee’s dedication of mathematical works. In his Compendious rehearsal (1592), and again in a letter to the Archbishop of Canterbury (1599), he claims to have completed a work in 1560 on the areas of plane triangles—De Triangulorum rectilineorum Areis35— dedicated to the ‘excellentissimum Mathematicum’, Pedro Nunes. In his essay in this volume, Stephen Johnston establishes an interesting connection between this work on triangle areas and another lost work, named Tyrocinium mathematicum, also dedicated to Nunes (Johnston, 2011). These dedications may point to a broader recognition of the Portuguese cosmographer’s mathematical work, perhaps even to a quest for some kind of ‘intellectual patronage.’ They may also indicate a shared interest in specific mathematical topics. A possible connection could be found with Nunes’ Libro de algebra. This book was only published in 1567, but Nunes began work on it about thirty years earlier.36 In this book, he studied the areas of several geometric figures, including plane triangles. If we assume that a common interest existed, we may also speculate that the Englishman already knew of Nunes’ algebraic texts, and worked on similar subjects. Furthermore, in 1584 Dee noted that he and Jacob Kurtz had examined instruments containing improvements (the nonius scale) also proposed by the Portuguese cosmographer: After dinner I went to Dr. Curtz home. . . . he showed divers his labours and inventions mathematical, and chiefly arithmetical tables, both for his invention by squares to have the minute and second of observations astronomical and so for the mending of Nonnius his invention of the quadrant dividing in 90, 91, 92, 93, etc.37 In the Mathematicall praeface, Dee tried to justify and promote the translation of Euclid’s Elements into the vernacular, giving several examples of similar translation projects undertaken abroad.38 These included an example from the Iberian Peninsula: Nor yet the Vniuersities of Spaine, or Portugall, thinke their reputation to be decayed: or suppose any their Studies to be hindred by the Excellent P. Nonnius, his Mathematicall workes, in vulgare speche by him put forth (Dee, 1570, sig. A.iiij.r).

31 For example, when Wright introduces the errors associated with common nautical charts he states: ‘These errors . . . have been much complained of by diverse, as namely by Martine Cortese . . . but specially by Petrus Nonius in his second book of Geometrical observations, rules, and instruments: And although Gerardus Mercator in his universal Map of the world seemed to correct them, by making the distances of the parallels . . . yet none of them taught any certain way how to amend such gross faults . . . ’, Edward Wright (1599), fol. C2v. 32 Dee (1570), sigs. d.iiij.v–A.j.r. The English Elements was of great importance for the English mathematical arts in the sixteenth century. In David Waters’ opinion: ‘Probably no other work in the English tongue has been so influential in stimulating the growth in England of mathematics, navigation, and hydrography, and in leading to the general application of mathematics to the daily problems of life . . . ’, Waters (1958, p. 131). 33 My translation. The original text is in Nunes (2008, p. 30). 34 It is possible to trace Nunes’ ideas and influence in works by some of the most important actors of European nautical science of the sixteenth and seventeenth centuries. As examples of printed vernacular books: in Spain, Céspedes (1606); in the Low Countries, Coignet (1581); in France, Fournier (1643); in England, Wright (1599). 35 See Dee (1851 [1592]) and (1599). In the 1599 letter, Dee lists De Triangulorum rectilineorum Areis-libri-3-demonstrati: ad excellentissimum Mathematicum Petrum Nonium conscripti-Anno-1560. 36 He states this in the dedication to Cardinal Henrique. See Nunes (2010, p. 8). 37 Dee (1998, p. 165). Jakob Kurtz also discussed improvements to the ‘nonius scale’ with Cristopher Clavius, a former student at Coimbra. It is not certain whether, as a student, Clavius attended Nunes’ classes at that university. Nevertheless, he had a good knowledge of the cosmographer’s works and also considered him to be one of the best mathematicians of his time. On Clavius’ contacts with Kurtz, see Clavius (1992, p. 64). 38 Taylor (1954, p. 314), notes that William Thomas suggested the translation of classical texts into English in the mid-1540s. He even left a translation of Sacrobosco’s Sphere: The sphere of Sacrobosco. Dedicated to The high and mightie Prince Harry, Duke of Suffolk, in London, British Library MS Egerton 837.

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This significant remark shows that Dee was aware of the cosmographer’s vernacular writings and that he was also familiar with the translation of his works and agreed with the principles behind this.39 At the time, the use of vernaculars for natural philosophical discourse was being widely defended throughout Europe. Nunes was also involved in such defences, having clear opinions about the value of translated works. In his 1537 book, he had stated: Science has no language, so it is possible to explain it by using any language. . . . And if, therefore, one can translate any non scientific text from one language to other, I do not know where, so much fear to put a science text in common language, comes from.40 Unfortunately, these sources cannot support more detailed conclusions, and we can only hope that future sources and studies may shed more light on this interesting connection. 6. Final remarks My intention in this paper has been to focus on the John Dee– Pedro Nunes connection, a subject typically not addressed by Dee scholars. First, I have aimed to strengthen the idea that the Portuguese cosmographer had an influence on Dee’s nautical interests. Second, by examining both new and previously known evidence, I suggest that a wider influence can be established, and that both men shared comparable scientific programmes. One important conclusion of this study is that John Dee was the first and most important vector of transmission and diffusion of the Portuguese cosmographer’s work on navigation in sixteenthcentury England.41 In fact, Nunes’ mathematical approach to navigation had an impact on the young Dee while he was developing his early studies and establishing contacts with prominent men of science with similar interests. As William Sherman observes, ‘These men [that is, Frisius, Mercator, Ortelius and Nunes] exercised an influence on Dee that extended far beyond the classroom and long past his visit to Louvain’ (Sherman, 1995, p. 5). Naturally, Dee ended up developing his own programme which, in its first stage, was based on the use of mathematical principles as tools to describe the natural world, something that he would later promote in his role as a consultant on nautical subjects, and continue to practice throughout his life. In this paper, I have not focused on the technical details that Dee transmitted to his collaborators while advising on nautical subjects. These remain a subject for further investigation.42 It is Heilbron’s opinion that, Dee’s contributions were promotional and pedagogical: he advertised the uses and beauties of mathematics, collected books and manuscripts, and assisted in saving and circulating ancient texts; he attempted to interest and instruct artisans, mechanics, and navigators, and strove to ease the beginner’s entry into arithmetic and geometry. It is in this last role, as pedagogue, that Dee displayed his competence, and made his occasional small contributions (which he classed as great and original discoveries) to the study of mathematics as a consultant on nautical subjects (Dee, 1978, p. 17).

While agreeing with these words insofar as they concern Dee’s contribution to pure mathematics, I would stress that Dee was also up to date with many of the mathematical developments of his time, and that he worked alongside some of the most influential mathematical practitioners in Elizabethan England. He would ultimately outline his own mathematical programme in the Mathematicall preface, which Frances Yates considered, in a broad sense, ‘the manifesto of Dee’s movement’ (1979, p. 94). The words of Peter French echo this claim: The essential point to be remembered about Dee’s preface is that it is a revolutionary manifesto calling for the recognition of mathematics as a key to all knowledge and advocating broad application of mathematical principles (French, 1972, p. 167). Another inference to take from this study, and from French’s words, is that, even while recognising Dee as an important figure within the Elizabethan sciences, this ‘revolutionary manifesto’, as French called it, was deeply influenced by other authorities—such as Pedro Nunes—who were involved in a broad movement concerned with the legitimation of mathematics. It therefore helps us to situate Dee’s writing within this context. While Nunes was particularly influential in shaping the English polymath’s ideas on nautical science, I have argued that he also contributed more generally to Dee’s broader vision of the sciences. I therefore suggest that in the historiography of John Dee, Pedro Nunes’ contribution deserves more than a brief reference to the presence of his name in Dee’s letter of 1558. Acknowledgements This research was made possible by a PhD scholarship from ~ncia e a Tecnologia (reference SFRH/BD/ Fundação para a Cie 22952/2005). I also have to thank the support by Centro Interuni~ncia e da Tecnologia, at Lisbon Univerversitário de História da Cie sity. I am most of all grateful to Henrique Leitão for his constant help with this paper and for his ever valuable recommendations. I am also deeply grateful to Ana Almeida for revising this text. I would like to thank Samuel Guessner for inspiring discussions, and António Lopes Andrade for helpful suggestions. I cannot forget Jennifer Rampling and her commitment, work and motivation while organizing the John Dee Quatercentenary Conference and for the honour of being invited to participate. I also thank Katie Taylor for her kind help. I thank all the participants for their comments and for all that I learned while attending the conference, particularly Stephen Johnson for his helpful insights and for the paper he presented at the conference, which was of great encouragement to my own work. Finally, I thank the staff of the Rare Books room at Cambridge University Library. References António Manuel Lopes Andrade, A. (2005). O Cato Minor de Diogo Pires e a poesia didáctica do século XVI. Unpublished Ph.D. dissertation. Universidade de Aveiro. Albuquerque, L. (2002). Pedro Nunes e Diogo de Sá. In J. Carvalho e Silva (Ed.), Antologia de textos essenciais sobre a história da matemática em Portugal (pp. 183–207). Lisboa: Sociedade Portuguesa de Matemática. Almeida, B. (2006). A curva loxodrómica de Pedro Nunes. In M. V. Maroto & M. E. Piñeiro (Eds.), La ciencia y el mar (pp. 149–181). Valladolid: Los autores.

39 Besides having published his first work in Portuguese, Nunes would also publish his last work Libro de Algebra in Spanish. Also, in De crepusculis Nunes announced that he was working on a translation of Vitruvius’ De architectura. 40 ‘ . . . a sciencia não tem lingoagem: e que per qualquer que seja se pode dar a entender . . . E pois de huma lingoagem em outra se pode tirar qualquer scriptura que não seja de sciencia sem se estranhar: nam sey entender donde veo tamanho receo de treladar na linguagem vulgar outra qualquer obra de sciencia . . . ’ Nunes (2002, p. 5). The translation to English is mine. 41 That is, prior to Wright’s Certaine erros in navigation (1599). A more detailed study of the impact of Nunes’ works on navigation in England will be available in my Ph.D. thesis: Almeida (2011). 42 Taylor leaves some clues when addressing the nautical instructions to Frobisher and to his master navigator Christopher Hall. She also claimed that these instructions had no practical application; see Taylor (1954, p. 35). Several manuscripts at the British Library should also be carefully studied with that in mind, for instance MS Harley 167.

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Further reading Thomas, W. (1551). The sphere of Sacrobosco. Dedicated to the high and mightie Prince Harry, Duke of Suffolk. British Library MS Egerton 837. London.